×

zbMATH — the first resource for mathematics

On almost invariant subspaces for implicit linear discrete-time systems. (English) Zbl 0666.93052
The concept of almost invariant subspace for an implicit linear discrete- time system is introduced and studied in detail. It is shown also that for regular homogeneous implicit systems the so-called deflating subspaces can be identified with almost invariant subspaces.

MSC:
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
47A15 Invariant subspaces of linear operators
34A99 General theory for ordinary differential equations
39A12 Discrete version of topics in analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Banaszuk, A.; Kociȩcki, M.; Przyłuski, K.M., On implicit linear discrete-time systems, Preprint 397 IM PAN, (1987), to be published in Mathematics of Control, Signals and Systems · Zbl 0715.93038
[2] Banaszuk, A.; Kociȩcki, M.; Przyłuski, K.M., Remarks on controllability of implicit linear discrete-time systems, Systems control lett., 10, 67-70, (1988) · Zbl 0628.93006
[3] Bart, H.; Bart, H., Meromorphic operator valued functions, (), Mathematical center tracts no. 44, (1973) · Zbl 0288.47022
[4] D.J. Bender, Generalized output-nulling subspaces recursions for generalized linear systems, submitted.
[5] Birkhoff, G.D., Lattice theory, (1967), AMS Providence, RI
[6] Cohn, P.M., ()
[7] Dieudonné, J., Sur la reduction canonique des couples des matrices, Bull. soc. math. France, 74, 130-146, (1946) · Zbl 0061.01307
[8] Fletcher, L.R.; Aasaraai, A., On disturbance decoupling in descriptor systems, (), Paris, Organized by INRIA · Zbl 0693.93016
[9] Jaffe, S.; Karcanias, N., Matrix pencil characterization of almost (A, B)-invariant subspaces: A classification of geometric concepts, Internat. J. control, 33, 51-93, (1981) · Zbl 0552.93018
[10] Kato, T., Perturbation theory for nullity, deficiency, and other quantities of linear operators, J. analyse math., 6, 261-322, (1958) · Zbl 0090.09003
[11] Lewis, F.L., Descriptor systems: decomposition into forward and backward subsystems, IEEE trans. automat. control, 29, 167-170, (1984) · Zbl 0534.93013
[12] Malabre, M., More geometry about singular systems, rapport interne LAN-ENSM no. 1.87, (), 1138-1139, See also
[13] Özçaldiran, K., Control of descriptor systems, () · Zbl 0606.93017
[14] Özçaldiran, K., A geometric characterization of the reachable and the controllable subspace of descriptor systems, Circuits systems signal process, 5, 37-48, (1986) · Zbl 0606.93017
[15] Özçaldiran, K., Geometric notes on descriptor systems, (), 1134-1137
[16] Schumacher, J.M., Algebraic characterization of almost invariance, Internat. J. control, 38, 107-124, (1983) · Zbl 0516.93013
[17] Shayman, M.A.; Zhou, Z., Feedback control and classification of generalized linear systems, IEEE trans. automatic control, 32, 483-494, (1987) · Zbl 0624.93028
[18] Stewart, G.W., On the sensitivity of the eigenvalue problem ax = λbx, SIAM J. numer. anal., 9, 669-686, (1972) · Zbl 0252.65026
[19] Tarski, A., A lattice theoretical fixpoint theorem and its applications, Pacific J. math., 5, 285-309, (1955) · Zbl 0064.26004
[20] Trentelman, H.L., Almost invariant subspaces and high gain feedback, (1986), CWI Amsterdam · Zbl 0605.93003
[21] Van Dooren, P., The generalized eigenstructure problem in linear system theory, IEEE trans. automat. control, 26, 111-129, (1981) · Zbl 0462.93013
[22] Willems, J.C., Almost A (mod B)-invariant subspaces, Astérisque, 75-76, 239-248, (1980) · Zbl 0459.93029
[23] Willems, J.C., Almost invariant subspaces: an approach to high gain feedback design - part I: almost controlled invariant subspaces, IEEE trans. automat. control, 26, 235-252, (1981) · Zbl 0463.93020
[24] Willems, J.C., Almost invariant subspaces: an approach to high gain feedback design - part II: almost conditionally invariant subspaces, IEEE trans. automat. control, 27, 1071-1084, (1982) · Zbl 0491.93022
[25] Willems, J.C., Feedforward control, PID control laws, and almost invariant subspaces, Systems control lett., 1, 277-282, (1982) · Zbl 0473.93032
[26] Wong, K.T., The eigenvalue problem λtx + sx, J. differential equations, 16, 270-280, (1974) · Zbl 0327.15015
[27] Wonham, W.M., Linear multivariable control: A geometric approach, (1979), Springer-Verlag New York · Zbl 0393.93024
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.