# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Observer-based finite-time $H_\infty$ control of singular Markovian jump systems. (English) Zbl 1244.93048
Summary: This paper addresses the problem of finite-time $H_\infty$ control via observer-based state feedback for a family of singular Markovian jump systems (SMJSs) with time-varying norm-bounded disturbance. Firstly, the concepts of singular stochastic finite-time boundedness and singular stochastic finite-time $H_\infty$ stabilization via observer-based state feedback are given. Then an observer-based state feedback controller is designed to ensure singular stochastic finite-time $H_\infty$ stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criteria are presented for the solvability of the problem, which can be reduced to a feasibility problem involving linear matrix inequalities with a fixed parameter. As an auxiliary result, we also discuss the problem of finite-time stabilization via observer-based state feedback of a class of SMJSs and give sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs. Finally, illustrative examples are given to demonstrate the validity of the proposed techniques.

##### MSC:
 93B36 $H^\infty$-control
Full Text:
##### References:
 [1] F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459-1463, 2001. · Zbl 0983.93060 · doi:10.1016/S0005-1098(01)00087-5 [2] A. El-Gohary and A. S. Al-Ruzaiza, “Optimal control of non-homogeneous prey-predator models during infinite and finite time intervals,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 495-508, 2003. · Zbl 1026.92044 · doi:10.1016/S0096-3003(02)00601-X [3] A. El-Gohary, “Optimal control of an angular motion of a rigid body during infinite and finite time intervals,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 541-551, 2003. · Zbl 1041.70022 · doi:10.1016/S0096-3003(02)00274-6 [4] H. J. Kushner, “Finite time stochastic stability and the analysis of tracking systems,” IEEE Transactions on Automatic Control, vol. 11, pp. 219-227, 1966. [5] L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Transactions on Automatic Control, vol. 12, pp. 54-59, 1967. · Zbl 0168.33903 [6] F. Amato, M. Ariola, and C. Cosentino, “Finite-time stabilization via dynamic output feedback,” Automatica, vol. 42, no. 2, pp. 337-342, 2006. · Zbl 1099.93042 · doi:10.1016/j.automatica.2005.09.007 [7] W. Zhang and X. An, “Finite-time control of linear stochastic systems,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 3, pp. 689-696, 2008. [8] Q. Meng and Y. Shen, “Finite-time H\infty control for linear continuous system with norm-bounded disturbance,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1043-1049, 2009. · Zbl 1221.93066 · doi:10.1016/j.cnsns.2008.03.010 [9] Y. Yang, J. Li, and G. Chen, “Finite-time stability and stabilization of nonlinear stochastic hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 338-345, 2009. · Zbl 1163.93033 · doi:10.1016/j.jmaa.2009.02.046 [10] G. Garcia, S. Tarbouriech, and J. Bernussou, “Finite-time stabilization of linear time-varying continuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 364-369, 2009. · doi:10.1109/TAC.2008.2008325 [11] R. Ambrosino, F. Calabrese, C. Cosentino, and G. De Tommasi, “Sufficient conditions for finite-time stability of impulsive dynamical systems,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 861-865, 2009. · doi:10.1109/TAC.2008.2010965 [12] F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354-1358, 2009. · Zbl 1162.93375 · doi:10.1016/j.automatica.2008.12.016 [13] S. Li, Z. Wang, and S. Fei, “Finite-time control of a bioreactor system using terminal sliding mode,” International Journal of Innovative Computing, Information and Control B, vol. 5, no. 10, pp. 3495-3504, 2009. [14] Z. Wang, S. Li, and S. Fei, “Finite-time tracking control of bank-to-turn missiles using terminalsliding mode,” ICIC Express Letters B, vol. 3, no. 4, pp. 1373-1380, 2009. [15] F. Amato, M. Ariola, and C. Cosentino, “Finite-time control of discrete-time linear systems: analysis and design conditions,” Automatica, vol. 46, no. 5, pp. 919-924, 2010. · Zbl 1191.93099 · doi:10.1016/j.automatica.2010.02.008 [16] S. He and F. Liu, “Robust finite-time stabilization of uncertain fuzzy jump systems,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 9, pp. 3853-3862, 2010. [17] S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 2631-2638, 2011. · Zbl 1219.93143 · doi:10.1016/j.apm.2010.11.050 [18] Y. Zhang, C. Liu, and X. Mu, “Stochastic finite-time guaranteed cost control of Markovian jumping singular systems,” Mathematical Problems in Engineering, vol. 2011, Article ID 431751, 20 pages, 2011. · Zbl 1235.93265 · doi:10.1155/2011/431751 [19] Y. Zhang, C. Liu, and X. Mu, “On stochastic finite-time control of discrete-time fuzzy systems with packet dropout,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 752950, 18 pages, 2012. · Zbl 1235.93257 · doi:10.1155/2012/752950 [20] S. He and F. Liu, “Observer-based finite-time control of time-delayed jump systems,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2327-2338, 2010. · Zbl 1207.93113 · doi:10.1016/j.amc.2010.07.031 [21] L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, NY, USA, 1989. · doi:10.1007/BFb0002475 [22] F. L. Lewis, “A survey of linear singular systems,” Circuits, Systems, and Signal Processing, vol. 5, no. 1, pp. 3-36, 1986. · Zbl 0613.93029 · doi:10.1007/BF01600184 [23] J. Y. Ishihara and M. H. Terra, “On the Lyapunov theorem for singular systems,” IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1926-1930, 2002. · doi:10.1109/TAC.2002.804463 [24] Z. Wu and W. Zhou, “Delay-dependent robust stabilization of uncertain singular systems with state delay,” ICIC Express Letters, vol. 1, no. 2, pp. 169-176, 2007. · Zbl 1164.93407 [25] G. Zhang, Y. Xia, and P. Shi, “New bounded real lemma for discrete-time singular systems,” Automatica, vol. 44, no. 3, pp. 886-890, 2008. · Zbl 1283.93179 · doi:10.1016/j.automatica.2007.07.017 [26] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H\infty control for descriptor systems: a matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669-673, 1997. · Zbl 0881.93024 · doi:10.1016/S0005-1098(96)00193-8 [27] Y. Xia, P. Shi, G. Liu, and D. Rees, “Robust mixed H\infty /H2 state-feedback control for continuous-time descriptor systems with parameter uncertainties,” Circuits, Systems, and Signal Processing, vol. 24, no. 4, pp. 431-443, 2005. · Zbl 1136.93338 · doi:10.1007/s00034-004-0917-2 [28] L. Zhang, B. Huang, and J. Lam, “LMI synthesis of H2 and mixed H2/H\infty controllers for singular systems,” IEEE Transactions on Circuits and Systems, vol. 50, no. 9, pp. 615-626, 2003. · Zbl 1157.93519 · doi:10.1016/S0167-6911(03)00133-6 [29] S. Ma, C. Zhang, and Z. Cheng, “Delay-dependent robust H\infty control for uncertain discrete-time singular systems with time-delays,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 194-211, 2008. · Zbl 1142.93011 · doi:10.1016/j.cam.2007.01.044 [30] E.-K. Boukas, Communications and Control Engineering, Control of Singular Systems with Random Abrupt Changes, Springer-Verlag, Berlin, Germany, 2008. · Zbl 1251.93001 [31] E. K. Boukas, “Stabilization of stochastic singular nonlinear hybrid systems,” Nonlinear Analysis. Theory, Methods & Applicationss, vol. 64, no. 2, pp. 217-228, 2006. · Zbl 1090.93048 · doi:10.1016/j.na.2005.05.066 [32] Y. Xia, E. K. Boukas, P. Shi, and J. Zhang, “Stability and stabilization of continuous-time singular hybrid systems,” Automatica, vol. 45, no. 6, pp. 1504-1509, 2009. · Zbl 1166.93365 · doi:10.1016/j.automatica.2009.02.008 [33] X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and their Applications, vol. 79, no. 1, pp. 45-67, 1999. · Zbl 0962.60043 · doi:10.1016/S0304-4149(98)00070-2 [34] P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-mode control for stochastic jump systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 97-103, 2006. · doi:10.1109/TAC.2005.861716 [35] L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of Markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213-1219, 2010. · doi:10.1109/TAC.2010.2042234 [36] C. E. de Souza, “Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 836-841, 2006. · doi:10.1109/TAC.2006.875012 [37] P. Shi, E.-K. Boukas, and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,” IEEE Transactions on Automatic Control, vol. 44, no. 11, pp. 2139-2144, 1999. · Zbl 1078.93575 · doi:10.1109/9.802932 [38] E. K. Boukas, “On robust stability of singular systems with random abrupt changes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 3, pp. 301-310, 2005. · Zbl 1089.34046 · doi:10.1016/j.na.2005.03.110 [39] J. Qiu, H. He, and P. Shi, “Robust stochastic stabilization and H\infty control for neutral stochastic systems with distributed delays,” Circuits, Systems, and Signal Processing, vol. 30, no. 2, pp. 287-301, 2011. · Zbl 1205.93135 · doi:10.1007/s00034-010-9222-4 [40] K.-C. Yao and F. Q. Cai, “Robustness and reliability of decentralized stochastic singularly- perturbedcomputer controlled systems with multiple time-varying delays,” ICIC Express Letters A, vol. 3, no. 3, pp. 379-384, 2009. [41] S. Xu, T. Chen, and J. Lam, “Robust H\infty filtering for uncertain Markovian jump systems with mode-dependent time delays,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 900-907, 2003. · doi:10.1109/TAC.2003.811277 [42] M. S. Mahmoud, P. Shi, and A. Ismail, “Robust Kalman filtering for discrete-time Markovian jump systems with parameter uncertainty,” Journal of Computational and Applied Mathematics, vol. 169, no. 1, pp. 53-69, 2004. · Zbl 1067.93059 · doi:10.1016/j.cam.2003.11.002 [43] S. Xu and J. Lam, “Reduced-order H\infty filtering for singular systems,” Systems & Control Letters, vol. 56, no. 1, pp. 48-57, 2007. · Zbl 1120.93321 · doi:10.1016/j.sysconle.2006.07.010 [44] S. Ma and E. K. Boukas, “Robust H\infty filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay,” IET Control Theory & Applications, vol. 3, no. 3, pp. 351-361, 2009. · doi:10.1049/iet-cta:20080091 [45] S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer-Verlag, Berlin, Germany, 2006. · Zbl 1114.93005 [46] P. V. Pakshin, “Dissipativity of diffusion itô processes with markovain switching and problems of robust stabilization,” Automation and Remote Control, vol. 68, pp. 1502-1518, 2007. · Zbl 1145.93044 · doi:10.1134/S0005117907090056 [47] J. Qiu and K. Lu, “New robust passive stability criteria for uncertain singularly Markov jump systems with time delays,” ICIC Express Letters, vol. 3, no. 3, pp. 651-656, 2009. [48] E.-K. Boukas and P. Shi, “Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters,” International Journal of Robust and Nonlinear Control, vol. 8, no. 13, pp. 1155-1167, 1998. · Zbl 0918.93060 · doi:10.1002/(SICI)1099-1239(1998110)8:13<1155::AID-RNC380>3.0.CO;2-F [49] Y. Yin, P. Shi, and F. Liu, “Gain-scheduled PI tracking control on stochastic nonlinear systems with partially known transition probabilities,” Journal of the Franklin Institute, vol. 348, no. 4, pp. 685-702, 2011. · Zbl 1227.93127 · doi:10.1016/j.jfranklin.2011.01.011 [50] Y. Q. Xia and Y. M. Jia, “H\infty output feedback controller design for linear singular systems with time-delay,” Control Theory & Applications, vol. 20, no. 3, pp. 323-328, 2003. [51] E. K. Boukas, “Static output feedback control for stochastic hybrid systems: LMI approach,” Automatica, vol. 42, no. 1, pp. 183-188, 2006. · Zbl 1121.93365 · doi:10.1016/j.automatica.2005.08.012 [52] H. H. Choi and M. J. Chung, “Robust observer-based H\infty controller design for linear uncertain time-delay systems,” Automatica, vol. 33, no. 9, pp. 1749-1752, 1997. · doi:10.1016/S0005-1098(97)82235-2 [53] J.-D. Chen, “Robust H\infty output dynamic observer-based control of uncertain time-delay systems,” Chaos, Solitons and Fractals, vol. 31, no. 2, pp. 391-403, 2007. · Zbl 1142.93329 · doi:10.1016/j.chaos.2005.09.076 [54] J.-D. Chen, C.-D. Yang, C.-H. Lien, and J.-H. Horng, “New delay-dependent non-fragile H\infty observer-based control for continuous time-delay systems,” Information Sciences, vol. 178, no. 24, pp. 4699-4706, 2008. · Zbl 1158.93017 · doi:10.1016/j.ins.2008.08.009 [55] Q. Li, Q. Zhang, Y. Zhang, and Y. An, “Observer-based passive control for descriptor systems with time-delay,” Journal of Systems Engineering and Electronics, vol. 20, no. 1, pp. 120-128, 2009. [56] Y. Zhang, C. Liu, and X. Mu, “Robust finite-time stabilization of uncertain singular Markovian jump systems,” Applied Mathematical Modelling. In press. · Zbl 1252.93130 · doi:10.1016/j.apm.2011.12.052 [57] Y. Zhang, C. Liu, and X. Mu, “Robust finite-time H\infty control of singular stochastic systems via static output feedback,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5629-5640, 2012. · Zbl 1238.93121 [58] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory. SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. · Zbl 0816.93004