若干依赖型不确定时滞系统的理论与方法的研究
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摘要
对于一个系统而言,稳定是被关注的首要问题.在工程实践中,时滞是普遍存在的.众所周知,时滞的存在会对系统的稳定性产生很大的影响,这是因为时滞的存在将会改变系统的特征方程,所以时滞成为系统不稳定的重要因素.因此对时滞系统加以专门的研究是非常重要的.
本文首先主要研究了依赖型的线性不确定时滞系统的渐进稳定性和鲁棒H∞控制器的设计问题.基于线性矩阵不等式和Lyapunov稳定性理论,通过构造合适的Lyapunov函数,讨论了具有一定的匹配条件及数值界的参数不确定性的时滞系统.利用积分不等式引入自由权矩阵的方法,提出了此类系统渐近稳定的充分条件及最大时滞,我们得到的结果具有较小的保守性和较少的矩阵变量；我们又给出了时滞不确定系统保持渐进稳定的控制器设计方法.本文并对该方法进行了验证,结果表明设计方法是有效的.
然后,对于系统的状态是不可测量时,我们讨论了依赖型的时滞不确定系统的无偏滤波器设计方法,给出了系统鲁棒渐近稳定的充分条件和独立型的时滞不确定系统的无偏滤波器设计相比较具有较小的保守性；又讨论了时滞不确定系统的常规滤波器设计问题,并比较了两种滤波器设计的优缺点.
最后,当系统是非线性的情况下,基于T-S模型,通过引入自由权矩阵的方法,我们研究了一类模糊不确定时滞系统的鲁棒渐进稳定性及状态反馈H∞模糊控制器设计的问题,给出了用线性矩阵不等式表示的状态反馈H∞模糊控制器存在的充分条件.
Stability is of primary impotance for a system, and time-delay can be found in the engineering practices. As we all know, the existence of time delay can be bring in a large number of exffects to the stability of the systems, because the existence of time delay will alter the characteristic equation of the systems, and time delay becomes the main factor to cause instability of the system. Therefore, the specialized research of time-delay is very important.
Firstly, this paper discussed principally the asympototic stability and robust H∞controller of problems of the dependent of linear uncertain tine-delay systems. Based on linear matrix inequality method and Lyapunov stability theory, and by constructing the appropriate Lyapunov function, we have studied the time-delay systems which possess definite matching conditions and valued-bounded uncertainty. Introducting some free weighting matrixes by using the integral inequality, we gived the sufficient conditions of the systems and the largest tine delay. The results we have obtained have less conservative and matrix variables. In this paper we also gived the design approach of controller keeping the uncertain time-delay systems asympototical stabilization. This method is proved, and the results show the effectiveness of the proposed method.
Seconedly, when the state is not measurable, we discussed the design approach of unbiased filter of linear uncertain tine-delay systems, giving the sufficient conditions of robust asympototic stability of the systems, comparing with the independent time-delay uncertain system, and our results of the unbiased filter design have less conservative. Meanwhile, we also discuss the problems of conventional filter of linear uncertain tine-delay systems, and compare advantages and disadvantages with the unbiased filter design.
Lastly, considering the nonlinear system, introducting some free matrixes by Newton-leibniz formula and based on T-S model, we discussed the problem of robust asympototic stability and state feedback H∞fuzzy controller design of a class of fuzzy uncertain time-delay systems, giving the sufficient conditions of state feedback H∞fuzzy controller in terms of a strict linear matrix inequality.
本文首先主要研究了依赖型的线性不确定时滞系统的渐进稳定性和鲁棒H∞控制器的设计问题.基于线性矩阵不等式和Lyapunov稳定性理论,通过构造合适的Lyapunov函数,讨论了具有一定的匹配条件及数值界的参数不确定性的时滞系统.利用积分不等式引入自由权矩阵的方法,提出了此类系统渐近稳定的充分条件及最大时滞,我们得到的结果具有较小的保守性和较少的矩阵变量；我们又给出了时滞不确定系统保持渐进稳定的控制器设计方法.本文并对该方法进行了验证,结果表明设计方法是有效的.
然后,对于系统的状态是不可测量时,我们讨论了依赖型的时滞不确定系统的无偏滤波器设计方法,给出了系统鲁棒渐近稳定的充分条件和独立型的时滞不确定系统的无偏滤波器设计相比较具有较小的保守性；又讨论了时滞不确定系统的常规滤波器设计问题,并比较了两种滤波器设计的优缺点.
最后,当系统是非线性的情况下,基于T-S模型,通过引入自由权矩阵的方法,我们研究了一类模糊不确定时滞系统的鲁棒渐进稳定性及状态反馈H∞模糊控制器设计的问题,给出了用线性矩阵不等式表示的状态反馈H∞模糊控制器存在的充分条件.
Stability is of primary impotance for a system, and time-delay can be found in the engineering practices. As we all know, the existence of time delay can be bring in a large number of exffects to the stability of the systems, because the existence of time delay will alter the characteristic equation of the systems, and time delay becomes the main factor to cause instability of the system. Therefore, the specialized research of time-delay is very important.
Firstly, this paper discussed principally the asympototic stability and robust H∞controller of problems of the dependent of linear uncertain tine-delay systems. Based on linear matrix inequality method and Lyapunov stability theory, and by constructing the appropriate Lyapunov function, we have studied the time-delay systems which possess definite matching conditions and valued-bounded uncertainty. Introducting some free weighting matrixes by using the integral inequality, we gived the sufficient conditions of the systems and the largest tine delay. The results we have obtained have less conservative and matrix variables. In this paper we also gived the design approach of controller keeping the uncertain time-delay systems asympototical stabilization. This method is proved, and the results show the effectiveness of the proposed method.
Seconedly, when the state is not measurable, we discussed the design approach of unbiased filter of linear uncertain tine-delay systems, giving the sufficient conditions of robust asympototic stability of the systems, comparing with the independent time-delay uncertain system, and our results of the unbiased filter design have less conservative. Meanwhile, we also discuss the problems of conventional filter of linear uncertain tine-delay systems, and compare advantages and disadvantages with the unbiased filter design.
Lastly, considering the nonlinear system, introducting some free matrixes by Newton-leibniz formula and based on T-S model, we discussed the problem of robust asympototic stability and state feedback H∞fuzzy controller design of a class of fuzzy uncertain time-delay systems, giving the sufficient conditions of state feedback H∞fuzzy controller in terms of a strict linear matrix inequality.
引文
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[31]Yong He, Qing-Guo Wang, Lihua Xie, Chong Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay [J]. IEEE Trans Automatic control, Vol,52, no.2,2007, pp.293-299.
[32]S. Xu, James Lam, Improved delay-dependent stability criteria for time-delay systems [J]. IEEE Trans. Automatic control, Vol,50, no.3,2005, pp. 384-387.
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[35]M. Wu, Y. He, J. She. New delay-dependent stability criterion for and stabilizing methods for neutral systems. IEEE Trans. Automatic control, Vol, 49, no.12,2004, pp.2266-2271.
[36]M.N.A. Parlakci. Robust stability of uncertain time-varying state-delayed systems [J], IEE Proc:Control. Theory Appl. Vol.153, no.4,2006, pp. 469-477.
[37]S. XU, J. Lam, and M. Zhong. New exponential estimates for time-delay systems [J]," IEEE Trans. Automatica Control, vol.51, no.9,2006, pp. 1321-1326.
[38]Chen Peng. Yu-Chu Tian. Delay-dependent robust stability criterion for uncertain systems with interval time-varying delay [J]. Journal of Computational and Applied Mathematics 214 (2008) 480-494.
[39]Suza Carlos E de, Xie Lihua. On the discrete-time bounded real lemma with application in the characterization of static state feed back H∞ controller [J]. Systems& Control Letters,1992,18:61-71.
[40]Yesh Isaac, Shaked Uri. A transfer function approach to the problems of discrete-time system:H∞ optimal linear control and filtering [J]. IEEE Transactions on Automatic Control,1991,36(11):1264-1271.
[41]Xie Lihua, Soh Yeng chai. Robust kalman filtering for uncertain systems [J]. Systems& Control Letters,1994,22:123-129.
[42]Xie Lihua, Soh Yeng chai, Carlos E de Souza. Robust kalman filtering for uncertain discrete time systems [J]. IEEE Transactions on Automatic Control,1994,39(6):1301-1314.
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[44]Rotstein Hector, Sznaier Mario, Idan Moshe. H2/H∞ filtering theory and an aerospace application [J]. International Journal of Robust and Nonlinear Control,1996,6:347-366.
[45]Shaked Vri, Carlos S de. Robust minimum variance filtering [J]. IEEE Trans. On Signal Processing,1996,44(2):181-189.
[46]刘诗娜,费树岷,冯纯伯.线性不确定系统鲁棒滤波器的设计[J].自动化学报,2002,20(3)：50-55.
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[2]褚健,俞立,苏宏业.鲁棒控制理论及应用[M].浙江,浙江大学出版社,2005.
[3]J Y Yu, S M Yu and H Q Wang. Survey on the Performance Analysis of Networked Control Systems. IEEE International Conference on Systems [J], Manand Cybernetics, Hague, Netherlands Institute of Electrical and Electronics Engineers Inc,2004,6:5068-5073.
[4]樊卫华,蔡骅,陈庆伟,胡维礼.时延网络控制系统的稳定性[J].控制理论与应用,2004,21(6)：880-884.
[5]Zames, G. Functional analysis applied to nonlinear feedback systems [J]. IEEE Trans, Circuits and Systems,1963, Vol.CS-10:392-404.
[6]Kalman, R.Z. When is a linear control systems optimal [J]. Trans. ASME, Ser. D, Basic Engr,1964, Vol.86:51-60.
[7]Rosenbrock, H.H. The stability of multivariable systems [J]. IEEE Trans. Automat. Contr.,1976, Vol. AC-17:105-107.
[8]Youla, D.C, J abr.H.A. and Bongiorno,J.J.. Modern Wiener Hopf design of optimal controller, part II:The multivariable case [J]. IEEE Trans. Automat. Contr.,1976, Vol. AC-21:318-338.
[9]Safonov M. G.. Stability and robustness of multivariable feedback systems [M].MIT, Press,1980.
[10]Zames, G.. Feedback and optimal sensitivity model reference transformations, multivariable seminorms and approximate inverses [J]. IEEE Trans. Automat. Contr.,1981, Vol. AC-326:301-320.
[11]Doyle, J.C., Glover, K., Khargonekor, P.P. and Francis, B.A. State-space solutions to standard H∞ and H2 control problems [J]. IEEE Trans. Automat. Contr.,1989, Vol. AC-34:831-847.
[12]Iwasaki, T. and Skelton, R.E. All controllers for the general control problems: LMI existence conditions and state space formulas [J]. Automatica,1994, Vol.30:1307-1317.
[13]Fridman E, Shaked U. Delay-dependent stability and H∞ control:constant and time-varying delays [J]. Int. J. Control,2003,76 (1):48-60.
[14]Park P. A delay-dependent stability criterion for systems with uncertain time-invariant delays [J]. IEEE Transactions on Automatic Control,1999, 44 (4):876-877.
[15]Moon Y S, Park P, Kwon W H, et al. Delay-dependent robust stabilization of uncertain state-delayed systems [J]. Int. J. Control,2001,74 (14): 1447-1455.
[16]Fridman, E. New Lyapunov-Krasovskii functional for stability of linear retarded and neutral type systems [J]. System and Control Letters,2001. 43(4),309-319.
[17]S.-I. Niculescu, C. E. de Souza, J.-M. Dion, and L. Dugard, "Robust stability and stabilization for uncertain linear systems with state delay:Single delay case [C],"in Proc. IFAC Workshop Robust Contr. Design, Riode Janeiro, Brazil,1994, pp.469-474.
[18]Gu K, Niculescu S I. Additional dynamics in transformed time delay systems [J]. IEEE Transactions on Automatic Control,2000,45 (3):572-57.
[19]Gu K, Niculescu S.I. Further remarks on additional dynamics in various model transformations of linear delay systems [J]. IEEE Transactions on Automatic Control,2001,46 (3):497-50.
[20]Fridman, E.,& Shaked, U. A descriptor system approach to Hoo control of linear time-delay systems [J]. IEEE Transactions on Automatic Control, 2002,47(2),253-270.
[21]Wu M., He Y, She J H, et al. Delay-dependent criteria for robust stability of time-varying delay systems [J]. Automatic,2004,40 (8):1435-1439.
[22]He Y, Wu M, She J H, et al. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays [J]. Systems& Control Letters, 2004,51(1):57-65.
[23]YU L, CHU J. An LMI approach to guaranteed cost control of linear uncertain time-delay systems [J]. Automatic 1999,35:1155-1159.
[24]MUKAIDANI H. An LMI approach to guaranteed cost control for uncertain delay systems [J]. IEEE Trans. Circuits and Systems-I:Fundamental Theory and Applications,2003,50(6):795-800.
[25]LIEN C H, HOU Y Y. Guaranteed cost observer-based control for a class of uncertain time delay systems [J]. Journal of Dynamic Systems, Measurement, and Control,2005,127(12):723-728.
[26]刘飞,苏宏业,蒋培刚等.不确定离散时滞系统具有H∞干扰抑制的保成本控制[J].控制与决策,2002,17(1)：103-106.
[27]金学波,孙优贤.参数不确定系统鲁棒H∞保代价滤波器设计方法[J].电路与系统学报,2005,10(4)：116-120.
[28]俞立.鲁棒控制-线性矩阵不等式[M].北京：清华大学出版社,2002.
[29]仝茂达.线性系统理论和设计[M].合肥：中国科学技术大学出版社,1998.
[30]Q.-L. Han. A new day-dependent stability criterion for linear neutral systems with norm-bounded uncertainties in all systems matrices [J]. International Journal of Systems Control, Vol,36,2005, pp.469-475.
[31]Yong He, Qing-Guo Wang, Lihua Xie, Chong Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay [J]. IEEE Trans Automatic control, Vol,52, no.2,2007, pp.293-299.
[32]S. Xu, James Lam, Improved delay-dependent stability criteria for time-delay systems [J]. IEEE Trans. Automatic control, Vol,50, no.3,2005, pp. 384-387.
[33]张先明,吴敏,何勇.不确定多变时滞系统的时滞相关鲁棒控制[J].控制与决策,2004,19(5)：496-500.
[34]M. Wu, Y. He, J. She, G. Liu. Delay-dependent criterion for robust stability of time-varying delay systems. Automatic, Vol,40,2004, pp,1435-1439.
[35]M. Wu, Y. He, J. She. New delay-dependent stability criterion for and stabilizing methods for neutral systems. IEEE Trans. Automatic control, Vol, 49, no.12,2004, pp.2266-2271.
[36]M.N.A. Parlakci. Robust stability of uncertain time-varying state-delayed systems [J], IEE Proc:Control. Theory Appl. Vol.153, no.4,2006, pp. 469-477.
[37]S. XU, J. Lam, and M. Zhong. New exponential estimates for time-delay systems [J]," IEEE Trans. Automatica Control, vol.51, no.9,2006, pp. 1321-1326.
[38]Chen Peng. Yu-Chu Tian. Delay-dependent robust stability criterion for uncertain systems with interval time-varying delay [J]. Journal of Computational and Applied Mathematics 214 (2008) 480-494.
[39]Suza Carlos E de, Xie Lihua. On the discrete-time bounded real lemma with application in the characterization of static state feed back H∞ controller [J]. Systems& Control Letters,1992,18:61-71.
[40]Yesh Isaac, Shaked Uri. A transfer function approach to the problems of discrete-time system:H∞ optimal linear control and filtering [J]. IEEE Transactions on Automatic Control,1991,36(11):1264-1271.
[41]Xie Lihua, Soh Yeng chai. Robust kalman filtering for uncertain systems [J]. Systems& Control Letters,1994,22:123-129.
[42]Xie Lihua, Soh Yeng chai, Carlos E de Souza. Robust kalman filtering for uncertain discrete time systems [J]. IEEE Transactions on Automatic Control,1994,39(6):1301-1314.
[43]Wang Zidong, Guo Zhi, Unbehauen H. Robust H2/H∞ state estimation for discrete-time systems with error variance constrains [J]. IEEE Transactions on Automatic Control,1997,42(10):1431-1435.
[44]Rotstein Hector, Sznaier Mario, Idan Moshe. H2/H∞ filtering theory and an aerospace application [J]. International Journal of Robust and Nonlinear Control,1996,6:347-366.
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