时滞系统的稳定性分析与滤波器设计
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摘要
本文讨论了几类线性/非线性时滞系统的稳定性与滤波器设计问题,其中包含线性时滞系统、Markov跳跃系统、T-S模糊系统、网络控制系统。由于时滞系统的稳定性分析与滤波器设计的研究具有重要的理论和实际意义,因此受到广泛关注,并得到了许多研究成果。本文发展了一些时滞依赖的处理方法和技巧,通过使用线性矩阵不等式技术、时滞分段分析方法、矩阵函数凸性和一些矩阵不等式放缩技巧,获得了具有更小保守性的时滞系统的稳定性分析判据和滤波器设计算法。具体而言,本文的结构框架描述如下：
●第一章给出了本课题研究的背景、动机、结构以及本课题的主要贡献,并阐述了本文各章节的主要研究内容。
●第二章分别讨论了一类具有时变时滞和随机时滞的连续系统的滤波器设计问题。在2.2节,基于时滞分段分析方法及矩阵函数凸性,得到了具有更小保守性的滤波器存在的充分性条件,并用数值算例来验证本节提出方法的有效性和优越性。在2.3节,对一类具有随机时滞的线性系统设计了H∞滤波器,其中此时的随机时滞假定满足某些随机特性,通过考虑时滞在不同区间的取值概率,引入了一个满足贝努力分布的随机变量,建立了一个新的包含时滞概率分布的数学模型,借以得到了具有更小保守性滤波器存在的判据,该判据不仅和时滞的大小有关还和时滞的概率分布有关。最后通过一个数值算子验证该方法的有效性和优越性。
·第三章研究了一类具有Markov参数跳跃的时滞系统的滤波器设计问题。在3.2节,对一类具有确定时延的Markov参数跳跃系统进行了滤波器设计。首先,采用对确定时延进行分段和Lyapunov稳定性理论,得到了具有更小保守性的滤波误差系统稳定的判据,然后,借以得到了用LMIs形式表达的滤波器增益。最后,通过数值例子验证了该方法的有效性和优越性。在3.3节,对类具有时变时延的Markov参数跳跃系统进行了滤波器设计。音先,通过使用Lyapunov(?)急定性理论和矩阵函数凸性,对滤波误差系统进行H∞性能分析,得到了具有更小保守性的稳定性判据,进一步得到了用LMIs形式表示的要设计的滤波器增益。最后,用数值例子验证了该方法的有效性和优越性。
●第四章对一类T-S模糊系统分别讨论了镇定、可靠性控制以及H∞滤波器设计等问题。在4.2节,对一类带有随机输入时滞的T-S模糊离散系统设计了有记忆控制器。其中,通过使用状态增广的方法来补偿随机输入时滞给系统带来的影响,然后,引入一系列的满足贝努力分布的随机变量,进而建立了新的更符合实际的状态空间模型。在新建模型的基础上,通过利用Lyapunov泛函稳定性理论,设计了目标系统的有记忆的控制器,并且得到了该类T-S模糊离散系统随机均方稳定的充分性条件。最后借助一个数值例子验证了该方法的有效性。在4.3节,考虑了一类带有随机传感器和执行器故障的T-S模糊系统的可靠性控制问题。这里,每个传感器和执行器的故障都是随机出现的,他们失真的概率用两个不相关的满足一定概率分布的随机变量来表示。通过考虑概率传感器和执行器故障,建立了一个新的系统模型。基于新的数学模型,设计该系统的可靠性控制器,并且利用Lyapunov(?)急定性理论和线性矩阵不等式技术得到了该类系统稳定的充分性条件。最后通过一个数值例子验证了本方法的有效性。在4.4节,通过使用分段分析的方法,对一类T-S模糊系统设计了H∞滤波器。通过将时滞区间划分成多个小区间,然后利用矩阵函数凸性,以及引入自由权矩阵,得到用LMIs表示的具有更小保守性的时滞依赖H∞滤波器存在可行解的判据。最后在数值算例中,通过与已有文献结果进行比较,验证了本节提出方法的有效性以及具有更小的保守性。
●第五章,讨论了一类网络控制系统的H∞滤波器设计。在5.2节,对一类具有随机传感器时滞的网络控制系统进行了H∞滤波器设计。通过引入一个满足贝努力分布的随机变量,建立了一个新的系统模型。进而借助线性矩阵不等式技术得到了滤波误差系统稳定性判据。最后通过数值例子验证该方法的有效性。在5.3节,研究了一类具有随机传感器失真的离散网络控制系统。同时,在考虑网络诱导下的非理想QoS,建立了更符合实际的数学模型,然后通过使用矩阵函数凸性和LMIs技术得到了目标系统稳定的判据。需要指出的是,此时的稳定性判据不仅和传感器网络诱导的时延有关,还和随机传感器失真的概率有关。最后通过数值例子验证方法的有效性。
●第六章,总结了本论文的主要研究内容,并指出需要进一步去研究的工作。
In this thesis, the problems of stability analysis and H∞filtering are discussed for several classed of linear/nonlinear time-delay systems, including linear systems, Markov jump systems, nonlinear systems represented by T-S fuzzy models and net-work control systems. Recently, filter design for time-delay systems is a research subject of great practical and theoretical significance, which has received consider-able attention, and much significant work has been carried out. In this thesis, some new approaches will be developed to solve the stability analysis and H∞filtering design problems for several kinds of time-delay systems. The merit of the proposed approaches lie in their less design conservatism, which is realized by utilizing some more advance techniques such as linear matrix inequality (LMI) approach, piecewise analysis method, convexity property of the matrix inequality and more powerful re-laxation techniques. More specifically, the frame and description of this thesis are given as follows:
●In Chapter 1, the introduction of the thesis are given, including the background and motivation, the outline and contribution, and the research contents to be introduced in each individual chapters.
●In Chapter 2, the H∞filter design problems are discussed for a class of continue-time systems with time-varying delay and stochastic delay. In Section 2.2, Based on a piecewise analysis method and using the convexity property of the matrix inequality, new criteria are derived for H∞filtering, which can lead to much less conservative analysis results. A numerical example is given to demon-strate the effectiveness and the merit of the proposed method. In Section 2.3, we design an H∞filter for a class of linear time delay systems with random delay. The delay considered here is assumed to be satisfying a certain stochas-tic characteristic. Corresponding to the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli random binary distribution is introduced and a new system model is established by employing the information of the probability distribution. Then, new criteria are derived for the filtering-error systems, which can lead to much leas conservative analysis results. It should be noted that the solvability of the obtained criteria depend on not only the size of the delay, but also on the probability distribution of it. A numerical example is given to demonstrate the effectiveness and the merit of the proposed method.
In Chapter 3, the H∞filter design problems are addressed for a class of Markov jump systems (MJSs) with time-varying delay. In Section 3.2, we propose an H∞filter design for MJSs with time delay. Firstly, by exploiting the delay partitioning-based Lyapunov function, new criteria are derived for the H∞per-formance analysis of the filtering-error systems, which can lead to much les conservative analysis results. Secondly, based on the obtained conditions, the filter gain can be obtained in terms of LMIs. Finally, numerical examples are given to demonstrate the effectiveness and the merit of the proposed method. In Section 3.3, we propose a class of H∞filter design for MJSs with time-varying delays. Firstly, by exploiting a new Lyapunov function and using the convexity property of the matrix inequality, new criteria are derived for the H∞performance analysis of the filtering-error systems, which can lead to much less conservative analysis results. Secondly, based on the obtained conditions, the gain of filter can be obtained in terms of LMIs. Finally, numerical examples are given to demonstrate the effectiveness and the merit of the proposed method.
In Chapter 4, we deal with the stabilization, reliable control and H∞filter design problems for a class of nonlinear described by T-S fuzzy model. In Sec-tion 4.2, we design a memory controller for T-S fuzzy discrete-time systems with random input delay. A novel state space model with the compensator for the effects of the stochastic input delays is derived by introducing stochas-tic variables satisfying Bernoulli random binary distribution and using state augmentation method. Based on the new built model, memory controller is designed and sufficient conditions for the stochastic mean square stable of T-S fuzzy discrete-time systems are obtained by using Lyapunov functional method. Finally, a numerical example is given to show the effectiveness of the proposed method. In Section 4.3, a reliable control problem for T-S fuzzy discrete system with stochastic sensors and actuators faults is investigated. The faults of each sensor or actuator occur randomly and its failure rates are governed by two sets of unrelated random variables satisfying certain probabilistic distribution. In terms of the probabilistic failures of every sensor or actuator, a new fault model is proposed. Based on the new fault model, reliable controller is de-signed and sufficient conditions for the exponentially mean square stability of T-S fuzzy systems are derived by using Lyapunov functional method and LMIs technique. Finally, a numerical example is given to show the effectiveness of the proposed method. In Section 4.4, H∞filter design for nonlinear systems with time-delay via T-S fuzzy model approach is investigated based on a piecewise analysis method.Based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, then the convexity property of the matrix inequality and the free weighting matrix method are fully used in this paper. Some novel delay-dependent H∞filtering criteria are expressed as a set of LMIs, which can lead to much less conservative analysis results. Finally, a numerical example is given to illustrate that the results in this paper are more effective and less conservative than some existing ones.
●In Chapter 5, the reliable H∞filtering are considered for a class of network control systems (NCSs) with randomly varying sensor delay and stochastic sensor-failure. In Section 5.2, we design an H∞filter for NCSs with randomly varying sensor delay. A stochastic variable satisfying Bernoulli random binary distribution is introduced and a new system model is established. By using LMIs technique, sufficient conditions are derived for ensuring the mean-square stochastic stability of the filtering error systems. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach. In Section 5.3, we study a class of network control systems with stochastic sensor-failure, a more general NCS model is established with considering non-ideal Quality of Service(QoS). By using the convexity property of the matrix inequality and LMIs technique, new criteria for the stochastic mean square stable of NCSs are derived for the target systems. It should be noted that the solvability of the obtained criteria depend on not only the sensor induced delay, but also on the probability sensor-failure. A numerical example is given to demonstrate the effectiveness of the proposed method.
●In Chapter 6, the main contents of this thesis are summarized and a few po-tential topics for further research are pointed out.
●第一章给出了本课题研究的背景、动机、结构以及本课题的主要贡献,并阐述了本文各章节的主要研究内容。
●第二章分别讨论了一类具有时变时滞和随机时滞的连续系统的滤波器设计问题。在2.2节,基于时滞分段分析方法及矩阵函数凸性,得到了具有更小保守性的滤波器存在的充分性条件,并用数值算例来验证本节提出方法的有效性和优越性。在2.3节,对一类具有随机时滞的线性系统设计了H∞滤波器,其中此时的随机时滞假定满足某些随机特性,通过考虑时滞在不同区间的取值概率,引入了一个满足贝努力分布的随机变量,建立了一个新的包含时滞概率分布的数学模型,借以得到了具有更小保守性滤波器存在的判据,该判据不仅和时滞的大小有关还和时滞的概率分布有关。最后通过一个数值算子验证该方法的有效性和优越性。
·第三章研究了一类具有Markov参数跳跃的时滞系统的滤波器设计问题。在3.2节,对一类具有确定时延的Markov参数跳跃系统进行了滤波器设计。首先,采用对确定时延进行分段和Lyapunov稳定性理论,得到了具有更小保守性的滤波误差系统稳定的判据,然后,借以得到了用LMIs形式表达的滤波器增益。最后,通过数值例子验证了该方法的有效性和优越性。在3.3节,对类具有时变时延的Markov参数跳跃系统进行了滤波器设计。音先,通过使用Lyapunov(?)急定性理论和矩阵函数凸性,对滤波误差系统进行H∞性能分析,得到了具有更小保守性的稳定性判据,进一步得到了用LMIs形式表示的要设计的滤波器增益。最后,用数值例子验证了该方法的有效性和优越性。
●第四章对一类T-S模糊系统分别讨论了镇定、可靠性控制以及H∞滤波器设计等问题。在4.2节,对一类带有随机输入时滞的T-S模糊离散系统设计了有记忆控制器。其中,通过使用状态增广的方法来补偿随机输入时滞给系统带来的影响,然后,引入一系列的满足贝努力分布的随机变量,进而建立了新的更符合实际的状态空间模型。在新建模型的基础上,通过利用Lyapunov泛函稳定性理论,设计了目标系统的有记忆的控制器,并且得到了该类T-S模糊离散系统随机均方稳定的充分性条件。最后借助一个数值例子验证了该方法的有效性。在4.3节,考虑了一类带有随机传感器和执行器故障的T-S模糊系统的可靠性控制问题。这里,每个传感器和执行器的故障都是随机出现的,他们失真的概率用两个不相关的满足一定概率分布的随机变量来表示。通过考虑概率传感器和执行器故障,建立了一个新的系统模型。基于新的数学模型,设计该系统的可靠性控制器,并且利用Lyapunov(?)急定性理论和线性矩阵不等式技术得到了该类系统稳定的充分性条件。最后通过一个数值例子验证了本方法的有效性。在4.4节,通过使用分段分析的方法,对一类T-S模糊系统设计了H∞滤波器。通过将时滞区间划分成多个小区间,然后利用矩阵函数凸性,以及引入自由权矩阵,得到用LMIs表示的具有更小保守性的时滞依赖H∞滤波器存在可行解的判据。最后在数值算例中,通过与已有文献结果进行比较,验证了本节提出方法的有效性以及具有更小的保守性。
●第五章,讨论了一类网络控制系统的H∞滤波器设计。在5.2节,对一类具有随机传感器时滞的网络控制系统进行了H∞滤波器设计。通过引入一个满足贝努力分布的随机变量,建立了一个新的系统模型。进而借助线性矩阵不等式技术得到了滤波误差系统稳定性判据。最后通过数值例子验证该方法的有效性。在5.3节,研究了一类具有随机传感器失真的离散网络控制系统。同时,在考虑网络诱导下的非理想QoS,建立了更符合实际的数学模型,然后通过使用矩阵函数凸性和LMIs技术得到了目标系统稳定的判据。需要指出的是,此时的稳定性判据不仅和传感器网络诱导的时延有关,还和随机传感器失真的概率有关。最后通过数值例子验证方法的有效性。
●第六章,总结了本论文的主要研究内容,并指出需要进一步去研究的工作。
In this thesis, the problems of stability analysis and H∞filtering are discussed for several classed of linear/nonlinear time-delay systems, including linear systems, Markov jump systems, nonlinear systems represented by T-S fuzzy models and net-work control systems. Recently, filter design for time-delay systems is a research subject of great practical and theoretical significance, which has received consider-able attention, and much significant work has been carried out. In this thesis, some new approaches will be developed to solve the stability analysis and H∞filtering design problems for several kinds of time-delay systems. The merit of the proposed approaches lie in their less design conservatism, which is realized by utilizing some more advance techniques such as linear matrix inequality (LMI) approach, piecewise analysis method, convexity property of the matrix inequality and more powerful re-laxation techniques. More specifically, the frame and description of this thesis are given as follows:
●In Chapter 1, the introduction of the thesis are given, including the background and motivation, the outline and contribution, and the research contents to be introduced in each individual chapters.
●In Chapter 2, the H∞filter design problems are discussed for a class of continue-time systems with time-varying delay and stochastic delay. In Section 2.2, Based on a piecewise analysis method and using the convexity property of the matrix inequality, new criteria are derived for H∞filtering, which can lead to much less conservative analysis results. A numerical example is given to demon-strate the effectiveness and the merit of the proposed method. In Section 2.3, we design an H∞filter for a class of linear time delay systems with random delay. The delay considered here is assumed to be satisfying a certain stochas-tic characteristic. Corresponding to the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli random binary distribution is introduced and a new system model is established by employing the information of the probability distribution. Then, new criteria are derived for the filtering-error systems, which can lead to much leas conservative analysis results. It should be noted that the solvability of the obtained criteria depend on not only the size of the delay, but also on the probability distribution of it. A numerical example is given to demonstrate the effectiveness and the merit of the proposed method.
In Chapter 3, the H∞filter design problems are addressed for a class of Markov jump systems (MJSs) with time-varying delay. In Section 3.2, we propose an H∞filter design for MJSs with time delay. Firstly, by exploiting the delay partitioning-based Lyapunov function, new criteria are derived for the H∞per-formance analysis of the filtering-error systems, which can lead to much les conservative analysis results. Secondly, based on the obtained conditions, the filter gain can be obtained in terms of LMIs. Finally, numerical examples are given to demonstrate the effectiveness and the merit of the proposed method. In Section 3.3, we propose a class of H∞filter design for MJSs with time-varying delays. Firstly, by exploiting a new Lyapunov function and using the convexity property of the matrix inequality, new criteria are derived for the H∞performance analysis of the filtering-error systems, which can lead to much less conservative analysis results. Secondly, based on the obtained conditions, the gain of filter can be obtained in terms of LMIs. Finally, numerical examples are given to demonstrate the effectiveness and the merit of the proposed method.
In Chapter 4, we deal with the stabilization, reliable control and H∞filter design problems for a class of nonlinear described by T-S fuzzy model. In Sec-tion 4.2, we design a memory controller for T-S fuzzy discrete-time systems with random input delay. A novel state space model with the compensator for the effects of the stochastic input delays is derived by introducing stochas-tic variables satisfying Bernoulli random binary distribution and using state augmentation method. Based on the new built model, memory controller is designed and sufficient conditions for the stochastic mean square stable of T-S fuzzy discrete-time systems are obtained by using Lyapunov functional method. Finally, a numerical example is given to show the effectiveness of the proposed method. In Section 4.3, a reliable control problem for T-S fuzzy discrete system with stochastic sensors and actuators faults is investigated. The faults of each sensor or actuator occur randomly and its failure rates are governed by two sets of unrelated random variables satisfying certain probabilistic distribution. In terms of the probabilistic failures of every sensor or actuator, a new fault model is proposed. Based on the new fault model, reliable controller is de-signed and sufficient conditions for the exponentially mean square stability of T-S fuzzy systems are derived by using Lyapunov functional method and LMIs technique. Finally, a numerical example is given to show the effectiveness of the proposed method. In Section 4.4, H∞filter design for nonlinear systems with time-delay via T-S fuzzy model approach is investigated based on a piecewise analysis method.Based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, then the convexity property of the matrix inequality and the free weighting matrix method are fully used in this paper. Some novel delay-dependent H∞filtering criteria are expressed as a set of LMIs, which can lead to much less conservative analysis results. Finally, a numerical example is given to illustrate that the results in this paper are more effective and less conservative than some existing ones.
●In Chapter 5, the reliable H∞filtering are considered for a class of network control systems (NCSs) with randomly varying sensor delay and stochastic sensor-failure. In Section 5.2, we design an H∞filter for NCSs with randomly varying sensor delay. A stochastic variable satisfying Bernoulli random binary distribution is introduced and a new system model is established. By using LMIs technique, sufficient conditions are derived for ensuring the mean-square stochastic stability of the filtering error systems. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach. In Section 5.3, we study a class of network control systems with stochastic sensor-failure, a more general NCS model is established with considering non-ideal Quality of Service(QoS). By using the convexity property of the matrix inequality and LMIs technique, new criteria for the stochastic mean square stable of NCSs are derived for the target systems. It should be noted that the solvability of the obtained criteria depend on not only the sensor induced delay, but also on the probability sensor-failure. A numerical example is given to demonstrate the effectiveness of the proposed method.
●In Chapter 6, the main contents of this thesis are summarized and a few po-tential topics for further research are pointed out.
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