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2 International Journal of Applied Mathematics and Computer Science

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Proper feedback compensators for a strictly proper plant by polynomial equations

Callier F. M., Kraffer F.

International Journal of Applied Mathematics and Computer Science

2005

Vol. 15, no 4

493-507

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) Dk is 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 a priori. Two examples enhance the tutorial value of the paper, revealing also a novel computational method.

Circle criterion and boundary control systems in factor form: input-output approach

Grabowski P., Callier F. M.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 6

1387-1403

A circle criterion is obtained for a SISO Lur'e feedback control system consisting of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the transfer function belongs to Hinfty(Pi+) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well-posed, i.e. for any initial state x0in H the truncated input and output signals uT, yT belong to L2(0,T) for any T>0. The technique of the proof adapts Desoer-Vidyasagar's circle criterion method (Desoer and Vidyasagar, 1975, Ch.3, Secs.1 and 2, pp.37-43, Ch.5, Sec.2, pp.139-142 and Ch.6, Secs.3 and 4, pp.172-174]), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two transmission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.