Sliding mode control for semi-Markovian jump systems via output feedback. (English) Zbl 1376.93029
Summary: This paper focuses on the output-feedback sliding mode controller design problem for uncertain continuous-time semi-Markovian Jump Systems (MJSs) in a descriptor system setup. The Transition Rates (TRs) of semi-MJSs rely on the random sojourn-time, which is different from the constant TRs in the conventional MJSs. By carefully exploiting the dynamical properties of the original system, combining with the switching functions, a descriptor system is firstly formulated to describe the holonomic dynamics of the sliding mode. Then, with the construction of a semi-Markovian Lyapunov function and the full utilization of the characteristics of cumulative distribution functions, a sufficient condition on the sliding surface synthesis is presented, which also guarantees the Stochastic Stability (SS) of sliding mode dynamical system. Furthermore, a sliding mode controller is synthesized to drive the underlying closed-loop system onto the sliding surface in finite time, locally for a given sliding region. Finally, an illustrative example is carried out to validate the effectiveness of the developed approach.
MSC:
| 93B12 | Variable structure systems |
| 60J75 | Jump processes (MSC2010) |
| 93C41 | Control/observation systems with incomplete information |
| 93E15 | Stochastic stability in control theory |
Keywords:
semi-Markovian jump systems; sliding mode control; output feedback control; sojourn-time-dependent transition ratesReferences:
| [1] | Basin, M. V.; Rodriguez-Ramirez, P. C., Sliding mode controller design for stochastic polynomial systems with unmeasured states, IEEE Transactions on Industrial Electronics, 61, 1, 387-396 (2014) |
| [2] | Campo, L.; Mookerjee, P.; Bar-Shalom, Y., State estimation for systems with sojourn-time-dependent Markov model switching, IEEE Transactions on Automatic Control, 36, 2, 238-243 (1991) · Zbl 0764.93074 |
| [3] | Chen, B.; Niu, Y.; Huang, H., Output feedback control for stochastic Markovian jumping systems via sliding mode design, Optimal Control Applications & Methods, 32, 1, 83-94 (2011) · Zbl 1213.93059 |
| [4] | Feng, X.; Loparo, K. A.; Ji, Y.; Chizeck, H. J., Stochastic stability properties of jump linear system, IEEE Transactions on Automatic Control, 37, 1, 38-52 (1992) · Zbl 0747.93079 |
| [5] | Foucher, Y.; Mathieu, E.; Saint-Pierre, P.; Durand, J.; Daurès, J., A semi-Markov model based on generalized Weibull distribution with an illustration for HIV disease, Biometrical Journal, 47, 6, 825-833 (2005) · Zbl 1442.62362 |
| [6] | He, S.; Liu, F., Robust peak-to-peak filtering for Markov jump systems, Signal Processing, 90, 2, 513-522 (2010) · Zbl 1177.93080 |
| [7] | He, S.; Liu, F., Stochastic finite-time stabilization for uncertain jump systems via state feedback, Journal of Dynamic Systems, Measurement, and Control, 132, 3, Article 034504 pp. (2010) |
| [8] | Hou, Z.; Luo, J.; Shi, P.; Nguang, S. K., Stochastic stability of Itô differential equations with semi-Markovian jump parameters, IEEE Transactions on Automatic Control, 51, 8, 1838-1842 (2006) |
| [9] | Huang, J.; Shi, Y.; Zhang, X., Active fault tolerant control systems by the semi-Markov model approach, International Journal of Adaptive Control and Signal Processing, 28, 9, 833-847 (2014) · Zbl 1327.93355 |
| [10] | Janssen, J.; Manca, R., Applied semi-markov processes (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1096.60002 |
| [11] | Lam, J.; Shu, Z.; Xu, S.; Boukas, E.-K., Robust \(\mathcal{H}_\infty\) control of descriptor discrete-time Markovian jump systems, International Journal of Control, 80, 3, 374-385 (2007) · Zbl 1120.93057 |
| [12] | Li, X.; Lam, J.; Gao, H.; Xiong, J., \( \mathcal{H}_\infty\) and \(\mathcal{H}_2\) filtering for linear systems with uncertain Markov transitions, Automatica, 67, 5, 252-266 (2016) · Zbl 1335.93126 |
| [13] | Li, Y.; Wu, X., A unified approach to time-aggregated Markov decision processes, Automatica, 67, 5, 77-84 (2016) · Zbl 1335.93149 |
| [14] | Ma, S.; Boukas, E.-K., A singular system approach to robust sliding mode control for uncertain Markov jump systems, Automatica, 45, 11, 2707-2713 (2009) · Zbl 1180.93025 |
| [15] | Mariton, M., Jump linear systems in automatic control (1990), M. Dekker: M. Dekker New York |
| [16] | Ouhbi, B.; Limnios, N., The rate of occurrence of failures for semi-Markov processes and estimation, Statistics & Probability Letters, 59, 3, 245-255 (2002) · Zbl 1017.62106 |
| [17] | Pisano, A.; Tanelli, M.; Ferrara, A., Switched/time-based adaptation for second-order sliding mode control, Automatica, 64, 2, 126-132 (2016) · Zbl 1329.93044 |
| [18] | Shi, P.; Xia, Y.; Liu, G.; Rees, D., On designing of sliding-mode control for stochastic jump systems, IEEE Transactions on Automatic Control, 51, 1, 97-103 (2006) · Zbl 1366.93682 |
| [19] | Shmerling, E.; Hochberg, K. J., Stability of stochastic jump-parameter semi-Markov linear systems of differential equations, Stochastics: An International Journal of Probability and Stochastic Processes: Formerly Stochastics and Stochastics Reports, 80, 6, 513-518 (2008) · Zbl 1153.60389 |
| [20] | Stone, L. D., Necessary and sufficient conditions for optimal control of semi-Markov jump processes, SIAM Journal on Control, 11, 2, 187-201 (1973) · Zbl 0258.93028 |
| [21] | Wu, L.; Shi, P.; Gao, H., State estimation and sliding-mode control of Markovian jump singular systems, IEEE Transactions on Automatic Control, 55, 5, 1213-1219 (2010) · Zbl 1368.93696 |
| [22] | Xu, S.; Lam, J.; Mao, X., Delay-dependent \(\mathcal{H}_\infty\) control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems. I. Regular Papers, 54, 9, 2070-2077 (2007) · Zbl 1374.93134 |
| [23] | Zhang, J.; Shi, P.; Lin, W., Extended sliding mode observer based control for Markovian jump linear systems with disturbances, Automatica, 70, 8, 140-147 (2016) · Zbl 1339.93119 |
| [24] | Zhang, J.; Xia, Y., Design of static output feedback sliding mode control for uncertain linear systems, IEEE Transactions on Industrial Electronics, 57, 6, 2161-2170 (2010) |
| [25] | Zhang, B.; Zheng, W.; Xu, S., Filtering of Markovian jump delay systems based on a new performance index, IEEE Transactions on Circuits and Systems. I. Regular Papers, 60, 5, 1250-1263 (2013) · Zbl 1468.94288 |
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