An alternative converse Lyapunov theorem for discrete-time systems

This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete–time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi–global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of systems, the developed converse theorem can be used to establish non–conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non–conservative for GES conewise linear systems is given and, as a by–product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.