A generating function approach to the stability of discrete-time switched linear systems

Exponential stability of switched linear systems under both arbitrary and proper switching is studied through two suitably defined families of functions called the strong and the weak generating functions. Various properties of the generating functions are established. It is found that the radii of convergence of the generating functions characterize the exponential growth rate of the trajectories of the switched linear systems. In particular, necessary and sufficient conditions for the exponential stability of the systems are derived based on these radii of convergence. Numerical algorithms for computing estimates of the generating functions are proposed and examples are presented for illustration purpose.