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Keywords:
descriptor systems; controllability; reachable controllability
descriptor systems; controllability; reachable controllability
Summary:
Contrary to state space systems, there are different notions of controllability for linear time invariant descriptor systems due to the non smooth inputs and inconsistent initial conditions. A comprehensive study of different notions of controllability for linear descriptor systems is performed. Also, it is proved that reachable controllability for general linear time invariant descriptor system is equivalent to the controllability of some matrix pair under an assumption milder than impulse controllability. The whole theory has been developed by coining two new decompositions for system matrices. Examples are given to illustrate the presented theory.
References:
[1] Berger, T., Reis, T.: Controllability of linear differential-algebraic systems: A survey. In: Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum, (A. Ilchman and T. Reis, eds.), Springer-Verlag, Berlin, Heildelberg 2013, pp. 1-61. DOI 10.1007/978-3-642-34928-7_1 | MR 3076031
[2] Berger, T., Trenn, S.: Kalman controllability decompositions for differential-algebraic systems. Syst. Control Lett. 71 (2014), 54-61. DOI 10.1016/j.sysconle.2014.06.004 | MR 3250381
[3] Bernstein, D. S.: Matrix Mathematics: Theory, Facts, And Formulas With Application To Linear Systems Theory. Princeton University Press 41, Princeton 2005. MR 2123424
[4] Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N. K.: Feedback design for regularizing descriptor systems. Linear Algebra Appl. 299 (1999), 119-151. DOI 10.1016/s0024-3795(99)00167-6 | MR 1723712
[5] Campbell, S.: Singular Systems Of Differential Equations. Pitman 40, San Francisco 1980. MR 0569589
[6] Campbell, S.: Singular Systems Of Differential Equations II. Pitman 61, San Francisco 1982. MR 0665426
[7] Campbell, S. L., Kunkel, P., Mehrmann, V.: Regularization of linear and nonlinear descriptor systems. In: Control and Optimization with Differential-Algebraic Constraints, Advances in Design and Control 2012, pp. 17-36. DOI 10.1137/9781611972252.ch2 | MR 2905713
[8] Chen, C.-T.: Linear System Theory And Design. Oxford University Press, Inc. 1995.
[9] Christodoulou, M., Paraskevopoulos, P.: Solvability, controllability, and observability of singular systems. J. Optim. Theory Appl. 45 (1985), 53-72. DOI 10.1007/bf00940813 | MR 0778157
[10] Chua, L. O., Desoer, C. A., Kuh, E. S.: Linear and Nonlinear Circuits. McGraw-Hill, New York 1987. MR 0356971
[11] Cobb, D.: Controllability, observability, and duality in singular systems. IEEE Trans. Automat. Control 29 (1984), 1076-1082. DOI 10.1109/tac.1984.1103451 | MR 0771396
[12] Dai, L.: The difference between regularity and irregularity in singular systems. Circuits Syst. Signal Process. 8 (1989), 435-444. DOI 10.1007/bf01599765 | MR 1027908
[13] Dai, L.: Singular Control Systems. Springer Verlag, Lecture Notes in Control and Information Sciences, Berlin 1989. DOI 10.1007/bfb0002475 | MR 0986970
[14] Duan, G.-R.: Analysis and Design Of Descriptor Linear Systems. Springer 10, Berlin 2010. MR 2723074
[15] Duan, G.-R., Nichols, N. K., Liu, G.-P.: Robust pole assignment in descriptor linear systems via state feedback. Eur. J. Control 8 (2002), 136-149. DOI 10.3166/ejc.8.136-149
[16] Gantmacher, F. R.: The Theory of Matrices: Vol. 2. Chelsea Publishing Company, New York 1959. MR 0107649
[17] Geerts, T.: Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems: the general case. Linear Algebra Appl. 181 (1993), 111-130. DOI 10.1016/0024-3795(93)90027-l | MR 1204345
[18] Golub, G. H., Loan, C. F. Van: Matrix Computations. Third edition. Johns Hopkins University Press, Baltimore, London 1996. MR 1417720
[19] Hautus, M.: Stabilization controllability and observability of linear autonomous systems. In: Proc. Indagationes Mathematicae 73, Elsevier 1970, pp. 448-455. DOI 10.1016/s1385-7258(70)80049-x | MR 0289170
[20] Hou, M.: Controllability and elimination of impulsive modes in descriptor systems. IEEE Trans. Automat. Control 49 (2004), 1723-1729. DOI 10.1016/s1385-7258(70)80049-x | MR 2091322
[21] Ishihara, J. Y., Terra, M. H.: Impulse controllability and observability of rectangular descriptor systems. IEEE Trans. Automat. Control 46 (2001), 991-994. DOI 10.1109/9.928613 | MR 1836508 | Zbl 1007.93006
[22] Karampetakis, N., Jones, J., Antoniou, E.: Forward, backward, and symmetric solutions of discrete ARMA representations. Circuits Syst. Signal Process. 20 (2001), 89-109. DOI 10.1007/bf01204924 | MR 1820533
[23] Kumar, A., Daoutidis, P.: Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes. MR 1681229
[24] Kunkel, P., Mehrmann, V. .L.: Differential-algebraic Equations: Analysis and Numerical Solution. Europ. Math. Society, Zürich 2006. DOI 10.4171/017 | MR 2225970
[25] Mishra, V. K., Tomar, N. K.: On complete and strong controllability for rectangular descriptor systems. Circuits Syst. Signal Process. 35 (2016), 1395-1406. DOI 10.1007/s00034-015-0111-8 | MR 3465880
[26] Mishra, V. K., Tomar, N. K., Gupta, M. K.: On controllability and normalizability for linear descriptor systems. J. Control Automat. Electr. Syst. 27 (2016), 19-28. DOI 10.1007/s40313-015-0218-y
[27] Piziak, R., Odell, P. L.: Matrix Theory: From Generalized Inverses to Jordan Form. CRC Press 2007. MR 2292386
[28] Stefanovski, J.: Transformation of optimal control problems of descriptor systems into problems with state-space systems. Kybernetika 48 (2012), 1156-1179. MR 3052879
[29] Verghese, G. C., Levy, B. C., Kailath, T.: A generalized state-space for singular systems. IEEE Trans. Automat. Control 26 (1981), 811-831. DOI 10.1109/tac.1981.1102763 | MR 0635842
[30] Yip, E., Sincovec, R.: Solvability, controllability, and observability of continuous descriptor systems. IEEE Trans. Automat. Control 26 (1981), 702-707. DOI 10.1109/tac.1981.1102699 | MR 0630799
[31] Zhang, Q., Liu, C., Zhang, X.: Complexity, Analysis and Control of Singular Biological Systems. MR 3014673
[32] Zubova, S. P.: On full controllability criteria of a descriptor system. The polynomial solution of a control problem with checkpoints. Automat. Remote Control 72 (2011), 23-37. DOI 10.1134/s0005117911010036 | MR 2808551
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