On feedback stabilisation of switched discrete-time systems via Lie-algebraic techniques

This paper studies the stabilisation of switched discrete-time linear control systems under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems. The existence of such CQLF can be ensured using Lie-algebraic techniques by the existence of a nonsingular similarity transformation that simultaneously triangularises the closed-loop evolution maps of the component subsystems. The present work formulates a Lie-algebraic feedback design problem in terms of invariant subspaces and proposes an iterative algorithm that seeks a set of feedback maps that guarantee the existence of a CQLF, and thus UGES of the switched feedback system. The main contribution of the paper is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution.