Optimal nonlinear—nonquadratic feedback control for nonlinear discrete-time systems with l 2 and l ∞ disturbances

In a recent paper [6] an optimality-based disturbance rejection nonlinear control framework for nonlinear continuous-time systems with bounded exogenous disturbances was developed. In this paper, analogous results are developed for the discrete-time case. Speci cally, to address the optimality-based discrete-time disturbance rejection nonlinear control problem we extend the nonlinear–nonquadratic controller analysis and synthesis framework developed in [5]. The basic underlying ideas of the results in [5] are based on the fact that the steady-state solution to the Bellman equation is a control Lyapunov function for the nonlinear controlled system thus guaranteeing both optimality and stability. In this paper we extend the framework developed in [5] to address the problem of optimal nonlinear–nonquadratic discrete-time control that additionally guarantees disturbance rejection to bounded input disturbances. Speci cally, using nonlinear discrete-time dissipation theory [1] with appropriate storage functions and supply rates we transform the nonlinear disturbance rejection problem into an optimal control problem. This is accomplished by properly modifying the cost functional to account for exogenous disturbances so that the solution of the modi ed optimal nonlinear control problem serves as the solution to the disturbance rejection problem.