PROPER FEEDBACK COMPENSATORS FOR A STRICTLY PROPER PLANT BY POLYNOMIAL EQUATIONS
Frank M. Callier, F. Kraffer · Digital library of Zielona Gora (University of Zielona Góra) · 2005
We review the polynomial matrix compensator equation XlDr + YlNr = Dk (COMP), e.g. (Callier and Desoer, 1982, Kučera, 1979; 1991), where (a) the right-coprime polynomial matrix pair (Nr,Dr) is given by the strictly proper rational plant right matrix-fraction P = NrD −1 r,(b)Dkis a given nonsingular stable closed-loop characteristic polynomial matrix, and (c) (Xl,Yl) is a polynomial matrix solution pair resulting possibly in a (stabilizing) rational compensator given by the left fraction C = X −1 l Yl. We recall first the class of all polynomial matrix pairs (Xl,Yl) 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 Dr is column-reduced, and (b) the closed-loop characteristic matrix Dk is row-column-reduced, e.g., monically diagonally degree-dominant. This allows us to get all solution pairs (Xl,Yl) giving a proper compensator with a row-reduced denominator Xl having (sufficiently large) row degrees prescribed apriori. Two examples enhance the tutorial value of the paper, revealing also a novel computational method.