线性切换系统的若干问题研究
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摘要
作为一类重要的混杂系统,线性切换系统由多个线性子系统和一条切换规则组成,该规则也称切换信号或切换律,它决定了子系统间如何切换。线性切换系统具有重要的理论意义和广泛的工程背景,因而受到了人们的重视。近年来,线性切换系统研究领域产生了很多重要的成果。然而,由于切换系统本身的复杂性,这一理论尚未完善,仍有很多问题亟待解决。
本文在总结前人工作的基础上,主要研究了线性切换系统的静态输出反馈、鲁棒H2控制和滤波、有限频滤波器设计以及故障检测等几类问题。目前,关于受限切换下线性切换系统的静态输出反馈的研究还比较少,针对这一情况,本文结合Finsler引理和平均驻留时间技术,给出了这类切换系统的静态输出反馈控制器设计方法,新方法通过引入两组松弛变量减少了设计的保守性。对于任意切换下线性切换系统的静态输出反馈,已有的方法只适用于几类具有特殊结构的线性切换系统,本文借助于Finsler引理和切换Lyapunov函数,给出了新的静态输出反馈设计条件,该条件可用于已有方法失效时,是已有方法的补充。参数依赖Lyapunov函数已用于解决离散不确定线性切换系统的鲁棒控制问题,而对连续不确定切换系统则不然,其原因是很难利用已有方法实现系统矩阵和Lyapunov变量的解耦,本文通过使用Schur补克服了这一困难,从而给出了连续不确定线性切换系统的鲁棒H2控制器和滤波器的设计方法。已有的线性切换系统滤波器设计都是在全频范围内进行的,当外部扰动的频率范围已知时,全频方法可能引入保守性。针对这种情况,本文首次提出了有限频l2增益概念处理线性切换系统的有限频问题,给出了离散不确定线性切换系统的有限频滤波器设计方法,该方法在已知扰动频率范围时,可以取得比全频方法更好的性能。目前,关于切换系统故障检测的研究还比较少,本文在最后部分给出了两类线性切换系统的故障检测方法。针对任意切换和受限切换两种情况,分别给出了故障检测滤波器设计方法以及阈值设计条件。
具体工作如下：
第1章系统地分析和总结了切换系统研究领域的发展状况和研究方法,并给出了与本文有关的一些预备知识。
第2章研究了离散线性切换系统的静态输出反馈控制问题。分为两个部分,第一部分研究了受限切换下离散线性切换系统的静态输出反馈问题。利用多Lyapunov函数和平均驻留时间技术,首先给出了保证闭环切换系统指数稳定且具有指定加权l2增益的充分条件。根据此条件,结合Finsler引理,给出了新的静态输出反馈控制器设计条件。对于给定的系统衰减度,最小平均驻留时间以及相应的控制器增益可以通过所给的条件求得。该方法通过引入两组松弛变量减少了设计保守性。本章第二部分研究了任意切换下离散线性切换系统的静态输出反馈问题。基于切换Lyapunov函数和Finsler引理,给出了新的静态输出反馈控制器综合条件,该条件在已有方法失效时仍可用。最后,仿真算例说明了所提出方法的有效性。
第3章研究了具有多胞不确定性的连续线性切换系统的鲁棒H2控制及滤波问题。本章分为两部分,第一部分研究了鲁棒H2状态反馈控制器设计问题。利用参数依赖的多Lyapunov函数和平均驻留时间技术,首先得到了切换系统指数稳定且满足H2性能要求的充分条件。之后,通过运用Schur补技术,实现了系统矩阵与Lyapunov变量的解耦,从而得到了鲁棒H2控制器设计的线性矩阵不等式条件。为了方便比较,同时给出了基于参数无关Lyapunov函数的控制器设计方法。第二部分研究了鲁棒H2滤波器设计问题。所用的技术手段与第一部分类似,得到了鲁棒H2滤波器设计的线性矩阵不等式条件。仿真算例进一步验证了所提出方法的有效性。
第4章研究了离散不确定线性切换系统的有限频滤波器设计问题。为了处理线性切换系统的有限频问题,首次提出了有限频l2增益的概念,并给出了离散线性切换系统具有有限频l2增益的充分条件,基于此条件,分别给出了低频、中频以及高频滤波器的设计条件。当外部扰动的频率范围已知时,所设计的有限频滤波器可以取得比全频更好的性能。最后,给出仿真算例验证所提出方法的有效性。
第5章研究了线性切换系统的故障检测问题。分为两个部分,第一部分研究了任意切换下离散线性切换系统的故障检测问题。基于切换Lyapunov函数和Finsler引理,首先得到了滤波误差系统稳定且具有指定H∞性能指标的线性矩阵不等式条件,然后根据此条件设计出了故障检测滤波器,最后给出了阈值的设计条件。本章第二部分研究了受限切换下连续线性切换系统的故障检测问题。结合多Lyapunov函数和平均驻留时间技术,首先得到了滤波误差系统指数稳定且具有指定H∞性能指标的充分条件。之后,给出了故障估计滤波器的设计方法,故障可根据滤波器的输出估计出来。仿真算例验证了本章所提出方法的有效性。
第6章总结了本文的主要工作,并展望了下一步的研究工作。
As an important class of hybrid systems, switched linear systems consist of several linear subsystems and a rule named switching signal or switching law that orchestrates the switching among them. Switched linear systems have attracted increasingly more at-tention since they are of theoretical importance and have wide engineering applications. Despite the rapid progress made so far, many fundamental problems are still either unex-plored or less well understood due to the complicated behavior of switched systems.
Based on previous works of others, this dissertation studies several problems of switched linear systems including static output feedback (SOF), robust H2 control and filtering, finite frequency filter design and fault detection. At present, there are few results on SOF control for switched linear systems under constrained switching. In response to this situation, this dissertation proposes a SOF control method for this class of switched systems based on Finsler's lemma combined with average dwell time technique. The pro-posed method reduces design conservatism by introducing two sets of slack variables. For switched linear systems under arbitrary switching, the existing SOF methods are only ap-plicable to several classes of switched linear systems with special structures. By the aid of Finsler's lemma and switched Lyapunov functions, this dissertation presents new SOF de-sign conditions which can work successfully in situations where the existing ones do not. The proposed method and the existing methods can be seen as alternative ones. Parameter-dependent Lyapunov functions have been employed to solve robust control problems of discrete-time uncertain switched linear systems. However, for continuous-time uncertain switched systems, this is not the case since it is hard to decouple system matrices with Lyapunov variables using the existing methods. In this dissertation, we overcome this difficulty by using Schur complement formula. Then a robust H2 state-feedback con-trol method and a robust H2 filtering method are proposed for continuous-time uncertain switched linear systems. The existing filters for switched linear systems are designed in full frequency domain. The full frequency approaches can increase conservatism when the frequency ranges of external disturbances are known beforehand. For this case, the con-cept of finite frequency l2 gain is first defined which is used to deal with finite frequency problems of switched linear systems. Based on it, a finite frequency filtering method is proposed for discrete-time uncertain switched linear systems. When the frequency ranges of disturbances are known, the finite frequency filters can achieve better performances than the full frequency ones. To my knowledge, fault detection for switched systems has not been fully investigated up to now. The last part of this dissertation studies the problem of fault detection for two classes of switched linear systems. The design conditions for fault detection filters and thresholds are presented for arbitrary and constrained switching, respectively.
The details of the dissertation are as follows:
Chapter 1 summarizes and analyzes the development and main research methods of switched systems. Preliminaries about the considered problems are also given.
Chapter 2 investigates the SOF control problem of discrete-time switched linear systems. The first part studies the SOF problem of discrete-time switched linear sys-tems under constrained switching. Based on multiple Lyapunov functions combined with the average dwell time technique, sufficient conditions which guarantee switched linear systems exponentially stable with a weighted l2 gain are given. Then, combined with Finsler's lemma, new SOF control design conditions are derived. The minimal average dwell time and the corresponding controller gains can be obtained from these conditions for a given system decay degree. The proposed method reduces design conservatism by introducing two sets of slack variables. The second part studies the SOF problem of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finslers lemma, new sufficient conditions for SOF controller synthesis are derived. The proposed approach can work successfully in situations where the existing ones fail. The simulation examples have shown the effectiveness of the pro-posed methods.
Chapter 3 focuses on the problems of robust H2 control and filtering for continuous-time switched linear systems with polytopic uncertainties. The first part studies the robust H2 state-feedback control problem. By using multiple parameter-dependent Lyapunov functions and the average dwell time technique, sufficient conditions are obtained such that switched systems are exponentially stable and satisfy a prescribed H2 performance index. Then, we decouple the system matrices and Lyapunov variables by Schur com-plement formula. LMI-based conditions for robust H2 controller synthesis are derived. Additionally, another controller design method based on multiple parameter-independent Lyapunov functions is also given for comparison. The second part studies the robust H2 filtering problem. By using similar techniques, LMI-based conditions for robust H2 fil- ter design are obtained. The simulation examples have shown the effectiveness of the proposed methods.
Chapter 4 studies the problem of finite frequency filter design for discrete-time un-certain switched linear systems. The concept of finite frequency l2 gain is first defined to deal with finite frequency problems of switched linear systems. Sufficient conditions are given such that discrete-time switched linear systems have a prescribed finite frequency l2 gain. Based on these conditions, low-frequency, middle-frequency and high-frequency filter design conditions are derived, respectively. When the frequency ranges of external disturbances are known beforehand, the proposed finite frequency filters can receive bet-ter results than the existing full frequency ones. The simulation examples have shown the effectiveness of the proposed methods.
Chapter 5 investigates the fault detection problem of switched linear systems. The first part of this chapter studies the problem of fault detection of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finsler's lemma, LMI-based conditions are first derived such that the filtering error system is stable with a prescribed H∞performance index. Then a fault detection filter is designed based on these conditions, and a threshold design condition is given. The second part studies the problem of continuous-time switched linear systems under constrained switching. Combined multiple Lyapunov functions with average dwell time technique, sufficient conditions are obtained which guarantee the filtering error system exponentially stable and satisfying the H∞performances. Then a fault detection filter design method is proposed. The fault signal can be estimated from the output of the filter. The simulation examples have shown the effectiveness of the proposed methods.
Finally, the results of the dissertation are summarized and further research topics are pointed out in Chapter 6.
本文在总结前人工作的基础上,主要研究了线性切换系统的静态输出反馈、鲁棒H2控制和滤波、有限频滤波器设计以及故障检测等几类问题。目前,关于受限切换下线性切换系统的静态输出反馈的研究还比较少,针对这一情况,本文结合Finsler引理和平均驻留时间技术,给出了这类切换系统的静态输出反馈控制器设计方法,新方法通过引入两组松弛变量减少了设计的保守性。对于任意切换下线性切换系统的静态输出反馈,已有的方法只适用于几类具有特殊结构的线性切换系统,本文借助于Finsler引理和切换Lyapunov函数,给出了新的静态输出反馈设计条件,该条件可用于已有方法失效时,是已有方法的补充。参数依赖Lyapunov函数已用于解决离散不确定线性切换系统的鲁棒控制问题,而对连续不确定切换系统则不然,其原因是很难利用已有方法实现系统矩阵和Lyapunov变量的解耦,本文通过使用Schur补克服了这一困难,从而给出了连续不确定线性切换系统的鲁棒H2控制器和滤波器的设计方法。已有的线性切换系统滤波器设计都是在全频范围内进行的,当外部扰动的频率范围已知时,全频方法可能引入保守性。针对这种情况,本文首次提出了有限频l2增益概念处理线性切换系统的有限频问题,给出了离散不确定线性切换系统的有限频滤波器设计方法,该方法在已知扰动频率范围时,可以取得比全频方法更好的性能。目前,关于切换系统故障检测的研究还比较少,本文在最后部分给出了两类线性切换系统的故障检测方法。针对任意切换和受限切换两种情况,分别给出了故障检测滤波器设计方法以及阈值设计条件。
具体工作如下：
第1章系统地分析和总结了切换系统研究领域的发展状况和研究方法,并给出了与本文有关的一些预备知识。
第2章研究了离散线性切换系统的静态输出反馈控制问题。分为两个部分,第一部分研究了受限切换下离散线性切换系统的静态输出反馈问题。利用多Lyapunov函数和平均驻留时间技术,首先给出了保证闭环切换系统指数稳定且具有指定加权l2增益的充分条件。根据此条件,结合Finsler引理,给出了新的静态输出反馈控制器设计条件。对于给定的系统衰减度,最小平均驻留时间以及相应的控制器增益可以通过所给的条件求得。该方法通过引入两组松弛变量减少了设计保守性。本章第二部分研究了任意切换下离散线性切换系统的静态输出反馈问题。基于切换Lyapunov函数和Finsler引理,给出了新的静态输出反馈控制器综合条件,该条件在已有方法失效时仍可用。最后,仿真算例说明了所提出方法的有效性。
第3章研究了具有多胞不确定性的连续线性切换系统的鲁棒H2控制及滤波问题。本章分为两部分,第一部分研究了鲁棒H2状态反馈控制器设计问题。利用参数依赖的多Lyapunov函数和平均驻留时间技术,首先得到了切换系统指数稳定且满足H2性能要求的充分条件。之后,通过运用Schur补技术,实现了系统矩阵与Lyapunov变量的解耦,从而得到了鲁棒H2控制器设计的线性矩阵不等式条件。为了方便比较,同时给出了基于参数无关Lyapunov函数的控制器设计方法。第二部分研究了鲁棒H2滤波器设计问题。所用的技术手段与第一部分类似,得到了鲁棒H2滤波器设计的线性矩阵不等式条件。仿真算例进一步验证了所提出方法的有效性。
第4章研究了离散不确定线性切换系统的有限频滤波器设计问题。为了处理线性切换系统的有限频问题,首次提出了有限频l2增益的概念,并给出了离散线性切换系统具有有限频l2增益的充分条件,基于此条件,分别给出了低频、中频以及高频滤波器的设计条件。当外部扰动的频率范围已知时,所设计的有限频滤波器可以取得比全频更好的性能。最后,给出仿真算例验证所提出方法的有效性。
第5章研究了线性切换系统的故障检测问题。分为两个部分,第一部分研究了任意切换下离散线性切换系统的故障检测问题。基于切换Lyapunov函数和Finsler引理,首先得到了滤波误差系统稳定且具有指定H∞性能指标的线性矩阵不等式条件,然后根据此条件设计出了故障检测滤波器,最后给出了阈值的设计条件。本章第二部分研究了受限切换下连续线性切换系统的故障检测问题。结合多Lyapunov函数和平均驻留时间技术,首先得到了滤波误差系统指数稳定且具有指定H∞性能指标的充分条件。之后,给出了故障估计滤波器的设计方法,故障可根据滤波器的输出估计出来。仿真算例验证了本章所提出方法的有效性。
第6章总结了本文的主要工作,并展望了下一步的研究工作。
As an important class of hybrid systems, switched linear systems consist of several linear subsystems and a rule named switching signal or switching law that orchestrates the switching among them. Switched linear systems have attracted increasingly more at-tention since they are of theoretical importance and have wide engineering applications. Despite the rapid progress made so far, many fundamental problems are still either unex-plored or less well understood due to the complicated behavior of switched systems.
Based on previous works of others, this dissertation studies several problems of switched linear systems including static output feedback (SOF), robust H2 control and filtering, finite frequency filter design and fault detection. At present, there are few results on SOF control for switched linear systems under constrained switching. In response to this situation, this dissertation proposes a SOF control method for this class of switched systems based on Finsler's lemma combined with average dwell time technique. The pro-posed method reduces design conservatism by introducing two sets of slack variables. For switched linear systems under arbitrary switching, the existing SOF methods are only ap-plicable to several classes of switched linear systems with special structures. By the aid of Finsler's lemma and switched Lyapunov functions, this dissertation presents new SOF de-sign conditions which can work successfully in situations where the existing ones do not. The proposed method and the existing methods can be seen as alternative ones. Parameter-dependent Lyapunov functions have been employed to solve robust control problems of discrete-time uncertain switched linear systems. However, for continuous-time uncertain switched systems, this is not the case since it is hard to decouple system matrices with Lyapunov variables using the existing methods. In this dissertation, we overcome this difficulty by using Schur complement formula. Then a robust H2 state-feedback con-trol method and a robust H2 filtering method are proposed for continuous-time uncertain switched linear systems. The existing filters for switched linear systems are designed in full frequency domain. The full frequency approaches can increase conservatism when the frequency ranges of external disturbances are known beforehand. For this case, the con-cept of finite frequency l2 gain is first defined which is used to deal with finite frequency problems of switched linear systems. Based on it, a finite frequency filtering method is proposed for discrete-time uncertain switched linear systems. When the frequency ranges of disturbances are known, the finite frequency filters can achieve better performances than the full frequency ones. To my knowledge, fault detection for switched systems has not been fully investigated up to now. The last part of this dissertation studies the problem of fault detection for two classes of switched linear systems. The design conditions for fault detection filters and thresholds are presented for arbitrary and constrained switching, respectively.
The details of the dissertation are as follows:
Chapter 1 summarizes and analyzes the development and main research methods of switched systems. Preliminaries about the considered problems are also given.
Chapter 2 investigates the SOF control problem of discrete-time switched linear systems. The first part studies the SOF problem of discrete-time switched linear sys-tems under constrained switching. Based on multiple Lyapunov functions combined with the average dwell time technique, sufficient conditions which guarantee switched linear systems exponentially stable with a weighted l2 gain are given. Then, combined with Finsler's lemma, new SOF control design conditions are derived. The minimal average dwell time and the corresponding controller gains can be obtained from these conditions for a given system decay degree. The proposed method reduces design conservatism by introducing two sets of slack variables. The second part studies the SOF problem of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finslers lemma, new sufficient conditions for SOF controller synthesis are derived. The proposed approach can work successfully in situations where the existing ones fail. The simulation examples have shown the effectiveness of the pro-posed methods.
Chapter 3 focuses on the problems of robust H2 control and filtering for continuous-time switched linear systems with polytopic uncertainties. The first part studies the robust H2 state-feedback control problem. By using multiple parameter-dependent Lyapunov functions and the average dwell time technique, sufficient conditions are obtained such that switched systems are exponentially stable and satisfy a prescribed H2 performance index. Then, we decouple the system matrices and Lyapunov variables by Schur com-plement formula. LMI-based conditions for robust H2 controller synthesis are derived. Additionally, another controller design method based on multiple parameter-independent Lyapunov functions is also given for comparison. The second part studies the robust H2 filtering problem. By using similar techniques, LMI-based conditions for robust H2 fil- ter design are obtained. The simulation examples have shown the effectiveness of the proposed methods.
Chapter 4 studies the problem of finite frequency filter design for discrete-time un-certain switched linear systems. The concept of finite frequency l2 gain is first defined to deal with finite frequency problems of switched linear systems. Sufficient conditions are given such that discrete-time switched linear systems have a prescribed finite frequency l2 gain. Based on these conditions, low-frequency, middle-frequency and high-frequency filter design conditions are derived, respectively. When the frequency ranges of external disturbances are known beforehand, the proposed finite frequency filters can receive bet-ter results than the existing full frequency ones. The simulation examples have shown the effectiveness of the proposed methods.
Chapter 5 investigates the fault detection problem of switched linear systems. The first part of this chapter studies the problem of fault detection of discrete-time switched linear systems under arbitrary switching. By the aid of switched Lyapunov functions and Finsler's lemma, LMI-based conditions are first derived such that the filtering error system is stable with a prescribed H∞performance index. Then a fault detection filter is designed based on these conditions, and a threshold design condition is given. The second part studies the problem of continuous-time switched linear systems under constrained switching. Combined multiple Lyapunov functions with average dwell time technique, sufficient conditions are obtained which guarantee the filtering error system exponentially stable and satisfying the H∞performances. Then a fault detection filter design method is proposed. The fault signal can be estimated from the output of the filter. The simulation examples have shown the effectiveness of the proposed methods.
Finally, the results of the dissertation are summarized and further research topics are pointed out in Chapter 6.
引文
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