滞后型时滞动力系统的α-稳定性和稳定性区域
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摘要
本文研究了滞后型时滞动力系统的α-稳定性及其稳定性区域。
对于α-稳定性分析,问题转化为具有与时滞相关的系数的时滞系统的渐近稳定性。我们采用的是稳定性切换思想,研究随着某个参数的变化,特征方程的根是否由左半复平面跨过虚轴到右半复平面或由右半复平面跨过虚轴到左半复平面;采用Beretta和Kuang提出的几何方法,引入S j=τ-τj确定临界参数值,进而判断特征根是否穿越虚轴。
在稳定性区域分析中,为了确定保证时滞系统α-稳定性的一个稳定性区域,本文提出了“稳定性切换点法”的计算方法。在一阶时滞系统情形,本文分别采用该方法和应用Hayes定理、LambertW函数法进行分析、比较,得到的结果是相同的。为了说明该方法的有效性,我们将其进一步应用到向日葵方程,比较方便地得到了稳定性的一个区域。
In this thesis, theα-Stability and its stable region in the parameter space are investigated for some delay differential equations of retarded type.
The first part of the thesis is devoted to theα-stability analysis, which is justified if a certain characteristic quasi-polynomial with delay-dependent coefficients has roots with negative real parts only. On the basis of the graphical test of stability switches for time-delayed systems, proposed by Bretta and Kuang, an auxiliary function S j=τ-τj is introduced to find the critical values of the control parameter, for which the characteristic quasi-polynomial has a pair of conjugate roots on the imaginary axis, and to determine whether this pair of roots passes through the imaginary axis as the control parameter varies.
In the second part of the thesis, a new method, based on the principal of stability switch, is proposed to determine the stable region for theα-stability of time-delay systems. Two examples are given to show the efficiency of the method. One is a first-order delay differential equation, and the other is the sunflower equation of second order. In the former example, a comparison is made between the proposed method and the available methods on the basis of Hayes Theorem and Lambert W function, the three routines yield the same stable region.
对于α-稳定性分析,问题转化为具有与时滞相关的系数的时滞系统的渐近稳定性。我们采用的是稳定性切换思想,研究随着某个参数的变化,特征方程的根是否由左半复平面跨过虚轴到右半复平面或由右半复平面跨过虚轴到左半复平面;采用Beretta和Kuang提出的几何方法,引入S j=τ-τj确定临界参数值,进而判断特征根是否穿越虚轴。
在稳定性区域分析中,为了确定保证时滞系统α-稳定性的一个稳定性区域,本文提出了“稳定性切换点法”的计算方法。在一阶时滞系统情形,本文分别采用该方法和应用Hayes定理、LambertW函数法进行分析、比较,得到的结果是相同的。为了说明该方法的有效性,我们将其进一步应用到向日葵方程,比较方便地得到了稳定性的一个区域。
In this thesis, theα-Stability and its stable region in the parameter space are investigated for some delay differential equations of retarded type.
The first part of the thesis is devoted to theα-stability analysis, which is justified if a certain characteristic quasi-polynomial with delay-dependent coefficients has roots with negative real parts only. On the basis of the graphical test of stability switches for time-delayed systems, proposed by Bretta and Kuang, an auxiliary function S j=τ-τj is introduced to find the critical values of the control parameter, for which the characteristic quasi-polynomial has a pair of conjugate roots on the imaginary axis, and to determine whether this pair of roots passes through the imaginary axis as the control parameter varies.
In the second part of the thesis, a new method, based on the principal of stability switch, is proposed to determine the stable region for theα-stability of time-delay systems. Two examples are given to show the efficiency of the method. One is a first-order delay differential equation, and the other is the sunflower equation of second order. In the former example, a comparison is made between the proposed method and the available methods on the basis of Hayes Theorem and Lambert W function, the three routines yield the same stable region.
引文
[1] J.K.Hale, Theory of Function Differential Equation, New York: Springer-Verlag,1977.
[2] Y.Kuang, Delay Differential Equation with Applications to Population Dynamics , New York: Academic Press, 1993.
[3]李森林,温立志,泛函微分方程,长沙:湖南科学技术出版社, 1987.
[4]秦元勋,刘永清,王联,郑祖庥,带有时滞的动力系统的运动稳定性(第二版),北京:科学出版社,1989.
[5]徐鉴,裴利军,时滞系统动力学近期研究进展与展望,力学进展,2006, 36(1):17-30.
[6] H.Y.Hu and Z.H.Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin:Springer-Verlag, 2002.
[7]胡海岩,王在华,非线性时滞动力系统的研究进展,力学进展, 1999,29(4):501-512.
[8] Heiden. U an der, Walther.H.O, Existence of chaos in control system with delayed feedback, Journal of Differential Equations, 1983, 47:273-295.
[9] V.Sree Kristanya, Stability analysis of structurally unstable man-machine system involving time delays, Nonlinear Analysis: Real World Applications ,2005, 6: 845-857.
[10] X.F.Liao, G.R.Chen, Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two Time delays, Int..J.Bifurcation and Chaos, 2001,11(8):2105- 2121.
[11] Z.H.Wang,H.Y.Hu, Stability switches of time-delayed dynamic systems with unknown param -eters, Journal of Sound and Vibration 2000,233(2):215-233.
[12] H.Y.Hu, E.H.Dowell , Resonance of harmonically forced Duffing oscillator with delay state feedback, Nonlinear Dynamics, 1998, 15:311-327.
[13] S.LDas,A.Chatterjee,Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics. 2002, 30:323-335.
[14] Z.H.Wang,H.Y.Hu, Pseudo-Oscillator Analysis of Scalar Nonlinear Systems with Time-Delay near a hopf Bifurcation. Int..J.Bifurcation and Chaos, 2007, 17(8): 2805-2814.
[15]黄琳,系统与控制论中的线性代数,北京:科学出版社,1984.
[16] J.L.Mozola ,G..Chen, Hopf Bifuecation Analysis: A Frequency Domain Approach, Singapore: World Science, 1996.
[17] P.Wahi, A.Chatterjee, Aspmptotics for the characteristic roots of delayed dynamics systems, Journal of Applied Mechanics, 2005,72: 475-483.
[18] C.Hwang , Yi-C. Cheng, A note on the use of the Lambert W function in the stability analysis of time-delay systems, Automatica, 2005, 41: 1979-1985.
[19] H.Shinozaki, T.Mori, Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results , Automatica 2006, 42;1791-1799.
[20] Y. Q. Chen, K.L.Moore , Analysical stability bounal for delayed second-order systems with repeating poles using Lambert W, Automatica, 2002,38: 891-895.
[21] K.L.Cooke, Z.Grossman, Discrete delay, distributed delay and stability switches,J.Math.Anal, Appl. 1982, 86: 592-627.
[22] H.Y.Hu, Using delayed state feedback to stabilitize periodic motions of an oscillator, Journal of Sound and Vibration, 2004, 275: 1009-1025.
[23] G..Stephen, Y.Kuang, A stage structured predator-prey model and its dependence on matura -tion delay and death rate, Math. Biol, 2004, 49: 188-200.
[24]马苏奇,陆启韶,具有非线性出生率的时滞模型稳定性分析。南京师范大学学报(自然科学版),2005,28(2):1-5.
[25] W.G.Aiello, H.I.Freedman, A time-delay model of single-spaces growth with stage structure , Math. Biosci.1990, 101: 139-153.
[26] K.L.Cooke, P.V.Diressche and X.Zou, Internation of maturation delay and nonlinear birth in population and epidemic models, J.Math.Biol, 1999, 39: 332-352.
[27] E.Beretta, Y.Kuang , Geometric stability switches criteria in delay differential systems with delay dependent parameters, SIAM.J.Math.Anal. 2002, 33:1144-1165.
[28] J.Q.Li, Z.E.Ma, Stability switches in a class of characteristic equations with delay dependent parameters, Nonlinear Analysis, 2004, 5:389-408.
[29] J.Q.Li, Z.E.Ma,Ultimate Stability of a Type of Characteristic Equation with Delay Dependent Parameters, Jrl.Syst.Sci & Complexity, 2006, 19: 137-144.
[30] Z.H.Wang,H.Y.Hu, H.L.Wang, Robust stabilization to non-linear delayed systems via delayed state feedback: the averaging method , Journal of Sound and Vibration , 2005, 279:937-953.
[31] L.Huang, Fundamentals of Stability and Robustness, Beijing: Science Press, 2003.
[32] M.Y.Fu, A.WOlbrot, M.P.Polis., Robust stability for time-delay systems:the edge theoremand graphicaltests, IEEE Transactions on Automatic Control,1989, 34: 813-820.
[33] Z.H.Wang,H.Y.Hu, Robust stability of time-delay systems with uncertain parameters, IUTAM Symposium on Dynamics and Control of Systems with Uncertainty,2007.363-372.
[34]李俊余,非线性复系数时滞系统的局部Hopf分岔, [南京航空航天大学硕士学位论文]江苏南京,南京航空航天大学,2007.
[35]唐风军,严迎建,参数对向日葵方程的Hopf分岔的影响,吉林大学自然科学学报, 2000, 2:29-31.
[36] T.Mori, E.Noldus, and M.Kuwahara, Automatica, 1983, 19: 571-572.
[37] T.Mori, M.Fukuma, and M.Kuwahara, On an estimate of the decay rate for stable linear delay systems, Int. J. Control, 1982, 36: 95.
[38] A.Hmamed,Comments on“On an estimate of the delay rate for stable linear delay system”,Int .J. Control, 1985, 42, 2: 539-540.
[39] T.Mori,Further comments on Comments on“On an estimate of the decay rate for stable linear delay systems”, Int. J. Control, 1986,43(5): 1613-1614.
[40] H.Bourles,α?Stability of systems governed by a functional equation—extension of results concerning linear delay systems,Int. J. Control, 1987, 45(6): 2233-2234.
[41] H.Bourles,α?Stability and robustness of large-scale interconnected systems, Int. J. Control, 1987, 45(6):2221-2232.
[42] Rong-Jyue Wang , Wen-June Wang,α?Stability analysis of rertubed systems with multiple multiple noncommensurate time delays,IEEE Transactions on Circuits and Systems-1:Funda -mental Theory and Applications, 1996, 43,(4):349-352.
[43] J.Li, Z H.Wang, Hopf bifurcation of a nonlinear Lasota-Wazwska-type population model with with maturation delay , Dynamics of Continuous, Discrete & Impulsive Systems, Ser B:Appli -cation & Algorithems, 2007,14:611-623.
[44] Chengjun Sun,Yiping Lin,MaoanHan,Analysis for a special first order characteristic Equation with delay dependent parameters, Chaos,Solitons and Fractals, 2007, 33:388-395.
[2] Y.Kuang, Delay Differential Equation with Applications to Population Dynamics , New York: Academic Press, 1993.
[3]李森林,温立志,泛函微分方程,长沙:湖南科学技术出版社, 1987.
[4]秦元勋,刘永清,王联,郑祖庥,带有时滞的动力系统的运动稳定性(第二版),北京:科学出版社,1989.
[5]徐鉴,裴利军,时滞系统动力学近期研究进展与展望,力学进展,2006, 36(1):17-30.
[6] H.Y.Hu and Z.H.Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin:Springer-Verlag, 2002.
[7]胡海岩,王在华,非线性时滞动力系统的研究进展,力学进展, 1999,29(4):501-512.
[8] Heiden. U an der, Walther.H.O, Existence of chaos in control system with delayed feedback, Journal of Differential Equations, 1983, 47:273-295.
[9] V.Sree Kristanya, Stability analysis of structurally unstable man-machine system involving time delays, Nonlinear Analysis: Real World Applications ,2005, 6: 845-857.
[10] X.F.Liao, G.R.Chen, Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two Time delays, Int..J.Bifurcation and Chaos, 2001,11(8):2105- 2121.
[11] Z.H.Wang,H.Y.Hu, Stability switches of time-delayed dynamic systems with unknown param -eters, Journal of Sound and Vibration 2000,233(2):215-233.
[12] H.Y.Hu, E.H.Dowell , Resonance of harmonically forced Duffing oscillator with delay state feedback, Nonlinear Dynamics, 1998, 15:311-327.
[13] S.LDas,A.Chatterjee,Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics. 2002, 30:323-335.
[14] Z.H.Wang,H.Y.Hu, Pseudo-Oscillator Analysis of Scalar Nonlinear Systems with Time-Delay near a hopf Bifurcation. Int..J.Bifurcation and Chaos, 2007, 17(8): 2805-2814.
[15]黄琳,系统与控制论中的线性代数,北京:科学出版社,1984.
[16] J.L.Mozola ,G..Chen, Hopf Bifuecation Analysis: A Frequency Domain Approach, Singapore: World Science, 1996.
[17] P.Wahi, A.Chatterjee, Aspmptotics for the characteristic roots of delayed dynamics systems, Journal of Applied Mechanics, 2005,72: 475-483.
[18] C.Hwang , Yi-C. Cheng, A note on the use of the Lambert W function in the stability analysis of time-delay systems, Automatica, 2005, 41: 1979-1985.
[19] H.Shinozaki, T.Mori, Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results , Automatica 2006, 42;1791-1799.
[20] Y. Q. Chen, K.L.Moore , Analysical stability bounal for delayed second-order systems with repeating poles using Lambert W, Automatica, 2002,38: 891-895.
[21] K.L.Cooke, Z.Grossman, Discrete delay, distributed delay and stability switches,J.Math.Anal, Appl. 1982, 86: 592-627.
[22] H.Y.Hu, Using delayed state feedback to stabilitize periodic motions of an oscillator, Journal of Sound and Vibration, 2004, 275: 1009-1025.
[23] G..Stephen, Y.Kuang, A stage structured predator-prey model and its dependence on matura -tion delay and death rate, Math. Biol, 2004, 49: 188-200.
[24]马苏奇,陆启韶,具有非线性出生率的时滞模型稳定性分析。南京师范大学学报(自然科学版),2005,28(2):1-5.
[25] W.G.Aiello, H.I.Freedman, A time-delay model of single-spaces growth with stage structure , Math. Biosci.1990, 101: 139-153.
[26] K.L.Cooke, P.V.Diressche and X.Zou, Internation of maturation delay and nonlinear birth in population and epidemic models, J.Math.Biol, 1999, 39: 332-352.
[27] E.Beretta, Y.Kuang , Geometric stability switches criteria in delay differential systems with delay dependent parameters, SIAM.J.Math.Anal. 2002, 33:1144-1165.
[28] J.Q.Li, Z.E.Ma, Stability switches in a class of characteristic equations with delay dependent parameters, Nonlinear Analysis, 2004, 5:389-408.
[29] J.Q.Li, Z.E.Ma,Ultimate Stability of a Type of Characteristic Equation with Delay Dependent Parameters, Jrl.Syst.Sci & Complexity, 2006, 19: 137-144.
[30] Z.H.Wang,H.Y.Hu, H.L.Wang, Robust stabilization to non-linear delayed systems via delayed state feedback: the averaging method , Journal of Sound and Vibration , 2005, 279:937-953.
[31] L.Huang, Fundamentals of Stability and Robustness, Beijing: Science Press, 2003.
[32] M.Y.Fu, A.WOlbrot, M.P.Polis., Robust stability for time-delay systems:the edge theoremand graphicaltests, IEEE Transactions on Automatic Control,1989, 34: 813-820.
[33] Z.H.Wang,H.Y.Hu, Robust stability of time-delay systems with uncertain parameters, IUTAM Symposium on Dynamics and Control of Systems with Uncertainty,2007.363-372.
[34]李俊余,非线性复系数时滞系统的局部Hopf分岔, [南京航空航天大学硕士学位论文]江苏南京,南京航空航天大学,2007.
[35]唐风军,严迎建,参数对向日葵方程的Hopf分岔的影响,吉林大学自然科学学报, 2000, 2:29-31.
[36] T.Mori, E.Noldus, and M.Kuwahara, Automatica, 1983, 19: 571-572.
[37] T.Mori, M.Fukuma, and M.Kuwahara, On an estimate of the decay rate for stable linear delay systems, Int. J. Control, 1982, 36: 95.
[38] A.Hmamed,Comments on“On an estimate of the delay rate for stable linear delay system”,Int .J. Control, 1985, 42, 2: 539-540.
[39] T.Mori,Further comments on Comments on“On an estimate of the decay rate for stable linear delay systems”, Int. J. Control, 1986,43(5): 1613-1614.
[40] H.Bourles,α?Stability of systems governed by a functional equation—extension of results concerning linear delay systems,Int. J. Control, 1987, 45(6): 2233-2234.
[41] H.Bourles,α?Stability and robustness of large-scale interconnected systems, Int. J. Control, 1987, 45(6):2221-2232.
[42] Rong-Jyue Wang , Wen-June Wang,α?Stability analysis of rertubed systems with multiple multiple noncommensurate time delays,IEEE Transactions on Circuits and Systems-1:Funda -mental Theory and Applications, 1996, 43,(4):349-352.
[43] J.Li, Z H.Wang, Hopf bifurcation of a nonlinear Lasota-Wazwska-type population model with with maturation delay , Dynamics of Continuous, Discrete & Impulsive Systems, Ser B:Appli -cation & Algorithems, 2007,14:611-623.
[44] Chengjun Sun,Yiping Lin,MaoanHan,Analysis for a special first order characteristic Equation with delay dependent parameters, Chaos,Solitons and Fractals, 2007, 33:388-395.