Conditions for the genericity of feedback stabilisability of discrete-time switching systems based on Lie-algebraic solvability

This paper addresses the stabilisation of switching discrete-time linear systems with control inputs under arbitrary switching, based on the existence of a common quadratic Lyapunov function (CQLF). While stabilising feedback gains may be directly designed by solving linear matrix inequalities (LMIs), LMI-based designs in general do not provide information about closed-loop system structure. The authors have begun a line of work dealing with control design based on the Lie-algebraic solvability property. In contrast to LMI-based design, although at the expense of generality, control design based on such Lie-algebraic property seeks to assign closed-loop structure approximating that required to satisfy Lie algebraic conditions that guarantee existence of a CQLF. The present paper expands on earlier work by deriving conditions under which the closed-loop system can be caused to satisfy the Lie-algebraic solvability property generically, i.e. for almost every set of system parameters, furthermore admitting straightforward and efficient numerical implementation.