几类时滞系统的保性能控制和鲁棒H∞滤波器设计
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摘要
由于工作环境的变化以及不可测量的干扰因素,在分析和设计系统时,要精确建立系统的数学模型几乎是不可能的,必定存在不确定性.另外,时滞现象大量存在于各种实际系统中,时滞和不确定性的存在常常导致系统不稳定或性能恶化.对于完全状态系统,不确定时滞系统的鲁棒控制理论已经比较成熟,但是对于非线性多时滞系统和时变时滞系统的研究成果还比较少.当系统呈现不完全状态时,出现快变状态和慢变状态,该系统就称为奇异系统.奇异系统广泛存在于各种实际系统中,如电力系统、生物化学过程、核反应堆、飞机和火箭系统等,它含有完全状态系统所不具有的不唯一解和脉冲解.因此,奇异系统是比完全状态系统更具有广泛形式的动力学系统,对不确定奇异时滞系统的鲁棒控制研究具有重要的理论意义和实际意义.
本文的主要结论有：
(1)针对一类带有多时变滞后的非线性系统,基于Lyapunov理论,讨论该系统的记忆和非记忆混合型反馈保性能控制问题.用线性矩阵不等式(LMI)方法给出系统的混合型反馈控制器存在的充分条件和相应的控制器设计方法,同时给出寻找最小性能指标的方法.最后用数值算例验证该方法的有效性和较小的保守性.
(2)针对一类不确定奇异时滞Lur'e系统,讨论该系统的非脆弱保性能控制器设计问题.对于非线性部分具有区间限制,而且带有不确定项的Lur'e系统,将非脆弱思想引入到这类系统中,使用线性矩阵不等式(LMI)方法,设计这类系统的非脆弱保性能控制器.针对控制器存在加法扰动的情况,给出奇异时滞Lur'e系统的非脆弱保性能控制器存在的充分条件,并且给出相应的控制器和系统的最小性能指标.最后,用数值算例验证该方法的有效性.
(3)针对一类带有时变状态滞后和时变中立滞后的不确定系统,研究该系统的鲁棒H∞滤波器设计问题.目的是设计一个满阶滤波器,使得对于所容许的不确定性和时变滞后,滤波误差动态系统是渐近稳定的且满足描述的H∞性能水平.最后用数值算例验证该方法的有效性.
Because of the changing of running environment and ot her immeasurable distur-bances, it is impossible to obtain the exact mathematics model of practical systems. The difference usually can be described as the uncertainties. Moreover, time de-lays are frequently encountered in a variety of practical systems. Uncertainties and time delays are often sources of instability and poor performance in systems. The robust control theory for uncertain time-delay systems is mature, but there are few contributions to the problems of nonlinear systems with multiple time-varying de-lays. When the system takes on non-full states, which include fast and slow states, it is named as singular system. Singular systems widely appear in the study of some naturally occurring systems than regular ones. e.g. in electric power systems, biochemical process, nuclear reactor, airplane and rocket systems. Therefore, the discussion on robust control for uncertain singular systems with delays is key both in theory and in practice.
The main conclusions in this paper are as follows:
(1)Based on the Lyapunov stability theory, the problem of mixed memory and memoryless guaranteed cost controller design for nonlinear systems with multiple time-varying delays is considered. A sufficient condition for the existence of mixed guaranteed cost controller and the design of the corresponding controller are pro-posed by the method of linear matrix inequalitics(LMIs). what's more, the method of finding the minimum performance specification is given. At last, a numerical example demonstrates that the designed method is effective and less conservative.
(2)The problem of non-fragile guaranteed cost controller design for a class of uncertain singular Lur'e systems with time-delay is discussed. The idea of non-fragile guaranteed cost controller is introduced to uncertain singular Lur'e systems with time-delay and sector restricted nonlinearities. Based on linear matrix inequal-ity(LMI) techniques, the non-fragile guaranteed cost controller is designed. For the additive perturbation, the sufficient conditions for the existence of non-fragile guar-anteed cost controller is addressed. Simultaneously, the corresponding controller and the minimum guaranteed cost specification are proposed. At last, a numerical example is illustrated to demonstrate the effectiveness of this method.
(3)The problem of robust H∞filtering for uncertain systems with time-varying state delays and time-varying neutral delays is considered. The uncertainties are norm bounded. The purpose is to design a full filter such that the filtering error dynamic system is asymptotically stable and satisfies a prescribed H∞performance level, for all admissible uncertainties and time-varying delays. At last the effective-ness of the proposed design method is demonstrated by a numerical example.
本文的主要结论有：
(1)针对一类带有多时变滞后的非线性系统,基于Lyapunov理论,讨论该系统的记忆和非记忆混合型反馈保性能控制问题.用线性矩阵不等式(LMI)方法给出系统的混合型反馈控制器存在的充分条件和相应的控制器设计方法,同时给出寻找最小性能指标的方法.最后用数值算例验证该方法的有效性和较小的保守性.
(2)针对一类不确定奇异时滞Lur'e系统,讨论该系统的非脆弱保性能控制器设计问题.对于非线性部分具有区间限制,而且带有不确定项的Lur'e系统,将非脆弱思想引入到这类系统中,使用线性矩阵不等式(LMI)方法,设计这类系统的非脆弱保性能控制器.针对控制器存在加法扰动的情况,给出奇异时滞Lur'e系统的非脆弱保性能控制器存在的充分条件,并且给出相应的控制器和系统的最小性能指标.最后,用数值算例验证该方法的有效性.
(3)针对一类带有时变状态滞后和时变中立滞后的不确定系统,研究该系统的鲁棒H∞滤波器设计问题.目的是设计一个满阶滤波器,使得对于所容许的不确定性和时变滞后,滤波误差动态系统是渐近稳定的且满足描述的H∞性能水平.最后用数值算例验证该方法的有效性.
Because of the changing of running environment and ot her immeasurable distur-bances, it is impossible to obtain the exact mathematics model of practical systems. The difference usually can be described as the uncertainties. Moreover, time de-lays are frequently encountered in a variety of practical systems. Uncertainties and time delays are often sources of instability and poor performance in systems. The robust control theory for uncertain time-delay systems is mature, but there are few contributions to the problems of nonlinear systems with multiple time-varying de-lays. When the system takes on non-full states, which include fast and slow states, it is named as singular system. Singular systems widely appear in the study of some naturally occurring systems than regular ones. e.g. in electric power systems, biochemical process, nuclear reactor, airplane and rocket systems. Therefore, the discussion on robust control for uncertain singular systems with delays is key both in theory and in practice.
The main conclusions in this paper are as follows:
(1)Based on the Lyapunov stability theory, the problem of mixed memory and memoryless guaranteed cost controller design for nonlinear systems with multiple time-varying delays is considered. A sufficient condition for the existence of mixed guaranteed cost controller and the design of the corresponding controller are pro-posed by the method of linear matrix inequalitics(LMIs). what's more, the method of finding the minimum performance specification is given. At last, a numerical example demonstrates that the designed method is effective and less conservative.
(2)The problem of non-fragile guaranteed cost controller design for a class of uncertain singular Lur'e systems with time-delay is discussed. The idea of non-fragile guaranteed cost controller is introduced to uncertain singular Lur'e systems with time-delay and sector restricted nonlinearities. Based on linear matrix inequal-ity(LMI) techniques, the non-fragile guaranteed cost controller is designed. For the additive perturbation, the sufficient conditions for the existence of non-fragile guar-anteed cost controller is addressed. Simultaneously, the corresponding controller and the minimum guaranteed cost specification are proposed. At last, a numerical example is illustrated to demonstrate the effectiveness of this method.
(3)The problem of robust H∞filtering for uncertain systems with time-varying state delays and time-varying neutral delays is considered. The uncertainties are norm bounded. The purpose is to design a full filter such that the filtering error dynamic system is asymptotically stable and satisfies a prescribed H∞performance level, for all admissible uncertainties and time-varying delays. At last the effective-ness of the proposed design method is demonstrated by a numerical example.
引文
[1]Jinhui Zhang,Peng Shi.Jiqing Qiu. Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems[J]. Journal of The Franklin Institute.2009,346(7):676-690.
[2]Jonghae Kim. Delay-dependent robust and non-fragile guaranteed cost control for uncertain singular systems with time-varying state and input delays[J]. International Journal of Control, Automation.and Systems,2009,7(3):357-364.
[3]Z.P.Du.Q.L.Zhang,Y.Li. Delay-dependent robust H∞ control for uncertain sin-gular systems with multiple state delays[J]. IET Control Theory Application. 2009.3(6):731-740.
[4]Xiaohong Nian. Juntao Feng. Guaranteed cost control of linear uncertain neu-tral type systems with multiple delays[J]. IEEE.International Conference on Control Applications,2004.9:1199-1204.
[5]Yanan Xu.Yuechao Ma,Junxian Shi.et al. Robust guaranteed cost H∞ control for time-delay singular systems with input saturation [C]. Chinese Control and Decision Conference,Shanghai.2009:1427-1431.
[6]Olga I.Kosmidou.Yiannis S.Boutalis. A linear matrix inequality approach for guaranteed cost control of systems with state and input delays[J]. IEEE Trans-actions on Systems.MAN.and Cybernetics-Part A:Systems And Humans.2006. 36(5):936-942.
[7]Y.F.Xie.W.H.Gui.Y.L.Wang.et al. Memory state feedback controller design for singular systems with multiple internal constant point delays[J]. IET Control Theory Application.2009.3(6):631-641.
[8]Changzhong Gong.Aihong Wang. Reliable guaranteed cost control for singu-lar systems with ucertainty and thue-delay[C]. Proceedings of the Seventh In-ternational Conference on Machine Learning and Cybernetics. Kunming.2008. 7:2342-2346.
[9]Changchun Shen.Shouming Zhong. New delay-dependent robust stability cri-terion for uncertain neutral systems with time-varying delay and nonlinear uncertainties[J]. Chaos.Solitons and Fractals.2009.40(5):2277-2285.
[10]俞立.鲁棒控制 -线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.
[11]Deyin Chen,Chaoyong Jin. Delay-dependent stability criteria for a class of uncertain neutral systems[J].Acta Automatica Sinica.2008.34(8):889-892.
[12]K.Gu.V.L.Kharitonor,J.Chen.Stability of time-delay systems[M].Boston: Birkhauser.2003.
[13]S.L.Campbell.Singular Systems of Differential Equations[M].Pitman,Marsh-field:MA,1980.
[14]Shuqian Zhu,Chenghui Zhang,Zhaolin Cheng,June Feng.Delay-dependent ro-bust criteria for two classes of uncertain singular time-delay systems[J].IEEE Transaction Automatic Control.2007.52(5):880-885.
[15]Fang Mei.Delay-dependent robust H∞ control for uncertain singular systems with state delary[J].Acta Automatica Sinica,2009,35(1):65-70.
[16]Zhengguang Wu,Wuneng Zhou.Delay-depenclent robust stabilization for un-certain singular systems with state delay[J].Acta Automatica Sinica,2007. 33(7):714-718.
[17]Renquan Lu.Junhong Wang,Anke Xue,Hongye Su.Jian Chu. Robust H∞ fil-tering for a class of uncertain lurie time-delay singular systems[J].Acta Auto-matica Sinica.2007.33(3):292-296.
[18]Shuqian Zhu.Zhaolin Cheng,June Feng.Delay-dependent robust stability cri-terion and robust stabilization for uncertain singular time-delay systems[C]. American Control Confernce.2005.6:2839-2844.
[19]Shengyuall Xu,Paulvan Doorell.Radu Stefan,James Lam.Robust stability and stabilization for singular systems with state delay and parameter uncert ainty[J]. IEEE Transaction Automatic Control.2002,47(7):1122-1128.
[20]Hongye Su,Xiaofu Ji.Jian Chu.Delay-dependent robust control for uncertain singular time-delay systems[J].Asian Journal of Control,2006.8(2):180-189.
[21]Zhengguang Wu.Hongye Su,Jian Chu.Improved results on delay-dependent H∞ control for singular time-delay systems[J].Acta Automatica Sinica 2009.35(8):1101-1106.
[22]Shuping Ma.Chenghui Zhang.Zhaolin Cheng.Delay-dependent robust stability and stabilization for uncertain discrete singular systems with delays[J].Asian Journal of Control.2009.11(3):309-318.
[23]D.H.Ji.J.H.Park.W.J.Yoo.S.C.Won.Robust memory state feedback model pre-dictive control for discrete-time state delayed systems[J].Applied Mathematies and Computation.2009.215(6):2035-2044.
[24]Huijiao Wang,Anke Xue. Delay-dependent absolute stability criteria for un-certain lur'e singular systems with time-varying delay[C]. American Control Conference,2009,6(4):4079-4084.
[25]张佳成,吴保卫,李新芳.不确定时滞Lur'e系统的非脆弱保性能控制[J].西北大学学报(自然科学版),2009.39：738-744.
[26]S.M.Lee,Ju H.Park. Delay-dependent criteria for absolute stability of un-certain time-delayed Lur'e dynamical systems[J]. Journal of The Franklin Institute,2010,347(1):146-153.
[27]Q.L.Han,D.Yue. Absolute stability of Lur'e systems with time-varying delay[J]. IET Control Theory Application.2007,1(3):854-859.
[28]苏宏业,褚健,鲁仁全,等.不确定时滞系统的鲁棒控制理论[M].北京：科学出版社,2007.
[29]Hiroaki Mukaidani. The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback[J]. Journal of Mathematical Analysis and Applications,2009,359(2):527-535.
[30]薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M].北京：清华大学出版社,2004.
[31]吴保卫,仝云旭,陈敏.不确定多时滞奇异系统鲁棒H∞降维滤波器的设计[J].陕西师范大学学报：自然科学版,2008,36(1)：7-14.
[32]Shengyuan Xu.Tongwen Chen. An LMI approach to the H∞ filter design for uncertain systems with distributed delays[J]. IEEE Transaction Circuits Sys tems:Express Briefs.2004.51(4):195-201.
[33]Shengyuan Xu.James Lam.Tongwen Chen.Yun Zou. A delay-dependent ap-proach to robust H∞ filtering for uncertain distributed delay systems[J]. IEEE Transaction Signal Process.2005.53(10):3764-3772.
[34]Xingang Yu. An LMI approach to robust H∞ filtering for uncertain systems with time-varying distributed delays[J]. Journal of The Franklin Institute.2008, 345(8):877-890.
[35]Magdi S.Mahmoud.Hosam E.Emara-Shabaik. New H2 filter for uncertain singular systems using strict LMIs[J]. Circuits System Signal Process.2009. 28(5):665-677.
[36]Shengyuan Xu.James Lam.Yun Zou. H∞ filtering for singular systems[J]. IEEE Transactions On Automatic Control.2003.48(12):2217-2222.
[37]Wei Qian,Shen Cong.Youxian Sun.Shumin Fei. Novel robust stability criteria for uncertain systems with time-varying delay[J]. Applied Mathematics and Computation.2009.215(2):866-872.
[38]Kerwei Yu,Changhua Lien. Stability criteria for uncertain neutral systems with interval time-varying delays[J]. Chaos.Solitons and Fractals,2008.38(3):650-657.
[39]王月娥,吴保卫.不确定时滞中立系统的鲁棒H∞滤波器设计[J].陕西师范大学学报：自然科学版,2010,38(2)：5-11.
[2]Jonghae Kim. Delay-dependent robust and non-fragile guaranteed cost control for uncertain singular systems with time-varying state and input delays[J]. International Journal of Control, Automation.and Systems,2009,7(3):357-364.
[3]Z.P.Du.Q.L.Zhang,Y.Li. Delay-dependent robust H∞ control for uncertain sin-gular systems with multiple state delays[J]. IET Control Theory Application. 2009.3(6):731-740.
[4]Xiaohong Nian. Juntao Feng. Guaranteed cost control of linear uncertain neu-tral type systems with multiple delays[J]. IEEE.International Conference on Control Applications,2004.9:1199-1204.
[5]Yanan Xu.Yuechao Ma,Junxian Shi.et al. Robust guaranteed cost H∞ control for time-delay singular systems with input saturation [C]. Chinese Control and Decision Conference,Shanghai.2009:1427-1431.
[6]Olga I.Kosmidou.Yiannis S.Boutalis. A linear matrix inequality approach for guaranteed cost control of systems with state and input delays[J]. IEEE Trans-actions on Systems.MAN.and Cybernetics-Part A:Systems And Humans.2006. 36(5):936-942.
[7]Y.F.Xie.W.H.Gui.Y.L.Wang.et al. Memory state feedback controller design for singular systems with multiple internal constant point delays[J]. IET Control Theory Application.2009.3(6):631-641.
[8]Changzhong Gong.Aihong Wang. Reliable guaranteed cost control for singu-lar systems with ucertainty and thue-delay[C]. Proceedings of the Seventh In-ternational Conference on Machine Learning and Cybernetics. Kunming.2008. 7:2342-2346.
[9]Changchun Shen.Shouming Zhong. New delay-dependent robust stability cri-terion for uncertain neutral systems with time-varying delay and nonlinear uncertainties[J]. Chaos.Solitons and Fractals.2009.40(5):2277-2285.
[10]俞立.鲁棒控制 -线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.
[11]Deyin Chen,Chaoyong Jin. Delay-dependent stability criteria for a class of uncertain neutral systems[J].Acta Automatica Sinica.2008.34(8):889-892.
[12]K.Gu.V.L.Kharitonor,J.Chen.Stability of time-delay systems[M].Boston: Birkhauser.2003.
[13]S.L.Campbell.Singular Systems of Differential Equations[M].Pitman,Marsh-field:MA,1980.
[14]Shuqian Zhu,Chenghui Zhang,Zhaolin Cheng,June Feng.Delay-dependent ro-bust criteria for two classes of uncertain singular time-delay systems[J].IEEE Transaction Automatic Control.2007.52(5):880-885.
[15]Fang Mei.Delay-dependent robust H∞ control for uncertain singular systems with state delary[J].Acta Automatica Sinica,2009,35(1):65-70.
[16]Zhengguang Wu,Wuneng Zhou.Delay-depenclent robust stabilization for un-certain singular systems with state delay[J].Acta Automatica Sinica,2007. 33(7):714-718.
[17]Renquan Lu.Junhong Wang,Anke Xue,Hongye Su.Jian Chu. Robust H∞ fil-tering for a class of uncertain lurie time-delay singular systems[J].Acta Auto-matica Sinica.2007.33(3):292-296.
[18]Shuqian Zhu.Zhaolin Cheng,June Feng.Delay-dependent robust stability cri-terion and robust stabilization for uncertain singular time-delay systems[C]. American Control Confernce.2005.6:2839-2844.
[19]Shengyuall Xu,Paulvan Doorell.Radu Stefan,James Lam.Robust stability and stabilization for singular systems with state delay and parameter uncert ainty[J]. IEEE Transaction Automatic Control.2002,47(7):1122-1128.
[20]Hongye Su,Xiaofu Ji.Jian Chu.Delay-dependent robust control for uncertain singular time-delay systems[J].Asian Journal of Control,2006.8(2):180-189.
[21]Zhengguang Wu.Hongye Su,Jian Chu.Improved results on delay-dependent H∞ control for singular time-delay systems[J].Acta Automatica Sinica 2009.35(8):1101-1106.
[22]Shuping Ma.Chenghui Zhang.Zhaolin Cheng.Delay-dependent robust stability and stabilization for uncertain discrete singular systems with delays[J].Asian Journal of Control.2009.11(3):309-318.
[23]D.H.Ji.J.H.Park.W.J.Yoo.S.C.Won.Robust memory state feedback model pre-dictive control for discrete-time state delayed systems[J].Applied Mathematies and Computation.2009.215(6):2035-2044.
[24]Huijiao Wang,Anke Xue. Delay-dependent absolute stability criteria for un-certain lur'e singular systems with time-varying delay[C]. American Control Conference,2009,6(4):4079-4084.
[25]张佳成,吴保卫,李新芳.不确定时滞Lur'e系统的非脆弱保性能控制[J].西北大学学报(自然科学版),2009.39：738-744.
[26]S.M.Lee,Ju H.Park. Delay-dependent criteria for absolute stability of un-certain time-delayed Lur'e dynamical systems[J]. Journal of The Franklin Institute,2010,347(1):146-153.
[27]Q.L.Han,D.Yue. Absolute stability of Lur'e systems with time-varying delay[J]. IET Control Theory Application.2007,1(3):854-859.
[28]苏宏业,褚健,鲁仁全,等.不确定时滞系统的鲁棒控制理论[M].北京：科学出版社,2007.
[29]Hiroaki Mukaidani. The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback[J]. Journal of Mathematical Analysis and Applications,2009,359(2):527-535.
[30]薛定宇,陈阳泉.高等应用数学问题的MATLAB求解[M].北京：清华大学出版社,2004.
[31]吴保卫,仝云旭,陈敏.不确定多时滞奇异系统鲁棒H∞降维滤波器的设计[J].陕西师范大学学报：自然科学版,2008,36(1)：7-14.
[32]Shengyuan Xu.Tongwen Chen. An LMI approach to the H∞ filter design for uncertain systems with distributed delays[J]. IEEE Transaction Circuits Sys tems:Express Briefs.2004.51(4):195-201.
[33]Shengyuan Xu.James Lam.Tongwen Chen.Yun Zou. A delay-dependent ap-proach to robust H∞ filtering for uncertain distributed delay systems[J]. IEEE Transaction Signal Process.2005.53(10):3764-3772.
[34]Xingang Yu. An LMI approach to robust H∞ filtering for uncertain systems with time-varying distributed delays[J]. Journal of The Franklin Institute.2008, 345(8):877-890.
[35]Magdi S.Mahmoud.Hosam E.Emara-Shabaik. New H2 filter for uncertain singular systems using strict LMIs[J]. Circuits System Signal Process.2009. 28(5):665-677.
[36]Shengyuan Xu.James Lam.Yun Zou. H∞ filtering for singular systems[J]. IEEE Transactions On Automatic Control.2003.48(12):2217-2222.
[37]Wei Qian,Shen Cong.Youxian Sun.Shumin Fei. Novel robust stability criteria for uncertain systems with time-varying delay[J]. Applied Mathematics and Computation.2009.215(2):866-872.
[38]Kerwei Yu,Changhua Lien. Stability criteria for uncertain neutral systems with interval time-varying delays[J]. Chaos.Solitons and Fractals,2008.38(3):650-657.
[39]王月娥,吴保卫.不确定时滞中立系统的鲁棒H∞滤波器设计[J].陕西师范大学学报：自然科学版,2010,38(2)：5-11.