A small-gain approach to inverse optimal adaptive control of nonlinear systems with unmodeled dynamics

Abstract

This work formulates and solves the problem of inverse optimal adaptive control for nonlinear systems with parametric and dynamic uncertainties. A concept and sufficient conditions for solving this problem are first introduced and derived based on adaptive control Lyapunov function method. One difficulty obstructing current works to solve this problem is that inverse optimality and stability of the closed-loop system are not necessarily achieved simultaneously in the presence of both parametric uncertainties and unmodeled dynamics. To this end, a new auxiliary system and a cost functional that puts integral penalty on the state, control, parameter estimate and unmodeled dynamics are proposed to develop a new inverse optimal adaptive feedback approach. Then a small-gain approach is proposed to establish the links between inverse optimality and stability of the closed-loop system. It is proved that if inverse optimality is achieved and the small-gain condition holds, the closed-loop system is uniformly bounded and the system state converges to an arbitrarily small neighborhood of the origin. Finally, an example is given to illustrate the obtained results.