Asymptotic optimal tracking control for an uncertain nonlinear Euler-Lagrange system: A RISE-based closed-loop Stackelberg game approach

Differential game theory is used to develop controllers for an uncertain nonlinear Euler-Lagrange system. A closed-loop Stackelberg strategy based on hierarchical characteristics of the system is employed. A Robust Integral Sign of the Error (RISE) controller is used to partially cancel uncertain nonlinearities in the system first, and the residual system is modeled as an infinite-horizon two-person Stackelberg differential game. Although the game is linear-quadratic (LQ) not all the nonlinearities are lost since the residual system is linear in errors but not in the original states. To alleviate time inconsistency a closed-loop strategy is sought such that the controller assumes the potential perturbation to the system and computes its strategy accordingly. An analytical solution is presented to allow of a real-time controller implementation. A Lyapunov analysis is provided to examine the stability of the developed controller.