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Evaluation of the impulsive solution space of linear multivariable homogeneous implicit systems. (English) Zbl 0832.93031
A linear homogeneous matrix differential equation is a differential equation $$A(d/dt) x(t)= 0$$, where $$A(s)$$ is a matrix polynomial. A classical formula is given which allows the determination of the impulsive solutions of linear homogeneous matrix differential equations directly in terms of finite and infinite spectral data of the associated polynomial matrix. The notions of finite and infinite Jordan pairs are defined for a general polynomial matrix and the strong relationship between them and the impulsive solutions of linear homogeneous matrix differential equations are pointed out.
##### MSC:
 93C35 Multivariable systems, multidimensional control systems 93C05 Linear systems in control theory 93B60 Eigenvalue problems
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##### References:
 [1] F. M. Callier, C. A. Desoer: Multivariable Feedback Systems. Springer-Verlag, New York 1982. · Zbl 0248.93017 [2] S. L. Cambell: Singular Systems of Differential Equations. Pitman, London 1980. [3] S. L. Cambell C. D. Meyer, N. Rose: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31 (1976), 3, 411-425. · Zbl 0341.34001 [4] D. Cobb: Feedback and Pole placement in descriptor-variable systems. Internat. J. Control 33 (1981), 6, 1135-1146. · Zbl 0464.93039 · doi:10.1080/00207178108922981 [5] D. Cobb: A further interpretation of inconsistent initial conditions in descriptor variable systems. IEEE Trans. Automat. Control AC-28 (1983), 9. · Zbl 0522.93037 · doi:10.1109/TAC.1983.1103344 [6] D. Cobb: On the solution of linear differential equations with singular coefficients. J. Differential Equations 46 (1982), 310-323. · Zbl 0489.34006 · doi:10.1016/0022-0396(82)90097-3 [7] N. Cohen: Spectral analysis of regular matrix polynomials. Integral Equations Operator Theory 6 (1983), 161-183. · Zbl 0521.15015 · doi:10.1007/BF01691894 [8] F. Lewis: A survey of linear singular systems. Circuits Systems Signal Process. 5 (1986), 1. · Zbl 0613.93029 · doi:10.1007/BF01600184 [9] D. G. Luenberger: Dynamic systems in descriptor form. IEEE Trans. Automat. Control AC-22 (1977), 312-321. · Zbl 0354.93007 · doi:10.1109/TAC.1977.1101502 [10] H. H. Rosenbrock: State-Space and Multivariable Theory. Nelson, London 1970. · Zbl 0246.93010 [11] H. H. Rosenbrock: Structural Properties of Linear dynamical systems. Internat. J. Control 20 (1974), 2, 191-202. · Zbl 0285.93019 · doi:10.1080/00207177408932729 [12] A. I. G. Vardulakis: Linear Multivariable Control: Algebraic Analysis and Synthesis Methods. Wiley, New York 1991. · Zbl 0751.93002 [13] A. I. G. Vardulakis, G. F. Fragulis: Infinite elementary divisors of polynomial matrices and impulsive solutions of linear homogeneous matrix differential equations. Circuits Systems Signal Process. 8 (1989), 3, 357-373. · Zbl 0678.34002 · doi:10.1007/BF01598420 [14] G. Verghese: Infinite-frequency Behaviour in Dynamical Systems. Ph.D. Dissertation, Dept. Electrical Engineering, Stanford Univ. 1978. [15] G. Verghese B. C. Levy, T. Kailath: A generalized state-space for singular systems. IEEE Trans. Automat. Control AC-26 (1981), 4, 811-830. · Zbl 0541.34040 · doi:10.1109/TAC.1981.1102763 [16] G. Verghese, T. Kailath: Impulsive behaviour in dynamical systems: Structure and significance. Proc. 4th Internat. Symp. Mathematical Theory of Networks and Systems, Delft 1979, The Netherlands, pp. 162-168. [17] L. Gohberg P. Lancaster, L. Rodman: Matrix Polynomials. Academic Press, New York 1982. · Zbl 1081.15007
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