A Separation Principle for Linear Switching Systems and Parametrization of All Stabilizing Controllers

Abstract

In this paper, we investigate the problem of designing a switching compensator for a plant switching amongst a (finite) family of given configurations ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ). We assume that switching is uncontrolled, namely governed by some arbitrary switching rule, and that the controller has the information of the current configuration <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> . As a first result, we provide necessary and sufficient conditions for the existence of a switching compensator such that the closed-loop plant is stable under arbitrary switching. These conditions are based on a separation principle, precisely, the switching stabilizing control can be achieved by separately designing an observer and an estimated state (dynamic) compensator. These conditions are associated with (non-quadratic) Lyapunov functions. In the quadratic framework, similar conditions can be given in terms of LMIs which provide a switching controller which has the same order of the plant. As a second result, we furnish a characterization of all the stabilizing switching compensators for such switching plants. We show that, if the necessary and sufficient conditions are satisfied then, given any arbitrary family of compensators K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (s), each one stabilizing the corresponding LTI plant ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) for fixed i, there exist suitable realizations for each of these compensators which assure stability under arbitrary switching.