Abstract
We consider a controlled linear system of differential-algebraic equations with infinitely differentiable coefficients that is allowed to have an arbitrarily high unsolvability index. It is assumed that the matrix multiplying the derivative of the desired vector function has a constant rank. We prove a theorem on the existence of a solution in the class of Sobolev–Schwartz type generalized functions and derive conditions for the existence of a feedback control such that the general solution of the closed-loop system does not contain singular terms. The relation of these conditions to impulse controllability is shown.
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REFERENCES
Cobb, D., On the solution of linear differential equations with singular coefficients, J. Differ. Equat., 1982, vol. 46, pp. 310–323.
Verghese, G.C., Levy, B., and Kailath, T., A generalized state-space for singular systems, IEEE Trans. Autom. Control, 1981, vol. AC-26, no. 4, pp. 811–831.
Boyarintsev, Yu.E. and Chistyakov, V.F., Algebro-differentsial’nye sistemy: metody resheniya i issledovaniya (Differential-Algebraic Systems: Solution and Research Methods), Novosibirsk: Nauka, 1998.
Cobb, D., Controllability, observability, and duality in singular systems, IEEE Trans. Autom. Control, 1984, vol. AC-29, no. 12, pp. 1076–1082.
Dai, L., Singular Control System. Lecture Notes in Control and Information Sciences. Vol. 118 , Berlin–Heidelberg–New York: Springer, 1989.
Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1988.
Cobb, D., State feedback impulse elimination for singular systems over a Hermite domain, SIAM J. Control Optim., 2006, vol. 44, no. 6, pp. 2189–2209.
Wang, C.-J., State feedback impulse elimination of linear time-varying singular systems, Automatica, 1996, vol. 32, no. 1, pp. 133–136.
Campbell, S.L. and Petzold, L.R., Canonical forms and solvable singular systems of differential equations, SIAM J. Algebraic Discrete Methods, 1983, no. 4, pp. 517–512.
Shcheglova, A.A., Controllability of differential-algebraic equations in the class of impulse effects, Sib. Math. J., 2018, vol. 59, no. 1, pp. 166–178.
Shcheglova, A.A., The solvability of the initial problem for a degenerate linear hybrid system with variable coefficients, Russ. Math., 2010, vol. 54, no. 9, pp. 49–61.
Chistyakov, V.F. and Shcheglova, A.A., Izbrannye glavy teorii algebro-differentsial’nykh sistem (Selected Chapters from the Theory of Differential-Algebraic Systems), Novosibirsk: Nauka, 2003.
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Translated by V. Potapchouck
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Shcheglova, A.A. Feedback Elimination of Impulse Terms from the Solutions of Differential-Algebraic Equations. Diff Equat 57, 41–59 (2021). https://doi.org/10.1134/S0012266121010043
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DOI: https://doi.org/10.1134/S0012266121010043