A polynomial solution to regulation and tracking. I: Deterministic problem. (English) Zbl 0542.93013

Summary: Recent results on polynomial techniques in solving the discrete-time linear-quadratic regulation and/or tracking problems are presented. Both deterministic and stochastic problems are considered in order to let appear their formal similarity and to contrast the inherent differences. The analysis is based on external polynomial models and the construction of the optimal controller or control sequence is reduced to the solution of linear polynomial equations, combined with spectral factorization. The existence of admissible controls that yield a finite performance index is studied and all such controls are specified in a parametric form. The optimal control then corresponds to the zero parameter and is shown to be recurrent, i.e. realizable by a linear finite dimensional system.


93B25 Algebraic methods
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
93B15 Realizations from input-output data
12E12 Equations in general fields
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[1] K. J. Åström: Introduction to Stochastic Control Theory. Academic Press, New York 1970.
[2] T. Kailath: Linear Systems. Prentice-Hall, Englewood Cliffs 1980. · Zbl 0454.93001
[3] V. Kučera: State space approach to discrete linear control. Kybernetika 8 (1972), 233 - 251.
[4] V. Kučera: Algebraic Theory of Discrete Linear Control. (in Czech). Academia, Praha 1978.
[5] V. Kučera: Discrete Linear Control - The Polynomial Equation Approach. Wiley, Chichester 1979.
[6] V. Kučera: Discrete stochastic regulation and tracking. Kybernetika 16 (1980), 263 - 272.
[7] V. Kučera: Linear quadratic control - State space vs. polynomial equations. Kybernetika 19 (1983), 185-195. · Zbl 0549.93030
[8] V. Peterka: On steady state minimum variance control strategy. Kybernetika 8 (1972), 219-232. · Zbl 0256.93070
[9] V. Peterka: Predictor-based self-tuning control. Automatica 20 (1984), 39-50. · Zbl 0539.93054
[10] M. Šebek: Optimal tracking via polynomial matrix equations. Internat. J. Systems Sci. 12 (1981), 357-369. · Zbl 0458.93040
[11] M. Šebek: Polynomial design of stochastic tracking systems. IEEE Trans. Automat. Control AC-27 (1982), 468-470. · Zbl 0488.93066
[12] M. Šebek, V. Kučera: Polynomial approach to quadratic tracking in discrete linear systems. IEEE Trans. Automat. Control AC-27 (1982), 1248-1250.
[13] L. N. Volgin: The Fundamentals of the Theory of Controlling Machines. (in Russian). Soviet Radio, Moscow 1962.
[14] W. A. Wolovich: Linear Multivariable Systems. Springer-Verlag, New York 1974. · Zbl 0291.93002
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