The Generation of Lyapunov Functions for Input-Output Stable Systems

This paper discusses the relationship between properties of input-output descriptions and state space models for dynamical systems. It is shown that a state space realization of an input-output stable dynamical system is globally asymptotically stable in the sense of Lyapunov if it is uniformly observable and if every state is reachable. This result is proved in the context of abstract dynamical systems and leads to the equivalence of input-output stability and asymptotic stability for uniformly controllable and uniformly observable linear finite-dimensional systems. The generation of Lyapunov functions is subsequently considered, and variational techniques for the construction of Lyapunov functions are presented. Passivity and related energy concepts are particularly exploited in this context. These results yield the Lyapunov functions used in the proofs of the circle criterion and the Popov criterion as particular cases. The generality of the approach, however, makes these ideas applicable to much more general situations. Examples illustrating the results and the unifying point of view are included.