Reza Kamyar

Solving Large-Scale Robust Stability Problems by Exploiting the Parallel Structure of Polya's Theorem

In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polyas theorem to convert the polynomial optimization problem to a set of highly structured linear matrix inequalities (LMIs). We then use a slight modification of a common interior-point primal-dual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

Decentralized computation for robust stability of large-scale systems with parameters on the hypercube

In this paper, we propose a parallel algorithm to solve the problem of robust stability of systems with large state-space and with large number of uncertain parameters. The dependence of the system on the parameters is polynomial and the parameters are assumed to lie in a hypercube. Although the parameters are assumed to be static, the method can also be applied to systems with time-varying parameters. The algorithm relies on a variant of Polyas theorem which is applicable to polynomials with variables inside a multi-simplex. The algorithm is divided into formulation and solution subroutines. In the formulation phase, we construct a large-scale semidefinite programming problem with structured elements. In the solution phase, we use a structured primal-dual approach to solve the structured semidefinite programming problem. In both subroutines, computation, memory and communication are efficiently distributed over hundreds and potentially thousands of processors. Numerical tests confirm the accuracy and scalability of the proposed algorithm.

Constructing piecewise-polynomial lyapunov functions for local stability of nonlinear systems using Handelman's theorem

In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelmans theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelmans theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polyas theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.

Decentralized Polya's algorithm for stability analysis of large-scale nonlinear systems

In this paper, we introduce an algorithm to decentralize the computation associated with the stability analysis of systems of nonlinear differential equations with a large number of states. The algorithm applies to dynamical systems with polynomial vector fields and checks the local asymptotic stability on hypercubes. We perform the analysis in three steps. First, by applying a multi-simplex version of Polyas theorem to some Lyapunov inequalities, we derive a sequence of stability conditions of increasing accuracy in the form of structured linear matrix inequalities. Then, we design a set-up algorithm to decentralize the computation of the coefficients of the LMIs, among the processing units of a parallel environment. Finally, we use a parallel primal-dual central path algorithm, specifically designed to solve the structured LMIs given by the set-up algorithm. For a sufficiently large number of available processors, the per-core computational complexity of the resulting algorithm is fixed with the accuracy. The algorithm demonstrates a near-linear speed-up in numerical experiments.

Optimal thermostat programming and optimal electricity rates for customers with demand charges

We consider the coupled problems of optimal thermostat programming and optimal pricing of electricity. Our framework consists of a single user and a single provider (a regulated utility). The provider sets prices for the user, who pays for both total energy consumed (

Optimal Thermostat Programming for Time-of-Use and Demand Charges With Thermal Energy Storage and Optimal Pricing for Regulated Utilities

/kWh, including peak and off-peak rates) and the peak rate of consumption in a month (a demand charge) (

Multi-objective dynamic programming for constrained optimization of non-separable objective functions with application in energy storage

/kW). The cost of electricity for the provider is based on a combination of capacity costs (

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

/kW) and fuel costs (

A Multi-objective Approach to Optimal Battery Storage in The Presence of Demand Charges

/kWh). In the optimal thermostat programming problem, the user minimizes the amount paid for electricity while staying within a pre-defined temperature range. The user has access to energy storage in the form of thermal capacitance of the interior structure of the building. The provider sets prices designed to minimize the total cost of producing electricity while meeting the needs of the user. To solve the user-problem, we use a variant of dynamic programming. To solve the provider-problem, we use a descent algorithm coupled with our dynamic programming code - yielding optimal on-peak, off-peak and demand prices. We show that thermal storage and optimal thermostat programming can reduce electricity bills using current utility prices from utilities Arizona Public Service (APS) and Salt River Project (SRP). Moreover, we obtain optimal utility prices which lead to significant reductions in the cost of generating electricity and electricity bills.

Constructing Piecewise-Polynomial Lyapunov Functions on Convex Polytopes Using Handelman's Basis

In this paper, we solve the optimal thermostat programming problem for consumers with combined demand (