Abstract
The set \(C^{n^2 + nm} = \{ A:X \to X,B:U \to X\} \) of all linear control systems over the field of complex numbers C with m=dim C U inputs and n=dim C X states is studied from the point of view of algebraic geometry. We give a classification of this set with respect to a simultaneous change of bases of the spaces X and U. We also find a subset \(W \subset C^{n^2 + nm} \) of all pre-stable systems. In the case where m=2 we construct the factors \(C^{n^2 + nm} /(f)\) of principal open sets covering W (the polynomial f is a relative invariant). Assuming the same conditions to hold, we find generators of the ring of invariants. These results are also partially generalized to the case of systems with outputs. We show an application of the theory of invariants to the problem of parametric identification of systems.
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Belozerov, V.E. On One Problem of Group Classification of Linear Dynamical Systems. Journal of Mathematical Sciences 109, 1686–1702 (2002). https://doi.org/10.1023/A:1014329830956
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DOI: https://doi.org/10.1023/A:1014329830956