时滞系统稳定性的若干研究
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摘要
近几十年来,时滞系统的稳定性研究引起了人们广泛关注。时滞系统的理论和实际重要性也得到了大家的普遍认可。可是,由于时滞系统的复杂性,时滞系统的稳定性问题还远远未被研究透彻。目前对这类问题的研究主要有两类方法:时域方法和特征根方法。本论文介绍了作者基于这两种方法(时域方法和特征根方法)所做的关于时滞系统稳定性的若干研究工作。
其中论文的第二章为作者基于简单Lyapunov泛函方法分析时滞系统稳定性的工作。这一章主要有三部分内容。1)提出一种新的时滞系统稳定性分析方法:积分等式方法。在这一部分首先提出一种积分等式。该结论在分析时滞系统时优于已有的积分不等式。这里还给出了积分等式中自由项的选取规则。积分等式和自由项的选取规则构成了一个有效分析时滞系统稳定性的方法。该方法较积分不等式方法保守性更低。2)分析了含分布时滞的中立系统稳定性。这一部分首先提出了一种改进Lyapunov-Krasovskii泛函。基于这个泛函的方法比已有方法保守性更低。3)分析了网络控制系统的镇定问题。在这部分,分析了一类被控对象含状态和输入时滞的网络控制系统镇定问题。
论文的第三章介绍了作者基于离散化Lyapunov泛函方法分析时滞系统稳定性的工作。这一章有四部分内容。1)提出一种新的针对中立型时滞系统的完全Lyapunov-Krasovskii泛函。基于这个新泛函的离散化Lyapunov泛函方法可较以往离散化Lyapunov泛函方法处理更广的时滞系统鲁棒稳定性问题。2)分析含混合时滞的中立系统稳定性。在这部分提出了一种新的针对混合型中立系统的离散化Lyapunov泛函方法。该方法较之前的方法保守性更低。且所得结论非常接近实际结果。3)提出一种新的时滞系统指数估计方法。该方法为基于离散化Lyapunov泛函的估计方法。从而该方法要优于基于简单Lyapunov泛函的估计方法。4)分析含时变时滞的时滞系统稳定性。这里提出了一种不需要引入输入-输出方法的分析方法。因此该方法要比已有方法更简单。同时该方法的保守性更低。
论文的第二章和第三章为时域分析方法。目前这种方法的一个主要研究动力是如何进一步降低结论的保守性。时域方法的优势为可以有效地分析鲁棒稳定性和设计控制器。而这种方法却不可避免的引入保守性。基于时域方法得到的稳定性结论为充分非必要的。因此,作者在论文的后半部分提出了两种非时域的研究方法。
在论文的第四章,提出了一种新的时滞系统稳定性分析方法:混合代数、频域方法。我们知道已有的代数方法不能给出区间型的稳定结果;而频域方法不能有效的分析给定时滞下的系统稳定性。而目前常用的时域方法又不可避免地具有保守性。当采用这些方法时,我们很难有效地分析非相称型多时滞系统的稳定性。这章提出的混合代数、频域方法克服了已有代数方法和频域方法的不足。这种方法可以准确地分析多时滞系统的稳定性并获得保证系统稳定的时滞区间。
在论文的第五章,提出一种新的方法分析时滞系统的完全稳定性。首先研究了一般形式的时滞系统虚根穿越情况并得到一些重要的性质。根据这些性质我们知道,时滞系统的虚根穿越情况可以直接由系统的结构信息得到。详细的说,这里的结构信息指哪些特征虚根属于同一个子准多项式。一个子准多项式所对应虚根的穿越方向根据这个子准多项式所含的虚根的大小顺序决定。进一步,我们可以断定一个一般形式的时不变参数时滞系统随时滞的增加将最终不稳定,如果该系统含有至少一个特征虚根。为得到这个结构信息,在这章提出了一种频域扫描方法。通过执行该频域扫描方法,可以得到想要的结构信息并求得系统的临界时滞值。这样,系统的完全稳定区间可以容易地求得。通过仿真可以看出,该方法比已有方法简单许多。
The stability of time-delay systems has received considerable attention in the last few decades. The theoretical and practical importance of this topic has been well recognized. However, the stability of time-delay systems is not fully investigated due to the complexity. Roughly speaking, there are two types of approaches to study the stability: namely, time domain ones and the eigenvalue-based ones. In this dissertation, some studies of the author based on these two approaches are introduced.
In Chapter 2, the researches based on simple Lyapunov functional methods are introduced. There are three parts in the chapter. 1) A new method for the stability analysis, an integral equality method, is proposed. First, an integral equality is presented in this part. This equality is better than the integral inequality in the stability analysis of time-delay systems. In addition, the selecting rules of the free terms are given. The integral equality and the selecting rules constitute a systematic method to study the stability. This new method is less conservative than the one based on the integral inequality. 2) The stability of neutral systems with distributed delays is studied. By constructing a modified Lyapunov-Krasovskii functional, a new and less conservative stability criterion is obtained. 3) The stabilization of networked control system, in which the plant has state and input delays, is addressed.
In Chapter 3, the studies based on discretized Lyapunov functional method are presented. Four contributions are introduced therein. 1) A new complete Lyapunov-Krasovskii functional for neutral systems is proposed. Based on this new complete functional, the discretized functional method is applicable to a wider class of robust analysis problem. 2) The stability of neutral systems with mixed delays is studied. A new discretized Lyapunov functional method is proposed. By the method, the results are less conservative than the existing ones and are very close to the analytical results. 3) A new exponential estimate method, based on discretized Lyapunov functional method, is proposed. The estimate results by the method are better than the ones obtained by the methods based simple Lyapunov functionals. 4) The stability of the systems with time-varying delays is discussed. The proposed method does not introduce the input-output method. As a consequence, the method is easier than the existing ones. In addition, the method is less conservative than the previous methods.
Chapter 2 and Chapter 3 are in the time domain framework. A main motivation for this framework is to further reduce the conservatism. The time domain methods can be used to effectively study the robustness and design controller. However, the results by the methods are inevitably conservative. In addition, the obtained stability conditions are sufficient though not necessary. Considering this, two new non-time-domain methods are proposed.
In Chapter 4, a new method, mixed algebra and frequency method is proposed. It is known that the algebra method cannot lead to delay interval which guarantees the stability while the frequency method cannot be utilized to study the system stability for given delay. As to the time-domain methods, the results are inevitably conservative. One can hardly apply these methods to address the stability of the systems with incommensurate multiple delays. The mixed algebra and frequency method can overcome the respective disadvantages of the algebra method and the frequency method. By this new method, one can accurately find the delay interval for system stability.
In Chapter 5, a new methodology is presented to study the complete stability of time-delay systems. First, we study the crossing directions of the imaginary roots and some important properties are obtained. According to these properties we can directly determine the crossing directions via the structure information. More specifically, here the structure information means which imaginary roots belong to one factored quasi-polynomial. The crossing direction of an imaginary root can be directly determined by the magnitude order of the imaginary roots of the factored quasi-polynomial where it belongs. Moreover, we can confirm that a time invariant delayed system will be ultimately unstable as delay increases if there exists at least one imaginary root. To obtain the desired structure information, a frequency sweeping method is presented. By performing the frequency sweeping method, the complete stability regions can be depicted. The method is easier than the existing ones.
其中论文的第二章为作者基于简单Lyapunov泛函方法分析时滞系统稳定性的工作。这一章主要有三部分内容。1)提出一种新的时滞系统稳定性分析方法:积分等式方法。在这一部分首先提出一种积分等式。该结论在分析时滞系统时优于已有的积分不等式。这里还给出了积分等式中自由项的选取规则。积分等式和自由项的选取规则构成了一个有效分析时滞系统稳定性的方法。该方法较积分不等式方法保守性更低。2)分析了含分布时滞的中立系统稳定性。这一部分首先提出了一种改进Lyapunov-Krasovskii泛函。基于这个泛函的方法比已有方法保守性更低。3)分析了网络控制系统的镇定问题。在这部分,分析了一类被控对象含状态和输入时滞的网络控制系统镇定问题。
论文的第三章介绍了作者基于离散化Lyapunov泛函方法分析时滞系统稳定性的工作。这一章有四部分内容。1)提出一种新的针对中立型时滞系统的完全Lyapunov-Krasovskii泛函。基于这个新泛函的离散化Lyapunov泛函方法可较以往离散化Lyapunov泛函方法处理更广的时滞系统鲁棒稳定性问题。2)分析含混合时滞的中立系统稳定性。在这部分提出了一种新的针对混合型中立系统的离散化Lyapunov泛函方法。该方法较之前的方法保守性更低。且所得结论非常接近实际结果。3)提出一种新的时滞系统指数估计方法。该方法为基于离散化Lyapunov泛函的估计方法。从而该方法要优于基于简单Lyapunov泛函的估计方法。4)分析含时变时滞的时滞系统稳定性。这里提出了一种不需要引入输入-输出方法的分析方法。因此该方法要比已有方法更简单。同时该方法的保守性更低。
论文的第二章和第三章为时域分析方法。目前这种方法的一个主要研究动力是如何进一步降低结论的保守性。时域方法的优势为可以有效地分析鲁棒稳定性和设计控制器。而这种方法却不可避免的引入保守性。基于时域方法得到的稳定性结论为充分非必要的。因此,作者在论文的后半部分提出了两种非时域的研究方法。
在论文的第四章,提出了一种新的时滞系统稳定性分析方法:混合代数、频域方法。我们知道已有的代数方法不能给出区间型的稳定结果;而频域方法不能有效的分析给定时滞下的系统稳定性。而目前常用的时域方法又不可避免地具有保守性。当采用这些方法时,我们很难有效地分析非相称型多时滞系统的稳定性。这章提出的混合代数、频域方法克服了已有代数方法和频域方法的不足。这种方法可以准确地分析多时滞系统的稳定性并获得保证系统稳定的时滞区间。
在论文的第五章,提出一种新的方法分析时滞系统的完全稳定性。首先研究了一般形式的时滞系统虚根穿越情况并得到一些重要的性质。根据这些性质我们知道,时滞系统的虚根穿越情况可以直接由系统的结构信息得到。详细的说,这里的结构信息指哪些特征虚根属于同一个子准多项式。一个子准多项式所对应虚根的穿越方向根据这个子准多项式所含的虚根的大小顺序决定。进一步,我们可以断定一个一般形式的时不变参数时滞系统随时滞的增加将最终不稳定,如果该系统含有至少一个特征虚根。为得到这个结构信息,在这章提出了一种频域扫描方法。通过执行该频域扫描方法,可以得到想要的结构信息并求得系统的临界时滞值。这样,系统的完全稳定区间可以容易地求得。通过仿真可以看出,该方法比已有方法简单许多。
The stability of time-delay systems has received considerable attention in the last few decades. The theoretical and practical importance of this topic has been well recognized. However, the stability of time-delay systems is not fully investigated due to the complexity. Roughly speaking, there are two types of approaches to study the stability: namely, time domain ones and the eigenvalue-based ones. In this dissertation, some studies of the author based on these two approaches are introduced.
In Chapter 2, the researches based on simple Lyapunov functional methods are introduced. There are three parts in the chapter. 1) A new method for the stability analysis, an integral equality method, is proposed. First, an integral equality is presented in this part. This equality is better than the integral inequality in the stability analysis of time-delay systems. In addition, the selecting rules of the free terms are given. The integral equality and the selecting rules constitute a systematic method to study the stability. This new method is less conservative than the one based on the integral inequality. 2) The stability of neutral systems with distributed delays is studied. By constructing a modified Lyapunov-Krasovskii functional, a new and less conservative stability criterion is obtained. 3) The stabilization of networked control system, in which the plant has state and input delays, is addressed.
In Chapter 3, the studies based on discretized Lyapunov functional method are presented. Four contributions are introduced therein. 1) A new complete Lyapunov-Krasovskii functional for neutral systems is proposed. Based on this new complete functional, the discretized functional method is applicable to a wider class of robust analysis problem. 2) The stability of neutral systems with mixed delays is studied. A new discretized Lyapunov functional method is proposed. By the method, the results are less conservative than the existing ones and are very close to the analytical results. 3) A new exponential estimate method, based on discretized Lyapunov functional method, is proposed. The estimate results by the method are better than the ones obtained by the methods based simple Lyapunov functionals. 4) The stability of the systems with time-varying delays is discussed. The proposed method does not introduce the input-output method. As a consequence, the method is easier than the existing ones. In addition, the method is less conservative than the previous methods.
Chapter 2 and Chapter 3 are in the time domain framework. A main motivation for this framework is to further reduce the conservatism. The time domain methods can be used to effectively study the robustness and design controller. However, the results by the methods are inevitably conservative. In addition, the obtained stability conditions are sufficient though not necessary. Considering this, two new non-time-domain methods are proposed.
In Chapter 4, a new method, mixed algebra and frequency method is proposed. It is known that the algebra method cannot lead to delay interval which guarantees the stability while the frequency method cannot be utilized to study the system stability for given delay. As to the time-domain methods, the results are inevitably conservative. One can hardly apply these methods to address the stability of the systems with incommensurate multiple delays. The mixed algebra and frequency method can overcome the respective disadvantages of the algebra method and the frequency method. By this new method, one can accurately find the delay interval for system stability.
In Chapter 5, a new methodology is presented to study the complete stability of time-delay systems. First, we study the crossing directions of the imaginary roots and some important properties are obtained. According to these properties we can directly determine the crossing directions via the structure information. More specifically, here the structure information means which imaginary roots belong to one factored quasi-polynomial. The crossing direction of an imaginary root can be directly determined by the magnitude order of the imaginary roots of the factored quasi-polynomial where it belongs. Moreover, we can confirm that a time invariant delayed system will be ultimately unstable as delay increases if there exists at least one imaginary root. To obtain the desired structure information, a frequency sweeping method is presented. By performing the frequency sweeping method, the complete stability regions can be depicted. The method is easier than the existing ones.
引文
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Michiels,W., & Vyhlídal,T.(2005).An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type. Automatica,41,991-998.
Mondié,S., & Kharitonov, V.L.(2005).Exponential estimates for retarded time-delay systems: An LMI approach. IEEE Trans. Autom. Control, vol.50,no.2, 268-273.
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Mori,T., Fukuma,N. & Kuwahara,M.(1982).On an estimate of the decay rate for stable linear delay systems. Int. J.Control,vol.36, 95-97.
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Olgac,N., & Sipahi,R. (2005).The cluster treatment of characteristic roots and the neutral type time-delayed systems. ASME J. Dyna., Syst., Measure. Control, vol.127, no.1,88-97.
Olgac, N., & Sipahi,R. (2006).An improved procedure in detecting the stability robustness of systems with uncertain delay. IEEE Trans. Autom.Control, vol.51, no.7, 1164-1165.
Papachristodoulou,A., Peet, M., & Sanjay,L.(2005). Constructing Lyapunov-Krasovskii functionals for linear time delay system. American Control Conferences. Portland, OR, USA, 2845-2850.
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Sipahi, R., & Olgac, N.(2005).Complete stability robustness of third-order LTI multiple time-delay systems. Automatica,41,1413-1422.
Sipahi, R., & Olgac, N.(2006a). A unique methodology for the stability robustness of multiple time delay systems. Syst. Control Lett.,55,819-825.
Sipahi, R.,& Olgac, N.(2006b). Stability robustness of retarded LTI systems with single delay and exhaustive determination of their imaginary spectra. SIAM J. Control Optim., 45,1680-1696.
Teel,A.R.,& Ne?i?,D.(2004).Input-to-state stability of networked control systems.Automatica, 40,2121-2128.
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