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Observer-based quantized sliding mode \(\mathcal {H}_{\infty}\) control of Markov jump systems. (English) Zbl 1398.93358

Summary: An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty}\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty}\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.

MSC:

93E15 Stochastic stability in control theory
93B36 \(H^\infty\)-control
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[1] Mariton, M.: Jump Linear Systems in Automatic Control. M. Dekker, New York (1990)
[2] Revathi, VM; Balasubramaniam, P; Park, JH; Lee, TH, \({\cal{H}}_ {∞ }\) filtering for sample data systems with stochastic sampling and Markovian jumping parameters, Nonlinear Dyn., 78, 813-830, (2014) · Zbl 1331.93058
[3] Ramasamy, S; Nagamani, G; Zhu, Q, Robust dissipativity and passivity analysis for discrete-time stochastic T-S fuzzy cohengrossberg Markovian jump neural networks with mixed time delays, Nonlinear Dyn., 85, 2777-2799, (2016) · Zbl 1349.93251
[4] Shen, H; Park, JH; Wu, Z, Finite-time synchronization control for uncertain Markov jump neural networks with input constraints, Nonlinear Dyn., 77, 1709-1720, (2014) · Zbl 1331.92019
[5] Samidurai, R; Manivannan, R; Ahn, CK; Karimi, HR, New criteria for stability of generalized neural networks including Markov jump parameters and additive time delays, IEEE Trans. Syst. Man Cybern. Syst., (2016)
[6] Gonçalves, APC; Fioravanti, AR; Geromel, JC, Markov jump linear systems and filtering through network transmitted measurements, Sig. Process., 90, 2842-2850, (2010) · Zbl 1197.94056
[7] Saravanakumar, R; Ali, MS; Ahn, CK; Karimi, HR; Shi, P, Stability of Markovian jump generalized neural networks with interval time-varying delays, IEEE Trans. Neural Netw. Learn. Syst., (2016)
[8] Zhao, S; Ahn, CK; Shmaily, YS; Shi, P; Agarwal, RK, An iterative filter with finite measurements for suddenly maneuvering targets, AIAA J. Guid. Control Dyn., 40, 2316-2322, (2017)
[9] Mathiyalagan, K; Park, JH; Sakthivel, R; Anthoni, SM, Robust mixed \(\cal{H}_∞ \) and passive filtering for networked Markov jump systems with impulses, Sig. Process., 101, 162-173, (2014)
[10] Orey, S.: Markov chains with stochastically stationary transition probabilities. Ann. Probab. 19(3), 907-928 (1991) · Zbl 0735.60040
[11] Shen, M; Yan, S; Zhang, G; Park, JH, Finite-time \(\cal{H}_∞ \) static output control of Markov jump systems with an auxiliary approach, Appl. Math. Comput., 273, 553-561, (2016) · Zbl 1410.93114
[12] Wu, Z-G; Shen, Y; Su, H; Lu, R; Huang, T, \(\cal{H}_2\) performance analysis and applications of two-dimensional hidden Bernoulli jump system, IEEE Trans. Syst. Man Cybern. Syst., (2017)
[13] Baik, H-S; Jeong, HS; Abraham, DM, Estimating transition probabilities in Markov chain-based deterioration models for management of wastewater systems, J. Water Resour. Plan. Manag., 132, 15-24, (2006)
[14] Zhang, L; Boukas, E-K, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45, 463-468, (2009) · Zbl 1158.93414
[15] Shen, M; Ye, D; Wang, Q, Mode-dependent filter design for Markov jump systems with sensor nonlinearities in finite frequency domain, Sig. Process., 134, 1-8, (2017)
[16] Luan, X; Zhao, S; Liu, F, \(\cal{H}_∞ \) control for discrete-time Markov jump systems with uncertain transition probabilities, IEEE Trans. Autom. Control, 58, 1566-1572, (2013) · Zbl 1369.93178
[17] Kim, SH, Control synthesis of Markovian jump fuzzy systems based on a relaxation scheme for incomplete transition probability descriptions, Nonlinear Dyn., 78, 691-701, (2014) · Zbl 1314.93053
[18] Edwards, C; Spurgeon, SK, On the development of discontinuous observers, Int. J. Control, 59, 1211-1229, (1994) · Zbl 0810.93009
[19] Almutairi, NB; Zribi, M, On the sliding mode control of a ball on a beam system, Nonlinear Dyn., 59, 221-238, (2010) · Zbl 1183.70066
[20] Shi, P; Xia, Y; Liu, GP; Rees, D, On designing of sliding-mode control for stochastic jump systems, IEEE Trans. Autom. Control, 51, 97-103, (2006) · Zbl 1366.93682
[21] Ligang, W; Shi, P; Gao, H, State estimation and sliding-mode control of Markovian jump singular systems, IEEE Trans. Autom. Control, 55, 1213-1219, (2010) · Zbl 1368.93696
[22] Li, J; Zhang, Q; Zhai, D; Zhang, Y, Sliding mode control for descriptor Markovian jump systems with mode-dependent derivative-term coefficient, Nonlinear Dyn., 82, 465-480, (2015) · Zbl 1348.93045
[23] Wei, Y; Park, JH; Qiu, J; Wu, L; Jung, HY, Sliding mode control for semi-Markovian jump systems via output feedback, Automatica, 81, 133-141, (2017) · Zbl 1376.93029
[24] Chen, B; Niu, Y; Zou, Y, Sliding mode control for stochastic Markovian jumping systems with incomplete transition rate, IET Control Theory Appl., 7, 1330-1338, (2013)
[25] Kao, Y; Xie, J; Zhang, L; Karimi, HR, A sliding mode approach to robust stabilisation of Markovian jump linear time-delay systems with generally incomplete transition rates, Nonlinear Anal. Hybrid Syst., 17, 70-80, (2015) · Zbl 1326.93111
[26] Khalili, A; Rastegarnia, A; Sanei, S, Quantized augmented complex least-mean square algorithm: derivation and performance analysis, Sig. Process., 121, 54-59, (2016)
[27] Liberzon, D, Hybrid feedback stabilization of systems with quantized signals, Automatica, 39, 1543-1554, (2003) · Zbl 1030.93042
[28] Zheng, B; Yang, G, Quantised feedback stabilisation of planar systems via switching-based sliding-mode control, IET Control Theory Appl., 6, 149-156, (2012)
[29] Zheng, B; Park, JH, Sliding mode control design for linear systems subject to quantization parameter mismatch, J. Frankl. Inst., 353, 37-53, (2016) · Zbl 1395.93162
[30] Song, G; Li, T; Kai, H; Zheng, B, Observer-based quantized control of nonlinear systems with input saturation, Nonlinear Dyn., 86, 1157-1169, (2016) · Zbl 1349.93185
[31] Zheng, B; Yang, G, Robust quantized feedback stabilization of linear systems based on sliding mode control, Opt. Control Appl. Methods, 34, 458-471, (2013) · Zbl 1417.93257
[32] Xiao, N; Xie, L; Minyue, F, Stabilization of Markov jump linear systems using quantized state feedback, Automatica, 46, 1696-1702, (2010) · Zbl 1204.93127
[33] Rasool, F; Nguang, SK, Quantized robust \(\cal{H}_∞ \) control of discrete-time systems with random communication delays, Int. J. Syst. Sci., 42, 129-138, (2011) · Zbl 1209.93046
[34] Shi, P; Liu, M; Zhang, L, Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements, IEEE Trans. Ind. Electron., 62, 5910-5918, (2015)
[35] Shen, M; Park, JH, \(\cal{H}_∞ \) filtering of Markov jump linear systems with general transition probabilities and output quantization, ISA Trans., 63, 204-210, (2016)
[36] Shen, M; Park, JH; Ye, D, A separated approach to control of Markov jump nonlinear systems with general transition probabilities, IEEE Trans. Cybern., 46, 2010-2018, (2016)
[37] Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) · Zbl 0816.93004
[38] Chen, B; Niu, Y; Huang, H, Output feedback control for stochastic Markovian jumping systems via sliding mode design, Opt. Control Appl. Methods, 32, 83-94, (2011) · Zbl 1213.93059
[39] Liu, M; Zhang, L; Shi, P; Zhao, Y, Sliding mode control of continuous-time Markovian jump systems with digital data transmission, Automatica, 80, 200-209, (2017) · Zbl 1370.93074
[40] Yang, D; Zhao, J, Robust finite-time output feedback \(\cal{H}_∞ \) control for stochastic jump systems with incomplete transition rates, Circuits Syst. Sig. Process., 34, 1799-1824, (2015) · Zbl 1341.93104
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