时滞系统若干问题的研究
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摘要
在各种实际的工业系统中,时滞是一种普遍存在的现象。其存在是引起系统不稳定和性能变差的重要原因;因此,分析时滞现象对系统动力学行为及控制性能的影响,以及如何利用或消除这种影响一直是控制理论与控制工程领域的研究热点。
本文从控制理论的基本概念与方法出发,利用Lyapunov-Krasovskii泛函方法,以线性矩阵不等式(LMI)为主要工具,在构造合适的Lyapunov-Krasovskii泛函的基础上,通过使用不同的分析方法,探讨了时滞系统的若干问题。本文的主要工作包括以下几个方面:
1.针对中立型时滞与离散时滞不同的中立型常时滞系统,考虑其的鲁棒稳定性问题。首先,构造了新颖的Lyapunov-Krasovskii泛函并证明了其正定性;然后,通过使用积分不等式方法和引入自由矩阵,得到了与中立型时滞及离散时滞均相关的稳定性结论;并通过理论分析和仿真例子说明了所得到的结论在保守性上优于现存的一些结果,同时还说明了中立型时滞与离散时滞之间的关系。
2.分析了一类中立型变时滞系统的鲁棒稳定性问题。通过构造合适的Lyapunov-Krasovskii泛函,并使用积分不等式、自由矩阵和凸组合条件等分析方法,得到了基于线性矩阵不等式的与中立型时滞、离散时滞及离散时滞导数均相关的充分条件,并通过理论分析和数值仿真说明了所得到的结论具有较小的保守性,同时还说明了中立型时滞、离散时滞及其导数三者之间的关系。
3.探讨了一般时滞系统的动态输出反馈控制镇定问题。通过引入自由矩阵对系统进行适当的变换、构造合适的Lyapunov-Krasovskii泛函,得到了系统时滞相关的控制器的存在性条件;然后在此基础上通过控制器的参数化方法,将控制器的参数与泛函参数的求解归结为线性矩阵不等式的形式,并给出了动态反馈控制器的具体表达式。
4.针对更具有一般性的中立型分布时滞系统模型,设计动态输出反馈控制器,使得闭环系统为渐近稳定的。通过对系统进行模型变换、构造合适的Lyapunov-Krasovskii泛函以及使用参数化方法,得到了基于线性矩阵不等式的控制器存在的充分条件,所得到的结论不仅与离散时滞而且与分布时滞相关。由于该系统模型的一般性,因此,一般时滞系统、中立型时滞系统和分布时滞系统的动态输出反馈控制器的设计问题,均可以作为本文的特例得到。
5.考虑了不确定时滞系统的鲁棒容错控制问题。针对一般时滞系统,基于一种更具有一般性的传感器故障模型,设计动态输出反馈控制器,使得闭环系统在传感器发生故障时仍然能保持渐近稳定。在使用一个保守性较小的稳定性定理的基础上,通过非线性变换,得到了控制器时滞相关的存在性条件;进一步通过锥补线性化算法,求解得到了泛函参数及控制器参数,并给出了控制器的具体表达式。
6.在范数有界不确定和非线性不确定性两种不同情况下,讨论了随机时滞系统的均方指数稳定性问题。基于伊藤微积分法则和布朗运动的基本性质,通过构造不同的Lyapunov-Krasovskii泛函、在一些独立变量的交叉项上引入泛函参数,并使用自由矩阵,得到了时滞相关的稳定性结论,并通过理论分析和数值仿真说明了所得到的稳定性判据在保守性上优于现存的结论。
Time delays are often encountered in various practical systems. It has been shown that the existence of delays in a dynamic system may result in instability, oscillations or poor performances. Therefore, the stability analysis and the synthesis of controllers for time-delay systems have received considerable attentions over the decades.
Based on the basic concepts and methods of control theory, with the help of LMI tools, through constructing proper Lyapunov-Krasovskii functional and using various derivation methods, several issues of time-delay systems are investigated in this dissertation. The major contributions of this thesis are as follows:
1. The robust stability of uncertain neutral systems with mixed delays is studied. Firstly, novel Lyapunov-Krasovskii functional is constructed and its positive definiteness is proved by using integral inequality, which relaxes the constraint on some parameters of Lyapunov-Krasovskii functional. Then, by introducing slack matrices, the stability criteria are obtained in terms of LMIs, which is dependent on sizes of neutral delay and discrete delay. Theory analysis and numerical examples are given to demonstrate that the obtained results are less conservative than some existing stability criteria. Furthermore, the relationship between neutral- and discrete delay is also illustrated.
2. The robust stability of uncertain neutral systems with time-varying delay is investigated. Through constructing proper Lyapunov-Krasovskii functional, using slack matrices and the convex combination condition, the stability criteria, which are dependent on the sizes of neutral delay, discrete delay and its derivative, are derived in terms of LMIs. The significant improvement on the conservativeness of the delay bound over some reported results are illustrated by theory analysis and numerical examples, and the relationships of neutral delay, discrete delay and its derivative are also presented.
3. The dynamical output feedback stabilization for retarded time-delay systems is considered. A proper transformation of the closed-loop system by introducing free matrices is used and the correspondent Lyapunov-Krasovskii functional is constructed, by which the delay-dependent stability is derived. Then, the parameterization of controller is used to es- tablish the design condition in terms of LMI with respect to all parameters of controller and Lyapunov-Krasovskii functional. The desired controller is also explicitly formulated.
4. The issue of stabilization for the linear neutral systems with mixed delays is concerned. Based on the model transformation of neutral type, the Lyapunov-Krasovskii functional method is employed to establish the stability criterion, which is dependent both on discrete delay and distributed delay. Then, through the controller parameterization, the desired parameters are determined under the design condition in terms of LMI, and the desired controller is also explicitly formulated. Since the considered system is a more general case, the design of controllers for some time-delay system, such as retarded time-delay system, neutral system without distributed delay, can be seen as a special case of this result.
5. The fault-tolerant control for uncertain linear time-delay systems is considered. Attention is focused on the design of the dynamical output feedback controller, which maintain the asymptotically stability of closed-loop systems with sensor failures. Based on a general sensor model of failure and a less conservative stability condition, nonlinear transformation and cone-complementary linearization iterative algorithm are used, and the desired controller is obtained.
6. The exponential stability in mean square for uncertain stochastic systems with time delay is discussed. The uncertainties under consideration are norm-bounded uncertainties and nonlinear uncertainties. Based on It(o|^) calculus rules and the properties of the Brown motion, through constructing proper Lyapunov-Krasovskii functional, introducing additional parameter of Lyapunov-Krasovskii functional and using slack matrices, the novel delay-dependent stability criteria are obtained in terms of LMIs. The significant improvement on the conservativeness of the delay bound over some reported results are illustrated by theory analysis and numerical examples.
本文从控制理论的基本概念与方法出发,利用Lyapunov-Krasovskii泛函方法,以线性矩阵不等式(LMI)为主要工具,在构造合适的Lyapunov-Krasovskii泛函的基础上,通过使用不同的分析方法,探讨了时滞系统的若干问题。本文的主要工作包括以下几个方面:
1.针对中立型时滞与离散时滞不同的中立型常时滞系统,考虑其的鲁棒稳定性问题。首先,构造了新颖的Lyapunov-Krasovskii泛函并证明了其正定性;然后,通过使用积分不等式方法和引入自由矩阵,得到了与中立型时滞及离散时滞均相关的稳定性结论;并通过理论分析和仿真例子说明了所得到的结论在保守性上优于现存的一些结果,同时还说明了中立型时滞与离散时滞之间的关系。
2.分析了一类中立型变时滞系统的鲁棒稳定性问题。通过构造合适的Lyapunov-Krasovskii泛函,并使用积分不等式、自由矩阵和凸组合条件等分析方法,得到了基于线性矩阵不等式的与中立型时滞、离散时滞及离散时滞导数均相关的充分条件,并通过理论分析和数值仿真说明了所得到的结论具有较小的保守性,同时还说明了中立型时滞、离散时滞及其导数三者之间的关系。
3.探讨了一般时滞系统的动态输出反馈控制镇定问题。通过引入自由矩阵对系统进行适当的变换、构造合适的Lyapunov-Krasovskii泛函,得到了系统时滞相关的控制器的存在性条件;然后在此基础上通过控制器的参数化方法,将控制器的参数与泛函参数的求解归结为线性矩阵不等式的形式,并给出了动态反馈控制器的具体表达式。
4.针对更具有一般性的中立型分布时滞系统模型,设计动态输出反馈控制器,使得闭环系统为渐近稳定的。通过对系统进行模型变换、构造合适的Lyapunov-Krasovskii泛函以及使用参数化方法,得到了基于线性矩阵不等式的控制器存在的充分条件,所得到的结论不仅与离散时滞而且与分布时滞相关。由于该系统模型的一般性,因此,一般时滞系统、中立型时滞系统和分布时滞系统的动态输出反馈控制器的设计问题,均可以作为本文的特例得到。
5.考虑了不确定时滞系统的鲁棒容错控制问题。针对一般时滞系统,基于一种更具有一般性的传感器故障模型,设计动态输出反馈控制器,使得闭环系统在传感器发生故障时仍然能保持渐近稳定。在使用一个保守性较小的稳定性定理的基础上,通过非线性变换,得到了控制器时滞相关的存在性条件;进一步通过锥补线性化算法,求解得到了泛函参数及控制器参数,并给出了控制器的具体表达式。
6.在范数有界不确定和非线性不确定性两种不同情况下,讨论了随机时滞系统的均方指数稳定性问题。基于伊藤微积分法则和布朗运动的基本性质,通过构造不同的Lyapunov-Krasovskii泛函、在一些独立变量的交叉项上引入泛函参数,并使用自由矩阵,得到了时滞相关的稳定性结论,并通过理论分析和数值仿真说明了所得到的稳定性判据在保守性上优于现存的结论。
Time delays are often encountered in various practical systems. It has been shown that the existence of delays in a dynamic system may result in instability, oscillations or poor performances. Therefore, the stability analysis and the synthesis of controllers for time-delay systems have received considerable attentions over the decades.
Based on the basic concepts and methods of control theory, with the help of LMI tools, through constructing proper Lyapunov-Krasovskii functional and using various derivation methods, several issues of time-delay systems are investigated in this dissertation. The major contributions of this thesis are as follows:
1. The robust stability of uncertain neutral systems with mixed delays is studied. Firstly, novel Lyapunov-Krasovskii functional is constructed and its positive definiteness is proved by using integral inequality, which relaxes the constraint on some parameters of Lyapunov-Krasovskii functional. Then, by introducing slack matrices, the stability criteria are obtained in terms of LMIs, which is dependent on sizes of neutral delay and discrete delay. Theory analysis and numerical examples are given to demonstrate that the obtained results are less conservative than some existing stability criteria. Furthermore, the relationship between neutral- and discrete delay is also illustrated.
2. The robust stability of uncertain neutral systems with time-varying delay is investigated. Through constructing proper Lyapunov-Krasovskii functional, using slack matrices and the convex combination condition, the stability criteria, which are dependent on the sizes of neutral delay, discrete delay and its derivative, are derived in terms of LMIs. The significant improvement on the conservativeness of the delay bound over some reported results are illustrated by theory analysis and numerical examples, and the relationships of neutral delay, discrete delay and its derivative are also presented.
3. The dynamical output feedback stabilization for retarded time-delay systems is considered. A proper transformation of the closed-loop system by introducing free matrices is used and the correspondent Lyapunov-Krasovskii functional is constructed, by which the delay-dependent stability is derived. Then, the parameterization of controller is used to es- tablish the design condition in terms of LMI with respect to all parameters of controller and Lyapunov-Krasovskii functional. The desired controller is also explicitly formulated.
4. The issue of stabilization for the linear neutral systems with mixed delays is concerned. Based on the model transformation of neutral type, the Lyapunov-Krasovskii functional method is employed to establish the stability criterion, which is dependent both on discrete delay and distributed delay. Then, through the controller parameterization, the desired parameters are determined under the design condition in terms of LMI, and the desired controller is also explicitly formulated. Since the considered system is a more general case, the design of controllers for some time-delay system, such as retarded time-delay system, neutral system without distributed delay, can be seen as a special case of this result.
5. The fault-tolerant control for uncertain linear time-delay systems is considered. Attention is focused on the design of the dynamical output feedback controller, which maintain the asymptotically stability of closed-loop systems with sensor failures. Based on a general sensor model of failure and a less conservative stability condition, nonlinear transformation and cone-complementary linearization iterative algorithm are used, and the desired controller is obtained.
6. The exponential stability in mean square for uncertain stochastic systems with time delay is discussed. The uncertainties under consideration are norm-bounded uncertainties and nonlinear uncertainties. Based on It(o|^) calculus rules and the properties of the Brown motion, through constructing proper Lyapunov-Krasovskii functional, introducing additional parameter of Lyapunov-Krasovskii functional and using slack matrices, the novel delay-dependent stability criteria are obtained in terms of LMIs. The significant improvement on the conservativeness of the delay bound over some reported results are illustrated by theory analysis and numerical examples.
引文
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