Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Abstract

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.