时滞系统的稳定性分析
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摘要
在实际的动力学系统中,时滞现象是普遍存在的,这常常是破坏系统稳定性,影响系统控制性能的主要因素。因此,对时滞系统的稳定性研究一直是控制领域的一个热点问题。到目前为止,尽管取得了很多的成果,但是都带有很大的保守性。本文通过引入时滞分割的思想,构造新的Lyapunov-Krasovskii泛函,得到了优越性随时滞分割的细化而明显的稳定性判据。
首先,我们应用时滞分割的思想来分析带有非线性项的时滞非线性系统的稳定性,具体的研究对象选择了工程应用背景很强的时滞神经网络,为时滞Hopfield神经网络建立了新的全局渐近稳定性准则,及为时滞细胞神经网络建立了新的全局指数稳定性定理。最终的定理由线性矩阵不等式形式给出,可由MATLAB工具箱求解。最后的应用算例证明了新的定理的优越性。
其次,在实际的动力学系统中,时滞往往是变化的,而且,时变时滞可能是在非零下界跟上界之间变化。在本文中,我们将这种区间变时滞分成不变部分和变化部分,并对下界应用时滞分割的思想,进而得到了保守性低的稳定性判据。此外,由于不确定性的广泛存在,我们也考虑了变时滞系统的结构参数不确定性和非线性扰动的情形,并用相应的处理方法得到了优越的稳定性判据。
最后,我们研究了时滞系统的几个热点问题,为时滞神经网络建立了新的无源性定理,也建立了带有耦合时滞的复杂网络的同步稳定性判据。并且,在稳定性分析的基础了上,为离散时滞神经网络设计了状态观测器,提出了一种Smith预估器设计的新思路。
本文应用的时滞分割的思想以及相关的先进的处理方法,对时滞系统的稳定性分析有着重要的意义,大量的应用算例也保证了该方法的可行性和与优越性。
In practical dynamic systems, time delays are general, which often lead to instability and bad performance. Then, the stability analysis of system with time delay is the hot topic in the control field. Based on Lyapunov stability theory, many results have been obtained, which still need improvement. By constructing new Lyapunov-Krasovskii functional with the idea of delay fractioning, this paper presents new stability theorems, which become less conservative as the fraction number increases.
First, we introduce the idea of delay partitioning to perform stability analysis for the delay systems with nonlinearity, such as neural networks, which have been extensively applied in engineering. We obtain the asymptotic stability condition for Hopfield neural networks and the exponential stability theorem for cellular neural networks. These results are in standard form of linear matrix inequalities and can be solved by MATLAB tools. Some examples are provided to prove the advantage of these theorems.
Second, in practice, time delays are often time varying, which usually vary from nonzero lower bound. This paper first partitions the time-varying delay into constant part and time-varying part, then, apply delay fractioning to the constant part. In addition, we consider the structure parameter uncertainty and nonlinear disturbance and apply proper methods to get the stability conditions.
Finally, this paper deals with some other hot-topic problems of time delay systems. For instance, we obtain new passivity theorem for neural networks with time-varying delays; present new synchronization stability of complex networks with coupling delays. Furthermore, based on the stability analysis, this paper proposes the method of state estimator design and the new idea of Smith predictor.
The delay fractioning idea plays an important role in the stability analysis of time delay systems。Its advantage can be shown by many examples.
首先,我们应用时滞分割的思想来分析带有非线性项的时滞非线性系统的稳定性,具体的研究对象选择了工程应用背景很强的时滞神经网络,为时滞Hopfield神经网络建立了新的全局渐近稳定性准则,及为时滞细胞神经网络建立了新的全局指数稳定性定理。最终的定理由线性矩阵不等式形式给出,可由MATLAB工具箱求解。最后的应用算例证明了新的定理的优越性。
其次,在实际的动力学系统中,时滞往往是变化的,而且,时变时滞可能是在非零下界跟上界之间变化。在本文中,我们将这种区间变时滞分成不变部分和变化部分,并对下界应用时滞分割的思想,进而得到了保守性低的稳定性判据。此外,由于不确定性的广泛存在,我们也考虑了变时滞系统的结构参数不确定性和非线性扰动的情形,并用相应的处理方法得到了优越的稳定性判据。
最后,我们研究了时滞系统的几个热点问题,为时滞神经网络建立了新的无源性定理,也建立了带有耦合时滞的复杂网络的同步稳定性判据。并且,在稳定性分析的基础了上,为离散时滞神经网络设计了状态观测器,提出了一种Smith预估器设计的新思路。
本文应用的时滞分割的思想以及相关的先进的处理方法,对时滞系统的稳定性分析有着重要的意义,大量的应用算例也保证了该方法的可行性和与优越性。
In practical dynamic systems, time delays are general, which often lead to instability and bad performance. Then, the stability analysis of system with time delay is the hot topic in the control field. Based on Lyapunov stability theory, many results have been obtained, which still need improvement. By constructing new Lyapunov-Krasovskii functional with the idea of delay fractioning, this paper presents new stability theorems, which become less conservative as the fraction number increases.
First, we introduce the idea of delay partitioning to perform stability analysis for the delay systems with nonlinearity, such as neural networks, which have been extensively applied in engineering. We obtain the asymptotic stability condition for Hopfield neural networks and the exponential stability theorem for cellular neural networks. These results are in standard form of linear matrix inequalities and can be solved by MATLAB tools. Some examples are provided to prove the advantage of these theorems.
Second, in practice, time delays are often time varying, which usually vary from nonzero lower bound. This paper first partitions the time-varying delay into constant part and time-varying part, then, apply delay fractioning to the constant part. In addition, we consider the structure parameter uncertainty and nonlinear disturbance and apply proper methods to get the stability conditions.
Finally, this paper deals with some other hot-topic problems of time delay systems. For instance, we obtain new passivity theorem for neural networks with time-varying delays; present new synchronization stability of complex networks with coupling delays. Furthermore, based on the stability analysis, this paper proposes the method of state estimator design and the new idea of Smith predictor.
The delay fractioning idea plays an important role in the stability analysis of time delay systems。Its advantage can be shown by many examples.
引文
1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 (1):1~7.
2胡海岩,王在华,王怀磊.非线性时滞系统的动力学与控制.现代数学和力学. 2005:1~5.
3 K. Zhou, P. P. Khargonekar. Robust Stabilization of linear systems with norm-bounded time-varyig uncertainty. Systems & Cotrol Letters. 1988, 10(1):17-20.
4俞立,鲁棒控制—线性矩阵不等式处理方法,清华大学出版社,2002
5 A. Hmamed. Stability Conditions of Delay Differential systems:new results. International Journal of Control, 1986, 46:455-463.
6 J. Su, I. Fong, and C. Tsen .Stability Analysis of Linear System with Time Delay. IEEE Trans. Automatic Control. 1994, 39(6):1341-1343.
7 K. Gu, V.L. Kharitonov and J. Chen. Stability of time-delay systems. Springer-Verlag, Berlin, Germany, 2003.
8 P. Park. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans. Automatic Control. 1999,44(4): 876-877.
9 Y.S. Moon, P. Park, W.H. Kwon and Y.S.Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control. 2001,74:1447-1455.
10 E. Fridman and U. Shaked. Stability of linear descriptor systems with delay: A Lyapunov-based approach. Jounal of Mathematical Analysis and Applications. 2002, 273(1):24-44.
11 S. Xu and J. Lam. Improved delay-dependent stability criteria for time-delay systems. IEEE Trans. Automatic Control. 2005.50(3):384-387.
12 Y. He, Q.Wang, L. Xie and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Trans. Automatic Control. 2007. 52(2):293-299.
13 Q. Han. On stability of linear neutral systems with mixed time delays: A discretized Lyapunov functional approach. Automatica, 2005. 41(7):1209–1218.
39 W. Michiel, S. Niculescu, On the delay sensitivity of Smith predictors. Int. J. Syst. Sci., 2003,34:543-551.
40 C. Morarescut, S.I. Niculescu, K. Gu. On the geometry of stability regions of Smith predictors subject to delay uncertainty. IMA Journal of Mathematical Control and Information. 2007, 24:411-423.
41 C. Li and X. Liao. Passivity analysis of neural networks with time delay. IEEE Trans. On Circuits and Systems. 2005, 52(8):471-475.
42 C. Li and and G. Chen. Synchronization in general complex dynamical networks with coupling delays. Physica-A, 2004, 343:263-278.
43 H. Gao, J. Lam and G. Chen. New critera for synchronization stability of general complex dynamical networks with coupling delays. Physics Letters A, 2006, 360:263-373.
2胡海岩,王在华,王怀磊.非线性时滞系统的动力学与控制.现代数学和力学. 2005:1~5.
3 K. Zhou, P. P. Khargonekar. Robust Stabilization of linear systems with norm-bounded time-varyig uncertainty. Systems & Cotrol Letters. 1988, 10(1):17-20.
4俞立,鲁棒控制—线性矩阵不等式处理方法,清华大学出版社,2002
5 A. Hmamed. Stability Conditions of Delay Differential systems:new results. International Journal of Control, 1986, 46:455-463.
6 J. Su, I. Fong, and C. Tsen .Stability Analysis of Linear System with Time Delay. IEEE Trans. Automatic Control. 1994, 39(6):1341-1343.
7 K. Gu, V.L. Kharitonov and J. Chen. Stability of time-delay systems. Springer-Verlag, Berlin, Germany, 2003.
8 P. Park. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans. Automatic Control. 1999,44(4): 876-877.
9 Y.S. Moon, P. Park, W.H. Kwon and Y.S.Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control. 2001,74:1447-1455.
10 E. Fridman and U. Shaked. Stability of linear descriptor systems with delay: A Lyapunov-based approach. Jounal of Mathematical Analysis and Applications. 2002, 273(1):24-44.
11 S. Xu and J. Lam. Improved delay-dependent stability criteria for time-delay systems. IEEE Trans. Automatic Control. 2005.50(3):384-387.
12 Y. He, Q.Wang, L. Xie and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Trans. Automatic Control. 2007. 52(2):293-299.
13 Q. Han. On stability of linear neutral systems with mixed time delays: A discretized Lyapunov functional approach. Automatica, 2005. 41(7):1209–1218.
39 W. Michiel, S. Niculescu, On the delay sensitivity of Smith predictors. Int. J. Syst. Sci., 2003,34:543-551.
40 C. Morarescut, S.I. Niculescu, K. Gu. On the geometry of stability regions of Smith predictors subject to delay uncertainty. IMA Journal of Mathematical Control and Information. 2007, 24:411-423.
41 C. Li and X. Liao. Passivity analysis of neural networks with time delay. IEEE Trans. On Circuits and Systems. 2005, 52(8):471-475.
42 C. Li and and G. Chen. Synchronization in general complex dynamical networks with coupling delays. Physica-A, 2004, 343:263-278.
43 H. Gao, J. Lam and G. Chen. New critera for synchronization stability of general complex dynamical networks with coupling delays. Physics Letters A, 2006, 360:263-373.