Arbitrary Switching

  • Zhendong SunEmail author
  • Shuzhi Sam Ge
Part of the Communications and Control Engineering book series (CCE)


Chapter 2 focuses on the guaranteed stability analysis of switched dynamical systems under arbitrary switching. As global uniform asymptotic stability is equivalent to the existence of a common Lyapunov function of the subsystems, the Lyapunov approach plays a dominant role in the stability analysis. For switched linear systems, emphasis is laid on the sets of functions that are universal in the sense that each asymptotically stable system admits a Lyapunov function from the function set. We also pay much attention to the algebraic theory of discrete-time switched linear systems, where the stability is elegantly characterized by the spectral radius of the matrix set, which generalizes the standard matrix spectral theory. To approximate the spectral radius numerically, the homogeneous polynomials are utilized to serve as common Lyapunov functions, where the sum of squares technique and the semi-definite programming are used to searching for suitable homogeneous polynomial Lyapunov functions. Finally, the more subtle issue of marginal stability is carefully examined, and its connection to the common weak Lyapunov function is established. We reveal that marginal stability admits a block triangular decomposition with clear spectral information, and this leads to an invariant set viewpoint for characterizing marginal stability and marginal instability.


Arbitrary Switching Common Lyapunov Function Joint Spectral Radius Marginal Instability Polytopic Systems 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.College Automation Science & Engineering, Center for Control and OptimizationSouth China University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of Electrical and Computer EngineeringThe National University of SingaporeSingaporeSingapore
  3. 3.Robotics Institue and Institute of Intelligent Systems and Information TechnologyUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

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