×

Generalized invariant subspaces for infinite-dimensional systems. (English) Zbl 0978.93017

The notion of invariant subspaces has been used successfully to study various control problems for linear finite-dimensional systems by G. Basile and G. Marro [ J. Optim. Theory Appl. 3, 306-315 (1969; Zbl 0172.12501)] and W. M. Wonham [“Linear multivariable control: A geometric approach”, Springer-Verlag, Berlin, New York (1985; Zbl 0609.93001)]. This notion has been extended to systems defined in Hilbert spaces, and some of the corresponding problems in Hilbert spaces have also been studied by some authors. After that, the simultaneous versions of these invariant subspaces for both finite- and infinite-dimensional linear systems have also been studied by various authors in order to investigate uncertain linear systems whose systems operators are represented as convex combinations of given operators, and then parameter-insensitive disturbance-rejection problems have also been studied.
Here, the authors study the infinite-dimensional versions of generalized invariant subspaces for infinite-dimensional systems. Further, they formulate the parameter-insensitive disturbance-rejection problems with state feedback and with static output feedback for infinite-dimensional systems and then they give some solvability conditions for these problems. Finally, they consider an illustrative example.

MSC:

93B27 Geometric methods
93C25 Control/observation systems in abstract spaces
93C73 Perturbations in control/observation systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balakrishnan, A. V., Applied Functional Analysis (1981), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0459.46014
[2] Basile, G.; Marro, G., Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl., 3, 306-315 (1969) · Zbl 0172.12501
[3] Bhattacharyya, S. P., Generalized controllability, \((A,B)\)-invariant subspaces and parameter invariant control, SIAM J. Algebraic Discrete Meth., 4, 529-533 (1983) · Zbl 0528.93013
[4] Curtain, R. F., \((C,A,B)\)-pairs in infinite dimensions, Syst. Control Lett., 5, 59-65 (1984) · Zbl 0553.93037
[5] Curtain, R. F., Invariance concepts in infinite dimensions, SIAM J. Control Optim., 24, 1009-1030 (1986) · Zbl 0602.93037
[6] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems. Infinite Dimensional Linear Systems, Lecture Notes in Control and Information Sciences (1978), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0389.93001
[7] Ghosh, B. K., A geometric approach to simultaneous system design: Parameter insensitive disturbance decoupling by state and output feedback, (Byrnes, C. I.; Lindquist, A., Modeling, Identification and Robust Control (1986), North-Holland: North-Holland Amsterdam), 476-484 · Zbl 0607.93015
[8] Otsuka, N., Simultaneous decoupling and disturbance-rejection problems for infinite-dimensional systems, IMA J. Math. Control Inf., 8, 165-178 (1991) · Zbl 0759.93019
[9] Otsuka, N., Generalized invariant subspaces and parameter insensitive disturbance-rejection problems with static output feedback, IEEE Trans. Automat. Control, 45 (2000) · Zbl 0990.93020
[10] Otsuka, N., Generalized \((C,A,B)\)-pairs and parameter insensitive disturbance-rejection problems with dynamic compensator, IEEE Trans. Automat. Control, 44, 2195-2200 (1999) · Zbl 0958.93024
[11] Otsuka, N.; Inaba, H., Parameter-insensitive disturbance-rejection for infinite-dimensional systems, IMA J. Math. Control Inf., 14, 401-413 (1997) · Zbl 0897.93035
[12] Otsuka, N.; Inaba, H., A note on robust disturbance-rejection problems for infinite-dimensional systems, Syst. Control Lett., 34, 33-41 (1998) · Zbl 0902.93013
[13] Otsuka, N.; Inaba, H., Simultaneous \((C,A,B)\)-pairs for infinite-dimensional systems, J. Math. Anal. Appl., 236, 415-437 (1999) · Zbl 0952.93016
[14] Otsuka, N.; Inaba, H.; Oide, T., Decoupling by state feedback in infinite-dimensional systems, IMA J. Math. Control Inf., 7, 125-141 (1990) · Zbl 0713.93014
[15] Otsuka, N., Decoupling by incomplete-state feedback for infinite dimensional systems, Japan J. Industrial Appl. Math., 11, 363-377 (1994) · Zbl 0811.93028
[16] Otsuka, N., Parameter insensitive disturbance-rejection problem with incomplete-state feedback, IEICE Trans. Fund. Electron. Comm. Computer Sci., E78-A, 1589-1594 (1995)
[17] Wonham, W. M., Linear Multivariable Control: A Geometric Approach (1984), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0556.93045
[18] Zwart, H., Geometric Theory for Infinite-Dimensional Systems. Geometric Theory for Infinite-Dimensional Systems, Lecture Notes in Control and Information Sciences (1989), Springer-Verlag: Springer-Verlag Berlin/New York · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.