Observability of Switched Linear Systems

Abstract

This work deals with finite time observability of switched linear systems ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SLS</i> ) when they are represented by a family of nonautonomous linear systems ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LS</i> ) and an interpreted Petri net ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IPN</i> ). Based on this <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SLS</i> representation, new detection of the commutation time and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LS</i> distinguishability characterizations in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SLS</i> extended to the non autonomous case are presented. Using these results, the novel concept of distinguishability between <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LS</i> sequences is presented and characterized. This concept together with the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IPN</i> input-output information is used to determine the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IPN</i> marking sequence. From the knowledge of this sequence, the conditions for the computation of the continuous state are presented. Also necessary and sufficient conditions for the observability in infinitesimal time are provided.