T-S模糊时滞系统的滤波研究
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摘要
滤波一词源于通信理论,它是从含干扰的可测接收信号中提取有用信号的一种技术。滤波技术在导航、目标跟踪、通信、声音信号处理等许多方面都有应用,一直受到广泛的关注。从上个世纪四十年代的Wiener滤波发展到现在的鲁棒滤波,根据不同的待测信号及性能需求,在理论及工业快速发展的进程中,滤波器理论也在逐步完善。
模糊系统由于能够在系统复杂、病态、含有各类不确定的情况下,利用专家的经验和知识,提供有效的控制方法而倍受广大学者的关注。尽管模糊理论在发展的过程中经历了很多的纷争,但是在工业应用方面,各类基于模糊理论产品的大量涌现,推动了模糊理论的前进。由Takagi和Sugeno提出的T-S模糊模型为非线性系统提供了一种描述方法,该模型将非线性系统的问题转化为多个线性子系统的问题,从而可以使用已有的线性系统理论和工具。因此T-S模糊系统这种方便有效的非线性系统处理方法具有很好的研究意义。本文在Lyapunov稳定性理论的框架下,对多类T-S模糊时滞系统进行了滤波问题的研究,主要内容概括如下：
1、研究了一类T-S模糊中立型系统的H_∞滤波器设计问题,首先在时滞相关稳定性的基础上,对T-S模糊中立型系统给出了时滞相关的有界实引理。基于该有界实引理,引入松弛矩阵不等式,给出了中立型时滞T-S模糊系统的H_∞滤波器存在的充分条件和设计方法,并通过仿真算例进行验证。
2、对非线性T-S模糊中立型系统,研究了观测器型滤波器的设计方法。由于在实际工业生产过程中,并不是所有的情况下状态都可以测量,所以当状态不可测时,使用外部变量来给出状态的估计值就是一种有效的方法。首先对研究不显含时间t,即时滞无关情况下,非线性T-S模糊中立型系统H_∞滤波器存在的充分条件,随后导出时滞相关结果。通过采用其它文献的模糊算例,可以看出所设计滤波器的有效性。
3、对时变时滞的不确定T-S模糊系统,设计了鲁棒L_2-L_∞。滤波器。基于全局Lyapunov函数和模糊加权Lyapunov函数,引入自由加权矩阵,分别给出了不确定变时滞T-S模糊系统的鲁棒滤波器存在界实引理。基于界实引理,使用松弛矩阵不等式,导出各自的鲁棒L_2-L_∞滤波器存在的充分条件,并给出设计方法。
4、考虑了一类不确定时变时滞T-S模糊系统的非脆弱鲁棒H_∞滤波器设计问题,即保证滤波器当参数发生一定变化时,仍能维持滤波误差系统的稳定且满足给定的H_∞性能要求。在设计过程中,使用自由权矩阵和松弛矩阵不等式的技巧,给出了保守性较小的结果,经过仿真算例的验证,所设计的非脆弱鲁棒H_∞滤波器具有良好的效果。
5、使用一个变时滞的普通工业模型验证之前所设计滤波器的有效性。采用连续搅拌反应釜(CSTR)模型,对所设计的非脆弱鲁棒H_∞滤波器、基于全局Lyapunov函数的鲁棒L_2-L_∞滤波器及基于模糊加权Lyaunov函数的鲁棒L_2-L_∞滤波器,进行了仿真得到良好的结果。
6、总结全文的研究结果并提出了可进一步研究并待解决的问题。
The idea of filer is rooted in the theory of communication, which is a technology to extract useful message from the received signals that contain noise disturbances. The filter-ing problem has attracted extensive attention because it has applications in many industrial environment, such as navigation, target tracking, communication, sound signal processing and so on. From the theory of Wiener filter in 40th of the last century to the robust filter in nowadays, the filtering theory is turning maturity.
Fuzzy systems offer an effective method to control the complex systems, bad systems or uncertain systems by using the experts'knowledge. Although the fuzzy theory endures a plenty of arguments, the rushing of the fuzzy productions in industry impulses its develop-ment. The T-S fuzzy model, first proposed by Takagi and Sugeno, is an effect way to describe nonlinear systems, in which a nonlinear system is transformed into a set of linear subsystems. So it provides a good opportunity to employ the well-established theory and tools of linear systems. Therefore, the T-S fuzzy model is powerful to investigate the complex nonlinear systems. The purpose of this thesis is to study several kinds of delayed T-S fuzzy systems and to find conditions for the fuzzy filtering problems on the basis of Lyapunov stability theory. The main results obtained in this thesis are as follows:
1. The robust H_∞fuzzy filtering problem of uncertain T-S fuzzy neutral systems is studied. Based on the delay-dependent stability condition in the second subsection, a delay-dependent bounded real lemma is obtained. Then, by using the relaxed LMI approach, a sufficient condition of the uncertain T-S fuzzy neutral systems is proposed. At last, two numerical examples are provided to demonstrate the effectiveness of the proposed approach.
2. the observer-based filtering problem for the nonlinear T-S neutral delay fuzzy sys-tems is considered. In the industrial producing process, it is impossible that all the states are measurable. And it is available to use outer variables to estimate the state when it unmea-surable. A delay-independent sufficient condition and a delay-dependent sufficient condition are proposed, guaranteeing the stability of the resulting filtering error systems stable.
3. Attention goes to the filtering design of robust L_2 - L_∞filters for uncertain T-S fuzzy systems with time-varying delay. Based on a common Lyapunov functional and a fuzzy Lyapunov functional, two robust fuzzy filters are designed. By applying the free weighting matrices and the relaxed LMI, less conservatism sufficient conditions are gained. At last, two numerical examples are given to prove the correctness of our method.
4. The problem of delay-dependent non-fragile robust H_∞filtering for uncertain time- varying T-S fuzzy delay systems is addressed. Attention is focused on the design of a filter when its parameters change it can keep the filter error systems stable and satisfy the H_∞per-formance. The approached of free weighting matrixes and relaxed LMI are used to further reduce the conservatism. The numerical examples are provided to demonstrate the effective-ness of the design method.
5. A time-varying industrial model is used to demonstrate the effectiveness of the proposed filters in the work before. Investigating a continuous stirred tank reactor(CSTR) model, we emulate the non-fragile robust H_∞fuzzy filter, the robust L2 - L_∞fuzzy filter un-der a normal Lyapunov function and the robust L2 - L_∞fuzzy filter under fuzzy-weighting Lyapunov function, and get good results.
6. We conclude the thesis and point out some open problems that should be further studied.
模糊系统由于能够在系统复杂、病态、含有各类不确定的情况下,利用专家的经验和知识,提供有效的控制方法而倍受广大学者的关注。尽管模糊理论在发展的过程中经历了很多的纷争,但是在工业应用方面,各类基于模糊理论产品的大量涌现,推动了模糊理论的前进。由Takagi和Sugeno提出的T-S模糊模型为非线性系统提供了一种描述方法,该模型将非线性系统的问题转化为多个线性子系统的问题,从而可以使用已有的线性系统理论和工具。因此T-S模糊系统这种方便有效的非线性系统处理方法具有很好的研究意义。本文在Lyapunov稳定性理论的框架下,对多类T-S模糊时滞系统进行了滤波问题的研究,主要内容概括如下：
1、研究了一类T-S模糊中立型系统的H_∞滤波器设计问题,首先在时滞相关稳定性的基础上,对T-S模糊中立型系统给出了时滞相关的有界实引理。基于该有界实引理,引入松弛矩阵不等式,给出了中立型时滞T-S模糊系统的H_∞滤波器存在的充分条件和设计方法,并通过仿真算例进行验证。
2、对非线性T-S模糊中立型系统,研究了观测器型滤波器的设计方法。由于在实际工业生产过程中,并不是所有的情况下状态都可以测量,所以当状态不可测时,使用外部变量来给出状态的估计值就是一种有效的方法。首先对研究不显含时间t,即时滞无关情况下,非线性T-S模糊中立型系统H_∞滤波器存在的充分条件,随后导出时滞相关结果。通过采用其它文献的模糊算例,可以看出所设计滤波器的有效性。
3、对时变时滞的不确定T-S模糊系统,设计了鲁棒L_2-L_∞。滤波器。基于全局Lyapunov函数和模糊加权Lyapunov函数,引入自由加权矩阵,分别给出了不确定变时滞T-S模糊系统的鲁棒滤波器存在界实引理。基于界实引理,使用松弛矩阵不等式,导出各自的鲁棒L_2-L_∞滤波器存在的充分条件,并给出设计方法。
4、考虑了一类不确定时变时滞T-S模糊系统的非脆弱鲁棒H_∞滤波器设计问题,即保证滤波器当参数发生一定变化时,仍能维持滤波误差系统的稳定且满足给定的H_∞性能要求。在设计过程中,使用自由权矩阵和松弛矩阵不等式的技巧,给出了保守性较小的结果,经过仿真算例的验证,所设计的非脆弱鲁棒H_∞滤波器具有良好的效果。
5、使用一个变时滞的普通工业模型验证之前所设计滤波器的有效性。采用连续搅拌反应釜(CSTR)模型,对所设计的非脆弱鲁棒H_∞滤波器、基于全局Lyapunov函数的鲁棒L_2-L_∞滤波器及基于模糊加权Lyaunov函数的鲁棒L_2-L_∞滤波器,进行了仿真得到良好的结果。
6、总结全文的研究结果并提出了可进一步研究并待解决的问题。
The idea of filer is rooted in the theory of communication, which is a technology to extract useful message from the received signals that contain noise disturbances. The filter-ing problem has attracted extensive attention because it has applications in many industrial environment, such as navigation, target tracking, communication, sound signal processing and so on. From the theory of Wiener filter in 40th of the last century to the robust filter in nowadays, the filtering theory is turning maturity.
Fuzzy systems offer an effective method to control the complex systems, bad systems or uncertain systems by using the experts'knowledge. Although the fuzzy theory endures a plenty of arguments, the rushing of the fuzzy productions in industry impulses its develop-ment. The T-S fuzzy model, first proposed by Takagi and Sugeno, is an effect way to describe nonlinear systems, in which a nonlinear system is transformed into a set of linear subsystems. So it provides a good opportunity to employ the well-established theory and tools of linear systems. Therefore, the T-S fuzzy model is powerful to investigate the complex nonlinear systems. The purpose of this thesis is to study several kinds of delayed T-S fuzzy systems and to find conditions for the fuzzy filtering problems on the basis of Lyapunov stability theory. The main results obtained in this thesis are as follows:
1. The robust H_∞fuzzy filtering problem of uncertain T-S fuzzy neutral systems is studied. Based on the delay-dependent stability condition in the second subsection, a delay-dependent bounded real lemma is obtained. Then, by using the relaxed LMI approach, a sufficient condition of the uncertain T-S fuzzy neutral systems is proposed. At last, two numerical examples are provided to demonstrate the effectiveness of the proposed approach.
2. the observer-based filtering problem for the nonlinear T-S neutral delay fuzzy sys-tems is considered. In the industrial producing process, it is impossible that all the states are measurable. And it is available to use outer variables to estimate the state when it unmea-surable. A delay-independent sufficient condition and a delay-dependent sufficient condition are proposed, guaranteeing the stability of the resulting filtering error systems stable.
3. Attention goes to the filtering design of robust L_2 - L_∞filters for uncertain T-S fuzzy systems with time-varying delay. Based on a common Lyapunov functional and a fuzzy Lyapunov functional, two robust fuzzy filters are designed. By applying the free weighting matrices and the relaxed LMI, less conservatism sufficient conditions are gained. At last, two numerical examples are given to prove the correctness of our method.
4. The problem of delay-dependent non-fragile robust H_∞filtering for uncertain time- varying T-S fuzzy delay systems is addressed. Attention is focused on the design of a filter when its parameters change it can keep the filter error systems stable and satisfy the H_∞per-formance. The approached of free weighting matrixes and relaxed LMI are used to further reduce the conservatism. The numerical examples are provided to demonstrate the effective-ness of the design method.
5. A time-varying industrial model is used to demonstrate the effectiveness of the proposed filters in the work before. Investigating a continuous stirred tank reactor(CSTR) model, we emulate the non-fragile robust H_∞fuzzy filter, the robust L2 - L_∞fuzzy filter un-der a normal Lyapunov function and the robust L2 - L_∞fuzzy filter under fuzzy-weighting Lyapunov function, and get good results.
6. We conclude the thesis and point out some open problems that should be further studied.
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