Polynomial equations giving a proper feedback compensator for a strictly proper plant

Our purpose is a review of the polynomial matrix compensator equation XlDr + YlNr = D-k (COMP), (Callier and Desoer, 1982, Section 6.2), (Kucera, 1979; Kucera, 1991), where a) the right-coprime polynomial matrix pair (N-r, D-r) is given by the strictly proper rational plant right matrix-fraction P = NrDr-1, b) Dk is a given nonsingular stable closed-loop characteristic polynomial matrix, and c) (X-l, Y-l) is a polynomial matrix solution pair resulting possibly in a (stabilizing) rational compensator given by the left fraction C = Xl-1Yl. We recall first the class of all polynomial matrix pairs (X-l, Y-l) solving (COMP) and then single out those pairs which result in a proper rational compensator. An important role is hereby played by the assumptions that a) the plant denominator D-r is column-reduced (Wolovich, 1974), (Kailath, 1980), and b) the closed-loop characteristic matrix Dk is row-column-reduced (Callier and Desoer, 1982) ( e.g. monically diagonally degree dominant (Rosenbrock and Hayton, 1978), (Zagalak and Kucera, 1985)). This (using the information of (Callier and Desoer, 1982, pp.187-192)) allows to get all solution pairs (X-l, Y-l) giving a proper compensator with row-reduced denominator X-l having a priori prescribed (sufficiently large) row degrees as in (Kucera and Zagalak, 1999) and (Callier, 2000). Copyright (C) 2001 IFAC.