不确定动态时滞系统的鲁棒滤波设计新方法
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摘要
状态估计是控制领域较为重要的问题之一。自从Kalman提出随机系统的最优滤波理论以及Luenberger提出确定系统的观测器理论之后,广大学者在状态估计这一领域做了大量的研究工作。应用于随机系统的Kalman滤波理论在航天、航空、通讯、工业过程控制等领域得到了广泛的应用,它建立在精确的数学模型基础之上,并假设噪声输入过程为严格的Gauss过程或者Gauss序列。然而,在很多实际的工程应用中,噪音的信息是未知的,或者系统的精确模型很难获取,以及系统存在漂移现象等等,所有这些都导致系统存在不确定性,从而导致滤波发散。因此,存在模型不确定性以及非Gauss噪声输入情况下的滤波算法具有重要意义。针对这些情况,广大学者提出了很多方法来改善传统Kalman滤波算法的鲁棒性,同时也提出了一些替代滤波算法,比如H_∞,广义H_2以及H_2/H_∞混合滤波算法等。另一方面,除了系统不确定性,时滞现象广泛存在各类实际的工程系统中,比如制造系统,电力系统,以及网络化控制系统等等。时滞现象是导致系统不稳定,降低系统性能,或引起系统出现混沌现象的一个重要因素。时滞系统的滤波设计具有很高的理论意义和工程价值。
本文针对几类不确定动态时滞系统提出了一些鲁棒滤波设计的新方法。这些不确定系统包括传统的凸多面体参数不确定系统,T-S模糊系统,以及切换系统。研究重点将集中到一类具有区间类型的时变时滞,其中时滞的上下界已知。在目前的工程系统中,传感器,控制器,以及系统模型之间往往是通过网络媒质连接,不可避免的将引起时滞现象。本文提出的滤波设计方法主要优点在于其具有较低的设计保守性,其主要通过以下几个方面来实现,比如新的时滞相关条件,针对不同滤波器结构以及滤波策略采用不用的矩阵松弛技术,以及新的矩阵不等式线性化方法等等。
首先,我们研究一类含有区间类型时变时滞以及凸多面体参数不确定性动态系统的时滞相关鲁棒H_∞滤波设计。基于一个新的参数依赖Lyapunov-Krasovskii泛函结合Finsler's引理以及Projection引理,首先提出一些新的时滞相关鲁棒H_∞性能分析充分条件。然后,假设系统的未知参数是否在线可测量,分别设计了参数依赖以及参数非依赖的鲁棒H_∞滤波器。设计结果表明,滤波器参数可以通过求解一组线性矩阵不等式得到。
其次,我们将研究重点集中到求解一类时滞非线性系统的鲁棒滤波设计问题。在信号处理领域,非线性滤波具有重要的理论意义和工程价值,近几十年来得到广大学者的广泛研究。同时,我们知道Takagi-Sugeno(T-S)模糊模型是用来逼近复杂非线性系统的一个很有效的工具。因此,基于T-S模糊模型,发展一些新的时滞非线性滤波设计方法具有重要的意义。基于一个新的模糊基依赖Lyapunov-Krasovskii泛函结合Finsler's引理以及一个改进的自由权矩阵方法,首先提出一些新的时滞相关鲁棒稳定性以及H_∞。性能分析充分条件,然后应用一个新的矩阵不等式线性化方法进行滤波器综合。设计结果表明,全阶滤波器以及降阶滤波器都可以在一个统一的框架下通过求解一组线性矩阵不等式得到。基于这个新的矩阵不等式线性化方法,所有的松弛变量都可以设置为模糊基依赖的,并且可以得到一个非PDC(parallel distributed compensation)类型的滤波器结构,这种滤波器结构包含了传统的PDC类型滤波器结构,具有更好的鲁棒性能。
然后,我们将研究重点转移到一类更为复杂系统的滤波器设计,即切换系统的滤波。切换系统是一类重要的混杂系统,它包含了一组子系统以及一个切换信号来引导这些子系统之间的切换机制。特别的,我们考虑了一类含有凸多面体参数不确定性离散切换的参数依赖鲁棒H_2/H_∞。混合滤波设计。切换信号假设事先未知并且其取值是实时可测量的。基于一个参数依赖切换Lyapunov函数,首先提出一些新的鲁棒H_2以及H_∞性能分析条件,然后采用一个新的矩阵不等式线性化方法及有界不等式进行滤波器设计。这个新的线性化方法不涉及到Lyapunov矩阵的逆运算,并且使得所有的松弛变量都可以设置为切换的。同时,在这个新的线性化方法下,可以为两个性能通道选择大部分不同的松弛变量。最后滤波器设计结果通过线性矩阵不等式给出。
最后,我们将前面几章的结果推广到一类具有平均驻留时间切换信号的时滞切换系统滤波设计。着重考虑了鲁棒能量-峰值滤波并给出指数稳定性分析条件。设计结果表明,当系统的指数下降速率给定的时候,系统的相应滤波性能依赖于切换信号的平均驻留时间,增加平均驻留时间也会相应提高系统的鲁棒滤波性能。
State estimation of dynamical systems has long been an interesting problem in the control and signal processing fields.Among various filtering schemes,one landmark design approach is the celebrated Kalman filtering,which minimizes the variance of the estimation error under the assumptions that system dynamics under consideration is exactly known and the external disturbances are stationary Gaussian noises with known statistical properties.However,in many practical situations,a priori information on external noises is not precisely known and/or an accurate system model is hard to obtain or the system may drift,all of which result in uncertainties.In such cases,various approaches have been developed to improve the robustness of the traditional Kalman filters and some alternatives, such as H_∞,generalized H_2 and mixed H_2/H_∞filtering schemes were introduced and have received a lot of attention over the past few decades.On the other hand,in addition to system parametric uncertainties,it is well known that timedelays are frequently encountered in various practical control systems,such as manufacturing systems,power systems,and networked control systems,etc.It has been well recognized that time-delay is an important source of instability and poor performance of a control system.Filtering design for dynamical systems with time-delays is a research subject of great practical and theoretical significance, which has received considerable attention in the past few years.
In this thesis,some new approaches will be developed to solve the robust filtering design problems for several kinds of uncertain dynamical systems with time-delays,including the classical polytopic-type uncertain systems,nonlinear systems represented by T-S fuzzy models,and switched systems.The main attention will be focused on the case of time-varying delay with an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available.It is noted that in modern engineering systems,the sensors,controllers, actuators and plants are usually connected via a common network medium which might include time-delays.It has also been pointed out that the systems over a network connection are essentially the systems with time-varying delays.The merit of the proposed approaches lies in their less design conservatism,which is realized by utilizing some more advanced techniques such as new delay-dependent criteria,more powerful relaxation techniques with different filter structures and filtering schemes,and new matrix inequality linearization methods.
Firstly,we revisit the problem of delay-dependent robust H_∞filtering design for a class of polytopic-type uncertain linear systems with interval-like timevarying delay.Based on a new parameter-dependent Lyapunov-Krasovskii functional combined with Finsler's lemma and Projection lemma,some novel sufficient conditions for delay-dependent H_∞performance analysis are derived.Moreover, under conditions whether the unknown parameters can be measured online or not, the parameter-dependent and parameter-independent filters are respectively developed which guarantee the asymptotic stability of the resulting filtering error systems with a prescribed robust H_∞performance level.
Secondly,attention goes to the filtering design for a class of nonlinear systems with time-varying state delay.Nonlinear filtering is of both theoretical and practical importance in signal processing community and has been receiving a lot of attention.It is also noted that the well known Takagi-Sugeno(T-S) fuzzy model has been well recognized to be effective in approximating a complex nonlinear system. Consequently,it is of great significance to develop some new approaches for the filtering design of nonlinear systems with time-varying delay via a T-S fuzzy model approach.By using a novel fuzzy-basis-dependent Lyapunov-Krasovskii functional combined with Finsler's lemma,a new robust H_∞performance analysis result is proposed and then the filter synthesis is developed by using a new and simple linearization technique incorporating a bounding inequality.A unified framework is developed such that both the full-order and reduced-order filters can be obtained by solving a set of linear matrix inequalities.It is noted that under this new linearization technique,all the slack variables can be set to be fuzzy-basis -dependent and a non-PDC(parallel distributed compensation) type filter structure is utilized.This kind of filter structure covers the traditional PDC type structure by choosing some slack variables to be common.
Then,attention is focused on the filtering design for a class of switched systems, which are an important class of hybrid systems consisting of a family of subsystems and a rule orchestrating the switching among them.Specially,the mode and parameter-dependent robust mixed H_2/H_∞filtering design for a class of discrete-time switched polytopic linear systems is considered.The switching signal is assumed to be unknown a priori,but its instantaneous valuable is available in real-time implementation.Based on a switched parameter-dependent Lyapunov function,some new conditions for robust H_2 and H_∞performance analysis are firstly proposed and in turn the filter synthesis is developed by using a new matrix inequality linearization approach and a bounding technique.Under this new linearization approach,no inverses of the Lyapunov matrices are involved and all the slack variables are set to be switched.It is also noted that when applying the new linearization technique to the mixed H_2/H_∞filtering scheme,only the slack variable associated with the filter gain variables is set to be the same for two different performance channels.
Finally,we extend the results given in the previous chapters to the switched polytopic linear time-delay systems with average dwell-time switching scheme.The robust energy-to-peak filtering scheme is considered and an exponential stability condition is presented.It is shown that the filtering performance is dependent on the parameterμfor a given system decay rate.Specially,it is observed that the largerμresults in the better performance,which is at the expense of longer average dwell-time in the system.
本文针对几类不确定动态时滞系统提出了一些鲁棒滤波设计的新方法。这些不确定系统包括传统的凸多面体参数不确定系统,T-S模糊系统,以及切换系统。研究重点将集中到一类具有区间类型的时变时滞,其中时滞的上下界已知。在目前的工程系统中,传感器,控制器,以及系统模型之间往往是通过网络媒质连接,不可避免的将引起时滞现象。本文提出的滤波设计方法主要优点在于其具有较低的设计保守性,其主要通过以下几个方面来实现,比如新的时滞相关条件,针对不同滤波器结构以及滤波策略采用不用的矩阵松弛技术,以及新的矩阵不等式线性化方法等等。
首先,我们研究一类含有区间类型时变时滞以及凸多面体参数不确定性动态系统的时滞相关鲁棒H_∞滤波设计。基于一个新的参数依赖Lyapunov-Krasovskii泛函结合Finsler's引理以及Projection引理,首先提出一些新的时滞相关鲁棒H_∞性能分析充分条件。然后,假设系统的未知参数是否在线可测量,分别设计了参数依赖以及参数非依赖的鲁棒H_∞滤波器。设计结果表明,滤波器参数可以通过求解一组线性矩阵不等式得到。
其次,我们将研究重点集中到求解一类时滞非线性系统的鲁棒滤波设计问题。在信号处理领域,非线性滤波具有重要的理论意义和工程价值,近几十年来得到广大学者的广泛研究。同时,我们知道Takagi-Sugeno(T-S)模糊模型是用来逼近复杂非线性系统的一个很有效的工具。因此,基于T-S模糊模型,发展一些新的时滞非线性滤波设计方法具有重要的意义。基于一个新的模糊基依赖Lyapunov-Krasovskii泛函结合Finsler's引理以及一个改进的自由权矩阵方法,首先提出一些新的时滞相关鲁棒稳定性以及H_∞。性能分析充分条件,然后应用一个新的矩阵不等式线性化方法进行滤波器综合。设计结果表明,全阶滤波器以及降阶滤波器都可以在一个统一的框架下通过求解一组线性矩阵不等式得到。基于这个新的矩阵不等式线性化方法,所有的松弛变量都可以设置为模糊基依赖的,并且可以得到一个非PDC(parallel distributed compensation)类型的滤波器结构,这种滤波器结构包含了传统的PDC类型滤波器结构,具有更好的鲁棒性能。
然后,我们将研究重点转移到一类更为复杂系统的滤波器设计,即切换系统的滤波。切换系统是一类重要的混杂系统,它包含了一组子系统以及一个切换信号来引导这些子系统之间的切换机制。特别的,我们考虑了一类含有凸多面体参数不确定性离散切换的参数依赖鲁棒H_2/H_∞。混合滤波设计。切换信号假设事先未知并且其取值是实时可测量的。基于一个参数依赖切换Lyapunov函数,首先提出一些新的鲁棒H_2以及H_∞性能分析条件,然后采用一个新的矩阵不等式线性化方法及有界不等式进行滤波器设计。这个新的线性化方法不涉及到Lyapunov矩阵的逆运算,并且使得所有的松弛变量都可以设置为切换的。同时,在这个新的线性化方法下,可以为两个性能通道选择大部分不同的松弛变量。最后滤波器设计结果通过线性矩阵不等式给出。
最后,我们将前面几章的结果推广到一类具有平均驻留时间切换信号的时滞切换系统滤波设计。着重考虑了鲁棒能量-峰值滤波并给出指数稳定性分析条件。设计结果表明,当系统的指数下降速率给定的时候,系统的相应滤波性能依赖于切换信号的平均驻留时间,增加平均驻留时间也会相应提高系统的鲁棒滤波性能。
State estimation of dynamical systems has long been an interesting problem in the control and signal processing fields.Among various filtering schemes,one landmark design approach is the celebrated Kalman filtering,which minimizes the variance of the estimation error under the assumptions that system dynamics under consideration is exactly known and the external disturbances are stationary Gaussian noises with known statistical properties.However,in many practical situations,a priori information on external noises is not precisely known and/or an accurate system model is hard to obtain or the system may drift,all of which result in uncertainties.In such cases,various approaches have been developed to improve the robustness of the traditional Kalman filters and some alternatives, such as H_∞,generalized H_2 and mixed H_2/H_∞filtering schemes were introduced and have received a lot of attention over the past few decades.On the other hand,in addition to system parametric uncertainties,it is well known that timedelays are frequently encountered in various practical control systems,such as manufacturing systems,power systems,and networked control systems,etc.It has been well recognized that time-delay is an important source of instability and poor performance of a control system.Filtering design for dynamical systems with time-delays is a research subject of great practical and theoretical significance, which has received considerable attention in the past few years.
In this thesis,some new approaches will be developed to solve the robust filtering design problems for several kinds of uncertain dynamical systems with time-delays,including the classical polytopic-type uncertain systems,nonlinear systems represented by T-S fuzzy models,and switched systems.The main attention will be focused on the case of time-varying delay with an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available.It is noted that in modern engineering systems,the sensors,controllers, actuators and plants are usually connected via a common network medium which might include time-delays.It has also been pointed out that the systems over a network connection are essentially the systems with time-varying delays.The merit of the proposed approaches lies in their less design conservatism,which is realized by utilizing some more advanced techniques such as new delay-dependent criteria,more powerful relaxation techniques with different filter structures and filtering schemes,and new matrix inequality linearization methods.
Firstly,we revisit the problem of delay-dependent robust H_∞filtering design for a class of polytopic-type uncertain linear systems with interval-like timevarying delay.Based on a new parameter-dependent Lyapunov-Krasovskii functional combined with Finsler's lemma and Projection lemma,some novel sufficient conditions for delay-dependent H_∞performance analysis are derived.Moreover, under conditions whether the unknown parameters can be measured online or not, the parameter-dependent and parameter-independent filters are respectively developed which guarantee the asymptotic stability of the resulting filtering error systems with a prescribed robust H_∞performance level.
Secondly,attention goes to the filtering design for a class of nonlinear systems with time-varying state delay.Nonlinear filtering is of both theoretical and practical importance in signal processing community and has been receiving a lot of attention.It is also noted that the well known Takagi-Sugeno(T-S) fuzzy model has been well recognized to be effective in approximating a complex nonlinear system. Consequently,it is of great significance to develop some new approaches for the filtering design of nonlinear systems with time-varying delay via a T-S fuzzy model approach.By using a novel fuzzy-basis-dependent Lyapunov-Krasovskii functional combined with Finsler's lemma,a new robust H_∞performance analysis result is proposed and then the filter synthesis is developed by using a new and simple linearization technique incorporating a bounding inequality.A unified framework is developed such that both the full-order and reduced-order filters can be obtained by solving a set of linear matrix inequalities.It is noted that under this new linearization technique,all the slack variables can be set to be fuzzy-basis -dependent and a non-PDC(parallel distributed compensation) type filter structure is utilized.This kind of filter structure covers the traditional PDC type structure by choosing some slack variables to be common.
Then,attention is focused on the filtering design for a class of switched systems, which are an important class of hybrid systems consisting of a family of subsystems and a rule orchestrating the switching among them.Specially,the mode and parameter-dependent robust mixed H_2/H_∞filtering design for a class of discrete-time switched polytopic linear systems is considered.The switching signal is assumed to be unknown a priori,but its instantaneous valuable is available in real-time implementation.Based on a switched parameter-dependent Lyapunov function,some new conditions for robust H_2 and H_∞performance analysis are firstly proposed and in turn the filter synthesis is developed by using a new matrix inequality linearization approach and a bounding technique.Under this new linearization approach,no inverses of the Lyapunov matrices are involved and all the slack variables are set to be switched.It is also noted that when applying the new linearization technique to the mixed H_2/H_∞filtering scheme,only the slack variable associated with the filter gain variables is set to be the same for two different performance channels.
Finally,we extend the results given in the previous chapters to the switched polytopic linear time-delay systems with average dwell-time switching scheme.The robust energy-to-peak filtering scheme is considered and an exponential stability condition is presented.It is shown that the filtering performance is dependent on the parameterμfor a given system decay rate.Specially,it is observed that the largerμresults in the better performance,which is at the expense of longer average dwell-time in the system.
引文
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