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On the well-posedness in the solution of the disturbance decoupling by dynamic output feedback with self bounded and self hidden subspaces. (English) Zbl 1429.93063

Summary: This paper studies the disturbance decoupling problem by dynamic output feedback with required closed-loop stability, in the general case of nonstrictly-proper systems. We will show that the extension of the geometric solution based on the ideas of self boundedness and self hiddenness presents structural differences with respect to the strictly proper case. The most crucial aspect that emerges in the general case is the issue of the well-posedness of the feedback interconnection, which obviously has no counterpart in the strictly proper case. A fundamental property of the feedback interconnection that has so far remained unnoticed in the literature is investigated in this paper: the well-posedness condition is decoupled from the remaining solvability conditions. An important consequence of this fact is that the well-posedness condition written with respect to the supremal output nulling and infimal input containing subspaces does not need to be modified when we consider the solvability conditions of the problem with internal stability (where one would expect the well-posedness condition to be expressed in terms of supremal stabilizability and infimal detectability subspaces), and also when we consider the solution which uses the dual lattice structures of G. Basile and G. Marro [J. Optim. Theory Appl. 3, 306–315 (1969; Zbl 0172.12501)].

MSC:

93B27 Geometric methods
93B52 Feedback control
93D15 Stabilization of systems by feedback

Citations:

Zbl 0172.12501
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References:

[1] Basile, G.; Marro, G., Controlled and conditioned invariant subspaces in linear system theory, Journal of Optimization Theory and Applications, 3, 5, 306-315 (1969) · Zbl 0172.12501
[2] Basile, G.; Marro, G., Self-bounded controlled invariant subspaces: a straightforward approach to constrained controllability, Journal of Optimization Theory and Applications, 38, 1, 71-81 (1982) · Zbl 0471.93008
[3] Basile, G.; Marro, G., Controlled and conditioned invariants in linear system theory (1992), Prentice Hall: Prentice Hall New Jersey · Zbl 0758.93002
[4] Basile, G.; Marro, G.; Piazzi, A., Revisiting the regulator problem in the geometric approach. Part I. Disturbance localization by dynamic compensation, Journal of Optimization Theory and Applications, 53, 1, 9-22 (1987) · Zbl 0594.93051
[5] Del-Muro-Cuéllar, B., Commande des systèmes lineaires: rejet de perturbations et pôles fixes (1997), Ecole Centrale, Univ. de Nantes: Ecole Centrale, Univ. de Nantes France, (Thèse de Doctorat)
[6] Del-Muro-Cuéllar, B.; Malabre, M., Fixed poles of disturbance rejection by dynamic measurement feedback: a geometric approach, Automatica, 37, 2, 231-238 (2001) · Zbl 0978.93018
[7] Del-Muro-Cuéllar, B.; Malabre, M., Fixed poles for the disturbance rejection by measurement feedback: the case without any controllability assumption, (Proc. of the 2003 European control conference (2003)), 1-4
[8] Imai, H.; Akashi, H., Disturbance localization and pole shifting by dynamic compensation, IEEE Transactions on Automatic Control, AC-26, 1, 226-235 (1981) · Zbl 0464.93045
[9] Malabre, M.; Martinez-Garcia, J. C.; Del-Muro-Cuéllar, B., On the fixed poles for disturbance rejection, Automatica, 33, 6, 1209-1211 (1997) · Zbl 0879.93007
[10] Ntogramatzidis, L., Self-bounded subspaces for non strictly proper systems and their application to the disturbance decoupling problem, IEEE Transactions on Automatic Control, 53, 1, 423-428 (2008) · Zbl 1367.93262
[11] Schumacher, J. M.H., Compensator synthesis using \((C, A, B)\)-pairs, IEEE Transactions on Automatic Control, AC-25, 6, 1133-1138 (1980) · Zbl 0483.93035
[12] Stoorvogel, A. A.; van der Woude, J. W., The disturbance decoupling problem with measurement feedback and stability for systems with direct feedthrough matrices, Systems & Control Letters, 17, 217-226 (1991) · Zbl 0762.93016
[13] Trentelman, H. L.; Stoorvogel, A. A.; Hautus, M., Control theory for linear systems (2001), Springer · Zbl 0963.93004
[14] Willems, J. C.; Commault, C., Disturbance decoupling with measurement feedback with stability or pole placement, SIAM Journal on Control and Optimization, 19, 4, 490-504 (1981) · Zbl 0467.93036
[15] Wonham, W. M.; Morse, A. S., Decoupling and pole assignment in linear multivariable systems: a geometric approach, SIAM Journal of Control, 8, 1, 1-18 (1970) · Zbl 0206.16404
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