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.

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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...

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

In many multi-robot applications, the specific assignment of goal configurations to robots is less important than the overall behavior of the robot formation. In such cases, it is convenient to define a permutation-invariant multi-robot formation as a set of robot configurations, without assigning specific configurations to specific robots. For the case of robots that translate in the plane, we can represent such a formation by the coefficients of a complex polynomial whose roots represent the robot configurations. Since these coefficients are invariant with respect to permutation of the roots of the polynomial, they provide an effective representation for permutation-invariant formations. In this paper, we extend this idea to build a full representation of a permutation-invariant formation space. We describe the properties of the representation, and show how it can be used to construct collision-free paths for permutationinvariant formations.

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A global optimization algorithm, αBB, for twice-differentiable NLPs is presented. It operates within a branch-and-bound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twice-differentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.

Abstract: We prove that, for low-order (n ≤ 4) stable polynomial segments or interval polynomials, there always exists a fixed polynomial such that their ratio is SPR-invariant, thereby providing a rigorous proof of Anderson’s claim on SPR synthesis for the fourth-order stable interval polynomials. Moreover, the relationship between SPR synthesis for low-order polynomial segments and SPR synthesis for low-order interval polynomials is also discussed.

This paper addresses the computation of frequency responses of plants with uncertain parameters using symbolic computation. A computationally tractable procedure is developed based on a Jocobian-like technique where a general uncertain structure including nonlinear can be dealt with. This procedure is applied to computing the frequency responses of a vehicle clutch system and the longitudinal dynamics of an aircraft where the underlying uncertain parameters perturb both plants nonlinearly. The developed procedure is also compared with the currently widely used parameter gridding method. 1 Introduction It has long been identified that the solution of the generic problem of calculating the roots of an uncertain polynomial for all possible perturbation vectors, would be a break through in the analysis and design of uncertain systems. This motivates the computation of the value set of uncertain polynomials. Considerable attention has recently been attracted and many significant r...