H^∞-optimal control for singularly perturbed systems. Part I: Perfect state measurements

We study the H^∞-optimal control of singularly perturbed linear systems under perfect state measurements, for both finite and infinite horizons. Using a differential game theoretic approach, we show that as the singular perturbation parameter ε{lunate} approaches zero, the optimal disturbance attenuation level for the full-order system under a quadratic performance index converges to a value that is bounded above by the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate "slow" and "fast" quadratic cost functions. Furthermore, we construct a composite controller based on the solution of the slow and fast games, which guarantees a desired achievable performance level for the full-order plant, as ε{lunate} approaches zero. A "slow" controller, however, is not generally robust in this sense, but still under some conditions, which are delineated in the paper, the fast dynamics can be totally ignored. The paper also studies optimality when the controller includes a feedforward term in the disturbance, and presents some numerical examples to illustrate the theoretical results.