\(L_1\) control for positive Markovian jump systems with time-varying delays and partly known transition rates. (English) Zbl 1341.93101
Summary: This paper deals with the problem of \(L_1\) control for positive Markovian jump systems with time-varying delays and partly known transition rates. Firstly, by the use of appropriate co-positive type Lyapunov function, sufficient conditions for stochastic stability of positive Markovian jump systems with time-varying delays and partly known transition rates are proposed. Then, \(L_1\)-gain performance of the system considered is analyzed. Based on the results obtained, a state feedback controller is constructed such that the closed-loop Markovian jump system is positive and stochastically stable with \(L_1\)-gain performance. All the proposed conditions are derived in linear programming. Finally, an example is given to demonstrate the validity of the main results.
MSC:
| 93E15 | Stochastic stability in control theory |
| 60J75 | Jump processes (MSC2010) |
| 93B52 | Feedback control |
| 90C05 | Linear programming |
Keywords:
positive Markovian jump systems; partly known transition rates; stochastic stability; linear programmingReferences:
| [1] | P. Bolzern, P. Colaneri, G. Nicolao, Stochastic stability of positive Markov jump linear systems. Automatica 50(4), 1181-1187 (2014) · Zbl 1298.93336 · doi:10.1016/j.automatica.2014.02.016 |
| [2] | L. Caccetta, V.G. Rumchev, A positive linear discrete-time model of capacity planning and its controllability properties. Math. Comput. Model. 40(1-2), 217-226 (2004) · Zbl 1112.93009 · doi:10.1016/j.mcm.2003.03.010 |
| [3] | X.M. Chen, L. James, P. Li, Z. Shu, \[L_1\] L1-induced norm and controller synthesis of positive systems. Automatica 49(5), 1377-1385 (2013) · Zbl 1319.93024 · doi:10.1016/j.automatica.2013.02.023 |
| [4] | J.W. Dong, W.J. Kim, Markov-chain-based output feedback control for stabilization of networked control systems with random time delays and packet losses. Int. J. Control Autom. Syst. 10(5), 1013-1022 (2012) · doi:10.1007/s12555-012-0519-x |
| [5] | Y. Ebihara, D. Peaucelle, D. Arzelier, LMI approach to linear positive system analysis and synthesis. Syst. Control Lett. 63, 50-56 (2014) · Zbl 1283.93119 · doi:10.1016/j.sysconle.2013.11.001 |
| [6] | L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000) · Zbl 0988.93002 · doi:10.1002/9781118033029 |
| [7] | E. Fornasini, M.E. Valcher, Stability and stabilizability criteria for discrete-time positive switched systems. IEEE Trans. Autom. Control 57(5), 1208-1221 (2012) · Zbl 1369.93526 · doi:10.1109/TAC.2011.2173416 |
| [8] | X.H. Ge, Q.L. Han, Distributed fault detection over sensor networks with Markovian switching topologies. Int. J. Gen. Syst. 43(3-4), 305-318 (2014) · Zbl 1302.93192 · doi:10.1080/03081079.2014.883715 |
| [9] | Y.G. Kao, J.F. Guo, C.H. Wang, X.Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen-Grossberg neural networks with mixed delays. J. Frankl. Inst. 349(6), 1972-1988 (2012) · Zbl 1300.93131 · doi:10.1016/j.jfranklin.2012.04.005 |
| [10] | P. Li, J. Lam, Z. Shu, \[H_\infty H\]∞ positive filtering for positive linear discrete-time systems: an augmentation approach. IEEE Trans. Autom. Control 55(10), 2337-2342 (2010) · Zbl 1368.93719 · doi:10.1109/TAC.2010.2053471 |
| [11] | J. Liu, J. Lian, Y. Zhuang, Output feedback \[L_1\] L1 finite-time control for switched positive delayed systems with MDADT. Nonlinear Anal. Hybrid Syst. 15, 11-22 (2015) · Zbl 1301.93086 · doi:10.1016/j.nahs.2014.06.001 |
| [12] | X. Liu, W. Yu, L. Wang, Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans. Autom. Control 55(4), 1024-1028 (2010) · Zbl 1368.93600 · doi:10.1109/TAC.2010.2041982 |
| [13] | X.R. Mao, Stability of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 79(1), 45-67 (1999) · Zbl 0962.60043 · doi:10.1016/S0304-4149(98)00070-2 |
| [14] | L.J. Shen, U. Buscher, Solving the serial batching problem in job shop manufacturing systems. Eur. J. Oper. Res. 221(1), 14-26 (2012) · Zbl 1253.90119 · doi:10.1016/j.ejor.2012.03.001 |
| [15] | R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. 14(3), 616-629 (2006) · doi:10.1109/TNET.2006.876178 |
| [16] | Y.J. Wang, Z.Q. Zuo, Y.L. Cui, Stochastic stabilization of Markovian jump systems with partial unknown transition probabilities and actuator saturation. Circuits Syst. Signal Process. 31(1), 371-383 (2012) · Zbl 1242.93136 · doi:10.1007/s00034-011-9297-6 |
| [17] | M. Xiang, Z.R. Xiang, Finite-time \[L_1\] L1 control for positive switched linear systems with time-varying delay. Commun. Nonlinear Sci. Numer. Simul. 18(11), 3158-3166 (2013) · Zbl 1329.93086 · doi:10.1016/j.cnsns.2013.04.014 |
| [18] | M. Xiang, Z.R. Xiang, Stability, \[L_1\] L1-gain and control synthesis for positive switched systems with time-varying delay. Nonlinear Anal. Hybrid Syst. 9, 9-17 (2013) · Zbl 1287.93078 · doi:10.1016/j.nahs.2013.01.001 |
| [19] | M. Xiang, Z.R. Xiang, Robust fault detection for switched positive linear systems with time-varying delays. ISA Trans. 53(1), 10-16 (2014) · doi:10.1016/j.isatra.2013.07.013 |
| [20] | Y. Yin, P. Shi, F. Liu, J.S. Pan, Gain-scheduled fault detection on stochastic nonlinear systems with partially known transition jump rates. Nonlinear Anal. Real World Appl. 13, 359-369 (2012) · Zbl 1238.93126 · doi:10.1016/j.nonrwa.2011.07.043 |
| [21] | L.X. Zhang, E.K. Boukas, Stability and stabilization for Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(2), 463-468 (2009) · Zbl 1158.93414 · doi:10.1016/j.automatica.2008.08.010 |
| [22] | L.X. Zhang, E.K. Boukas, Mode-dependent \[H_\infty H\]∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica 45(6), 463-468 (2009) · Zbl 1158.93414 · doi:10.1016/j.automatica.2008.08.010 |
| [23] | J.F. Zhang, Z.Z. Han, F. Zhu, Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Anal. Hybrid Syst. 12, 147-155 (2014) · Zbl 1291.93323 · doi:10.1016/j.nahs.2013.12.002 |
| [24] | Y. Zhang, Y. He, M. Wu, J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47(1), 79-84 (2011) · Zbl 1209.93162 · doi:10.1016/j.automatica.2010.09.009 |
| [25] | X.D. Zhao, X.W. Liu, S. Yin, H.Y. Li, Improved results on stability of continuous-time switched positive linear systems. Automatica 50(2), 614-621 (2014) · Zbl 1364.93583 · doi:10.1016/j.automatica.2013.11.039 |
| [26] | J.J. Zhao, H. Shen, B. Li, J. Wang, Finite-time \[H_\infty H\]∞ control for a class of Markovian jump delayed systems with input saturation. Nonlinear Dyn. 73(1-2), 1099-1110 (2013) · Zbl 1281.93041 · doi:10.1007/s11071-013-0855-2 |
| [27] | X.D. Zhao, Q.S. Zeng, Delay-dependent \[H_\infty H\]∞ performance analysis for Markovian jump linear systems with mode-dependent time varying delays and partially known transition rates. Int. J. Control Autom. Syst. 8(2), 482-489 (2010) · doi:10.1007/s12555-010-0238-0 |
| [28] | X.D. Zhao, L.X. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132-1137 (2012) · Zbl 1244.93129 · doi:10.1016/j.automatica.2012.03.008 |
| [29] | X.D. Zhao, L.X. Zhang, P. Shi, Stability of a class of switched positive linear time-delay systems. Int. J. Robust Nonlinear Control 23(5), 578-589 (2013) · Zbl 1284.93208 · doi:10.1002/rnc.2777 |
| [30] | S.Q. Zhu, Q.L. Han, C.H. Zhang, \[L_1\] L1-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach. Automatica 50(8), 2098-2107 (2014) · Zbl 1297.93168 · doi:10.1016/j.automatica.2014.05.022 |
| [31] | S. Zhu, Z. Li, C. Zhang, Exponential stability analysis for positive systems with delays. IET Control Theory Appl. 6(6), 761-767 (2012) · doi:10.1049/iet-cta.2011.0133 |
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.
