On complete and strong controllability for rectangular descriptor systems. (English) Zbl 1346.93074
Summary: Algebraic criteria have been developed to check the complete and strong controllability for rectangular descriptor systems. Under the assumption of controllability at infinity, the complete controllability for a descriptor system has been proved to be equivalent to the controllability for a normal system. The essence of the technique to design the proposed normal system is based on the row and column compression operations of basic matrix theory. The strong controllability concept for a descriptor system is related with the complete controllability concept for another descriptor system. Examples are provided to illustrate the theory.
References:
| [1] | R.A. Adams, J.J. Fournier, Sobolev Spaces, vol. 140 (Academic Press, London, 2003) · Zbl 1098.46001 |
| [2] | A. Banaszuk, M. Kociecki, F. Lewis, Kalman decomposition for implicit linear systems. IEEE Trans. Autom. Control 37(10), 1509-1514 (1992) · Zbl 0770.93041 · doi:10.1109/9.256370 |
| [3] | A. Banaszuk, M. Kociecki, K. Przyłuski, On Hautus-type conditions for controllability of implicit linear discrete-time systems. Circuits Syst. Signal Process. 8(3), 289-298 (1989) · Zbl 0681.93007 · doi:10.1007/BF01598416 |
| [4] | Berger, T.; Reis, T.; Ilchman, A. (ed.); Reis, T. (ed.), Controllability of linear differential-algebraic systems a survey, 1-61 (2013), Berlin · Zbl 1266.93001 · doi:10.1007/978-3-642-34928-7_1 |
| [5] | T. Berger, S. Trenn, Kalman controllability decompositions for differential-algebraic systems. Syst. Control Lett. 71, 54-61 (2014) · Zbl 1296.93021 · doi:10.1016/j.sysconle.2014.06.004 |
| [6] | H. Boughari, N. Radhy, Regulation of linear continuous-time singular systems with constrained states and controls. Int. J. Syst. Sci. 38(9), 689-698 (2007) · Zbl 1160.93343 · doi:10.1080/00207720701594602 |
| [7] | K.E. Brenan, S.L. Campbell, L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, vol. 14 (SIAM, Philadelphia, 1996) · Zbl 0844.65058 |
| [8] | A. Bunse-Gerstner, V. Mehrmann, N.K. Nichols, Regularization of descriptor systems by derivative and proportional state feedback. SIAM J. Matrix Anal. Appl. 13(1), 46-67 (1992) · Zbl 0743.93026 · doi:10.1137/0613007 |
| [9] | R. Byers, T. Geerts, V. Mehrmann, Descriptor systems without controllability at infinity. SIAM J. Control Optim. 35(2), 462-479 (1997) · Zbl 0871.93021 · doi:10.1137/S0363012994269818 |
| [10] | S. Campbell, Singular Systems of Differential Equations, vol. 40 (Pitman, San Francisco, 1980) · Zbl 0419.34007 |
| [11] | S. Campbell, Singular Systems of Differential Equations II, vol. 61 (Pitman, San Francisco, 1982) · Zbl 0482.34008 |
| [12] | M. Christodoulou, P. Paraskevopoulos, Solvability, controllability, and observability of singular systems. J. Optim. Theory Appl. 45(1), 53-72 (1985) · Zbl 0537.93017 · doi:10.1007/BF00940813 |
| [13] | D. Chu, V. Mehrmann, Disturbance decoupling for descriptor systems by state feedback. SIAM J. Control Optim. 38(6), 1830-1858 (2000) · Zbl 0963.93060 · doi:10.1137/S0363012900331891 |
| [14] | L.O. Chua, C.A. Desoer, E.S. Kuh, Linear and Nonlinear Circuits (McGraw-Hill, New York, 1987) · Zbl 0631.94017 |
| [15] | D. Cobb, Controllability, observability, and duality in singular systems. IEEE Trans. Autom. Control 29(12), 1076-1082 (1984) · doi:10.1109/TAC.1984.1103451 |
| [16] | L. Dai, Lecture Notes in Control and Information Sciences Singular Control Systems (Springer, Berlin, 1989) · Zbl 0669.93034 |
| [17] | X. Dong, M. Xiao, Admissible control of linear singular delta operator systems. Circuits Syst. Signal Process. 33(7), 2043-2064 (2014) · doi:10.1007/s00034-013-9732-y |
| [18] | G.-R. Duan, Analysis and Design of Descriptor Linear Systems, vol. 10 (Springer, Berlin, 2010) · Zbl 1227.93001 · doi:10.1007/978-1-4419-6397-0 |
| [19] | E. Eich-Soellner, C. Führer, Numerical Methods in Multibody Dynamics, vol. 45 (Springer Fachmedien Wiesbaden, Berlin, 1998) · Zbl 0899.70001 · doi:10.1007/978-3-663-09828-7 |
| [20] | L. Fletcher, M. Goodwin, An intermediate algorithm for pole placement by output feedback in descriptor systems. Circuits Syst. Signal Process. 13(2-3), 329-345 (1994) · Zbl 0805.93021 · doi:10.1007/BF01188114 |
| [21] | H. Frankowska, On controllability and observability of implicit systems. Syst. Control Lett. 14(3), 219-225 (1990) · Zbl 0699.93003 · doi:10.1016/0167-6911(90)90016-N |
| [22] | F.R. Gantmacher, The Theory of Matrices, vol. 2 (Chelsea Publishing Company, New York, 1959) · Zbl 0085.01001 |
| [23] | T. Geerts, Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems: the general case. Linear Algebra Appl. 181, 111-130 (1993) · Zbl 0772.34002 · doi:10.1016/0024-3795(93)90027-L |
| [24] | M. Hautus, Stabilization controllability and observability of linear autonomous systems. In: Indagationes Mathematicae, vol. 73 (Elsevier, Amsterdam, 1970), pp. 448-455 · Zbl 0212.47302 |
| [25] | M. Hou, Controllability and elimination of impulsive modes in descriptor systems. IEEE Trans. Autom. Control 49(10), 1723-1729 (2004) · Zbl 1365.93201 · doi:10.1109/TAC.2004.835392 |
| [26] | J.Y. Ishihara, M.H. Terra, Impulse controllability and observability of rectangular descriptor systems. IEEE Trans. Autom. Control 46(6), 991-994 (2001) · Zbl 1007.93006 · doi:10.1109/9.928613 |
| [27] | A. Kumar, P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems with Applications to Chemical Processes, vol. 397 (CRC Press, Boca Raton, 1999) · Zbl 0916.93001 |
| [28] | P. Kunkel, V.L. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution (European Mathematical Society, Zürich, 2006) · Zbl 1095.34004 · doi:10.4171/017 |
| [29] | V. Lovass-Nagy, D. Powers, R. Schilling, On regularizing descriptor systems by output feedback. IEEE Trans. Autom. Control 39(7), 1507-1509 (1994) · Zbl 0800.93542 · doi:10.1109/9.299645 |
| [30] | D.G. Luenberger, Dynamic equations in descriptor form. IEEE Trans. Autom. Control 22(3), 312-321 (1977) · Zbl 0354.93007 · doi:10.1109/TAC.1977.1101502 |
| [31] | B. Mertzios, M. Christodoulou, Decoupling and pole-zero assignment of singular systems with dynamic state feedback. Circuits Syst. Signal Process. 5(1), 49-68 (1986) · Zbl 0634.93026 · doi:10.1007/BF01600186 |
| [32] | K. Özçaldiran, A geometric characterization of the reachable and the controllable subspaces of descriptor systems. Circuits Syst. Signal Process. 5, 37-48 (1986) · Zbl 0606.93017 · doi:10.1007/BF01600185 |
| [33] | K. Özçaldiran, F. Lewis, On the regularizability of singular systems. IEEE Trans. Autom. Control 35(10), 1156-1160 (1990) · Zbl 0724.93011 · doi:10.1109/9.58561 |
| [34] | R. Riaza, Differential-Algebraic Systems: Analytical Aspects and Circuit Applications (World Scientific, Singapore, 2008) · Zbl 1184.34004 · doi:10.1142/6746 |
| [35] | H.H. Rosenbrock, Structural properties of linear dynamical systems. Int. J. Control 20(2), 191-202 (1974) · Zbl 0285.93019 · doi:10.1080/00207177408932729 |
| [36] | S.J. Small, Runge-Kutta Type Methods for Differential-Algebraic Equations in Mechanics (University of Iowa, Iowa, 2011) |
| [37] | G.C. Verghese, B.C. Levy, T. Kailath, A generalized state-space for singular systems. IEEE Trans. Autom. Control 26(4), 811-831 (1981) · Zbl 0541.34040 · doi:10.1109/TAC.1981.1102763 |
| [38] | X.-R. Yang, R.-Z. Zhang, X. Zhang, Regulation of rectangular descriptor systems with constrained states and controls. In: Chinese Control and Decision Conference (IEEE, 2010), pp. 1005-1010 · Zbl 0724.93011 |
| [39] | E. Yip, R. Sincovec, Solvability, controllability, and observability of continuous descriptor systems. IEEE Trans. Autom. Control 26(3), 702-707 (1981) · Zbl 0482.93013 · doi:10.1109/TAC.1981.1102699 |
| [40] | G. Zhang, Regularizability, controllability and observability of rectangular descriptor systems by dynamic compensation. In: American Control Conference (IEEE, 2006), pp. 4393-4398 |
| [41] | Q. Zhang, C. Liu, X. Zhang, Complexity, Analysis and Control of Singular Biological Systems, vol. 421 (Springer, New York, 2012) · Zbl 1402.92009 · doi:10.1007/978-1-4471-2303-3 |
| [42] | S.P. Zubova, On full controllability criteria of a descriptor system. The polynomial solution of a control problem with checkpoints. Autom. Remote Control 72(1), 23-37 (2011) · Zbl 1235.93049 · doi:10.1134/S0005117911010036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
