In this chapter we formalize the notion of continuous-time switched linear system adopted in the monograph, emphasizing the main classes of switching signals. Various definitions of stability, as well as the related notions of maximal Lyapunov exponent and uniform exponential rate, are presented. Basic properties of the dynamics of switched linear systems are established, such as the equivalence between most stability notions under the assumption of weak-* compactness of the class of signals, expressed by Fenichel lemma, and the characterization of the maximal Lyapunov exponent via block reduction. The notion of long-time approximation, allowing to prove the equality between maximal Lyapunov exponent and uniform exponential rate, is also introduced.

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Switched Linear Systems: Definitions and Basic Facts

  • Yacine Chitour,
  • Paolo Mason,
  • Mario Sigalotti

摘要

In this chapter we formalize the notion of continuous-time switched linear system adopted in the monograph, emphasizing the main classes of switching signals. Various definitions of stability, as well as the related notions of maximal Lyapunov exponent and uniform exponential rate, are presented. Basic properties of the dynamics of switched linear systems are established, such as the equivalence between most stability notions under the assumption of weak-* compactness of the class of signals, expressed by Fenichel lemma, and the characterization of the maximal Lyapunov exponent via block reduction. The notion of long-time approximation, allowing to prove the equality between maximal Lyapunov exponent and uniform exponential rate, is also introduced.