H^∞-Optimal Control for Singularly Perturbed Systems—Part II: Imperfect State Measurements

In this paper we study the H_∞ -optimal control of singularly perturbed linear systems under general imperfect measurements, for both finite- and infinite-horizon formulations. Using a differential game theoretic approach, we first show that as the singular perturbation parameter (say, e > 0) 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 (and in some cases equal to) the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate “slow” and “fast” quadratic cost functions, with the bound being computable independently of e and knowing only the slow and fast dynamics of the system. We then construct a controller based on the slow subsystem only and obtain conditions under which it delivers a desired performance level even though the fast dynamics are completely neglected. The ultimate performance level achieved by this “slow” controller can be uniformly improved upon, however, by a composite controller that uses some feedback from the output of the fast subsystem. We construct one such controller, via a two-step sequential procedure, which uses static feedback from the fast output and dynamic feedback from an appropriate slow output, each one obtained by solving appropriate 6-independent lower dimensional H°°- optimal control problems under some informational constraints. We provide a detailed analysis of the performance achieved by this lower-dimensional 6-independent composite controller when applied to the full-order system and illustrate the theory with some numerical results on some prototype systems.