# zbMATH — the first resource for mathematics

Structure at infinity, model matching and disturbance rejection for linear systems with delays. (English) Zbl 0805.93008
The structur at infinity is studied for rational transfer matrices with two unknows $$T(s,v)$$, where $$s$$ is the classical Laplace variable, and $$v= e^ s$$ is the unitary shift operator; $$v^{-1}$$ is interpreted as the unit delay. Denote by $$D$$ the set of all such matrices. Using the Smith-Macmillan form at infinity for $$T\in D$$, the following Model Matching Problem is solved: Given proper matrices $$T\in D$$, $$T_ m\in D$$, determine if there is a proper $$T_ c\in D$$ that solves the equation $$T\cdot T_ c= T_ m$$. Consider now the class of systems with one delay and disturbance vector $$d(t)$$: $x'(t)= A_ 0 x(t)+ A_ 1 x(t-1)+ Bu(t)+ Ed(t);\quad y(t)= Cx(t).\tag{1}$ The subscripts “$$m$$” and “$$c$$” stand for “model” and “compensator”, respectively. The Disturbance Rejection Problem with Dynamic Precompensator for systems (1) reads as follows: Does there exists a proper (or strictly proper) precompensator $$u= T_ c d$$, where $$T_ c\in D$$, that rejects the effect of $$d(t)$$ on the output $$y(t)$$? The solution is given in terms of the supremal frequency invariant subspaces. This notion is well-known for the linear time invariant systems, and it is suitably generalized in the reviewed paper for the delay systems (1). The structure at infinity is also used to obtain a criterion for the partial disturbance rejection property of the system (1); namely, there exists a nonanticipative state feedback law $$u(t)$$ that depends linearly on $x(t),x(t- 1),\dots, x(t-k),\;d(t),\dots,d(t- k)$ and is such that the output $$y(t)$$ of the compensated system is not affected by $$d(t)$$ over the interval $$[0,k+ 1]$$.

##### MSC:
 93B17 Transformations 93C05 Linear systems in control theory
Full Text:
##### References:
 [1] G. Basile, G. Marro: Controlled and Conditioned Invariants in Linear System Theory. Prentice Hall, N. J. 1992. · Zbl 0758.93002 [2] C. Commault J. F. Lafay, M. Malabre: Structure on linear systems: Geometric and transfer matrix approaches. Kybernetika 27(1991), 170-185. · Zbl 0746.93036 · www.kybernetika.cz · eudml:28967 [3] R. Curtain: Invariance concepts in infinite dimensions. SIAM J. Control Optim. 24 (1986), 1009-1031. · Zbl 0602.93037 · doi:10.1137/0324059 [4] J. Descusse, J. M. Dion: On the structure at infinity of linear square decoupled systems. IEEE Trans. Automat. Control AC-27 (1982), 4, 971-974. · Zbl 0485.93042 · doi:10.1109/TAC.1982.1103041 [5] E. Emre, M. L. J. Hautus: A polynomial characterization of (A, B) invariant and reachability subspaces. SIAM J. Control Optim. 18 (1980), 4, 402-436. · Zbl 0452.93011 · doi:10.1137/0318031 [6] E. Emre, L. M. Silverman: Partial model matching of linear systems. IEEE Trans. Automat. Control AC-25 (1980), 4, 280-281. · Zbl 0432.93026 · doi:10.1109/TAC.1980.1102321 [7] G. Lebret: Contribution a l’etude des systemes lineaires generalises: approches geometrique et structurelle. These de Doctorat, Universite de Nantes et Ecole Centrale de Nantes, Nantes 1991. [8] M. Malabre: Structure a Pinfini des triplets invariants. Analysis and Optimization of Systems - Proceedings of the Fifth International Conference on Analysis and Optimization of Systems, Versailles, December 1982 (A. Bensoussan and J.L. Lions, Lecture Note in Control and Information Sciences 44), Springer-Verlag, Berlin 1982, pp. 43-54. [9] M. Malabre, V. Kučera: Infinite structure and exact model matching problem: polynomial and geometric approaches. Rapport Interne L.A.N. No. 01.83, 1983. · Zbl 0534.93014 [10] M. Malabre, V. Kučera: Infinite structure and exact model matching problem: a geometric approach. IEEE Trans. Automat. Control AC-29 (1984), 3, 266-268. · Zbl 0534.93014 · doi:10.1109/TAC.1984.1103502 [11] M. Malabre, R. Rabah: On infinite zeros for infinite dimensional systems. Progress in Systems and Control Theory 3, Realization and Modelling in System Theory (Kaashoek, Van Schuppen and Ran, Vol. 1, Birhauser, Boston, pp. 199-206. · Zbl 0726.93041 [12] C. Moog: Inversion, decouplage, poursuite de modele des systemes non lineaires. These de Doctorat es Sciences, ENSM, Nantes 1987. [13] M. Morf B.C. Levy, S. Y. Kung: New results in 2-D systems theory. Part 1: 2-D polynomial matrices, factorization, and coprimeness. Proc. IEEE 65 (1977), 6, 861-872. [14] A. W. Olbrot: Algebraic criteria of controllability to zero function for linear constant time-lag systems. Control Cybernet. 2 (1973), 59-77. · Zbl 0332.93011 [15] L. Pandolfi: Disturbance decoupling and invariant subspaces for delay systems. Appl. Math. Optim. 14 (1986), 55-72. · Zbl 0587.93039 · doi:10.1007/BF01442228 [16] A. M. Perdon, G. Conte: The disturbance decoupling problem for systems over a principal ideal domain. Proc. New Trends in Systems Theory, Genova, Progress in Systems and Control Theory 7 (1991), Birkhauser, Boston, pp. 583-590. · Zbl 0736.93013 [17] L. M. Silverman, A. Kitapci: System structure at infinity. Colloque National CNRS, Developpement et Utilisation d’Outils et Modeles Mathematiques en Automatique, Analyse des Systemes et Traitement du Signal, Belle-Ile 1982, (CNRS, 3 (1983), pp. 413-424. · Zbl 0529.93018 [18] A. C. Tsoi: Recent advances in the algebraic system theory of delay-differential equations. Recent Theoretical Developments in Control (M. J. Gregson, Chapter 5, Academic Press, New York 1987, pp. 67-127. [19] A. I. G. Vardulakis: Linear Multivariate Control: Algebraic Analysis and Synthesis Methods. Wiley, New York. · Zbl 0751.93002 [20] W. M. Wonham: Linear Multivariable Control: a Geometric Approach. Second edition. Springer-Verlag, New York 1979. · Zbl 0424.93001 [21] H. J. Zwart: Geometric theory for infinite dimensional systems. (Lecture Notes in Control and Information Sciences 115.) Springer-Verlag, Berlin 1989. · Zbl 0667.93063
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.