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A Survey of Computational Complexity Results in Systems and Control
, 2000
"... 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 fi ..."
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Cited by 118 (20 self)
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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, NPcompleteness, 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, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
Branch and Bound Algorithm for Computing the Minimum Stability Degree of Parameterdependent Linear Systems
, 1991
"... 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 ..."
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Cited by 21 (5 self)
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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...
Path planning for permutationinvariant multirobot formations
, 2002
"... In many multirobot 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 permutationinvariant multirobot formation as a set of robot configurations, without assigning ..."
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Cited by 12 (0 self)
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In many multirobot 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 permutationinvariant multirobot 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 permutationinvariant formations. In this paper, we extend this idea to build a full representation of a permutationinvariant formation space. We describe the properties of the representation, and show how it can be used to construct collisionfree paths for permutationinvariant formations.
Global Optimization in Control System Analysis and Design
 CONTROL AND DYNAMIC SYSTEMS: ADVANCES IN THEORY AND APPLICATIONS
, 1992
"... 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 ..."
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Cited by 11 (2 self)
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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
A Global Optimization Method, αBB, for Process Design
 COMPUT. CHEM. ENG
, 1996
"... A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions ..."
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Cited by 9 (1 self)
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A global optimization algorithm, αBB, for twicedifferentiable NLPs is presented. It operates within a branchandbound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twicedifferentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.
Let's Get Real
 In Robust Control Theory, IMA Proceedings
, 1995
"... 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 &qu ..."
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Cited by 8 (2 self)
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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...
Robust SPR synthesis for loworder polynomial segments and interval polynomials
 Proceedings of the American Control Conference (ACC 2001), Crystal Gateway Marriott
, 2001
"... Abstract: We prove that, for loworder (n ≤ 4) stable polynomial segments or interval polynomials, there always exists a fixed polynomial such that their ratio is SPRinvariant, thereby providing a rigorous proof of Anderson’s claim on SPR synthesis for the fourthorder stable interval polynomials. ..."
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Cited by 3 (1 self)
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Abstract: We prove that, for loworder (n ≤ 4) stable polynomial segments or interval polynomials, there always exists a fixed polynomial such that their ratio is SPRinvariant, thereby providing a rigorous proof of Anderson’s claim on SPR synthesis for the fourthorder stable interval polynomials. Moreover, the relationship between SPR synthesis for loworder polynomial segments and SPR synthesis for loworder interval polynomials is also discussed.
Improved Results on Robust Stability of Multivariable Interval Control Systems 1
"... Abstract: For interval polynomial matrices, we identify the minimal testing set, whose stability can guarantee that of the whole uncertain set. Our results improve the conclusions given by Kamal and Dahleh. ..."
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Cited by 3 (2 self)
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Abstract: For interval polynomial matrices, we identify the minimal testing set, whose stability can guarantee that of the whole uncertain set. Our results improve the conclusions given by Kamal and Dahleh.