Componentwise ultimate bound computation for switched linear systems

We present a novel ultimate bound computation method for switched linear systems with disturbances and arbitrary switching. We consider both discrete-time and continuous-time systems. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds in the form of polyhedral sets and/or mixed ellipsoidal/polyhedral sets, and it is completely systematic once the aforementioned transformation is obtained. We show that the transformation can be found in the well-known case where the matrices of the switched linear system generate a solvable Lie algebra. In the latter case, our results also constitute a new sufficient condition for practical stability. An example comparing the bounds obtained by the proposed ultimate bound computation method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.