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.