摘要
切换中立型时滞系统是一类重要的混杂系统。由于这类系统中既有连续动态,又有离散动态,还有中立型时滞的交互作用,所以许多工程实际问题可以用这样的系统模型来描述,具有广泛的工程实际背景和理论研究意义。本文以Lyapunov稳定性理论为基础,采用线性矩阵不等式方法,研究了切换中立型时滞系统的稳定性分析与设计的一些问题,涉及的主要工作如下:
第一,根据时滞大小和切换间隔的关系,研究切换中立型时滞系统在任意切换序列情况下的渐近稳定性问题。通过对系统进行等效变换,利用Lyapunov泛函方法,研究各个子系统的结构对整个切换系统稳定性的影响,然后以线性矩阵不等式的形式给出使系统渐近稳定的充分条件,同时给出系统带记忆的状态反馈控制器的设计方法。
第二,研究了一类固定切换律的切换中立型系统的指数稳定性问题。根据现实世界里许多切换控制系统的切换信号是状态依赖型的情形,对一类切换域固定的切换中立型系统,通过Lyapunov泛函方法和不等式分析技巧研究了其等效变换系统的E-指数稳定性,然后根据两系统稳定性之间的等效性,得到原系统指数稳定的充分条件,在此基础上给出时滞状态反馈控制器的设计方法。
第三,研究了一类切换中立型系统的切换律设计问题。对于系数矩阵存在Hurwitz线性凸组合的一组中立型时滞子系统,可以通过状态空间的合理划分,找到子系统渐近稳定的区域,据此构造切换域并得到切换律的设计方案,并以线性矩阵不等式的形式给出自治切换系统渐近稳定的充分条件。这里研究的子系统不一定是稳定的,只需要满足一定的约束,而切换律起到了稳定系统的关键作用。对于不满足约束的情况,可以通过设计切换状态反馈控制器和相应的切换律,使闭环系统渐近稳定。
第四,研究了一类不确定切换中立型系统的鲁棒非脆弱H_(∞)控制问题。通过研究确定切换中立型系统的H_(∞)性能问题,给出系统内稳定的条件以及无记忆状态反馈控制器的设计方法。在此结论的基础上,将其推广到系统结构中存在有界不确定的情形,并兼顾考虑控制器增益扰动的影响,设计出带有H_(∞)性能的鲁棒非脆弱控制器。
第五,研究了切换中立型系统的保成本可靠控制问题。利用连续故障模型,针对实际控制系统中执行器非理想化的事实,构造出更接近实际的系统模型,利用Lyapunov泛函法,考虑二次成本函数的约束,给出了系统保成本可靠控制器的设计方法,并通过优化方法得到成本函数的上界值。
最后,研究了切换中立型系统的鲁棒滑模控制问题。对存在线性分式不确定性的切换中立型系统,利用Lyapunov方法,结合Finsler引理对每个子系统分别设计滑模面和控制器,使系统在任意切换律作用下均能满足到达条件和滑模运动稳定条件。在此基础上研究了设计单一滑模面和切换律来共同稳定系统的方法。
论文的结束部分对全文所做的工作进行了总结,并指出了下一步研究的方向。
A switched neutral delay system is an important category of hybrid systems. Due to interaction among the continuous dynamics and discrete dynamics and neutral time-delay is common in this kind of systems, many practical engineering issues can be described by switched neutral delay systems, the research of switched neutral delay systems is significant for both practice and theory. Based on Lyapunov stability theory, this dissertation mainly investigates the stability analysis and design of switched neutral delay systems by linear matrix inequalities method. The main contributions of this dissertation are as follows.
Firstly, based on the relationship between time-delay and switching interval, the asymptotic stability of switched neutral delay systems under arbitrary switching sequence is discussed. The equivalent transformation is done to research the effect of configuration of subsystems to the stability of the whole system by Lyapunov functional method. Then sufficient conditions are given to guarantee asymptotic stability of switched neutral delay systems in term of linear matrix inequalities. Also, the design of state feedback controllers with memory is established.
Secondly, exponential stability of switched neutral delay systems with fixed switching law is addressed. Since it is ture that state-dependent switching signals are usual in many switched control systems in real world, E-exponential stability of equivalent systems with fixed switching region is investigated by Lyapunov functional methods and inequalities analysis technique. Owning to the stability equivalence between the two systems, exponential stability of original system is obtained. Corresponding design of state feedback controllers are also accomplished on this basis.
Thirdly, switching law design of a class of switched neutral delay systems is studied. If there is a Hurwitz linear convex combination of coefficient matrices for a class of neutral delay subsystems, asymptoticly stable regions of each subsystem can be found by dividing the state space properly. Based on the divisions of state space, switching law can be obtained by setting switching regions. Sufficient conditions of this kind of switched neutral systems are provided in term of linear matrix inequalities. Here, the stability of subsystems is not necessary, they are only required to satisfy certain conditions. Switching laws are crucial to stabilize the whole system in these problems. Switching state feedback controllers and switching laws can also be designed to make the close-loop system stable for those which do not meet the constraint conditions.
Fouthly, the problem of robust non-fragile H_∞control is discussed for a class of uncertain switched neutral delay systems. H_∞performance of definite switched neutral delay systems is discussed in advance, and the condition of interior stability and design of corresponding state feedback controllers without memory are presented. These results are extended to uncertain switched neutral delay systems, influence of gain disturbance of controllers is also considered. And then, controllers of robust non-fragile with H_∞performance are obtained.
Next, reliable guaranteed cost control problem of switched neutral delay systems is concerned. Based on the fact that actuators are nonideal in real control systems, a more practical system is constructed by using continuous faults model. Reliable guaranteed cost controllers with constraint of quadratic cost function are obtained by the method of Lyapunov functional. Upper bound of the cost function can be derived by optimal methods.
Finally, the problem of robust sliding control of switched neutral delay systems is studied. Combined the Lyapunov method with Finsler's lemma, sliding surfaces and corresponding controllers of each subsystems are designed, which meet reching conditions and sliding mode conditions for switched neutral delay systems with linear fractional uncertainty under arbitrary switching laws. Based on these results, the design of single sliding surface and switching laws are also presented to make systems stable.
At the end of the dissertation, the results are summarized and further research problems are pointed out.
引文
[1]Xie G M,Wang L.Stabilization of switched linear systems with time-delay in detection of switching signal.Journal of Mathematical Analysis and Applications,2005,305(1):277-290
[2]Liberzon D,Morse A S.Basic problems in stability and design of switched systems,IEEE Contr.Syst.Mag.,1999,19(5):59-70
[3]Cheng D Z.Stabilization of planar switched systems.Systems and Control Letters,2004,51(2):79-88
[4]程代展,郭宇骞.切换系统进展.控制理论与应用,2005,22(6):954-960
[5]Hai L,Panos J A.Stability and stabilization of switched linear systems:A short survey of recent results.Proceeding of the 2005 IEEE International Symposium on Intelligent Control Limassol,Cyprus,2005:24-29
[6]Wicks M A,Peleties P,Decarlo R.Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems.European Journal of Control,1998,4:140-147
[7]Skafindas E,Evans R J,Savkin A V,Petersen I R.Stability results for switched controller systems.Automatiea,1999,35(4):553-564
[8]Brockett R W.Asymptotic stability and feedback stabilization.Differential Geometric Control Theory,Boston:Birkhauser,1983
[9]Vu L,Liberzon D.Common Lyapunov function for a families of commuting nonlinear systems.Systems & Control Letters,2005,54(5):405-416
[10]Balluehi A,Di Benedetto M D,Ponello C,et al.Hybrid control in automotive applications:the cut-off control.Automatiea,1999,35(3):519-535
[11]Oishi M,Mitchell I,Bayen A M,et al.Hybrid verification of an interface for an automatic landing.In Proceeding of the IEEE Conference on Decision and Control,2002:1607-1613
[12]Amonlirdviman K,Khare N A,Tree D R P,et al.Mathematical modeling of planar cell polarity to understand domineering nonautonomy.Science,2005,307(5708):423-426
[13]Astrom K J,Furuta K.Swinging up a pendulum by energy control.Automatica,2000,36:287-295
[14]Zhao J,Dimirovski G M.Quadratic stability of a class of switched nonlinear systems.IEEE Tram.Automat.Contr.,2004,49:574-578
[15]刘玉忠,赵军.一类非线性开关系统的二次稳定的充要条件.自动化学报,2002,28(3):596-600
[16]王仁明,关治洪,刘新芝.一类非线性切换系统的稳定性分析.系统工程与电子技术,2004,49:574-578
[17]孙文安,赵军.凸锥型不确定线性切换系统的二次镇定.控制理论与应用,2005,22(5):790-793
[18]Dayawansa W P,Mattin C F.A Lyapunov theorem for a class of dynamical systems which undergo switching.IEEE Tran on Automatic Control,2004,44(4):751-760
[19]Bdondel V,Megretski A.Unsolved problem in mathematical systems and control theory.New Jersey:Princeton University Press,2004.
[20]Mosca E.Predictive switching supervisory control of persistently disturbed input-saturated planets.Automatica,394(1),2005,91-107
[21]Mehmet Akar,Ayanendu Paul,Michael Safonov G.,et al.Conditions on the stability of a class of second-order switched systems.IEEE Transactions on Automatic control,2006,51(2):338-340
[22]Shorten R,Narendra K.Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems.Proceeding of 1999 American Control Conference,1999:1410-1414
[23]Shorten R,Narendra K.Necessary and sufficient conditions for the existence of a CQLF for a finite number of stable LTI systems.International Journal of Adaptive Control and Signal Processing,2002,16(10):709-728
[24]Cheng D Z.Stabilization of planar switched systems.System & Control Letters,2004,51:79-88
[25]Branicky M S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Trans.On Automatic Control,1998,43(4):475-482
[26]Wicks M A,DeCarlo R A.Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems.Proceeding of 1997 American Control Conference,1997:1709-1713
[27]Mignone D,Ferrari-Trecate G,Morari M.Stability and stabilization of piecewise affine and hybrid systems:An LMI approach.Proceeding of 39~(th) Conference on Decision and Control,2000:504-509
[28]Pettersson S,Lennartson B.LMI for stability and robustness of hybrid systems.In Proc.Amer.Control.Conf.,1997:1714-1718
[29]Peleties P,DeCarlo R.Asymptotic stability of m-switched systems using Lyapunov-like functions.Proceeding of the American Control Conference,1991,:1697-1684
[30]Koutsoukos X D.Antsaklis P.J.Stabilizing supervisory control of hybrid systems based on piecewise linear Lyapunov functions,Proc.IEEE Med.Conference on Control and Automation,2000:456-481
[31]DeCarlo R A,Branicky M S,Pettersson S,et al.Perspectives and results on the stability and stabilizability of hybrid systems.Proceedings of the IEEE,2000,88(7):1069-1082
[32]Branicky M S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Trans.Automat.Contr.,1998,43(4):475-482
[33]Michel A N.Recent trends in the stability analysis of hybrid dynamical systems.IEEE Trans.Circuits Syst.I,1999,46(1):120-134
[34]Ye H,Michel A N,Hou L.Stability analysis of systems with impulse effects.IEEE Trans.Automat.Contr.,1998,43(12):1719-1723
[35]Johansson M,Rantzer A.Computation of piecewise quadratic Lyapunov functions for hybrid systems.IEEE Trans.Automat.Contr.,1998,43(4):555-559
[36]Jamal D,Peirre R,Claude I.Stability analysis and control synthesis for switched systems:a switched Lyapunov function approach.IEEE Transactions on Automatic Control,2002,47(11):1883-1887
[37]Morse A S.Supervisory control of families of linear set-point controllers-part 1:exact matching.IEEE Trans.Automat.Contr.,1996,41(10):1413 - 1431
[38]Hespanha J P,Morse A S.Stability of switched systems with average dwell-time.In Proceedings of the 38~(th) Conference on Decision & Control,Phoenix,Arizona,USA,1999:2655-2660
[39]Zhai G,Hu B,Yasuda K,et al.Stability analysis of switched systems with stable and unstable subsystems:an average dwell time approach.International journal.of systems science,2001,32(8):1055-1061
[40]Zhai G,Hu B.,Yasuda K,et al.Piecewise Lyapunov functions for switched systems with average dwell time.Proc.Asian Control Conference 2000:2655-2660
[41]Liberzon D,Hepspanha J P,Morse A S.Stability of switched systems:A Lie-algebraic condition.Systems & Control Letters,1999,37:117-122
[42]秦元勋,刘永清,王联,等.带有时滞的动态系统的运动稳定性.北京,科学出版社,1989
[43]Richard J P.Time delay systems:an overview of some recent advances and open problems.Automatica,2003,39:1667-1694
[44]郑祖麻.泛函微分方程理论.合肥,安徽教育出版社,1984
[45]褚健,俞立,苏宏业.鲁棒控制理论及应用.杭州,浙江大学出版社,2000
[46]李宏飞.中立型时滞系统的鲁棒控制.西北工业大学出版社,2006.10
[47]Olbrot A W.A sufficiently large time delay in feedback loop must destroy exponential stability of any decay rete.IEEE Trans.Automat.Control,1984,29:367-368
[48]Kolmanovskii V B,Myshkis A.Applied theory of functional differential equations.Boston,Kluwer Academic Publishers,1992
[49]Kolmanovskii V B,Nosov V R.Stability of functional differential equations.London,Academic Press,1986
[50]Kuang Y.Delay differential equations with applications in population dynamics.Boston,Academic Press,1993
[51]Brayton R,Willoughby R A.On the numerical integration of a symmetric system of differential difference equations of neutral type.J.Math.Anal.,1967,18:182-279
[52]Bellen A,Guglielmi N,Ruechli A E.Methods for linear systems of circuit delay deferential equations of neutral type.IEEE Trans.Circuit Syst.I,1999,46:212-216
[53]Han Q L.On stability of linear neutral systems with mixed time delays:A discretized Lyapunov functional approach.Automatica,2005,41:1209-1218
[54]Cao D Q,He P.Sufficient conditions for stability of linear neutral systems with a single delay.Applied Mathematics Letters,2004,17:139-144
[55]He Y,Wu M,She J H,et al.Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,2004,51:57-65
[56]Park J H.Dynamic output guaranteed cost controller for neutral systems with input delay.Chaos,Solitons and Fractals,2005,23:1819-1828
[57]Catherine B,Jonathan R P.Analysis of fractional delay systems of retarded and neutral type.Automatica,2002,38:1133-1138
[58]Hu G D,Hu G D,Cahlon B.Algebraic criteria for stability of linear neutral systems with single delay.J.Compu.&Appl.Math,2001,135:125-133
[59]俞立.鲁棒控制-线性矩阵不等式处理方法.北京,清华大学出版社,2002
[60]Chen J.On computing the maximal delay intervals for stability of linear delay systems.IEEE Trans.Automat.Control,1995,40:1087-I093
[61]Hu G D,Hu G D.Some simple stability criteria of neutral delay-differential systems.Appl.Math.Comput.,1996,80:257-271
[62]Hui G D,Hu G D.Simple criteria for stability of neutral systems with multiple delays.Internat.J.Systems Sci.,1997,28:1325-1328
[63]Mahmoud.Robust H- control of linear neutral systems.Automatica,2000,36:757-764
[64]Chen J D,Lien C H,Fan K K,et al.Criteria for asymptotic stability of a class of neutral systems via LMI approach,IEE Proc.Control Theory Appl.,2001,148:442-447
[65]Fddman E.New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems.Systems Control Letters,2001,43:309-319
[66]Lien C H,Yu K W,Hsieh J G;Stability conditions for a class of neutral systems with multiple delays.J.Math.Anal.Appl.,2000,245:20-27
[67]Lien C H.New stability criteria for a class of uncertain nonlinear neutral time-delay systems.J.Systems Sci.,2001,32:215-219
[68]Nieulescu S I.On delay-dependent stability under model transformations of some neutral linear systems.Internat.J.Control,2001,74:609-617
[69]Park J H.A new delay-dependent criterion for neutral systems with multiple delays.J.Comput.Appl.Math.,2001,136:177-184
[70]Park J H.Stability criterion for neutral differential systems with mixed multiple time-varying delay argument.Math.Comput.Simulation,2002,59:401-412
[71]Fan K K,Chen J D,Lien C H,et al.Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach.J.Math.Anal.Appl.,2002,273:580-589
[72]Chen D J.New Stability criteria for a class of neutral systems with discrete and distributed time-delay:an LMI approach.Appl.Math.Comput.,2004,150:719-736
[73]Yue D,Han Q L.A delay-dependent stability criterion of nentral systems and it's application to a partial element equivalent circuit model,IEEE Transaction on Circuits and Systems-Ⅱ,2004,51(12):685-689
[74]Sen M D L,Malaina J L,Gallego A,et al.Stability of non-neutral and neutral dynamic switched systems subjected to internal delays.American Journal of Applied Science,2005,2(10):1481-1490
[75]Sen M D L.Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays.Appl.Math.Comput.,2007,185:508-526
[76]孙希明,付俊,孙洪飞,赵军.一类切换线性中立时滞系统稳定性的分析.中国电机工程学报,2005,25(23):42-46
[77]Liberzon D.Switching in systems and control.Birkhauser,Boston,2003
[78]Dayawansa W P,Martin C F.A converse Lyapunov theorem for a class of dynamical systems which undergo switching.IEEE Trans.On Automatic Control,1999,44:751-760
[79]Gu,K.An integral inequality in the stability problem of time-delay systems.In Proceedings of the 39th IEEE conference on decision and control,Sydney,Australia,December 2000:2805-2810.
[80]Wang C H,Zhang L X,Gao H J,et al.Delay-dependent stability and stabilization of a class of linear switched time-varying delay systems.Proceeding of the fourth international conference on machine learning and cybernetics,Guangzhou,2005:917-922
[81]Johansson M,Rantzer A.Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems.IEEE Trans.Automatic Control 1998,43(4):555-559
[82]Yuping Zhang,Xinzhi Liu,Hong Zhu,Shouming Zhong.Analysis and synthesis of a class of cell partitions switched neutral control systems.Proceeding on 3~(rd) International Conference on Impulsive Dynamic Systems and Applications,2006:1580-1584
[83]孙洪飞,赵军,高晓东.带有时滞摄动的线性切换系统的稳定性.控制与决策,2002,17(4):431-434
[84]Yuping Zhang,Xinzhi Liu,Hong Zhu,Shouming Zhong.Stability analysis and control synthesis for a class of switched neutral systems.Applied mathematics and computation,2007,190:1258-1266
[85]王德进.H_2和H_∞优化控制理论.哈尔滨,哈尔滨工业大学出版社,2001
[86]孙文安,赵军.基于LMI_s的不确定线性切换系统H_∞鲁棒控制.控制与决策,2005,20(6):650-655
[87]郑连伟,刘晓平,张庆灵.具有时变不确定线性的线性时滞系统的鲁棒H_∞控制.自动化学报,2001,27(3):377-380
[88]姜偕富,费树岷,冯纯伯.线性时滞系统时滞相关H_∞输出反馈控制器设计.系统科学与数学,2001,21(1):115-122
[89]姜偕富,费树岷,冯纯伯.线性时滞系统依赖于时滞的H_∞状态反馈控制.自动化学报,2001,27(1):108-114
[90]Dorato P.Non-fragile controller design:An overview.Proceedings of American Control Conference,Philadelphia,Pennsylvania,1998:2829-2831
[91]Keel L,Bhattacharyya S.Robust,fragile,or optimal.IEEE Trans.On Automatic Control,1997,42:1098-1105
[92]王武,杨富文.不确定时滞系统的时滞依赖鲁棒非脆弱H_∞控制.控制理论与应 用,2003,20(3):473-476
[93]Xie L H,De Souza C E.Robust H control for linear systems with norm-boundede time-varying uncertainty.IEEE Trans on Automatic Control,1992,37(2):1188-1191
[94]Yuping Zhang,Xinzhi Liu,Hong ghu,Shouming Zhong.Robust non-fragile H- Control for a Class of Switched Neutral Systems.2~(nd) IEEE Conference on Industrial Electronics and Applications,2007:1003-1008
[95]周东华,孙优贤.控制系统故障检测与诊断技术.北京,清华大学出版社,1994
[96]王福忠,姚波,张嗣瀛.线性系统区域稳定的可靠控制.控制理论与应用,2004,21(5):835-839
[97]Yang G H,Wang J L,Soh Y C.Reliable controller design for linear systems.Automatica,2001,37(5):717-725
[98]张榆平,朱宏.具有传感器故障的不确定系统的区域稳定可靠控制.信息与控制,2006,35(6):679-683
[99]张榆平,朱宏。具有传感器和执行器故障的不确定系统区域稳定可靠控制.华东理工大学学报(自然科学版),2006,32(7):868-871
[100]汪锐,赵军.一类非线性不确定切换系统的鲁棒可靠控制.东北大学学报,2005,26(8):715-717
[101]汪锐,孙文安,齐丽,赵军。一类线性不确定切换系统的保成本鲁棒控制.东北大学学报,2005,26(4):213-215
[102]王福忠,姚波,张嗣瀛.线性系统区域稳定的可靠控制.控制理论与应用,2004,21(5):835-839
[103]Kolmanovskii V,Myshkis A.Introduction to the Theory and Applications of Functional Differential Equations.Dordrecht,Kluwer Academic Publishers,1999
[104]Man Z H,Paplinski A P,Wu H R.A robust MIMO terminal sliding mode control scheme for rigid robot manipulators.IEEE Transactions on Automatic Control,1994,39:2464-2469
[105]Ficola A,Cave M L.A sliding mode controller for a two-joint robot with an elastic link.Mathematics and computers in Simulation,1996,41:559-569
[106]Utkin A,Guldner J,Shi J X.Sliding Mode Control in Electromechanical Systems.Taylor &Francis,1999
[107]姚琼荟,黄继起,吴汉松.变结构控制系统.重庆,重庆大学出版社,1999
[108]田宏奇.滑模控制理论及其应用.武汉出版社,1995
[109]刘金琨.滑模变结构控制MATLAB仿真.北京,清华大学出版社,2005
[110]Zhang Y,Duan G R,He L.Robust stability of a class of uncertain discrete-time switched systems with time-delay.lst International Symposium on System and Control in Aerospace and Astronautics,2006:849-852