Proper feedback compensators for a strictly proper plant by polynomial equations

• Autorzy: Frank Callier , Ferdinand Kraffer
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

polynomial matrix systems; linear time-invariant feedback control systems; flexible belt device; row-column-reduced polynomial matrices; feedback compensator design

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2005
• Tom: 15
• Numer: 4
• Strony: 493-507

Daty

wydano

2005

otrzymano

2005-05-18

poprawiono

2005-10-08

Twórcy

autor

Frank Callier

• Department of Mathematics, University of Namur (FUNDP), Rempart de la Vierge 8, B-5000 Namur, Belgium

autor

Ferdinand Kraffer

• UTIA, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, POB 18, 18208 Prague, Czech Republic

Bibliografia

• Antoniou E.N. and Vardulakis A.I.G. (2005): On the computation and parametrization of proper denominator assigning compensators for strictly proper plants. - IMA J. Math. Contr. Inf., Vol. 22, pp. 12-25.
• Ataman E., Aatre V.K. and Wong K.M. (1981): Some statistical properties of median filters. - IEEE Trans. Acoust. Speech Signal. Process., Vol. ASSP-29, No. 5, pp. 1073- 1075
• M. Athans and P. L. Falb (1966): Optimal Control. -New York: McGraw-Hill.
• Bitmead R.R., Kung S.-Y., Anderson B.D.O. and Kailath T. (1978): Greatest common divisors via generalized Sylvester and Bezout matrices. - IEEE Trans. Automat. Contr., Vol. 23, No. 7, pp. 1043-1047.
• Callier F.M. (2000): Proper feedback compensators for a strictly proper plant by solving polynomial equations. - Proc. Conf. Math. Models. Automat. Robot., MMAR, Miedzyzdroje, Poland, Vol. 1, pp. 55-59.
• Callier F.M. (2001): Polynomial equations giving a proper feedback compensator for a strictly proper plant. - Prep. 1st IFAC/IEEE Symp. System Structure and Control, Prague, (CD-ROM)
• Callier F.M. and Desoer C.A. (1982): Multivariable Feedback Systems. -New York: Springer.
• Emre E. (1980): The polynomial equation QQ_c + RP_c = Φ with applications to dynamic feedback. - SIAM J. Contr. Optim., Vol. 18, No. 6, pp. 611-620.
• Francis B.A. (1987): A Course in H? Control Theory. -New York: Springer.
• Fuhrmann P.A. (1976): Algebraic system theory: An analysts point of view. -J. Franklin Inst., Vol. 301, No. 6, pp. 521- 540
• Hagadoorn H. and Readman M. (2004): Coupled Drives, Part 1: Basics, Part 2: Control and Analysis. - Available at www.control-systems-principles.co.uk
• Kailath T. (1980): Linear Systems. - Englewood Cliffs, N.J.: Prentice-Hall.
• Kraffer F. and Zagalak P. (2002): Parametrization and reliable extraction of proper compensators. - Kybernetika, Vol. 38, No. 5, pp. 521-540.
• Kucera V. (1979): Discrete Linear Control: The Polynomial Equation Approach. -Chichester, UK: Wiley.
• Kucera V. (1991): Analysis and Design of Discrete Linear Control Systems. -London: Prentice-Hall.
• Kucera V. and Zagalak P. (1999): Proper solutions of polynomial equations. - Prep. 14th IFAC World Congress, Beijing, Vol. D, pp. 357-362.
• Messner W. and Tilbury D. (1999): Example: DC Motor Speed Modeling, In: Control Tutorials for MATLAB and Simulink: A Web-Based Approach (W. Messner and D. Tilbury, Eds.). - Englewood Cliffs, N.J.: Prentice-Hall, Available at www.engin.umich.edu/group/ctm/examples/motor/motor.html
• ProTyS, Inc. (2003): Personal communication.
• Rosenbrock H.H. and Hayton G.E. (1978): The general problem of pole assignment. - Int. J. Contr., Vol. 27, No. 6, pp. 837-852.
• Vidyasagar M. (1985): Control System Synthesis. -Cambridge MA: MIT Press.
• Wolovich W.A. (1974): Linear Multivariable Systems. - New York: Springer.
• Zagalak P. and Kucera V. (1985): The general problem of pole assignment. - IEEE Trans. Automat. Contr., Vol. 30, No. 3, pp. 286-289.