Feedback control and classification of generalized linear systems

We present a unified theory of control synthesis for generalized linear (i.e., descriptor) systems using constant-ratio proportional and derivative (CRPD) feedback. Our framework includes the theory of static state feedback and output feedback for regular state-space systems as a special case. The main elements of this theory include 1) a covering of the space of all systems, both regular and singular, by a family of open and dense subsets indexed by the unit circle; 2) a group of transformations which may be viewed as symmetries of the cover; 3) an admissible class of feedback transformations on each subset which is specifically adapted to that subset. We obtain a general procedure of control synthesis of CRPD feedback for generalized linear systems which uses the symmetry transformations to systematically reduce each synthesis problem to an ordinary static-state feedback (or output feedback) synthesis problem for a corresponding regular system. We apply this approach to obtain natural generalizations of the disturbance decoupling theorem, the pole assignment theorem, and the Brunovsky classification theorem.