Alexandre Megretski

System analysis via integral quadratic constraints

This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time variations, and uncertain parameters. From an original idea by Yakubovich (1967), the approach has been developed under a combination of influences from the Western and Russian traditions of control theory. It is shown how a complex system can be described, using integral quadratic constraints (IQC) for its elementary components. A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality. A systematic computational approach is described, and relations to other methods of stability analysis are discussed. Last, but not least, the paper contains a summarizing list of IQCs for important types of system components.

Unsolved problems in mathematical systems and control theory

Among the signs of developing maturity in a field of research is willingness to share research problems and encourage collaborative efforts toward resolving them. There are now two important examples of this trend in control theory. In Liège, Belgium, in 1997, a workshop put together 53 expositions of open problems in our field which appeared in print as [1]; a web site [2] keeps track of comments and attempts at solutions of these problems. The book under review presents 63 new and stimulating papers, some with several distinct questions; see [3] to keep in touch with their progress. Some of these papers were presented at Open Problem sessions organized for Oberwohlfach Control Theory Days, February 2002, and for MTNS, Notre Dame, August 2002. Editors V. D. Blondel and A. Megretski were assisted by Associate Editors R. W. Brockett, J. M. Coron, R. Hildebrand, M. Krstic, A. Rantzer, J. Rosenthal, E. D. Sontag, M. Vidyasagar, and J. Willems. These editors, with over 100 authors from four continents, have produced an attractive volume, modestly priced, that will remain interesting for a long time; even the dust-jacket is admirable. My only complaint about the book is the absence of an index. The book has ten parts, a list of which follows: 1) Linear Systems: 13 problems, 60 pages; 2) Stochastic Systems: four problems, 20 pages; 3) Nonlinear Systems: nine problems, 38 pages; 4) Discrete-Event, Hybrid Systems: four problems, 20 pages; 5) Distributed Parameter Systems: six problems, 33 pages; 6) Stability, Stabilization: ten problems, 55 pages; 7) Controllability, Observability: 4 problems, 17 pages; 8) Robustness, Robust Control: four problems, 16 pages; 9) Identification, Signal Processing: two problems, ten pages; 10) Algorithms, Computation: seven problems, 21 pages.

Brief New results for analysis of systems with repeated nonlinearities

This work presents new results for analysis of systems with repeated monotonic or slope restricted nonlinearities. We first give new integral quadratic constraints (IQCs) that are satisfied by the inputs and outputs of diagonal operators with equal, monotonic or slope restricted, diagonal entries. Then, we present new analysis results for systems with repeated nonlinearities, obtained using the new IQCs. For the type of nonlinearities considered in this work, the new results are less conservative than previously known results. The computation of the new conditions is discussed. The results in this paper apply also to problems with nonrepeated nonlinearities whenever these are representable as the interconnection of multiple repeated nonlinearities and a constant matrix.

Designing optimal quantum detectors via semidefinite programming

We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.

A Framework for Robust Stability of Systems Over Finite Alphabets

Systems over finite alphabets are discrete-time systems whose input and output signals take their values in finite sets. Three notions of input/output stability (gain stability, incremental stability and external stability) that are particularly applicable to this class of systems are proposed and motivated through examples. New formulations for generalized small gain and incremental small gain theorems are presented, thus showing that gain stability and incremental stability are useful robustness measures. The paper then focuses on deterministic finite state machine (DFM) models. For this class, the problems of verifying gain stability, incremental stability, and corresponding gain bounds are shown to reduce to searching for an appropriate storage function. These problems are also shown to be related to the problem of verifying the nonexistence of negative cost cycles in an appropriately constructed network. Using this insight and based on a solution approach for discrete shortest path problems, a strongly polynomial algorithm is proposed. Finally, incremental stability and external stability are shown to be equivalent notions for this class of systems.

Optimal detection of symmetric mixed quantum states

We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.

Controller design for a class of underactuated nonlinear systems

We design a controller for a class of underactuated nonlinear systems. First, we try to find an appropriate global change of coordinates to transform the dynamics of the system into a desired form which consists of a lower order nonlinear subsystem plus a chain of integrators. Then, we find a control Lyapunov function (CLF) and its associated control law for the lower order subsystem. Finally, using a backstepping procedure we derive the control Lyapunov function and the controller for the whole system. The obtained controller renders the origin semiglobally asymptotically stable. As an special case, we demonstrate this procedure for the Acrobot example which is a two-link planar robot with a single actuator at the elbow.

Finite Approximations of Switched Homogeneous Systems for Controller Synthesis

In this note, we demonstrate the use of a control oriented notion of finite state input/output approximation to synthesize correct-by-design controllers for hybrid plants under sensor limitations. Specifically, we consider the problem of designing stabilizing switching controllers for a pair of unstable homogeneous second order systems with binary output feedback. In addition to yielding a deterministic finite state approximate model of the hybrid plant, our approach allows one to efficiently establish a useable upper bound on the quality of approximation, and leads to a discrete optimization problem whose solution immediately provides a certified finite state controller for the plant. The resulting controller consists of a deterministic finite state observer and a corresponding full state feedback control law.

A quasi-convex optimization approach to parameterized model order reduction

In this paper, an optimization-based model order reduction (MOR) framework is proposed. The method involves setting up a quasi-convex program that solves a relaxation of the optimal Hinfin norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexible in imposing additional constraints such as exact matching of specific frequency response samples. The proposed optimization-based approach is also extended to solve the parameterized model-reduction problem (PMOR). The proposed method is compared to existing moment matching and optimization-based MOR methods in several examples. PMOR models for large RF inductors over substrate and power-distribution grid are also constructed.

Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically non-existent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that guarantee global asymptotic stability of a limit cycle induced by a relay with hysteresis in feedback with an LTI stable system. The analysis is based on finding global quadratic Lyapunov functions for a Poincare map associated with the RFS. We found that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex. The search for globally quadratic Lyapunov functions is then done by solving a set of LMIs. Most examples of RFS analyzed by the authors were proven globally stable. Systems analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. This leads us to believe that quadratic stability of associated Poincare maps is common in RFS.