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68
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 114 (8 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture
 J. of the American Mathematical Society
, 2001
"... A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterate ..."
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Cited by 71 (5 self)
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A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterated Function System (IFS),
Incremental search methods for reachability analysis of continuous and hybrid systems
 In Hybrid Systems: Computation and Control
, 2004
"... Abstract. In this paper we present algorithms and tools for fast and efficient reachability analysis, applicable to continuous and hybrid systems. Most of the work on reachability analysis and safety verification concentrates on conservative representations of the set of reachable states, and conseq ..."
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Cited by 58 (6 self)
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Abstract. In this paper we present algorithms and tools for fast and efficient reachability analysis, applicable to continuous and hybrid systems. Most of the work on reachability analysis and safety verification concentrates on conservative representations of the set of reachable states, and consequently on the generation of safety certificates; however, inability to prove safety with these tools does not necessarily result in a proof of unsafety. In this paper, we propose an alternative approach, which aims at the fast falsification of safety properties; this approach provides the designer with a complementary set of tools to the ones based on conservative analysis, providing additional insight into the characteristics of the system under analysis. Our algorithms are based on algorithms originally proposed for robotic motion planning; the key idea is to incrementally grow a set of feasible trajectories by exploring the state space in an efficient way. The ability of the proposed algorithms to analyze the reachability and safety properties of general continuous and hybrid systems is demonstrated on examples from the literature. 1
The Generalized Spectral Radius and Extremal Norms
, 2000
"... The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm alway ..."
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Cited by 58 (10 self)
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The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generalized spectral radius. We prove that the generalized spectral radius is locally Lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius. Sufficient conditions for the existence of extremal norms are obtained.
An Elementary Counterexample to the Finiteness Conjecture
 SIAM JOURNAL ON MATRIX ANALYSIS
, 2001
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Undecidable Problems for Probabilistic Automata of Fixed Dimension
 Theory of Computing Systems
, 2001
"... We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matr ..."
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Cited by 43 (3 self)
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We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 &times; 47 matrices with nonnegative rational entries is bounded is undecidable.
Computationally efficient approximations of the joint spectral radius
 SIAM J. Matrix Anal
, 2005
"... Abstract. The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approxima ..."
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Cited by 38 (7 self)
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Abstract. The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed. For the special case of matrices with nonnegative entries we give elementary proofs of simple inequalities that we then use to obtain approximations of arbitrary high accuracy. From these inequalities it follows that the spectral radius of matrices with nonnegative entries is given by the simple expression ρ(A1,...,Am) = lim k→ ∞ ρ1/k (A ⊗k 1 + ···+ A⊗k m), where it is somewhat surprising to notice that the righthand side does not directly involve any mixed product between the matrices. (A ⊗k denotes the kth Kronecker power of A.)
Automata based interfaces for control and scheduling
 In Proc. 10th Int. Workshop on Hybrid Systems: Computation and Control, LNCS 4416
, 2007
"... Abstract. We propose the use of formal languages of infinite words over the alphabet of task identifiers as an interface between control designs and software implementations. We argue that this approach is more flexible than the classical realtime scheduling framework based on periodic tasks, and ..."
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Cited by 24 (6 self)
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Abstract. We propose the use of formal languages of infinite words over the alphabet of task identifiers as an interface between control designs and software implementations. We argue that this approach is more flexible than the classical realtime scheduling framework based on periodic tasks, and allows composition of interfaces by languagetheoretic operations. We show that finite automata over infinite words offer analyzable representation and can capture many interesting interface specifications such as exponential stability of switched linear systems. 1
Joint spectral characteristics of MATRICES: A CONIC PROGRAMMING APPROACH
, 2010
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