Robust nonlinear-nonquadratic feedback control via parameter-dependent 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 (parameter-dependent) 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-Jacobi-Bellman 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.