Robust control of singular systems with time delay. Part I: Continuous time case

In this paper for the class of continuous time nonlinear uncertain singular time delay dynamical systems we present robust stability analyze results. Next, we consider a control problem for nonlinear continuous time uncertain singular time delay dynamical systems involving a notion of optimality with respect to an auxiliary cost which guarantees a bound on the worst-case value of a nonlinear-nonquadratic cost criterion over a prescribed uncertainty set. Further we specialize result to affine uncertain systems to obtain controllers predicated on an inverse optimal control problem. In particular, to avoid the complexity in solving the steady-state Hamilton-Jacobi-Bellman equation we parameterize a family of stabilizing controllers that minimize some derived cost functional that provides flexibility in specifying the control law. The performance integrand is shown to explicitly depend on the continuous time nonlinear singular time delay system dynamics, the Lyapunov function of the closed-loop system, and the stabilizing feedback control law wherein the coupling is introduced via the Hamilton-Jacobi-Bellman equation. By varying the parameters in the Lyapunov function and the performance integrand, the proposed framework can be used to characterize a class of globally stabilizing controllers that can meet the closed-loop system response constraints. Obtained results for nonlinear case are further specialized to continuous time linear singular time delay dynamical systems.