Reduced-Order Proper $H_\infty$ Controllers for Descriptor Systems: Existence Conditions and LMI-Based Design Algorithms

In this note, we present a new approach to investigate the existence and design of reduced-order proper ${H_{\infty}}$ controllers that provide the same level of performance as that of full-order controllers. By revealing some special features of the LMI-based solvability conditions for the ${H_{\infty}}$ control problem for descriptor systems, we obtain a refined bound on the order of ${H_{\infty}}$ controllers, which is independent of (invariant under the allowed transformations on) a descriptor realization of the generalized plant. Moreover, we provide LMI-based algorithms to design the reduced-order controllers. This note not only extends in a satisfying way the results on reduced-order ${H_{\infty}}$ controllers for state-space systems to descriptor systems, but also provides insight into the mechanism by which the order of ${H_{\infty}}$ controllers for descriptor systems can be reduced through a consideration of the unstable finite zeros or infinite zeros.