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Proper feedback compensators for a strictly proper plant by polynomial equations

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We review the polynomial matrix compensator equation X_lD_r + Y_lN_r = Dk (COMP), e.g. (Callier and Desoer, 1982, Kučera, 1979; 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 = N_rD_r, (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 = (X_l)^{−1}Y_l . 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, 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 (X_l, Y_l) giving a proper compensator with a row-reduced denominator X_l having (sufficiently large) row degrees prescribed a priori. Two examples enhance the tutorial value of the paper, revealing also a novel computational method.








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  • Department of Mathematics, University of Namur (FUNDP), Rempart de la Vierge 8, B-5000 Namur, Belgium
  • UTIA, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, POB 18, 18208 Prague, Czech Republic


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