马尔科夫跳变系统的有限时间稳定与镇定
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摘要
马尔科夫跳变系统是一类具有多模态的随机系统。在实际生产应用过程中,由于系统的状态方程往往具有一定的随机性,因而这类工业系统一般不能通过线性时不变运动方程来描述,这时我们就引入了马尔科夫跳变系统。该系统通过一组随机过程马尔科夫链来刻画系统在不同模态间的跳变转移规律,从而精确地描述了这类状态方程具有随机性的工业控制系统。另一方面,在实际应用过程中,人们越来越关注系统状态在一个有限时间区域内的运行轨迹。我们通常将对系统的这种要求称为有限时间稳定性。本文的主要目的就是讨论在马尔科夫跳变系统下的有限时间稳定与控制问题,具体内容如下:
首先我们给出了在马尔科夫跳变系统下的有限时间随机稳定性的定义。基于这个定义,我们分别针对连续时间和离散时间马尔科夫跳变系统,通过合理的选取Lyapunov函数,得到了系统在转移概率完全已知情况下的稳定性分析结果。紧接着,在分析结果的基础上,我们给出了使系统可镇定的控制器设计方法。
其次针对实际应用大量出现系统转移概率部分未知的情况,通过对系统转移概率的分析,我们得出了该条件下的有限时间稳定性分析结果。在这里我们给出了三种不同的方法,即固定权、自由权和非放缩的方法并分别比较了它们的区别与优劣,丰富了分析手段并同时给出了控制器的设计方法。
最后我们将前面两部分的结果拓展到了更一般的系统上面,即具有时滞的马尔科夫跳变系统的有限时间控制问题。针对系统出现的时变延迟,我们通过选取适当的Lyapunov函数,利用Razumikhin方法,最终得出了其稳定性分析结果以及控制器的设计方法。需要说明的是,本文结论的计算结果完全基于线性矩阵不等式。
Markov jump system is a class of stochastic systems with multiple modes. In practical applications, the industrial control systems cannot be represented by linear time-invariant equations due to the fact that the state equations of these systems are changing stochastically. As a result, the Markov jump system is adopted which accurately describe these industrial systems and the transition probabilities between different modes are governed by the stochastic Markov chain. On the other hand, researchers pay more attention to the operating trajectory of system’s state in a prescribed finite-time interval. This kind of requirement is defined as finite-time stability. The objective of this paper is to investigate the finite-time stability and control problems for Markov jump systems. Details are as follows:
The finite-time stochastical stability of Markov jump systems is firstly given. Based on it, by appropriately choosing Lyapunov functions, sufficient conditions for finite-time stochastical stability are derived for both continuous-time and discrete-time systems with complete knowledge on transition probabilities.
Consequently, the design procedure of the stabilizing controller is presented. Secondly, due to the fact that the transition probabilities are partly unknown for most of the systems, the stability results are derived by carefully studying the properties of transition probabilities. Three techniques, including fixed-weighting matrices, free-weighting matrices and non-magnifying and reducing are suggested. The comparision between these methods has been made and the design procedure is given.
Finally, the above results have been extended to systems under more general framework, that is, the finite-time control of Markov jump systems with time-varying delays. By choosing Lyapunov-Razumikhin functions, the stability and stabilization results are derived. It is noticed that all the computational results are based on linear matrix inequalities.
首先我们给出了在马尔科夫跳变系统下的有限时间随机稳定性的定义。基于这个定义,我们分别针对连续时间和离散时间马尔科夫跳变系统,通过合理的选取Lyapunov函数,得到了系统在转移概率完全已知情况下的稳定性分析结果。紧接着,在分析结果的基础上,我们给出了使系统可镇定的控制器设计方法。
其次针对实际应用大量出现系统转移概率部分未知的情况,通过对系统转移概率的分析,我们得出了该条件下的有限时间稳定性分析结果。在这里我们给出了三种不同的方法,即固定权、自由权和非放缩的方法并分别比较了它们的区别与优劣,丰富了分析手段并同时给出了控制器的设计方法。
最后我们将前面两部分的结果拓展到了更一般的系统上面,即具有时滞的马尔科夫跳变系统的有限时间控制问题。针对系统出现的时变延迟,我们通过选取适当的Lyapunov函数,利用Razumikhin方法,最终得出了其稳定性分析结果以及控制器的设计方法。需要说明的是,本文结论的计算结果完全基于线性矩阵不等式。
Markov jump system is a class of stochastic systems with multiple modes. In practical applications, the industrial control systems cannot be represented by linear time-invariant equations due to the fact that the state equations of these systems are changing stochastically. As a result, the Markov jump system is adopted which accurately describe these industrial systems and the transition probabilities between different modes are governed by the stochastic Markov chain. On the other hand, researchers pay more attention to the operating trajectory of system’s state in a prescribed finite-time interval. This kind of requirement is defined as finite-time stability. The objective of this paper is to investigate the finite-time stability and control problems for Markov jump systems. Details are as follows:
The finite-time stochastical stability of Markov jump systems is firstly given. Based on it, by appropriately choosing Lyapunov functions, sufficient conditions for finite-time stochastical stability are derived for both continuous-time and discrete-time systems with complete knowledge on transition probabilities.
Consequently, the design procedure of the stabilizing controller is presented. Secondly, due to the fact that the transition probabilities are partly unknown for most of the systems, the stability results are derived by carefully studying the properties of transition probabilities. Three techniques, including fixed-weighting matrices, free-weighting matrices and non-magnifying and reducing are suggested. The comparision between these methods has been made and the design procedure is given.
Finally, the above results have been extended to systems under more general framework, that is, the finite-time control of Markov jump systems with time-varying delays. By choosing Lyapunov-Razumikhin functions, the stability and stabilization results are derived. It is noticed that all the computational results are based on linear matrix inequalities.
引文
[1] N.N.Krasovskii and E.A.Lidskii. Analysis design of controller in systems with random attributes-Part 1. Automatic Remote Control. 1961, 22: 1021-1025.
[2] D.D.Sworder. Feedback control of a class of linear systems with jump parameters.IEEE Transactions on Automatic Control. 1969, 14(1): 9-14.
[3] M.Mariton. Jump linear systems in automatic control. Marcel Dekker, New Youk. 1990.
[4] Y.Ji and H.J.Chizeck. Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control. 1990, 35(7): 777-788.
[5] W.H.Chen, Z.H.Guan and Xu.Lu. Delay-dependent output feedback stabilization of Markovian jump systems with time-delay. IEE Proceedings-Control Theory and Applications. 2004, 151(5): 561-566.
[6] E.K.Boukas. Stochastic switching systems: analysis and design. Basel, Berlin: Birkhauser. 2005.
[7] H.J.Kushner. Stochastic stability and control. New York: Academic Press. 1967.
[8] X.Y.Lou and B.T.Cui. Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays. IEEE Transactions on Systems. 2007, 37(3): 713-719.
[9] H.Li, B.Chen, Q.Zhou and W.Y.Qian. Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters. IEEE Transactions on Systems. 2009, 39(1): 94-102.
[10] Z.Mao, B.Jiang and P.Shi. H∞fault detection filter design for networked control systems modeled by discrete Markovian jump systems. IET Control Theory and Applications. 2007, 1(5): 1336-1343.
[11] R.Krtolica and U.Ozguner. Stability of linear feedback systems with random communication delays. International Journal of Control. 1994, 59(4): 925-953.
[12] P.Seiler and R.Sengupta. An H∞approach to networked control. IEEE Transactions on Automatic Control. 2005, 50(3): 356-364.
[13] L.Zhang, Y.Shi, T.Chen and B.Huang. A new method for stabilization of networked control systems with random delays. IEEE Transactions on Automatic Control. 2005, 50(8): 1177-1181.
[14] L.Zhang and E.K.Boukas. Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica. 2009, 45: 463-468.
[15] Z.Wu, H.Su and J.Chu. H∞model reduction for discrete singular Markovian jump systems. Proceedings of the Institution of Mechanical Engineers. 2009, 223(7): 1017-1025.
[16] L.X.Zhang, E.K.Boukas. H∞control for discrete-time Markovian jump linear systems with partly unknown transition probabilities. International Journal of Robust and Nonlinear Control. 2009, 19(8): 868-883.
[17] L.X.Zhang, E.K.Boukas. Mode-dependent H∞filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica. 2009, 45(6): 1462-1467.
[18] L.X.Zhang, E.K.Boukas and J.Lam. Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Transactions on Automatic Control. 2008, 53(10): 2458-2464.
[19] O.L.V.Costa, M.D.Fragoso and R.P.Marques. Discrete-time markovian jump linear systems.London: Springer-Verlag. 2005.
[20] M.Karan, P.Shi and C.Y.Kaya. Transition probability bounds for the stochastic syability robustness of continuous- and discrete-time Markovian jump linear systems. Automatica. 2006, 42(12): 2159-2168.
[21] G.Kamenkov. On stability of motion over a finite interval of time. Journal of Applied Math and Mechanics. 1953. 17: 529-540.
[22] P.Dorato. Short-time stability in linear time-varying systems. IRE International Convention Record. 1961: 83-87.
[23] P.Dorato. Short-time stability in linear time-varying systems. PHD thesis, Polytechnic Institute of Brooklyn. 1961.
[24] P.Dorato. Short-time stability. IRE Transactions on Automatic Control. 1961: 6-86.
[25] L.Weiss and E.F.Infante. On the stability of systems defined over a finite time interval. Proceedings of the National Academy of Sciences. 1965, 54(1): 440-448.
[26] L.Weiss and E.Infante. Finite time stability under perturbing forces and on product spaces. IEEE Transactions on Automatic Control. 1967, 12(1): 54-59.
[27] H.Kushner. Finite-time stochastic stability and the analysis of tracking systems. IEEE Transactions on Automatic Control. 1966, 11(2): 219-227.
[28] H.Kushner. Stochastic stability and control. Academic Press, New York, 1967.
[29] A.Michel and S.Wu. Stability of discrete-time systems over a finite interval of time. International Journal of Control. 1969, 9: 679-694.
[30] W.Garrard. Further results on the synthesis of finite-time stable systems. IEEE Transactions on Automatic Control. 1972, 17(1): 142-144.
[31] W.Garrard. Finite-time stability in control system synthesis. Proceedings 4th IFAC Congress, Warsaw, Poland. 1969.
[32] L.Van Mellaert and P.Dorato. Numerical solution of an optimal control problem with a probability criterion. IEEE Transactions on Automatic Control. 1972, 17(4): 543-546.
[33] L.Van Mellaert. Inclusion-probability-optimal control. PHD thesis, Polytechnic Institute of Brooklyn. 1967.
[34] F.San Filippo and P.Dorato. Short-time parameter optimization with flight control applications. Automatica. 1974, 10(4): 425-430.
[35] P.Dorato, C.Abdallah and D.Famularo. Robust finite-time stability design via linear matrix inequalities. Proceedings of IEEE Conference on Decision and Control. San Diego. 1997.
[36] F.Amato, M.Ariola and C.Abdallah. Application of finite-time stability concepts to control of ATM networks. 40th Allerton Conference on Communication. 2003.
[37] F.Amato, M.Ariola and C.Abdallah. Dynamic output feedback finite-time control of LTI systems subject to parametric uncertainties and disturbances. European Control Conference, Karlsruhe, Germany. 1999.
[38] F.Amato, M.Ariola and C.Abdallah. Finite-time control for uncertain linear systems with disturbance inputs. Proceedings of American Control Conference. 1999: 1776-1780.
[39] F.Amato, M.Ariola and C.Abdallah. Necessary and sufficient conditions for finite-time stability of linear systems. Proceedings of American Control Conference. 2003: 4452-4456.
[40] F.Amato, M.Ariola and P.Dorato. Robust finite-time stabilization of linear systems depending on parameter uncertainties. Proceedings of IEEE Conference on Decision and Control. 1998: 1207-1208.
[41] M.S.Mahmoud. Delay-dependent H∞filtering of a class of switched discrete-time state delay systems. Signal Processing. 2008, 88(11): 2709-2719.
[42] F.Martinelli. Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate. IEEE Transactions on Automatic Control. 2007, 52(10): 1937-1942.
[43] P.Shi, M.Mahmoud and S.K.Nguang. Robust filtering for jumping systems with mode-dependent delays. Signal Processing. 2006, 86(1): 140-152.
[44] Z.Shu, J.Lam and S.Y.Xu. Robust stabilization of Markovian delay systems with delay-dependent exponential estimates. Automatica. 2006, 42(11): 2001-2008.
[45] A.V.Skorohod. Asymptotic methods in the theory of stochastic differential equation. Providence, RI: American Mathematical Society. 1989.
[46] G.L.Wang, Q.L.Zhang and V.Sreeram. Design of reduced-order filtering for Markovian jump systems with mode-dependent time delays. Signal Processing. 2009, 89(2): 187-196.
[47] F.Tao, Q.Zhao. Synthesis of active fault-tolerant control based on Markovian jump system models. IET Control Theory and Applications. 2007, 1(4): 1160-1168.
[48] S.Y.Xu, J.Lam and X.R.Mao. Delay-dependent H∞control and filtering for uncertain Markovian jump system with time-varying delay. IEEE Transactions on Circuits and Systems, Part 1: Regular Papers. 2007, 54(9): 2070-2077.
[49] H.G.Zhang and Y.C.Wang. Stability analysis of Markovian jumping stochastic Cohen-Grossbery neural networks with mixed time delays. IEEE Transactions on Neural Networks. 2008, 19(2): 366-370.
[50] J.Daafouz, P.Riedinger and C.Iung. Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Transactions on Automatic Control. 2002, 47(11): 1883-1887.
[51] R.Krtolica, U.Ozguner and H.Chan. Stability of linear feedback systems with random communication delays. International Journal of Control. 1994, 59(4): 925-953.
[52] D.Liberzon.Switching in systems and control. Berlin: Birkhauser. 2003.
[53] P.Seiler and R.Sengupta. An H∞approach to networked control. IEEE Transactions on Automatic Control. 2005, 50(3): 356-364.
[54] J.L.Xiong, H.J.Gao and W.C.Daniel. On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica. 2005, 41(5): 897-903.
[55] L.Zhang, Y.Shi and T.Chen. A new method for stabilization of networked control systems with random delays. IEEE Transactions on Automatic Control. 2005, 50(8): 1177-1181.
[56] Z.G.Yan, G.S.Zhang and J.K.Wang. Finite-time stability and stabilization of linear stochastic systems. Proceedings of the 29th Chinese Control Conference. 2010: 1115-1120.
[57] S.P.Bhat, D.S.Bernstein. Finite time stability of continuous autonomous systems. SIAM J.Control Optim. 2000, 38(3): 751-766.
[58] V.T.Haimo. Finite time controllers.SIAM J.Control Optim. 1986, 24(4): 760-770.
[59] E.Moulay, W.Perruquetti. Finite time stability and stabilization of a class of continuous systems. J.Math.Anan.Appl. 2006, 323(2): 1430-1443.
[60] Y.Hong, Y.Xu and J.Huang. Finite-time control for robot manipulators. Systems Control Letters. 2002, 46(4): 243-253.
[61] S.P.Bhat and D.S.Bernstein. Continuous finite-time stabilization of the translational and rotational double integrator. IEEE Transactions on Automatic Control. 1998, 43(5): 678-682.
[62] I.Karafyllis. Finite-time global stabilization using time-varying distributed delay feedback. SIAM J.Control Optim. 2006, 45(1): 320-342.
[63] Z.Artstein. Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control. 1982, 27(4): 869-879.
[2] D.D.Sworder. Feedback control of a class of linear systems with jump parameters.IEEE Transactions on Automatic Control. 1969, 14(1): 9-14.
[3] M.Mariton. Jump linear systems in automatic control. Marcel Dekker, New Youk. 1990.
[4] Y.Ji and H.J.Chizeck. Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control. 1990, 35(7): 777-788.
[5] W.H.Chen, Z.H.Guan and Xu.Lu. Delay-dependent output feedback stabilization of Markovian jump systems with time-delay. IEE Proceedings-Control Theory and Applications. 2004, 151(5): 561-566.
[6] E.K.Boukas. Stochastic switching systems: analysis and design. Basel, Berlin: Birkhauser. 2005.
[7] H.J.Kushner. Stochastic stability and control. New York: Academic Press. 1967.
[8] X.Y.Lou and B.T.Cui. Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays. IEEE Transactions on Systems. 2007, 37(3): 713-719.
[9] H.Li, B.Chen, Q.Zhou and W.Y.Qian. Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters. IEEE Transactions on Systems. 2009, 39(1): 94-102.
[10] Z.Mao, B.Jiang and P.Shi. H∞fault detection filter design for networked control systems modeled by discrete Markovian jump systems. IET Control Theory and Applications. 2007, 1(5): 1336-1343.
[11] R.Krtolica and U.Ozguner. Stability of linear feedback systems with random communication delays. International Journal of Control. 1994, 59(4): 925-953.
[12] P.Seiler and R.Sengupta. An H∞approach to networked control. IEEE Transactions on Automatic Control. 2005, 50(3): 356-364.
[13] L.Zhang, Y.Shi, T.Chen and B.Huang. A new method for stabilization of networked control systems with random delays. IEEE Transactions on Automatic Control. 2005, 50(8): 1177-1181.
[14] L.Zhang and E.K.Boukas. Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica. 2009, 45: 463-468.
[15] Z.Wu, H.Su and J.Chu. H∞model reduction for discrete singular Markovian jump systems. Proceedings of the Institution of Mechanical Engineers. 2009, 223(7): 1017-1025.
[16] L.X.Zhang, E.K.Boukas. H∞control for discrete-time Markovian jump linear systems with partly unknown transition probabilities. International Journal of Robust and Nonlinear Control. 2009, 19(8): 868-883.
[17] L.X.Zhang, E.K.Boukas. Mode-dependent H∞filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica. 2009, 45(6): 1462-1467.
[18] L.X.Zhang, E.K.Boukas and J.Lam. Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities. IEEE Transactions on Automatic Control. 2008, 53(10): 2458-2464.
[19] O.L.V.Costa, M.D.Fragoso and R.P.Marques. Discrete-time markovian jump linear systems.London: Springer-Verlag. 2005.
[20] M.Karan, P.Shi and C.Y.Kaya. Transition probability bounds for the stochastic syability robustness of continuous- and discrete-time Markovian jump linear systems. Automatica. 2006, 42(12): 2159-2168.
[21] G.Kamenkov. On stability of motion over a finite interval of time. Journal of Applied Math and Mechanics. 1953. 17: 529-540.
[22] P.Dorato. Short-time stability in linear time-varying systems. IRE International Convention Record. 1961: 83-87.
[23] P.Dorato. Short-time stability in linear time-varying systems. PHD thesis, Polytechnic Institute of Brooklyn. 1961.
[24] P.Dorato. Short-time stability. IRE Transactions on Automatic Control. 1961: 6-86.
[25] L.Weiss and E.F.Infante. On the stability of systems defined over a finite time interval. Proceedings of the National Academy of Sciences. 1965, 54(1): 440-448.
[26] L.Weiss and E.Infante. Finite time stability under perturbing forces and on product spaces. IEEE Transactions on Automatic Control. 1967, 12(1): 54-59.
[27] H.Kushner. Finite-time stochastic stability and the analysis of tracking systems. IEEE Transactions on Automatic Control. 1966, 11(2): 219-227.
[28] H.Kushner. Stochastic stability and control. Academic Press, New York, 1967.
[29] A.Michel and S.Wu. Stability of discrete-time systems over a finite interval of time. International Journal of Control. 1969, 9: 679-694.
[30] W.Garrard. Further results on the synthesis of finite-time stable systems. IEEE Transactions on Automatic Control. 1972, 17(1): 142-144.
[31] W.Garrard. Finite-time stability in control system synthesis. Proceedings 4th IFAC Congress, Warsaw, Poland. 1969.
[32] L.Van Mellaert and P.Dorato. Numerical solution of an optimal control problem with a probability criterion. IEEE Transactions on Automatic Control. 1972, 17(4): 543-546.
[33] L.Van Mellaert. Inclusion-probability-optimal control. PHD thesis, Polytechnic Institute of Brooklyn. 1967.
[34] F.San Filippo and P.Dorato. Short-time parameter optimization with flight control applications. Automatica. 1974, 10(4): 425-430.
[35] P.Dorato, C.Abdallah and D.Famularo. Robust finite-time stability design via linear matrix inequalities. Proceedings of IEEE Conference on Decision and Control. San Diego. 1997.
[36] F.Amato, M.Ariola and C.Abdallah. Application of finite-time stability concepts to control of ATM networks. 40th Allerton Conference on Communication. 2003.
[37] F.Amato, M.Ariola and C.Abdallah. Dynamic output feedback finite-time control of LTI systems subject to parametric uncertainties and disturbances. European Control Conference, Karlsruhe, Germany. 1999.
[38] F.Amato, M.Ariola and C.Abdallah. Finite-time control for uncertain linear systems with disturbance inputs. Proceedings of American Control Conference. 1999: 1776-1780.
[39] F.Amato, M.Ariola and C.Abdallah. Necessary and sufficient conditions for finite-time stability of linear systems. Proceedings of American Control Conference. 2003: 4452-4456.
[40] F.Amato, M.Ariola and P.Dorato. Robust finite-time stabilization of linear systems depending on parameter uncertainties. Proceedings of IEEE Conference on Decision and Control. 1998: 1207-1208.
[41] M.S.Mahmoud. Delay-dependent H∞filtering of a class of switched discrete-time state delay systems. Signal Processing. 2008, 88(11): 2709-2719.
[42] F.Martinelli. Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate. IEEE Transactions on Automatic Control. 2007, 52(10): 1937-1942.
[43] P.Shi, M.Mahmoud and S.K.Nguang. Robust filtering for jumping systems with mode-dependent delays. Signal Processing. 2006, 86(1): 140-152.
[44] Z.Shu, J.Lam and S.Y.Xu. Robust stabilization of Markovian delay systems with delay-dependent exponential estimates. Automatica. 2006, 42(11): 2001-2008.
[45] A.V.Skorohod. Asymptotic methods in the theory of stochastic differential equation. Providence, RI: American Mathematical Society. 1989.
[46] G.L.Wang, Q.L.Zhang and V.Sreeram. Design of reduced-order filtering for Markovian jump systems with mode-dependent time delays. Signal Processing. 2009, 89(2): 187-196.
[47] F.Tao, Q.Zhao. Synthesis of active fault-tolerant control based on Markovian jump system models. IET Control Theory and Applications. 2007, 1(4): 1160-1168.
[48] S.Y.Xu, J.Lam and X.R.Mao. Delay-dependent H∞control and filtering for uncertain Markovian jump system with time-varying delay. IEEE Transactions on Circuits and Systems, Part 1: Regular Papers. 2007, 54(9): 2070-2077.
[49] H.G.Zhang and Y.C.Wang. Stability analysis of Markovian jumping stochastic Cohen-Grossbery neural networks with mixed time delays. IEEE Transactions on Neural Networks. 2008, 19(2): 366-370.
[50] J.Daafouz, P.Riedinger and C.Iung. Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Transactions on Automatic Control. 2002, 47(11): 1883-1887.
[51] R.Krtolica, U.Ozguner and H.Chan. Stability of linear feedback systems with random communication delays. International Journal of Control. 1994, 59(4): 925-953.
[52] D.Liberzon.Switching in systems and control. Berlin: Birkhauser. 2003.
[53] P.Seiler and R.Sengupta. An H∞approach to networked control. IEEE Transactions on Automatic Control. 2005, 50(3): 356-364.
[54] J.L.Xiong, H.J.Gao and W.C.Daniel. On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica. 2005, 41(5): 897-903.
[55] L.Zhang, Y.Shi and T.Chen. A new method for stabilization of networked control systems with random delays. IEEE Transactions on Automatic Control. 2005, 50(8): 1177-1181.
[56] Z.G.Yan, G.S.Zhang and J.K.Wang. Finite-time stability and stabilization of linear stochastic systems. Proceedings of the 29th Chinese Control Conference. 2010: 1115-1120.
[57] S.P.Bhat, D.S.Bernstein. Finite time stability of continuous autonomous systems. SIAM J.Control Optim. 2000, 38(3): 751-766.
[58] V.T.Haimo. Finite time controllers.SIAM J.Control Optim. 1986, 24(4): 760-770.
[59] E.Moulay, W.Perruquetti. Finite time stability and stabilization of a class of continuous systems. J.Math.Anan.Appl. 2006, 323(2): 1430-1443.
[60] Y.Hong, Y.Xu and J.Huang. Finite-time control for robot manipulators. Systems Control Letters. 2002, 46(4): 243-253.
[61] S.P.Bhat and D.S.Bernstein. Continuous finite-time stabilization of the translational and rotational double integrator. IEEE Transactions on Automatic Control. 1998, 43(5): 678-682.
[62] I.Karafyllis. Finite-time global stabilization using time-varying distributed delay feedback. SIAM J.Control Optim. 2006, 45(1): 320-342.
[63] Z.Artstein. Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control. 1982, 27(4): 869-879.