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Generalized linear systems: Geometric and structural approaches. (English) Zbl 0679.93048

Summary: This paper presents a new way of introducing invariant subspaces for generalized systems. This comes from somewhat known geometric algorithms but, in most cases, with initial conditions different from those usually seen in this context. These definitions are shown to be consistent with other ones directly deduced from “classical” approaches like matrix pencil tools or discrete algebraic formalism.
The key starting point for their introduction is the extension of structural descriptions for these generalized systems such as have been available for proper systems. Many results from the proper case can thus be extended, for instance the geometric definition for controllability indices and something similar to the famous Morse canonical decomposition of the strictly proper case. Some links with the inversion algorithm are also sketched.

MSC:

93C05 Linear systems in control theory
93B10 Canonical structure
93B27 Geometric methods
93C15 Control/observation systems governed by ordinary differential equations
34A99 General theory for ordinary differential equations
Full Text: DOI

References:

[1] Gantmacher, G., The Theory of Matrices (1959), Chelsea, New York
[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] Rosenbrock, H. H., State Space and Multivariable Theory (1970), Nelson Wiley: Nelson Wiley London · Zbl 0246.93010
[4] Morse, A. S., Structural invariants of linear multivariable systems, SIAM J. Control Optim., 11, 3, 446-465 (1973) · Zbl 0259.93011
[5] Thorp, J. S., The singular pencil of a linear dynamical system, Internat. J. Control, 18, 3, 577-596 (1973) · Zbl 0262.93020
[6] Wong, K. T., The eigenvalue problem λTx+Sx, J. Differential Equations, 16, 270-281 (1974) · Zbl 0327.15015
[7] Wonham, W. M., Linear Multivariable Control: A Geometric Approach (1979), Springer-Verlag: Springer-Verlag Paris · Zbl 0393.93024
[8] Verghese, G. C., Further notes on singular descriptions, (JACC (1981), Charlottesville), presented at TA4
[9] Part 2: Almost conditionally invariant subspaces, IEEE Trans. Automat. Control, AC-27, 1071-1085 (1982) · Zbl 0491.93022
[10] Bernhard, P., On singular implicit dynamical systems, SIAM J. Control Optim., 20, 612-633 (1982) · Zbl 0491.93004
[11] Commault, C.; Dion, J. M., Structure at infinity of linear multivariable systems: A geometric approach, IEEE Trans. Automat. Control, AC-27, 3, 693-696 (1982) · Zbl 0485.93041
[12] Malabre, M., Almost invariant subspaces, transmission and infinite zeros: A lattice interpretation, Systems Control Lett., 1, 6, 347-355 (1982) · Zbl 0483.93038
[13] Schumacher, J. M., Algebraic characterization of almost invariance, Internat. J. Control, 38, 1, 107-124 (1983) · Zbl 0516.93013
[14] Van der Weiden, A. J.J., The Use of Structural Properties in Linear Multivariable Control System Design, (Ph.D. Thesis (1983), Delft Univ. of Technology) · Zbl 0366.93014
[15] Silverman, L. M.; Kitapci, A., Systems structure at infinity, Systems Control Lett., 3, 123-131 (1983) · Zbl 0529.93018
[16] Lewis, F. L., Inversion of descriptor systems, Proceedings of A.C.C., 1153-1158 (June 1983), San Francisco
[17] Malabre, M.; Kučera, V., Infinite structure and exact model matching problem: A geometric approach, IEEE Trans. Automat. Control, Ac-29, 3, 266-268 (1984) · Zbl 0534.93014
[18] Aplevich, J. D., Minimal representations of implicit linear systems, Automatica, 21, 3, 259-269 (1985) · Zbl 0565.93012
[19] Ozcaldiran, K., Control of descriptor systems, (Ph.D. Thesis (May 1985), Georgia Inst. of Tech) · Zbl 0606.93017
[20] Loiseau, J. J., Some geometric considerations about the Kronecker normal form, Internat. J. Control, 42, 6, 1411-1431 (1985) · Zbl 0609.93014
[21] Armentano, V. A., The pencil \(( sE -A)\) and controllability-observability for generalized linear systems: A geometric approach, SIAM J. Control Optim., 24, 4, 616-638 (1986) · Zbl 0606.93014
[22] Descusse, J.; Lafay, J. F.; Malabre, M., A survey on Morgan’s problem, 25th I.E.E.E. C.D.C., 1289-1294 (10-12 Dec. 1986), Athens
[23] Banaszuk, A.; Kociecki, M.; Przyluski, K. M., On implicit linear discrete-time systems, (Math. Control Signals Systems (May 1987), Inst. of Mathematics, Polish Academy of Sciences), to appear. · Zbl 0715.93038
[24] Lewis, F. L.; Beauchamp, G., Computation of subspaces for singular systems, MTNS’87 (June 1987), Phoenix
[25] Kučera, V.; Zagalak, P., Towards a fundamental theorem of state feedback for singular systems, 10th World Congress on Automatic Control, IFAC’87 (27-31 July 1987), Munich
[26] Malabre, M., Geometric characterization of “complete controllability indexes” for singular systems, Systems Control Lett., 9, 323-327 (1987) · Zbl 0636.93020
[27] Karcanias, N.; Kalogeropoulous, G., A matrix pencil approach to the study of singular systems: Algebraic and geometric aspects, International Minisymposium on Singular Systems (4-6 Dec. 1987), Atlanta
[28] Malabre, M., A structural approach for linear singular systems, International Minisymposium on Singular Systems (4-6 Dec. 1987), Atlanta
[29] Malabre, M., More geometry about singular systems, 26th IEEE C.D.C. (9-11 Dec. 1987), Los Angeles
[30] Lebret, G., Approche géométrique des systémes singuliers, (D.E.A. Automatique (29 Sept. 1987), E.N.S.M: E.N.S.M Nantes), presented at
[31] Malabre, M., Geometric algorithms and structural invariants for linear singular systems, 12th IMACS World Congress (July 1988), Paris
[32] Moog, C. H., Nonlinear decoupling and structure at infinity, Math. Control Signals Systems, 1, 3, 257-268 (1988) · Zbl 0657.93027
[33] K. Ozcaldiran and F.L. Lewis, Generalized reachability subspaces for singular systems, SIAM J. Control Optim.; K. Ozcaldiran and F.L. Lewis, Generalized reachability subspaces for singular systems, SIAM J. Control Optim. · Zbl 0689.93004
[34] Loiseau, J. J.; Ozcaldiran, K.; Malabre, M.; Karcanias, N., A feedback classification of singular systems, IFAC Workshop on “System Structure and Control: State Space and Polynomial Methods,”, 25-27 (Sept. 1989), Prague
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