Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation fields for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the dow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems. A slow controller, however, is not generally as robust as the composite controller, but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.