Nice reachability for planar bilinear control systems with applications to planar linear switched systems

Michael Margaliot, Michael S. Branicky

Research output: Contribution to journalArticlepeer-review

Abstract

We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a "nice" control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.

Original languageEnglish
Pages (from-to)900-905
Number of pages6
JournalIEEE Transactions on Automatic Control
Volume54
Issue number4
DOIs
StatePublished - 2009

Keywords

  • Lie algebra
  • Lie brackets
  • Maximum principle
  • Metzler matrices
  • Optimal control
  • Positive linear systems
  • Stability under arbitrary switching
  • Switched systems

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