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11 - Long products of matrices
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- By V. D. Blondel, Université catholique de Louvain, R. M. Jungers, Université catholique de Louvain
- Edited by Valérie Berthé, Université de Paris VII, Michel Rigo, Université de Liège, Belgium
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- Book:
- Combinatorics, Automata and Number Theory
- Published online:
- 05 March 2013
- Print publication:
- 12 August 2010, pp 530-562
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Summary
The joint spectral radius of a set of matrices is the maximal growth rate that can be obtained by forming long products of matrices taken in the set. This quantity and its minimal growth counterpart, the joint spectral subradius, have proved useful for studying several problems from combinatorics and number theory. For instance, they characterise the growth of certain classes of languages, the capacity of forbidden difference constraints on languages, and the trackability of sensor networks. In Section 11.2 we describe some of these applications.
While the joint spectral radius and related notions have applications in combinatorics and number theory, these disciplines have in turn been helpful to improve our understanding of problems related to the joint spectral radius. As an example, we present in Section 11.3 a central result that has been proved with the help of techniques from combinatorics on words: the falseness of the finiteness conjecture.
In practice, computing a joint spectral radius is not an easy task. As we will see, this quantity is NP-hard to approximate in general, and the simple question of knowing, given a set of matrices, if its joint spectral radius is larger than one is even algorithmically undecidable. However, in recent years, approximation algorithms have been proposed that perform well in practice. Some of these algorithms run in exponential time while others provide no accuracy guarantee. In practice, by combining the advantages of the different algorithms, it is often possible to obtain satisfactory estimates.
4 - Switched and piecewise affine systems
- from Part I - Theory
- Edited by Jan Lunze, Ruhr-Universität, Bochum, Germany, Françoise Lamnabhi-Lagarrigue, Centre National de la Recherche Scientifique (CNRS), Paris
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- Book:
- Handbook of Hybrid Systems Control
- Published online:
- 21 February 2011
- Print publication:
- 15 October 2009, pp 87-138
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Summary
Switched systems are described by a set of continuous state-space models together with conditions that decide which model of this set is valid for the current continuous state. As an extension of the classical linear or affine state-space representations of dynamical systems, this modelling formalism has been thoroughly investigated, as this chapter shows. The identification of the model parameters, observability, and stability analysis as well as methods for stabilization and control of switched systems are surveyed. As shown in the last section, many analysis and design problems for switched systems have a high computational complexity or are even undecidable.
Definition of the system class
Switched systems represent a type of model of hybrid systems that has been studied extensively. The reason for this research activity is given by the fact that this class of systems is very close to “non-hybrid” systems and an extension of the theory of continuous systems towards hybrid systems is, therefore, rather straightforward. Nevertheless, this system class already exhibits several important phenomena of hybrid dynamical systems.
The basic representation format is the state-space model
which describes the dynamical behavior of the system for the input u ∈ ℝm and the operation mode q ∈ Q. The vector field f and the output function g are assumed to be Lipschitz continuous with respect to x and u so that for a fixed operation mode q solutions to the state-space model exist.