Stability analysis of two-dimensional Markovian jump state-delayed systems in the Roesser model with uncertain transition probabilities. (English) Zbl 1429.93404
Summary: This paper is concerned with the problem of stochastic stability analysis of discrete-time two-dimensional (2-D) Markovian jump systems (MJSs) described by the Roesser model with interval time-varying delays. The transition probabilities of the jumping process/Markov chain are assumed to be uncertain, that is, they are not exactly known but can be estimated. A Lyapunov-like scheme is first extended to 2-D MJSs with delays. Based on some novel 2-D summation inequalities proposed in this paper, delay-dependent stochastic stability conditions are derived in terms of linear matrix inequalities (LMIs) which can be computationally solved by various convex optimization algorithms. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.
MSC:
| 93E15 | Stochastic stability in control theory |
| 60J76 | Jump processes on general state spaces |
| 93C43 | Delay control/observation systems |
| 93C55 | Discrete-time control/observation systems |
Keywords:
Markovian jump systems; Roesser model; time-varying delay; stochastic stability; uncertain transition probabilitiesReferences:
| [1] | Bors, D.; Walczak, S., Application of 2D systems to investigation of a process of gas filtration, Multidim. Syst. Signal Process., 23, 119-130 (2012) · Zbl 1256.49051 |
| [2] | Chen, B.; Li, H.; Shi, P.; Lin, C.; Zhou, Q., Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays, Inf. Sci., 179, 2851-2860 (2009) · Zbl 1165.93341 |
| [3] | Chen, C. W.; Tsai, J. S.H.; Shieh, L. S., Modeling of variable coefficient roesser model for systems described by second-order partial differential equation, Circ. Syst. Signal Process., 22, 423-463 (2003) · Zbl 1044.93031 |
| [4] | Chen, Y.; Zheng, W. X., Exponential \(h_∞\) filtering for stochastic Markovian jump systems with time delays, Int. J. Robust Nonl. Control, 24, 625-643 (2014) · Zbl 1284.93231 |
| [5] | Dey, A.; Kar, H., LMI-based criterion for robust stability of 2D discrete systems with interval time-varying delays employing quantisation/overflow nonlinearities, Multidim. Syst. Signal Process., 25, 473-492 (2014) · Zbl 1319.93067 |
| [6] | Du, B.; Lam, J.; Zou, Y.; Shu, Z., Stability and stabilization for Markovian jump time-delay systems with partially unknown transition rates, IEEE Trans. Circ. Syst.-I Reg. Pap., 60, 341-351 (2013) · Zbl 1468.93134 |
| [7] | Du, D.; Qi, B.; Fei, M.; Peng, C., Multiple event-triggered \(h_2/h_∞\) filtering for hybrid wired-wireless networked systems with random network-induced delays, Inf. Sci., 325, 393-408 (2015) · Zbl 1391.94201 |
| [8] | Freeman, C. T.; Rogers, E.; Hughes, A. M.; Burridge, J. H.; Meadmore, K. L., Iterative learning control in healthcare electrical stimulation and robotic-assisted upper limb stroke rehabilitation, IEEE Control Syst. Mag., 32, 18-43 (2012) · Zbl 1395.93388 |
| [9] | Geromel, J. C.; Gonçalves, A. P.; Fioravanti, A. R., Dynamic output feedback control of discrete-time Markov jump linear systems through linear matrix inequalities, SIAM J. Control Optim., 48, 573-593 (2009) · Zbl 1194.93070 |
| [10] | Ghous, I.; Xiang, Z.; Karimi, H. R., State feedback \(h_∞\) control for 2-D switched delay systems with actuator saturation in the second FM model, Circ. Syst. Signal Process., 34, 2167-2192 (2015) · Zbl 1341.93025 |
| [11] | Guo, G., Linear systems with medium-access constraint and Markov actuator assignment, IEEE Trans. Circ. Syst.-I: Reg. Pap., 57, 2999-3010 (2010) · Zbl 1468.93184 |
| [12] | Guo, G.; Lu, Z.; Han, Q.-L., Control with Markov sensors/actuators assignment, IEEE Trans. Autom. Control, 57, 1799-1804 (2012) · Zbl 1369.93691 |
| [13] | Guo, Y.; Wang, Z., Stability of Markovian jump systems with generally uncertain transition rates, J. Frankl. Inst., 350, 2826-2836 (2013) · Zbl 1287.93106 |
| [14] | Hien, L. V.; An, N. T.; Trinh, H., New results on state bounding for discrete-time systems with interval time-varying delay and bounded disturbance inputs, IET Control Theory Appl., 8, 1405-1414 (2014) |
| [15] | Hien, L. V.; Dzung, N. T.; Minh, H. B., A novel approach to state bounding for discrete-time Markovian jump systems with interval time-varying delay, IMA Math. Control Info. (2014) |
| [16] | Hien, L. V.; Trinh, H., Refined Jensen-based inequality approach to stability analysis of time-delay systems, IET Control Theory Appl., 9, 2188-2194 (2015) |
| [17] | Hien, L. V.; Dzung, N. T.; Trinh, H., Stochastic stability of nonlinear discrete-time Markovian jump systems with time-varying delay and partially unknown transition rates, Neurocomputing, 175, 450-458 (2016) |
| [18] | Huang, S.; Xiang, Z., Delay-dependent robust \(h_∞\) control for 2-D discrete nonlinear systems with state delays, Multidim. Syst. Signal Process., 25, 775-794 (2014) · Zbl 1317.93094 |
| [19] | Kaczorek, T., Two-dimensional Linear Systems (1985), Springer: Springer Berlin · Zbl 0593.93031 |
| [20] | Kao, Y.; Xie, J.; Wang, C., Stabilization of singular Markovian jump systems with generally uncertain transition rates, IEEE Trans. Autom. Control, 59, 2604-2610 (2014) · Zbl 1360.93743 |
| [21] | Kao, Y.; Xie, J.; Wang, C.; Karimi, H. R., A sliding mode approach to \(h_∞\) non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems, Automatica, 52, 218-226 (2015) · Zbl 1309.93038 |
| [22] | Li, Z.; Sun, G.; Gao, H., Guaranteed cost control for discrete-time Markovian jump linear system with time delay, Int. J. Syst. Sci., 44, 1312-1324 (2013) · Zbl 1278.93288 |
| [23] | Long, S.; Zhong, S., Mean-square exponential stability for a class of discrete-time nonlinear singular Markovian jump systems with time-varying delay, J. Frankl. Inst., 351, 4688-4723 (2014) · Zbl 1395.93567 |
| [24] | Qi, W.; Gao, X., State feedback controller design for singular positive Markovian jump systems with partly known transition rates, Appl. Math. Lett., 46, 111-116 (2015) · Zbl 1319.93079 |
| [25] | Qi, W.; Gao, X., \(l_1\) control for positive Markovian jump systems with time-varying delays and partly known transition rates, Circ. Syst. Signal Process., 34, 2711-2726 (2015) · Zbl 1341.93101 |
| [26] | Rogers, E.; Gałkowski, K.; Paszke, W.; Moore, K. L.; Bauer, P. H.; Hladowski, L.; Dabkowski, P., Multidimensional control systems: case studies in design and evaluation, Multidim. Syst. Signal Process., 26, 895-939 (2015) · Zbl 1367.93293 |
| [27] | Shen, H.; Park, J. H.; Wu, Z. G., Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dyn., 77, 1709-1720 (2014) · Zbl 1331.92019 |
| [28] | Sipahi, R.; Niculescu, S.-I.; Abdallah, C. T.; Michiels, W.; Gu, K., Stability and stabilization of systems with time delay, IEEE Control Syst., 31, 38-65 (2011) · Zbl 1395.93271 |
| [29] | Tadepalli, S. K.; Kandanvli, V. K.R.; Kar, H., A new delay-dependent stability criterion for uncertain 2-D discrete systems described by Roesser model under the influence of quantization/overflow nonlinearities, Circ. Syst. Signal Process., 34, 2537-2559 (2015) · Zbl 1341.93072 |
| [30] | Wang, J. L.; Wu, H. N.; Huang, T., Passivity-based synchronization of a class of complex dynamical networks with time-varying delay, Automatica, 56, 105-112 (2015) · Zbl 1323.93012 |
| [31] | Wang, Y.; Zhang, S.; Zhang, X.; Wang, X., Fault detection for a class of non-linear networked systems with Markovian transmission delays, IMA J. Math. Control Inf., 32, 141-159 (2015) · Zbl 1395.94412 |
| [32] | Wei, Y.; Qiu, J.; Karimi, H. R.; Wang, M., Filtering design for two-dimensional Markovian jump systems with state-delays and deficient mode information, Inf. Sci., 269, 316-331 (2014) · Zbl 1339.93111 |
| [33] | Wu, L.; Shi, P.; Gao, H.; Wang, C., \(h_∞\) filtering for 2-D Markovian jump systems, Automatica, 44, 7, 1849-1858 (2008) · Zbl 1149.93346 |
| [34] | Wu, Z. G.; Shi, P.; Su, H.; Chu, J., \(l_2 - l_\infty\) filter design for discrete-time singular Markovian jump systems with time-varying delays, Inf. Sci., 181, 5534-5547 (2011) · Zbl 1243.93116 |
| [35] | Wu, L.; Su, X.; Shi, P., Sliding mode control with bounded \(l_2\) gain performance of Markovian jump singular time-delay systems, Automatica, 48, 1929-1933 (2012) · Zbl 1268.93037 |
| [36] | Wu, L.; Yang, R.; Shi, P.; Su, X., Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica, 59, 206-215 (2015) · Zbl 1326.93110 |
| [37] | Wu, L.; Yao, X.; Zheng, W. X., Generalized \(h_2\) fault detection for two-dimensional Markovian jump systems, Automatica, 48, 1741-1750 (2012) · Zbl 1268.93096 |
| [38] | Xu, J.; Nan, Y.; Zhang, G.; Ou, L.; Ni, H., Delay-dependent \(h_∞\) control for uncertain 2-D discrete systems with state delay in the Roesser model, Circ. Syst. Signal Process., 32, 1097-1112 (2013) |
| [39] | Xu, R.; Kao, Y.; Gao, C., Exponential synchronization of delayed Markovian jump complex networks with generally uncertain transition rates, Appl. Math. Comput., 271, 682-693 (2015) · Zbl 1410.93140 |
| [40] | Yao, J.; Wang, W.; Zou, Y., The delay-range-dependent robust stability analysis for 2-D state-delayed systems with uncertainty, Multidim. Syst. Signal Process., 24, 87-103 (2013) · Zbl 1272.93091 |
| [41] | Zhang, L.; Boukas, E. K.; Baron, L.; Karimi, H. R., Fault detection for discrete-time Markov jump linear systems with partially known transition probabilities, Int. J. Control, 83, 1564-1572 (2010) · Zbl 1200.93134 |
| [42] | Zhang, Y.; He, Y.; Wu, M.; Zhang, J., Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica, 47, 79-84 (2011) · Zbl 1209.93162 |
| [43] | Zhang, J.; Lam, J.; Xia, Y., Output feedback delay compensation control for networked control systems with random delays, Inf. Sci., 265, 154-166 (2014) · Zbl 1327.93363 |
| [44] | Zhang, Y., Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities, Nonlinear Dyn., 75, 101-111 (2014) · Zbl 1281.60064 |
| [45] | Zhang, R.; Zhang, Y.; Hu, C.; Meng, M. Q.H.; He, Q., Delay-range-dependent \(h_∞\) filtering for two-dimensional Markovain jump systems with interval delays, IET Control Theory Appl., 5, 2191-2199 (2011) |
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.
