zbMATH — the first resource for mathematics

External reachability (reachability with pole assignment by p. d. feedback) for implicit descriptions. (English) Zbl 0802.93025
Summary: Even for flat implicit linear systems (i.e. having more state components than state equations), reachability is a well defined concept in terms of the set of state trajectories: it characterizes the property that from any initial state can start a smooth state trajectory which reaches any final state. What can happen, however, for this general class of systems, is that a system with no control input can be completely reachable. We introduce here the notion of “external reachability” which expresses the fact that trajectories can actually be controlled through the input (by proportional and derivative state feedback). Geometric necessary and sufficient conditions are given for external reachability. A new design method is proposed for pole assignment which uses this concept and relies upon right inversion techniques.

93B55 Pole and zero placement problems
93B03 Attainable sets, reachability
93B51 Design techniques (robust design, computer-aided design, etc.)
Full Text: Link EuDML
[1] A. Banaszuk M. Kociecki, K. M. Przyluski: Remarks on the theory of implicit linear discrete-time systems. Proc. Internat. Symp. on Singular Systems, Atlanta 1987, pp. 44-47.
[2] P. Bernhard: On singular implicit linear dynamical systems. SIAM J. Control Optim. 20 (1982), 5, 612-633. · Zbl 0491.93004 · doi:10.1137/0320046
[3] M. Bonilla: Etude de proprietes structurelles des systemes singuliers vues en particulier a partir de l’algorithme d’inversion. These de Doctorat, Universite de Nantes et E.N.S.M., Nantes 1991.
[4] M. Bonilla M. Fonseca, M. Malabre: Implementing non proper control laws for proper systems. Proc. Internat. Symp. on Implicit and Nonlinear Systems, SINS’92 (F. Lewis, Ft. Worth 1992, pp. 163-169.
[5] M. Bonilla G. Lebret, M. Malabre: Some robust control via implicit descriptions. Proc. 2nd Internat. Symp. on Implicit and Robust Systems, ISIRS’92, Warsaw 1991, pp. 37-40.
[6] M. Bonilla, M. Malabre: One side invertibility for implicit descriptions. Proc. 29th IEEE-CDC, Vol. 6, Hawaii 1990, pp. 3601-3602.
[7] M. Bonilla, M. Malabre: Variable structure systems via implicit descriptions. Proc. 1st European Control Conference, Vol. 1, Grenoble, Hermes 1991, pp. 403-408.
[8] M. Bonilla, M. Malabre: Redundant and non observable spaces Iot implicit descriptions. Proc. 30th IEEE-CDC, Vol. 2, Brighton 1991, pp. 1425-1430.
[9] M. Bonilla, M. Malabre: Geometric characterization of Lewis’ Structure Algorithm. Proc. 2nd Int. Symp. on Implicit and Robust Systems, ISIRS’91, Warsaw 1991, pp. 41-46.
[10] M. Bonilla, M. Malabre: External minimality for implicit description. Internal Report, LAN-ENSM, No. 90.02, 1990.
[11] M. Bonilla M. Malabre, J. A. Cheang Wong: On the use of derivators (and approximations) for solving basic control problems. Submitted.
[12] D. Cobb: Controllability, observability and duality in singular systems. IEEE Trans. Automat. Control AC-29 (1984), 12, 1076-1082.
[13] H. Frankowska: On controllability and observability of implicit systems. Systems Control Lett. 14 (1990), 219-225. · Zbl 0699.93003 · doi:10.1016/0167-6911(90)90016-N
[14] F. R. Gantmacher: The Theory of Matrices. Vol. II. Chelsea, New York 1959. · Zbl 0085.01001
[15] J. Grimm: Application de la theorie des systemes implicites a l’inversion des systemes. Analysis and Optimization of Systems: Proceedings of the Sixth Internatational Conference on Analysis and Optimization of Systems (Lecture Notes in Control and Information Sciences 63), Springer-Verlag, Berlin 1984, pp. 142-156. · Zbl 0556.93026
[16] S. Jaffe, N. Karcanias: Matrix pencil characterization of almost (A, B)-invariant subspaces: A classification of geometric concepts. Internat. J. Control 33 (1981), 1, 51-93. · Zbl 0552.93018 · doi:10.1080/00207178108922907
[17] M. Kuijper: Descriptor representations without direct feedthrough term. CWI, BS-R9103, Amsterdam 1991. · Zbl 0766.93026
[18] M. Kuijper, J. M. Schumacher: Realization of autoregressive equations in pencil and descriptor form. SIAM J. Control Optim. 28 (1990), 5, 1162-1189. · Zbl 0721.93016 · doi:10.1137/0328063
[19] G. Lebret: Contribution a l’etude des systemes lineaires generalises: approches geometrique et structurelle. These de Doctorat, Univ. de Nantes et E.C.N, Nantes 1991.
[20] F. Lewis: A tutorial on the properties of linear time-invariant singular systems. Presented at the IFAC Workshop on System Structure and Control, State Space and Polynomial Methods, Prague 1989. Automatica, to appear.
[21] F. Lewis, G. Beauchamp: Computation of subspaces for singular systems. Proc. MTNS’87, Phoenix 1987.
[22] J. J. Loiseau K. Ozcaldiran M. Malabre, N. Karcanias: Feedback canonical forms of singular systems. Kybernetika 27(1991), 4, 289-305. · Zbl 0767.93007
[23] M. Malabre: Generalized linear systems: geometric and structural approaches. Linear Algebra Appl. (1980), 122/123/124, 591-621.
[24] M. Malabre: On infinite zeros for generalized systems. Proc. MTNS’89, in PSCT3, Birkhauser, Boston, pp. 271-278. · Zbl 0726.93036
[25] K. Ozcaldiran: A geometric characterization of the reachable and controllable sub-spaces of descriptor systems. Circuits Systems Signal Process. 5 (1986), 1, 37-48. · Zbl 0606.93017 · doi:10.1007/BF01600185
[26] K. Ozcaldiran: Geometric notes on descriptor systems. Proc. 26th IEEE-CDC, Los Angeles 1987. · Zbl 0623.93031
[27] K. Ozcaldiran: A complete classification of controllable singular systems. Submitted. · Zbl 0606.93017
[28] G. C. Verghese B. C. Levy, T. Kailath: A generalized state-space for singular systems. IEEE Trans. Automat. Control AC-26 (1981), 4, 811-831. · Zbl 0541.34040
[29] W. M. Wonham: Linear Multivariate Control: a Geometric Approach. Springer-Verlag, New York 1979. · 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.