×

Stochastic multivariable tracking: A polynomial equation approach. (English) Zbl 0531.93069

The paper presents a general solution to the following stochastic tracking problem. Given the following input output equation for the plant \(y(s)=R(s)u(s)+S(s)w(s)\) and the reference \(r(s)=\bar s(s)\bar w(s),\) where R, S and \(\bar S\) are rational matrices (not necessarily proper) and w and \(\bar w\) independent stationary white noises with specified covariances (not necessarily positive definite), find the optimal controller of the following form \(u(s)=-P(s)(y(s)+v(s))+Q(s)(r(s)+\bar v(s))\) (where v and \(\bar v\) are independent measurement noises), such that a weighted sum of steady state covariances of the input and the tracking error is minimized, leaving the closed-loop system asymptotically stable. The design procedure allows to obtain also necessary and sufficient condition for the existence of the solution and it is based on polynomial matrices techniques. Such techniques allows to handle the problem in such a generality, but the optimal P and Q are not restricted to be proper, so that in general the optimal controller is not realizable. The procedure generalizes a construction made by the same author in the SISO case [IEEE Trans. Autom. Control AC-27, 468-470 (1982; Zbl 0488.93066)] and essentially follows the steps of the paper by V. Kučera [ibid. AC-25, 913-919 (1980; Zbl 0447.93031)] about regulation problems. Three spectral factorization and the solution of two linear equations in polynomial matrices are required. Moreover, the author shows that the feedback part P is independent of the reference, whose influence is restricted only to the feedforward part Q.
Reviewer: M.Piccioni

MSC:

93E20 Optimal stochastic control
12E12 Equations in general fields
93B50 Synthesis problems
93C35 Multivariable systems, multidimensional control systems
12D05 Polynomials in real and complex fields: factorization
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] E. Emre: The polynomial equation \(QQ_C + RP_C = \Phi\) with application to dynamic feedback. SIAM J. Control Optim. 18 (1980), 6, 611-620. · Zbl 0505.93016
[2] E. Emre, L. M. Silvermann: The equation \(XR + QY = \Phi\): A characterization of solutions. SIAM J. Control Optim. 19 (1981), 1, 33-38. · Zbl 0466.93017
[3] M. J. Grimble: Design of stochastic optimal feedback control systems. Proc. IEEE 125 (1978), II, 1275-1284.
[4] T. Kailath: Linear Systems. Prentice-Hall, Englewood Cliffs, N. J. 1980. · Zbl 0454.93001
[5] V. Kučera: Discrete Linear Control - The Polynomial Equation Approach. Wiley, Chichester 1979.
[6] V. Kučera: Stochastic multivariable control: A polynomial equation approach. IEEE Trans. Automat. Control AC-25 (1980), 5, 913-919.
[7] H. Kwakernaak, R. Sivan: Linear Optimal Control Systems. Wiley, New York 1972. · Zbl 0276.93001
[8] M. Šebek: Polynomial design of stochastic tracking systems. IEEE Trans. Automat. Control AC-27 0982), 2, 468-470. · Zbl 0488.93066
[9] M. Šebek: Direct polynomial approach to discrete-time stochastic tracking. Problems Control Inform. Theory 12 (1983), 4, 293-300. · Zbl 0517.93067
[10] Z. Vostrý: New algorithm for polynomial spectral factorization with quadratic convergence. Kybernetika 12 (1976), 4, 248-259. · Zbl 0353.65029
[11] W. A. Wolovich: Linear Multivariable Systems. Springer-Verlag, New York 1974. · Zbl 0291.93002
[12] D. C. Youla J. J. Bongiorno, H. A. Jabr: Modern Wiener-Hopf design of optimal controllers. II: The multivariable case. IEEE Trans. Automat. Control AC-21 (1976), 3, 319-338. · Zbl 0339.93035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.