A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µ-based methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests

The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

In this paper, the deterministic global optimization algorithm, αBB, (α-based Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...

. In order to generate valid convex lower bounding problems for nonconvex twice--differentiable optimization problems, a method that is based on second-- order information of general twice--differentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the ffBB, a branch and bound algorithm which relies on this underestimation procedure [3]. Key words: convex underestimators; twice--differentiable; interval anlysis; eigenvalues 1. Introduction The mathematical description of many physical phenomena, such as phase equilibrium, or of chemical processes generally requires the introduction of nonconvex functions. As the number of local solutions to a nonconvex optimization problem cannot be predicted a priori, the identifi...

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We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree. 1 Introduction 1.1 Notation R (C) denotes the set of real (complex) numbers. For c 2 C, Re c is the real part of c. The set of n \Theta n matrices with real (complex) entries is denoted R n\Thetan (C n\Thetan ). P T stands for the transpose of P , and P , the complex conjugate transpose. I denotes the identity matrix, with size determined from context. For a matrix P 2 R n\Thetan (or C n\Thetan ), i (P ); 1 i n denotes the ith eigenvalue of P (with no particular ordering). oe max (P ) denotes the maximum singular value (or spectral norm) of P , define...

In this note we study the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices. Different perturbation norms are considered. The aim is to derive robustness criteria which are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials [13] are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulae are obtained whereas for perturbations with multiple input matrices structured singular values are involved. 1 Introduction If the equations of a physical system are written down, they will often not be in state--space form but involve higher order derivatives of some variables, see [18], [21]. In these cases it is useful to have results available which can be directly applied without prior transformation into state space form. In a recent paper [20] Fuhrmann and Willems ...

This paper gives an overview of promising new developments in robust stability and performance analysis of linear control systems with real parametric uncertainty. The goal is to develop a practical algorithm for medium size problems, where medium size means less than 100 real parameters, and "practical" means avoiding combinatoric (nonpolynomial) growth in computation with the number of parameters for all of the problems which arise in engineering applications. We present an algorithm and experimental evidence to suggest that this goal has, for the first time, been achieved. We also place these results in context by comparing with other approaches to robustness analysis and considering potential extensions, including controller synthesis. 1 Introduction Robust stability and performance analysis with real parametric uncertainty can be naturally formulated as a Structured Singular Value, or , problem, where the block structured uncertainty description is allowed to contain both...

Many problems in control system analysis and design can be posed in a setting where a system with a fixed model structure and nominal parameter values is affected by parameter variations. An example is parametric robustness analysis, where the parameters might represent physical quantities that are known only to within a certain accuracy, or vary depending on operating conditions etc. Frequently asked questions here deal with performance issues: "How bad can a certain performance measure of the system be over all possible values of the parameters?" Another example is parametric controller design, where the parameters represent degrees of freedom available to the control system designer. A typical question here would be: "What is the best choice of parameters, one that optimizes a certain design objective?" Many of the questions above may be directly restated as optimization problems: If q denotes the vector of parameters, Q

This paper studies the problem of robust absolute stability of a class of nonlinear discrete-time systems with time-varying matrix uncertainties of polyhedral type and multiple time-varying sector nonlinearities. By using the variational method and the Lyapunov second method, criteria for robust absolute stability are obtained in different forms for the class of systems under consideration. Specifically, we determine the parametric classes of Lyapunov functions which define the necessary and sufficient conditions of robust absolute stability. We apply the piecewise-linear Lyapunov functions of the infinity vector norm type to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Some simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for several special cases. An example is presented as an application of these results to a specific class of systems with time-varying interval matrices in the linear part.