Parallel Optimization of Polynomials for Large-scale Problems in Stability and Control
PParallel Optimization of Polynomials for Large-scale Problems inStability and ControlbyReza KamyarA Dissertation Presented in Partial Fulfillmentof the Requirements for the DegreeDoctor of PhilosophyApproved January 2016 by theGraduate Supervisory Committee:Matthew Peet, ChairDaniel RiveraGeorgios FainekosPanagiotis ArtemiadisSpring BermanARIZONA STATE UNIVERSITYMay 2016 a r X i v : . [ m a t h . O C ] F e b BSTRACTIn today’s world, optimal operation of ever-growing industries and markets oftenrequires solving optimization problems with unprecedented sizes. Economic dispatchof generating units in power companies, frequency assignment in large mobile commu-nication networks, profit maximization in competitive markets, and optimal operationof smart grids are few examples of many real-world problems which can be closelymodeled as optimization over a large number (tens of thousands) of integer- andreal-valued decision variables. Unfortunately, majority of the existing commercialoff-the-shelf software are not designed to scale to optimization problems of this size.Moreover, in theory, these optimization problems often fall into the class NP-hard -meaning that despite the tremendous effort towards modernization of optimizationalgorithms, it is widely suspected that no algorithm can find exact solutions to theseproblems in a reasonable amount of time.In this thesis, we focus on some of the NP-hard problems in control theory. Thanksto the converse Lyapunov theory, these problems can often be modeled as optimiza-tion over polynomials. To avoid the problem of intractability, we establish a trade offbetween accuracy and complexity. In particular, we develop a sequence of tractableoptimization problems - in the form of Linear Programs (LPs) and/or Semi-DefinitePrograms (SDPs) - whose solutions converge to the exact solution of the NP-hardproblem. However, the computational and memory complexity of these LPs andSDPs grow exponentially with the progress of the sequence - meaning that improvingthe accuracy of the solutions requires solving SDPs with tens of thousands of deci-sion variables and constraints. Setting up and solving such problems is a significantchallenge. Unfortunately, the existing optimization algorithms and software are onlydesigned to use desktop computers or small cluster computers - machines which donot have sufficient memory for solving such large SDPs. Moreover, the speed-up ofihese algorithms does not scale beyond dozens of processors. This in fact is the reasonwe seek parallel algorithms for setting-up and solving large SDPs on large cluster-and/or super-computers.We propose parallel algorithms for stability analysis of two classes of systems: 1)Linear systems with a large number of uncertain parameters; 2) Nonlinear systemsdefined by polynomial vector fields. First, we develop a distributed parallel algorithmwhich applies Polya’s and/or Handelman’s theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex.The result is a sequence of SDPs which possess a block-diagonal structure. We thendevelop a parallel SDP solver which exploits this structure in order to map the com-putation, memory and communication to a distributed parallel environment. We pro-duce a Message Passing Interface (MPI) implementation of our parallel algorithmsand provide a comprehensive theoretical and experimental analysis on its complexityand scalability. Numerical tests on a supercomputer demonstrate the ability of thealgorithm to efficiently utilize hundreds and potentially thousands of processors andanalyze systems with 100+ dimensional state-space. We then apply our algorithmsto two real-world problems: Stability of plasma in a Tokamak reactor, and optimalelectricity pricing in a smart grid environment. Finally, we extend our algorithms toanalyze robust stability over more complicated geometries such as hypercubes andarbitrary convex polytopes. Our algorithms can be readily extended to address awide variety of problems in control; e.g., H / H ∞ control synthesis for systems withparametric uncertainty, computing control Lyapunov functions for optimal controlproblems, and analysis and control of switched/hybrid systems.iiABLE OF CONTENTS PageLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiCHAPTER1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Sum of Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Moments Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Quantifier Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Reformulation Linear Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Groebner Basis Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Blossoming Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Bernstein, Polya and Handelman Theorems . . . . . . . . . . . . . . . . . . . . . . . 81.8 Motivations and Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . 92 FUNDAMENTAL RESULTS FOR OPTIMIZATION OF POLYNOMI-ALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Background on Positivity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Polynomial Optimization and Optimization of Polynomials . . . . . . . . 202.3 Algorithms for Optimization of Polynomials . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Case 1: Optimization over the Standard Simplex ∆ n . . . . . . . 232.3.2 Case 2: Optimization over The Hypercube Φ n . . . . . . . . . . . . . . 242.3.3 Case 3: Optimization over The Convex Polytope Γ K . . . . . . . . 272.3.4 Case 4: Optimization over Compact Semi-algebraic Sets . . . . 292.3.5 Case 5: Tests for Non-negativity on R n : . . . . . . . . . . . . . . . . . . . 333 SEMI-DEFINITE PROGRAMMING AND INTERIOR-POINT ALGO-RITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35iiiHAPTER Page3.1 Convex Optimization and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Descent Algorithms for Convex Optimization . . . . . . . . . . . . . . . . . . . . . 373.3 Interior-point Algorithms for Convex Problems with Inequality Con-straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Semi-definite Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 A Primal-dual Interior-point Algorithm for Semi-definite Program-ming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 PARALLEL ALGORITHMS FOR ROBUST STABILITY ANALYSISOVER SIMPLEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 Background and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Notation and Preliminaries on Homogeneous Polynomials . . . . . . . . . . 604.3 Setting-up the Problem of Robust Stability Analysis over a Simplex 624.3.1 General Formulae for Calculating Coefficients β and H . . . . . . 654.3.2 Number of Coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) . . . . . . . . . . . . . . . . . 664.3.3 The Elements of the SDP Problem Associated with Polya’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.4 A Parallel Algorithm for Setting-up the SDP . . . . . . . . . . . . . . . 714.4 Complexity Analysis of the Set-up Algorithm . . . . . . . . . . . . . . . . . . . . . 714.4.1 Computational Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . 744.4.2 Communication Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . 754.5 A Parallel SDP Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5.1 Structure of the SDP Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5.2 A Parallel Implementation for the SDP Solver . . . . . . . . . . . . . . 83ivHAPTER Page4.6 Computational complexity analysis of the SDP algorithm . . . . . . . . . . 874.6.1 Complexity Analysis for Systems with Large Number of States 874.6.2 Complexity of Increasing Accuracy/Decreasing Conserva-tiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6.3 Analysis of Scalability/Speed-up . . . . . . . . . . . . . . . . . . . . . . . . . . 904.6.4 Synchronization and Load Balancing Analysis . . . . . . . . . . . . . . 914.6.5 Communication Graph of the Algorithm . . . . . . . . . . . . . . . . . . . 924.7 Testing and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.7.1 Example 1: Application to Control of a Discretized PDEModel in Fusion Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.7.2 Example 2: Accuracy and Convergence . . . . . . . . . . . . . . . . . . . . 984.7.3 Example 3: Evaluating Speed-up . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.7.4 Example 4: Maximum State-space and Parameter Dimen-sions for a 9-Node Linux-based Cluster Computer . . . . . . . . . . 1035 PARALLEL ALGORITHMS FOR ROBUST STABILITY ANALYSISOVER HYPERCUBES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1.1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2 Notation and Preliminaries on Multi-homogeneous Polynomials . . . . 1065.3 Setting-up the Problem of Robust Stability Analysis over Multi-simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3.1 General Formulae for Calculating Coefficients β and H . . . . . 1135.3.2 The SDP Elements Associated with the Multi-simplex Ver-sion of Polya’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114vHAPTER Page5.3.3 A Parallel Algorithm for Setting-up the SDP . . . . . . . . . . . . . . . 1165.4 Computational Complexity Analysis of the Set-up Algorithm . . . . . . 1165.4.1 Computational Cost of the Set-up Algorithm: . . . . . . . . . . . . . . 1165.4.2 Communication Cost of the Set-up Algorithm: . . . . . . . . . . . . . 1185.4.3 Speed-up and Memory Requirement of the Set-up Algorithm:1195.5 Testing and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5.1 Example 1: Evaluating Speed-up . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5.2 Example 2: Verifying Robust Stability over a Hypercube . . . . 1215.5.3 Example 2: Evaluating Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 1236 PARALLEL ALGORITHMS FOR NONLINEAR STABILITY ANALYSIS1256.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.1.1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.3 Statement of the Stability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.4 Expressing the Stability Problem as a Linear Program . . . . . . . . . . . . . 1336.5 Computational Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5.1 Complexity of the LP Associated with Handelman’s Repre-sentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.5.2 Complexity of the SDP Associated with Polya’s Algorithm . . 1426.5.3 Complexity of the SDP Associated with SOS Algorithm . . . . 1436.5.4 Comparison of the Complexities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457 OPTIMIZATION OF SMART GRID OPERATION: OPTIMAL UTIL-ITY PRICING AND DEMAND RESPONSE . . . . . . . . . . . . . . . . . . . . . . . . . . 151viHAPTER Page7.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.1.1 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.2 Problem Statement: User-level and Utility Level Problems . . . . . . . . . 1577.2.1 A Model for the Building Thermodynamics . . . . . . . . . . . . . . . . 1577.2.2 Calibrating the Thermodynamics Model . . . . . . . . . . . . . . . . . . . 1597.2.3 User-level Problem I: Optimal Thermostat Programming . . . 1607.2.4 User-level Problem II: 4-Setpoint Thermostat Program . . . . . 1627.2.5 Utility-level Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 1637.3 Solving User- and Utility-level Problems by Dynamic Programming 1667.4 Policy Implications and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.4.1 Effect of Electricity Prices on Peak Demand and ProductionCosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.4.2 Optimal Thermostat Programming with Optimal Electric-ity Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1797.4.3 Optimal Thermostat Programming for Solar Customers -Impact of Distributed Solar Generation on Non-solar Cus-tomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818 SUMMARY, CONCLUSIONS AND FUTURE DIRECTIONS OF OURRESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.2 Future Directions of Our Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.2.1 A Parallel Algorithm for Nonlinear Stability Analysis UsingPolya’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187viiHAPTER Page8.2.2 Parallel Computation for Parameter-varying H ∞ -optimalControl Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.2.3 Parallel Computation of Value Functions for ApproximateDynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198viiiIST OF TABLESTable Page4.1 Per Processor, Per Iteration Computational Complexity of the Set-upAlgorithm. L is the Number of Monomials Is P ( α ); L Is the Numberof Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ); M Is the Number of Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Per Processor, Per Iteration Communication Complexity of the Set-upAlgorithm. L is the Number of Monomials Is P ( α ); L Is the Numberof Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ); M Is the Number of Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Data for Example 1: Nominal Values of the Plasma Resistivity . . . . . . . . 954.4 Upper Bounds Found for L opt by the SOS Algorithm Using DifferentDegrees for x and α (inf: Infeasible, O.M.: Out of Memory) . . . . . . . . . . . 1005.1 The Lower-bounds on r ∗ Computed by Algorithm 7 Using DifferentDegree Vector D p and Using Methods in Bliman (2004a) and Chesi(2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.1 Building’s Parameters as Determined in Section 7.2.1 . . . . . . . . . . . . . . . . . 1757.2 On-peak, Off-peak & Demand Prices of Arizona Utility APS . . . . . . . . . . 1767.3 CASE I: Electricity Bills (or Three Days) and Demand Peaks for Dif-ferent Strategies. Electricity Prices Are from APS. . . . . . . . . . . . . . . . . . . . 1777.4 CASE I: Costs of Production (for Three Days) and Demand Peaks forVarious Prices and Strategies. Prices Are Non-regulated and SRP’sCoefficients of Utility Cost Are: τ =0.00401 $/(MWh) , ν =4.54351$/(MWh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177ixable Page7.5 CASE II: Production Costs (for Three Days) and Demand Peaks As-sociated with Regulated Optimal Electricity Prices (Calculated by Al-gorithm 10) and SRP’s Electricity Prices. SRP’s Marginal Costs: a = 0 . $ kW h , b = 59 . $ kW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.6 CASE III: Optimal Electricity Prices, Bills (for Three Days) and De-mand Peaks for Various Customers. Marginal osts from SRP: a =0 . $ kW h , b = 59 . $ kW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182xIST OF FIGURESFigure Page4.1 Various Interconnections of Nodes in a Cluster Computer (Top), Typ-ical Memory Hierarchies of a GPU and a Multi-core CPU (bottom) . . . . 564.2 Number of β (cid:104) h (cid:105) , (cid:104) γ (cid:105) Coefficients vs. the Number of Uncertain Parametersfor Different Polya’s Exponents and for d p = 2 . . . . . . . . . . . . . . . . . . . . . . . 674.3 Number of H (cid:104) h (cid:105) , (cid:104) γ (cid:105) Coefficients vs. the Number of Uncertain Parame-ters for Different Polya’s Exponents and for d p = d a = 2 . . . . . . . . . . . . . . 674.4 Memory Required to Store the Coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) vs.Number of Uncertain Parameters, for Different d , d and d p = d a = 2 . . 694.5 Graph Representation of the Network Communication of the Set-upAlgorithm. (a) Communication Directed Graph for the Case α ∈ ∆ , d p = 2. (b) Communication Directed Graph for the Case α ∈ ∆ , d p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Theoretical Speed-up vs. No. of Processors for Different System Di-mensions n for l = 10, d p = 2, d a = 3 and d = d = 4, Where L + M = 53625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.7 The Number of Blocks of the SDP Elements Assigned to Each Proces-sor. An Illustration of Load Balancing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.8 The Communication Graph of the SDP Algorithm . . . . . . . . . . . . . . . . . . . 934.9 Speed-up of Set-up and SDP Algorithms vs. Number of Processors fora Discretized Model of Magnetic Flux in Tokamak . . . . . . . . . . . . . . . . . . . . 984.10 Upper Bound on Optimal L vs. Polya’s Exponents d and d , forDifferent Degrees of P ( α ). ( d = d ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.11 Error of the Approximation for the Optimal Value of L vs. Degrees of P ( α ), for Different Polya’s Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101xiigure Page4.12 Computation Time of the Parallel Set-up Algorithm vs. Number ofProcessors for Different Dimensions of Linear System n and Numbersof Uncertain Parameters l - Executed on Blue Gene Supercomputer ofArgonne National Labratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.13 Computation Time of the Parallel SDP Algorithm vs. Number of Pro-cessors for Different Dimensions of Primal Variable ( L + M ) n and ofDual Variable K - Executed on Karlin Cluster Computer . . . . . . . . . . . . . . 1034.14 Comparison Between the Speed-up of the Present SDP Solver and SD-PARA 7.3.1, Executed on Karlin Cluster Computer . . . . . . . . . . . . . . . . . . 1044.15 Largest Number of Uncertain Parameters of n -Dimensional Systemsfor Which the Set-up Algorithm (Left) and SDP Solver (Right) CanSolve the Robust Stability Problem of the System Using 24 and 216GB of RAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1 Number of Operations vs. Dimension of the Hypercube, for DifferentPolya’s Exponents d = d = d . (H): Hypercube and (S): Simplex. . . . . . 1195.2 Required Memory for the Calculation of SDP Elements vs. Number ofUncertain Parameters in Hypercube and Simplex, for Different State-space Dimensions and Polya’s Exponents d = d . (H): Hypercube,(S): Simplex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 Execution Time of the Set-up Algorithm vs. Number of Processors,for Different State-space Dimensions n and Polya’s Exponents . . . . . . . . . 1226.1 An Illustration of a D-decomposition of a 2D Polytope. λ i ( x ) := h Ti,j x + g i,j for j = 1 , · · · , m i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Decomposition of the Hypercube in 1 − ,2 − and 3 − Dimensions . . . . . . . . 141xiiigure Page6.3 Number of Decision Variables and Constraints of the OptimizationProblems Associated with Algorithm 1, Polya’s Algorithm and SOSAlgorithm for Different Degrees of the Lyapunov Function and theVector Field f ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.4 The Largest Level-set of Lyapunov Function (6.21) Inscribed in Poly-tope (6.20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.5 Largest Level-sets of Lyapunov Functions of Different Degrees andTheir Associated Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.6 Largest Level-sets of Lyapunov Functions of Different Degrees andTheir Associated Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.1 Effect of Solar Power on Demand: Net Loads for Typical Summer andWinter Days in Arizona in 2014 and for 2029 (Projected), from Ari-zona Public Service (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.2 Peak to Average Demand of Electricity and Its Trend-line in Californiaand New England from 1993 to 2012, Data Adopted from Shear (2014) 1537.3 A Schematic View of Our Thermal Mass Model . . . . . . . . . . . . . . . . . . . . . . 1597.4 Simulated and Measured Power Consumptions . . . . . . . . . . . . . . . . . . . . . . . 1607.5 An Illustration for the Programming Periods of the 4-Setpoint Ther-mostat Problem, Switching Times t i , Pricing Function r , L i and ∆ t i . . . 1637.6 External Temperature of Three Typical Summer Days in Phoenix, Ari-zona. Shaded Areas Correspond to On-peak Hours. . . . . . . . . . . . . . . . . . . . 1757.7 CASE I: Power Consumption and Temperature Settings for VariousProgramming Strategies Using APS’s Rates. . . . . . . . . . . . . . . . . . . . . . . . . . 178xiiiigure Page7.8 CASE I: Power Consumption and Optimal Temperature Settings forHigh, Medium and Low Demand Penalties. Shaded Areas Correspondto On-peak Hours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.9 CASE I: Power Consumption and Temperature Settings for High, Mediumand Low Demand Penalties Using 4-Setpoint Thermostat Programming.1797.10 CASE III: Power Consumption, Solar Generated Power and OptimalTemperature Settings for the Non-solar and Solar Users. . . . . . . . . . . . . . . 182xivhapter 1INTRODUCTIONConsider problems such as portfolio optimization, path-planning, structural de-sign, local stability of nonlinear ordinary differential equations, control of time-delaysystems and control of systems with uncertainties. These problems can all be formu-lated as polynomial optimization and/or optimization of polynomials . In this disser-tation, we show how computation can be applied in a variety of ways to solve theseclasses of problems. A simple example of polynomial optimization is β ∗ = min x ∈ Q p ( x ),where p : R n → R is a multi-variate polynomial and Q ⊂ R n . In general, since p ( x )and Q are not convex, this is not a convex optimization problem. In fact, it has beenproved that polynomial optimization is NP-hard (L. Blum and Smale (1998)). Fortu-nately, algorithms such as branch-and-bound can find arbitrarily precise solutions topolynomial optimization problems by repeatedly partitioning Q into subsets Q i andcomputing lower and upper bounds on p ( x ) over each Q i . To find an upper boundfor p ( x ) over each Q i , one could use a local optimization algorithm such as sequentialquadratic programming. To find a lower bound on p ( x ) over each Q i , one can solvethe following optimization problem. β ∗ = max y ∈ R y subject to p ( x ) − y ≥ x ∈ Q i . (1.1)This problem is in fact an instance of the problem of optimization of polynomials.Optimization of polynomials is convex, yet again NP-hard. We will discuss optimiza-tion of polynomials in more depth in Chapter 2. In the following, we discuss some of1he state-of-the-art methods for solving optimization of polynomials - hence findinglower bounds on β ∗ . 1.1 Sum of Squares MethodOne approach to find lower bounds on the optimal objective β ∗ is to apply Sumof Squares (SOS) programming (Parrilo (2000), Papachristodoulou et al. (2013)). Apolynomial p is SOS if there exist polynomials q i such that p ( x ) = (cid:80) ri =1 q i ( x ) . Theset { q i ∈ R [ x ] , i = 1 , · · · , r } is called an SOS decomposition of p ( x ), where R [ x ] is thering of real polynomials. An SOS program is an optimization problem of the formmin x ∈ R m c T x subject to A i, ( y ) + m (cid:88) j =1 x j A i,j ( y ) is SOS, i = 1 , · · · , k, (1.2)where c ∈ R m and A i,j ∈ R [ y ] are given. If p ( x ) is SOS, then clearly p ( x ) ≥ R n . While verifying p ( x ) ≥ R n is NP-hard, verifying whether p ( x ) is SOS -hence non-negative - can be done in polynomial time (Parrilo (2000)). It was firstshown in Parrilo (2000) that verifying the existence of a SOS decomposition is aSemi-Definite Program (SDP). Fortunately, there exist several algorithms (Monteiro(1997); Helmberg et al. (1996); Alizadeh et al. (1998)) and solvers (Yamashita et al. (2010); Sturm (1999); Tutuncu et al. (2003)) which solve SDPs to arbitrary precisionin polynomial time. To find lower bounds on β ∗ = min x ∈ R n p ( x ), consider the SOSprogram y ∗ = max y ∈ R y subject to p ( x ) − y is SOS . Clearly y ∗ ≤ β ∗ . One can compute y ∗ by performing a bisection search on y and usingsemi-definite programming to verify p ( x ) − y is SOS. SOS programming can also be2sed to find lower bounds on global minimum of polynomials over a semi-algebraic set S := { x ∈ R n : g i ( x ) ≥ , i = 1 , · · · , m } generated by g i ∈ R [ x ]. Given Problem (1.1)with x ∈ S , Positivstellensatz results (Stengle (1974), Putinar (1993), Schmudgen(1991)) define a sequence of SOS programs whose objective values form a sequence oflower bounds on the global minimum β ∗ . For instance, Putinar’s Positivstellensatzdefines the optimization problem y d := max y ∈ R y subject to p ( x ) − y = s ( x ) + m (cid:88) i =1 s i ( x ) g i ( x ) , s i ∈ Σ d , (1.3)where Σ d denotes the cone of SOS polynomials of degree 2 d . Putinar (1993) hasshown that under certain conditions (verifiable by semi-definite programming) on S and for sufficiently large d , y d = β ∗ . See Laurent (2009) for a comprehensive discussionon the Positivstellensatz results.1.2 Moments MethodAs a dual to SOS program, Lasserre (2001) used the theory of moments to definea sequence of lower bounds for global optima of polynomials. Let β ∗ := min x ∈ S p ( x ),where S := { x ∈ R n : g i ( x ) ≥ , i = 1 , · · · , m } is compact and p ( x ) := (cid:80) α ∈ W p p α x α with the index set W p := { α ∈ N n : (cid:107) α (cid:107) ≤ p } . Let us denote the degree of g i by e i .Then, Lasserre (2001) showed that z d defined as z d := min z (cid:88) α ∈ W p p α z α subject to M d ( z ) ≥ M d − e i ( g i z ) ≥ i = 1 , · · · , m, (1.4)3s a lower bound on β ∗ . In Equation (1.4), z := { z α } α ∈ I d , where z α := (cid:82) S x α µ ( dx ) iscalled the moment of order α and is represented by any probability measure µ on R n such that µ ( R \ S ) = 0. Moreover, M d ( z ) is called the moment matrix associatedwith sequence z and in two dimensions is defined as M d ( z )= z [1 , z [0 , z [2 , z [1 , z [0 , · · · z [ d, · · · z [0 ,d ] z [1 , z [2 , z [1 , z [3 , z [2 , z [1 , · · · z [ d +1 , · · · z [1 ,d ] z [0 , z [1 , z [0 , z [2 , z [1 , z [0 , · · · z [ d, · · · z [0 ,d +1] z [2 , z [3 , z [2 , z [4 , z [3 , z [2 , · · · z [ d +2 , · · · z [2 ,d ] z [1 , z [2 , z [1 , z [3 , z [2 , z [1 , · · · z [ d +1 , · · · z [1 ,d +1] z [0 , z [1 , z [0 , z [2 , z [1 , z [0 , · · · z [ d, · · · z [0 ,d +2] ... ... ... ... ... ... . . . · · · · · · · · · z [ d, z [ d +1 , z [ d, z [ d +2 , z [ d +1 , z [ d, · · · z [2 d, · · · z [ d,d ] ... ... ... ... ... ... ... ... . . . ... z [0 ,d ] z [1 ,d ] z [0 ,d +1] z [2 ,d ] z [1 ,d +1] z [0 ,d +2] · · · z [ d,d ] · · · z [0 , d ] . It can be shown that the SDPs in (1.4) are duals to the SDPs in (1.3) - implyingthat y d ≤ z d . Indeed, if S has a non-empty interior, then for all sufficiently large d ,the duality gap is zero, i.e., y d = z d . See Laurent (2009) and Jeyakumar et al. (2014)for conditions on convergence of the lower bounds to global minima and extension ofmoments method to polynomial optimization over non-compact semi-algebraic sets.In the sequel, we explore the merits of some of the alternatives to SOS pro-gramming and moments method. There exist several results in the literature thatcan be applied to polynomial optimization; e.g., Quantifier Elimination (QE) algo- Let X be a set and M be a σ − algebra over X . Then µ : M → [0 ,
1] is a probability measure if1. µ ( ∅ ) = 0 and µ ( X ) = 1.2. For all countable collections { S i } i ∈ N of pairwise disjoint subsets of M , µ ( (cid:83) i ∈ N S i ) = (cid:80) i ∈ N µ ( S i ). et al. (2008); Leroy(2012)) and Handelman’s (Handelman (1988a)) theorems for positivity over simplicesand convex polytopes, and other results based on Groebner bases (Adams and Lous-taunau (1994)) and Blossoming (Ramshaw (1987)) techniques. In particular, we willfocus on Polya’s, Bernstein’s and Handelman’s results in more depth and elaborateon the computational advantages of these results over the others. The discussion ofthe other results are beyond the scope of this dissertation, however the ideas behindthese results can be summarized as follows.1.3 Quantifier EliminationQE algorithms apply to First-Order Logic formulae, e.g., ∀ x ∃ y ( f ( x, y ) ≥ ⇒ (( g ( a ) < xy ) ∧ ( a > , to eliminate the quantified variables x and y (preceded by quantifiers ∀ , ∃ ) and con-struct an equivalent formula in terms of the unquantified variable a . The key resultunderlying QE algorithms is Tarski-Seidenberg theorem (Tarski (1951)). The theo-rem implies that for every formula of the form ∀ x ∈ R n ∃ y ∈ R m ( f i ( x, y, a ) ≥ f i ∈ R [ x, y, a ], there exists an equivalent quantifier-free formula of the form ∧ i ( g i ( a ) ≥ ∨ j ( h j ( a ) ≥
0) with g i , h j ∈ R [ a ]. QE implementations (e.g., Brown(2003) and Dolzmann and Sturm (1997)) with a bisection search yields the exactsolution to optimization of polynomials, however the complexity scales double expo-nentially in the dimension of variables x, y .5.4 Reformulation Linear TechniquesRLT was initially developed to find the convex hull of feasible solutions of zero-onelinear programs (Sherali and Adams (1990)). It was later generalized by Sherali andTuncbilek (1992) to address polynomial optimizations of the form min x p ( x ) subjectto x ∈ [0 , n ∩ S . RLT constructs a δ − hierarchy of linear programs by performingtwo steps. In the first step (reformulation), RLT introduces the new constraints (cid:81) i x i (cid:81) j (1 − x j ) ≥ i, j : i + j = δ . In the second step (linearization),RTL defines a linear program by replacing every product of variables x i by a newvariable. By increasing δ and repeating the two steps, one can construct a δ − hierarchyof lower bounding linear programs. A combination of RLT and branch-and-boundpartitioning of [0 , n was developed by Sherali and Tuncbilek (1997) to achieve tighterlower bounds on the global minimum. For a survey of different extensions of RLTsee Sherali and Liberti (2009).1.5 Groebner Basis TechniqueGroebner bases can be used to reduce a polynomial optimization over a semi-algebraic set S := { x ∈ R n : g i ( x ) ≥ , h j ( x ) = 0 } to the problem of finding the rootsof univariate polynomials (Chang and Wah (1994)). First, one needs to construct thesystem of polynomial equations[ ∇ x L ( x, λ, µ ) , ∇ λ L ( x, λ, µ ) , ∇ µ L ( x, λ, µ )] = 0 , (1.5)where L := p ( x )+ (cid:80) i λ i g i ( x )+ (cid:80) j µ j h j ( x ) is the Lagrangian function. It is well-knownthat the set of solutions to (1.5) is the set of extrema of the polynomial optimizationmin x ∈ S p ( x ). Let[ f ( x, λ, µ ) , · · · , f N ( x, λ, µ )] := [ ∇ x L ( x, λ, µ ) , ∇ λ L ( x, λ, µ ) , ∇ µ L ( x, λ, µ )] . f , · · · , f N defines a triangular-formsystem of polynomial equations. This system can be solved by calculating one variableat a time and back-substituting into other polynomials. The most computationallyexpensive part is the calculation of the Groebner basis, which in the worst case scalesdouble-exponentially in the number of decision variables.1.6 Blossoming TechniqueThe blossoming technique involves a bijective map between the space of poly-nomials p : R n → R and the space of multi-affine functions q : R d + d + ··· + d n → R (polynomials that are affine in each variable), where d i is the degree of p in variable x i . For instance, the blossom of a cubic polynomial p ( x ) = ax + bx + cx + d is themulti-affine function q ( z , z , z ) = az z z + b z z + z z + z z ) + c z + z + z ) + d. It can be shown that the blossom, q , of any polynomial p ∈ R [ x ] with degree d i invariable x i satisfies the so-called diagonal property (Ramshaw (1987)), i.e., p ( z , z , · · · , z n ) = q ( z , · · · , z (cid:124) (cid:123)(cid:122) (cid:125) d times , · · · , z n , · · · , z n (cid:124) (cid:123)(cid:122) (cid:125) d n times ) for all z ∈ R . By using this property, one can reformulate any polynomial optimization min x ∈ S p ( x )as min z ∈ Q q ( z )subject to z φ ( i ) = z φ ( i ) − j for i = 1 , · · · , n and for j = 1 , · · · , d i − , (1.6)where φ ( i ) := i (cid:80) k =1 d i and Q is the semi-algebraic set defined by the blossoms of thegenerating polynomials of S . In the special case, where S is a hypercube, Sassi and Gi-rard (2012) showed that the Lagrangian dual optimization problem to Problem (1.6)7s a linear program. Hence, the optimal objective value of this linear program is alower bound on the minimum of p ( x ) over the hypercube. Application of blossomingin estimation of reachability sets of discrete-time dynamical systems can be foundin Sassi et al. (2012).1.7 Bernstein, Polya and Handelman TheoremsWhile QE, RLT, Groebner bases and blossoming are all useful techniques withadvantages and disadvantages (such as exponential complexity), we focus on Polya’s,Bernstein’s and Handelman’s theorems - results which yield polynomial-time testsfor positivity of polynomials. Polya’s theorem yields a basis to represent the coneof polynomials that are positive over the positive orthant. Bernstein’s and Handel-man’s theorems yield bases which represent the cones of polynomials that are positiveover simplices and convex polytopes, respectively. Similar to SOS programming, onecan find certificates of positivity using Polya’s, Bernstein’s and Handelman’s repre-sentations by solving a sequence of Linear Programs (LPs) and/or SDPs. However,unlike the SDPs associated with SOS programming, the SDPs associated with thesetheorems have a block-diagonal structure. In this dissertation, we exploit this struc-ture to design parallel algorithms for optimization of polynomials of high degreeswith several independent variables. See Kamyar and Peet (2012a), Kamyar and Peet(2012b), Kamyar and Peet (2013) and Kamyar et al. (2013) for parallel implementa-tions of variants of Polya’s theorem applied to various Lyapunov inequalities.Unfortunately, unlike the SOS methodology, the bases given by Polya’s theorem,Bernstein’s theorem and Handelman’s theorem cannot be used to represent the coneof non-negative polynomials which have zeros in the interior of simplices and poly-topes. This is indeed a barrier against using these theorems to compute polynomialLyapunov functions, since Lyapunov functions, by definition, have a zero at the ori-8in. There do, however, exist some variants of Polya’s theorem which consider zerosat the corners Powers and Reznick (2006) and edges Castle et al. (2011) by con-structing local certificates of non-negativity over closed subsets, C i , of the simplexsuch that ∪ C i is the simplex. These results apply to non-negative polynomials whosezeros are on the corners and/or edges of the simplex. Moreover, Oliveira et al. (2008)and Kamyar and Peet (2012b) propose versions of Polya’s theorem which prove pos-itivity over hypercubes by: 1) Providing certificates of positivity on the Cartesianproduct of unit simplices; and 2) Introducing a one-to-one map between products ofunit simplices (multi-simplex) and hypercubes. A generalization of Polya’s theoremfor proving positivity on the entire R n was introduced by de Loera and Santos (1996).This generalization first applies Polya’s theorem to each orthant of R n to computea certificate of positivity over each orthant. Then, it uses the merging techniquein Lombardi (1991) to obtain a unified certificate - in the form of SOS of rationals -over R n . A recent extension of Polya’s theorem by Dickinson and Pohv (2014) can beused to prove positivity over an intersection of a semi-algebraic set with the positiveorthant. Finally, positivity of polynomials with rational exponents can be verified bya weak version of Polya’s theorem in Delzell (2008).1.8 Motivations and Summary of ContributionsThe novelty of our research centers on the areas of: computation and energy.In the realm of computation, we observed that processors speeds are not growingat the rate they once were. The entire controls community seems to have ignoredthis fact, since everyone speaks of polynomial-time algorithms as the gold standardfor what the solution to a control problem should look like. But what good is apolynomial-time algorithm when the degree of the polynomial is bounded by thecurrent state-of-the-art computers. Our solution was to look at the only area where9he computing world was getting faster (growing) - supercomputers. Surprisingly,there have been no studies on the use of parallel computers for controls since the1970’s. The reason was that the mathematical machinery for analysis and control isbased on Semidefinite Programming, which is inherently sequential (NC-hard). Ouridea, however, was that if the SDP problem has special structure, then this structurecan be exploited to distribute computation among processors. With this in mind, wedecided to seek out alternatives to the classical Sum-of-Squares approach to nonlinearand robust stability analyses. We identified more than seven different alternatives tothe Sum-of-Squares approach. In the end, not all of these had usable structures forparallelization. However, we identified three which did: polynomial positivity resultsby Handelman, Polya and Bernstein. To demonstrate how well this approach worksin practice, we developed a Message Passing Interface code for Polya’s theorem. Theresult enabled stability analysis for systems three times larger (in terms of numberof states) than any other algorithm. As a real-world application, we further used ourcode to analyze robust stability of plasma in the Tore Supra Tokamak reactor.In the realm of energy, we noticed that the two electrical utility companies ofArizona (APS and SRP) have recently started charging their customers for theirmaximum rate of electricity usage. This intrigued us as a mathematical problemof how to optimize the thermostat settings of HVAC systems (the major sources ofelectricity consumption in Arizona) in order to minimize the electricity bill. Thisproblem is interesting in that the time of peak electricity use is not usually at thehottest time of day, but rather a couple of hours after - a behaviour which is usuallyassociated with a diffusion PDE. We used the heat equation to model the thermostatprogramming problem as an optimal control problem and it turned out to be unsolved.The mathematical reason being that the cost function is not separable in time - aproperty which is necessary for optimal control algorithms to converge to an optimal10olution. We noticed that an arbitrarily precise approximation of the cost functionhowever, satisfy certain properties which make it solvable on a Pareto-optimal front.The result is an optimal thermostat which can significantly reduce the electricity billsand peak demand of both solar and nonsolar customers under the current pricingplans. Expanding this approach, we started thinking about related topics, such ashow to set the demand price on order to influence customers’ behavior in an optimalmanner. Based on that, we proposed an optimal pricing algorithm which resultedin a moderate reduction in the cost of generating, transmission and distribution ofelectricity at SRP.We highlight our contributions as follows. In Chapter 4, we propose a parallel set-up algorithm which applies Polya’s theorem to the parameter-dependent Lyapunovinequalities P ( α ) > A T ( α ) P ( α ) + P ( α ) A ( α ) < α belonging to the stan-dard simplex. Feasibility of these inequalities implies robust stability of the systemof linear Ordinary Differential Equations (ODEs) ˙ x ( t ) = A ( α ) x ( t ) over the simplex.The output of our set-up algorithm is a sequence of SDPs of increasing size and pre-cision. A solution to any of these SDPs yield a Lyapunov function which is quadraticin the states and depends polynomially on the uncertain parameters. An interestingproperty of these SDPs is that they possess a block-diagonal structure. We show howthis structure can be exploited to design a parallel interior-point primal-dual SDPsolver which distributes the computation of search direction among a large number ofprocessors. We then produce a Message Passing Interface (MPI) implementation ofour set-up and solver algorithms. Through numerical experiments, we show that thesealgorithms achieve a near-linear theoretical and experimental speed-up (the increasein processing speed per additional processor). Moreover, our numerical experimentson cluster computers demonstrate the ability of our algorithms in utilizing hundredsand potentially thousands of processors to analyze systems with 100+ states.11n Chapter 5, we generalize our methodology to perform robust stability analy-sis over hypercubes. We first propose an extended version of Polya’s theorem. Thistheorem parameterizes every homogeneous polynomial which is positive over a hyper-cube. We then propose an extended set-up algorithm which maps the computationand memory - associated with applying the extended Polya’s theorem to stabilityanalysis problems - to parallel machines. This set-up algorithm has no centralizedcomputation and its per-core communication complexity scales polynomially withthe state-space dimension and the number of uncertain parameters. As the result, itdemonstrates a near-linear speed-up.In Chapter 6, we further extend our analysis to address stability of nonlinearODEs defined by a polynomial vector field f . Our proposed solution to this prob-lem is to reformulate the nonlinear stability problem using only strictly positiveforms. Specifically, we use our extended version of Polya’s theorem in Chapter 5to compute a matrix-valued homogeneous polynomial P ( x ) such that P ( x ) > (cid:104)∇ ( x T P ( x ) x ) , f ( x ) (cid:105) < x inside a hypercube containing the origin in its in-terior. This yields a Lyapunov function of the form V ( x ) = x T P ( x ) x for the system˙ x ( t ) = f ( x ( t )). To do this, we design a new parallel set-up algorithm which appliesPolya’s theorem to the inequalities P ( x ) > (cid:104)∇ ( x T P ( x ) x ) , f ( x ) (cid:105) <
0. The resultis a sequence of SDPs with coefficients of P as decision variables. Again, we showthat these SDPs have a block-diagonal structure - thus can be solved in parallel usingour SDP solver in Chapter 4. As an extension to stability analysis over arbitraryconvex polytopes, we then propose an algorithm which applies Handelman’s theo-rem to the aforementioned Lyapunov inequalities. Unfortunately, as in the case ofPolya’s theorem, Handelman’s theorem is incapable of parameterizing polynomialswhich possess zeros in the interior of a polytope. However, we show that this is notthe case if the zeros are on the vertices of the polytope. By using this property,12e propose the following methodology: 1) Decompose the polytope into several con-vex sub-polytopes with a common vertex on the equilibrium; 2) Apply Handelman’stheorem to Lyapunov inequalities defined on each sub-polytope. The result is a se-quence of linear programs whose solutions define a piecewise polynomial Lyapunovfunction V - hence proving asymptotic stability over the sublevel-set of V inscribed inthe original polytope. We provide a comprehensive comparison between the compu-tational complexities of SOS algorithm, our Polya’s algorithms and our Handelmanalgorithm. Our analysis shows that by using a certain decomposition scheme, ouralgorithm (based on Handelman’s theorem) has the lowest computational complexitycompared to the SOS and Polya’s algorithms.13hapter 2FUNDAMENTAL RESULTS FOR OPTIMIZATION OF POLYNOMIALSIn this chapter, we first provide an overview of fundamental theorems on posi-tivity of polynomials over various sets. Then, we show how applying these theoremsto optimization of polynomials problems of the Form (1.1) yields tractable convexoptimization problems in the forms of LPs and/or SDPs. Any solution to these LPsand/or SDPs yields a lower-bound on the global minimum of the polynomial opti-mization problem min x ∈ Q p ( x ).2.1 Background on Positivity ResultsIn 1900, Hilbert published a list of mathematical problems, one of which is: Forevery non-negative f ∈ R [ x ], does there exist any non-zero q ∈ R [ x ] such that q f is asum of squares? In other words, is every non-negative polynomial a sum of squares ofrational functions? This question was motivated by his earlier works (Hilbert (1888,1893)), in which he proved: 1) Every non-negative bi-variate degree 4 homogeneous polynomial (A polynomial whose monomials all have the same degree) is a SOS ofthree polynomials; 2) Every bi-variate non-negative polynomial is a SOS of four ra-tional functions; 3) Not every non-negative homogeneous polynomial with more thantwo variables and degree greater than 5 is SOS of polynomials. While there existsystematic ways (e.g., semi-definite programming) to prove that a non-negative poly-nomial is SOS, proving that a non-negative polynomial is not a SOS of polynomialsis not straightforward. Indeed, the first example of a non-negative non-SOS polyno-mial was published eighty years after Hilbert posed his 17 th problem. Motzkin (1967)14onstructed a PSD degree 6 polynomial with three variables which is not SOS: M ( x , x , x ) = x x + x x − x x x + x . (2.1)Non-negativity of M follows directly from the inequality of arithmetic and geometricmeans, i.e., ( a + · · · + a n ) /n ≥ n √ a · · · a n , by letting n = 3 , a = x x , a = x x and a = x . To show that M is not SOS, first by contradiction suppose that there existsome N ∈ N and coefficients b i,j ∈ R such that M ( x , x , x ) = N (cid:88) i =1 (cid:32)(cid:104) b i, · · · b i, (cid:105)(cid:104) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x (cid:105) T (cid:33) . (2.2)By substituting (2.1) in (2.2) and equating the coefficients of both sides of (2.2), itfollows that (cid:80) Ni =1 b i, = −
3. This is a contradiction, thus M is not SOS of polyno-mials. A generalization of Motzkin’s example is given by Robinson (Reznick (2000)).Polynomials of the form ( (cid:81) ni =1 x i ) f ( x , · · · , x n ) + 1 are not SOS if polynomial f ofdegree < n is not SOS. Hence, although the non-homogeneous Motzkin polynomial M ( x , x ,
1) = x x ( x + x −
3) + 1 is non-negative it is not SOS.Artin (1927) answered Hilbert’s problem in the following theorem.
Theorem 1. (Artin’s theorem) A polynomial f ∈ R [ x ] satisfies f ( x ) ≥ on R n ifand only if there exist SOS polynomials N and D (cid:54) = 0 such that f ( x ) = N ( x ) D ( x ) . Although Artin settled Hilbert’s problem, his proof was neither constructive norgave a characterization of the numerator N and denominator D . In 1939, Habichtprovided some structure on N and D for a certain class of polynomials f . Habicht(1939) showed that if a homogeneous polynomial f is positive definite and can be ex-pressed as f ( x , · · · , x n ) = g ( x , · · · , x n ) for some polynomial g , then one can choosethe denominator D = (cid:80) ni =1 x i . Moreover, he showed that by using D = (cid:80) ni =1 x i ,15he numerator N can be expressed as a sum of squares of monomials. Habicht usedPolya’s theorem (Hardy et al. (1934), Theorem 56) to obtain the above characteriza-tions for N and D . Theorem 2. (Polya’s theorem) Suppose a homogeneous polynomial p satisfies p ( x ) > for all x ∈ { x ∈ R n : x i ≥ , (cid:80) ni =1 x i (cid:54) = 0 } . Then p ( x ) can be expressed as p ( x ) = N ( x ) D ( x ) , where N ( x ) and D ( x ) are homogeneous polynomials with all positive coefficients. Fur-thermore, for every homogeneous p ( x ) and some e ≥ , the denominator D ( x ) can bechosen as ( x + · · · + x n ) e . To see Habicht’s result, suppose f is homogeneous and positive on the positiveorthant and can be expressed as f ( x , · · · , x n ) = g ( x , · · · , x n ) for some homogeneouspolynomial g . By using Polya’s theorem, g ( y ) = N ( y ) D ( y ) , where y := ( y , · · · , y n ) andpolynomials N and D have all positive coefficients. Furthermore, from Theorem 2we may choose D ( y ) = ( (cid:80) ni =1 y i ) e . Then, ( (cid:80) ni =1 y i ) e g ( y ) = N ( y ). Now let x i = √ y i ,then ( (cid:80) ni =1 x i ) e f ( x , · · · , x n ) = N ( x , · · · , x n ). Since N has all positive coefficients, N ( x , · · · , x n ) is a sum of squares of monomials.Similar to the case of positive definite polynomials, ternary positive semi-definitepolynomials of the form g ( x , x , x ) can be parameterized using the denominator D = ( x + x + x ) N (Scheiderer (2006)). However, in any dimension higher thanthree, there exist positive semi-definite polynomials f such that if h f is SOS, then h has a zero other than the origin. Thus, for such polynomials f , Df cannot beSOS. Indeed, it has been shown by Reznick (2005) that there exists no single SOSpolynomial D (cid:54) = 0 which satisfies f = ND for every positive semi-definite f and someSOS polynomial N . 16s in the case of positivity on R n , there has been an extensive research regardingpositivity of polynomials on bounded sets. A pioneering result on local positivity isBernstein’s theorem (Bernstein (1915)). Bernstein’s theorem uses the polynomials h i,j = (1 + x ) i (1 − x ) j as a basis to parameterize univariate polynomials which arepositive on [ − , Theorem 3. (Bernstein’s theorem) If a polynomial f ( x ) > on [ − , , then thereexist c i,j > such that f ( x ) = (cid:88) i,j ∈ N : i + j = d c i,j (1 + x ) i (1 − x ) j for some d > . Powers and Reznick (2000) used Goursat’s transformation of f to find an up-per bound on d . Unfortunately, the bound itself is a function of the minimum of f on [ − , et al. (2008) proposed a decomposition scheme for breaking [ − , f over each sub-interval to find a certificate of positivity over each sub-interval. Anextension of this technique was proposed in Leroy (2012) to verify positivity oversimplices (a simplex is the convex hull of n + 1 vertices in R n ). Moreover, Leroy(2012) provided a degree bound as a function of the minimum of f over the simplex,the number of variables in f , the degree of f and the maximum of certain affinecombinations of the coefficients c i,j .Handelman (1988b) also used products of affine functions as a basis (the Han-delman basis) to extend Bernstein’s theorem to multi-variate polynomials which arepositive on convex polytopes. Theorem 4. (Handelman’s Theorem) Given w i ∈ R n and u i ∈ R , define the polytope Γ K := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } . If a polynomial f ( x ) > on Γ K , then here exist b α ≥ , α ∈ N K such that for some d ∈ N , f ( x ) = (cid:88) α ∈ N K α + ··· + α K ≤ d b α ( w T x + u ) α · · · ( w TK x + u K ) α K . (2.3)Recently, S. Sankaranarayanan and Abrahm (2013) combined the Handelman ba-sis with positive basis functions x α · · · x α n n − l α and u α − x α · · · x α n n to compute Lyapunov functions over a hypercube Φ, where l α and u α are the minimumand maximum of x α · · · x α n n over the hypercube Φ. A generalization of Handelman’stheorem was made by Schweighofer (2005) to verify non-negativity of polynomialsover compact semi-algebraic sets. Schweighofer used the cone of polynomials definedin (2.5) to parameterize any polynomial f which has the following properties:1. f is non-negative over the compact semi-algebraic set S defined in (2.4)2. f = q p + q p + · · · for some q i in the Cone (2.5) and for some p i > S ∩ { x ∈ R n : f ( x ) = 0 } Theorem 5. (Schweighofer’s theorem) Suppose S := { x ∈ R n : g i ( x ) ≥ , g i ∈ R [ x ] for i = 1 , · · · , K } (2.4) is compact. Define the following set of polynomials which are positive on S . Θ d := (cid:88) λ ∈ N K : λ + ··· + λ K ≤ d s λ g λ · · · g λ K K : s λ are SOS (2.5) If f ≥ on S and there exist q i ∈ Θ d and polynomials p i > on S ∩ { x ∈ R n : f ( x ) =0 } such that f = (cid:80) i q i p i for some d , then f ∈ Θ d . A set of polynomials S ⊂ R [ x , · · · , x n ] is a cone if: 1) f ∈ S and f ∈ S imply f f ∈ S and f + f ∈ S ; and 2) f ∈ R [ x , · · · , x n ] implies f ∈ S .
18n the assumption that g i are affine functions, p i = 1 and s λ are constant,Schweighofer’s theorem gives the same parameterization of f as in Handelman’s the-orem. Another special case of Schweighofer’s theorem is when λ ∈ { , } K . In thiscase, Schweighofer’s theorem reduces to Schmudgen’s Positivstellensatz (Schmudgen(1991)). Schmudgen’s Positivstellensatz states that the coneΛ g := (cid:88) λ ∈{ , } K s λ g λ · · · g λ K K : s λ are SOS ⊂ Θ d (2.6)is sufficient to parameterize every f > S generated by { g , · · · , g K } . Unfortunately, the cone Λ g contains 2 K products of g i , thus finding arepresentation of Form (2.6) for f requires a search for at most 2 K SOS polynomials.Putinar’s Positivstellensatz (Putinar (1993)) reduces the complexity of Schmudgen’sparameterization in the case where the quadratic module M g (as defined in (2.8)) ofpolynomials g i is Archimedean , i.e., there exists N ∈ N such that N − n (cid:88) i =1 x i ∈ M g . (2.7)Equivalently, if there exists some f ∈ M g such that { x ∈ R n : f ( x ) ≥ } is compact,then M g is Archimedean. Theorem 6. (Putinars’s Positivstellensatz) Let S := { x ∈ R n : g i ( x ) ≥ , g i ∈ R [ x ] for i = 1 , · · · , K } and define M g := (cid:40) s + K (cid:88) i =1 s i g i : s i are SOS (cid:41) . (2.8) If there exist some
N > such that N − (cid:80) ni =1 x i ∈ M g , then M g is Archimedean. If M g is Archimedean and f > over S , then f ∈ M g . Finding a representation of Form (2.8) for f , only requires a search for K + 1 SOSpolynomials using SOS programming. Verifying the Archimedian Condition (2.7)19s also an SOS program. Observe that if M g is not Archimedean, one can add aredundant constraint r − (cid:80) ni =1 x i ≥ r ∈ R ) to S in order tomake M g Archimedean. Archimedean condition clearly implies compactness of thesemi-algebraic set S because for any f ∈ M g , S ⊂ { x ∈ R n : f ( x ) ≥ } . The followingtheorem lifts the compactness requirement for the semi-algebraic set S . Theorem 7. (Stengle’s Positivstellensatz) Let S := { x ∈ R n : g i ( x ) ≥ , g i ∈ R [ x ] for i = 1 , · · · , K } and define the cone Λ g := (cid:88) λ ∈{ , } K s λ g λ · · · g λ K K : s λ are SOS . If f > on S , then there exist p, g ∈ Λ g such that qf = p + 1 . Notice that the Parameterziation (2.3) in Handelman’s theorem is affine in f and the coefficients b α . Likewise, the parameterizations in Theorems 5 and 6, i.e., f = (cid:80) λ s λ g λ · · · g λ K K and f = s + (cid:80) i s i g i are affine in f, s λ and s i . Thus, one canuse convex optimization to find b α , s λ , s i and f efficiently. Unfortunately, since theparameterization qf = p + 1 in Stengle’s Positivstellensatz is non-convex (bilinearin q and f ), it is more difficult to verify qf = p + 1 compared to Handelman’s andPutinar’s parameterizations.For a comprehensive discussion on the Positivstellensatz and other results onpolynomial positivity in algebraic geometry see Laurent (2009); Scheiderer (2009),and Prestel and Delzell (2004).2.2 Polynomial Optimization and Optimization of PolynomialsGiven f, g i , h j ∈ R [ x ] for i = 1 , · · · , m and j = 1 , · · · , r , define a semi-algebraicset S as S := { y ∈ R n : g i ( y ) ≥ , h j ( y ) = 0 for i = 1 , · · · , m and j = 1 , · · · , r } . (2.9)20e then define polynomial optimization problems as β ∗ = min x ∈ S f ( x ) . (2.10)For example, the integer programmin x ∈ R n p ( x )subject to a Ti x ≥ b i for i = 1 , · · · , m,x ∈ {− , } n , (2.11)with given a i ∈ R n , b i ∈ R and p ∈ R [ x ], can be formulated as a polynomial optimiza-tion problem by setting f = p in (2.10) and setting g i ( x ) = a Ti x − b i for i = 1 , · · · , mh j ( x ) = x j − j = 1 , · · · , n. in the definition of S in (2.9).Given c ∈ R n and g i , h j ∈ R [ x ] for i = 1 , · · · , m and j = 1 , · · · , r , we define Optimization of polynomials problems as γ ∗ = max x ∈ R q c T x subject to F ( x, y ) := F ( y ) + q (cid:88) i =1 x i F i ( y ) ≥ y ∈ S, (2.12)where S is defined in (2.9) and F i ( y ) := (cid:88) α ∈ E di F i,α y α · · · y α n n with E d i := { α ∈ N n : (cid:80) ni =1 α i ≤ d i } , where coefficients F i,α ∈ R t × t , i = 0 , · · · , q are given. If the goal is to optimize over a polynomial variable, p ( y ), this may beachieved using a basis of monomials for F i ( y ) so that the polynomial variable becomes21 ( y ) = (cid:80) i x i F i ( y ). Optimization of polynomials can be used to find β ∗ in (2.10). Forexample, we can compute the optimal objective value η ∗ of the polynomial optimiza-tion problem η ∗ = min x ∈ R n p ( x )subject to a Ti x − b i ≥ i = 1 , · · · , m,x j − j = 1 , · · · , n, by solving the problem η ∗ = max η ∈ R η subject to p ( y ) ≥ η for y ∈ { y ∈ R n : a Ti y ≥ b i , y j − i = 1 , · · · , m and j = 1 , · · · , n } , (2.13)where Problem (2.13) can be expressed in the Form (2.12) by setting c = 1 , q = 1 , t = 1 , F = p F = − ,S := { y ∈ R n : a Ti y ≥ b i , y j − i = 1 , · · · , m, and j = 1 , · · · , n } . Optimization of polynomials (2.12) can be reformulated as the feasibility problem γ ∗ = min γ γ subject to S γ := (cid:8) x ∈ R q : c T x > γ, F ( x, y ) ≥ y ∈ S (cid:9) = ∅ , (2.14)where c and F are given and S := { y ∈ R n : g i ( y ) ≥ , h j ( y ) = 0 for i = 1 , · · · , m and j = 1 , · · · , r } , where polynomials g i and h j are given. The question of feasibility of a semi-algebraicset is NP-hard (L. Blum and Smale (1998)). However, if we have a test to verify22 γ = ∅ , we can find γ ∗ by performing a bisection on γ . In the following section, weuse the results of Section 2.1 to provide sufficient conditions, in the form of LinearMatrix Inequalities (LMIs), for S γ = ∅ .2.3 Algorithms for Optimization of PolynomialsIn this section, we discuss how to find lower bounds on β ∗ for different classes ofpolynomial optimization problems. The results in this section are primarily expressedas methods for verifying S γ = ∅ and can be used with bisection to solve polynomialoptimization problems. ∆ n Define the standard unit simplex as∆ n := { x ∈ R n : n (cid:88) i =1 x i = 1 , x i ≥ } . (2.15)Consider the polynomial optimization problem γ ∗ = min x ∈ ∆ n f ( x ) , where f is a homogeneous polynomial of degree d . If f is not homogeneous, wecan homogenize it by multiplying each monomial x α · · · x α n n in f by ( (cid:80) ni =1 x i ) d −(cid:107) α (cid:107) .Notice that since (cid:80) ni =1 x i = 1 for all x ∈ ∆ n , the homogenized f is equal to f forevery x ∈ ∆ n . To find γ ∗ , one can solve the following optimization of polynomialsproblem. γ ∗ = max γ ∈ R γ subject to f ( x ) ≥ γ for all x ∈ ∆ n (2.16)23learly, Problem (2.16) can be re-stated as the following feasibility problem γ ∗ = min γ ∈ R γ subject to S γ := { x ∈ R n : f ( x ) − γ < , n (cid:88) i =1 x i = 1 , x i ≥ } = ∅ . For a given γ , we can use the following version of Polya’s theorem to verify S γ = ∅ . Theorem 8. (Polya’s theorem, simplex version) If a homogeneous matrix-valued poly-nomial F satisfies F ( x ) > for all x ∈ ∆ n := { x ∈ R n : (cid:80) ni =1 x i = 1 , x i ≥ } , thenthere exists e ≥ such that all the coefficients of (cid:32) n (cid:88) i =1 x i (cid:33) e F ( x ) are positive definite. See pages 57-59 of G. Hardy and Polya (1934) for a proof. The converse of thetheorem only implies F ≥ γ ∈ R , it follows from theconverse of Theorem 8 that S γ = ∅ if there exists some e ≥ (cid:32) n (cid:88) i =1 x i (cid:33) e f ( x ) − γ (cid:32) n (cid:88) i =1 x i (cid:33) d (2.17)has all positive coefficients, where recall that d is the degree of f . We can computelower bounds on γ ∗ by performing a bisection on γ . For each γ of the bisection, ifthere exists some e ≥ γ ≤ γ ∗ . We have detailed this procedure in Algorithm 1.In Chapter 4, we will propose a decentralized version of Algorithm 1 to performrobust stability analysis over a simplex. Φ n Given r i ∈ R , define the hypercubeΦ n := { x ∈ R n : | x i | ≤ r i , i = 1 , · · · , n } . (2.18)24 nput: Polynomial f ; maximum polya’s exponent e max ; lower-bound γ l andupper-bound γ u for bisection search; number of bisection iterations b max ; Initialization:
Set Polya’s exponent e = 0.Set k = 0. Main Loop: while d ≤ b max do Set γ = γ u + γ l . while Eq. (2.17) has some negative coefficient or e ≤ e max do Set e = e + 1.Calculate the Product (2.17). endif Eq. (2.17) has all positive coefficients then
Set γ l = γ . else Set γ u = γ . end Set k = k + 1. end Output: γ : a lower bound on the minimum of f over the standard simplex. Algorithm 1:
Polya’s algorithm for polynomial optimization over the simplex25efine the set of n -variate multi-homogeneous polynomials of degree vector d ∈ N n as p ∈ R [ x, y ] : p ( x, y ) = (cid:88) h,g ∈ N n h + g = d p h,g x h y g · · · x h n n y g n n , p h,g ∈ R . (2.19)In a more general case, if the coefficients p h,g are matrices, we call p a matrix-valuedmulti-homogeneous polynomial. Now consider the polynomial optimization problem γ ∗ = min x ∈ Φ n f ( x ) . To find γ ∗ , one can solve the following feasibility problem. γ ∗ = min γ ∈ R γ subject to S γ,r := { x ∈ R n : f ( x ) − γ < , | x i | ≤ r i , i = 1 , · · · , n } = ∅ (2.20)For a given γ , we propose the following version of Polya’s theorem (Kamyar and Peet(2012b)) to verify S γ,r = ∅ . Theorem 9. (Polya’s theorem: multi-simplex version) A matrix-valued multi-homogeneouspolynomial F satisfies F ( x, y ) > for all ( x i , y i ) ∈ ∆ , i = 1 , · · · , n , if there exist e ≥ such that all the coefficients of (cid:32) n (cid:89) i =1 ( x i + y i ) e (cid:33) F ( x, y ) are positive definite. We will prove this result in Section 5.2. The converse of Theorem 9 only impliesnon-negativity of F over the hypercube. To find lower bounds on γ , we first obtainthe multi-homogeneous form p of the polynomial f in (2.20). In 5.2 we have provideda procedure to construct p . Given γ ∈ R and r ∈ R n , it follows from the converse ofTheorem 9 that S γ,r defined in (2.20) is empty if there exists some e ≥ (cid:32) n (cid:89) i =1 ( x i + y i ) e (cid:33) (cid:32) p ( x, y ) − γ (cid:32) n (cid:89) i =1 ( x i + y i ) d i (cid:33)(cid:33) (2.21)26as all positive coefficients, where d i is the degree of x i in p ( x, y ). We can computelower bounds on γ ∗ , as defined in (2.20), by performing a bisection on γ . For each γ of the bisection, if there exists some e ≥ γ ≤ γ ∗ . By replacing (2.17) with (2.21) in Algorithm 1, this algorithmcomputes γ . In Chapter 5, we will propose a parallel algorithm to perform robuststability analysis for systems with uncertain parameters inside a hypercube. Γ K Given w i ∈ R n and u i ∈ R , define the convex polytopeΓ K := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } . (2.22)Suppose Γ K is bounded. Consider the polynomial optimization problem γ ∗ = min x ∈ Γ K f ( x ) , where f is a polynomial of degree d f . To find γ ∗ , one can solve the feasibility problem γ ∗ = min γ ∈ R γ subject to S γ,K := { x ∈ R n : f ( x ) − γ < , w Ti x + u i ≥ , i = 1 , · · · , K } = ∅ . Given γ , one can use Handelman’s theorem to verify S γ,K = ∅ . Theorem 10. (Handelman’s Theorem) Given w i ∈ R n and u i ∈ R , define the polytope Γ K := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } . If a polynomial f ( x ) > on Γ K , thenthere exist b α ≥ , α ∈ N K such that for some d ∈ N , f ( x ) = (cid:88) α ∈ N K α + ··· + α K ≤ d b α ( w T x + u ) α · · · ( w TK x + u K ) α K . (2.23)Consider the Handelman basis associated with polytope Γ K defined as H s := (cid:40) λ α ∈ R [ x ] : λ α ( x ) = K (cid:89) i =1 (cid:0) w Ti x + u i (cid:1) α i , α ∈ N K , K (cid:88) i =1 α i ≤ s (cid:41) . H s spans the space of polynomials of degree s or less, however it is not minimal.As a special case, if we take Γ K to be the standard unit simplex of R K , i.e.,Γ K := { x ∈ R K : 1 − K (cid:88) i =1 x i ≥ , x i ≥ i = 1 , · · · , K } , then the following set of polynomials is called the Bernstein basis associated with Γ K . B s := (cid:40) λ α ∈ R [ x ] : λ α ( x ) = s ! α ! · · · α K +1 ! (cid:32) K (cid:89) i =1 x α i i (cid:33)(cid:32) − K (cid:88) i =1 x i (cid:33) α K +1 , α ∈ N K +1 , K (cid:88) i =1 α i = s (cid:41) . Unlike H s , B s is a minimal basis for the vector space of polynomials of degree ≤ s .Given γ ∈ R , polynomial f ( x ) of degree d f and d max ∈ N , if there exist c α ≥ α ∈ I d := { α ∈ N K : (cid:107) α (cid:107) ≤ d } (2.24)such that f ( x ) − γ = (cid:88) α ∈ I d c α K (cid:89) i =1 ( w Ti x + u i ) α i (2.25)for some d ≥ d f , then f ( x ) − γ ≥ x ∈ Γ K . Thus S γ,K = ∅ . Since H s is not a minimal basis, if (2.24) is feasible, then c α are not unique. Feasibility ofConditions (2.24) and (2.25) can be determined using linear programming. To set-upthe linear program, we first represent the right and left hand side of (2.25) in thecanonical basis as f ( x ) − γ = (cid:20) b b · · · b M (cid:21) z n,d ( x ) (2.26) (cid:88) α ∈ I d c α K (cid:89) i =1 ( w Ti x + u i ) α i = (cid:20) l ( c α , w, u ) l ( c α , w, u ) · · · l M ( c α , w, u ) (cid:21) z n,d ( x ) , (2.27) This follows from the fact that every polynomial can be uniquely represented in the canonicalbasis and every member of the canonical basis is a unique linear combination of λ α ∈ B s . Aderivation for these linear combinations can be found in Farin (2002) l i : R N × R n × K × R K → R are affine in c α and N := (cid:18) K + dd (cid:19) is the cardinality of the index set { α ∈ N K : (cid:107) α (cid:107) ≤ d } . In (2.27), w := (cid:104) w w · · · w K (cid:105) and u := (cid:104) u u · · · u K (cid:105) T , where w i ∈ R n and u i ∈ R define the polytope Γ K in (2.22). Recall that in (2.27), z n,d ( x ) denotes the vector of all n − variate monomials of degree d or less. By equat-ing (2.26) and (2.27) and cancelling z n,d ( x ) from both sides, the problem of finding alower bound γ d on γ ∗ can be expressed as the following linear program. γ d := max γ ∈ R ,c α ≥ γ subject to l i ( c α , w, u ) = b i for i = 1 , · · · , M. (2.28)If Linear Program (2.28) is infeasible for some d , then one can increase d and repeatsetting-up and solving Linear Program (2.28). From Handelman’s theorem, if f ( x ) − γ > x ∈ Γ K , then for some d ≥ d f , Conditions (2.24) and (2.25) holdand Linear Program (2.28) will have a solution. We have outlined this procedure inAlgorithm 2. Unfortunately, to this date all the proposed upper-bounds on d (seee.g., Powers and Reznick (2001) and Leroy (2012)) are functions of the minimumof f ( x ) − γ over the polytope Γ K . In Chapter 6, we will combine this algorithmwith a polytope decomposition scheme to construct Lyapunov functions for nonlinearsystems with polynomial vector fields. Recall that we defined a semi-algebraic set as S := { x ∈ R n : g i ( x ) ≥ , i = 1 , · · · , m, h j ( x ) = 0 , j = 1 , · · · , r } . (2.29)29 nput: Polynomial f of degree d f ; maximum degree d max for Handelman’sbasis; polytope data: w i ∈ R n and u i ∈ R in (2.22); tolerance (cid:15) > Initialization:
Set d = d f .Set γ old = − .Set γ new = − + 2 (cid:15) . Main Loop:while ( d < d max ) and ( γ new − γ old < (cid:15) ) do Express f ( x ) − γ new in the canonical basis as in (2.26).Express (cid:80) α ∈ I d c α (cid:81) Ki =1 ( w Ti x + u i ) α i in the canonical basis as in (2.27).Set-up LP (2.28). if LP (2.28) is feasible then Set γ old = γ new .Set γ new = γ d . end Set d = d + 1. endOutput: γ new : a lower bound on the minimum of f over polytope Γ K . Algorithm 2:
Polynomial optimization over convex polytopes using Handel-man’s theorem 30uppose S is bounded. Consider the polynomial optimization problem γ ∗ = min x ∈ R n f ( x )subject to g i ( x ) ≥ i = 1 , · · · , mh j ( x ) = 0 for j = 1 , · · · , r. Define the following cone of polynomials which are positive over S . M g,h := (cid:40) m ∈ R [ x ] : m ( x ) − m (cid:88) i =1 s i ( x ) g i ( x ) − r (cid:88) i =1 t i ( x ) h i ( x ) is SOS , s i ∈ Σ d , t i ∈ R [ x ] (cid:41) , (2.30)where Σ d denotes the cone of SOS polynomials of degree 2 d . From Putinar’s Posi-tivstellensatz (Theorem 6) it follows that if the Cone (2.30) is Archimedean, then thesolution to the following SOS program is a lower bound on γ ∗ . Given d ∈ N , define γ d := max γ ∈ R ,s i ,t i γ subject to f ( x ) − γ − m (cid:88) i =1 s i ( x ) g i ( x ) − r (cid:88) i =1 t i ( x ) h i ( x ) is SOS , t i ∈ R [ x ] , s i ∈ Σ d . (2.31)On the other hand, every F ∈ Σ d has a quadratic representation with a positive semi-definite matrix. To see this, suppose F ( x ) = (cid:80) i q i ( x ) , where q i are polynomials ofdegree d . Each q i can be written in the canonical basis as q i ( x ) = c Ti z n,d ( x ), where z n,d ( x ) is the vector of all n − variate monomials of degree d or less. Hence, we canwrite F as F ( x ) = (cid:88) i q i ( x ) = (cid:88) i z n,d ( x ) T c i c Ti z n,d ( x )= z n,d ( x ) T (cid:32)(cid:88) i c i c Ti (cid:33) z n,d ( x ) = z n,d ( x ) T Qz n,d ( x ) , Q ≥
0. Therefore, for given γ ∈ R and d ∈ N , Problem (2.31) can beformulated as the following linear matrix inequality.Find Q i ≥ P j for i = 0 , · · · , m and j = 1 , · · · , r such that f ( x ) − γ = z Tn,d ( x ) (cid:32) Q + m (cid:88) i =1 Q i g i ( x ) + r (cid:88) j =1 P j h j ( x ) (cid:33) z n,d ( x ) , (2.32)where Q i and P j ∈ S N , where S N is the subspace of symmetric matrices in R N × N and N := (cid:18) n + dd (cid:19) . See G. Blekherman and Thomas (2013) for methods of solving SOSprograms. Also Papachristodoulou et al. (2013) provide a MATLAB package calledSOSTOOLs for solving SOS programs.If the Cone (2.30) is not Archimedean, then we can use Schmudgen’s Positivstel-lensatz to obtain the following SOS program with solution γ d ≤ γ ∗ . γ d = max γ ∈ R ,s i ∈ Σ d ,t i ∈ R [ x ] γ subject to f ( x ) − γ = 1 + (cid:88) λ ∈{ , } m s λ ( x ) g ( x ) λ · · · g m ( x ) λ m + r (cid:88) i =1 t i ( x ) h i ( x ) . (2.33)The Positivstellensatz and SOS programming can also be applied to polynomialoptimization over a more general form of semi-algebraic sets defined as T := { x ∈ R n : g i ( x ) ≥ , i = 1 , · · · , m, h j ( x ) = 0 , j = 1 , · · · , r, q k ( x ) (cid:54) = 0 , k = 1 , · · · , l } . It can be shown that T = ∅ if and only ifˆ T := { ( x, y ) ∈ R n + l : g i ( x ) ≥ , i = 1 , · · · , m, h j ( x ) = 0 , j = 1 , · · · , r,y k q k ( x ) = 1 , k = 1 , · · · , l } = ∅ . Thus, for any f ∈ R [ x ], we havemin x ∈ T f ( x ) = min ( x,y ) ∈ ˆ T f ( x ) . Therefore, to find lower bounds on min x ∈ T f ( x ), one can apply SOS programmingand Putinar’s Positivstellensatzs to min ( x,y ) ∈ ˆ T f ( x ).32 .3.5 Case 5: Tests for Non-negativity on R n : The following result from Habicht (1939) defines a test for non-negativity of ho-mogeneous polynomials over R n . Theorem 11. (Habicht theorem) For every homogeneous polynomial f that satisfies f ( x , · · · , x n ) > for all x ∈ R n \ { } , there exists some e ≥ such that all of thecoefficients of (cid:32) n (cid:88) i =1 x i (cid:33) e f ( x , · · · , x n ) (2.34) are positive. In particular, the product is a sum of squares of monomials. Using this theorem, one can verify non-negativity of any homogeneous polyno-mial f over R n by multiplying F repeatedly by (cid:80) ni =1 x i . If for some e ∈ N , theProduct (2.34) has all positive coefficients, then f ≥
0. We can define an alternativetest for non-negativity over R n using the following theorem (de Loera and Santos(1996)). Theorem 12.
Define E n := {− , } n . Suppose a polynomial f ( x , · · · , x n ) of degree d satisfies f ( x , · · · , x n ) > for all x ∈ R n and its homogenization is positive definite.Then1. there exist λ e ≥ and coefficients c α ∈ R such that (cid:0) e T x (cid:1) λ e f ( x , · · · , x n ) = (cid:88) α ∈ I e c α x α · · · x α n n for all e ∈ E n , (2.35) where I e := { α ∈ N n : (cid:107) α (cid:107) ≤ d + λ e } and sgn ( c α ) = e α · · · e α n n .2. there exist positive N, D ∈ R [ x , · · · , x n , f ] such that f = ND . Associated to every polynomial f ( x , · · · , x n ) , x ∈ R n of degree d , there exists a degree e homogeneous polynomial h ( x , · · · , x n , y ) := y e f ( x y , · · · , x n y ), where e ≥ d . e := { x ∈ R n : sgn ( x i ) = e i , i = 1 , · · · , n } for some e ∈ E n . Multiply a given polynomial f repeatedly by 1 + e T x for some e ∈ E n . If there exists some λ e ≥ sgn ( c α ) = e α · · · e α n n , then (2.35)clearly implies that f ( x ) ≥ x ∈ Λ e . Since R n = ∪ e ∈ E n Λ e , we can repeat thetest 2 n times to obtain a test for non-negativity of f over R n .The second part of Theorem 12 gives a solution to Hilbert’s 17 th problem (seeSection 2.1). For a construction of this solution (i.e., numerator N and denominator D ) see de Loera and Santos (1996). 34hapter 3SEMI-DEFINITE PROGRAMMING AND INTERIOR-POINT ALGORITHMSAs discussed in Chapter 2, Polya’s theorem, Handelman’s theorem and the Posi-tivstellensatz results can be used to approximate the minimum of a polynomial oversimplicies, hypercubes, polytopes and semi-algebraic sets. We showed that thesetheorems define sequences of Linear/Semi-Definite Programs (SDPs) whose solutionsdefine lower bounds on the objective of the polynomial optimization problem. In thissection, we focus on solving these SDPs. In particular, we discuss the primal anddual forms of semi-definite programming problems and introduce a state-of-the-artprimal-dual interior-point algorithm for solving SDPs. In Section 4.5, we will proposea new parallel version of this algorithm - an algorithm which is specifically designedto solve the SDPs defined by applying Polya’s theorem to optimization of polynomialsarising in robust stability and control problems.3.1 Convex Optimization and DualityLet us define the constrained optimization problem f ∗ := min x ∈ R n f ( x )subject to f i ( x ) ≤ , i = 1 , · · · , ph i ( x ) = 0 , i = 1 , · · · , q, (3.1)where f i : R n → R and h i : R n → R . For every problem of Form (3.1), one can definethe Lagrangian function L : R n × R p × R q as L ( x, λ, ν ) := f ( x ) + p (cid:88) i =1 λ i f i ( x ) + q (cid:88) i =1 ν i h i ( x ) , (3.2)35here λ i ∈ R and ν i ∈ R are called the Lagrange multipliers associated with theinequality constraints and the equality constraints in (3.1), respectively. The vectors λ = [ λ , · · · , λ p ] and ν = [ ν , · · · , ν q ] are called the dual variables of Problem (3.1).Let us define the Lagrange dual function g : R p × R q → R as g ( λ, ν ) := inf x (cid:32) f ( x ) + p (cid:88) i =1 λ i f i ( x ) + q (cid:88) i =1 ν i h i ( x ) (cid:33) . The Lagrange dual functions have some interesting properties. First, because the La-grangian is affine in λ i and ν i and the pointwise infimum of a family of affine functionsis concave (Boyd and Vandenberghe (2004)), g is a concave function. Second, it iseasy to show that the dual functions yield lower bounds on f ∗ as define in (3.1), i.e., g ( λ, ν ) ≤ p ∗ . To find the best lower bound on f ∗ using the Lagrange dual function,one can solve the Lagrange dual problem defined as d ∗ := max λ,ν g ( λ, ν )subject to λ ≥ . (3.3)Every pair ( λ, ν ) which satisfies λ ≥ g ( λ, ν ) > −∞ is called a dual feasible point for Problem (3.3). Likewise, every x ∈ R n satisfying f i ( x ) ≤ i = 1 , · · · , p and h i ( x ) ≤ i = 1 , · · · , q is a primal feasible point for Problem (3.1). Dualfeasible points can be used to bound sub-optimality of a primal feasible point. Inparticular, for every primal feasible point x and dual feasible point ( λ, ν ), f ( x ) − f ∗ ≤ f ( x ) − g ( λ, ν ) , where f ( x ) − g ( λ, ν ) is called the duality gap associated with x and ( λ, ν ). For certainproblems, the duality gap associated with primal optimal point x ∗ and dual optimalpoint ( λ ∗ , ν ∗ ) is zero, i.e., f ( x ∗ ) = f ∗ = d ∗ = g ( λ ∗ , ν ∗ ) . strong duality . One important class of problems whichusually posses this property is convex optimization problems. A convex optimizationproblem is an optimization problem of Form (3.1), where the functions f i , i = 0 , · · · , p are convex and h i , i = 1 , · · · , q are affine. For example, the Lagrange dual prob-lem (3.3) is by definition a convex problem (it is a maximization of a concave function)whether or not its primal (Eq. (3.1)) is convex. It can be shown that (Slater (2014))if the primal problem (3.1) is convex and there exists some x ∈ R n such that f i ( x ) < i = 1 , · · · , p and h i ( x ) = 0 for i = 1 , · · · , q, (3.4)then strong duality holds. Strong duality can be exploited to solve the primal problemvia its dual. This is useful specially when the dual is easier or computationally lessexpensive to solve. Suppose a dual optimal solution ( λ ∗ , ν ∗ ) is known and strongduality holds. If x ∗ := arg min L ( x, λ ∗ , ν ∗ )is unique and primal feasible, x ∗ is the primal optimal solution.3.2 Descent Algorithms for Convex OptimizationSuppose f : R n → R is differentiable. For ˆ x to be a minimum of f , the nec-essary condition is that [ ∇ x f ( x )] x =ˆ x = 0. The Karush-Kuhn-Tucker (KKT) con-ditions (Kuhn et al. (1951)) generalize this necessary condition for the constrainedoptimization problem (3.1), under the assumption that the functions f i and g i aredifferentiable. The KKT conditions can be stated as follows: Suppose x ∗ ∈ R n is a pri-mal optimal point for (3.1) and λ ∗ i , i = 1 , · · · , p and ν ∗ i , i = 1 , · · · , q are dual optimal A function f : R n → R is convex if f ( αx + βy ) ≤ αf ( x ) + βf ( y ) for all x, y ∈ R n and for all α, β ∈ R such that α + β = 1 and α, β ≥ f ( x ∗ ) = g ( λ ∗ , ν ∗ ).Then, the optimal primal and dual points satisfy the following. (cid:34) ∇ x f ( x ) + p (cid:88) i =1 λ ∗ i ∇ x f i ( x ) + q (cid:88) i =1 ν ∗ i ∇ x h i ( x ) (cid:35) x = x ∗ = 0 f i ( x ∗ ) ≤ i = 1 , · · · , ph i ( x ∗ ) = 0 for i = 1 , · · · , qλ ∗ i ≥ i = 1 , · · · , pλ ∗ i f i ( x ∗ ) = 0 for i = 1 , · · · , p. (3.5)The first line follows from the fact that x ∗ is a minimizer of the Lagrangian L ( x, λ ∗ , ν ∗ ).The second, third and fourth lines indicate that x ∗ and ( λ ∗ , ν ∗ ) are primal and dualfeasible. The last line is called the complementary slackness and follows from strongduality. This condition implies that for i = 1 , · · · , p , either the i th primal constraintmust be active at x ∗ (i.e., f i ( x ∗ ) = 0) or its corresponding optimal dual variable λ ∗ i must be zero.In general, the KKT conditions are only necessary conditions for optimality. In-deed, under certain regularity conditions, local minima of the primal Problem (3.1)satisfy the KKT conditions. However, when the primal problem is convex and thereexists x ∈ R n which satisfies (3.4), the KKT conditions become necessary and suf-ficient. Motivated by this result, many of the existing convex optimization algo-rithms are in principle algorithms for solving the KKT conditions iteratively. Thesealgorithms are often called descent algorithms because they generate a sequence { x k } k =1 , , ··· of primal feasible solutions which satisfy f ( x k ) > f ( x k +1 ) for k = 1 , , , · · · , (3.6)unless x k is optimal. One example of descent algorithms is the Newton’s algorithm.Given a primal feasible starting point x , Newton’s algorithm finds a sequence of38 earch directions ∆ x k ∈ R n and step length t k ∈ R + such that all the iterates x k +1 = x k + t k ∆ x k k = 1 , , , · · · are feasible and satisfy (3.6). Given a primal feasible point x k , Newton’s algorithmcalculates the search directions ∆ x k by first defining the convex optimization problemˆ f k := min v ∈ R (cid:20) f ( x ) + ∇ x f ( x ) T v + 12 v T ∇ x f ( x ) v (cid:21) x = x k subject to h i ( x k + v ) = 0 for i = 1 , · · · , m. (3.7)The objective function of Problem (3.7) is the second-order Taylor’s approximationof the objective function f ( x ) at x = x k . Then, the KKT optimality conditions forProblem (3.7) yield the following system of linear equations. [ ∇ x f ( x )] x = x k D x h ( x ) T D x h ( x ) ∆ x k ν k = − [ ∇ x f ( x )] x = x k , (3.8)where D x h ( x ) := [ ∇ x h ( x ) , · · · , ∇ x h m ( x )] T . If the coefficient matrix in (3.8) is non-singular, then there exist a unique Newton’s search direction ∆ x k and optimal dualpoint ν k for the dual to Problem (3.8). Finally, Newton’s algorithm calculates thenew iterate as x k +1 = x k + t k ∆ x k , where a step length t k can be obtained using a linesearch method such as backtracking (Dennis Jr and Schnabel (1996)) or bisection. Atypical stopping criterion for Newton’s algorithm is f ( x k ) − ˆ f k ≤ (cid:15) for some desired (cid:15) >
0, where recall that ˆ f k is the minimum of the second-order Taylor’s approxima-tion of f at x k , subject to the equality constraints in (3.7). The difference between f ( x k ) and ˆ f k can also be interpreted as the size of Newton’s search direction definedby the following weighted norm of ∆ x k : f ( x k ) − ˆ f k = (cid:107) ∆ x k (cid:107) ∇ x f ( x ) | x = xk := 2(∆ x k ) T ∇ x f ( x ) | x = x k ∆ x k . (3.9)For a comprehensive discussion on the complexity and convergence of Newton’s algo-rithm, refer to Boyd and Vandenberghe (2004).39.3 Interior-point Algorithms for Convex Problems with Inequality ConstraintsSuppose in Problem (3.1), f i are convex and differentiable and h i are affine anddifferentiable. One of the most successful class of algorithms for solving this type ofproblems is interior-point algorithms. Typically, interior-point algorithms solve thisproblem in two steps: 1- Reducing the problem to a sequence of convex optimizationprograms with only linear equality constraints; and 2- Applying a descent algorithm,e.g., Newton’s algorithm, to solve the equality constrained problem. One way todefine this sequence of equality constrained problems is to incorporate the inequalityconstraints into the objective function using barrier functions . For example, by usinglogarithmic barrier functions one can approximate Problem (3.1) asmin x ∈ R n f ( x ) − p (cid:88) i =1 (cid:18) b (cid:19) log( − f i ( x ))subject to h i ( x ) = 0 , i = 1 , · · · , q. (3.10)for some b >
0. Clearly, if any of the inequality constraints becomes active ( f i ( x ) → { x ∈ R n : f i ( x ) ≤ , h j ( x ) = 0 , for i = 1 , · · · , p and j = 1 , · · · , q } . Since Problem (3.10) is convex, one can use Newton’s algorithm to find the optimalsolution x ∗ b for any b >
0. In particular, given b > x , Newton’s40lgorithm finds a sequence { x k } k =1 , , ··· → x ∗ b by solving the modified KKT conditions (cid:20) b ∇ x f ( x ) − ∇ x p (cid:80) i =1 log( − f i ( x )) (cid:21) x = x k D x h ( x ) T D x h ( x ) ∆ x k ν k = − (cid:20) ∇ x f ( x ) + ∇ p (cid:80) i =1 log( − f i ( x )) (cid:21) x = x k (3.11)for ∆ x k and ν k and setting x k +1 = x k + t k x k . The set of optimal solutions x ∗ b for all b > central path . Corresponding to any x ∗ b in the central path, one canverify that λ ∗ i = − b f i ( x ∗ b ) for i = 1 , · · · , p and ν ∗ = ν k b ∈ R q , are dual feasible and together with x ∗ b yield the duality gap pb . This indicates that as b → ∞ , x ∗ b converges to the optimal solution of Problem 3.1 under the assumptionthat f i are convex and differentiable and h i are affine and differentiable. Basedon this result, we can summarize the interior-point barrier algorithm for inequalityconstrained problems in Algorithm 3.An alternative subclass of interior-point algorithms for solving inequality con-strained problems is the primal-dual algorithms. Similar to the barrier algorithm,primal-dual algorithms find their search direction by solving the KKT optimalityconditions. However, instead of incorporating the inequality constraints into theobjective function (equivalently, eliminating the dual variable λ from the KKT con-dition (3.5)), primal-dual algorithms simultaneously solve the primal problem and itsdual by computing independent Newton’s search directions ∆ x ∈ R n , ∆ λ ∈ R p and∆ ν ∈ R q for primal and dual variables x, λ and ν . Given a feasible point ( x k , λ k , ν k )for Problem (3.1) and b >
0, the basic version of primal-dual algorithms computes the41 nput:
Convex functions f , · · · , f p ; affine functions h , · · · , h q ; a feasiblestarting point x ; initial barrier parameter b ; tolerances (cid:15) b > (cid:15) N > Initialization:
Set b = b .Choose µ > Barrier Algorithm:while pb > (cid:15) b do Set x = x .Set ∆ x = 10 · n . Newton’s Algorithm:while (cid:107) ∆ x (cid:107) as defined in (3.9) is greater than or equal to (cid:15) N do Calculate Newton’s search direction ∆ x by solving the system of linearequations in (3.11).Choose step length t using backtracking line search.Update Newton’s iterate as x := x + t ∆ x . end Set x ∗ b = x .Update the barrier parameter as b := µb . endOutput: x ∗ b : A pb -suboptimal solution to Problem (3.1). Algorithm 3:
Barrier algorithm for inequality constrained convex optimizationproblems 42earch directions (∆ x k , ∆ λ k , ∆ ν k ) by approximating the modified KKT conditions R ( x, λ, ν, b ) = ∇ f ( x ) + D x f ( x ) T λ + D x h ( x ) T νλ f ( x ) − b ... λ p f p ( x ) − bh ( x )... h q ( x ) = 0at the point ( x k , λ k , ν k ) as R ( x k + ∆ x k , λ k + ∆ λ k ,ν k + ∇ ν k , b ) ≈ R ( x, λ, ν, b ) + (cid:2) ∇ R ( x, λ, ν, b ) T (cid:3) x = x k λ = λ k ν = ν k ... (cid:2) ∇ R n + p + q ( x, λ, ν, b ) T (cid:3) x = x k λ = λ k ν = ν k ∆ x k ∆ λ k ∆ ν k = 0 , (3.12)and solving for (∆ x k , ∆ λ k , ∆ ν k ). The primal-dual iterates are then updated accordingto x k +1 = x k + t k ∆ x k , λ k +1 = λ k + t k ∆ λ k , ν k +1 = ν k + t k ∆ ν k . Similar to the barrier algorithm, the duality gap corresponding to any feasible primal-dual iterate ( x k , λ k , ν k ) is pb . Thus, as b → ∞ in (3.12), the resulting iterates convergeto the optimal solution of Problem (3.1), assuming that f i are convex and h i areaffine. In the sequel, we describe a primal-dual algorithm for solving semi-definiteprograms - a class of convex optimization problems which has several applications incontrol theory. 43.4 Semi-definite ProgrammingConsider the delay-differential equation˙ x ( t ) = Ax ( t ) + N (cid:88) i =1 A i x ( t − τ i ) (3.13)where x ( t ) ∈ R n and τ i > , i = 1 , · · · , N . From Repin (1965), a sufficient conditionfor asymptotic stability of this system is existence of P > , · · · , P N > V ( x, t ) = x T ( t ) P x ( t ) + N (cid:88) i =1 (cid:90) τ i x ( t − s ) T P i x ( t − s ) ds satisfies ˙ V ( x, t ) < x ( t ) ∈ R n \ { } and t >
0. The derivative ˙ V ( x, t ) can beexpanded as˙ V ( x, t ) = x ( t ) T (cid:32) A T P + P A + N (cid:88) i =1 P i (cid:33) x ( t ) + x ( t ) T (cid:32) N (cid:88) i =1 P A i x ( t − τ i ) (cid:33) + (cid:32) N (cid:88) i =1 x ( t − τ i ) T A Ti P (cid:33) x ( t ) − N (cid:88) i =1 x ( t − τ i ) T P i x ( t − τ i ) . Thus, ˙ V ( x, t ) = z ( t ) T Q ( P , · · · , P N ) z ( t ), where z ( t ) := [ x ( t ) x ( t − τ ) · · · x ( t − τ N )] T and Q ( P , · · · , P N ) := A T P + P A + (cid:80) Ni =1 P i P A · · · P A N A T P − P · · · A TN P · · · − P N . Thus, stability of System (3.13) can be verified by solving the following feasibilityproblem: Find P > , · · · , P N > Q ( P , · · · , P N ) < . (3.14)44ow let us parameterize each P i as P i ( y iL +1 , · · · , y ( i +1) L ) = y iL +1 y iL +2 · · · y iL + n y iL +2 y iL + n +1 · · · y iL +2 n − ... . . . ... y iL + n y iL +2 n − · · · y ( i +1) L for i = 0 , · · · , N , where L := n ( n + 1) / y j ∈ R for j = 0 , · · · , ( N + 1) L . Then,we can formulate the problem of stability of System (3.13) as the convex optimizationproblem min y ∈ R ( N +1) L Z ∈ S ( N +2) n (cid:104) ( N +1) L , y (cid:105) subject to ( N +1) L (cid:88) i =1 F i y i = ZZ ≥ , (3.15)where the matrices F i ∈ S ( N +2) n for i = 1 , · · · , ( N + 1) L are defined as F i = diag { P ( x , · · · , x L ) , Q ( P ( x , · · · , x L ) , · · · , P N ( x NL +1 , · · · , x ( N +1) L )) } , (3.16)where x j = j = i j (cid:54) = i for j = 1 , · · · , ( N + 1) L. Problem (3.15) is an example of the dual form of the Semi-Definite Programming(SDP) problem. We define SDP as the optimization of a linear objective functionover the cone of positive definite matrices subject to linear matrix equality and linearmatrix inequality constraints. Given C ∈ S n , B i ∈ S n for i = 1 , · · · , k , G i ∈ S n for45 = 1 , · · · , l , a ∈ R k and b ∈ R l , the primal SDP problem is p ∗ := max X ∈ S n tr( CX )subject to B ( X ) = aG ( X ) ≤ bX ≥ , (3.17)where the linear maps B : S n → R k and G : S n → R l are defined as B ( X ) = tr( B X )tr( B X )...tr( B k X ) and G ( X ) = tr( G X )tr( G X )...tr( G l X ) . (3.18)To derive the dual SDP to Problem (3.17), we employ Lagrange multipliers t ∈ R l + and y ∈ R k as follows. p ∗ = max X ≥ min y ∈ R k ,t ∈ R l + tr( CX ) + t T ( b − G ( X )) + y T ( a − B ( x ))Then, from the min-max inequality , i.e.,max u ∈ U min v ∈ V f ( u, v ) ≤ min v ∈ V max u ∈ U f ( u, v )it follows that p ∗ ≤ max y ∈ R k ,t ∈ R l + min X ≥ tr( CX ) + t T ( b − G ( X )) + y T ( a − B ( x ))= min y ∈ R k ,t ∈ R l + max X ≥ tr( C − k (cid:88) i =1 B i y i − l (cid:88) i =1 G i t i ) X + a T y + b T t. Note that max X ≥ tr( C − k (cid:88) i =1 B i y i − l (cid:88) i =1 G i t i ) X < ∞ C − (cid:80) ki =1 B i y i − (cid:80) li =1 G i t i ≤
0. In this case, clearly the maximum occurswhen C − k (cid:88) i =1 B i y i − l (cid:88) i =1 G i t i = 0 . Therefore, we can the write dual
SDP problem asmax y ∈ R k ,t ∈ R l + a T y + b T t subject to k (cid:88) i =1 B i y i + l (cid:88) i =1 G i t i − C = ZZ ≥ . (3.19)From (3.15) and (3.19) it is clear that the problem of stability of the delay-differentialEquation (3.13) can be formulated as the dual SDP defined by the elements a := ( N +1) L , b := , G i = , C = , B i = F i , where we have defined F i in (3.16).SDPs are popular among controls community because not only they can be solvedefficiently using convex optimization algorithms, but also a wide variety of problems incontrols can be formulated as SDPs; e.g., robust stability (Bliman (2004a); Oliveiraand Peres (2007)) and robust performance (Peaucelle and Arzelier (2001); Scherer(2006)) of uncertain systems, H /H ∞ -optimal filter design (Li and Fu (1997); Geromeland de Oliveira (2001)), estimation of regions of attraction (Wang et al. (2005);Tan and Packard (2008); Topcu et al. (2010)) and reachability sets (Wang et al. (2013)) of nonlinear systems, stability and control of hybrid systems (Boukas (2006);Papchristodoulou and Prajna (2009)) and game theory (Parrilo (2006)). In the nextsection, we describe a state-of-the-art primal-dual algorithm by Helmberg et al. (2005)for solving SDPs. 47.5 A Primal-dual Interior-point Algorithm for Semi-definite ProgrammingFortunately, there exists several interior-point algorithms in the literature for solv-ing SDPs; e.g., dual scaling (Benson (2001); Benson et al. (1998)), primal-dual (Al-izadeh et al. (1998); Monteiro (1997); Helmberg et al. (1996)) and cutting-plane/spectralbundle (Helmberg and Rendl (2000); Sivaramakrishnan (2010); Nayakkankuppam(2007)) algorithms. In our study, we are particularly interested in a state-of-the-artprimal-dual algorithm proposed by Helmberg et al. (2005) mainly because at eachiteration, it preserves a certain property (see (4.47)) of the primal and dual searchdirections. In Section 4.5, we will exploit this property to propose a distributed par-allel version of this algorithm for solving large-scale SDPs in robust and/or nonlinearstability analysis. In the following, we briefly discuss the original version of thisalgorithm algorithm.Similar to the barrier method described in Section 3.3, we can incorporate theinequality constraints in the dual SDP (3.19) using logarithmic barrier functions andthe barrier parameter µ > y ∈ R k ,t ∈ R l a T y + b T t − µ (cid:32) log det Z + l (cid:88) i =1 log t i (cid:33) subject to k (cid:88) i =1 B i y i + l (cid:88) i =1 G i t i − C = Z. (3.20)The Lagrangian for Problem (3.20) is defined as L ( X, y, t, Z ) := a T y + b T t − µ (cid:32) log det Z + l (cid:88) i =1 log t i (cid:33) + tr (cid:32)(cid:32) Z + C − k (cid:88) i =1 B i y i − l (cid:88) i =1 G i t i (cid:33) X (cid:33) . Then, the KKT optimality conditions for Problem (3.20) is ∇ L ( X, y, t, Z ) = 0, which48an be expanded as ∇ X L ( X, y, t, Z ) = Z + C − k (cid:88) i =1 B i y i − l (cid:88) i =1 G i t i = 0 (3.21) ∇ y L ( X, y, t, Z ) = a − B ( X ) = 0 (3.22) ∇ t L ( X, y, t, Z ) = b − G ( X ) − µ [1 /t , · · · , /t l ] T = 0 (3.23) ∇ Z L ( X, y, t, Z ) = X − µZ − = 0 , (3.24)where B ( X ) and G ( x ) are defined in (3.18).Given a barrier parameter µ >
0, at each iteration, the primal-dual algorithm findsa search direction ∆ s := [∆ X, ∆ y, ∆ t, ∆ Z ] such that the new iterate [ X + ∆ X, y +∆ y, t + ∆ t, Z + ∆ Z ] belongs to the central path, i.e., { [ X µ , y µ , t µ , Z µ ] : µ ∈ [0 , ∞ ] and X µ , y µ , t µ , Z µ satisfy Conditions (3.21)-(3.24) } . Conversely, given a point [
X, y, t, Z ], one can use (3.23) and (3.24) to find its corre-sponding barrier parameter as µ = tr( ZX ) + [1 /t , · · · , /t l ] ( b − G ( X )) n + l . (3.25)The search direction ∆ s of the primal-dual algorithm is the sum of two steps: the predictor step ∆ˆ s := [∆ ˆ X, ∆ˆ y, ∆ˆ t, ∆ ˆ Z ] and the corrector or centering step ∆¯ s :=[∆ ¯ X, ∆¯ y, ∆¯ t, ∆ ¯ Z ]. The predictor step is defined as the Newton’s step for solvingthe optimality conditions (3.21)-(3.24) with µ = 0, starting at any point ( X, y, t, Z )which satisfies
X > , Z > , t > , G ( X ) < b. (3.26)Similar to the Taylor’s approximation in (3.12), we find the Newton’s step by solving ∇ L ( X, y, t, Z ) + ∇ L ( X, y, t, Z )∆ˆ s T = 0 (3.27)49or ∆ˆ s . Substituting for ∇ L from (3.21)-(3.24) into (3.27) yields the following systemof equations for the predictor step. Λ Λ Λ Λ ∆ˆ y ∆ˆ t = B ( Z − T X ) − a ) G ( Z − T X ) − b ) , (3.28)∆ ˆ X = Z − T X − Z − (cid:32) k (cid:88) i =1 B i ∆ˆ y + l (cid:88) i =1 G i ∆ˆ t (cid:33) X − X (3.29)∆ ˆ Z = − T + k (cid:88) i =1 B i ∆ˆ y + l (cid:88) i =1 G i ∆ˆ t (3.30)where T = − k (cid:88) i =1 B i y + l (cid:88) i =1 G i t + C + Z Λ = (cid:2) B ( Z − B X ) · · · B ( Z − B k X ) (cid:3) Λ = (cid:2) B ( Z − G X ) · · · B ( Z − G l X ) (cid:3) Λ = (cid:2) G ( Z − B X ) · · · G ( Z − B k X ) (cid:3) Λ = (cid:2) G ( Z − G X ) · · · G ( Z − G l X ) (cid:3) + diag (cid:26) b − tr( G X ) t , · · · , b l − tr( G l X ) t l (cid:27) . The corrector step is defined as the Newton’s step for solving the KKT condi-tions (3.21)-(3.24), using the barrier parameter µ as defined in (3.25) and startingat [ X + ∆ ˆ X, y + ∆ˆ y, t + ∆ˆ t, Z + ∆ ˆ Z ] , where [ X, y, t, Z ] can be any point satisfying (3.26) and [∆ ˆ X, ∆ˆ y, ∆ˆ t, ∆ ˆ Z ] can becalculated using (3.28)-(3.30). Thus, to derive the corrector step, we substitute for ∇ L from KKT conditions (3.21)-(3.24) into (cid:2) ∇ L ( ¯ X, ¯ y, ¯ t, ¯ Z ) + ∇ L ( ¯ X, ¯ y, ¯ t, ¯ Z ) (cid:3) ¯ X = X +∆ ˆ X ¯ Z = Z +∆ ˆ Z ¯ y = y +∆ˆ y ¯ t = t +∆ˆ t ∆¯ s T = 050his yields the following set of equations for the corrector step. Λ Λ Λ Λ ∆¯ y ∆¯ t = µB ( Z − ) − B ( Z − ∆ ˆ Z ∆ ˆ X ) µ (cid:20) t · · · t l (cid:21) + G (cid:16) X + ∆ ˆ X + µZ − − Z − ∆ ˆ Z ∆ ˆ X (cid:17) (3.31)∆ ¯ X = Z − (cid:16) − ∆ ˆ Z ∆ ˆ X + µI − ∆ ¯ ZX (cid:17) (3.32)∆ ¯ Z = k (cid:88) i =1 B i ∆¯ y + l (cid:88) i =1 G i ∆¯ t (3.33)By solving (3.28)-(3.30) for the predictor step and solving (3.31)-(3.33) for the cor-rector step, we can calculate the search direction as∆ s = (cid:104) Sym(∆ ˆ X + ∆ ¯ X ) , ∆ˆ y + ∆¯ y, ∆ˆ t + ∆¯ t, ∆ ˆ Z + ∆ ¯ Z (cid:105) , (3.34)where Sym( W ) := ( W + W T ) / W . We have providedan outline of the discussed primal-dual algorithm in Algorithm 4.51 nput: SDP elements
C, a, b, B i for i = 1 , · · · , k and G i for i = 1 , · · · , l ;starting point satisfying (3.26); tolerance (cid:15) > Initialization:
Set the duality gap γ = 2 (cid:15) . while duality gap γ > (cid:15) doCalculating the predictor step: Solve ∆ˆ y and ∆ˆ t by solving system of the equations in (3.28).Calculate ∆ˆ y and ∆ˆ t using (3.29) and (3.30). Calculating the corrector step:
Calculate the barrier parameter µ using (3.25).Solve ∆¯ y and ∆¯ t by solving system of the equations in (3.31).Calculate ∆¯ y and ∆¯ t using (3.32) and (3.33). Updating the primal and dual variables:
Calculate the search direction as∆ X := Sym(∆ ˆ X +∆ ¯ X ) , ∆ y := ∆ˆ y +∆¯ y, ∆ t := ∆ˆ t +∆¯ t, ∆ Z := ∆ ˆ Z +∆ ¯ Z. Calculate primal and dual step lengths α p and α d using an appropriateline-search algorithm.Set the primal and dual variables as X := X + α p ∆ X, y := y + ∆ y, t := t + ∆ t, Z := Z + ∆ Z. Calculate the duality gap as γ = tr( CX ) − ( a T y + b T t ). endOutput: [ X ∗ , y ∗ , t ∗ , Z ∗ ]: A γ -suboptimal solution to Problems (3.17)and (3.19). Algorithm 4:
An interior-point central-path primal-dual algorithm for SDP52hapter 4PARALLEL ALGORITHMS FOR ROBUST STABILITY ANALYSIS OVERSIMPLEX4.1 Background and MotivationsControl system theory when applied in practical situations often involves the useof large state-space models, typically due to inherent complexity of the system, theinterconnection of subsystems, or the reduction of an infinite-dimensional or PDEmodel to a finite-dimensional approximation. One approach to dealing with suchlarge scale models has been to use model reduction techniques such as balanced trun-cation (Gugercin and Antoulas (2004)). However, the use of model reduction tech-niques are not necessarily robust and can result in arbitrarily large errors. In additionto large state-space, practical problems often contain uncertainty in the model dueto modeling errors, linearization, or fluctuation in the operating conditions. Theproblem of stability and control of systems with uncertainty has been widely studied.See, e.g. the texts Ackermann et al. (2001); Bhattacharyya et al. (1995); Green andLimebeer (1995); Zhou and Doyle (1998); Dullerud and Paganini (2000). Famousresults such as the small-gain theorem,
Popov’s criterion, passivity theorems and
Kharitonov’s theorem have been widely used to find tractable solutions to certainrobust stability problems of a single and/or interconnected uncertain systems. As anexample, Kharitonov’s theorem reduces the stability problem of an infinite family ofdifferential equations a d n dt n x ( t ) + a d n − dt n − x ( t ) + · · · + a n − ddt x ( t ) + a n +1 x ( t ) + a n +2 = 0 , a i ∈ [ u i , ¯ u i ] ⊂ R (4.1)53o verifying whether the following four characteristic polynomials k ( s ) = u n +2 + u n +1 s + ¯ u n s + ¯ u n − s + u n − s + u n − s + · · · k ( s ) = ¯ u n +2 + ¯ u n +1 s + u n s + u n − s + ¯ u n − s + ¯ u n − s + · · · k ( s ) = u n +2 + ¯ u n +1 s + ¯ u n s + u n − s + u n − s + ¯ u n − s + · · · k ( s ) = ¯ u n +2 + u n +1 s + u n s + ¯ u n − s + ¯ u n − s + u n − s + · · · have all their roots in the open left half-plane - a problem which can be tractablysolved (in O ( n ) operations) using the Routh-Hurwitz criterion. Despite all theprogress in robust control theory during the past few decades, a drawback of ex-isting computational methods for analysis and control of systems with uncertainty ishigh computational complexity. This is a consequence of the fact that a wide rangeof problems in robust stability and control of systems with parametric uncertaintyare known to be NP-hard. For example, even the classical problem of stability of˙ x ( t ) = A ( a ) x ( t ) for all a inside a hypercube (the matrix analog of System (4.1)) isNP-hard . Other examples are calculation of structured singular values for robust per-formance analysis and µ -synthesis (Zhou et al. (1996)), deciding null-controllability of x ( k + 1) = f ( x ( k ) , u ( k )) for a given f : R n × R m → R n (Blondel and Tsitsik-lis (1999)), and computing arbitrarily precise bounds on the joint spectral radius ofmatrices for stability analysis of systems with time-varying uncertainty (Gripenberg(1996)). See Blondel and Tsitsiklis (2000) for a comprehensive survey on NP-hardproblems in control theory. The result of such complexity is that for systems with Nemirovskii (1993) proves that the {− , +1 } -integer linear programming problem (a well-knownNP-complete problem) admits a polynomial-time reduction to the problem of verifying positive semi-definiteness of a family of symmetric matrices with entries belonging to an interval on R . A system x ( k + 1) = f ( x ( k ) , u ( k )) is called null-controllable if for every initial state x (0), thereexist some T > u ( k ) , k = 0 , · · · , T − x ( T ) = 0 et al. (1996); Oliveira and Peres(2005, 2001) and Ramos and Peres (2001). The use of polynomial QITS Lyapunov55 igure 4.1: Various Interconnections of Nodes in a Cluster Computer (Top), TypicalMemory Hierarchies of a GPU and a Multi-core CPU (bottom)variables can be motivated by Bliman (2004b), wherein it is shown that any feasibleparameter-dependent LMI with parameters inside a compact set has a polynomialsolution or Peet (2009) wherein it is shown that local stability of a nonlinear vectorfield implies the existence of a polynomial Lyapunov function.There are several results which use polynomial QITS Lyapunov functions to proverobust stability. In most cases, the stability problem is reduced to the general prob-lem of optimization of polynomial variables subject to LMI constraints - an NP-hardproblem (Ben-Tal and Nemirovski (1998)). To avoid NP-hardness, the optimizationof polynomials problem is usually solved in an asymptotic manner by posing a se-quence of sufficient conditions of increasing accuracy and decreasing conservatism.For example, building on the result in Bliman (2004b), Bliman (2004a) proposes a56equence of increasingly precise LMIs for robust stability analysis of linear systemswith affine dependency on uncertain parameters on the complex unit ball. Necessaryand sufficient stability conditions for linear systems with one uncertain parameter arederived in Zhang and Tsiotras (2003), providing an explicit bound on the degree ofthe polynomial-type Lyapunov function. This result is extended to multi-parameter-dependent linear systems in Zhang et al. (2005). Another important approach tooptimization of polynomials is the SOS methodology which replaces the polynomialpositivity constraint with the constraint that the polynomial admits a representationas a sum of squares of polynomials. See Sections 2.3.4 and 1.1 for a review of thisapproach. Applications of the SOS methodology in robust stability analysis of linearand nonlinear systems can be found in Scherer and Hol (2006); Lavaei and Aghdam(2008) and Tan and Packard (2008). While the SOS methodology have been exten-sively utilized in the literature, we have not, as of yet, been able to adapt algorithmsfor solving the resulting LMI conditions to a parallel-computing environment. Finally,there have been multiple results in recent years on the use of Polya’s theorem to solveoptimization of polynomials problems (Oliveira and Peres (2007)) on the simplex. Anextension of Polya’s theorem for uncertain parameters on the multisimplex or hyper-cube can be found in Oliveira et al. (2008). In this section, we propose an extensionof Polya’s theorem and its use for solving optimization of polynomials problems in aparallel computing environment.Our goal is to create algorithms which explicitly map computation, communica-tion and storage to existing parallel processing architectures. This goal is motivatedby the failure of existing general-purpose Semi-Definite Programming (SDP) solversto efficiently utilize platforms for large-scale computation. Specifically, it is well-established that linear programming and semi-definite programming both belong tothe complexity class P-Complete, also known as the class of inherently sequential57roblems. Although there have been several attempts to map certain SDP solversto a parallel computing environment (Borchers and Young (2007); Yamashita et al. (2003)), certain critical steps cannot be distributed. The result is that as the numberof processors increases, certain computational and communication bottlenecks dom-inate - leading to a saturation in the speed-up (the increase in processing speed peradditional processor) of these solvers (Amdahl’s law (Amdahl (1967))). We avoidthese bottlenecks by exploiting the particular structure of the LMI conditions asso-ciated with Polya’s theorem. Note that, in principle, a perfectly designed general-purpose SDP algorithm could identify the structure of the SDP, as we have, and mapthe communication, computation and memory constraints to a parallel architecture.Indeed, there has been a great deal of research on creating programming languageswhich attempt to do just this (Kal´e et al. (1994); Deitz (2005)). However, at presentsuch languages are mostly theoretical and have certainly not been incorporated intoexisting SDP solvers.In addition to parallel SDP solvers, there have been some efforts to exploit struc-ture in certain polynomial optimization algorithms to reducing the size and complex-ity of the resulting LMI’s. For example, for the case of finding SOS representationsfor symmetric polynomials , Gatermann and Parrilo (2004) exploited symmetry toreduce the number of decision variables and constraints in the associated SDPs. An-other example is the use of an specific sparsity structure in Parrilo (2005); Kim et al. (2005) and Waki et al. (2008) to reduce the complexity of the linear algebra calcula-tions associated with the SOS methodology. The use of generalized Lagrangian dualsand Groebner basis techniques for reducing the complexity of the SDPs associated A symmetric polynomial is a polynomial which is invariant under all permutations of its vari-ables, e.g., f ( x, y, z ) = x + y + z − xyz + x + y + z . et al. (2005) and Permenter and Parrilo (2012). In this section, we focus on robust stability analysis of: 1- Systems with paramet-ric uncertainty inside a simplex; and 2- Systems with parametric uncertainty insidea hypercube. We solve each problem in two phases by proposing the following algo-rithms: 1- A decentralized algorithm for Setting up the sequence of structured SDPsassociated with Polya’s theorem; and 2- A parallel SDP solver to solve the SDPs.Note that the problem of decentralizing the set-up algorithm is significant in thatfor large-scale systems, the instantiation of the problem may be beyond the memoryand computational capacity of a single processing node. For the set-up problem, thealgorithm that we propose has no centralized memory/computational requirementswhatsoever. Furthermore, we show that for a sufficiently large number of availableprocessors, the communication complexity is independent of the size of the state-spaceor the number of Polya’s iterations.In the second phase, we propose a variant of Helmberg’s primal-dual algorithm(Helmberg et al. (2005)) and map the computational, memory and communicationrequirements to a parallel computing environment. Unlike the set-up algorithm, theprimal-dual algorithm does have a “relatively small” centralized computation associ-ated with the update of the dual variables. However, we have structured the algorithmso that the size of this centralized computation is solely a function of the degree ofthe polynomial Lyapunov function and does not depend on the number of Polya’siterations. In addition, there is no point-to-point communication between the proces-sors, which means that the algorithm is compatible with most of the existing parallel59omputing architectures. We will provide a graph representation of the communica-tion architecture of both the set-up and SDP algorithms.By linking the set-up and SDP algorithms and conducting tests on various clustercomputers, we demonstrate the ability of our algorithms in performing robust stabilityanalysis on systems with 100+ states and several uncertain parameters. Specifically,we ran a series of numerical experiments using the Linux-based cluster computerKarlin at Illinois Institute of Technology and the Blue Gene supercomputer (with 200processor allocation). First, we applied the algorithm to a current problem in robuststability analysis of magnetic confinement fusion using a discretized PDE model.Next, we examine the accuracy of the algorithm as Polya’s iterations progress andcompare this accuracy with the SOS approach. We show that unlike the general-purpose parallel SDP solver SDPARA Yamashita et al. (2003), the speed-up of ouralgorithm shows no evidence of saturation. Finally, we calculate the envelope ofthe algorithm on the cluster computer Karlin in terms of the maximum state-spacedimension, number of processors and Polya’s iterations.4.2 Notation and Preliminaries on Homogeneous PolynomialsLet us denote an l − variate monomial as α γ = (cid:81) li =1 α γ i i , where α ∈ R l is thevector of variables, γ ∈ N l is the vector of exponents and l (cid:80) i =1 γ i = d is the degree ofthe monomial. We define W d := (cid:40) γ ∈ N l : l (cid:88) i =1 γ i = d (cid:41) (4.2)as the totally ordered set of the exponents of l − variate monomials of degree d , wherethe ordering is lexicographic. Recall that in lexicographical ordering γ ∈ W d precedes η ∈ W d , if the left most non-zero entry of γ − η is positive. The lexicographical indexof every γ ∈ W d can be calculated using the map (cid:104)·(cid:105) : N l → N defined as (Peet and60eet (2010)) (cid:104) γ (cid:105) = l − (cid:88) j =1 γ i (cid:88) i =1 f (cid:32) l − j, d + 1 − j − (cid:88) k =1 γ k − i (cid:33) + 1where f ( l, d ) := l = 0 (cid:18) l + d − l − (cid:19) = ( d + l − d !( l − l > , (4.3)is the cardinality of W d , i.e., the number of l − variate monomials of degree d . For con-venience, we also denote the index of a monomial α γ by (cid:104) γ (cid:105) . We represent l − variatehomogeneous polynomials of degree d p as P ( α ) = (cid:88) γ ∈ W dp P (cid:104) γ (cid:105) α γ , where P (cid:104) γ (cid:105) ∈ R n × n is the matrix coefficient of the monomial α γ .Now consider the linear system˙ x ( t ) = A ( α ) x ( t ) , (4.4)where A ( α ) ∈ R n × n and α ∈ Q ⊂ R l is a vector of uncertain parameters. We assumethat A ( α ) is a homogeneous polynomial and Q = ∆ l ⊂ R l , where ∆ l is the unitsimplex, i.e., ∆ l = (cid:40) α ∈ R l : l (cid:88) i =1 α i = 1 , α i (cid:62) (cid:41) . If A ( α ) is not homogeneous, we can homogenize it in the following manner. Suppose A ( α ) with α ∈ ∆ l is a non-homogeneous polynomial of degree d a and has N a mono-mials with non-zero coefficients. Define D = (cid:0) d a , · · · , d a Na (cid:1) , where d a i is the degreeof the i th monomial of A ( α ) according to the lexicographical ordering. Now definethe polynomial B ( α ) as per the following:1. Let B = A . 61. For i = 1 , · · · , N a , multiply the i th monomial of B ( α ), according to lexicograph-ical ordering, by (cid:32) l (cid:80) j =1 α j (cid:33) d a − d ai .Then, since l (cid:80) j =1 α j = 1, B ( α ) = A ( α ) for all α ∈ ∆ l and hence all properties of ˙ x ( t ) = A ( α ) x ( t ) for any α ∈ ∆ l are retained by the homogeneous system ˙ x ( t ) = B ( α ) x ( t ).To further clarify the homogenization procedure, we provide the following example. Example: Construction of the homogeneous system ˙ x ( t ) = B ( α ) x ( t ) . Consider the non-homogeneous polynomial A ( α ) = Cα + Dα + Eα + F ofdegree d a = 2, where [ α , α , α ] ∈ ∆ . Using the above procedure, the homogeneouspolynomial B ( α ) can be constructed as B ( α ) = Cα + Dα ( α + α + α ) + Eα ( α + α + α ) + F ( α + α + α ) = ( C + F ) (cid:124) (cid:123)(cid:122) (cid:125) B α + ( D + 2 F ) (cid:124) (cid:123)(cid:122) (cid:125) B α α + ( E + 2 F ) (cid:124) (cid:123)(cid:122) (cid:125) B α α + ( D + F ) (cid:124) (cid:123)(cid:122) (cid:125) B α + ( D + E + 2 F ) (cid:124) (cid:123)(cid:122) (cid:125) B α α + ( E + F ) (cid:124) (cid:123)(cid:122) (cid:125) B α = (cid:88) γ ∈ W B (cid:104) γ (cid:105) α γ . (4.5)4.3 Setting-up the Problem of Robust Stability Analysis over a SimplexIn this section, we show that applying Polya’s Theorem to the robust stabilityproblem, i.e., the inequalities in Theorem 13 yields a semi-definite program with ablock-diagonal structure - hence can be an efficiently distributed among processingunits. We start by stating the following well-known Lyapunov result on stability ofSystem (4.4). Theorem 13.
System (4.4) is stable if and only if there exists a polynomial matrix P ( α ) such that P ( α ) (cid:31) for all α ∈ ∆ l and A T ( α ) P ( α ) + P ( α ) A ( α ) ≺ for all α ∈ ∆ l . (4.6)62 similar condition also holds for discrete-time linear systems. The conditions as-sociated with Theorem 13 are infinite-dimensional LMIs, meaning they must hold atinfinite number of points. Such problems are known to be NP-hard (Ben-Tal and Ne-mirovski (1998)). Our goal is to derive a sequence of polynomial-time algorithms suchthat their outputs converge to a solution of the parameter-dependent LMI in (5.8).Key to this result is Polya’s Theorem (Hardy et al. (1934)). A variation of thistheorem for matrices is given as follows. Theorem 14. (Polya’s theorem, simplex version) If a homogeneous matrix-valuedpolynomial F satisfies F ( α ) > for all α ∈ ∆ l , then there exists d ≥ such that allthe coefficients of (cid:32) l (cid:88) i =1 α i (cid:33) d F ( α ) (4.7) are positive definite. See Chapter 2 for a more detailed discussion on this result.Consider the stability of the system described by Equation (4.4). We are interestedin finding a P ( α ) which satisfies the conditions of Theorem 13. According to Polya’stheorem, the constraints of Theorem 13 are satisfied if for some sufficiently large d and d , the polynomials (cid:32) l (cid:88) i =1 α i (cid:33) d P ( α ) and (4.8) − (cid:32) l (cid:88) i =1 α i (cid:33) d (cid:0) A T ( α ) P ( α ) + P ( α ) A ( α ) (cid:1) (4.9)have all positive definite coefficients.Let P ( α ) be a homogeneous polynomial of degree d p which can be represented as P ( α ) = (cid:88) γ ∈ W dp P (cid:104) γ (cid:105) α γ , (4.10)63here the coefficients P (cid:104) γ (cid:105) ∈ S n . Recall that W d p := (cid:110) γ ∈ N l : (cid:80) li =1 γ i = d p (cid:111) is the setof the exponents of all l -variate monomials of degree d p . Since A ( α ) is a homogeneouspolynomial of degree d a , we can write it as A ( α ) = (cid:88) γ ∈ W da A (cid:104) γ (cid:105) α γ , (4.11)where the coefficients A (cid:104) γ (cid:105) ∈ R n × n . By substituting (4.10) and (4.11) into (4.8)and (4.9) and defining d pa as the degree of P ( α ) A ( α ), the conditions of Theorem 14can be represented in the form (cid:32) l (cid:88) i =1 α i (cid:33) d (cid:88) h ∈ W dp P (cid:104) h (cid:105) α h = (cid:88) g ∈ W dp + d (cid:88) h ∈ W dp β (cid:104) h (cid:105) , (cid:104) γ (cid:105) P (cid:104) h (cid:105) α γ and − (cid:32) l (cid:88) i =1 α i (cid:33) d (cid:88) h ∈ W da A T (cid:104) h (cid:105) α h (cid:88) h ∈ W dp P (cid:104) h (cid:105) α h + (cid:88) h ∈ W dp P (cid:104) h (cid:105) α h (cid:88) h ∈ W da A (cid:104) h (cid:105) α h = (cid:88) γ ∈ W dpa + d (cid:88) h ∈ W dp H T (cid:104) h (cid:105) , (cid:104) γ (cid:105) P (cid:104) h (cid:105) + P (cid:104) h (cid:105) H (cid:104) h (cid:105) , (cid:104) γ (cid:105) α γ have all positive coefficients. This means that (cid:88) h ∈ W dp β (cid:104) h (cid:105) , (cid:104) γ (cid:105) P (cid:104) h (cid:105) > γ ∈ W d p + d and (4.12) (cid:88) h ∈ W dp ( H T (cid:104) h (cid:105) , (cid:104) γ (cid:105) P (cid:104) h (cid:105) + P (cid:104) h (cid:105) H (cid:104) h (cid:105) , (cid:104) γ (cid:105) ) < γ ∈ W d pa + d . (4.13)Here we have defined β (cid:104) h (cid:105) , (cid:104) γ (cid:105) to be the scalar coefficient which multiplies P (cid:104) h (cid:105) in the (cid:104) γ (cid:105) -th monomial of the homogeneous polynomial (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) using the lexico-graphical ordering. Likewise, H (cid:104) h (cid:105) , (cid:104) γ (cid:105) ∈ R n × n is the term which left or right multiplies P (cid:104) h (cid:105) in the (cid:104) γ (cid:105) -th monomial of (cid:16)(cid:80) li =1 α i (cid:17) d (cid:0) A T ( α ) P ( α ) + P ( α ) A ( α ) (cid:1) using the lex-icographical ordering. For an intuitive explanation as to how these β and H termsare calculated, we consider a simple example. Precise formulae for these terms willfollow the example. 64 xample: Calculating the β and H coefficients. Consider A ( α ) = A α + A α and P ( α ) = P α + P α . By expanding Equa-tion (4.8) for d = 1 we have ( α + α ) P ( α ) = P α + ( P + P ) α α + P α . Thecoefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) are then extracted as β , = 1 , β , = 0 , β , = 1 , β , = 1 , β , = 0 , β , = 1 . Next, by expanding Equation (4.9) for d = 1 we have( α + α ) (cid:0) A T ( α ) P ( α ) + P ( α ) A ( α ) (cid:1) = (cid:0) A T P + P A (cid:1) α + (cid:0) A T P + P A + A T P + P A + A T P + P A (cid:1) α α + (cid:0) A T P + P A + A T P + P A + A T P + P A (cid:1) α α + (cid:0) A T P + P A (cid:1) α . The coefficients H (cid:104) h (cid:105) , (cid:104) γ (cid:105) are then extracted as H , = A , H , = , H , = A + A , H , = A ,H , = A , H , = A + A , H , = , H , = A . β and H The set { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } of coefficients can be formally defined recursively as follows. Letthe initial values for β (cid:104) h (cid:105) , (cid:104) γ (cid:105) be defined as β (0) (cid:104) h (cid:105) , (cid:104) γ (cid:105) = h = γ γ ∈ W d p and h ∈ W d p . (4.14)Then, iterating for i = 1 , . . . d , we let β ( i ) (cid:104) h (cid:105) , (cid:104) γ (cid:105) = (cid:88) λ ∈ W β ( i − (cid:104) h (cid:105) , (cid:104) γ − λ (cid:105) for all γ ∈ W d p + i and h ∈ W d p . (4.15)Finally, we set { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } = { β d (cid:104) h (cid:105) , (cid:104) γ (cid:105) } . 65o obtain the set { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } of coefficients, set the initial values as H (0) (cid:104) h (cid:105) , (cid:104) γ (cid:105) = (cid:88) λ ∈ W da : λ + h = γ A (cid:104) λ (cid:105) for all γ ∈ W d p + d a and h ∈ W d p . (4.16)Then, iterating for i = 1 , . . . d , we let H ( i ) (cid:104) h (cid:105) , (cid:104) γ (cid:105) = (cid:88) λ ∈ W H ( i − (cid:104) h (cid:105) , (cid:104) γ − λ (cid:105) for all γ ∈ W d pa + i and h ∈ W d p . (4.17)Finally, set { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } = { H d (cid:104) h (cid:105) , (cid:104) γ (cid:105) } .For the case of large-scale systems, computing and storing { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } and { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } is a significant challenge due to the number of these coefficients. Specifically, thenumber of terms increases with l (number of uncertain parameters in System (4.4)), d p (degree of P ( α )), d pa (degree of P ( α ) A ( α )) and d , d (Polya’s exponents) asfollows. β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) Given l, d p and d , since h ∈ W d p and γ ∈ W d p + d , the number of coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) is the product of L := card( W d p ) and L := card( W d p + d ). Recall that card( W d p ) isthe number of all l -variate monomials of degree d p and can be calculated using (4.3)as follows. L = f ( l, d p ) = l = 0 (cid:18) d p + l − l − (cid:19) = ( d p + l − d p !( l − l > . (4.18)Likewise, card( W d p + d ), i.e., the number of all l − variate monomials of degree d p + d is calculated using (4.3) as follows. L = f ( l, d p + d ) = l = 0 (cid:18) d p + d + l − l − (cid:19) = ( d p + d + l − d p + d )!( l − l > . (4.19)66 Number of uncertain parameters l N u m b e r o f β c o e ff i c i e n t s d =d =0d =d =2d =d =4d =d =6d =d =8d =d =10 Figure 4.2:
Number of β (cid:104) h (cid:105) , (cid:104) γ (cid:105) Coefficients vs. the Number of Uncertain Parametersfor Different Polya’s Exponents and for d p = 2 Number of uncertain parameters l N u m b e r o f H c o e ff i c i e n t s d =d =0d =d =2d =d =4d =d =6d =d =8d =d =10 Figure 4.3:
Number of H (cid:104) h (cid:105) , (cid:104) γ (cid:105) Coefficients vs. the Number of Uncertain Parametersfor Different Polya’s Exponents and for d p = d a = 2The number of coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) is L · L . In Figure 4.2, we have plotted the numberof coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) in terms of the number of uncertain parameters l and for differentpolya’s exponents.Given l, d p , d a and d , since h ∈ W d p and γ ∈ W d pa + d , the number of coefficients H (cid:104) h (cid:105) , (cid:104) γ (cid:105) is the product of L := card( W d p ) and M := card( W d pa + d ). By using (4.3),67e have M = f ( l, d pa + d ) = l = 0 (cid:18) d pa + d + l − l − (cid:19) = ( d pa + d + l − d pa + d )!( l − l > . (4.20)The number of H (cid:104) h (cid:105) , (cid:104) γ (cid:105) coefficients is L · M . In Figure 4.3, we have plotted thenumber of coefficients H (cid:104) h (cid:105) , (cid:104) γ (cid:105) in terms of the number of uncertain parameters l andfor different polya’s exponents.We have shown the required memory to store the coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) inFigure 4.4 in terms of the number of uncertain parameters l and for different Polya’sexponents. It is observed from Figure 4.4 that even for small degree d p of P ( α ) andsmall degree d a of the system matrix A ( α ), the required memory is in the Terabyterange. Peet and Peet (2010) proposed a decentralized computing approach to thecalculation of { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } on a cluster computer. In the work, we extend this methodto the calculation of { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } and the SDP elements which will be discussed in thefollowing section. We express the LMIs associated with conditions (4.12) and (4.13)as an SDP in both primal and dual forms. We will also discuss the structure of theprimal and dual SDP variables and the constraints. Recall from Section 3.4 that a semi-definite program can be stated either in primalor in dual format. Given C ∈ S m , a ∈ R K and B i ∈ S m , here we considermax X ∈ S m tr( CX )subject to B ( X ) = aX ≥ , (4.21)68 −8 −6 −4 −2 Number of uncertain parameters l M e m o r y r e qu i r e d t o s t o r e β a nd H c o e ff s ( G b y t es ) d =d =1,n=1d =d =10,n=1d =d =1,n=10d =d =10,n=10d =d =1,n=100d =d =10,n=100d =d =1,n=1000d =d =10,n=1000 Figure 4.4:
Memory Required to Store the Coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) vs. Num-ber of Uncertain Parameters, for Different d , d and d p = d a = 2as the primal SDP form, where the linear operator B : S m → R K is defined in (3.18).The associated dual problem is min y,Z a T y subject to K (cid:88) i =1 B i y i − C = ZZ ≥ , y ∈ R K . (4.22)The elements C , B i and a of the SDP problem associated with the LMIs in (4.12)and (4.13) are defined as follows. We define the element C as C := diag( C , · · · C L , C L +1 , · · · C L + M ) , (4.23)where C i := δI n · (cid:16)(cid:80) h ∈ W dp β (cid:104) h (cid:105) ,i d p ! h ! ··· h l ! (cid:17) , for 1 ≤ i ≤ L n , for L + 1 ≤ i ≤ L + M, (4.24)where recall that L = card( W d p + d ) is the number of monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ), M = card( W d pa + d ) is the number of monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α ), n is the69imension of System (4.4), l is the number of uncertain parameters and δ is a smallpositive parameter.For i = 1 , · · · , K , define B i elements as B i := diag( B i, , · · · B i,L , B i,L +1 , · · · B i,L + M ) , (4.25)where K is the number of dual variables in (4.22) and is equal to the product ofthe number of upper-triangular elements in each P γ ∈ S n (the coefficients in P ( α ))and the number of monomials in P ( α ) (i.e. the cardinality of W d p ). Since there are f ( l, d p ) = (cid:18) d p + l − l − (cid:19) coefficients in P ( α ) and each coefficient has ˜ N := n ( n + 1)upper-triangular elements, we find K as K = ( d p + l − d p !( l − N . (4.26)To define the B i,j blocks, first we define the map V (cid:104) h (cid:105) : Z K → Z n × n , V (cid:104) h (cid:105) ( x ) := ˜ N (cid:88) j =1 E j x j + ˜ N ( (cid:104) h (cid:105)− for all h ∈ W d p , (4.27)which maps each variable to E j , where E j , j = 1 , · · · ˜ N define the canonical basis for S n (subspace of symmetric matrices) as follows.[ E j ] i,k := i = k = j , for j ≤ n and (4.28)[ E j ] i,k := [ F j ] i,k + [ F j ] Ti,k , for j > n, (4.29)where [ F j ] i,k := i = k − j − n . (4.30)70ote that a different choice of basis would require a different function V (cid:104) h (cid:105) . Then, for i = 1 , · · · , K , we define B i,j matrices as B i,j := (cid:80) h ∈ W dp β (cid:104) h (cid:105) ,j V (cid:104) h (cid:105) ( e i ) , for 1 ≤ j ≤ L ( I ) − (cid:80) h ∈ W dp (cid:16) H T (cid:104) h (cid:105) ,j − L V (cid:104) h (cid:105) ( e i ) + V (cid:104) h (cid:105) ( e i ) H (cid:104) h (cid:105) ,j − L (cid:17) , for L + 1 ≤ j ≤ L + M, ( II )(4.31)where we have denoted the canonical basis for R n by e i = [0 ... i th (cid:122)(cid:125)(cid:124)(cid:123) ... , i =1 , · · · , n . Finally, to complete the SDP problem associated with Polya’s algorithm,we choose a as a = (cid:126) ∈ R K . (4.32) In this section, we propose a decentralized, iterative algorithm for calculating theterms { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } , { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } , C and B i as defined in (4.15), (4.17), (4.23) and (4.25). Wehave provided an MPI implementation of this algorithm in C++. The source codeis available at . InAlgorithm 5, we have presented a pseudo-code for this algorithm, wherein N is thenumber of available processors.4.4 Complexity Analysis of the Set-up AlgorithmSince verifying the positive definiteness of all representatives of a square matrixwith entries on proper real intervals is intractable (Nemirovskii (1993)), the questionof feasibility of (5.8) is also intractable. To solve the problem of inherent intractabilitywe establish a trade off between accuracy and complexity. In fact, we develop asequence of decentralized polynomial-time algorithms whose solutions converge to theexact solution of the NP-hard problem. In other words, the translation of a polynomial71 nputs : d p : degree of P ( α ), d a : degree of A ( α ), n : number of states, l : No. ofuncertain parameters, d , d : number of Polya’s iterations, Coefficients of A ( α ). Initialization : Set ˆ d = ˆ d = 0 and d pa = d p + d a .Calculate L as the No. of monomials in P ( α ) using (4.18). Set L = L .Calculate M as the No. of monomials in P ( α ) A ( α ) using (4.20).Calculate L (cid:48) = floor ( LN ) as No. of monomials in P assigned to each processor.Calculate M (cid:48) = floor ( MN ) as the No. of monomials in P ( α ) A ( α ) assigned toeach processor. for i = 1 , · · · , N , processor i do Initialize β k,j for j = ( i − L (cid:48) + 1 , · · · , iL (cid:48) and k = 1 , · · · L using (4.14).Initialize H k,m for m = ( i − M (cid:48) + 1 , · · · , iM (cid:48) & k = 1 , · · · L using (4.16). end Calculating β and H coefficients: while ˆ d ≤ d or ˆ d ≤ d doif ˆ d ≤ d thenfor i = 1 , · · · , N , processor i do Set d p = d p + 1. Set ˆ d = ˆ d + 1.Update L using (4.19). Update L (cid:48) as L (cid:48) = floor ( LN ).Calculate β k,j , j = ( i − L (cid:48) + 1 , · · · , iL (cid:48) & k = 1 , · · · L using (4.15). endendif ˆ d ≤ d thenfor i = 1 , · · · , N , processor i do Set d pa = d pa + 1 and ˆ d = ˆ d + 1.Update M using (4.20). Update M (cid:48) as M (cid:48) = floor ( MN ).Calculate H k,m for m = ( i − M (cid:48) + 1 , · · · , iM (cid:48) and k = 1 , · · · L .using (4.17). endendend alculating the SDP elements :for i = 1 , · · · , N , processor i do Calculate the number of dual variables K using (5.24).Set T (cid:48) = floor ( L + MN ).Calculate the blocks of the SDP element C as C j using (4.24) for j = ( i − L (cid:48) + 1 , · · · , iL (cid:48) C j = 0 n for j = L + ( i − M (cid:48) + 1 , · · · , L + iM (cid:48) . Set the sub-blocks of the SDP element C as C i = diag (cid:0) C ( i − T (cid:48) +1 , · · · , C iT (cid:48) (cid:1) . (4.33) for j = 1 , · · · , K do Calculate the blocks of the SDP elements B j as B j,k using (5.25)- I for k = ( i − L (cid:48) + 1 , · · · , iL (cid:48) B j,k using (5.25)- II for k = L + ( i − M (cid:48) + 1 , · · · , L + iM (cid:48) . Set the sub-blocks of the SDP element B j as B j,i = diag (cid:0) B j, ( i − T (cid:48) +1 , · · · , B j,iT (cid:48) (cid:1) . (4.34) endend Outputs : Sub-blocks C i and B j,i of the SDP elements for i = 1 , · · · , N and j = 1 , · · · , K . Algorithm 5:
A parallel set-up algorithm for robust stability analysis over thestandard simplex 73ptimization problem to an LMI problem is the main source of complexity. This highcomplexity is unavoidable and, in fact, is the reason we seek parallel algorithms.Algorithm 5 distributes the computation and storage of coefficients { β (cid:104) h (cid:105) , (cid:104) γ (cid:105) } and { H (cid:104) h (cid:105) , (cid:104) γ (cid:105) } among the processors and their dedicated memories, respectively. In anideal case, where the number of available processors is sufficiently large (equal to thenumber of monomials in P ( α ) A ( α ), i.e. M ) only one monomial (that corresponds to L of coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and L of coefficients H (cid:104) h (cid:105) , (cid:104) γ (cid:105) ) is assigned to each processor. The most computationally expensive part of the set-up algorithm is the calculationof the B i,j blocks in (5.25). Considering that the cost of matrix-matrix multiplicationis ∼ n , the cost of calculating each B i,j block is ∼ card( W d p ) · n . According to (4.25)and (5.25), the total number of B i,j blocks is K ( L + M ). Hence, as per Algorithm 5,each processor processes K (cid:0) floor ( LN ) + floor ( MN ) (cid:1) of the B i,j blocks, where N isthe number of available processors. Therefore, the per processor computational costof calculating the B i,j at each Polya’s iteration is ∼ card( W d p ) · n · K (cid:18) floor (cid:18) LN (cid:19) + floor (cid:18) MN (cid:19)(cid:19) . (4.35)By substituting for K from (5.24), card( W d p ) from (4.18), L from (4.19) and M from (4.20), the per processor computation cost at each iteration is ∼ (cid:18) ( d p + l − d p !( l − (cid:19) n ( n + 1)2 floor ( d p + d + l − d p + d )!( l − N + floor ( d pa + d + l − d pa + d )!( l − N , umber of processors L L M
Computationalcost per processor ∼ ( l d p + d + l d p + d a + d ) n ∼ ( l d p + d + l d p + d a + d ) n ∼ l d p + d a + d − d n Table 4.1:
Per Processor, Per Iteration Computational Complexity of the Set-upAlgorithm. L is the Number of Monomials Is P ( α ); L Is the Number of Monomialsin (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ); M Is the Number of Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α ).assuming that l > N ≤ M , i.e., the number of monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α )is at least as large as the number of available processors. Under the assumption thatthe dynamical systems has large numbers of states and uncertain parameters (large n and l ), Table (4.1) presents the computational cost per processor of each Polya’s iter-ation for three different numbers of available processors. For the case where d p ≥ n than in l . Communication between processors can be modeled by a directed graph G ( V, E ),where the set of nodes V = { , · · · , N } is the set of indices of the available processorsand the set of edges E = { ( i, j ) : i, j ∈ V } is the set of all pairs of processors thatcommunicate with each other. For every directed graph, we can define an adjacencymatrix T G as follows. If processor i communicates with processor j , then [ T G ] i,j = 1,otherwise [ T G ] i,j = 0. Here we only define the adjacency matrix for the part ofthe algorithm that performs Polya’s iterations on P ( α ). For Polya’s iterations on P ( α ) A ( α ), the adjacency matrix can be defined in a similar manner. For simplicity,we assume that at each iteration, the number of available processors is equal tothe number of monomials in ( (cid:80) li =1 α i ) d P ( α ). Using (4.19), let us define r d and r d +1 as the numbers of monomials in ( (cid:80) li =1 α i ) d P ( α ) and ( (cid:80) li =1 α i ) d +1 P ( α ). For75 = 1 , · · · , r d , define E I := { lexicographical indices of monomials in (cid:32) l (cid:88) i =1 α i (cid:33) α γ : γ ∈ W d p + d and (cid:104) γ (cid:105) = I } . Then, for i = 1 , · · · , r d +1 and j = 1 , · · · , r d +1 , the adjacency matrix of the commu-nication graph is [ T G ] i,j := i ≤ r d and j ∈ E i and i (cid:54) = j . Note that this definition implies that the communication graph of the set-up algorithmchanges at every iteration. To help visualize the graph, the adjacency matrix for thecase where α ∈ ∆ is T G := · · · · · ·
00 0 1 0 · · · · · · · · · · · · · · · · · · · · · · · · ∈ R r d × r d , where the nonzero sub-block of T G lies in R r d × r d . We can also illustrate the com-munication graphs for the cases α ∈ ∆ and α ∈ ∆ with d p = 2 as seen in Figure 4.5.For a given algorithm, the communication complexity is defined as the sum ofthe size of all communicated messages. For simplicity, let us consider the worst casescenario, where each processor is assigned more than one monomial and sends allof its assigned coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) and H (cid:104) h (cid:105) , (cid:104) γ (cid:105) to other processors. In this case, thealgorithm assigns floor ( LN ) · card( W d p ) of coefficients β (cid:104) h (cid:105) , (cid:104) γ (cid:105) , each of size 1, and (cid:0) floor ( LN ) + floor ( MN ) (cid:1) · card( W d p ) of coefficients H (cid:104) h (cid:105) , (cid:104) γ (cid:105) , each of size n , to each76 a) (b) Figure 4.5:
Graph Representation of the Network Communication of the Set-upAlgorithm. (a) Communication Directed Graph for the Case α ∈ ∆ , d p = 2. (b)Communication Directed Graph for the Case α ∈ ∆ , d p = 2.Number of processors L L M communication cost per processor ∼ l d pa + d n ∼ l d pa + d − d n ∼ l d p n Table 4.2:
Per Processor, Per Iteration Communication Complexity of the Set-upAlgorithm. L is the Number of Monomials Is P ( α ); L Is the Number of Monomialsin (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ); M Is the Number of Monomials in (cid:16)(cid:80) li =1 α i (cid:17) d P ( α ) A ( α ).processor. Thus, the communication complexity of the algorithm per processor andper iteration becomescard( W d p ) (cid:18) floor (cid:18) LN (cid:19) + floor (cid:18) MN (cid:19) n (cid:19) . (4.36)This indicates that increasing the number of processors (up to M ) actually leads to lesscommunication overhead per processor and improves the scalability of the algorithm.By substituting for card( W d p ) from (4.18), L from (4.19) and M from (4.20) andconsidering large l and n , the communication complexity per processor of each Polya’siteration can be represented as in Table 4.2.77.5 A Parallel SDP SolverCurrent state-of-the-art interior-point algorithms for solving linear and semi-definiteprograms are: dual-scaling, primal-dual, cutting-plane/spectral bundle method. Al-though we found it possible to use dual-scaling algorithms, we chose to pursue acentral-path following primal-dual algorithm. One reason that we prefer primal-dualalgorithms is because in general, primal-dual algorithms converge faster than dual-scaling algorithms. This assertion is motivated by experience as well as bounds onthe convergence rate, such as those found in the literature (Helmberg et al. (1996);Benson et al. (2000)). More importantly, we prefer primal-dual algorithms becausethey have the property of preserving the structure (see (4.47)) of the solution at eachiteration. We will elaborate on this property in Theorem 15.We prefer primal-dual algorithms over cutting plane/spectral bundle algorithmsbecause, as we show in Section 4.6, the centralized part of our primal-dual algorithmconsists of solving a symmetric system of linear equations (see (4.61)), whereas for thecutting plane/spectral bundle algorithm, the centralized computation would consist ofsolving a constrained quadratic program (see Sivaramakrishnan (2010), Nayakkankup-pam (2007)) with the number of variables equal to the size of the system of linearequations. Because centralized computation is a limiting factor in a parallel algo-rithm (Amdahl’s law), and because solving symmetric linear equations is simplerthan solving a quadratic programming problem, we chose the primal-dual approach.The choice of a central path-following primal-dual algorithm as in Helmberg et al. (1996) and F. Alizadeh (1994) was motivated by results in Alizadeh et al. (1998)which demonstrated better convergence, accuracy and robustness over the other typesof primal-dual algorithms. More specifically, we chose the approach in Helmberg et al. (1996) over F. Alizadeh (1994) because unlike the Schur Complement Matrix et al. (1996) is symmetric and only the upper-triangular elements need to be sent/receivedby the processors. This leads to less communication overhead. The other reasonfor choosing Helmberg et al. (1996) is that the symmetric SCM of the algorithmin Helmberg et al. (1996) can be factorized using Cholesky factorization, whereas thenon-symmetric SCM of F. Alizadeh (1994) must be factorized by LU factorization(LU factorization is roughly twice as expensive as Cholesky factorization). Sincefactorization of SCM comprises the main portion of the centralized computation in ouralgorithm, it is crucial for us to use computationally-cheaper factorization methodsto achieve a better scalability.Recall from Section 3.5 that in the primal-dual algorithm, both primal and dualproblems are solved by iteratively calculating primal and dual search directions andstep sizes, and applying these to the primal and dual variables. Let X be the primalvariable and y and Z be the dual variables. At each iteration, the variables areupdated as X k +1 = X k + t p ∆ X (4.37) y k +1 = y k + t d ∆ y (4.38) Z k +1 = Z k + t d ∆ Z, (4.39)where ∆ X , ∆ y , and ∆ Z are the search directions defined in (3.34) and t p and t d areprimal and dual step sizes. For the SDPs associated with Polya’s theorem (see (4.21)and (4.22)), because the map G (defined in (3.18)) is zero, the predictor search di-79ections defined in (3.28)-(3.30) reduce to the following:∆ (cid:98) y = Ω − (cid:0) − a + B ( Z − GX ) (cid:1) (4.40)∆ (cid:98) X = − X + Z − G (cid:32) K (cid:88) i =1 B i ∆ (cid:98) y i (cid:33) X (4.41)∆ (cid:98) Z = (cid:32) K (cid:88) i =1 B i y i (cid:33) − Z − C + (cid:32) K (cid:88) i =1 B i ∆ (cid:98) y i (cid:33) , (4.42)where G = − K (cid:88) i =1 B i y i + Z + C, (4.43)and Ω = [ B ( Z − B X ) · · · B ( Z − B K X )]. Similarly, the corrector search directionsdefined in (3.31)-(3.33) reduce to∆ y = Ω − (cid:16) B ( µZ − ) − B ( Z − ∆ (cid:98) Z ∆ (cid:98) X ) (cid:17) (4.44)∆ X = µZ − − Z − ∆ (cid:98) Z ∆ (cid:98) X − Z − ∆ ZX (4.45)∆ Z = K (cid:88) i =1 B i ∆ y i . (4.46)In the following section, we discuss the structure of the decision variables of the SDPdefined by the Elements (4.23), (4.25) and (4.32). The key algorithmic insight of this study which allows us to use the primal-dualapproach presented in Algorithm 4 is that by choosing an initial value for the primalvariable with a certain block structure corresponding to the distributed structureof the processors, the algorithm will preserve this structure on the primal and dualvariables at every iteration. Specifically, we define the following structured block-diagonal subspace, where each block corresponds to a single processor. S l,m,n := (cid:8) Y ⊂ R ( l + m ) n × ( l + m ) n : Y = diag( Y , · · · Y l , Y l +1 , · · · Y l + m ) for Y i ∈ R n × n (cid:9) (4.47)80ccording to the following theorem, the subspace S l,m,n is invariant under the pre-dictor and corrector iterations in the sense that when Algorithm 4 is applied to theSDP problem defined by the Elements (4.23), (4.25) and (4.32) with a primal startingpoint X ∈ S l,m,n , then the primal and dual variables remain in the subspace at everyiteration. Theorem 15.
Consider the SDP problem defined in (4.21) and (4.22) with elementsgiven by (4.23) , (4.25) and (4.32) . Suppose L and M are the cardinalities of W d p + d and W d pa + d as defined in (4.19) and (4.20) . If (4.37) , (4.38) and (4.39) are initializedby X ∈ S L,M,n , y ∈ R K , Z ∈ S L,M,n , then for all k ∈ N , X k ∈ S L,M,n , Z k ∈ S L,M,n . Proof.
We proceed by induction. First, suppose for some k ∈ N ,X k ∈ S L,M,n and Z k ∈ S L,M,n . (4.48)We would like to show that this implies X k +1 , Z k +1 ∈ S L,M,n . To see this, observethat according to (4.37), X k +1 = X k + t p ∆ X k for all k ∈ N . From (3.34), ∆ X k canbe written as ∆ X k = ∆ (cid:98) X k + ∆ X k for all k ∈ N . (4.49)To find the structure of ∆ X k , we focus on the structures of ∆ (cid:98) X k and ∆ X k individually.Using (4.41), ∆ (cid:98) X k is∆ (cid:98) X k = − X k + Z − k G k (cid:32) K (cid:88) i =1 B i ∆ (cid:98) y k (cid:33) X k for all k ∈ N , (4.50)where according to (4.43), G k is G k = C − K (cid:88) i =1 B i y i + Z k for all k ∈ N . (4.51)81irst, we examine the structure of G k . According to the definition of C and B i in (4.23) and (4.25), we know that C ∈ S L,M,n and K (cid:88) i =1 B i y i ∈ S L,M,n for any y ∈ R K . (4.52)Since all the terms on the right hand side of (4.51) are in S L,M,n and S L,M,n is asubspace, we conclude G k ∈ S L,M,n . (4.53)Returning to (4.50), using our assumption in (4.48) and noting that the structureof the matrices in S L,M,n is also preserved through multiplication and inversion, weconclude ∆ (cid:98) X k ∈ S L,M,n . (4.54)According to (4.45), the second term in (4.49) is∆ X k = µZ − k − Z − k ∆ (cid:98) Z k ∆ (cid:98) X k − Z − k ∆ Z k X k for all k ∈ N . (4.55)To determine the structure of ∆ X k , first we investigate the structure of ∆ (cid:98) Z k and∆ Z k . According to (4.42) and (4.46) we have∆ (cid:98) Z k = K (cid:88) i =1 B i y k i − Z k − C + K (cid:88) i =1 B i ∆ (cid:98) y k i for all k ∈ N (4.56)∆ Z k = K (cid:88) i =1 B i ∆ y k i for all k ∈ N . (4.57)Because all the terms in the right hand side of (4.56) and (4.57) are in S L,M,n , itfollows that ∆ (cid:98) Z k ∈ S L,M,n , ∆ Z k ∈ S L,M,n . (4.58)Recalling (4.54), (4.55) and our assumption in (4.48), we have∆ X k ∈ S L,M,n . (4.59)82ccording to (4.54), (4.58) and (4.59), the total step directions are in S L,M,n ,∆ X k = ∆ (cid:98) X k + ∆ X k ∈ S L,M,n ∆ Z k = ∆ (cid:98) Z k + ∆ Z k ∈ S L,M,n , and it follows that X k +1 = X k + t p ∆ X k ∈ S L,M,n Z k +1 = Z k + t p ∆ Z k ∈ S L,M,n . Thus, for any y ∈ R K and k ∈ N , if X k , Z k ∈ S L,M,n , we have X k +1 , Z k +1 ∈ S L,M,n .Since we have assumed that the initial values X , Z ∈ S L,M,n , we conclude by induc-tion that X k ∈ S L,M,n and Z k ∈ S L,M,n for all k ∈ N . In this section, we propose a parallel algorithm for solving the SDP problemsassociated with Polya’s algorithm. In particular, we show how to map the block-diagonal structure of the primal variable and the primal-dual search directions de-scribed in Section 4.5 to a parallel computing structure consisting of a central rootprocessor with N slave processors. Note that processor steps are simultaneous andtransitions between root and processor steps are synchronous. Processors are idlewhen root is active and vice-versa. A C++ implementation of this algorithm us-ing MPI and numerical linear algebra libraries CBLAS and CLAPACK is providedat: . Let N be the number ofavailable processors and J := floor (cid:0) L + MN (cid:1) . As per Algorithm 6, we assume pro-cessor i has access to the sub-blocks C i and B j,i defined in (4.33) and (4.34) for j = 1 , · · · , K . Be aware that minor parts of Algorithm 6 have been abridged in orderto simplify the presentation. 83 nputs : C i , B j,i for i = 1 , · · · , N and j = 1 , · · · , K : the sub-blocks of the SDPelements provided to processor i by the set-up algorithm; Stopping criterion (cid:15) . Processors Initialization step: for i = 1 , · · · , N , processor i do Initialize primal and dual variables X i , Z i and y as X i = I ( J +1) n , ≤ i < L + M − N JI Jn , L + M − N J ≤ i < N, , Z i = X i and y = (cid:126) ∈ R K , Calculate the complementary slackness as S i = tr ( Z i X i ).Send S i to the processor root. end Root Initialization step:
Root processor do Calculate the barrier parameter µ = N (cid:80) i =1 S i . Set SDP element a = (cid:126) ∈ R K . Processors step 1: for i = 1 , · · · , N , processor i dofor k = 1 , · · · , K do Calculate the elements of Ω (R-H-S of System (4.61)) ω i,k = tr B k,i ( Z i ) − − K (cid:88) j =1 y j B j,i + Z i + C i X i for l = 1 , · · · , K do Calculate the elements of the SCM as λ i,k,l = tr (cid:0) B k,i ( Z i ) − B l,i X i (cid:1) (4.60) endend Send ω i,k and λ i,k,l , k = 1 , · · · , K and l = 1 , · · · , K to the root processor. end oot step 1: Root processor do Construct the R-H-S of System (4.61) and the SCM as Ω = (cid:80) Ni =1 ω i, (cid:80) Ni =1 ω i, ... (cid:80) Ni =1 ω i,K − a and Λ = (cid:80) Ni =1 λ i, , (cid:80) Ni =1 λ i, , ... (cid:80) Ni =1 λ i,K, , · · · , (cid:80) Ni =1 λ i, ,K (cid:80) Ni =1 λ i, ,K ... (cid:80) Ni =1 λ i,K,K Solve the following system of equations for the predictor dual step ∆ (cid:98) y .Λ∆ (cid:98) y = Ω (4.61)Send ∆ (cid:98) y to all processors. Processors step 2: for i = 1 , · · · , N , processor i do Calculate the predictor step directions∆ (cid:98) X i = − X i + ( Z i ) − (cid:32) − K (cid:88) j =1 y j B j,i + Z i + C i (cid:33) K (cid:88) j =1 ∆ (cid:98) y j B j,i X i , ∆ (cid:98) Z i = K (cid:88) j =1 y j B j,i − Z i − C i + K (cid:88) j =1 ∆ (cid:98) y j B j,i . for k = 1 , · · · , K do Calculate the elements of Ω (R-H-S of (4.62)) δ i,k = tr( B k,i ( Z i ) − ) , τ i,k = tr ( B k,i ( Z i ) − ∆ (cid:98) Z i ∆ (cid:98) X i ) end Send δ i,k and τ i,k , k = 1 , · · · , K to the root processor. end Root step 2:
Construct the R-H-S of (4.62) asΩ = µ (cid:20) N (cid:80) i =1 δ i, N (cid:80) i =1 δ i, · · · N (cid:80) i =1 δ i,K (cid:21) T − (cid:20) N (cid:80) i =1 τ i, N (cid:80) i =1 τ i, · · · N (cid:80) i =1 τ i,K (cid:21) T Solve the following system of equations for the corrector dual variable ∆ y .Λ∆ y = Ω (4.62)Send ∆ y to all processors. 85 rocessors step 3: for i = 1 , · · · , N , processor i do Calculate the corrector step directions as follows.∆ Z i = K (cid:88) j =1 ∆ y j B j,i ∆ X i = − ( Z i ) − (∆ Z i X i + ∆ (cid:98) Z i ∆ (cid:98) X i ) + µ ( Z i ) − Calculate the primal and dual total search directions as∆ X i = ∆ (cid:98) X i + ∆ X i , ∆ Z i = ∆ (cid:98) Z i + ∆ Z i , ∆ y = ∆ (cid:98) y + ∆ y. Set the primal step size t p and dual step size t d using a line search method.Update the primal and dual variables as X i ≡ X i + t p ∆ X i , Z i ≡ Z i + t d ∆ Z i , y ≡ y + t d ∆ y end Processors step 4: for i = 1 , · · · , N , processor i do Calculate the contribution of X i to primal cost and complementaryslackness as ˜ φ i = tr (cid:0) C i X i (cid:1) and S i = tr ( Z i X i ) . Send S i and ˜ φ i to the root processor. end Root step 4:
Update the barrier parameter as µ = (cid:80) Ni =1 S i .Calculate the primal and dual costs as φ = (cid:80) Ni =1 ˜ φ i and ψ = a T y . if | φ − ψ | > (cid:15) then go to Processors step 1 endelse Calculate the coefficients of P ( α ) as P i = (cid:80) ˜ Nj =1 E j y ( j + ˜ Ni − for i = 1 , · · · , L . end Output:
Coefficients P i of a polynomial P ( α ) such that P ( α ) > α ∈ ∆ l and satisfies the Lyapunov inequalities in (5.8). Algorithm 6:
A parallel SDP algorithm86.6 Computational complexity analysis of the SDP algorithmComplexity theory for parallel computation has been studied in some depth (Green-law et al. (1995)). The class NC ⊂ P is often considered to be the class of problemsthat can be parallelized efficiently. More precisely, a problem is in NC if there ex-ist integers c and d such that the problem can be solved in O (log( n ) c ) steps using O ( n d ) processors. On the other hand, the class P-complete is a class of problemswhich are equivalent up to an NC reduction, but contains no problem in NC andis thought to be the simplest class of “inherently sequential” problems. It has beenproven that Linear Programming (LP) is P-complete Greenlaw et al. (1995) and SDPis P-hard (at least as hard as any P-complete problem) and thus is unlikely to admit ageneral-purpose parallel solution. Given this fact and given the observation that theproblem we are trying to solve is NP-hard, it is important to thoroughly understandthe complexity of the algorithms we are proposing and how this complexity scaleswith various parameters which define the size of the stability analysis problem. Tobetter understand these issues, we have broken our complexity analysis down intoseveral cases which should be of interest to the control community. Note that thecases below do not discuss memory complexity. This is because in the cases when asufficient number of processors are available, for a system with n states, the memoryrequirements per block are simply proportional to n . Suppose we are considering a problem with n states. For this case, the mostcomputationally expensive part of the algorithm is the calculation of the Schur com-plement matrix Λ by the processors in Processors step 1 (and summed by the rootin Root step 1, although we neglect this part). In particular, the computational87omplexity of the algorithm is determined by the number of operations required tocalculate (4.60), restated here. λ i,k,l = tr (cid:0) B k,i ( Z i ) − B l,i X i (cid:1) for k = 1 , · · · , K and l = 1 , · · · , K. Since the cost of n × n matrix-matrix multiplication requires O ( n ) steps and eachof X i , Z i , B l,i has floor ( L + MN ) number of blocks in R n × n , the number of operationsperformed by the i th processor to calculate λ i,k,l for k = 1 , · · · , K and l = 1 , · · · , K is proportional to floor (cid:18) L + MN (cid:19) K n N < L + MK n N ≥ L + M (4.63)at each iteration, where i = 1 , · · · , N . By substituting K in (4.63) from (5.24), for N ≥ L + M , each processor performs ∼ (( d p + l − ( d p !) (( l − n (4.64)operations per iteration. Therefore, for systems with large number n of states andfixed degree d p of P ( α ) and number l of uncertain parameters, the number of opera-tions per processor required to solve the SDP associated with parameter-dependentfeasibility problem A ( α ) T P ( α ) + P ( α ) A ( α ) < , is proportional to n . Solving theLMI associated with the parameter-independent problem A T P + P A < et al. (2003) also requires O ( n ) operations per processor. Therefore, ifwe have a sufficient number of available processors (at least L + M ), the proposedalgorithm solves both the stability and robust stability problems by performing O ( n )operations per processor. 88 .6.2 Complexity of Increasing Accuracy/Decreasing Conservativeness We now consider the effect of raising Polya’s exponent. Consider the definition ofsimplex as follows. ˜∆ lr = (cid:40) α ∈ R l : l (cid:88) i =1 α i = r, α i (cid:62) (cid:41) Suppose we now define the accuracy of the algorithm as the largest value of r foundby the algorithm (if it exists) such that if the uncertain parameters lie inside thecorresponding simplex, the stability of the system is verified. Typically, increasingPolya’s exponent d in (4.7) improves the accuracy of the algorithm. If we again onlyconsider Processor step 1, according to (4.64), the number of processor operations isindependent of the Polya’s exponent d and d . Because this part of the algorithmdoes not vary with Polya’s exponent, we look at the root processing requirementsassociated with solving the systems of equations in (4.61) and (4.62) in Root step 1using Cholesky factorization. Each of these systems consists of K equations. Thecomputational complexity of Cholesky factorization is O ( K ). Thus, the number ofoperations performed by the root processor is proportional to K = (( d p + l − ( d p !) (( l − n . (4.65)In terms of communication complexity, the most significant operation between theroot and other processors is sending and receiving λ i,k,l for i = 1 , · · · , N , k = 1 , · · · , K and l = 1 , · · · , K in Processors step 1 and Root step 1. Thus, the total communicationcost for N processors per iteration is ∼ N · K = N (( d p + l − ( d p !) (( l − n . (4.66)From (4.64), (4.65) and (4.66) it is observed that the number of processors operations,root operations and communication operations are independent of Polya’s exponent d and d . Therefore, we conclude that for a fixed d p and sufficiently large number89f processors N ( N ≥ L + M ), improving the accuracy by increasing d and d doesnot add any computation per processor or communication overhead. The speed-up of a parallel algorithm is defined as SP N = T s T N , where T s is theexecution time of the algorithm on a single processor and T N is the execution time ofthe parallel algorithm using N processors. The speed-up is governed by SP N = ND + N S , (4.67)where D is the decentralization ratio and is defined as the ratio of the total opera-tions performed by all processors except the root to total operations performed byall processors and root. S is the centralization ratio and is defined as the ratio ofthe operations performed by the root processor to total operations performed by allprocessors and the root. Suppose that the number of available processors is equal tothe number of sub-blocks in C defined in (4.23), i.e, equal to L + M . Using the abovedefinitions for D and S , Equation (4.64) as the decentralized computation and (4.65)as the centralized computation, D and S can be approximated as D (cid:39) N (( d p + l − ( d p !) (( l − n N (( d p + l − ( d p !) (( l − n + (( d p + l − ( d p !) (( l − n (4.68)and S (cid:39) (( d p + l − ( d p !) (( l − n N (( d p + l − ( d p !) (( l − n + (( d p + l − ( d p !) (( l − n . (4.69)According to (4.19) and (4.20) the number of processors N = L + M is independentof n . Therefore, lim n →∞ D = 1 and lim n →∞ S = 0 .
10 20 30 40 50 60 7001020304050607080
No. of Processors S p ee d − up n=5n=10n=25n=50n=100n=1000 Figure 4.6:
Theoretical Speed-up vs. No. of Processors for Different System Di-mensions n for l = 10, d p = 2, d a = 3 and d = d = 4, Where L + M = 53625By substituting D and S in (4.67) with their limit values, we have lim n →∞ SP N = N .Thus, for large n , by using L + M processors, the presented decentralized algorithmsolves large robust stability problems L + M times faster than the sequential algo-rithms. For different values of the state-space dimension n , the theoretical speed-upof the algorithm versus the number of processors is illustrated in Figure 4.6. As shownin Figure 4.6, for problems with large n , by using N ≤ L + M processors the paral-lel algorithm solves the robust stability problems approximately N times faster thanthe sequential algorithm. As n increases, the trend of speed-up becomes increasinglylinear. Therefore, for problems with a large number of states, our algorithm becomesincreasingly efficient in terms of processor utilization. The proposed algorithm is synchronous in that all processors must return valuesbefore the centralized step can proceed. However, in the case where we have fewerprocessors than blocks, some processors may be assigned one block more than otherprocessors. In this case, some processors may remain idle while waiting for the moreheavily loaded blocks to complete. In the worst case, this can result in a 50% decrease91n the execution speed. We have addressed this issue in the following manner:1. We allocate almost the same number ( ±
1) of blocks of the SDP elements C and B i to all processors, i.e., floor ( L + MN )+1 blocks to r processors and floor ( L + MN )blocks to the other N − r processors, where r is the remainder of dividing L + M by N .2. We assign the same routine to all of the processors in the Processors steps ofAlgorithm 6.If L + M is a multiple of N , then the algorithm assigns the same amount of data,i.e., L + MN blocks of C and B i to each processor. In this case, the processors areperfectly synchronized. If L + M is not a multiple of N , then according to (4.63), r of the N processors perform K n extra operations per iteration. This fractionis 11 + floor ( L + MN ) ≤ . r processors. Thus in the worst case, we have a 50% reduction, although this situationis rare. As an example, the load balancing (distribution of data and calculation) forthe case of solving an SDP of the size L + M = 24 using different numbers of availableprocessors N is demonstrated in Figure 4.7. This figure shows the number of blocksthat are allocated to each processor. According to this figure, for N = 2 ,
12 and 24,the processors are perfectly balanced, whereas for the case where N = 18, twelveprocessors perform 50% fewer calculations. The communication directed graph of the SDP algorithm (see Figure 4.8) is static(fixed for all iterations). At each iteration, root sends messages (dual predictor andcorrector search directions ∆ (cid:98) y and ∆ y ) to all of the processors and receives messages(elements λ i,k,l of the SCM defined in (4.60)) from all of the processors. The adjacency92 N o . o f b l o cks N=2 1 2 3 4 50246 Index of processors N o . o f b l o cks N=5 1 2 3 4 5 6 7 8 9100123 Index of processors N o . o f b l o cks N=101 2 3 4 5 6 7 8 9101112012 Index of processors N o . o f b l o cks N=12 0 5 10 15 18012 Index of processors N o . o f b l o cks N=18 N o . o f b l o cks N=24
Figure 4.7:
The Number of Blocks of the SDP Elements Assigned to Each Processor.An Illustration of Load Balancing.
Figure 4.8:
The Communication Graph of the SDP Algorithmmatrix of the communication directed graph is defined as follows. For i = 1 , · · · , N and j = 1 , · · · , N , [ T G ] i,j := (cid:0) i = 1 or j = 1 (cid:1) and (cid:0) i (cid:54) = j (cid:1) . The goal of this example is to use the proposed algorithm to solve a real-worldstability problem. A simplified model for the poloidal magnetic flux gradient in aTokamak reactor (Witrant et al. (2007)) is ∂ψ x ( x, t ) ∂t = 1 µ a ∂∂x (cid:18) η ( x ) x ∂∂x ( xψ x ( x, t )) (cid:19) (4.70)with Dirichlet boundary conditions ψ x (0 , t ) = 0 and ψ x (1 , t ) = 0 for all t ∈ R + , where ψ x is the deviation of the flux gradient from a reference flux gradient profile, µ isthe permeability of free space, η ( x ) is the plasma resistivity and a is the radius ofthe Last Closed Magnetic Surface (LCMS). To obtain the finite-dimensional state-space representation of the PDE, we discretize the PDE in the spatial domain [0 , N = 7 points. The state-space model is then˙ ψ x ( t ) = A ( η ( x )) ψ x ( t ) , (4.71)94
12 32 52 72 92 112 132 152 x j (cid:98) η ( x j ) 1 . e − . e − . e − . e − . e − . e − . e − . e − Table 4.3:
Data for Example 1: Nominal Values of the Plasma Resistivitywhere A ( η ( x )) ∈ R N × N has the following non-zero entries. a = − µ ∆ x a (cid:32) η ( x ) x + 2 η ( x ) x (cid:33) ,a = 43 µ ∆ x a (cid:32) η ( x ) x x (cid:33) ,a j,j − = 1∆ x µ a (cid:32) η ( x j − ) x j − x j − (cid:33) for j = 2 , · · · , N − ,a j,j = − x µ a (cid:32) η ( x j + ) x j + + η ( x j − ) x j − (cid:33) x j for j = 2 , · · · , N − ,a j,j +1 = 1∆ x µ a (cid:32) η ( x j + ) x j + x j +1 (cid:33) for j = 2 , · · · , N − ,a N,N − = 43∆ xµ a η ( x N − ) x N − x N − ∆ x ,a N,N = − xµ a (cid:32) η ( x N + ) x N x N + ∆ x + η ( x N − ) x N x N − ∆ x (cid:33) , where ∆ x = 1 N and x j := ( j − )∆ x . Typically η ( x j ) are not precisely known (theydepend on other state variables), so we substitute for η ( x j ) in (4.71) with (cid:98) η ( x j ) + α k ,where (cid:98) η ( x j ) are the nominal values of η ( x j ) and α k for k = 1 , · · · , x , · · · , x j and their corresponding values of (cid:98) η ( x j ) arepresented in Table 4.3. Note that we have used data from the Tore Supra reactor toestimate the nominal values (cid:98) η ( x j ).The uncertain system is then written as˙ ψ x ( t ) = A ( α ) ψ x ( t ) , (4.72)95here A is affine, A ( α ) = A + (cid:80) i =1 A i α i , where A = − .
09 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
71 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e A = − .
86 1 .
66 0 0 0 0 01 . − .
15 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
72 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e A = − .
25 6 .
47 0 0 0 0 00 . − .
35 6 .
84 0 0 0 00 4 . − .
25 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
71 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e A = − .
25 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
25 11 .
77 0 0 00 0 8 . − .
27 16 .
06 0 00 0 0 12 . − .
72 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e A = − .
25 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
57 18 .
76 0 00 0 0 14 . − .
42 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e A = − .
25 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
88 31 .
56 00 0 0 0 25 . − .
61 86 .
330 0 0 0 0 97 . − . e = − .
25 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
71 28 .
91 00 0 0 0 23 . − .
17 88 .
940 0 0 0 0 100 . − . e A = − .
25 6 .
47 0 0 0 0 00 . − .
54 3 .
83 0 0 0 00 2 . − .
24 8 .
97 0 0 00 0 6 . − .
46 16 .
06 0 00 0 0 12 . − .
71 28 .
91 00 0 0 0 23 . − .
96 86 .
330 0 0 0 0 97 . − . e . For a given ρ , we restrict the uncertain parameters α k to S ρ , defined as S ρ := { α ∈ R : (cid:88) i =1 α i = − | ρ | , −| ρ | ≤ α i ≤ | ρ |} , which is a simplex translated to the origin. We would like to determine the maximumvalue of ρ such that the system is stable by solving the following optimization problem. ρ ∗ := max ρ subject to System (4.72) is stable for all α ∈ S ρ . (4.73)To represent S ρ using the standard unit simplex defined in (2.15), we define theinvertible map g : ∆ → S ρ as g ( α ) = [ g ( α ) · · · g ( α )] , g i ( α ) := 2 | ρ | ( α i − . . (4.74)Then, if we let A (cid:48) ( α ) = A ( g ( α )), since g is one-to-one, { A ( α (cid:48) ) : α (cid:48) ∈ S ρ } = { A ( g ( α )) : α ∈ ∆ } = { A (cid:48) ( α ) : α ∈ ∆ } . Thus, stability of ˙ ψ x ( t ) = A (cid:48) ( α ) ψ x ( t ) , for all α ∈ ∆ l is equivalent to stability ofEquation (4.72) for all α ∈ S ρ . 97 S p ee d − up Figure 4.9:
Speed-up of Set-up and SDP Algorithms vs. Number of Processors fora Discretized Model of Magnetic Flux in TokamakWe solve the optimization problem in (4.73) using bisection. For each trial value of ρ , we use the proposed parallel SDP solver in Algorithm 6 to solve the associated SDPobtained by our parallel set-up Algorithm 5. The SDP problems have 224 constraintswith the primal variable X ∈ R × . The normalized value of ρ ∗ , i.e., ρ ∗ (cid:98) η ( x / ) isfound to be 0 . (cid:98) η ( x / ) = 8 . · − from Table 4.3. In this particularexample, the optimal value of ρ does not change with the degrees of P ( α ) and Polya’sexponents d and d , primarily because the model is affine. The SDPs are constructedand solved on a parallel Linux-based cluster Cosmea at Argonne National Laboratory.Figure 4.9 shows the algorithm speed-up vs. the number of processors. Note thatsolving this problem by SOSTOOLS (Papachristodoulou et al. (2013)) on the samemachine is impossible due to the lack of unallocated memory. The goal of this example is to investigate the effect of the degree d p of P ( α ) and thePolya’s exponents, d , d on the accuracy of our algorithms. Given a computer with98 fixed amount of RAM, we compare the accuracy (as we defined in Section 4.6.2) ofthe proposed algorithms to the SOS algorithm. Consider the system ˙ x ( t ) = A ( α ) x ( t )where A is a polynomial degree 3 defined as A ( α ) = A α + A α α + A α α α + A α α + A α + A α (4.75)with the constraint α ∈ S L := (cid:40) α ∈ R : (cid:88) i =1 α i = 2 L + 1 , L ≤ α i ≤ (cid:41) , where A i matrices are defined as A = − . − .
56 0 . − . − .
550 0 . − . − .
918 0 . , A = − . − .
86 1 . − . − . − . .
685 0 .
305 0 . , A = − .
357 0 . − . − . − .
505 0 . .
268 0 . − . ,A = − . − .
436 0 . . − .
812 0 . − .
012 0 . − . , A = − . − . − . . − . − . − .
010 0 . − . , A = − . − . − . . − . − . − .
440 0 . − . . Defining g as in Example 1, the problem ismin L s.t. ˙ x ( t ) = A ( g ( α )) x ( t ) is stable for all α ∈ ∆ . (4.76)Using bisection in L , as in Example 1, we varied the parameters d p , d and d . Thecluster computer Karlin at Illinois Institute of Technology with 24 Gbytes/node ofRAM (216 Gbytes total memory) was used to run our algorithm. The upper boundson the optimal L are shown in Figure 4.10 in terms of d and d and for different d p . Considering the optimal value of L to be L opt = − . d p and/or d , d - when they are still relatively small - improves theaccuracy of the algorithm. Figure 4.11 demonstrates how the error in our upperbound for L opt decreases by increasing d p and/or d , d .99 able 4.4: Upper Bounds Found for L opt by the SOS Algorithm Using DifferentDegrees for x and α (inf: Infeasible, O.M.: Out of Memory) (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) Degree in x Degree in α et al. (2013)) using only a single node of the same cluster computerand 24 Gbytes of RAM. We used Putinar’s Positivstellensatz (see Section 2.3.4) toimpose the constraints (cid:80) i =1 α i = 2 L + 1 and L ≤ α i ≤
1. Table 4.4 shows the upperbounds on L given by the SOS algorithm using different degrees for x and α . Byconsidering a Lyapunov function of degree two in x and degree one in α , the SOSalgorithm gives − .
102 as an upper bound on L opt as compared with our value of − . α in the Lyapunov function beyond two resulted ina failure due to lack of memory. In this example, we evaluate the efficiency of the algorithm in using additional pro-cessors to decrease computation time. As mentioned in Section 4.6 on computationalcomplexity, the measure of this efficiency is termed speed-up and in Section 4.6.3,we gave a formula for this number. To evaluate the true speed-up, we first ran theset-up algorithm on the Blue Gene supercomputer at Argonne National Laboratoryusing three random linear systems with different state-space dimensions and numbersof uncertain parameters. Figure 4.12 shows a log-log plot of the computation time100
Polya’s exponents d and d A pp r o x i m a t i on f o r op t i m a l va l u e o f L d p =0d p =1d p =2d p =3d p =4d p =5d p =6d p =7 Figure 4.10:
Upper Bound on Optimal L vs. Polya’s Exponents d and d , forDifferent Degrees of P ( α ). ( d = d ). α ), d p | L − L op t | / | L op t | d =d =0d =d =1d =d =2d =d =3d =d =4d =d =5d =d =6d =d =7d =d =8 Figure 4.11:
Error of the Approximation for the Optimal Value of L vs. Degrees of P ( α ), for Different Polya’s Exponents 101f the set-up algorithm vs. the number of processors. One can be observed thatthe scalability of the algorithm is practically ideal for several different state-spacedimensions and numbers of uncertain parameters.To evaluate the speed-up of the SDP portion of the algorithm, we solved threerandom SDP problems with different dimensions using the Karlin cluster computer.Figure 4.13 gives a log-log plot of the computation time of the SDP algorithm vs. thenumber of processors for three different dimensions of the primal variable X and thedual variable y . As indicated in the figure, the three dimensions of the primal variable X are 200 ,
385 and 1092, and the dimensions of the dual variable y are K = 50 , d p = 2 and d = d = 1. The linearity of the Timevs. Number of Processors curves in all three cases demonstrates the scalability of theSDP algorithm.For comparison, we plot the speed-up of our algorithm vs. that of the general-purpose parallel SDP solver SDPARA 7.3.1 as illustrated in Figure 4.14. Althoughsimilar for a small number of processors, for a larger number of processors, SDPARAsaturates, while our algorithm remains approximately linear. −1 No. of processors C o m pu t a t i on t i m e ( s ) n=10, l=5n=10, l=10n=10, l=15n=10, l=20n=20, l=5n=20, l=10n=20, l=15n=20, l=20 Figure 4.12:
Computation Time of the Parallel Set-up Algorithm vs. Number ofProcessors for Different Dimensions of Linear System n and Numbers of UncertainParameters l - Executed on Blue Gene Supercomputer of Argonne National Labratory102
10 20 40 60 8010 −2 −1 No. of processors C o m pu t a t i on t i m e ( s ) (L+M)n=200, K= 50(L+M)n=385, K=90(L+M)n=1092, K=224 Figure 4.13:
Computation Time of the Parallel SDP Algorithm vs. Number ofProcessors for Different Dimensions of Primal Variable ( L + M ) n and of Dual Variable K - Executed on Karlin Cluster Computer The goal of this example is to show that given moderate computational resources,the proposed decentralized algorithms can solve robust stability problems for systemswith 100+ states. We used the Karlin cluster computer with 24 Gbytes/node ofRAM and nine nodes. We ran the set-up and the SDP algorithms to solve the robuststability problem with dimension n and l uncertain parameters on one and nine nodesof Karlin cluster computer. Thus, the total accessible memory was 24 Gbytes and216 Gbytes, respectively. Using trial and error, for different n and d , d we foundthe largest l for which the algorithms do not terminate due to insufficient memory(Figure 4.15). In all of the runs d a = d p = 1. Figure 4.15 shows that by using216 Gbytes of RAM, the algorithms can solve the stability problem of size n = 100with 4 uncertain parameters in d = d = 1 Polya’s iteration and with 3 uncertainparameters in d = d = 4 Polya’s iterations.103
20 40 60 80051015202530354045 Number of processors N S p ee d − up (L+M)n=200, K=50(L+M)n=385, K=90(L+M)n=1092, K=224(L+M)n=200, K=50, SDPARA(L+M)n=385, K=90, SDPARA(L+M)n=1092, K=224, SDPARA Figure 4.14:
Comparison Between the Speed-up of the Present SDP Solver andSDPARA 7.3.1, Executed on Karlin Cluster Computer
20 30 40 50 60 70 80 90 100051015202530 L a r g es t N o . o f un ce r t a i n p a r a m e t e r s l Dimension of uncertain system n d=1, 216 Gigd=2, 216 Gigd=3, 216 Gigd=4, 216 Gigd=1, 24 Gigd=2, 24 Gigd=3, 24 Gigd=4, 24 Gig 20 30 40 50 60 70 80 90 1000510152025
Dimension of uncertain system n L a r g es t N o . o f un ce r t a i n p a r a m e t e r s l d=1, 216 Gigd=2, 216 Gigd=3, 216 Gigd=4, 216 Gigd=1, 24 Gigd=2, 24 Gigd=3, 24 Gigd=4, 24 Gig Figure 4.15:
Largest Number of Uncertain Parameters of n -Dimensional Systemsfor Which the Set-up Algorithm (Left) and SDP Solver (Right) Can Solve the RobustStability Problem of the System Using 24 and 216 GB of RAM104hapter 5PARALLEL ALGORITHMS FOR ROBUST STABILITY ANALYSIS OVERHYPERCUBES5.1 Background and MotivationIn Chapter 4, we proposed a distributed parallel algorithm for stability analysisover a simplex. Unfortunately, simplices are rather restrictive forms of uncertainty setin that they do not allow for parameters which take values on intervals or polytopes.Additionally, we hope to eventually extend our algorithms to the problem of nonlinearstability, which requires search over positive polynomials defined over a set whichcontains the origin. Since simplicies do not include the origin, our algorithms cannotbe readily applied to such problems.In this chapter, our goal is to generalize our algorithms in Chapter 4 in order toperform robust stability analysis on linear systems with uncertain parameters definedover hypercubes. Several recent papers such as Chesi et al. (2005) and Bliman (2004a),have proposed LMI-based techniques to construct parameter-dependent quadratic-in-state Lyapunov functions for this class of systems. In particular, researchers (Chesi(2005)) have recently turned to SOS methods and the Positivstellensatz results (seeSection 2.1) to construct increasingly accurate and increasingly complex LMI-basedtests for stability over hypercubes. Unfortunately, due to the inherent intractabilityof the problem of polynomial optimization, SOS based algorithms typically run outof memory for even relatively small-sized problems; see e.g., Table 4.4 of Section 4.7.This makes it difficult to solve SOS-based algorithms on desktop computers. In thischapter, we seek for a parallel methodology to distribute the required memory and105omputation among hundreds of processors - each processor possessing a dedicatedmemory.
We start by proposing an extension to Polya’s theorem. This new result param-eterizes every multi-homogeneous polynomial which is positive over a given multi-simplex/hypercube. Based on this result, we propose a parallel algorithm to set-up asequence of block-structured LMIs (similar to the case of a single simplex). Solutionsto these LMIs define parameter-dependent Lyapunov functions for the system. Fi-nally, we use our parallel SDP solver in Section 4.5 to efficiently solve these structuredLMIs. Similar to Algorithm 7, the proposed set-up algorithm in this chapter has nocentralized computation, memory or communication, hence resulting in a near-idealspeed-up. Specifically, we show that the communication operations per processor isproportional to N c , where N c is the number of processors used by the algorithm.This implies that by increasing the number of processors, we actually decrease thecommunication overhead per processor and improve the speed-up. Naturally, thereexists an upper-bound for the number of processors which can be used by the algo-rithm, beyond which, no speed-up is gained. This upper-bound is proportional to thenumber of uncertain parameters in the system and for practical problems will be farlarger than the number of available processors.5.2 Notation and Preliminaries on Multi-homogeneous PolynomialsRecall from Section 4.2 that we denote a monomial by α γ = (cid:81) li =1 α γ i i , where α ∈ R l is the vector of variables and γ ∈ W d is the vector of exponents, were W d isthe set of exponents defined in (4.2). Now consider the case where α = [ α , · · · , α n ]with α i ∈ R l i , and h = [ h , · · · , h n ], where h i ∈ W d pi . Then, we define the set of106 -variate multi-homogeneous polynomials of degree vector D = [ d , · · · , d n ] ∈ N n as(a generalization of (2.19)) P ∈ R [ α , · · · , α n ] : P ( α ) = (cid:88) h ∈ W d · · · (cid:88) h ∈ W dn P { h , ··· ,h n } α h · · · α h n n . (5.1)Note that for any i ∈ { , · · · , n } , the element d i of the degree vector D is the degree of α h i i in P . For brevity, we denote the index set { h , · · · , h n } by H n and { h j , · · · , h n j } by H n,j , where h i j is defined as the j th element of h i ∈ W d i using lexicographicalordering. We define the unit multi-simplex ˜∆ { l , ··· ,l N } as the Cartesian product of N unit simplexes; i.e., ˜∆ { l , ··· ,l N } := ∆ l × · · · × ∆ l N . Given r i ∈ R , let us define thehypercube Φ n ⊂ R n asΦ n := { α ∈ R n : | α i | ≤ r i , i = 1 , · · · , n } . Claim 1:
For every non-homogeneous polynomial F ( α ) with α ∈ ˜∆ { l , ··· ,l n } , thereexists a multi-homogeneous polynomial P such that (cid:110) F ( α ) ∈ R : α ∈ ˜∆ { l , ··· ,l n } (cid:111) = (cid:110) P ( β ) ∈ R : β ∈ ˜∆ { l , ··· ,l n } (cid:111) . To construct P , first let N F be the number of monomials in F . Define t ( k ) := (cid:104) t ( k )1 , · · · , t ( k ) n (cid:105) for k = 1 , · · · , N F , where t ( k ) i is the sum of the exponents of the vari-ables inside ∆ l i , in the k th monomial of F . Then, one can construct P by multiplyingthe k th monomial of F (according to lexicographical ordering) for k = 1 , · · · , N F by n (cid:89) i =1 (cid:32) l i (cid:88) j =1 α i j (cid:33) T i − t ( k ) i , T i := max k ∈{ , ··· ,N F } t ( k ) i . For more clarification, we provide the following example of constructing the multi-homogeneous polynomial P . Example:
Consider the non-homogeneous polynomial F ( α ) = F ( α , + α ) α , + F α , + F α , , α , α ) , ( α , α ) ∈ ∆ , t (1) = t (2) = [1 , , t (3) = [2 ,
0] and t (4) = [0 , P ( α ) is P ( α ) = F ( α + α ) α + F α ( α + α ) + F ( α + α ) α = F { (2 , , (1 , } α α + F { (2 , , (0 , } α α + F { (1 , , (1 , } α α α + F { (1 , , (0 , } α α α + F { (0 , , (1 , } α α + F { (0 , , (0 , } α α . Thus, the coefficients of the multi-homogeneous polynomial P are P { (2 , , (1 , } = F , P { (2 , , (0 , } = F , P { (1 , , (1 , } = 2 F P { (1 , , (0 , } = 2 F , P { (0 , , (1 , } = F + F , P { (0 , , (0 , } = F + F . Claim 2:
For every polynomial F ( x ) with x ∈ Φ n , there exists a multi-homogeneouspolynomial P such that { F ( x ) ∈ R : x ∈ Φ n } = (cid:8) P ( α, β ) ∈ R : α, β ∈ R n and ( α i , β i ) ∈ ∆ for i = 1 , · · · , n (cid:9) . (5.2)To construct P , we propose the following steps.1. Define new variables α i := x i + r i r i ∈ [0 ,
1] for i = 1 , · · · , n .2. Define Q ( α , · · · , α n ) := F (2 r α − r , · · · , r n α n − r n ).3. Define a new set of variables β i := 1 − α i for i = 1 , · · · , n .4. Let N Q be the number of monomials in Q . Define t ( k ) := (cid:104) t ( k )1 , · · · , t ( k ) n (cid:105) for k = 1 , · · · , N Q , where t ( k ) i is the sum of the exponents of the variables inside∆ , in the k th monomial of Q . Then, for k = 1 , · · · , N Q , multiply the k th monomial of Q (according to lexicographical ordering) by n (cid:89) i =1 ( α i + β i ) T i − t ( k ) i , T i := max k ∈{ , ··· ,N Q } t ( k ) i . Example:
Suppose F ( x , x ) = x + x , with x ∈ [ − ,
2] and x ∈ [ − , α := x +24 ∈ [0 ,
1] and α := x +12 ∈ [0 , Q ( α , α ) := f (4 α − , α −
1) = 16 α − α + 2 α + 3By homogenizing Q we obtain the multi-homogeneous polynomial P ( α, β ) =16 α ( α + β ) − α ( α + β )( α + β ) + 2 α ( α + β ) + 3( α + β ) ( α + β ) , ( α , β ) , ( α , β ) ∈ ∆ with the degree vector D = [2 , d = 2 is the sum of exponents of α and β in every monomial of P , and d = 1 is the sum of exponents of α and β in everymonomial of P .In the following theorem (Kamyar and Peet (2012b)), we parameterize all of themulti-homogeneous polynomials which are positive over a multi-simplex. Theorem 16. (Polya’s theorem, multi-simplex version) A matrix-valued multi-homogeneouspolynomial F satisfies F ( α, β ) > for all ( α i , β i ) ∈ ∆ , i = 1 , · · · , n , if there exist e ≥ such that all the coefficients of (cid:32) n (cid:89) i =1 ( α i + β i ) e (cid:33) F ( α, β ) are positive definite.Proof. We use induction as follows.
Basis step:
Suppose n = 1. Then, from the simplex version of Polya’s theorem(Theorem 2) it follows that for every F ( α, β ) > α, β ) ∈ ∆ , there exists some e ≥ α + β ) e F ( α, β ) are positive definite.109 nduction hypothesis: Suppose for every F ( α, β ) > α i , β i ) ∈ ∆ , i =1 , · · · , k there exists some e ≥ (cid:32) k (cid:89) i =1 ( α i + β i ) e (cid:33) F ( α, β )are positive definite.We need to prove that for every F ( α, β ) > α i , β i ) ∈ ∆ , i = 1 , · · · , k + 1 thereexists some e ∗ ≥ (cid:32) k +1 (cid:89) i =1 ( α i + β i ) e ∗ (cid:33) F ( α, β )are positive definite. From the induction hypothesis it follows that for any fixed( ˆ α, ˆ β ) ∈ ∆ , if F ( α , · · · , α k , ˆ α, β , · · · , β k , ˆ β ) > α i , β i ) ∈ ∆ , i = 1 , · · · , k ,then there exists some e ≥ (cid:32) k (cid:89) i =1 ( α i + β i ) e (cid:33) F ( α , · · · , α k , ˆ α, β , · · · , β k , ˆ β ) (5.3)are positive definite. Using our notation in (2.19), we can expand (5.3) as (cid:32) k (cid:89) i =1 ( α i + β i ) e (cid:33) F ( α , · · · , α k , ˆ α, β , · · · , β k , ˆ β ) = (cid:88) h,g ∈ N k h + g = d + e · k f h,g ( ˆ α, ˆ β ) α h β g · · · α h k k β g k k , (5.4)in which we have denoted the coefficients of Product (5.3) by f h,g ( ˆ α, ˆ β ) and we havedenoted the degree vector of F by d . Also k ∈ N k denotes the vector of ones.Because F is a homogeneous polynomial, f h,g are also homogeneous polynomials.Since f h,g ( ˆ α, ˆ β ) > h, g ) ∈ M d,e := { ( h, g ) ∈ N k × N k : h + g = d + e · } ,Polya’s theorem implies that there exist l g,h ≥ h, g ∈ M d,e such that all ofthe coefficients of ( ˆ α + ˆ β ) l g,h f g,h ( ˆ α, ˆ β ) are positive definite. Let us define e ∗ := max (cid:26) max h,g ∈ M d,e { l g,h } , e (cid:27) . α + ˆ β ) e ∗ f g,h ( ˆ α, ˆ β ) are also positive definite. Bymultiplying both sides of (5.4) by ( ˆ α + ˆ β ) e ∗ we have (cid:32) k (cid:89) i =1 ( α i + β i ) e (cid:33) ( ˆ α + ˆ β ) e ∗ F ( α , · · · , α k , ˆ α, β , · · · , β k , ˆ β )= (cid:88) h,g ∈ N k h + g = d + e · k ( ˆ α + ˆ β ) e ∗ f h,g ( ˆ α, ˆ β ) α h β g · · · α h k k β g k k . (5.5)Since all of the coefficients of ( ˆ α + ˆ β ) e ∗ f h,g ( ˆ α, ˆ β ) are positive definite, all of the coeffi-cients of the monomials on the right hand side of (5.5) are positive definite. Moreover,because e ∗ ≥ e (cid:32) k (cid:89) i =1 ( α i + β i ) e ∗ (cid:33) ( ˆ α + ˆ β ) e ∗ F ( α , · · · , α k , ˆ α, β , · · · , β k , ˆ β ) (5.6)will also have all positive definite coefficients. Since we chose ( ˆ α, ˆ β ) arbitrarily fromthe simplex ∆ , by replacing ˆ α and ˆ β with α k +1 and β k +1 in (5.6), (cid:32) k +1 (cid:89) i =1 ( α i + β i ) e ∗ (cid:33) F ( α, β ) with ( α i , β i ) ∈ ∆ , i = 1 , · · · , k + 1will have all positive definite coefficients.5.3 Setting-up the Problem of Robust Stability Analysis over Multi-simplexIn this section, we focus on the problem of robust stability of a system the form˙ x ( t ) = A ( α ) x ( t ) , (5.7)where A ( α ) ∈ R n × n is a multi-homogeneous polynomial of degree vector D a and α ∈ ˜∆ { l , ··· ,l N } denotes the parametric uncertainty in the system. Note that if A isnot homogeneous, one can use Claim 1 to find a multi-homogeneous representationfor A over the multi-simplex. Furthermore, if α ∈ Φ N , then one can use Claim 2 tofind an equivalent representation for A over the multi-simplex ˜∆ { l , ··· ,l N } .111he following theorem gives necessary and sufficient conditions for asymptoticstability of System (5.7). Theorem 17.
The linear system (5.7) is stable if and only if there exists a polynomialmatrix P ( α ) such that P ( α ) > and A T ( α ) P ( α ) + P ( α )A( α ) < for all α ∈ ˜∆ { l , ··· ,l N } . (5.8)Unfortunately, the question of feasibility of the inequalities in Theorem 17 isNP-hard. In this section, we show that applying Theorem 16 yields a sequence ofSDPs of increasing size (and precision) whose solutions converge to a solution ofthe inequalities in Theorem 17. Motivated by the result in Bliman et al. (2006),we consider P ( α ) to be homogeneous. In particular, let P be a multi-homogeneousmatrix-valued polynomial of form P ( α ) = (cid:88) h N ∈ W dpN · · · (cid:88) h ∈ W dp P H N α h · · · α h N N , (5.9)with degree d p = (cid:80) Ni =1 d p i and unknown coefficients P H N ∈ S n . Moreover, let A ( α )be of the form A ( α ) = (cid:88) h ∈ W da · · · (cid:88) h N ∈ W daN A H N α h · · · α h N N , (5.10)with degree d a = (cid:80) Ni =1 d a i . It follows from Theorem 16 that the Lyapunov inequalitiesin Theorem 17 hold for all α ∈ ˜∆ { l , ··· ,l N } if there exist some d ≥ d ≥ N (cid:89) i =1 (cid:32) l i (cid:88) j =1 α i j (cid:33) d P ( α ) and (5.11) − N (cid:89) i =1 (cid:32) l i (cid:88) j =1 α i j (cid:33) d (cid:0) A T ( α ) P ( α ) + P ( α )A( α ) (cid:1) (5.12)have all positive definite coefficients. By substituting for A ( α ) and P ( α ) in (5.11)and (5.12) from (5.10) and (5.9), we find that the inequalities of Theorem 17 hold if112here exists d , d ≥ (cid:88) h ∈ W d · · · (cid:88) h N ∈ W dN β {H N , Γ N } P H N > γ ∈ W d p + d , · · · , γ N ∈ W d pN + d , and (cid:88) h ∈ W d · · · (cid:88) h N ∈ W dN (cid:0) H T {H N , Γ N } P H N + P H N H {H N , Γ N } (cid:1) < γ ∈ W d pa + d , · · · , γ N ∈ W d paN + d , where recall that H N denotes { h , · · · , h N } ,Γ N denotes { γ , · · · , γ N } and d pa i = d p i + d a i for i = 1 , · · · , N . β and H To calculate the (cid:8) β {H N , Γ N } (cid:9) coefficients and (cid:8) H {H N , Γ N } (cid:9) we provide the follow-ing recursive formulae. These formulae are generalization of the recursive formulaein 4.3.1 for the case of a single simplex. First, for all γ ∈ W d p , · · · , γ N ∈ W d pN , andfor all h ∈ W d p , · · · , h N ∈ W d pN set β (0) {H N , Γ N } = h = γ , · · · , h N = γ N . (5.15)Then, for i = 1 , · · · , d , for all γ ∈ W d p + i, · · · , γ N ∈ W d pN + i and for all h ∈ W d p , · · · , h N ∈ W d pN , β ( i ) {H N , Γ N } can be calculated using β ( i ) {H N , Γ N } = (cid:88) λ N ∈ W · · · (cid:88) λ ∈ W β ( i − {H N , { γ − λ , ··· ,γ N − λ N }} . (5.16)Finally, set β {H N , Γ N } = β ( d ) {H N , Γ N } , where γ ∈ W d p + d .To calculate (cid:8) H {H N , Γ N } (cid:9) , first let H (0) {H N , Γ N } = (cid:88) λ N ∈ W daN : λ N + h N = γ N · · · (cid:88) λ ∈ W da : λ + h = γ A { λ , ··· ,λ N } . (5.17)113or γ ∈ W d pa , · · · , γ N ∈ W d paN and h ∈ W d p , · · · , h N ∈ W d pN . Then, for i =1 , . . . , d , γ ∈ W d pa + i , · · · , γ ∈ W d paN + i and h ∈ W d p , · · · , h N ∈ W d pN we have H ( i ) {H N , Γ N } = (cid:88) λ N ∈ W · · · (cid:88) λ ∈ W H ( i − {H N , { γ − λ , ··· ,γ N − λ N }} . (5.18)Finally, set H {H N , Γ N } = H ( d ) {H N , Γ N } , where γ ∈ W d pa + d , · · · , γ N ∈ W d paN + d . To solve the LMI conditions in (5.13) and (5.14), we express them in the form of adual Semi-Definite Programming (SDP) problem with a block-diagonal structure thatis suitable for parallel computation. Define the element C of the SDP formulation ofConditions (5.13) and (5.14) as C := diag( C , · · · C L , C L +1 , · · · C L + M ) , (5.19)where for given Polya’s exponents d and d , L = N (cid:89) i =1 ( d p i + d + l i − d p i + d )!( l i − N (cid:81) i =1 (cid:32) l i (cid:80) j =1 α i,j (cid:33) d P ( α ) and M = N (cid:89) i =1 ( d p i + d a i + d + l i − d p i + d a i + d )!( l i − N (cid:89) i =1 (cid:32) l i (cid:88) j =1 α i,j (cid:33) d ( A T ( α ) P ( α ) + P ( α ) A ( α )) , and for j = 1 , · · · , L + M , C j := (cid:15)I n ζ ( j ) , if 1 ≤ j ≤ L n , if L + 1 ≤ j ≤ L + M, (5.22)114here (cid:15) >
0. In (5.22), we define ζ ( j ) ∈ N L recursively as follows. First, let ζ (0) = ( d p N + d )! l N (cid:81) i =1 h ( N,i, ! , · · · , ( d p N + d )! l N (cid:81) i =1 h ( N,i,f ( l N ,d PN + d )) ! , where we have denoted the exponent of the i th variable in the j th (according tolexicographical ordering) element of W d PN by h ( N,i,j ) . Recall that f ( l, g ) := ( l + g − g !( l − g with l variables. Then, for k = 1 , · · · , N , define ζ ( k ) := ζ ( k − ⊗ ( d p r ( k ) + d )! l r ( k ) (cid:81) i =1 h ( r ( k ) ,i, ! , · · · , ( d p r ( k ) + d )! l r ( k ) (cid:81) i =1 h ( r ( k ) ,i,s ( k )) ! , where r ( k ) := N − k + 1 and s ( k ) := f ( l r ( k ) , d p r ( k ) + d ). Finally, set ζ = ζ ( N ) .For i = 1 , · · · , K , define the elements B i of the SDP as B i = diag ( B i, , · · · , B i,L , B i,L +1 , · · · , B i,L + M ) , (5.23)where K = n ( n + 1)2 N (cid:89) i =1 ( d p i + l i − d p i !( l i − , (5.24)is the total number of dual variables in the SDP problem (i.e., the total number ofupper-triangular elements in all of the coefficients of P ( α )) and where B i,j = (cid:80) h N ∈ W dpN · · · (cid:80) h ∈ W dp β {H N , Γ N,j } V H N ( e i ) , if 1 ≤ j ≤ L − (cid:80) h N ∈ W dpN · · · (cid:80) h ∈ W dp H T {H N , Γ N,j − L } V H N ( e i ) + V H N ( e i ) H {H N , Γ N,j − L } if L + 1 ≤ j ≤ L + M, (5.25) where recall from Section 5.2 that Γ N,j = { γ j , · · · , γ N j } , where γ i j is the j th elementof W d pi + d using lexicographical ordering, and V H N ( x ) = ˜ N (cid:88) k =1 E k x k + N ( I H N − , E k is the canonical basis for S n defined in (4.28), I H N is the lexicographicalindex of monomial α h · · · α h N N , and ˜ N := n ( n +1)2 . Finally, we complete the definitionof the SDP problem by setting a = (cid:126) ∈ R K . In the following section, we propose aparallel set-up algorithm to calculate the SDP elements defined in this section.
In this section, we propose a parallel set-up algorithm for computing the SDPelements in (5.19) and (5.23). An abridged description of the algorithm is presentedin Algorithm 7, wherein we suppose the algorithm is executed on N c number ofprocessors. A C++ parallel implementation of the algorithm is available at: .5.4 Computational Complexity Analysis of the Set-up AlgorithmIn this section, we discuss the performance of Algorithm 7 in terms of speed-up,computation cost, communication cost and memory requirement. The most computationally expensive part of the algorithm is calculation of theelements B i,j elements for i = 1 , · · · , K and j = 1 , · · · , L + M . If the number ofavailable processors is N c = L := N (cid:89) i =1 ( d p i + l i − d p i )!( l i − , then the number of operations per processor at each Polya’s iteration of Algorithm 7is ∼ K · L (cid:18) floor (cid:18) LN c (cid:19) + floor (cid:18) MN c (cid:19)(cid:19) n ∼ n N (cid:89) i =1 l d pi + d ai + d i , (5.27)116 nputs: N : dimension of multi-simplex; l , · · · , l N : dimensions of simplexes; D p , D a : degreevectors of P and A ; coefficients of A ; ˆ d , ˆ d : Polya’s exponents. Initialization: for i = 1 , · · · , N c , processor i do Set d = d = 0 and d pa = d p + d a .Calculate the number of monomials in P ( α ), i.e., L using (5.20).Calculate the number of monomials in P ( α ) A ( α ), i.e., M using (5.21).Calculate the per-processor number of monomials in P ( α ) and P ( α ) A ( α ), i.e., L (cid:48) = floor ( L/N c ) and M (cid:48) = floor ( M/N c ) . (5.26) for γ , h ∈ W d p , · · · , γ N , h N ∈ W d pN do Calculate β {H N , Γ N } using (5.15). endfor γ ∈ W d pa , · · · , γ N ∈ W d paN and h ∈ W d p , · · · , h N ∈ W d pN do Calculate H {H N , Γ N } using (5.17). endend Polya’s iterations: for i = 1 , · · · , N c , processor i dofor d = 1 , · · · , ˆ d do Set d p = d p + 1. Update L and L (cid:48) according to (5.20) and (5.26). for h ∈ W d p , · · · , h N ∈ W d pN do Update β {H N , Γ N } for γ ( i − L (cid:48) +1 ∈ W d p ( i − L (cid:48) +1 , · · · , γ iL (cid:48) ∈ W d piL (cid:48) asin (5.16). endendfor d = 1 , · · · , ˆ d do Set d pa = d pa + 1. Update M and M (cid:48) according to (5.21) and (5.26). for h ∈ W d p , · · · , h N ∈ W d pN do Update H {H N , Γ N } for γ ( i − M (cid:48) +1 ∈ W d pa ( i − M (cid:48) +1 , · · · , γ iM (cid:48) ∈ W d paiM (cid:48) using (5.18). endendend alculating the SDP elements: for i = 1 , · · · , N c , processor i dofor j = ( i − L (cid:48) + 1 , · · · , iL (cid:48) , L + ( i − M (cid:48) + 1 , · · · , iM (cid:48) do Calculate C j using (5.22).Calculate B j for using (5.23). endend Outputs:
The SDP elements C and B i for i = 1 , · · · , K . Algorithm 7:
A parallel set-up algorithm for robust stability analysis over themulti-simplexwhere recall that d p i is the degree of α h i i in polynomial P ( α ) (see (5.9)), d a i is thedegree of the variable α h i i in the polynomial A ( α ) (see (5.10)), and d is the Polya’sexponent. Note that for the case of systems with uncertain parameters inside asimplex, (5.27) reduces to our results in Table 4.2. The number of operations versusthe dimension of hypercube N is plotted in Figure 5.1 for different Polya’s exponents d := d = d . The figure shows that for the case of analysis over a hypercube,the number of operations grows exponentially with the dimension of the hypercube,whereas in analysis over a simplex, the number of operations grows polynomially. Thisis due to the fact that an N -dimensional hypercube is represented by the Cartesianproduct of N two-dimensional simplices. In the worst case scenario, where each processor sends all of its assigned coefficients { H {H N ,γ N } } to other processors (a very rare situation), the communication cost per118 Dimension of hypercube and simplex N N u m b e r o f op e r a t i on s d=0, (H)d=0, (S)d=1, (H)d=1, (S)d=2, (H)d=2, (S)d=3, (H)d=3, (S) Figure 5.1:
Number of Operations vs. Dimension of the Hypercube, for DifferentPolya’s Exponents d = d = d . (H): Hypercube and (S): Simplex.processor at each Polya’s iteration is ∼ L (cid:18) floor (cid:18) LN c (cid:19) + floor (cid:18) MN c (cid:19) n (cid:19) ∼ n N (cid:89) i =1 l d pai + d i , (5.28)assuming the number of processors N c = L . Therefore, in this case, by increasingthe number of processors, the communication cost per processor decreases and thescalability of the algorithm improves. for the case where the uncertain parametersbelong to a simplex, (5.28) reduces to our results in Table 4.2. Again, it can beshown that the communication cost increases exponentially with the dimension ofthe hypercube, whereas in analysis over a simplex, the communication cost increasespolynomially. In the proposed set-up algorithm (Algorithm 7), calculation of the coefficients { β } and { H } is distributed among all of the available processors such that there exists nocentralized computation. As a result, the algorithm can theoretically achieve ideal119 −5 No. of uncertain parameters R e qu i r e d m e m o r y ( G i g a b y t e ) n=10,d=2 (H)n=10,d=5 (H)n=40,d=2 (H)n=40,d=5 (H)n=100,d=2 (H)n=100,d=5 (H)n=10,d=2 (S)n=10,d=5 (S)n=40,d=2 (S)n=40,d=5 (S)n=100,d=2 (S)n=100,d=5 (S) Figure 5.2:
Required Memory for the Calculation of SDP Elements vs. Number ofUncertain Parameters in Hypercube and Simplex, for Different State-space Dimen-sions and Polya’s Exponents d = d . (H): Hypercube, (S): Simplex.(linear) speed-up. In other words, the speed-up SP N = ND + N S = N N, Where D = 1 is the ratio of the operations performed by all processors simultane-ously to the total operations performed simultaneously and sequentially, and S isthe ratio of the operations performed sequentially to the total operations performedsimultaneously and sequentially.In Figure 5.2, we have shown the amount of memory required for storing the SDPelements versus the number of uncertain parameters in the unit hypercube and theunit simplex. The figure shows the required memory for different dimensions of thestate-space n and Polya’s exponents d . In all of the cases, we use d p i = d a i = 1 for i = 1 , · · · , N . The figure shows that for the case analysis over the hypercube, therequired memory increases exponentially with the number of uncertain parameters,whereas for the case of the standard simplex the required memory grows polynomially.120his is again because an N -dimensional hypercube is the Cartesian product of N two-dimensional simplices, i.e., ∆ × · · · × ∆ (cid:124) (cid:123)(cid:122) (cid:125) N times .5.5 Testing and ValidationIn this section, we evaluate the scalability and accuracy of our algorithm throughnumerical examples. In example 1, we evaluate the speed-up of our algorithm throughnumerical experiments. In examples 2 and 3, we evaluate the conservativeness of ouralgorithm and compare it to other methods in the literature.
A parallel algorithm is scalable, if by using N c processors it can solve a problem N c times faster than solving the same problem using one processor. Thus, the speed-up ofthe ideal scalable algorithm is linear. To test the scalability of our algorithm, we runthe algorithm using two random uncertain systems with state-space dimensions n = 5and n = 10. The tests were performed on a linux-based Karlin cluster computer atIllinois Institute of Technology. In all of the runs, D p = [2 , , , , D a = [1 , , ,
1] and α ∈ Φ . Figure 5.3 shows the computation time of the algorithm versus the numberof processors, for two different state-space dimensions and two different number ofPolya’s iterations (Polya’s exponents d = d = d ). The linearity of the curves in allof the executions implies near-perfect scalability of the algorithm. Consider the system ˙ x ( t ) = A ( α ) x ( t ), where A ( α ) = A + A α + A α α α + A α α α ,α ∈ [ − , , α ∈ [ − . , . , α ∈ [ − . , . , C o m pu t a t i on t i m e ( s ) n=5, d=2n=5, d=5n=10, d=2n=10, d=5 Figure 5.3:
Execution Time of the Set-up Algorithm vs. Number of Processors, forDifferent State-space Dimensions n and Polya’s Exponentswhere A = − . − . . − . − . − . − . . . − . − . − . . − . − . A = . − . − . − . . . − . . − . − . . . − . − . − . . A = − . − . − . − . . − . . − . . − . . . . . . − . A = . . . − . . . . . − . − . . . − . − . . . . The problem is to investigate asymptotic stability of this system for all α in the givenintervals using Algorithm 7 and our solver in Algorithm 6. We first represented A ( α )defined over the hypercube [ − , × [ − . , . × [ − . , .
1] by a multi-homogeneouspolynomial B ( β, η ) with ( β i , η i ) ∈ ∆ and with the degree vector D b = [2 , , d = d = 1) our algorithm found the following Lyapunovfunction as a certificate for asymptotic stability of the system. V ( x, β, η ) = x T P ( β, η ) x = x T ( β ( P β β + P β η + P η β + P η η )+ η ( P β β + P β η + P η β + P η η )) x, β = 0 . α +0 . , β = α +0 . , β = 5 α +0 . , η = 1 − β , η = 1 − β , η = 1 − β and P = .
807 0 . − . − . .
010 5 . − .
369 0 . − . − .
369 8 . − . − .
186 0 . − .
824 8 . P = . − .
803 1 . − . − .
803 6 .
016 0 . − . .
804 0 .
042 7 . − . − . − . − .
118 8 . P = . − .
873 0 . − . − .
873 5 . − .
161 0 . . − .
161 7 . − . − .
125 0 . − .
538 6 . P = .
388 0 . − . − . .
130 5 . − . − . − . − .
113 6 . − . − . − . − .
236 5 . P = . − .
803 1 . − . − .
803 6 .
016 0 . − . .
804 0 .
042 7 . − . − . − . − .
118 8 . P = .
807 0 . − . − . .
010 5 . − .
369 0 . − . − .
369 8 . − . − .
186 0 . − .
824 8 . P = .
388 0 . − . − . .
130 5 . − . − . − . − .
113 6 . − . − . − . − .
236 5 . P = . − .
873 0 . − . − .
873 5 . − .
161 0 . . − .
161 7 . − . − .
125 0 . − .
538 6 . . In this example, we used our algorithm to find lower bounds on r ∗ = max r suchthat ˙ x ( t ) = A ( α ) x ( t ) with A ( α ) = A + (cid:88) i =1 A i α i , A = − . − . . − . − . − . − . . . − . − . − . . − . − . , A = . − . − . − . . . − . . − . − . . . − . − . − . − . , A = . . . . − . . − . . − . − . − . − . − . − . . , = − . . . − . . − . . . − . − . − . . − . − . . − . , A = − . − . − . . . . . . . . − . . . − . − . . is asymptotically stable for all α ∈ { α ∈ R : | α i | ≤ r } . In Table 5.1, we haveshown the computed lower bounds on r ∗ for different degree vectors D p (degree vec-tor of polynomial P in Theorem 17). In all of the cases, we set the Polya’s exponents d = d = 0. For comparison, we have also included the lower-bounds computed bythe methods in Bliman (2004a) and Chesi (2005) in Table 5.1. Table 5.1:
The Lower-bounds on r ∗ Computed by Algorithm 7 Using DifferentDegree Vector D p and Using Methods in Bliman (2004a) and Chesi (2005). D p =[0,0,0,0] D p =[0,1,0,1] D p =[1,0,1,0] D p =[1,1,1,1] D p =[2,2,2,2] Bliman (2004a) Chesi (2005)Bound on r ∗ et al. (2009)).125he downside to the use of SOS (with Positivstellensatz multipliers) for stabilityanalysis of nonlinear systems with many states is computational complexity. Specif-ically, this approach requires us to set up and solve large SDPs. As an example,applying SOS method to find a degree 8 Lyapunov function for a nonlinear systemwith 10 states requires at least 900 GB of memory and more than 116 days as compu-tation time on a single-core 2.5 GHz processor. Although Polya’s algorithm impliessimilar complexity to SOS, as we showed in Section 4.3.3, the SDPs associated withPolya’s algorithm possess a block-diagonal structure. This allowed us to develop par-allel algorithms (see Algorithms 5, 6, and 7) for robust stability analysis of linearsystems. Unfortunately, Polya’s theorem cannot be used to represent polynomialswhich have zeros in the interior of the unit simplex (see Powers and Reznick (2006)for an elementary proof of this). From the same reasoning as in Powers and Reznick(2006) it follows that our multi-simplex version of Polya’s theorem (See theorem 16)cannot be used to represent polynomials which have zeros in the interior of a multi-simplex/hypercube. Our proposed solution to this problem is to reformulate thenonlinear stability problem using only strictly positive forms. Specifically, we con-sider Lyapunov functions of the form V ( x ) = x T P ( x ) x , where P is a strictly positivematrix-valued polynomial on the hypercube. This way, we can use our multi-simplexversion of Polya’s theorem to search for a polynomial P ( x ) such that P ( x ) > x ∈ Φ \ { } and (cid:104)∇ x T P ( x ) x, f ( x ) (cid:105) < x ∈ Φ - hence proving asymptoticlocal stability of ˙ x ( t ) = f ( x ( t )) for some f ∈ R [ x ].Although Polya’s algorithm has been generalized to positivity over simplices andhypercubes; as yet no further generalization to arbitrary convex polytopes exists. Inorder to perform analysis on more complicated geometries such as arbitrary convexpolytopes, in this chapter, we look into Handelman’s theorem (see Theorem 19).Some preliminary work on the use of Handelman’s theorem and interval evaluation126or Lyapunov functions on the hypercube has been suggested in Sankaranarayanan et al. (2013) and has also been applied to robust stability of positive linear systemsin Briat (2013). One difficulty in using Handelman’s theorem in stability analysis isthat then theorem cannot be readily used to represent polynomials which have zerosin the interior of a given polytope. To see this, suppose a polynomial g ( g is notidentically zero) is zero at x = a , where a is in the interior of a polytopeΓ K := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } . Suppose there exist b α ≥ , α ∈ N K such that for some d ∈ N , g ( x ) = (cid:88) α ∈ N K : (cid:107) α i (cid:107) ≤ d b α ( w T x + u ) α · · · ( w TK x + u K ) α K . Then, g ( a ) = (cid:88) α ∈ N K : (cid:107) α i (cid:107) ≤ d b α ( w T a + u ) α · · · ( w TK a + u K ) α K = 0 . From the assumption a ∈ int(Γ K ) it follows that w Ti a + u i > i = 1 , · · · , K .Hence b α < α ∈ { α ∈ N K : (cid:107) α (cid:107) ≤ d } . This contradicts withthe assumption that all b α ≥
0. Based on the this reasoning, one cannot readilyuse Handelman’s theorem to search for a polynomial V such that V ( x ) > x ∈ Γ K \ { } and V (0) = 0. In this chapter, we consider a new approach to the use of Handelman’s theoremfor computing regions of attraction of stable equilibria by constructing piecewise-polynomial Lyapunov functions over arbitrary convex polytopes. Specifically, wedecompose a given convex polytope into a set of convex sub-polytopes that sharea common vertex at the origin. Then, on each sub-polytope, we use Handelman’sconditions to define linear programming constraints. Additional constraints are then127roposed which ensure continuity of the Lyapunov function over the entire polytope.We then show the resulting algorithm has polynomial complexity in the number ofstates and compare this complexity with algorithms based on SOS and Polya’s theo-rem. Finally, we evaluate the accuracy of our algorithm by numerically approximatingthe domain of attraction of two nonlinear dynamical systems.6.2 Definitions and NotationIn this section, we present/review notations and definitions of convex polytopes,facets of polytopes, decompositions and Handelman bases.
Definition 1. (Convex Polytope) Given the set of vertices P := { p i ∈ R n , i =1 , · · · , K } , define the convex polytope Γ P as Γ P := { x ∈ R n : x = K (cid:88) i =1 µ i p i : µ i ∈ [0 , and K (cid:88) i =1 µ i = 1 } . Every convex polytope can be represented asΓ K := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } , for some w i ∈ R n , u i ∈ R , i = 1 , · · · , K . Throughout the chapter, every polytope thatwe use contains the origin. Moreover, for brevity, we will drop the superscript K inΓ K . Definition 2.
Given a bounded polytope of the form
Γ := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } , we call ζ i (Γ) := (cid:8) x ∈ R n : w Ti x + u i = 0 and w Tj x + u j ≥ for j ∈ { , · · · , K } (cid:9) the i − th facet of Γ if ζ i (Γ) (cid:54) = ∅ . λ λ λ Figure 6.1:
An Illustration of a D-decomposition of a 2D Polytope. λ i ( x ) := h Ti,j x + g i,j for j = 1 , · · · , m i . Definition 3. ( D − decomposition) Given a bounded polytope of the form Γ := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } , we call D Γ := { D i } i =1 , ··· ,L a D − decomposition of Γ if D i := (cid:8) x ∈ R n : h Ti,j x + g i,j ≥ , j = 1 , · · · , m i (cid:9) for some h i,j ∈ R n , g i,j ∈ R , such that ∪ Li =1 D i = Γ , ∩ Li =1 D i = { } and int ( D i ) ∩ int ( D j ) = ∅ . In Figure 6.1, we have illustrated a D -decomposition of a two-dimensional polytope. Definition 4. (The Handelman basis associated with a polytope) Given a polytope ofthe form
Γ := (cid:8) x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K (cid:9) , we define the set of Handelman bases , indexed by α ∈ E d,K := (cid:8) α ∈ N K : | α | ≤ d (cid:9) (6.1) as Θ d (Γ) := (cid:40) ρ α ( x ) : ρ α ( x ) = K (cid:89) i =1 ( w Ti x + u i ) α i , α ∈ E d,K (cid:41) . efinition 5. (Restriction of a polynomial to a facet) Given a polytope of the form Γ := { x ∈ R n : w Ti x + u i , i = 1 , · · · , K } , and a polynomial P ( x ) of the form P ( x ) = (cid:88) α ∈ E d,K b α K (cid:89) i =1 ( w Ti x + u i ) α i , define the restriction of P ( x ) to the k -th facet of Γ as the function P | k ( x ) := (cid:88) α ∈ E d : α k =0 b α K (cid:89) i =1 ( w Ti x + u i ) α i . We will use the maps defined below in future sections.
Definition 6.
Given w i , h i,j ∈ R n and u i , g i,j ∈ R , let Γ be a convex polytope asdefined in Definition 1 with D − decomposition D Γ := { D i } i =1 , ··· ,L as defined in Def-inition 3, and let λ ( k ) , k = 1 , · · · , B be the elements of E d,n , as defined in (6.1) , forsome d, n, ∈ N . For any λ ( k ) ∈ E d,n , let p { λ ( k ) ,α,i } be the coefficient of b i,α x λ ( k ) in P i ( x ) := (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j . (6.2) Let N i be the cardinality of E d,m i , and denote by b i ∈ R N i the vector of all coefficients b i,α .Define F i : R N i × N → R B as F i ( b i , d ) := (cid:88) α ∈ E d,mi p { λ (1) ,α,i } b i,α , · · · , (cid:88) α ∈ E d,mi p { λ ( B ) ,α,i } b i,α T (6.3) for i = 1 , · · · , L . In other words, F i ( b i , d ) is the vector of the coefficients of P i ( x ) after expansion.Define H i : R N i × N → R Q as H i ( b i , d ) := (cid:88) α ∈ E d,mi p { δ (1) ,α,i } b i,α , · · · , (cid:88) α ∈ E d,mi p { δ ( Q ) ,α,i } b i,α T (6.4)130 or i = 1 , · · · , L , where we have denoted the elements of { δ ∈ N n : δ = 2 e j for j =1 , · · · , n } by δ ( k ) , k = 1 , · · · , Q , where e j are the canonical basis for N n . In otherwords, H i ( b i , d ) is the vector of coefficients of square terms of P i ( x ) after expansion.Define J i : R N i × N × { , · · · , m i } → R B as J i ( b i , d, k ) := (cid:88) α ∈ E d,mi α k =0 p { λ (1) ,α,i } b i,α · · · , (cid:88) α ∈ E d,mi α k =0 p { λ ( B ) ,α,i } b i,α T (6.5) for i = 1 , · · · , L . In other words, J i ( b i , d, k ) is the vector of coefficients of P i | k ( x ) after expansion.Given a polynomial vector field f ( x ) of degree d f , define G i : R N i × N → R Z as G i ( b i , d ) := (cid:88) α ∈ E d,mi s { η (1) ,α,i } b i,α , · · · , (cid:88) α ∈ E d,mi s { η ( P ) ,α,i } b i,α T (6.6) for i = 1 , · · · , L , and where we have denoted the elements of E d + d f − ,n by η ( k ) , k =1 , · · · , Z . For any η ( k ) ∈ E d + d f − ,n , we define s { η ( k ) ,α,i } as the coefficient of b i,α x η ( k ) in (cid:104)∇ P i ( x ) , f ( x ) (cid:105) , where P i ( x ) is defined in (6.2) . In other words, G i ( b i , d ) is the vectorof coefficients of (cid:104)∇ P i ( x ) , f ( x ) (cid:105) .Define R i ( b i , d ) : R N i × N → R C as R i ( b i , d ) := (cid:2) b i,β (1) , · · · , b i,β ( C ) (cid:3) T , (6.7) for i = 1 , · · · , L , where we have denoted the elements of S d,m i := { β ∈ E d,m i : β j = 0 for j ∈ { j ∈ N : g i,j = 0 }} by β ( k ) , k = 1 , · · · , C . Consider P i in the Handelman basis Θ d (Γ) . Then, R i ( b i , d ) isthe vector of coefficients of monomials of P i which are nonzero at the origin.It can be shown that the maps F i , H i , J i , G i and R i are affine in b i . efinition 7. (Upper Dini Derivative) Let f : R n → R n be a continuous map. Then,define the upper Dini derivative of a function V : R n → R in the direction f ( x ) as D + ( V ( x ) , f ( x )) = lim sup h → + V ( x + hf ( x )) − V ( x ) h . It can be shown that for a continuously differentiable V ( x ), D + ( V ( x ) , f ( x )) = (cid:104)∇ V ( x ) , f ( x ) (cid:105) . x ( t ) = f ( x ( t )) , (6.8)about the zero equilibrium, where f : R n → R n . We use the following Lyapunovstability condition. Theorem 18.
For any Ω ⊂ R n with ∈ Ω , suppose there exists a continuous function V : R n → R and continuous positive definite functions W , W , W , W ( x ) ≤ V ( x ) ≤ W ( x ) for x ∈ Ω and D + ( V ( x ) , f ( x )) ≤ − W ( x ) for x ∈ Ω , then System (6.8) is asymptotically stable on { x : { y : V ( y ) ≤ V ( x ) } ⊂ Ω } . In this paper, we construct piecewise-polynomial Lyapunov functions which maynot have classical derivatives. As such, we use Dini derivatives which are known toexist for piecewise-polynomial functions.
Problem statement : Given the vertices p i ∈ R n , i = 1 , · · · , K , we would liketo find the largest positive s such that there exists a polynomial V ( x ) where V ( x )satisfies the conditions of Theorem 18 on the convex polytope (cid:40) x ∈ R n : x = K (cid:88) i =1 µ i p i : µ i ∈ [0 , s ] and K (cid:88) i =1 µ i = s (cid:41) . Theorem 19. (Handelman’s Theorem) Given w i ∈ R n , u i ∈ R , i = 1 , · · · , K , let Γ be a convex polytope as defined in definition 1. If polynomial P ( x ) > for all x ∈ Γ ,then there exist b α ≥ , α ∈ N K such that for some d ∈ N , P ( x ) := (cid:88) α ∈ E d,K b α K (cid:89) ji =1 ( w Ti x + u i ) α i . Given a D-decomposition D Γ := { D i } i =1 , ··· ,L of the form D i := (cid:8) x ∈ R n : h Ti,j x + g i,j ≥ , j = 1 , · · · , m i (cid:9) of some polytope Γ, we parameterize a cone of piecewise-polynomial Lyapunov func-tions which are positive on Γ as V ( x ) = V i ( x ) := (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j for x ∈ D i and i = 1 , · · · , L. We will use a similar parameterization of piecewise-polynomials which are negativeon Γ in order to enforce negativity of the derivative of the Lyapunov function. We willalso use linear equality constraints to enforce continuity of the Lyapunov function.6.4 Expressing the Stability Problem as a Linear ProgramWe first present some lemmas necessary for the proof of our main result. Thefollowing lemma provides a sufficient condition for a polynomial represented in theHandelman basis to vanish at the origin ( V (0) = 0). Lemma 1.
Let D Γ := { D i } i =1 , ··· ,L be a D-decomposition of a convex polytope Γ , where D i := (cid:8) x ∈ R n : h Ti,j x + g i,j ≥ , j = 1 , · · · , m i (cid:9) . or each i ∈ { · · · , L } , let P i ( x ) := (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j ,N i be the cardinality E d,m i as defined in (6.1) , and let b i ∈ R N i be the vector of thecoefficients b i,α . Consider R i : R N i × N → R C as defined in (6.7) . If R i ( b i , d ) = ,then P i ( x ) = 0 for all i ∈ { · · · , L } .Proof. We can write P i ( x ) = (cid:88) α ∈ E d,mi \ S d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α i + (cid:88) α ∈ S d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α i , where S d,m i := { α ∈ E d,m i : α j = 0 for j ∈ { j ∈ N : g i,j = 0 }} . By the definitions of E d,m i and S d,m i , we know that for each α ∈ E d,m i \ S d,m i for i ∈ { , · · · , L } , there exists at least one j ∈ { , · · · , m i } such that g i,j = 0 and α k >
0. Thus, at x = 0, (cid:88) α ∈ E d,mi \ S d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α i = 0 for all i ∈ { , · · · , L } . Recall the definition of the map R i from (6.7). Since R i ( b i , d ) = for each i ∈{ , · · · , L } , it follows from that b i,α = 0 for each α ∈ S d,m i and i ∈ { , · · · , L } . Thus, (cid:88) α ∈ S d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α i = 0 for all i ∈ { , · · · , L } . Thus, P i (0) = 0 for all i ∈ { , · · · , L } .This Lemma provides a condition which ensures that a piecewise-polynomial func-tion on a D-decomposition is continuous.134 emma 2. Let D Γ := { D i } i =1 , ··· ,L be a D-decomposition of a polytope Γ , where D i := { x ∈ R n : h Ti,j x + g i,j ≥ , j = 1 , · · · , m i } . For each i ∈ { · · · , L } , let P i ( x ) := (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j ,N i be the cardinality of E d,m i as defined in (6.1) , and let b i ∈ R N i be the vector of thecoefficients b i,α . Given i, j ∈ { , · · · , L } , i (cid:54) = j , let Λ i,j ( D Γ ) := (cid:8) k, l ∈ N : k ∈ { , · · · , m i } , l ∈ { , · · · , m j } : ζ k ( D i ) (cid:54) = ∅ and ζ k ( D i ) = ζ l ( D j ) (cid:9) . (6.9) Consider J i : R N i × N × { · · · , m i } → R B as defined in (6.5) . If J i ( b i , d, k ) = J j ( b j , d, l ) for all i, j ∈ { , · · · , L } , i (cid:54) = j and k, l ∈ Λ i,j ( D Γ ) , then the piecewise-polynomialfunction P ( x ) = P i ( x ) , for x ∈ D i , i = 1 , · · · , L is continuous for all x ∈ Γ .Proof. From (6.5), J i ( b i , d, k ) is the vector of coefficients of P i | k ( x ) after expansion.Therefore, if J i ( b i , d, k ) = J j ( b j , d, l ) for all i, j ∈ { , · · · , L } , i (cid:54) = j and ( k, l ) ∈ Λ i,j ( D Γ ), then P i | k ( x ) = P j | l ( x ) for all i, j ∈ { , · · · , L } , i (cid:54) = j and ( k, l ) ∈ Λ i,j ( D Γ ) . (6.10)135n the other hand, from definition 5, it follows that for any i ∈ { , · · · , L } and k ∈ { , · · · , m i } , P i | k ( x ) = P i ( x ) for all x ∈ ζ k ( D i ) . (6.11)Furthermore, from the definition of Λ i,j ( D Γ ), we know that ζ k ( D i ) = ζ l ( D j ) ⊂ D i ∩ D j (6.12)for any i, j ∈ { · · · , L } , i (cid:54) = j and any ( k, l ) ∈ Λ i,j ( D Γ ). Thus, from (6.10), (6.11)and (6.12), it follows that for any i, j ∈ { , · · · , L } , i (cid:54) = j , we have P i ( x ) = P j ( x ) forall x ∈ D i ∩ D j . Since for each i ∈ { , · · · , L } , P i ( x ) is continuous on D i and forany i, j ∈ { · · · , L } , i (cid:54) = j , P i ( x ) = P j ( x ) for all x ∈ D i ∩ D j , we conclude that thepiecewise polynomial function P ( x ) = P i ( x ) x ∈ D i , i = 1 , · · · , L is continuous for all x ∈ Γ. Theorem 20. (Main Result) Let d f be the degree of the polynomial vector field f ( x ) of System (6.8) . Given w i , h i,j ∈ R n and u i , g i,j ∈ R , define the polytope Γ := { x ∈ R n : w Ti x + u i ≥ , i = 1 , · · · , K } , with D-decomposition D Γ := { D i } i =1 , ··· ,L , where D i := { x ∈ R n : h Ti,j x + g i,j ≥ , j = 1 , · · · , m i } . Let N i be the cardinality of E d,m i , as defined in (6.1) and let M i be the cardinalityof E d + d f − ,m i . Consider the maps R i , H i , F i , G i , and J i as defined in definition 6,and Λ i,j ( D Γ ) as defined in (6.9) for i, j ∈ { , · · · , L } . If there exists d ∈ N such that γ in the linear program (LP), max γ ∈ R ,b i ∈ R Ni ,c i ∈ R Mi γ subject to b i ≥ for i = 1 , · · · , Lc i ≤ for i = 1 , · · · , LR i ( b i , d ) = for i = 1 , · · · , LH i ( b i , d ) ≥ for i = 1 , · · · , LH i ( c i , d + d f − ≤ − γ · for i = 1 , · · · , LG i ( b i , d ) = F i ( c i , d + d f − for i = 1 , · · · , LJ i ( b i , d, k ) = J j ( b j , d, l ) for i, j = 1 , · · · , L and k, l ∈ Λ i,j ( D Γ ) (6.13) is positive, then the origin is an asymptotically stable equilibrium for System 6.8.Furthermore, V ( x ) = V i ( x ) = (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j for x ∈ D i , i = 1 , · · · , L with b i,α as the elements of b i , is a piecewise polynomial Lyapunov function provingstability of System (6.8) .Proof. Let us choose V ( x ) = V i ( x ) = (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j for x ∈ D i , i = 1 , · · · , L In order to show that V ( x ) is a Lyapunov function for system 6.8, we need to provethe following:1. V i ( x ) ≥ x T x for all x ∈ D i , i = 1 , · · · , L ,2. D + ( V i ( x ) , f ( x )) ≤ − γ x T x for all x ∈ D i , i = 1 , · · · , L and for some γ > V (0) = 0,4. V ( x ) is continuous on Γ.Then, by Theorem 18, it follows that System (6.8) is asymptotically stable at the ori-gin. Now, let us prove items (1)-(4). For some d ∈ N , suppose γ > , b i and c i for i =1 , · · · , L is a solution to linear program (6.13). Item 1.
First, we show that V i ( x ) ≥ x T x for all x ∈ D i , i = 1 , · · · , L . From thedefinition of the D-decomposition in the theorem statement, h Ti,j x + g i,j ≥
0, for all x ∈ D i , j = 1 , · · · , m i . Furthermore, b i ≥ . Thus, V i ( x ) := (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j ≥ x ∈ D i \ , i = 1 , · · · , L . From (6.4), H i ( b i , d ) ≥ for each i = 1 , · · · , L implies that all the coefficients of the expansion of x T x in V i ( x ) are greater than 1 for i = 1 , · · · , L . This, together with (6.14), prove that V i ( x ) ≥ x T x for all x ∈ D i , i =1 , · · · , L . Item 2.
Next, we show that D + ( V i ( x ) , f ( x )) ≤ − γx T x for all x ∈ D i , i = 1 , · · · , L .For i = 1 , · · · , L , let us refer the elements of c i as c i,β , where β ∈ E d + d f − ,m i . From(6.13), c i ≤ for i = 1 , · · · , L . Furthermore, since h Ti,j x + g i,j ≥ x ∈ D i , itfollows that Z i ( x ) = (cid:88) β ∈ E d + df − c β,i m i (cid:89) j =1 ( h Ti,j x + g i,j ) β j ≤ x ∈ D i , i = 1 , · · · , L . From (6.4), H i ( c i , d + d f − ≤ − γ · for i = 1 , · · · , L implies that all the coefficients of the expansion of x T x in Z i ( x ) are less than − γ for i = 1 , · · · , L . This, together with (6.15), prove that Z i ( x ) ≤ − γx T x for all x ∈ D i ,for i = 1 , · · · , L . Lastly, by the definitions of the maps G i and F i in (6.6) and (6.3),138f G i ( b i , d ) = F i ( c i , d + d f − (cid:104)∇ V i ( x ) , f ( x ) (cid:105) = Z i ( x ) ≤ − γx T x for all x ∈ D i and i ∈ { · · · , L } . Since D + ( V i ( x ) , f ( x )) = (cid:104)∇ V i ( x ) , f ( x ) (cid:105) for all x ∈ D i , it follows that D + ( V i ( x ) , f ( x )) ≤ − γx T x for all x ∈ D i , i ∈ { · · · , L } . Item 3.
Now, we show that V (0) = 0. By Lemma 1, R i ( b i , d ) = implies V i (0) = 0for each i ∈ { , · · · , L } . Item 4.
Finally, we show that V ( x ) is continuous for x ∈ Γ. By Lemma 2, J i ( b i , d, k ) = J j ( b j , d, l ) for all i, j ∈ { , · · · , L } , k, l ∈ Λ i,j ( D Γ ) implies that V ( x )is continuous for all x ∈ Γ.Using Theorem 20, we define Algorithm 8 to search for piecewise-polynomial Lya-punov functions to verify local stability of system (6.8) on convex polytopes. We haveprovided a Matlab implementation for Algorithm 8 at: .6.5 Computational Complexity AnalysisIn this section, we analyze and compare the complexity of the LP in (6.13) withthe complexity of the SDPs associated with Polya’s algorithm in Kamyar and Peet(2013) and an SOS approach using Positivstellensatz multipliers. For simplicity, weconsider Lyapunov functions defined on a hypercube centered at the origin. Notethat we make frequent use of the formula N vars := d (cid:88) i =0 ( i + K − i !( K − , which gives the number of basis functions in Θ d (Γ) for a convex polytope Γ with K facets. 139 nputs: Vertices of the polytope: p i for i = 1 , · · · , K . h i,j and g i,j for i = 1 , · · · , K and j = 1 , · · · , m i .Coefficients and degree of the polynomial vector field in (6.8).Maximum degree of the Lyapunov function: d max while d < d max doif the LP defined in (6.13) is feasible then Break the while loop. else
Set d = d + 1˙ endend Outputs:
If the LP in (6.13) is feasible, then the output is the coefficients b i,α of theLyapunov function V ( x ) = V i ( x ) = (cid:88) α ∈ E d,mi b i,α m i (cid:89) j =1 ( h Ti,j x + g i,j ) α j for x ∈ D i , i = 1 , · · · , L Algorithm 8:
Search for piecewise polynomial Lyapunov functions using Han-delman’s theorem 140 igure 6.2:
Decomposition of the Hypercube in 1 − ,2 − and 3 − Dimensions
Consider the following assumption on our D − decomposition. Assumption 1.
We perform the analysis on an n − dimensional hypercube, centeredat the origin. The hypercube is decomposed into L = 2 n sub-polytopes such thatthe i -th sub-polytope has m = 2 n − facets. Figure 6.2 shows the − , − and − dimensional decomposed hypercube. Let n be the number of states in System (6.8). Let d f be the degree of thepolynomial vector field in System (6.8). Suppose we use Algorithm 1 to search for aLyapunov function of degree d V . Then, the number of decision variables in the LP is N Hvars = L d V (cid:88) d =0 ( d + m − d !( m − d V + d f − (cid:88) d =0 ( d + m − d !( m − − ( d V + 1) (6.16)where the first term is the number of b i,α coefficients, the second term is the numberof c i,β coefficients and the third term is the dimension of R i ( b i , d ) in (6.13). Bysubstituting for L and m in (6.16), from Assumption 1 we have N Hvars = 2 n d V (cid:88) d =0 ( d + 2 n − d !(2 n − d V + d f − (cid:88) d =0 ( d + 2 n − d !(2 n − − d V − . Then, for large number of states, i.e., large n , N Hvars ∼ n (cid:0) (2 n − d V + (2 n − d V + d f − (cid:1) ∼ n d V + d f . N Hcons = N Hvars + L d V (cid:88) d =0 ( d + n − d !( n − d V + d f − (cid:88) d =0 ( d + n − d !( n − , (6.17)where the first term is the total number of inequality constraints associated withthe positivity of b i and negativity of c i , the second term is the number of equalityconstraints on the coefficients of the Lyapunov function required to ensure continuity( J i ( b i , d, k ) = J j ( b j , d, l ) in the LP (6.13)) and the third term is the number of equalityconstraints associated with negativity of the Lie derivative of the Lyapunov function( G i ( b i , d ) = F i ( c i , d + d f −
1) in the LP (6.13)). By substituting for L in (6.17), fromAssumption 1 for large n we get N Hcons ∼ n d V + d f + 2 n ( n d V + n d V + d f − ) ∼ n d V + d f . The complexity of an LP using interior-point algorithms is approximately O ( N vars N cons )(Boyd and Vandenberghe (2004)). Therefore, the computational cost of solving theLP (6.13) is ∼ n d V + d f ) . Recall our approach in Section 5.3 for applying Polya’s algorithm to analyze sta-bility over hypercubes. In Kamyar and Peet (2013), we used the same approachto construct Lyapunov functions for nonlinear ODEs with polynomial vector fields.In particular, this approach uses semi-definite programming to search for the coef-ficients of a matrix-valued polynomial P ( x ) which defines a Lyapunov function as V ( x ) = x T P ( x ) x . Using a similar complexity analysis as in 5.4, we determine thatthe number of decision variables in the associated SDP is N Pvars = n ( n + 1)2 d V − (cid:88) d =0 ( d + n − d !( n − . N Pcons = n ( n + 1)2 (( d V + e − n + ( d V + d f + e − n ) , where here we have denoted Polya’s exponent by e . Then, for large n , N Pvars ∼ n d V and N Pcons ∼ ( d V + d f + e − n . Since solving an SDP with an interior-pointalgorithm typically requires O ( N cons + N var N cons + N var N cons ) operations (Boyd andVandenberghe (2004)), the computational cost of solving the SDP associated withPolya’s algorithm is estimated as ∼ ( d V + d f + e − n . To find a Lyapunov function for (6.8) over the polytopeΓ = (cid:8) x ∈ R n : w Ti x + u i ≥ , i ∈ { , · · · , K } (cid:9) using the SOS approach with Positivstellensatz multipliers Stengle (1974), we searchfor a polynomial V ( x ) and SOS polynomials s i ( x ) and t i ( x ) such that for any (cid:15) > V ( x ) − (cid:15)x T x − K (cid:88) i =1 s i ( x )( w Ti x + u i ) is SOS −(cid:104)∇ V ( x ) , f ( x ) (cid:105) − (cid:15)x T x − K (cid:88) i =1 t i ( x )( w Ti x + u i ) is SOS . Suppose we choose the degree of the s i ( x ) to be d V − t i ( x ) tobe d V + d f −
2. Then, it can be shown that the total number of decision variables inthe SDP associated with the SOS approach is N Svars = N ( N + 1)2 + K N ( N + 1)2 + K N ( N + 1)2 , (6.18) where N is the number of monomials in a polynomial of degree d V / N is thenumber of monomials in a polynomial of degree ( d V − / N is the number of143onomials in a polynomial of degree ( d V + d f − / N = d V / (cid:88) d =1 ( d + n − d )!( n − , N = ( d V − / (cid:88) d =0 ( d + n − d )!( n − N = ( d V + d f − / (cid:88) d =0 ( d + n − d )!( n − . The first terms in (6.18) is the number of scalar decision variables associated withthe polynomial V ( x ). The second and third terms are the number of scalar variablesin the polynomials s i and t i , respectively. The number of constraints in the SDP is N Scons = N + K N + K N + N , (6.19)where N = ( d V + d f ) / (cid:88) d =0 ( d + n − d )!( n − . The first term in (6.19) is the number of constraints associated with positivity of V ( x ),the second and third terms are the number of constraints associated with positivity ofthe polynomials s i and t i , respectively. The fourth term is the number of constraintsassociated with negativity of the Lie derivative. By substituting K = 2 n (For thecase of a hypercube), for large n we have N Svars ∼ N ∼ n d V + d f − and N Scons ∼ KN + N ∼ n N + N ∼ n . d V + d f ) . Finally, using an interior-point algorithm with complexity O ( N cons + N var N cons + N var N cons ) to solve the SDP associated the SOS algorithm requires ∼ n . d V + d f ) − operations. As an additional comparison, we also consider the SOS algorithm forglobal stability analysis, which does not use Positivstellensatz multipliers. For a largenumber of states, we have N Svars ∼ n . d V and N Scons ∼ n . d V + d f ) . In this case, thecomplexity of the SDP is ∼ n . d V + d f ) + n d V + d f . .5.4 Comparison of the Complexities We draw the following conclusions from our complexity analysis.1. For large number of states, the complexity of the LP defined in (6.13) andthe SDP associated with SOS are both polynomial in the number of states,whereas the complexity of the SDP associated with Polya’s algorithm grows exponentially in the number of states. For a large number of states andlarge degree of the Lyapunov polynomial, the LP has the least computationalcomplexity.2. The complexity of the LP defined in (6.13) scales linearly with the number ofsub-polytopes L .3. In Figure 6.3, we show the number of decision variables and constraints for theLP and SDPs using different degrees of the Lyapunov function and differentdegrees of the vector field. The figure shows that in general, the SDP associ-ated with Polya’s algorithm has the least number of variables and the greatestnumber of constraints, whereas the SDP associated with SOS has the greatestnumber of variables and the least number of constraints.6.6 Numerical ResultsIn this section, we first use our algorithm to construct a Lyapunov function fora nonlinear system. we then assess the accuracy of our algorithm in estimating theregion of attraction of the equilibrium point using different types of convex polytopes.145 igure 6.3: Number of Decision Variables and Constraints of the OptimizationProblems Associated with Algorithm 1, Polya’s Algorithm and SOS Algorithm forDifferent Degrees of the Lyapunov Function and the Vector Field f ( x ) Numerical Example 1:
Consider the following nonlinear system (G. Chesi and Vicino (2005)).˙ x = x , ˙ x = − x − x + x x − x + x x + x . Using the polytopeΓ = { x , x ∈ R : 1 . x + x − . ≥ , − . x + x + 0 . ≥ , . x + x + 0 . ≥ , − . x + x − . ≥ } , (6.20)146nd D − decomposition D := { x , x ∈ R : − x ≥ , x ≥ , − . x + x − . ≥ } D := { x , x ∈ R : x ≥ , x ≥ , . x + x + 0 . ≥ } D := { x , x ∈ R : x ≥ , − x ≥ , − . x + x + 0 . ≥ } D := { x , x ∈ R : − x ≥ , − x ≥ , . x + x + 0 . ≥ } , we set-up the LP in (6.13) with d = 4. The solution to the LP certified asymp-totic stability of the origin and yielded the following piecewise polynomial Lyapunovfunction. Figure 6.4 shows the largest level set of V ( x ) inscribed in the polytope Γ. −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 x x D D D D Figure 6.4:
The Largest Level-set of Lyapunov Function (6.21) Inscribed in Poly-tope (6.20) 147 ( x ) = . x + 0 . x + 0 . x − . x x − . x +0 . x − . x x + 0 . x x + 0 . x if x ∈ D . x + 0 . x x + 0 . x + 0 . x + 0 . x x +0 . x x + 0 . x + 0 . x + 0 . x x + 0 . x x +0 . x x + 0 . x if x ∈ D . x + 0 . x − . x + 0 . x x + 0 . x +0 . x − . x x + 0 . x x + 0 . x if x ∈ D . x + 0 . x x + 0 . x − . x − . x x − . x x − . x + 0 . x + 0 . x x + 0 . x x +0 . x x + 0 . x if x ∈ D (6.21)(6.22) Numerical Example 2:
In this example, we test the accuracy of our algorithm in approximating the regionof attraction of a locally-stable nonlinear system known as the reverse-time Van DerPol oscillator. The system is defined as˙ x = − x , ˙ x = x + x ( x − . (6.23)We considered the following convex polytopes:1. Parallelogram Γ P s , P s := { sp i } i =1 , ··· , , where p = [ − . , . , p = [0 . , . , p = [ − . , − . , p = [1 . , − . Q s , Q s := { sq i } i =1 , ··· , , where q = [ − , , q = [1 , , q = [1 , − , q = [ − , − R s , R s := { sr i } i =1 , ··· , , where r = [ − . , , r = [0 , . , r = [1 . , , r = [0 , − . s ∈ R + is a scaling factor. We decompose the parallelogram and the diamondinto 4 triangles and decompose the square into 4 squares. We solved the followingoptimization problem for Lyapunov functions of degree d = 2 , , , max s ∈ R + s subject to max γ in LP (6.13) is positive, whereΓ = Γ P s := { x ∈ R : x = (cid:88) i =1 µ i sp i : µ i ≥ K (cid:88) i =1 µ i = 1 } . To solve this problem, we use a bisection search on s in an outer-loop and an LPsolver in the inner loop. Figure 6.5 illustrates the largest Γ P s , i.e.Γ P s ∗ := { x ∈ R n : x = (cid:88) i =1 µ i s ∗ p i : µ i ≥ (cid:88) i =1 µ i = 1 } and the largest level-set of V i ( x ) inscribed in Γ P s ∗ , for different degrees of V i ( x ). Sim-ilarly, we solved the same optimization problem replacing Γ P s with the square Γ Q s and diamond Γ R s . In all cases, increasing d resulted in a larger maximum inscribedsub-level set of V ( x ) (see Figure 6.6). We obtained the best results using the parallel-ogram Γ P s which achieved the scaling factor s ∗ = 1 . Q s was s ∗ = 1 .
800 and the maximum scaling factor for Γ R s was s ∗ = 1 . igure 6.5: Largest Level-sets of Lyapunov Functions of Different Degrees and TheirAssociated Parallelograms −3 −2 −1 0 1 2 3−3−2−10123 x x d=8d=4d=2d=6 (a) Square polytopes −3 −2 −1 0 1 2 3−3−2−10123 x x d=4d=2d=8d=6 (b) Diamond polytopes Figure 6.6:
Largest Level-sets of Lyapunov Functions of Different Degrees and TheirAssociated Polytopes 150hapter 7OPTIMIZATION OF SMART GRID OPERATION: OPTIMAL UTILITYPRICING AND DEMAND RESPONSE7.1 Background and MotivationReliable and efficient production and transmission of electricity are essential to theprogress of modern industrial societies. Engineers have strived for years to operatepower generating systems in a way to achieve the following objectives: 1) Reliability:maintaining an uninterrupted balance between the generated power and demand; 2)Minimizing the cost of generation and transmission of electricity; 3) Reducing theadverse effects of the system on the environment by increasingly the use of renewablesources such as solar energy. Unfortunately, the first two objectives are in conflict:increasing reliability (often by increasing the maximum capacity of generation) resultsin higher costs. Moreover, the dependence of reliability of power networks and costson integration of renewables is not yet well-understood.One concern of electric utilities is that rapid increase in distributed solar gener-ation may change customers’ consumption pattern in ways that current generatingunits cannot accommodate for these changes. One example of such a change is shownin Figure 7.1 (Arizona Public Service (2014)). In this figure, we have compared thedaily net demand profile of Arizona’s customers in 2014 with its projection in 2029.Because of the misalignment between the solar generation peak (at noon) and the de-mand peak (at 6 PM), as the solar penetration increases, the resulting demand profilewill reshape to a double-peak curve (see Figure 7.1). To respond to such variabilityin the demand profile, utilities will be required to re-structure their generating ca-151
10 15 20100020003000400050006000700080009000 Time (hr) N e t l o a d ( M W ) Summer 2014Summer 2029Winter 2014Winter 2029
Figure 7.1:
Effect of Solar Power on Demand: Net Loads for Typical Summer andWinter Days in Arizona in 2014 and for 2029 (Projected), from Arizona Public Service(2014)pacity by installing generating units which possess a shorter start-up time and highergeneration ramp rates. Moreover, as solar generation by users increases, the total en-ergy provided by the utility will decrease - implying a reduction in revenue for utilitycompanies which charge users based on their total energy consumption. This typeof change in the demand can indeed already be seen in a report by the US EnergyInformation Administration (EIA) as a significant increase in the ratio of the annualdemand peak to annual average demand (see Figuew 7.2). Because utilities mustpay to build and maintain generating capacity as determined by peak demand, theincreasing use of solar will thus result in a decrease in revenue, yet no decrease in thisform of cost. Ultimately, utilities might have a significant fraction of solar users withnegative energy consumption (kWh) during the day and positive consumption duringthe evening and morning. Due to net metering, such users might pay nothing forelectricity while contributing substantially to the costs of building and maintaininggenerating capacity. 152
995 2000 2005 20101.51.61.71.81.92 Year P eak t o ave r a g e d e m a nd Real data − CaliforniaTrendline − New EnglandReal data − New EnglandTrendline − California
Figure 7.2:
Peak to Average Demand of Electricity and Its Trend-line in Californiaand New England from 1993 to 2012, Data Adopted from Shear (2014)Recently, there has been extensive research on how to exploit smart grid fea-tures such as smart metering, energy storage, thermostat programming and vari-able/dynamic pricing in order to reduce peak demands and cost of generation, en-hance monitoring and security of networks, and prevent unintended events such ascascade failures and blackouts. Smart metering enables two-way communications be-tween consumers and utilities. It provides utilities with real-time data of consumption- hence enables them to directly control the load and/or apply prices as a function ofconsumption. Naturally, utilities have been studying this problem for some time andwith the widespread adoption of smart-metering (95% in Arizona), have begun to im-plement various pricing strategies at scale. Examples of this include on-peak, off-peakand super-peak pricing - rate plans wherein the energy price ($/kWh) depends on thetime of day. By charging more during peak hours, utilities encourage conservationor deferred consumption during hours of peak demand. Quite recently, some utili-ties have introduced demand charges for residential customers (SRP (2015),Rumolo(2013)). These charges are not based on energy consumption, but rather the max-imum rate of consumption ($/kW) over a billing period. While such charges more153ccurately reflect the cost of generation for the utilities, in practice the effects of suchcharges on consumption are not well-understood - meaning that the magnitude of thedemand charge must be set in an ad-hoc manner (typically proportional to marginalcost of adding generating capacity).An alternative approach to reducing peaks in demand is to use energy storage. Inthis scenario, batteries, pumping and retained heat are used during periods of low de-mand to create reservoirs of energy which can then be tapped during periods of highdemand - thus reducing the need to increase maximum generating capacity. Indeed,the optimal usage of energy storage in a smart-grid environment with dynamic pricinghas been recently studied in, for example, Li et al. (2011). See Ma et al. (2014) foroptimal distributed load scheduling in the presence of network capacity constraints.However, to date the high marginal costs of storage infrastructure relative to incen-tives/marginal cost of additional generating capacity have limited the widespread useof energy storage by consumers/utilities (EPRI-DOE (2003)). As a cost-free alter-native to direct energy storage, it has been demonstrated experimentally by Braun et al. (2002), Braun (2003), and in-silico by Braun et al. (2001) and Keeney and Braun(1997) that the interior structure of buildings and appliances can be exploited as a passive thermal energy storage system to reduce the peak-load of HVAC. A typicalstrategy - known as pre-cooling - is to artificially cool the interior thermal mass (e.g.,walls and floor) during periods of low demand. Then, during periods of high demand,heat absorption by these cool interior structures supplements or replaces electricitywhich would otherwise be consumed by the HVAC. Quantitative assessments of theeffect of pre-cooling on demand peak and electricity bills can be found in Braun andLee (2006) and Sun et al. (2013). Furthermore, there is an extensive literature onthermostat programming for HVAC systems for on-peak/off-peak pricing (Lu et al. (2005); Arguello-Serrano and Velez-Reyes (1999)) as well as real-time pricing (Old-154wurtel and Morari (2010); Henze et al. (2004); Chen (2001)) using Model PredictiveControl (MPC). Kintner-Meyer and Emery (1995) consider optimal thermostat pro-gramming with passive thermal energy storage and on-peak/off-peak rates. Braun andLee (2006) use the concept of deep and shallow mass to create a simplified analoguecircuit model of the thermal dynamics of the structure. By using this model andcertain assumptions on the gains of the circuit elements, Braun and Lee (2006) derivean analytical optimal temperature set-point for the demand limiting period whichminimizes the demand peak. This scenario would be equivalent to minimizing thedemand charge while ignoring on-peak or off-peak rates. Finally, Henze et al. (2004)use the heat equation to model the thermal energy storage in the walls and applyMPC to minimize monthly electricity bill in the presence of on-peak and off-peakcharges.
In this chapter, we design a computational framework to achieve the three ob-jectives of a modern power network: reliability, cost minimization and integrationof renewables to promote sustainability. This framework relies on smart metering,thermal-mass energy storage, distributed solar generation and on-peak, off-peak anddemand pricing. This framework consists of two nested optimization problems: 1)Optimal thermostat programming ( user-level problem ); 2) Optimal utility pricing( utility-level problem ). In the first problem, we consider optimal HVAC usage for aconsumer with fixed on-peak, off-peak and demand charges and model passive ther-mal energy storage using the heat equation. We address both solar and non-solarconsumers. For a given range of acceptable temperatures and using typical data forexterior temperature, we pose the optimal thermostat programming problem as a con-strained optimization problem and present a Dynamic Programming (DP) algorithm155hich is guaranteed to converge to the solution. This yields the temperature set-points which minimize the monthly electricity bill for the consumer. For the benefitof the consumers who do not have access to continuously adjustable thermostats, wealso develop thermostat programming solutions which include only four programmingperiods, where each period has a constant interior temperature.After solving the thermostat programming problem, we use this solution as amodel of user behaviour in order to quantify the consumer response to changes inon-peak rates, off-peak rates, and demand charges. Then in the second optimizationproblem, we apply a descent algorithm to this model in order to determine the priceswhich minimize the cost-of-generation for the utility. Through several case studies, weshow that the optimal prices are NOT necessarily proportional to the marginal costs ofgeneration - meaning that current pricing strategies may be inefficient. Furthermore,we show that in a network of solar and non-solar customers who use our optimalthermostat, the influence of solar generated power on the electricity bills of non-solar customers is NOT significant. Finally, we conclude that although the policyof calculating the demand charge based on the peak consumption over a full-day(rather than the on-peak hours) can substantially reduce the demand peak, it maynot reduce optimal cost of production. Our study differs from existing literature (inparticular Braun and Lee (2006), Braun (1990), Henze et al. (2004) and Kintner-Meyer and Emery (1995)) in that it: 1) Considers demand charges (demand chargesare far more effective at reducing demand peaks than dynamic pricing) 2) Uses a PDEmodel for thermal storage (yields a more accurate model of thermal storage) 3) Usesa regulated model for the utility (although unregulated utility models are popular,the fact is that most US utilities remain regulated) 4) Considers the effect of solargeneration on the electricity prices and cost of production.156.2 Problem Statement: User-level and Utility Level ProblemsIn this section, we first define a model of the thermodynamics which govern heatingand cooling of the interior structures of a building. We then use this model to posethe user-level (optimal thermostat programming) problem in Sections 7.2.3 and 7.2.4as minimization of a monthly electricity bill (with on/peak, off-peak and demandcharges) subject to constraints on the interior temperature of the building. Finally,we use this map of on-peak, off-peak and demand prices to electricity consumption todefine the utility-level problem in Section 7.2.5 as minimizing the cost of generation,transmission and distribution of electricity.
In 1822, J. Fourier proposed a PDE to model the dynamics of temperature andenergy in a solid mass. Now known as the classical one-dimensional unsteady heatconduction equation, this PDE can be applied to an interior wall as ∂T ( t, x ) ∂t = α ∂ T ( t, x ) ∂x , (7.1)where T : R + × [0 , L in ] → R represents the temperature distribution in the interiorwalls/floor with nominal width L in , and where α = k in ρC p is the coefficient of thermaldiffusivity. Here k in is the coefficient of thermal conductivity, ρ is the density and C p is the specific heat capacity. The wall is coupled to the interior air temperature usingDirichlet boundary conditions, i.e., T ( t,
0) = T ( t, L in ) = u ( t ) for all t ∈ R + , where u ( t ) represents the interior temperature which we assume can be controlled instanta-neously by the thermostat. In the Fourier model, the heat/energy flux through thesurface of the interior walls is modelled as q in ( T ( t, x )) := 2 C in ∂T∂x ( t, , (7.2)157here C in = k in A in is the thermal capacitance of the interior walls and A in is thenominal area of the interior walls. We assume that all energy storage occurs in theinterior walls and surfaces and that energy transport through exterior walls can bemodelled using a steady-state version of the heat equation. This implies that the heatflux q loss through the exterior walls is the linear sink q loss ( t, u ( t )) := T e ( t ) − u ( t ) R e , (7.3)where T e ( t ) is the outside temperature and R e = L e / ( k e A e ) is the thermal resistanceof the exterior walls, where L e is the nominal width of exterior walls, k e is the coef-ficient of thermal conductivity and A e is the nominal area of the exterior walls. Byconservation of energy, the power required from the HVAC to maintain the interiorair temperature is q ( t, u ( t ) , T ( t, x )) = q loss ( u ( t ) , T e ( t )) + q in ( T ( x, t )) . (7.4)See Figure 7.3 for an illustration of the model.Eqn. (7.1) is a PDE. For optimization purposes, we discretize (7.1) in space, using T ( t ) ∈ R M to replace T ( t, x ) ∈ R , where T i ( t ) denotes T ( t, i ∆ x ), where ∆ x := L in M +1 .Then ˙ T ( t ) = A T ( t ) + B u ( t ) , (7.5)where A = α ∆ x − − , B = α ∆ x ∈ R M . We then discretize in time, using ˙ T ( t ) ≈ ( T ( t + ∆ t ) − T ( t )) / ∆ t to rewrite Equa-158 e (t) T e (t)u(t) u(t)T(t,x) Interior wall(thermal storage)Exterior wall Exterior wall L in Figure 7.3:
A Schematic View of Our Thermal Mass Modeltion (7.5) as a difference equation. T k +1 = T k +11 ... T k +1 M = f ( T k , u k ) = f ( T k , u k )... f M ( T k , u k ) = ( I + A ∆ t ) T k + B ∆ t u k (7.6)for k = 0 , · · · , N f −
1, where T k = T ( k ∆ t ) and u k = u ( k ∆ t ). To find empirical values for the parameters α, C in , R e and L in in the thermody-namic model in Section 7.2.1, we collected data from a 4600 sq ft residential buildingin Scottsdale, Arizona. The building was equipped with a 5 ton two-stage and three2.5 ton single-stage RHEEM/RUUD heat pumps, 4-set point thermostats, and 5-min data metering for energy consumption and interior and exterior temperature. Inthis experiment, we applied two different thermostat programming sequences for twonon-consecutive summer days. On the first day, we applied a pre-cooling strategywhich lowers the interior temperature to 23.9 ◦ C during the off-peak hours and allowsthe temperature to increase to 27.8 ◦ C during the on-peak hours, i.e., 12:00 PM to7:00 PM. In the second day, we applied the same pre-cooling strategy except thatthe temperature is again lowered to 23.9 ◦ C between 2:00 PM and 4:00 PM. We thenused Matlab’s least squares optimization algorithm to optimize the parameters such159 ime (hr) P o w e r c on s u m p ti on ( k W ) ExperimentalSimulation (a) Power Consumption Corresponding toa Pre-cooling Strategy for the InteriorTemperature Setting
Time (hr) P o w e r c on s u m p ti on ( k W ) ExperimentalSimulation (b) Power Consumption Correspondingto a Pre-cooling Strategy with Addi-tional Cooling from 14:00-16:00
Figure 7.4:
Simulated and Measured Power Consumptionsthat the root-mean-squared error between the measured power consumption and thesimulated power consumption during the entire two days is minimized. The resultwas the following values for the parameters: L in = 0 . m ), α = 8 . × − ( m /s ), R e = 0 . K/W ) and C in = 45( W m/K ). In Figure 7.4, we have compared theresulting simulated and measured power consumption for the entire two days.
In this section, we define the problem of optimal thermostat programming. Wefirst divide each day into three periods: off-peak hours from 12 AM to t on withelectricity price p off ($ /kW h ); on-peak hours beginning at t on and ending at t off >t on with electricity price p on ($ /kW h ); and off-peak hours from t off to 12 AM withelectricity price p off ($ /kW h ). In addition to the on-peak and off-peak charges, weconsider a monthly charge which is proportional to the maximum rate of consumptionduring the peak hours. The proportionality constant is called the demand price p d ($ /kW ). Given p := [ p on , p off , p d ], the total cost of consumption (daily electricity160ill) is divided as J t ( u , T , p ) = J e ( u , T , p on , p off ) + J d ( u , T , p d ) , (7.7)where J e is the energy cost, J d is the demand cost and u := [ u , · · · , u N f − ] ∈ R N f is the vector temperature settings. The energy cost is J e ( u , T , p on , p off ) = (cid:16) p off (cid:88) k ∈ S off g ( k, u k , T k ) + p on (cid:88) k ∈ S on g ( k, u k , T k ) (cid:17) ∆ t, (7.8)where k ∈ S on if k ∆ t ∈ [ t on , t off ] and k ∈ S off otherwise. That is, S on and S off correspond to the set of on-peak and off-peak sampling times, respectively. Thefunction g is a discretized version of q (Eqn. (7.4)): g ( k, u k , T k ) := T ke − u k R e + 2 C in T k − u k ∆ x . (7.9)i.e., g the power consumed by the HVAC at time step k , where T ke denotes the externaltemperature at time-step k . If demand charges are calculated monthly, the demandcost, J d , for a single day can be considered as J d ( u , T , p d ) := p d
30 max k ∈ S on g ( k, u k , T k ) . (7.10)We now define the optimal thermostat programming (or user-level) problem asminimization of the total cost of consumption, J t , as defined in (7.7), subject to thebuilding thermodynamics in (7.6) and interior temperature constraints: J (cid:63) ( p ) = min u k ,γ ∈ R ,T k ∈ R M J e ( u , T , p on , p off ) + p d γ subject to g ( k, u k , T k ) ≤ γ for k ∈ S on T k +1 = f ( T k , u k ) for k ∈ S on ∪ S off T min ≤ u k ≤ T max for k ∈ S on ∪ S off T = [ T init (∆ x ) , · · · , T init ( M ∆ x )] T , (7.11)161here T min , T max are the acceptable bounds on interior temperature. Note that thisoptimization problem depends implicitly on the external temperature through thetime-varying function g . Most of the commercially available programmable thermostats only include fourprogramming periods per-day, each period possessing a constant temperature. In thissection, we account for this contraint. First, we partition the day into programmingperiods: P i := [ t i − , t i ] , i = 1 , · · · , (cid:91) i =1 P i = [0 , , t i − ≤ t i , t = 0 and t = 24 . We call t , · · · , t switching times. Similar to the previous model, u i ∈ [ T min , T max ]denotes the temperature setting corresponding to the programming period P i .To simplify the mathematical formulation of our problem, we introduce someadditional notation. Define the set S i by k ∈ S i if k ∆ t ∈ P i . Denote L i := max k ∈ S i k .For clarity, we have depicted L i in Figure 7.5. Moreover, for each P i , we define ∆ t i asthe period between the last time-step of P i and the end of P i , i.e., ∆ t i := t i − L i ∆ t. See Figure 7.5 for an illustration of ∆ t i . In this framework, the daily consumptioncharge is I t ( u , T , p ) = I e ( u , T , p on , p off ) + I d ( u , T , p d )where I e is the energy cost I e ( u , T , p on , p off ) = (cid:88) i =1 (cid:32)(cid:88) k ∈ S i (cid:0) r ( k ) g ( k, u i , T k )∆ t (cid:1) + r ( L i ) g ( k, u i , T L i )∆ t i (cid:33) , (7.12)and I d is the demand cost I d ( u , T , p d ) = max k ∈ S on g ( k, u, T k ) , where L = 0 and r isdefined as r ( k ) := p on t off ≤ k ∆ t < t on p off otherwise . igure 7.5: An Illustration for the Programming Periods of the 4-Setpoint Thermo-stat Problem, Switching Times t i , Pricing Function r , L i and ∆ t i .Assuming that the demand cost for a single day is p d max k ∈ S on g ( k, u, T k ), we definethe 4-setpoint thermostat programming problem asmin u , ··· ,u ∈ R t ,t ,t ,γ ∈ R , T k ∈ R M I e ( u , T , p on , p off ) + p d γ subject to g ( k, u i , T k ) ≤ γ for k ∈ S on , i ∈ { , , , } T k +1 = f ( T k , u i ) for k ∈ S i and i ∈ { , , , } T min ≤ u i ≤ T max for i ∈ { , , , } ≤ t i − ≤ t i ≤
24 for i ∈ { , , , } T = [ T init (∆ x ) , · · · , T init ( M ∆ x )] T , (7.13)where t = 0 and t = 24. Regulated utilities must meet expected load while maintaining a balance betweenrevenue and costs. Therefore, we define the utility-level optimization problem as min-imization of the total cost of generation, transmission and distribution of electricitysuch that generation is equal to consumption, and the total cost is a fixed percent-163ge of the revenue of the utility company. Note that in this dissertation, we focuson vertically integrated utility companies - meaning that the company provides allaspects of electric services including generation, transmission, distribution, meteringand billing services as a single firm. Let s ( t ) be the amount of electricity produced asa function of time and let s := [ s , · · · , s N f − ], where s k = s ( k ∆ t ). The vector s is de-termined by the electricity consumed by the users, which we model as a small numberof user groups which are lumped according to different building models, temperaturelimits, and solar generating capacity so that aggregate user group i has N i members.Next, we define u (cid:63),ik ( p ) to be the minimizing user temperature setting for user i attime k with prices p and T i,(cid:63),kj ( p ) to be the minimizing interior wall temperatures foraggregate user i at time k and discretization point j for prices p , where minimiza-tion is with respect to the user-level problem defined in (7.11). Then the model ofelectricity consumption by the rational user i at time step k for prices p is given by g ( k, u (cid:63),ik ( p ) , T (cid:63),k,i ( p )). Thus the constraint that production equals consumption at alltime implies s k = (cid:88) i N i g ( k, u (cid:63),ik ( p ) , T i,(cid:63),k ( p )) for all k = 0 , · · · , N f − . (7.14)Now, since utility’s revenue equals the amount paid by the users, the model for revenuefrom rational user i becomes J t ( u (cid:63),i ( p ) , T i,(cid:63) ( p ) , p ), where J t is defined in (7.7). Wemay now define the utility-level optimization problem as minimization of the totalcost subject to equality of generation and consumption and proportionality of revenueand total costs.min p on ,p off ,p d ∈ R C ( s )subject to s k = (cid:88) i N i g ( k, u (cid:63),ik ( p ) , T i,(cid:63),k ( p )) k = 0 , · · · , N f − C ( s ) = λ (cid:88) i N i J t ( u (cid:63),i ( p ) , T i,(cid:63) ( p ) , p ) , (7.15)164here λ ≤ p (cid:63) on , p (cid:63) off , p (cid:63)d which solveProblem (7.15) as optimal electricity prices . Model of total cost, C ( s ) , to utility company The algorithm defined in thefollowing section was chosen so that only a black-box model of utility costs is required.However, for the case studies included in Section 7.4, we use two models of utilitycosts based on ongoing discussions and collaboration with Arizona’s utility companySRP. In the first model, we consider a linear representation of both fuel and capacitycosts. C ( s ) := a (cid:88) k ∈ S on ∪ S off s k ∆ t + b max k ∈ S on s k , (7.16)where a ($ /kW h ) is the marginal cost of producing the next kW h of energy and b ($ /kW ) is the marginal cost of installing and maintaining the next kW of capacity.Estimated values of the coefficients a and b for SRP can be found in SRP (2014) as a = 0 . /kW h and b = 59 . /kW . According to SRP (2014), these marginalcosts include fuel, building, operation and maintenance of facilities, transmission anddistribution costs. The advantage of this model is that the solution to the utilityoptimization problem does not depend on the number of users, but rather the fractionof users in each group.Our second model for utility costs includes a quadratic term to represent fuel costs.The quadratic term reflects the increasing fuel costs associated with the required useof older, less-efficient generators when demand increases. C ( s ) := τ (cid:16) (cid:88) k ∈ S on ∪ S off s k ∆ t (cid:17) + ν (cid:88) k ∈ S on ∪ S off s k ∆ t + b max k ∈ S on s k (7.17)This model was calibrated using artificially modified fuel, operation and maintenance165ata provided by SRP, yielding estimated coefficients τ =0.00401 $/(MWh) and ν =4.54351 $/(MWh).7.3 Solving User- and Utility-level Problems by Dynamic ProgrammingFirst, we solve the optimal thermostat programming problem using a variant ofdynamic programming. This yields consumption as a function of prices p on , p off , p d .Next, we embed this implicit function in the Nelder-Mead simplex algorithm in orderto find prices which minimize the production cost in the utility-level optimizationproblem as formulated in (7.15). We start the user-level problem by fixing the variable γ ∈ R + and defining a cost-to-go function, V k . At the final time N f ∆ t = 24, we have V N f ( x ) := p d γ. (7.18)Here for simplicity, we use x = T ∈ R M to represent the discretized temperaturedistribution in the wall. We define the dilated vector of prices by p j = p off if j ∈ S off and p j = p on otherwise. Then, we construct the cost-to-go function inductively as V j − ( x ) := min u ∈ W γ,j − ( x ) ( p j − g ( j − , u, x )∆ t + V j ( f ( x, u ))) (7.19)for j = 1 , · · · , N f , where W γ,j ( x ) is the set of allowable inputs (interior air tempera-tures) at time j and state x : W γ,j ( x ) := { u ∈ R : T min ≤ u ≤ T max , g ( j, u, x ) ≤ γ } , j ∈ S on { u ∈ R : T min ≤ u ≤ T max } , j ∈ S off . Now we present the main result. 166 heorem 21.
Given γ ∈ R + , suppose that V i satisfies (7.18) and (7.19) . Then V ( T ) = J ∗ , where J ∗ ( p ) = min u k ,T k ∈ R M J e ( u , T , p on , p off ) + p d γ subject to g ( k, u k , T k ) ≤ γ for k ∈ S on T k +1 = f ( T k , u k ) for k ∈ S on ∪ S off T min ≤ u k ≤ T max for k ∈ S on ∪ S off T = [ T init (∆ x ) , · · · , T init ( M ∆ x )] T . (7.20)To prove Theorem 21, we require the following definitions. Definition 8.
Given p off , p on , p d , γ ∈ R + , N f ∈ N + , and t off , t on , ∆ t ∈ R + such that t on ∆ t , t off ∆ t ∈ N , define the cost-to-go functions Q j : R N f − j × R N f − j +1 × R + × R + → R for j = 0 , · · · , N f as Q j ( x, y, p on , p off ) := p off (cid:88) k ∈ S off k / ∈{ , ··· ,j − } g ( k, x k , y k ) + p on (cid:88) k ∈ S on g ( k, x k , y k ) ∆ t + (cid:88) i =1 Γ i if ≤ j < N on Γ = p on (cid:80) k ∈ S on k / ∈{ N on , ··· ,j − } g ( k, x k , y k ) + p off (cid:80) k ∈ S off k / ∈{ , ··· ,N on − } g ( k, x k , y k ) ∆ t + Γ + Γ if N on ≤ j < N off Γ = p off (cid:80) k ∈{ j, ··· ,N f − } g ( k, x k , y k )∆ t + Γ if N off ≤ j < N f Γ = p d γ if j = N f , (7.21) where g is defined as in (7.9) , and N on := t on ∆ t and N off := t off ∆ t are the time-stepscorresponding to start and end of the on-peak hours. Note that from (7.8), it is clear that Q = J e + p d γ .167 efinition 9. Given γ, T min , T max ∈ R and N f , M ∈ N + , define the set U j ( x ) := { ( u j , · · · , u N f − ) ∈ R N f − j : g ( k, u k , T k ) ≤ γ for all k ∈ S on ,T j = x and T k +1 = f ( T k , u k ) for all k ∈ { j, · · · , N f − } ,T min ≤ u k ≤ T max for all k ∈ S on ∪ S off } (7.22) for any x ∈ R M and for every j ∈ { , · · · , N f − } , where f and g are defined asin (7.6) and (7.9) . Definition 10.
Given N f , M ∈ N + , j ∈ { , · · · , N f − } , let µ j := [ µ j , · · · , µ N f − ] where µ k : R M → R for k = j, · · · , N f − . Consider U j as defined in (7.22) and f asdefined in (7.6) . If µ j ( w ) := [ µ j ( w ) , µ j +1 ( T j +1 ) · · · , µ N f − ( T N f − )] ∈ U j ( T j ) for any w ∈ R M , where T k +1 = f ( T k , µ k ( T k )) , T j = w for k = j, · · · , N f − , then we call µ j an admissible control law for the system T k +1 = f ( T k , µ k ( T k )) , k = j, · · · , N f − for any w ∈ R M . We now present a proof for Theorem 21.
Proof.
Since the cost-to-go function Q = J e + p d γ , if we show thatmin µ j ( T j ) ∈ U j ( T j ) Q j ( µ j ( T j ) , T , p on , p off ) = V j ( T j ) (7.23)168or j = 0 , · · · , N f and for any T j ∈ R M , where T := [ T j , f ( T j , µ j ( T j )) , · · · , f ( T N f − , µ N f − ( T N f − ))] , then it will follow that J ∗ = V ( T ). For brevity, we denote µ j ( T j ) by µ j , U j ( T j ) by U j and we drop the last two arguments of Q j . To show (7.23), we use induction asfollows. Basis step : If j = N f , then from (7.18) and (7.21) we have V N f ( T N f ) = p d γ . Induction hypothesis : Supposemin µ k ∈ U k Q k ( µ k , T ) = V k ( T k )for some k ∈ { , · · · , N f } and for any T k ∈ R M . Then, we need to prove thatmin µ k − ∈ U k − Q k − ( µ k − , T ) = V k − ( T k − ) (7.24)for any T k ∈ R M . Here, we only prove (7.24) for the case which N off < k ≤ N f − ≤ k ≤ N on and N on < k ≤ N off follow the same exact logic.Assume that N off < k ≤ N f −
1. Then, from Definition 8min µ k − ∈ U k − Q k − ( µ k − , T )= min µ k − , ··· ,µ Nf − ∈ R p off N f − (cid:88) j = k − g (cid:0) j, µ j , T j (cid:1) ∆ t = min µ k − , ··· ,µ Nf − ∈ R p off g (cid:0) k − , µ k − , T k − (cid:1) + N f − (cid:88) j = k g (cid:0) j, µ j , T j (cid:1) ∆ t, (7.25)where R := { x ∈ R : T min ≤ x ≤ T max } . From the principle of optimality (Bellman and169reyfus (1962)) it follows thatmin µ k − , ··· ,µ Nf − ∈ R p off g (cid:0) k − , µ k − , T k − (cid:1) + N f − (cid:88) j = k g (cid:0) j, µ j , T j (cid:1) ∆ t = min µ k − ∈ R p off g (cid:0) k − , µ k − , T k − (cid:1) ∆ t + min µ k , ··· ,µ Nf − ∈ R p off N f − (cid:88) j = k g (cid:0) j, µ j , T j (cid:1) ∆ t, (7.26)By combining (7.25) and (7.26) we havemin µ k − ∈ U k − Q k − ( µ k − , T ) = min µ k − ∈ R (cid:0) p off g (cid:0) k − , µ k − ) , T k − (cid:1) ∆ t + min µ k , ··· ,µ Nf − ∈ R p off N f − (cid:88) j = k g (cid:0) j, µ j , T j (cid:1) ∆ t. (7.27)From Definition 8, we can writemin µ k , ··· ,µ Nf − p off ∆ t N f − (cid:88) j = k g (cid:0) j, µ j , T j (cid:1) = min µ k ∈ U k Q k ( µ k , T ) . (7.28)Then, by combining (7.27) and (7.28) and using the induction hypothesis it followsthatmin µ k − ∈ U k − Q k − ( µ k − , T ) = min µ k − ∈ R (cid:18) p off g (cid:0) k − , µ k − , T k − (cid:1) ∆ t + min µ k ∈ U k Q k ( µ k , T ) (cid:19) = min µ k − ∈ R (cid:0) p off g (cid:0) k − , µ k − , T k − (cid:1) ∆ t + V k ( T k ) (cid:1) for any T k ∈ R M . By substituting for T k from (7.6) and using the definition of V in (7.19) we havemin µ k − ∈ U k − Q k − ( µ k − , T ) = min µ k − ∈ R (cid:0) p off g (cid:0) k − , µ k − , T k − (cid:1) ∆ t + V k (cid:0) f (cid:0) T k − , µ k − (cid:0) T k − (cid:1)(cid:1)(cid:1)(cid:1) = V k − (cid:0) T k − (cid:1) for any T k − ∈ R M . By using the same logic it can be shown thatmin µ k − ∈ U k − Q k − ( µ k − , T ) = V k − ( T k − )170or any k ∈ { , · · · , N off − } and for any T k − ∈ R M . Therefore, by induction, (7.23)is true. Thus, J ∗ = V ( T ).The optimal temperature set-points for Problem (7.20) can be found as the se-quence of minimizing arguments in the value function (7.19). However, this is nota solution to the original user-level optimization problem in (7.11), as the solutiononly applies for a fixed consumption bound, γ . However, as this consumption boundis scalar, we may apply a bisection on γ to solve the original optimization problemas formulated in (7.11). Details are presented in Algorithm 9. The computationalcomplexity of this algorithm is proportional to N f · n Ms · n u , where N f is the number ofdiscretization points in time, M is the state-space dimension of the discretized systemin (7.6), n s is the number of possible discrete values for each state, T and n u is thenumber of possible discrete values for the control input (interior air temperature). Inall of the case studies in Section 7.4, we use N f = 73 , M = 3 , n s = n u = 13. Theexecution time of our Matlab implementation of Algorithm 9 for solving the three-day user-level problem on a Core i7 processor with 8 GB of RAM was less than 4.5minutes.Finding a solution to the 4-Setpoint thermostat programming problem (7.13) issignificantly more difficult due to the presence of the switching times t , t , t asdecision variables. However, for this specific problem, a simple approach is to useAlgorithm 9 as an inner loop for fixed t i and then use a Monte Carlo search over t i . For fixed t i , our Matlab implementation for Algorithm 9 solves the 4-Setpointthermostat programming problem in less than 17 seconds on a Core i7 processor with8 GB of RAM. Our experiments on the same machine show that the total executiontime for a Monte Carlo search over 300 valid (i.e., t i ≤ t i +1 ) random combinations of t , t , t is less than 1.41 hours. 171o solve the utility-level problem in (7.15), we used Algorithm 9 as an inner loopfor the Nelder-Mead simplex algorithm (Olsson and Nelson (1975)). The Nelder-Meadsimplex algorithm is a heuristic optimization algorithm which is typically applied toproblems where the derivatives of the objective function and/or constraint functionsare unknown. Each iteration is defined by a reflection step and possibly a contractionor expansion step. The reflection begins by evaluation of the inner loop (Algorithm 9)at each of 4 vertices of a polytope. The polytope is then reflected about the hyperplanedefined by the vertices with the best three objective values. The polytope is theneither dilated or contracted depending on the objective value of the new vertex. In allof our case studies in Section 7.4, this hybrid algorithm achieved an error convergenceof < − in less than 15 iterations. Using a Core i7 machine with 8 GB of RAM,the execution time of the hybrid algorithm for solving the utility-level problem wasless than 2.25 hours. 7.4 Policy Implications and AnalysisIn this section, we use Algorithms 9 and 10 in three case studies to assess the effectsof passive thermal storage, solar power and various cooling strategies on utility prices,peak demand and cost to the utility company.In Case I, we compare our optimal thermostat program with other HVAC pro-gramming strategies and analyze the resulting peak demands and electricity bills fora set of electricity prices.In Case II, we apply the Nelder-Mead simplex and Algorithm 9 to the user-levelproblem in (7.11) and the utility-level problem in (7.15) to compute optimal electricityprices and optimal cost of production.In Case III, we first define an optimal thermostat program for solar users. Then,we examine the effect of solar power generation on the electricity prices of non-solar172 nputs: p on , p off , p d , T e , t on , t off , R e , C in , T init , ∆ t , ∆ x , T min , T max , maximum number ofbisection iterations b max , lower bound γ l and upper bound γ u for bisection search. Main loop:
Set k = 0. while k < b max do Set γ = γ u + γ l . if V in (7.19) exists then Calculate u , · · · , u N f − as the minimizers of the RHS of (7.19) using a policyiteration technique.Set γ u = γ . Set u (cid:63)i = u i for i = 0 , · · · , N f − . else Set γ l = γ . end Set k = k + 1. endOutputs: Optimal interior temperature setting: u (cid:63) , · · · , u (cid:63)N f − . Algorithm 9:
A bisection/dynamic programming algorithm for optimal ther-mostat programmingusers by solving a two-user single-utility optimization problem. We ran all cases forthree consecutive days prorated from a one month billing cycle with the time-step∆ t = 1 hr , spatial-step ∆ x = 0 . m and with the building parameters in Table 7.1.These parameters were determined using the model calibration procedure describedin Section 7.2.2. We used an external temperature profile measured for three typicalsummer days in Phoenix, Arizona (see Figure 7.6). For each day, the on-peak periodstarts at 12 PM and ends at 7 PM. We used min and max interior temperatures as T min = 22 ◦ C and T max = 28 ◦ C . 173 nputs: a , b , reflection parameter θ , expansion parameter κ , contraction parameter ζ ,reduction parameter τ , initial prices p d i , p i such that p d i + p i < i = 1 , · · · , (cid:15) and inputs to Algorithm 9. Initialization:
Set p d i = p d i , p on i = p i , p off i = 1 − p on i − p d i , p old i = [10 , , ] for i = 1 , · · · , Main loop:while (cid:80) i =1 (cid:107) p old i − p i (cid:107) > (cid:15) dofor i = 1 , · · · , do Calculate opt. temp. setting u (cid:63) ,i , · · · , u (cid:63)N f − ,i associated with prices p i using Alg. 9.Calculate the cost C i associated with u (cid:63) ,i , · · · , u (cid:63)N f − ,i using (7.16) and (7.14). end Re-index the costs and their associated prices and temperature settings such that C ≤ C ≤ C ≤ C . Set p (cid:63) = p .Calculate the centroid of all the prices as ¯ p = (cid:104) (cid:80) i =1 p off i , (cid:80) i =1 p on i , (cid:80) i =1 p d i (cid:105) .Calculate reflected prices as p r = ¯ p + θ (¯ p + p ).Calculate optimal temperature setting u (cid:63) ,r , · · · , u (cid:63)N f − ,r and the cost C r associated withthe reflected prices using Algorithm 9. if C ≤ C r < C then Set p old = p and p = p r . Go to the beginning of the loop. else if C r < C then Calculate expanded prices as p e = ¯ p + κ (¯ p + p ).Calculate optimal temperature setting u (cid:63) ,e , · · · , u (cid:63)N f − ,e and the cost C e associatedwith expanded prices using Algorithm 9. if C e < C r then Set p old = p and p = p e . Set p (cid:63) = p e . else Set p old = p and p = p r . Set p (cid:63) = p r .Go back to While loop. else Calculate contraction prices p c = ¯ p + ξ (¯ p + p ). Calculate optimal temp. setting u (cid:63) ,c , · · · , u (cid:63)N f − ,c & the cost C c associated with expanded prices using Algorithm 9. if C c < C then Set p old = p and p = p c . Go back to While loop. else Set p old i = p i for i = 2 , ,
4. Update prices as p i = p + τ ( p i − p ) for i = 2 , , endendOutputs: Optimal electricity prices p (cid:63) . Algorithm 10:
An algorithm for computing optimal electricity prices174
10 20 30 40 50 60 7030354045 E x t e r i o r t e m p e r a t u r e ( o C ) T i m e ( h r) Figure 7.6:
External Temperature of Three Typical Summer Days in Phoenix, Ari-zona. Shaded Areas Correspond to On-peak Hours.
Table 7.1:
Building’s Parameters as Determined in Section 7.2.1 L in ( m ) α ( m /s ) R e ( K/W ) C in ( W m/K ) ∆ x ( m )0.4 8 . × − In this case, we first applied Algorithm 9 to the optimal and 4-Setpoint thermostatprogramming problems (See (7.11) and (7.13)) for a non-solar customer using theelectricity prices determined by APS in Table 7.2 (Rumolo (2013)). The resultingelectricity bills are given in Table 7.3 as the total cost paid for three days prorated froma one month billing cycle with the external temperature profile shown in Figure 7.6.Prorated in this case means that for a 30-day month, the bill is one-tenth of themonthly bill based on repetition of the three-day cycle ten times. Practically, whatthis means is that the period in Problems (7.11) and (7.13) is tripled while the demandcharge in Problems (7.11) and (7.13) uses a demand price p d = 1 . $ kW . Forcomparison, we have solved Problem (7.11) using the general-purpose optimizationsolver GPOPS (Patterson and Rao (2013)). We have also compared our results with anaive strategy of setting the temperature to T max (constant) and a pre-cooling strategywith the temperature setting: u = 25 ◦ C from 12 AM to 8 AM; u = T min = 22 ◦ C from8 AM to 12 PM; u = T max = 28 ◦ C from 12 PM to 8 PM; u = 25 ◦ C from 8 PM to12 AM. As can be seen from Table 7.3, our algorithm offers significant improvement175 able 7.2:
On-peak, Off-peak & Demand Prices of Arizona Utility APSOn-peak ( $ kW h ) Off-peak ( $ kW h ) Demand ( $ kW h )APS 0.089 0.044 13.50over heuristic approaches. The power consumption and the temperature setting as afunction of time for each strategy can be found in Figure 7.7. For convenience, theon-peak and off-peak intervals are indicated on the figure.To examine the impact of changes in electricity prices on peak demand, we nextchose several different prices corresponding to high, medium and low penalties forpeak demand. Again, in each case, our algorithms (optimal and 4-setpoint) arecompared to GPOPS and the same pre-cooling strategy. The results are summarizedin Table 7.4. Note that for brevity, in this section, we refer to the total cost ofgeneration, transmission and distribution as simply production cost. For each price,the smallest computed production cost and associated demand peak are listed inbold. The power consumption and the temperature settings as a function of timefor the optimal and 4-Setpoint strategies can be found in Figures 7.8 and 7.9. Forthe optimal strategy, notice that by increasing the demand penalty, relative to thelow-penalty case, the peak consumption is reduced by 14% and 23% in the mediumand high penalty cases respectively. Furthermore, notice that by using the optimalstrategy and the high demand-limiting prices, we have reduced the demand peakby 29% with respect to the constant strategy in Table 7.3. Of course, a moderatereduction in peak demand at the expense of large additional production costs maynot be desirable. Indeed, the question of optimal distribution of electricity prices forminimizing the production cost is discussed in Case II.176 able 7.3: CASE I: Electricity Bills (or Three Days) and Demand Peaks for DifferentStrategies. Electricity Prices Are from APS.Temperature setting Electricity bill ($) Demand peak ( kW )Optimal (Theorem 21) Constant 39.42 10.462
Table 7.4:
CASE I: Costs of Production (for Three Days) and Demand Peaks forVarious Prices and Strategies. Prices Are Non-regulated and SRP’s Coefficients ofUtility Cost Are: τ =0.00401 $/(MWh) , ν =4.54351 $/(MWh)Prices [ p off , p on , p d ] Demand-limiting Production cost Demand peak[0 . , . , . $ (0.086 $ kW h ) kW O p t i m a l [0 . , . , . $ (0.116 $ kW h ) kW [0 . , . , . $ (0.168 $ kW h ) 9.6749 kW Prices [ p off , p on , p d ] Demand-limiting Production cost Demand peak[0 . , . , . $ kW h ) 8.5914 kW - S e t p o i n t [0 . , . , . $ kW h ) 8.910 kW [0 . , . , . $ kW h ) 9.974 kW Prices [ p off , p on , p d ] Demand-limiting Production cost Demand peak[0 . , . , . $ kW h ) 7.9440 kW G P O P S [0 . , . , . $ kW h ) 9.1486 kW [0 . , . , . $ kW h ) 9.6221 kW Prices [ p off , p on , p d ] Demand-limiting Production cost Demand peak[0 . , . , . $ kW h ) 8.8031 kW P r ec oo li n g [0 . , . , . $ kW h ) 8.8031 kW [0 . , . , . $ (0.116 $ kW h ) kW
10 20 30 40 50 60 70222426280 10 20 30 40 50 60 700500010000 P o w e r c on s u m p ti on ( W ) −
202 x 10 Theorem 1
Precooling Constant GPOPS 4-Setpoint T i m e ( h r) T i m e ( h r) I n t e r i o r t e m p e r a t u r e ( o C ) Figure 7.7:
CASE I: Power Consumption and Temperature Settings for VariousProgramming Strategies Using APS’s Rates. P o w e r c on s u m p ti on ( W ) Time (hr) I n t e r i o r t e m p e r a t u r e ( o C ) Time (hr) prices=[0.007,0.01,13.616] prices=[0.007,0.01,13.616] prices=[0.065,0.095,13.473]
Figure 7.8:
CASE I: Power Consumption and Optimal Temperature Settings forHigh, Medium and Low Demand Penalties. Shaded Areas Correspond to On-peakHours. 178
10 20 30 40 50 60 700500010000 P o w e r c on s u m p ti on ( W ) Time (hr) I n t e r i o r t e m p e r a t u r e ( o C ) Time (hr) prices=[0.007,0.01,13.616] prices=[0.015,0.045,13.573] prices=[0.065,0.095,13.473]
Figure 7.9:
CASE I: Power Consumption and Temperature Settings for High,Medium and Low Demand Penalties Using 4-Setpoint Thermostat Programming.
In this case, we consider the quadratic model of fuel cost defined in Section (7.17).A typical pricing strategy for SRP and other utilities is to set prices proportionalto marginal production costs. SRP estimates the mean marginal fuel cost at a =0 . /kW h (See (7.16)). Linearizing our quadratic model of fuel cost and equatingto this estimate of the marginal cost yields an estimate of the mean load. Dividingthis mean load by the aggregate user defined in Case I yields an estimate of the meannumber of users of this class at N = 24 , λ = 1, meaning that the regulated utilitydoes not make any profit from generation, transmission and distribution.From Table 7.5, optimal pricing results in a slight reduction ($82,000) in pro-duction costs. The discrepancy between optimal prices and marginal costs may besurprising given that both the user and utility are trying to minimize the cost ofelectricity. However, there are several reasons for this difference. The first and mostobvious reason is that the price structure for the user and the cost structure for theutility are not perfectly aligned. In the first place, the utility has a quadratic inconsumption model for costs, where the user has a linear model. The second mis-alignment is that the capacity cost for the utility is calculated as a maximum over 24hours and the demand charge for the user is calculated only during peak hours. Anadditional reason that marginal costs will not always be optimal prices is nonlinearityof the cost function and heterogeneity of the users. To see this, suppose that costfunction exactly equaled the price function for each user. The problem in this case isthat the sum of the individual bills is NOT equal to the total production cost. Thiscan be seen in the demand charge, where sup x f ( x ) + sup x g ( x ) (cid:54) = sup x ( f ( x ) + g ( x )). Table 7.5:
CASE II: Production Costs (for Three Days) and Demand Peaks Asso-ciated with Regulated Optimal Electricity Prices (Calculated by Algorithm 10) andSRP’s Electricity Prices. SRP’s Marginal Costs: a = 0 . $ kW h , b = 59 . $ kW Strategy [ p off ( $ kW h ) , p on ( $ kW h ) , p d ( $ kW )] Production cost Demand peakOptimal [0 . , . , . $ 195.607 M W
SRP [0 . , . , . M W .4.3 Optimal Thermostat Programming for Solar Customers - Impact ofDistributed Solar Generation on Non-solar Customers
We now evaluate the impact of solar power on the bills of non-solar users in aregulated electricity market. We consider a network consisting of a utility companyand two aggregate users - one solar and one non-solar. For the non-solar user, wedefine optimal thermostat programming as in (7.11). For the solar user, the optimalthermostat programming problem is as defined in (7.11), where we have now redefinedthe consumption function as g ( k, u k , T k ) := T ke − u k R e + 2 C in T k − u k ∆ x − Q k , (7.29)where Q k is the power supplied locally by solar panels. We assume that solar penetra-tion is 50%, so that both aggregate users contribute equally to revenue and costs tothe utility. For Q k , we used data generated on June 4-7 from a typical 13kW south-facing rooftop PV array in Scottsdale, AZ. We applied Algorithm 10 separately toeach user, while considering (7.16) as the utility cost model. The results are presentedin Table 7.6. For comparison, we have also included optimal prices, prorated electric-ity bills over three days and demand peaks of both users. From Table 7.6 we observethat the difference between the electricity bill of a non-solar user in a single-user net-work and the bill of a non-solar user in a two-user network (solar and non-solar) is < <
10 20 30 40 50 60 70 − P o w e r c on s u m p ti on - S o l a r po w e r Time (hr) S o l a r g e n e r a t e d po w e r ( W ) I n t e r i o r t e m p e r a t u r e ( o C ) SolarNon-solar
Time (hr)Time (hr)
Figure 7.10:
CASE III: Power Consumption, Solar Generated Power and OptimalTemperature Settings for the Non-solar and Solar Users.
Table 7.6:
CASE III: Optimal Electricity Prices, Bills (for Three Days) and DemandPeaks for Various Customers. Marginal osts from SRP: a = 0 . $ kW h , b = 59 . $ kW Customers [ p (cid:63) off ( $ kW h ) , p (cid:63) on ( $ kW h ) , p (cid:63)d ( $ kW )] Elect. Bill Demand peakSolar & [0 . , . , . kW Non-solar $ 84.717 8.6787 kW Single Non-solar [0 . , . , . kW Single Solar [0 . , . , . kW for positive polynomials - a propertythat our parallel algorithms exploit to achieve near-linear speed-up and scalability.In Chapter 2, we discuss how variants of Polya’s theorem, Handelman’s theo-rem and the Positivstellensatz results can be applied to optimization of polynomialsdefined over various compact sets, e.g., simplex, hypercube, convex polytopes andsemi-algebraic sets. We show that applying these theorems to an optimization ofpolynomials problem yields convex optimization problems in the form of LPs and/orSDPs. By solving these LPs and SDPs, one can find asymptotic solutions to the op-timization of polynomials problems (as defined in (2.12)). Subsequently, by combining A detailed discussion on the structure of these parameterizations can be found in Section 4.3.3 et al. (2005)) for solving semi-definite programs. We later decentralize the computation of the search directions ofthis algorithm to design a parallel SDP solver in Chapter 4.In Chapter 4, we propose a parallel-computing approach to stability analysis oflarge-scale linear systems of the form ˙ x ( t ) = A ( α ) x ( t ), where A is a real-valued poly-nomial, α ∈ ∆ l ⊂ R l and x ∈ R n . This approach is based on mapping the structureof the SDPs associated with Polya’s theorem to a parallel computing environment.We first design a parallel set-up algorithm with no centralized computation to con-struct the SDPs associated with Polya’s theorem. We then show that by choosing ablock-diagonal starting point for the SDP algorithm in Helmberg et al. (2005), theprimal and dual search directions will preserve their block-diagonal structure at everyiteration. By exploiting this property, we decentralize the computation of the searchdirections - the most computationally expensive step of an SDP algorithm. The resultis a parallel algorithm which under certain conditions, can solve the NP-hard prob-lem of robust stability of linear systems at the same per-core computational cost assolving the Lyapunov inequality for a linear system with no parametric uncertainty.Theoretical and experimental results verify near-linear speed-up and scalability of ouralgorithm for up to 200 processors. In particular, our numerical tests on cluster com-puters show that our MPI/C++ implementation of the SDP algorithm outperformsthe existing state-of-the-art SDP solvers such as SDPARA (Yamashita et al. (2003))in terms of speed-up. Moreover, our experimental tests on a mid-size (9-node) Linux-based cluster computer demonstrate the ability of our algorithm in performing robust184tability analysis of systems with 100+ states and several uncertain parameters. Acomprehensive complexity analysis of both set-up and solver algorithms can be foundin Sections 4.4 and 4.6.1.In Chapter 5, we further extend our analysis to consider linear systems with un-certain parameters inside hypercubes. We propose an extended version of Polya’stheorem for positivity over a multi-simplex (Cartesian product of standard simpli-cies). We claim that every polynomial defined over a hypercube has an equivalenthomogeneous representation over the multi-simplex. Therefore, our the multi-simplexversion of Polya’s theorem can be used to verify positivity over hypercubes. In thenext step, we generalize our parallel set-up algorithm from Chapter 4 to constructthe SDPs associated with our multi-simplex version of Polya’s theorem. Our com-plexity analysis shows that for sufficiently large number of available processors, ateach Polya’s iteration, the per processor computation and communication cost ofthe algorithm scales polynomially with the number of states and uncertain parame-ters. Through numerical experiments on a large cluster computer, we show that thealgorithm can achieve a near-perfect speed-up.In Chapter 6, we extend our approach to consider optimization of polynomialsdefined over more complicated geometries such as convex polytopes. Specifically, weapply Handelman’s theorem to construct piecewise polynomial Lyapunov functionsfor nonlinear dynamical systems defined by polynomial vector fields. Unfortunately,neither Polya’s theorem nor Handelman’s theorem can readily certify non-negativityof polynomials which have zeros in the interior of a simplex/polytope. Our proposedsolution to this problem is to decompose the domain of analysis (in this case a poly-tope) into several convex sub-polytopes with a common vertex at the equilibrium.Then, by using Handelman’s theorem, we derive a new set of affine feasibility condi-tions - solvable by linear programming - on each sub-polytope. Any solution to this185easibility problem yields a piecewise polynomial Lyapunov function on the entirepolytope. In a computational complexity analysis, we show that for large number ofstates and large degrees of the Lyapunov function, the complexity of the proposedfeasibility problem is less than the complexity of certain semi-definite programs asso-ciated with alternative methods based on Sum-of-Squares and Polya’s theorem.Finally, in chapter 7, we address a real-world optimization problem in energy plan-ning and smart grid control. We consider the coupled problems of optimal control ofHVAC systems for residential customers and optimal pricing of electricity by utilitycompanies. Our framework consists of multiple users (solar and non-solar customers)and a single regulated utility company. The utility company sets prices for the users,who pay for both total energy consumed ($/kWh, including peak and off-peak rates)and the peak rate of consumption in a month (a demand charge) ($/kW). The cost ofelectricity for the utility company is based on a combination of capacity costs ($/kW)and fuel costs ($/kWh). On the demand side, the users minimize the amount paidfor electricity while staying within a pre-defined temperature range. The users haveaccess to energy storage in the form of thermal capacitance of interior structures.Meanwhile, the utility sets prices designed to minimize the total cost of generation,transmission and distribution of electricity. To solve the user-level problem, we usea variant of dynamic programming. To solve the utility-level problem, we use theNelder-Mead simplex algorithm coupled with our dynamic programming code - yield-ing optimal on-peak, off-peak and demand prices. We then apply our algorithms to avariety of scenarios in which show that: 1) Thermal storage and optimal thermostatprogramming can reduce electricity bills using current rates from utilities ArizonaPublic Service (APS) and Salt River Project (SRP). 2) Our optimal pricing can re-duce the total cost to the utility companies. 3) In the presence of demand charges,186he impact of distributed solar generation on the electricity bills of the non-solar usersis not significant ( < %).8.2 Future Directions of Our ResearchIn the following sections, we discuss how the proposed algorithms in this dis-sertation can be extended to solve three well-known problems in controls: 1) Robuststability analysis of nonlinear systems; 2) Synthesis of parameter-varying H ∞ -optimalcontroller; 3) Computing value functions in approximate dynamic programming prob-lems. Consider the problem of local stability analysis of a nonlinear system of the form˙ x ( t ) = A ( x, α ) x ( t ) , (8.1)where A : R n × R m → R n × n is a matrix-valued polynomial and A (0 , (cid:54) = 0. Fromconverse Lyapunov theory, this problem can be expressed as a search for a polynomial V : R n × R m → R which satisfies the Lyapunov inequalities W ( x, α ) ≤ V ( x, α ) ≤ W ( x, α ) (cid:104)∇ x V, f (cid:105) ≤ − W ( x, α )for all x, α ∈ Ω ⊂ R , where 0 ∈ Ω. However, as we discussed in Section 6.1, Polya’stheorem (simplex and multi-simplex versions) cannot certify positivity of polynomialswhich have zeros in the interior of the unit- and/or multi-simplex. Moreover, if F ( x )in (5.2) has a zero in the interior of Φ n , then any multi-homogeneous polynomial P ( x, y ) that satisfies (5.2) has a zero in the interior of the multi-simplex ∆ × · · · × ∆ - hence cannot be parameterized by Polya’s theorem. One way to enforce the187ondition V ( , ) = 0 is to search for coefficients of a matrix-valued polynomial P which defines a Lyapunov function of the form V ( x, α ) = x T P ( x, α ) x . It can beshown that V ( x, α ) = x T P ( x, α ) x is a Lyapunov function for System (8.1) if and onlyif γ ∗ in the following optimization of polynomials problem is positive. γ ∗ = max γ ∈ R ,α,P ∈ R [ x,α ] γ subject to (cid:34) P ( x, α ) 00 − Q ( x, α ) (cid:35) − γI ≥ x ∈ Φ n , α ∈ Φ m , (8.2)where Q ( x, α ) = A T ( x, α ) P ( x, α )+ P ( x, α ) A ( x, α )+ 12 A T ( x, α ) x T ∂P ( x,α ) ∂x ... x T ∂P ( x,α ) ∂x n + x T ∂P ( x,α ) ∂x ... x T ∂P ( x,α ) ∂x n T A ( x, α ) . As we discussed in Section 2.3.2, by applying bisection search on γ and using themulti-simplex version of Polya’s theorem (Theorem 16) as a test for feasibility ofConstraint (8.2), we can compute lower bounds on γ ∗ . Suppose there exists amatrix-valued multi-homogeneous polynomial S of degree vector d s ∈ N n ( d s =[ d s , · · · , d s n , d s n +1 , · · · , d s n + m ], where for i ∈ { , · · · , n } , d s i is the degree of y i andfor i ∈ { n + 1 , · · · , m } , d s i is the degree of β i ) such that { P ( x, α ) ∈ S n : x ∈ Φ n , α ∈ Φ m } = { S ( y, z, β, η ) ∈ S n : ( y i , z i ) , ( β j , η j ) ∈ ∆ , i = 1 , · · · , n, and j = 1 , · · · , m } . Likewise, suppose there exist matrix-valued multi-homogeneous polynomials B and C of degree vectors d b ∈ N n and d c = d s ∈ N n such that { A ( x, α ) ∈ R n × n : x ∈ Φ n } = { B ( y, z, β, η ) ∈ R n × n : ( y i , z i ) , ( β j , η j ) ∈ ∆ , i = 1 , · · · , n, and j = 1 , · · · , m } (cid:110)(cid:104) ∂P ( x,α ) ∂x x, · · · , ∂P ( x,α ) ∂x n x (cid:105) ∈ R n × n : x ∈ Φ n , and α ∈ Φ m (cid:111) = (cid:8) C ( y, z, β, η ) ∈ R n × n : ( y i , z i ) , ( β j , η j ) ∈ ∆ , i = 1 , · · · , n, and j = 1 , · · · , m (cid:9) . Given γ ∈ R , it follows from Theorem 16 that the inequality condition in (8.2) holdsfor all α ∈ Φ l if there exist e ≥ (cid:32) n (cid:89) i =1 ( y i + z i ) e · m (cid:89) j =1 ( β j + η j ) e (cid:33) (cid:32) S ( y, z, β, η ) − γI (cid:32) n (cid:89) i =1 ( y i + z i ) d pi · m (cid:89) j =1 ( β i + η i ) ˆ d pi (cid:33)(cid:33) (8.3)and (cid:32) n (cid:89) i =1 ( y i + z i ) e · m (cid:89) j =1 ( β j + η j ) e (cid:33) (cid:0) B T ( y, z, β, η ) S ( y, z, β, η ) + S ( y, z, β, η ) B ( y, z, β, η )+ 12 (cid:0) B T ( y, z, β, η ) C T ( y, z, β, η ) + C ( y, z, β, η ) B ( y, z, β, η ) (cid:1) − γI (cid:32) n (cid:89) i =1 ( y i + z i ) d qi · m (cid:89) j =1 ( β i + η i ) ˆ d qi (cid:33)(cid:33) (8.4)have all positive coefficients, where d p i and ˆ d p i are the degrees of x i and α i in P ( x, α ),and d q i and ˆ d q i are the degrees of x i and α i in Q ( x, α ). Now, let S, B and C be of thefollowing forms. S ( y, z, β, η ) = (cid:88) h,g ∈ N n + m h + g = d s S h,g y h z g · · · y h n n z g n n β h n +1 η g n +1 · · · β h n + m m η g n + m m (8.5) B ( y, z, β, η ) = (cid:88) h,g ∈ N n + m h + g = d b B h,g y h z g · · · y h n n z g n n β h n +1 η g n +1 · · · β h n + m m η g n + m m (8.6) C ( y, z, β, η ) = (cid:88) h,g ∈ N n + m h + g = d c C h,g y h z g · · · y h n n z g n n β h n +1 η g n +1 · · · β h n + m m η g n + m m (8.7)Note that the coefficients C h,g can be written as linear combinations of S h,g . Forbrevity we have denoted C h,g ( S h,g ) as C h,g . By combining (8.5), (8.6) and (8.7)189ith (8.3) and (8.4) it follows that for a given γ ∈ R , the inequality condition in (8.2)holds for all α ∈ Φ n if there exist some e ≥ (cid:88) h,g ∈ N n + m h + g = d s f { q,r } , { h,g } S h,g > q, r ∈ N n + m : q + r = d s + 2 e · n + m (8.8)and (cid:88) h,g ∈ N n + m h + g = d s M T { u,v } , { h,g } S h,g + S h,g M { u,v } , { h,g } + N T { u,v } , { h,g } C Th,g + C h,g N { u,v } , { h,g } < u, v ∈ N n + m : u + v = d s + d b + 2 e · n + m , where we define f { q,r } , { h,g } to be thecoefficient of S h,g y q z r · · · y q n n z r n n β q n +1 η r n +1 · · · β q n + m m η r n + m m after substituting (8.5) into (8.3). Likewise, we define M { u,v } , { h,g } to be the coefficientof S h,g y u z v · · · y u n n z v n n β u n +1 η v n +1 · · · β u n + m m η v n + m m and N { u,v } , { h,g } to be the coefficient of C h,g y u z v · · · y u n n z v n n β u n +1 η v n +1 · · · β u n + m m η v n + m m after substituting (8.6) and (8.7) into (8.4). For any γ ∈ R , if there exist e ≥ { S h,g } such that Conditions (8.8) and (8.9) hold, then γ is a lower bound for γ ∗ asdefined in (8.2). Furthermore, if γ is positive, then origin is an asymptotically stableequilibrium for System (8.1). Fortunately, Conditions (8.8) and (8.9) form an SDPwith a block-diagonal structure - hence an algorithm similar to Algorithm 7 can bedeveloped to set-up the SDP in parallel. Furthermore, our parallel SDP solver inSection 4.5 can be used to efficiently solve the SDP.190 .2.2 Parallel Computation for Parameter-varying H ∞ -optimal Control Synthesis Algorithm 5 can be generalized to consider a more general class of feasibility prob-lems, i.e., N (cid:88) i =1 (cid:0) A i ( α ) X ( α ) B i ( α ) + B Ti ( α ) X ( α ) A Ti ( α ) + R i ( α ) (cid:1) < − γI for all α ∈ ∆ l , where A i , B i and R i are polynomials. Formulations such as this can be used to solvea wide variety of problem in systems analysis and control such as H ∞ -optimal controlsynthesis for systems with parametric uncertainty. To see this, consider a plant G with the state-space formulation˙ x ( t ) = A ( α ) x ( t ) + (cid:104) B ( α ) B ( α ) (cid:105) (cid:34) ω ( t ) u ( t ) (cid:35) , (cid:34) z ( t ) y ( t ) (cid:35) = (cid:34) C ( α ) C ( α ) (cid:35) x ( t ) + (cid:34) D ( α ) D ( α ) D ( α ) 0 (cid:35) (cid:34) ω ( t ) u ( t ) (cid:35) , (8.10)where α ∈ Q ⊂ R l , x ( t ) ∈ R n , u ( t ) ∈ R m is the control input, ω ( t ) ∈ R p is the externalinput and z ( t ) ∈ R q is the external output. Suppose ( A ( α ) , B ( α )) is stabilizable and( C ( α ) , A ( α )) is detectable for all α ∈ Q . According to P. Gahinet (1994) there existsa state feedback gain K ( α ) ∈ R m × n such that (cid:107) S ( G, K ( α )) (cid:107) H ∞ ≤ γ, for all α ∈ Q, if and only if there exist P ( α ) > R ( α ) ∈ R m × n such that K ( α ) = R ( α ) P − ( α )and (cid:104) A ( α ) B ( α ) (cid:105)(cid:34) P ( α ) R ( α ) (cid:35) + (cid:104) P ( α ) R T ( α ) (cid:105)(cid:34) A T ( α ) B T ( α ) (cid:35) (cid:63) (cid:63)B T ( α ) − γI (cid:63) (cid:104) C ( α ) D ( α ) (cid:105) (cid:34) P ( α ) R ( α ) (cid:35) D ( α ) − γI < , (8.11)191or all α ∈ Q , where γ > S ( G, K ( α )) is the map from the external input ω to the external output z of the closed loop system with a static full state feedbackcontroller. The symbol (cid:63) denotes the symmetric blocks in the matrix inequality. Tofind a solution to the robust H ∞ -optimal static state-feedback controller problem withoptimal feedback gain K ( α ) = P ( α ) R − ( α ), one can solve the following optimizationof polynomials problem. γ ∗ = min P,R ∈ R [ α ] ,γ ∈ R γ subject to − P ( α ) (cid:63) (cid:63) (cid:63) (cid:104) A ( α ) B ( α ) (cid:105)(cid:34) P ( α ) R ( α ) (cid:35) + (cid:104) P ( α ) R T ( α ) (cid:105)(cid:34) A T ( α ) B T ( α ) (cid:35) (cid:63) (cid:63) B T ( α ) 0 (cid:63) (cid:104) C ( α ) D ( α ) (cid:105) (cid:34) P ( α ) R ( α ) (cid:35) D ( α ) 0 − γ I
00 0 0 I < α ∈ Q. (8.12)In Problem (8.12), if Q = ∆ l as defined in (2.15), then we can apply Polya’s theoremas described in Section 2.3.1 to find a γ ≤ γ ∗ and P and R which satisfy the inequal-ity in (8.12). Suppose P, A, B , B , C , D and D are homogeneous polynomials(otherwise use the procedure in Section 4.2 to homogenize them). Let F ( P ( α ) , R ( α )) := − P ( α ) (cid:63) (cid:63) (cid:63) (cid:104) A ( α ) B ( α ) (cid:105)(cid:34) P ( α ) R ( α ) (cid:35) + (cid:104) P ( α ) R T ( α ) (cid:105)(cid:34) A T ( α ) B T ( α ) (cid:35) (cid:63) (cid:63) B T ( α ) 0 (cid:63) (cid:104) C ( α ) D ( α ) (cid:105) (cid:34) P ( α ) R ( α ) (cid:35) D ( α ) 0 , F by d f . Given γ ∈ R , the inequality in (8.12) holds if thereexist e ≥ (cid:32) l (cid:88) i =1 α i (cid:33) e F ( P ( α ) , R ( α )) − γ I
00 0 0 I (cid:32) l (cid:88) i =1 α i (cid:33) d f (8.13)are negative-definite. Let P and R be of the forms P ( α ) = (cid:88) h ∈ W dp P h α h · · · α h l l , P h ∈ S n (8.14)and R ( α ) = (cid:88) h ∈ W dr R h α h · · · α h l l , R h ∈ R n × n , (8.15)where W d p and W d r are the exponent sets defined in (4.2). By combining (8.14)and (8.15) with (8.13) it follows from Polya’s theorem that for a given γ , the inequalityin (8.12) holds, if there exist e ≥ (cid:88) h ∈ W dp (cid:0) M Th,q P h + P h M h,q (cid:1) + (cid:88) h ∈ W dr (cid:0) N Th,q R Th + R h N h,q (cid:1) < q ∈ W d f + e , (8.16)where we define M h,q ∈ R n × n as the coefficient of P h α q · · · α q l l after substituting (8.14)and (8.15) into (8.13). Likewise, N h,q ∈ R n × n is the coefficient of R h α q · · · α q l l aftersubstituting (8.14) and (8.15) into (8.13). For given γ >
0, if there exist e ≥ P h , h ∈ W d p and R g , g ∈ W d r , then K ( α ) = (cid:88) h ∈ W dp P h α h · · · α h l l (cid:88) g ∈ W dr R g α g · · · α g l l − is a feedback law of an H ∞ -suboptimal static state-feedback controller for System (8.10).By performing bisection search on γ and solving (8.16) for each γ of the bisection,one may find an H ∞ -optimal controller for System (8.10).193 .2.3 Parallel Computation of Value Functions for Approximate DynamicProgramming Consider the discrete-time optimal control problem J ∗ := min u k ∈ U ∞ (cid:88) k =0 β k g ( x k , u k )subject to x k +1 = f ( x k , u k ) for k = 1 , , , · · · x = z, x k ∈ X for k = 1 , , , · · · , (8.17)where f : R n × R m → R n and g : R n × R m → R n are given polynomials, β ∈ (0 ,
1] isa discount factor, U ⊂ R m , X ⊂ R n , and z ∈ R n is a given initial condition for thedynamical system. It is well-known that dynamic programming approach (Bertsekas et al. (1995)) provides sufficient conditions for existence of a solution to the optimalcontrol problem in (8.17). The key idea underlying dynamic programming is thatoptimization over-time can often be considered as optimization in stages. In suchframework, optimal control is any decision which minimizes the sum of: 1. cost oftransition from current stage k to the next stage k + 1; and 2. cost of all stagessubsequent to k + 1, incurred by the decision made at stage k . This is referred to asthe principle of optimality and was first formulated by Bellman (Bellman and Kalaba(1965)) as J ∗ = V ∗ ( z ) = ( P V ∗ )( z ) := inf v ∈ U { g ( z, v ) + β V ∗ ( f ( z, v )) } for all z ∈ X. (8.18)The unique solution to Bellman’s equation is called the value function - can be thoughtof as the minimum cost-to-go from the current state. Existence of the value functionsis a sufficient condition for existence of an optimal control. In fact, an optimal policy µ ∗ : X → U can be expressed in terms of the value function V ∗ : µ ∗ ( z ) = arg min u ∈ U { g ( z, u ) + β V ∗ ( f ( z, u )) } x ∈ X . Thus, Bellman’s equation solves the optimal control problem byproviding a closed-loop feedback law for every initial condition.It is shown that the Bellman’s operator P defined in (8.18) possesses the followingtwo properties:1. Iteratively applying of Bellman’s operator P on any function h : X → R resultsin a pointwise convergence to a value function, i.e., V ∗ ( x ) = lim k →∞ ( P k h )( x ) for all x ∈ X. (8.19)2. Bellman’s operator is monotonic : If V satisfies the Bellman’s inequality V ( x ) ≤ ( P V )( x ) for all x ∈ X , then V ( x ) ≤ ( P k V )( x ) for all x ∈ X and for any k ≥ V ≤ P k V for some k ≥ ⇒ V ≤ V ∗ . Unfortunately, for k >
1, the constraint V ≤ P k V is non-convex in the coefficientsof polynomial V . A sufficient condition for V ≤ P k V is to search for polynomials V and W i , i = 1 , · · · , k such that V ≤ P W , W ≤ P W , · · · , W k − ≤ P V. Note that all of these constraints are convex in the coefficients of V and W i . Let V and W i be polynomials of forms V ( x ) = (cid:88) α ∈ I ( d V ) V α x α and W i ( x ) = (cid:88) α ∈ I ( d Wi ) W i,α x α , where I ( d ) := { α ∈ N n : (cid:107) α (cid:107) ≤ d } . Then, any polynomial V which solves the convex195ptimization problem J k := max V α ,W i,α (cid:88) α ∈ I ( d V ) V α z α subject to (cid:88) α ∈ I ( d V ) V α x α ≤ P (cid:88) α ∈ I ( d W ) W ,α x α for all x ∈ X (cid:88) α ∈ I ( d Wi ) W i,α x α ≤ P (cid:88) α ∈ I ( d Wi +1 ) W i +1 ,α x α for all x ∈ X and i = 1 , · · · , k − (cid:88) α ∈ I ( d Wk − ) W k − ,α x α ≤ P (cid:88) α ∈ I ( d V ) V α x α for all x ∈ X (8.20)for any initial condition z ∈ X and some k ≥
1, is an under-estimator for the valuefunction V ∗ . Moreover, from monotonicity of P it follows that J ≤ J ≤ · · · ≤ J k ≤ · · · ≤ V ∗ ( z ) . In other words, by increasing k , the lower bound J k defined in (8.20) can only improveor remain constant. By substituting for P in (8.20) from (8.18), removing the infimumand enforcing the constraints of Problem 8.20 for all control inputs u ∈ U , we get thefollowing optimization of polynomials problem.max V α ,W i,α (cid:88) α ∈ I ( d V ) V α z α subject to (cid:88) α ∈ I ( d V ) V α x α ≤ g ( x, u ) + β (cid:88) α ∈ I ( d W ) ( W ,α f ( x, u ) α ) for x ∈ X, u ∈ U (cid:88) α ∈ I ( d Wi ) W i,α x α ≤ g ( x, u ) + β (cid:88) α ∈ I ( d Wi +1 ) ( W i +1 ,α f ( x, u ) α ) for x ∈ X, u ∈ U, i ∈ Θ (cid:88) α ∈ I ( d Wk − ) W k − ,α x α ≤ g ( x, u ) + β (cid:88) α ∈ I ( d V ) ( V α f ( x, u ) α ) for x ∈ X, u ∈ U, (8.21)where for brevity, we have denoted f ( x, u ) α · · · f n ( x, u ) α n by f ( x, u ) α and we havedefined Θ := { , · · · , k − } . 196roblem (8.21) has some interesting computational properties. Since all of theconstraints in this problem have the same structure, if we choose the same degree for V and W i , it is then sufficient to set-up only one of the constraints in order to set-upthe entire Problem (8.21). If X and U are simplicies, Algorithm 5 can be used toperform Polya’s iterations on the constraints of Problem (8.21). The result is a linearprogram whose solution yields an under-estimator for the value function V ∗ definedin (8.18). Likewise, if X and U are hypercubes (or polytopes), then Algorithm 7 (orAlgorithm 2) can be used to perform Polya’s iterations (or Handelman’s iterations)on the constraints of Problem (8.21). Another interesting property of Problem (8.21)is that increasing the accuracy of the under-estimations (by increasing k ) amounts toa linear growth in the number of decision variables and number of constraints.197EFERENCES“Appendix schedule B - summary of marginal costs”, available at: (2014).“Standard electric price plans”, Salt River Project Agricultural Improvement andPower District Corporate Pricing (November 2015).Ackermann, J., A. Bartlett, D. Kaesbauer, W. Sienel and R. Steinhauser, RobustControl: Systems with Uncertain Physical Parameters (Springer-Verlag New York,Inc., Secaucus, NJ, USA, 2001).Adams, W. and P. Loustaunau,
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