Designed Switching

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


Chapter 4 is devoted to the stabilizing switching design for switched dynamical systems under controlled switching. It is proven that a switched Lyapunov function exists if the system is globally asymptotically stabilizable. However, counterexamples exhibit that even stabilizable planar switched linear systems may not admit any convex switched Lyapunov function. To overcome the intrinsic difficulty, we introduce a class of nonconvex functions known as min functions that are piecewise quadratic and prove that each stabilizable switched linear system admits a min function as a switched Lyapunov function. To further address the stabilizability and robustness of switched linear system, we propose a pathwise state-feedback switching strategy, which accounts to concatenating a finite number of switching paths based on appropriate partitions of the state space. By aggregating the overall system into a discrete-time piecewise linear system, we are able to prove that the switching strategy exponentially stabilizes the original switched linear system whenever it is asymptotically stabilizable. We develop a computational procedure to calculate a stabilizing pathwise state-feedback switching law for an asymptotically stabilizable switched linear system. To further investigate the robustness of the pathwise state-feedback switching strategy, we define a (relative) distance between two switching signals and prove that the closed-loop system is robust against structural/unstructural/switching perturbations.


Lyapunov Function Switching Signal Quadratic Lyapunov Function Switch Linear System State Partition 
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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Copyright information

© 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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