Characterization and Optimization of l Gains of Linear Switched Systems

Mohammad Naghnaeian, Petros G. Voulgaris

Research output: Contribution to journalArticlepeer-review


In this paper, we consider the lgain characterizations of linear switched systems (LSS) and present various relevant results on their exact computation and optimization. Depending on the role of the switching sequence, we study two broad cases: first, when the switching sequence attempts to maximize, and second, when it attempts to minimize the lgain. The first, named as worst-case throughout the paper, can be related to robustness of the system to uncontrolled switching; the second relates to situations when the switching can be part to the overall decision making. Although, in general, the exact computation of lgains is difficult, we provide specific classes, the input-output switching systems, for which it is shown that linear programming can be used to obtain the worst-case lgain. This is a sufficiently rich class of systems as any stable LSS can be approximated by one. Certain applications to robust control design are provided where we show that a switched compensation independently of the plant has no advantage over a linear time invariant (LTI) compensation, and further, if the plant is strictly causal, even a switched compensation which has a matched switching with the plant does not provide a better performance over an LTI compensation. Also, we present a new necessary and sufficient condition to check the stability of LSS in form of a model matching problem. On the other hand, if one is interested in minimizing the lgain over the switching sequences, we show that, for finite impulse response (FIR) switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching an exact, readily computable, characterization of the minimal lgain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching.

Original languageEnglish (US)
Article number7307135
Pages (from-to)2203-2218
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - Aug 2016


  • Linear programming
  • l performace
  • linear switched systems
  • robust control

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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