时滞系统的有限频域H_∞滤波分析与综合
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摘要
滤波或者状态估计是利用可以测量的外部信号对不可测量的信号进行估计,是控制理论、信号处理等领域的基本问题之一。由于需要精确已知的噪声统计数据和系统状态空间模型,经典的Kalman滤波在工程实际应用中往往存在鲁棒性差的问题。因此,过去几十年,广大学者一直致力于研究非高斯噪声扰动输入下的鲁棒滤波问题,H∞滤波便是其中广受关注的问题之一。值得注意的是,经过二十多年发展的H∞滤波理论绝大多数建立在标准H∞性能指标的基础上,要求滤波误差系统对全频域内的噪声都具有同样的抑制水平。由于未能充分利用噪声的频率信息,导致标准H∞滤波器很难达到理想的性能要求。另外,时滞现象不可避免地存在于实际的动力学系统中,特别是随着网络技术的发展,时滞常常成为影响系统性能的关键因素。随着广义KYP引理的提出,时滞系统的有限频域H∞滤波分析和综合问题成为近年来滤波理论研究领域的研究热点。本论文综合应用Lyapunov稳定性理论和广义KYP引理,深入地研究连续和离散时滞系统的有限频域H∞性能分析、有限频域H∞滤波器设计等问题,在线性矩阵不等式框架下提出有限频域H∞滤波器的分析和综合方法。
第一、总结时滞系统H∞滤波理论研究领域的主要成果,指出目前时滞系统的有限频域H∞滤波研究成果不足之处在于:1)绝大多数的研究成果都是时滞无关的,能够处理的时滞系统范围很有限;2)未能充分利用能够大大降低时滞结果保守性的最新技术,比如时滞分割技术;3)所有的研究成果都是从传递函数角度导出的,很难在降低保守性方面进一步推广。在此基础上,研究具有状态常时滞的连续和离散线性定常系统的有限频域H∞性能分析问题,给出新的时滞系统有限频域有界实引理,并分别从时域和频域角度证明新的有限频域有界实引理。
第二、在提出的有界实引理基础上,研究具有状态常时滞的连续和离散线性定常系统的有限频域H∞滤波问题,利用投影定理,提出基于线性矩阵不等式的时滞相关和时滞无关的有限频域H∞滤波器设计方法。理论分析和数值结果都验证提出的滤波器设计方法比文献中已有的方法具有更低的保守性。
第三、作为时滞系统有限频域H∞问题的扩展,研究一类特殊时滞系统即具有多面体参数不确定性的2D Roesser系统的鲁棒有限频域H∞滤波问题,结合已有的2D Roesser系统的广义KYP引理和投影定理,获得2D系统的有限频域H∞滤波器设计方法。通过使用更加一般的线性化步骤,使已有的滤波研究成果仅仅是提出的2D系统滤波器设计方法的特殊情形。
Filtering or state estimation is using the external measured signal to estimatethe unmeasurable signal, which is one of the fundamental problems in control theory,signal processing and other areas. Due to the necessity of both precisely known sta-tistical data of noise and the system state space model, the classic Kalman filteringoften encounters the problem of poor robustness in practical engineering application-s. Hence, over the past decades, many scholars have been studying robust filteringunder nongaussian noise disturbance input, among which, H∞filtering is one ofthe most widely concerned problems. It is worth noting that, after twenty yearsdevelopment, H∞filtering theory is mostly based on the standard H∞performanceindex, which requires that the filtering error system has the same noise disturbanceattenuation level over the entire frequency domain. Since this index does not makefull use of the frequency information of the noise, it is difficult for standard H∞filters to achieve the ideal performance requirements. In addition, time-delay in-evitably exists in many practical dynamic systems; especially with the developmentof network technologies, time-delay often becomes the key factor that affects systemperformances. With the establishment of the generalized KYP lemma, the prob-lems of analysis and synthesis of finite frequency (FF) H∞filtering for time-delaysystems have been the research hotspots in recent years in the field of H∞filteringtheory. By comprehensively applying the Lyapunov stability theory and the gen-eralized KYP lemma, this thesis in deep investigates the problems of analysis ofthe FF H∞performance, design of FF H∞filters and so on for continuous- anddiscrete-time time-delay systems, and in the framework of linear matrix inequality(LMI), proposes the analysis and synthesis methods for FF H∞filters.
First, this thesis summarizes the main achievements in the research field ofH∞filtering theory for time-delay systems, and points out the shortages of thecurrent main achievements of FF H∞filtering threefold: 1) the majority of resultsare independent of delay, only able to handle quite limited time-delay systems; 2)they have not made full use of the latest techniques that reduce the conservatismof results for time-delay systems, such as delay-partitioning; 3) all the results are derived from the perspective of the transfer function and it is hard to further promotethis method in the aspect of conservatism reduction. Based on these analysis, theproblem of analyzing the FF H∞performance for continuous- and discrete-timelinear time-invariant (LTI) systems with constant state delay is studied, and newFF bounded real lemmas (BRLs) for time-delay systems are proposed. Moreover,time-domain method and frequency-domain method for the proof of the new BRLsare respectively provided.
Second, based on the proposed BRLs, the problems of FF H∞filtering forcontinuous- and discrete-time LTI systems with constant state delay are researched,and by virtue of projection lemma, LMI-based delay-dependent and -independentmethods are proposed for the design of FF H∞filters. Theoretical analysis andsimulation results verify that the proposed filter design methods are less conservativethan the existing ones in the literature.
Third, as the extension of the problem of FF H∞filtering for time-delay system-s, the thesis still investigates the problem of robust FF H∞filtering for 2D Roessersystems with polyhedron parameter uncertainty, a special class of time-delay sys-tems. By combining the existing generalized KYP lemma for 2D Roesser systemswith projection lemma, an FF H∞filter design method is obtained. Through theuse of a more general linearization procedure, the existing filter design methods for2D systems are only special cases of the proposed one.
第一、总结时滞系统H∞滤波理论研究领域的主要成果,指出目前时滞系统的有限频域H∞滤波研究成果不足之处在于:1)绝大多数的研究成果都是时滞无关的,能够处理的时滞系统范围很有限;2)未能充分利用能够大大降低时滞结果保守性的最新技术,比如时滞分割技术;3)所有的研究成果都是从传递函数角度导出的,很难在降低保守性方面进一步推广。在此基础上,研究具有状态常时滞的连续和离散线性定常系统的有限频域H∞性能分析问题,给出新的时滞系统有限频域有界实引理,并分别从时域和频域角度证明新的有限频域有界实引理。
第二、在提出的有界实引理基础上,研究具有状态常时滞的连续和离散线性定常系统的有限频域H∞滤波问题,利用投影定理,提出基于线性矩阵不等式的时滞相关和时滞无关的有限频域H∞滤波器设计方法。理论分析和数值结果都验证提出的滤波器设计方法比文献中已有的方法具有更低的保守性。
第三、作为时滞系统有限频域H∞问题的扩展,研究一类特殊时滞系统即具有多面体参数不确定性的2D Roesser系统的鲁棒有限频域H∞滤波问题,结合已有的2D Roesser系统的广义KYP引理和投影定理,获得2D系统的有限频域H∞滤波器设计方法。通过使用更加一般的线性化步骤,使已有的滤波研究成果仅仅是提出的2D系统滤波器设计方法的特殊情形。
Filtering or state estimation is using the external measured signal to estimatethe unmeasurable signal, which is one of the fundamental problems in control theory,signal processing and other areas. Due to the necessity of both precisely known sta-tistical data of noise and the system state space model, the classic Kalman filteringoften encounters the problem of poor robustness in practical engineering application-s. Hence, over the past decades, many scholars have been studying robust filteringunder nongaussian noise disturbance input, among which, H∞filtering is one ofthe most widely concerned problems. It is worth noting that, after twenty yearsdevelopment, H∞filtering theory is mostly based on the standard H∞performanceindex, which requires that the filtering error system has the same noise disturbanceattenuation level over the entire frequency domain. Since this index does not makefull use of the frequency information of the noise, it is difficult for standard H∞filters to achieve the ideal performance requirements. In addition, time-delay in-evitably exists in many practical dynamic systems; especially with the developmentof network technologies, time-delay often becomes the key factor that affects systemperformances. With the establishment of the generalized KYP lemma, the prob-lems of analysis and synthesis of finite frequency (FF) H∞filtering for time-delaysystems have been the research hotspots in recent years in the field of H∞filteringtheory. By comprehensively applying the Lyapunov stability theory and the gen-eralized KYP lemma, this thesis in deep investigates the problems of analysis ofthe FF H∞performance, design of FF H∞filters and so on for continuous- anddiscrete-time time-delay systems, and in the framework of linear matrix inequality(LMI), proposes the analysis and synthesis methods for FF H∞filters.
First, this thesis summarizes the main achievements in the research field ofH∞filtering theory for time-delay systems, and points out the shortages of thecurrent main achievements of FF H∞filtering threefold: 1) the majority of resultsare independent of delay, only able to handle quite limited time-delay systems; 2)they have not made full use of the latest techniques that reduce the conservatismof results for time-delay systems, such as delay-partitioning; 3) all the results are derived from the perspective of the transfer function and it is hard to further promotethis method in the aspect of conservatism reduction. Based on these analysis, theproblem of analyzing the FF H∞performance for continuous- and discrete-timelinear time-invariant (LTI) systems with constant state delay is studied, and newFF bounded real lemmas (BRLs) for time-delay systems are proposed. Moreover,time-domain method and frequency-domain method for the proof of the new BRLsare respectively provided.
Second, based on the proposed BRLs, the problems of FF H∞filtering forcontinuous- and discrete-time LTI systems with constant state delay are researched,and by virtue of projection lemma, LMI-based delay-dependent and -independentmethods are proposed for the design of FF H∞filters. Theoretical analysis andsimulation results verify that the proposed filter design methods are less conservativethan the existing ones in the literature.
Third, as the extension of the problem of FF H∞filtering for time-delay system-s, the thesis still investigates the problem of robust FF H∞filtering for 2D Roessersystems with polyhedron parameter uncertainty, a special class of time-delay sys-tems. By combining the existing generalized KYP lemma for 2D Roesser systemswith projection lemma, an FF H∞filter design method is obtained. Through theuse of a more general linearization procedure, the existing filter design methods for2D systems are only special cases of the proposed one.
引文
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