Asymptotic stability in probability of singular stochastic systems with Markovian switchings. (English) Zbl 1379.93095
Summary: This paper investigates the problem of asymptotic stability in probability for singular stochastic systems with Markovian switchings. A stochastic Lyapunov theorem on asymptotic stability in probability for the considered systems is provided. Also, we show that the original system has the same stability property as its difference-algebraic form based on singular value decomposition. By utilizing the earlier results, a sufficient condition is obtained in terms of linear matrix inequalities, which is easy to check by using standard software.
MSC:
| 93E15 | Stochastic stability in control theory |
| 93D20 | Asymptotic stability in control theory |
| 60J75 | Jump processes (MSC2010) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 93C55 | Discrete-time control/observation systems |
Keywords:
singular stochastic systems; Markovian switchings; asymptotic stability in probability; singular value decompositionReferences:
| [1] | DaiL. Singular Control Systems. Springer‐Verlag: Berlin, 1989. · Zbl 0669.93034 |
| [2] | IshiharaJY, TerraMH. On the Lyapunov theorem for singular systems. IEEE Transactions on Automatic Control2002; 47:1926-1930. · Zbl 1364.93570 |
| [3] | XuSY, LamJ. Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Transactions on Automatic Control2004; 49:568-574. · Zbl 1365.93375 |
| [4] | ChenWH, YuanG, ZhengWX. Robust stability of singularly perturbed impulsive systems under nonlinear perturbation. IEEE Transactions on Automatic Control2013; 58:168-174. · Zbl 1369.93457 |
| [5] | MahmoudC, MahmoudD. Further enhancement on robust \(H_\infty\) control design for discrete‐time singular systems. IEEE Transactions on Automatic Control2014; 59:494-499. · Zbl 1360.93382 |
| [6] | BanuLJ, BalasubramaniamP. Admissibility analysis for discrete‐time singular systems with randomly occurring uncertainties via delay‐divisioning approach. ISA Transactions2015; 59:354-362. |
| [7] | ZamaniI, ShafieeM. Stability analysis of uncertain switched singular time‐delay systems with discrete and distributed delays. Optimal Control Applications and Methods2015; 36:1-28. · Zbl 1315.93069 |
| [8] | NiamsupP, PhatVN. A new result on finite‐time control of singular linear time‐delay systems. Applied Mathematics Letters2016; 60:1-7. · Zbl 1339.93090 |
| [9] | XuSY, LamJ. Robust Control and Filtering of Singular Systems. Springer‐Verlag: Berlin, 2006. · Zbl 1114.93005 |
| [10] | XiaYQ, BoukasEK, ShiP, ZhangJH. Stability and stabilization of continuous‐time singular hybrid systems. Automatica2009; 45:1504-1509. · Zbl 1166.93365 |
| [11] | MaSP, ZhangCH. \( H_\infty\) control for discrete‐time singular Markov jump systems subject to actuator saturation. Journal of the Franklin Institute2012; 349:1011-1029. · Zbl 1273.93066 |
| [12] | WuZG, ParkJH, SuHY, ChuJ. Stochastic stability analysis for discrete‐time singular Markov jump systems with time‐varying delay and piecewise‐constant transition probabilities. Journal of the Franklin Institute2012; 349:2889-2902. · Zbl 1264.93267 |
| [13] | LuHQ, ZhouWN, DuanCM, QiXG. Delay‐dependent robust \(H_\infty\) control for uncertain singular time‐delay system with Markovian jumping parameters. Optimal Control Applications and Methods2013; 34:296-307. · Zbl 1270.93118 |
| [14] | ZhangYQ, ShiP, NguangSK, SongYD. Robust finite‐time \(H_\infty\) control for uncertain discrete‐time singular systems with Markovian jumps. IET Control Theory and Applications2014; 8:1105-1111. |
| [15] | WuZG, ShiP, SuHY, ChuJ. Asynchronous \(l_2 - l_\infty\) filtering for discrete‐time stochastic Markov jump systems with randomly occurred sensor nonlinearities. Automatica2014; 50:180-186. · Zbl 1417.93317 |
| [16] | MaYC, GuNN, ZhangQL. Non‐fragile robust \(H_\infty\) control for uncertain discrete‐time singular systems with time‐varying delays. Journal of the Franklin Institute2014; 351:3163-3181. · Zbl 1290.93063 |
| [17] | WangGL, ZhangQL, YangCY. Stabilization of singular Markovian jump systems with time‐varying switchings. Information Sciences2015; 297:254-270. · Zbl 1360.60148 |
| [18] | WangJH, ZhangQL, YanXG, ZhaiD. Stochastic stability and stabilization of discrete‐time singular Markovian jump systems with partially unknown transition probabilities. International Journal of Robust and Nonlinear Control2015; 25:1423-1437. · Zbl 1317.93266 |
| [19] | ShenH, SuL, ParkJH. Extended passive filtering for discrete‐time singular Markov jump systems with time‐varying delays. Signal Process2016; 128:68-77. |
| [20] | LiL, ZhangQL. Finite‐time \(H_\infty\) control for singular Markovian jump systems with partly unknown transition rates. Applied Mathematical Modelling2016; 40:302-314. · Zbl 1443.93014 |
| [21] | MaoXR. Stochastic Differential Equations and Applications (2nd edn). Horwood Publishing: Chichester, 2008. |
| [22] | KhasminskiiR. Stochastic Stability of Differential Equations (2nd edn). Springer‐Verlag: Berlin, 2012. |
| [23] | YinJ. Asymptotic stability in probability and stabilization for a class of discrete‐time stochastic systems. International Journal of Robust and Nonlinear Control2015; 25:2803-2815. · Zbl 1328.93221 |
| [24] | NishimuraY. Conditions for local almost sure asymptotic stability. Systems & Control Letters2016; 94:19-24. · Zbl 1344.93085 |
| [25] | CongS. On almost sure stability conditions of linear switching stochastic differential systems. Nonlinear Analysis: Hybrid Systems2016; 22:108-115. · Zbl 1346.60081 |
| [26] | ZhangWH, ZhaoY, ShengL. Some remarks on stability of stochastic singular systems with state‐dependent noise. Automatica2015; 51:273-277. · Zbl 1309.93182 |
| [27] | HuangLR, MaoXR. Stability of singular stochastic systems with Markovian switching. IEEE Transactions on Automatic Control2011; 56:424-429. · Zbl 1368.93767 |
| [28] | WuLG, HoDWC. Sliding mode control of singular stochastic hybrid systems. Automatica2010; 46:779-783. · Zbl 1193.93184 |
| [29] | DengH, KrsticM, WilliamsRJ. Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Transactions on Automatic Control2001; 46:1237-1253. · Zbl 1008.93068 |
| [30] | ZhaoY, ZhangWH. New results on stability of singular stochastic Markov jump systems with state‐dependent noise. International Journal of Robust and Nonlinear Control2016; 26:2169-2186. · Zbl 1342.93119 |
| [31] | LuenbergerDG. Dynamic equations in descriptor form. IEEE Transactions on Automatic Control1977; 22:312-321. · Zbl 0354.93007 |
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