×

Geometric analysis of reachability and observability for impulsive systems on complex field. (English) Zbl 1243.34093

Summary: The issue of reachability and observability is addressed for a class of linear impulsive systems on complex field, for simplicity, complex linear impulsive systems. This kind of time-driven impulsive systems allows free impulsive instants, which leads to the limitation of using traditional definitions of reachability and observability directly. New notations about the span reachable set and unobservable set are proposed. Sufficient and necessary conditions for span reachability and observability of such systems are established. Moreover, the explicit characterization of span reachable set and unobservable set is presented by geometric analysis. It is pointed out that the geometric conditions are equivalent to the algebraic ones in known results for special cases. Numerical examples are also presented to show the effectiveness of the proposed methods.

MSC:

34H05 Control problems involving ordinary differential equations
34A37 Ordinary differential equations with impulses
34A30 Linear ordinary differential equations and systems
93B05 Controllability
93B07 Observability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Q. Zhang, W.-Z. Gong, and X. H. Tang, “Existence of subharmonic solutions for a class of second-order p-Laplacian systems with impulsive effects,” Journal of Applied Mathematics, vol. 2012, Article ID 434938, 18 pages, 2012. · Zbl 1230.34030 · doi:10.1155/2012/434938
[2] Z. Ji, L. Wang, and X. Guo, “On controllability of switched linear systems,” IEEE Transactions on Automatic Control, vol. 53, no. 3, pp. 796-801, 2008. · Zbl 1367.93071 · doi:10.1109/TAC.2008.917659
[3] D. Cheng, “Controllability of switched bilinear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 4, pp. 511-515, 2005. · Zbl 1365.93039 · doi:10.1109/TAC.2005.844897
[4] H. Lin and P. J. Antsaklis, “Switching stabilizability for continuous-time uncertain switched linear systems,” IEEE Transactions on Automatic Control, vol. 52, no. 4, pp. 633-646, 2007. · Zbl 1366.93580 · doi:10.1109/TAC.2007.894515
[5] V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 0719.34002
[6] H. Yang and F. Jiang, “Analytic approximation of the solutions of stochastic differential delay equations with Poisson jump and Markovian switching,” Journal of Applied Mathematics, vol. 2012, Article ID 305945, 14 pages, 2012. · Zbl 1227.60082 · doi:10.1155/2012/305945
[7] E. A. Medina and D. A. Lawrence, “Reachability and observability of linear impulsive systems,” Automatica, vol. 44, no. 5, pp. 1304-1309, 2008. · Zbl 1283.93050 · doi:10.1016/j.automatica.2007.09.017
[8] G. Xie and L. Wang, “Necessary and sufficient conditions for controllability and observability of switched impulsive control systems,” IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 960-966, 2004. · Zbl 1365.93049 · doi:10.1109/TAC.2004.829656
[9] Z. Yan, “Geometric analysis of impulse controllability for descriptor system,” Systems & Control Letters, vol. 56, no. 1, pp. 1-6, 2007. · Zbl 1120.93010 · doi:10.1016/j.sysconle.2006.07.003
[10] S. Zhao and J. Sun, “Controllability and observability for impulsive systems in complex fields,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1513-1521, 2010. · Zbl 1188.93019 · doi:10.1016/j.nonrwa.2009.03.009
[11] S. Zhao and J. Sun, “Controllability and observability for a class of time-varying impulsive systems,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1370-1380, 2009. · Zbl 1159.93315 · doi:10.1016/j.nonrwa.2008.01.012
[12] G. Xie and L. Wang, “Controllability and stabilizability of switched linear-systems,” Systems & Control Letters, vol. 48, no. 2, pp. 135-155, 2003. · Zbl 1134.93403 · doi:10.1016/S0167-6911(02)00288-8
[13] W. M. Wonham, Linear multivariable control: a geometric approach, vol. 10 of Applications of Mathematics, Springer, New York, NY, USA, 2nd edition, 1979. · Zbl 0424.93001
[14] L. Wang, “Approximate boundary controllability for semilinear delay differential equations,” Journal of Applied Mathematics, vol. 2011, Article ID 587890, 10 pages, 2011. · Zbl 1235.34205 · doi:10.1155/2011/587890
[15] N. Abada, M. Benchohra, and H. Hammouche, “Existence and controllability results for impulsive partial functional differential inclusions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2892-2909, 2008. · Zbl 1160.34068 · doi:10.1016/j.na.2007.08.060
[16] B. Liu, “Controllability of impulsive neutral functional differential inclusions with infinite delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 8, pp. 1533-1552, 2005. · Zbl 1079.93008 · doi:10.1016/j.na.2004.11.022
[17] H. J. Sussmann and V. Jurdjevic, “Controllability of nonlinear systems,” Journal of Differential Equations, vol. 12, pp. 95-116, 1972. · Zbl 0242.49040 · doi:10.1016/0022-0396(72)90007-1
[18] J. Xin and H. Lu, “Random attractors for the stochastic discrete long wave-short wave resonance equations,” Journal of Applied Mathematics, vol. 2011, Article ID 452087, 13 pages, 2011. · Zbl 1223.37102 · doi:10.1155/2011/452087
[19] R.-B. Wu, T.-J. Tarn, and C.-W. Li, “Smooth controllability of infinite-dimensional quantum-mechanical systems,” Physical Review A, vol. 73, no. 1, Article ID 012719, 11 pages, 2006. · doi:10.1103/PhysRevA.73.012719
[20] M. Mirrahimi and R. Van Handel, “Stabilizing feedback controls for quantum systems,” SIAM Journal on Control and Optimization, vol. 46, no. 2, pp. 445-467, 2007. · Zbl 1136.81342 · doi:10.1137/050644793
[21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. · Zbl 1049.81015
[22] H. I. Nurdin, M. R. James, and I. R. Petersen, “Coherent quantum LQG control,” Automatica, vol. 45, no. 8, pp. 1837-1846, 2009. · Zbl 1185.49037 · doi:10.1016/j.automatica.2009.04.018
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.