Robust nonlinear-nonquadratic feedback control via parameterdependent Lyapunov functions

In this paper we develop a unified framework to address the problem of optimal nonlinear-nonquadratic robust control for systems with nonlinear time-invariant real parameter uncertainty. Specifically, we transform a given robust nonlinear control problem into an optimal control problem by modifying the performance functional to account for the system uncertainty. Robust stability of the closed loop nonlinear system is guaranteed by means of a parameter-dependent Lyapunov function composed of a fixed (parameter-independent) and variable (parameterdependent) part. The fixed part of the Lyapunov function can clearly be seen to be the solution to the steady-state Hamilton-Jacobi-Bellman equation for the nominal system. The overall framework generalizes the classical Hamilton-JacobiBellman conditions to address the design of robust optimal controllers for uncertain nonlinear systems via parameter-dependent Lyapunov functions and provides the foundation for extending robust linear-quadratic controller synthesis to robust nonlinear-nonquadratic problems.