中立型时滞系统的鲁棒性分析
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摘要
在对实际控制系统建模时,由于不可避免地存在着测量误差、各种干扰以及未建模动态等,导致系统模型与实际问题之间存在着误差,一般称这些误差为系统的不确定性。除此之外,系统中还常常存在着时滞,时滞不仅存在于状态中,而且还可能存在于状态的变化率中,这就是中立型时滞系统。不确定性和时滞是影响中立型系统性能的重要因素,因此对不确定中立型时滞系统的研究是系统控制问题研究的重要内容,也是难点内容。本文针对中立型时滞系统,在不确定性满足一定的假设条件下,研究其鲁棒稳定性。
1.研究具有离散时滞与分布时滞的中立型系统的鲁棒稳定性问题。首先,通过构造适当的Lyapunov泛函,结合不等式分析技巧,给出了系统与分布时滞相关的鲁棒稳定性条件;然后,通过构造另外的Lyapunov泛函,引入含有自由矩阵的恒等式,再结合矩阵不等式,推出了系统与离散时滞和分布时滞相关的鲁棒稳定性条件。上述条件均能转化为线性矩阵不等式。最后,构造了相应的算例对所得结果的有效性进行验证。
2.研究中立型时变时滞系统的指数稳定性问题。针对两种不同的初始状态空间,即连续的初始状态空间和连续可微的初始状态空间,运用Razuminkhin-Lyapunov技巧,获得了系统时滞无关的指数稳定性充分条件。
During establishing models for practical control systems, there are discrepancies between the system model and the practical problems which can be described as uncertainties of systems resulting from inevitable measurement error, various interference and unmodelled dynamics, and so on. In addition, there always exist time delays in the system. Time delays not only exist in the state, but also may appear in the state change rate, that is so called neutral systems with time delays. Uncertainty and time delays are important factors which affect the performance of neutral systems. Therefore, the research on uncertain neutral systems with time delays is an important and difficult part in the research on the problems of system control. In this paper the robust stability of neutral systems with time delays is investigated, assumed that the uncertainties in the system satisfy some conditions.
1. The problem of the robust stability for neutral systems with discrete and distribution time delays is investigated. Firstly, by constructing appropriate Lyapunov functional combined with the analysis technique of the inequality, the condition of distributed-delay-dependent robust stability of the system is present. Then by constructing appropriate Lyapunov functional and introducing zero equations with free-weighting matrices, together with using linear matrix inequalities, discrete-delay-dependent and distributed-delay-dependent robust stability condition of the system is deduced. Those conditions can be translated into linear matrix inequalities. Finally, a number of examples are presented to validate the effectiveness of the proposed results.
2. The problem of exponential stability for neutral systems with time-varying delays is investigated. By employing the Razumikhin-Lyapunov technique, the sufficient conditions of delay-independent exponential stability are obtained for both different initial state space which are continuous initial state space and continuously differentiable initial state space.
1.研究具有离散时滞与分布时滞的中立型系统的鲁棒稳定性问题。首先,通过构造适当的Lyapunov泛函,结合不等式分析技巧,给出了系统与分布时滞相关的鲁棒稳定性条件;然后,通过构造另外的Lyapunov泛函,引入含有自由矩阵的恒等式,再结合矩阵不等式,推出了系统与离散时滞和分布时滞相关的鲁棒稳定性条件。上述条件均能转化为线性矩阵不等式。最后,构造了相应的算例对所得结果的有效性进行验证。
2.研究中立型时变时滞系统的指数稳定性问题。针对两种不同的初始状态空间,即连续的初始状态空间和连续可微的初始状态空间,运用Razuminkhin-Lyapunov技巧,获得了系统时滞无关的指数稳定性充分条件。
During establishing models for practical control systems, there are discrepancies between the system model and the practical problems which can be described as uncertainties of systems resulting from inevitable measurement error, various interference and unmodelled dynamics, and so on. In addition, there always exist time delays in the system. Time delays not only exist in the state, but also may appear in the state change rate, that is so called neutral systems with time delays. Uncertainty and time delays are important factors which affect the performance of neutral systems. Therefore, the research on uncertain neutral systems with time delays is an important and difficult part in the research on the problems of system control. In this paper the robust stability of neutral systems with time delays is investigated, assumed that the uncertainties in the system satisfy some conditions.
1. The problem of the robust stability for neutral systems with discrete and distribution time delays is investigated. Firstly, by constructing appropriate Lyapunov functional combined with the analysis technique of the inequality, the condition of distributed-delay-dependent robust stability of the system is present. Then by constructing appropriate Lyapunov functional and introducing zero equations with free-weighting matrices, together with using linear matrix inequalities, discrete-delay-dependent and distributed-delay-dependent robust stability condition of the system is deduced. Those conditions can be translated into linear matrix inequalities. Finally, a number of examples are presented to validate the effectiveness of the proposed results.
2. The problem of exponential stability for neutral systems with time-varying delays is investigated. By employing the Razumikhin-Lyapunov technique, the sufficient conditions of delay-independent exponential stability are obtained for both different initial state space which are continuous initial state space and continuously differentiable initial state space.
引文
[1] KOLMANOVSKII V B, MYSHKIS A. Applied Theory of Functional Differential Equations[M]. Kluwer Academic Publishers, Boston, 1992: 220-291.
[2] KOLMANOVSKII V B, NOSOV V R. Stability of Functional Differential Equations [M]. Academic Press, London, 1986: 189-273.
[3] NICULESCU S I. Delay Effects on Stability: A Robust Control Approach[M]. Spring Berlin, 2001: 283-326.
[4] KUANG Y. Delay Differential Equations with Applications in Population Dynamics[M]. Academic Press, Boston, 1993: 34-67.
[5] BONNET C, PARTINGTON J R. Analysis of Fractional Delay Systems of Retraded and Neutral Type[J]. Automatica, 2002, 38: 1133-1138.
[6] HU G D, CAHLON B. Algebraic Criteria for Stability of Linear Neutral Systems with A Single Delay[J]. Compu. &Appl. Math., 2001, 135: 125-133.
[7] XU B G, LIU X X, MA X J. A Stability Criterion of Linear Neutral Delay-differential Systems Based on Frequency-domain[C]. IEEE 2004 8th International Conference on Control, 2004: 2006-2009.
[8] AGARWAR R P, GRACE S R. Asymptotic Stability of Differential Systems of Neutral Type[J]. Applied Mathematics Letters, 2000, 13: 15-19.
[9] PARK JU H, WON S. Stability Analysis for Neutral Delay-differential Systems[J]. Journal of the Franklin Institute, 2000, 337: 1-9.
[10] PARK JU H, KWON Q. On New Stability Criterion for Delay-differential Systems of Neutral Type[J]. Applied Mathematics and Computation, 2005, 162: 627-637.
[11] GU K. An Integral Inequality in the Stability Problem of Time-delay Systems[J]. Proceedings of 39th IEEE CDC, Sydney, Australia, 2000: 2805-2810.
[12] YUE D, WON S, WON O. Delay-dependent Stability of Neutral Systems with Time Delay: A LMI Approach[J]. IEE Proceedings Control Theory and Applications, 2003, 150: 23-27.
[13] LIEN C H, YU K W, HSIEH J G. Stability Conditions for A Class of Neutral Systems with Multiple Time Delays[J]. Math. Anal. Appl. 2000, 245: 20-27.
[14] NICULESCU S L. Further Remarks on Delay-dependent Stability of LinearNeutral Systems[J]. Proc. of MTUS, 2000, 37: 4368-4372.
[15]张先明,吴敏,何勇.中立型时滞系统的时滞相关稳定性[J].自动化学报,2004,30(4): 624-628.
[16] HAN QING LONG. Robust Stability of Uncertain Delay-differential Systems of Neutral Type[J]. Automatica, 2002, 38: 719-723.
[17] HE YONG, WU MIN, SHE JIN HUA, et al. Delay-dependent Robust Stability Criteria for Uncertain Neutral Systems with Mixed Delays[J]. Systems & Control Letters, 2004, 51: 57-65.
[18] HAN QINGLONG. On Stability of Linear Neutral Systems with Mixed Time Delays: A Discretized Lyapunov Functional Approach[J]. Automatica, 2005, 47: 1209-1218.
[19] HALE J K, INFANTE E F, TSEN F S P. Stability of Linear Delay Equations[J]. Journal of Mathematical Analysis and Applications, 1985, 105: 533-555.
[20] HUI G D, HU G D. Simple Criteria for Stability of Neutral Systems with Multiple Delays[J]. International Journal of Systems Science, 1996, 28: 1325-1328.
[21] PARK JU H, WON S. Asymptotic Stability of Neutral Systems with Multiple Delays[J]. Journal of Optimization Theory and Applications, 1999, 103: 183-200.
[22] PARK JU H. A New Delay-dependent Criterion for Neutral Systems with Multiple Delays[J]. Computational and Applied Mathematics, 2001, 136: 177-184.
[23] MOON Y S. Robust Control of Time-delay System Using Linear Matrix Inequalities[J]. Systems & Control Letters, 1998, 15: 32-36.
[24] YANG BIN, LI TAO, JIANG YAN HONG. On the Stability of Neutral Systems with Multiple Delays[C]. American Control Conference, 2003: 5066-5067.
[25] LIU XIN GE, WU MIN, MARTIN RALPH et al. Stability Analysis for Neutral Systems with Mixed Delays[J]. Journal of Computational and Applied Mathematics, 2006, 137: 198-217.
[26]曾铭涛,桂卫华,唐朝晖.一类变时滞中立型系统的全局指数稳定性研究[J].计算机工程与科学,2005, 27(10): 77-81.
[27] PARK JU H. Novel Robust Stability Criterion for A Class of Neutral Systems with Mixed Delays and Nonlinear Perturbations[J]. Applied Mathematics and Computation, 2005, 161: 413-421.
[28] CHEN JENQ DER. Robust Control for Uncertain Neutral Systems with Time-delays in State and Control Input via LMI and Gas[J]. Applied Mathematics and Computation, 2004, 157: 535-548.
[29]李宏飞,罗学波.具有非线性扰动的中立型系统的鲁棒稳定性[J].西南师范大学学报,2004, 4(29): 539-545.
[30] HE YONG, WU MIN. On Delay-dependent Stability of Neutral Systems with Nonlinear Perturbations. Mathematical Theory and Applications, 2003, 3(23): 91-96.
[31] HAN Q L, YU L. Robust Stability of Linear Neutral Systems with Nonlinear Parameter Perturbations[J]. IEEE Proc. Control Theory Appl., 2004, 5(151): 539-547.
[32] PARK JU H, JUNG HO Y. On the Exponential Stability of A Class of Nonlinear Systems Including Delayed Perturbations[J]. Journal of Computational and Applied Mathematics, 2003, 159: 467-471.
[33]廖晓晰.稳定性的理论、方法和应用[M].武汉:华中科技大学出版社, 2002: 42-65.
[34] KHUSAIN D Y, YUNKOVA E A. Investigation of the Stability of Linear Systems of Neutral Type by the Lyapunov Function Method[J]. Differential Equations, 1988, 24: 424-431.
[35]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: 55-68.
[36] EI GHAOUI, BOYD S, FERON L, et al. Linear Matrix Inequalities in Systems and Control Theory[M]. Philadelphia: SIAM, 1994: 120-133.
[37] GU K. An Integer Inequality in the Stability Problem of Time-delay Systems[J]. In Proc. 39th IEEE CDC Sydney, Australia, 2000: 2805-2810.
[38] XIE L. Output Feedback H∞Control of Systems with Parameter Uncertainty[J]. International Journal Control, 1996, 63: 741-750.
[39] HALE J. Theory of Functional Differential Equations[M]. New York: Springer, 1977: 42-43.
[40] CHEN J D, LIEN C H, et al. Criteria for Asymptotic Stability of A Class of Neutral Systems: A LMI Approach[J]. IEE Proc. Control Theory Appl., 2001, 6(148): 44-47.
[41] HAN Q H. A Descriptor Systems Approach to Robust Stability of UncertainNeutral Systems with Discrete and Distributed Delays[J]. Automatica, 2004, 40: 1791-1796.
[42] YUE D, WON S, KWON O. Delay-dependent Stability of Neutral Systems with Time Delay: A LMI Approach[J]. IEE Proceedings Control Theory and Applications, 2003, 150: 23-27.
[43] BELLMAN R, COOKE K. Differential Equations[M]. New York: Academic, 1963: 102-110.
[2] KOLMANOVSKII V B, NOSOV V R. Stability of Functional Differential Equations [M]. Academic Press, London, 1986: 189-273.
[3] NICULESCU S I. Delay Effects on Stability: A Robust Control Approach[M]. Spring Berlin, 2001: 283-326.
[4] KUANG Y. Delay Differential Equations with Applications in Population Dynamics[M]. Academic Press, Boston, 1993: 34-67.
[5] BONNET C, PARTINGTON J R. Analysis of Fractional Delay Systems of Retraded and Neutral Type[J]. Automatica, 2002, 38: 1133-1138.
[6] HU G D, CAHLON B. Algebraic Criteria for Stability of Linear Neutral Systems with A Single Delay[J]. Compu. &Appl. Math., 2001, 135: 125-133.
[7] XU B G, LIU X X, MA X J. A Stability Criterion of Linear Neutral Delay-differential Systems Based on Frequency-domain[C]. IEEE 2004 8th International Conference on Control, 2004: 2006-2009.
[8] AGARWAR R P, GRACE S R. Asymptotic Stability of Differential Systems of Neutral Type[J]. Applied Mathematics Letters, 2000, 13: 15-19.
[9] PARK JU H, WON S. Stability Analysis for Neutral Delay-differential Systems[J]. Journal of the Franklin Institute, 2000, 337: 1-9.
[10] PARK JU H, KWON Q. On New Stability Criterion for Delay-differential Systems of Neutral Type[J]. Applied Mathematics and Computation, 2005, 162: 627-637.
[11] GU K. An Integral Inequality in the Stability Problem of Time-delay Systems[J]. Proceedings of 39th IEEE CDC, Sydney, Australia, 2000: 2805-2810.
[12] YUE D, WON S, WON O. Delay-dependent Stability of Neutral Systems with Time Delay: A LMI Approach[J]. IEE Proceedings Control Theory and Applications, 2003, 150: 23-27.
[13] LIEN C H, YU K W, HSIEH J G. Stability Conditions for A Class of Neutral Systems with Multiple Time Delays[J]. Math. Anal. Appl. 2000, 245: 20-27.
[14] NICULESCU S L. Further Remarks on Delay-dependent Stability of LinearNeutral Systems[J]. Proc. of MTUS, 2000, 37: 4368-4372.
[15]张先明,吴敏,何勇.中立型时滞系统的时滞相关稳定性[J].自动化学报,2004,30(4): 624-628.
[16] HAN QING LONG. Robust Stability of Uncertain Delay-differential Systems of Neutral Type[J]. Automatica, 2002, 38: 719-723.
[17] HE YONG, WU MIN, SHE JIN HUA, et al. Delay-dependent Robust Stability Criteria for Uncertain Neutral Systems with Mixed Delays[J]. Systems & Control Letters, 2004, 51: 57-65.
[18] HAN QINGLONG. On Stability of Linear Neutral Systems with Mixed Time Delays: A Discretized Lyapunov Functional Approach[J]. Automatica, 2005, 47: 1209-1218.
[19] HALE J K, INFANTE E F, TSEN F S P. Stability of Linear Delay Equations[J]. Journal of Mathematical Analysis and Applications, 1985, 105: 533-555.
[20] HUI G D, HU G D. Simple Criteria for Stability of Neutral Systems with Multiple Delays[J]. International Journal of Systems Science, 1996, 28: 1325-1328.
[21] PARK JU H, WON S. Asymptotic Stability of Neutral Systems with Multiple Delays[J]. Journal of Optimization Theory and Applications, 1999, 103: 183-200.
[22] PARK JU H. A New Delay-dependent Criterion for Neutral Systems with Multiple Delays[J]. Computational and Applied Mathematics, 2001, 136: 177-184.
[23] MOON Y S. Robust Control of Time-delay System Using Linear Matrix Inequalities[J]. Systems & Control Letters, 1998, 15: 32-36.
[24] YANG BIN, LI TAO, JIANG YAN HONG. On the Stability of Neutral Systems with Multiple Delays[C]. American Control Conference, 2003: 5066-5067.
[25] LIU XIN GE, WU MIN, MARTIN RALPH et al. Stability Analysis for Neutral Systems with Mixed Delays[J]. Journal of Computational and Applied Mathematics, 2006, 137: 198-217.
[26]曾铭涛,桂卫华,唐朝晖.一类变时滞中立型系统的全局指数稳定性研究[J].计算机工程与科学,2005, 27(10): 77-81.
[27] PARK JU H. Novel Robust Stability Criterion for A Class of Neutral Systems with Mixed Delays and Nonlinear Perturbations[J]. Applied Mathematics and Computation, 2005, 161: 413-421.
[28] CHEN JENQ DER. Robust Control for Uncertain Neutral Systems with Time-delays in State and Control Input via LMI and Gas[J]. Applied Mathematics and Computation, 2004, 157: 535-548.
[29]李宏飞,罗学波.具有非线性扰动的中立型系统的鲁棒稳定性[J].西南师范大学学报,2004, 4(29): 539-545.
[30] HE YONG, WU MIN. On Delay-dependent Stability of Neutral Systems with Nonlinear Perturbations. Mathematical Theory and Applications, 2003, 3(23): 91-96.
[31] HAN Q L, YU L. Robust Stability of Linear Neutral Systems with Nonlinear Parameter Perturbations[J]. IEEE Proc. Control Theory Appl., 2004, 5(151): 539-547.
[32] PARK JU H, JUNG HO Y. On the Exponential Stability of A Class of Nonlinear Systems Including Delayed Perturbations[J]. Journal of Computational and Applied Mathematics, 2003, 159: 467-471.
[33]廖晓晰.稳定性的理论、方法和应用[M].武汉:华中科技大学出版社, 2002: 42-65.
[34] KHUSAIN D Y, YUNKOVA E A. Investigation of the Stability of Linear Systems of Neutral Type by the Lyapunov Function Method[J]. Differential Equations, 1988, 24: 424-431.
[35]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: 55-68.
[36] EI GHAOUI, BOYD S, FERON L, et al. Linear Matrix Inequalities in Systems and Control Theory[M]. Philadelphia: SIAM, 1994: 120-133.
[37] GU K. An Integer Inequality in the Stability Problem of Time-delay Systems[J]. In Proc. 39th IEEE CDC Sydney, Australia, 2000: 2805-2810.
[38] XIE L. Output Feedback H∞Control of Systems with Parameter Uncertainty[J]. International Journal Control, 1996, 63: 741-750.
[39] HALE J. Theory of Functional Differential Equations[M]. New York: Springer, 1977: 42-43.
[40] CHEN J D, LIEN C H, et al. Criteria for Asymptotic Stability of A Class of Neutral Systems: A LMI Approach[J]. IEE Proc. Control Theory Appl., 2001, 6(148): 44-47.
[41] HAN Q H. A Descriptor Systems Approach to Robust Stability of UncertainNeutral Systems with Discrete and Distributed Delays[J]. Automatica, 2004, 40: 1791-1796.
[42] YUE D, WON S, KWON O. Delay-dependent Stability of Neutral Systems with Time Delay: A LMI Approach[J]. IEE Proceedings Control Theory and Applications, 2003, 150: 23-27.
[43] BELLMAN R, COOKE K. Differential Equations[M]. New York: Academic, 1963: 102-110.