Proper feedback compensators for a strictly proper plant by polynomial equations

Frank Callier; Ferdinand Kraffer

International Journal of Applied Mathematics and Computer Science (2005)

  • Volume: 15, Issue: 4, page 493-507
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Callier, Frank, and Kraffer, Ferdinand. "Proper feedback compensators for a strictly proper plant by polynomial equations." International Journal of Applied Mathematics and Computer Science 15.4 (2005): 493-507. <http://eudml.org/doc/207761>.

@article{Callier2005,
abstract = {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.},
author = {Callier, Frank, Kraffer, Ferdinand},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {polynomial matrix systems; linear time-invariant feedback control systems; flexible belt device; row-column-reduced polynomial matrices; feedback compensator design},
language = {eng},
number = {4},
pages = {493-507},
title = {Proper feedback compensators for a strictly proper plant by polynomial equations},
url = {http://eudml.org/doc/207761},
volume = {15},
year = {2005},
}

TY - JOUR
AU - Callier, Frank
AU - Kraffer, Ferdinand
TI - Proper feedback compensators for a strictly proper plant by polynomial equations
JO - International Journal of Applied Mathematics and Computer Science
PY - 2005
VL - 15
IS - 4
SP - 493
EP - 507
AB - 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.
LA - eng
KW - polynomial matrix systems; linear time-invariant feedback control systems; flexible belt device; row-column-reduced polynomial matrices; feedback compensator design
UR - http://eudml.org/doc/207761
ER -

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