Feedback equivalence of constant linear systems. (English) Zbl 0628.93007
Let O(A,B) be the orbit of the feedback equivalence (A,B)\(\approx (S(A+BT^{-1}K)S^{-1},SBT^{-1})\) of pairs of matrices. The author establishes criteria for \(O(A,B)=O(A',B')\) and O(A’,B’)\(\subset \overline{O(A,B)}\) in terms of Kronecker indices of the pairs and relates his result to the Gerstenhaber-Hesse-link theorem [M. Gerstenhaber, Ann. Math., II. Ser. 70, 167-205 (1959; Zbl 0168.281)].
Reviewer: P.Brunovsky
MSC:
| 93B10 | Canonical structure |
| 15A21 | Canonical forms, reductions, classification |
| 93C05 | Linear systems in control theory |
| 93B05 | Controllability |
Citations:
Zbl 0168.281References:
| [1] | Brunovsky, P. A.A., A classification of linear controllable systems, Kybernetica (Praha), 3, 173-187 (1970) · Zbl 0199.48202 |
| [2] | Gerstenhaber, M., On nilalgebras and linear varieties of nilpotent matrices, III, Annals of Math., 70, 167-205 (1959) · Zbl 0168.28103 |
| [3] | Hazewinkel, M.; Martin, C. F., Representations of the symmetric group, the specialization order, systems and Grassman manifolds, Enseign. Math., 29, 53-87 (1983) · Zbl 0536.20009 |
| [4] | Kalman, R. E., Kronecker invariants and feedback (+ errata), (Weiss, L., Ordinary Differential Equations (1972), Academic Press: Academic Press New York), 459-471 · Zbl 0308.93008 |
| [5] | Tannenbaum, A., Invariance and System Theory: Algebraic and Geometric Aspects, (Lecture Notes in Math. No. 845 (1981), Springer: Springer Berlin-New York) · Zbl 0456.93001 |
| [6] | Wonham, W. A.; Morse, A. S., Feedback invariants of linear multivariable systems, Automatica, 8, 93-100 (1972) · Zbl 0235.93007 |
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