Optimal control of uncertain nonlinear systems using RISE feedback

A Hamilton-Jacobi-Bellman optimization scheme is used along with a RISE feedback structure to minimize a quadratic performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite general uncertainty in the dynamics, such as additive bounded disturbances and parametric uncertainty. Motivated by recent previous results that use a neural network structure to approximate the dynamics (i.e., the state space model is approximated with a residual function reconstruction error), the result in this paper uses the implicit learning capabilities of the RISE control structure to learn the dynamics asymptotically. Specifically, a Lyapunov stability analysis is performed to show that the RISE feedback term asymptotically identifies the unknown dynamics, yielding semi-global asymptotic tracking. In addition, it is shown that the system converges to a state space system that has a quadratic performance index which has been optimized by an additional control element. Simulation results are included to demonstrate the performance of the developed controller.