基于解析结构的模糊控制系统设计及稳定性分析
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摘要
自从1965年美国教授Zadeh提出模糊集的概念、1975年英国工程师Mamdani成功的将模糊理论应用于蒸汽机控制以来,模糊控制技术在化工、冶金、航天航空、交通运输、机器人和家用电器等各种领域得到了广泛应用。模糊控制是智能控制的一个重要分支,已经证明,它对于非线性、大滞后、无精确或无法建立数学模型的被控系统具有良好的控制效果。
与经典控制和现代控制方法相比,模糊控制的结构具有特殊性和复杂性,因此缺乏系统化的分析和设计方法,比如模糊论域的划分、模糊子集隶属函数的设计、模糊规则的获取等问题,尤其是与系统可行性密切相关的模糊系统稳定性分析等问题。这些问题使得模糊控制器的设计处于“黑箱”状态,这样既不能对模糊系统的特性进行有效的数学分析,也不能说明模糊控制优于常规控制的本质。因此,模糊控制领域的一个研究重点是如何从模糊控制的本质出发,完善模糊控制理论,建立系统化的模糊控制设计和分析方法,特别是建立一般性的、便于工程应用的模糊控制系统化方法是摆在研究人员面前的一项迫切任务。
模糊控制解析结构建立了模糊控制器输入-输出的明晰表达式,揭示了模糊控制的本质,为模糊控制的理论和应用研究提供了强有力的平台,是发展模糊控制理论的一条有效途径。目前,许多学者从不同的角度,研究了各种模糊控制器的解析结构,并得到了重要结论。但是,由于模糊控制器结构和参数设置的多样性和复杂性,关于模糊控制器的解析结构的研究还不够完善,特别是关于不同类型模糊控制器的解析结构研究、基于解析结构的模糊控制器系统化分析和设计以及模糊系统稳定性分析等问题需要进一步深入研究。本文从建立模糊控制系统的解析分析和设计方法、简化模糊控制器的设计过程和发挥模糊控制的工程应用价值的角度出发,基于模糊控制器的解析结构推导,探讨了模糊控制器的本质和工作机理,运用成熟的经典控制理论研究了模糊控制器设计和稳定性分析等关键问题。首先,研究了模糊系统的稳定性问题,将模糊Lyapunov函数用于T-S模糊系统的状态观测器和最优控制器的设计,并基于典型模糊控制器的解析结构,分析了Mamdani模糊控制系统的H∞鲁棒稳定性;然后,基于输入、输出隶属函数的系统化分析,提出了一种广义线性隶属函数并研究了其通用逼近性;最后,结合工业生产中常用的PID控制技术,提出了一种新模糊控制器系统化设计方法。总结全文,本文的主要研究工作如下:
(1)将模糊Lyapunov方法用于T-S和Mamdani模糊系统的稳定性分析和设计。通过构造模糊Lyapunov函数,基于并行分布补偿原理和线性矩阵不等式,推导了T-S模糊系统的状态观测器、最优控制器的稳定性充分条件和参数化设计方法,并提出了一种带有补偿量的并行分布补偿控制器的设计方法。在此基础上,基于典型模糊控制器的解析结构,将模糊Lyapunov方法应用于模糊系统的稳定性分析,得出了一个新的判断Mamdani模糊系统H∞稳定性的充分条件。
(2)提出了一种关于输入、输出隶属函数的系统化设计方法。系统分析了输入、输出隶属函数各个设计要素对控制系统性能的影响,为隶属函数的设计提供了理论基础。
(3)在总结工程中常用的三角形和梯形隶属函数的基础上,提出了一种将这两种隶属函数作为特例的广义线性隶属函数。基于模糊控制器的解析结构,分析了广义线性隶属函数的结构特性,证明了输入为广义线性隶属函数的模糊控制器的解析结构是一个带有控制补偿量的全局多值继电器和局部PD(PI)控制器之和,并推导了其极限特性和非线性特点。
(4)研究了具有广义线性输入隶属函数的Mamdani模糊控制器的通用逼近性,证明了该类模糊控制器在论域范围内能以任意精度逼近任意连续实函数,并提出了该类模糊控制器作为通用逼近器的一个充分条件。从模糊控制器通用逼近性的角度,分析和验证了所设计隶属函数的有效性。
(5)设计了一种具有加权因子的PID型模糊控制器并提出了基于普通PID控制器参数整定模糊控制器参数的方法。首先,推导了模糊控制器的解析结构,揭示了具有加权因子PID型模糊控制器的控制本质,并基于模糊Lyapunov方法分析了具有加权因子模糊控制系统的闭环稳定性,从稳定性分析的角度证明了所设计模糊控制器的控制性能优于常规PID控制器。在此基础上,针对大惯性、大滞后等被控对象易出现积分饱和的现象,提出了一种变积分PID型模糊控制器。最后,将所设计两种模糊控制器用于过程控制实验系统的温度控制和液位控制,仿真和实验结果表明了所设计的模糊控制器设计简单、易于实现、具有较强的鲁棒性和稳定性。
Since American professor Zadeh firstly proposed the concept of fuzzy set in 1965 and English engineer Mamdani applied fuzzy theory in steamer control successfully in 1975, fuzzy control has been widely used in many fields such as chemical engineering, metallurgy, aerospace engineering, communication and transportation etc. Compared with the traditional control, it is proved that fuzzy control has a better control effect on those systems which has the features of nonlinear, big time delay, and non-mathematical model etc. It has become an important branch of intelligence control.
Compared with the traditional control, fuzzy control lacks enough research on the systematic analysis and design, due to its special control theory and complicated structure, for example, the dividing of fuzzy universe, the design of membership function, and acquisition of fuzzy rules etc., specially, the stability analysis of fuzzy system which is relative to the feasibility of system. These problems make fuzzy controller be a“black-box”. So it is hard to mathematically analyze the characters of fuzzy systems and to explain the better nature of fuzzy control. As a result, these above shortages make the development of fuzzy control theory fall behind its application. Thus, how to make a research on fuzzy theory and systematic design method of fuzzy controller, has become an important research issue in fuzzy control field. Specially, how to establish a kind of design method for the general, applicable and engineering application of fuzzy controller, is an challengable job for designers.
The analytical structure of fuzzy controller builds the explicit expression between input and output of fuzzy controller, and establishes the relationship between fuzzy controller and traditional PID controller. So with the analytical structure of fuzzy controller, it is possible to reveal the essence of fuzzy control. The analytical structure of fuzzy controller provides a useful theoretical tool to design and analyze fuzzy controller. At present, many researchers have proposed many kinds of analytical structures of fuzzy controllers at different perspectives and attain a lot of important results. It is valuable to deeply study on the analytical structures of the different fuzzy controllers, and specially, on the systematic analysis and design of fuzzy controllers with their analytical structures.
In order to develop analytical theory, simplify design process and make full use of engineering application of fuzzy controller, some key issues on the design and stability analysis of fuzzy controller are researched in detail in the paper, based on the derivation of analytical structure of fuzzy controllers. Firstly, the fuzzy Lyapunov function is used in stability analysis of T-S fuzzy control systems and H∞stability of Mamdani fuzzy control systems by the analytical structure of a typical fuzzy controller. Secondly, with the discussion of input and output membership function, a kind of general linear membership function is put forward and its universal approximation is discussed. Finally, a new method for the design of fuzzy controller is presented, whose parameters are defined according to the traditional PID control technology. Simulation and experiment results manifest the effect of the researches in the paper. As a conclusion, the main contents of the paper are as follows:
(1) Apply fuzzy Lyapunov function in the stability analysis and design of T-S and Mamdani fuzzy system. For T-S fuzzy control system, the sufficient conditions and parameters design method, for the state observer, optimal controller and a parallel distributed compensation controller with a compensator, are derived by formulating the fuzzy Lyapunov function of the fuzzy system, the parallel distributed compensation strategy and linear matrix inequality. Based on the above researches, the fuzzy Lyapunov function is adopted in the H∞stability analysis of Mamdani fuzzy system, and a new sufficient condition is derived on the basis of the analytical structure of the fuzzy controller.
(2) A kind of systematic design method is put forward about the input and output membership function. The design factors of membership function are analyzed in detail on the basis of analytical structure of a typical fuzzy controller. And the research provides theoretical foundation for the design of fuzzy controller.
(3) Based on the summary of the characteristic of triangle and trapezoid membership function, a kind of general linear membership function is put forward, and triangle and trapezoid membership function can be look as its special cases. It is proved that the controller with general linear membership function is the sum of a global multi-delay controller with an offset and a local PD(or PI) controller. Meanwhile, the limiting structure and nonlinear character are derived.
(4) The universal approximation and sufficient condition of Mamdani fuzzy controller with general linear membership function is derived and proved that the controller can approximate any continuous real function with arbitrary precision in discourse. The effect of the designed membership function is verified by the research of universal approximation of fuzzy controller. (5) A kind of PID fuzzy controller with a weighted factor is designed and a systematic design method is proposed to set parameters of fuzzy controller according to the ordinary PID controller. On the basis of analytical structure of fuzzy controller, the control essence of PID fuzzy controller is revealed and the stability of fuzzy system is researched with the fuzzy Lyapunov method. These results manifest that the performance of the designed fuzzy controller is better than that of ordinary PID controller. Secondly, a kind of PID fuzzy controller with a changeable integral parameter is designed to control these systems with big inertial and delay, etc. Finally, the designed fuzzy controllers are used in temperature and level control of the process control experiment system. Simulation and experiment results show that the designed fuzzy controllers are simple and easy to be realized, and meanwhile, they have some advantages of robustness and stability.
与经典控制和现代控制方法相比,模糊控制的结构具有特殊性和复杂性,因此缺乏系统化的分析和设计方法,比如模糊论域的划分、模糊子集隶属函数的设计、模糊规则的获取等问题,尤其是与系统可行性密切相关的模糊系统稳定性分析等问题。这些问题使得模糊控制器的设计处于“黑箱”状态,这样既不能对模糊系统的特性进行有效的数学分析,也不能说明模糊控制优于常规控制的本质。因此,模糊控制领域的一个研究重点是如何从模糊控制的本质出发,完善模糊控制理论,建立系统化的模糊控制设计和分析方法,特别是建立一般性的、便于工程应用的模糊控制系统化方法是摆在研究人员面前的一项迫切任务。
模糊控制解析结构建立了模糊控制器输入-输出的明晰表达式,揭示了模糊控制的本质,为模糊控制的理论和应用研究提供了强有力的平台,是发展模糊控制理论的一条有效途径。目前,许多学者从不同的角度,研究了各种模糊控制器的解析结构,并得到了重要结论。但是,由于模糊控制器结构和参数设置的多样性和复杂性,关于模糊控制器的解析结构的研究还不够完善,特别是关于不同类型模糊控制器的解析结构研究、基于解析结构的模糊控制器系统化分析和设计以及模糊系统稳定性分析等问题需要进一步深入研究。本文从建立模糊控制系统的解析分析和设计方法、简化模糊控制器的设计过程和发挥模糊控制的工程应用价值的角度出发,基于模糊控制器的解析结构推导,探讨了模糊控制器的本质和工作机理,运用成熟的经典控制理论研究了模糊控制器设计和稳定性分析等关键问题。首先,研究了模糊系统的稳定性问题,将模糊Lyapunov函数用于T-S模糊系统的状态观测器和最优控制器的设计,并基于典型模糊控制器的解析结构,分析了Mamdani模糊控制系统的H∞鲁棒稳定性;然后,基于输入、输出隶属函数的系统化分析,提出了一种广义线性隶属函数并研究了其通用逼近性;最后,结合工业生产中常用的PID控制技术,提出了一种新模糊控制器系统化设计方法。总结全文,本文的主要研究工作如下:
(1)将模糊Lyapunov方法用于T-S和Mamdani模糊系统的稳定性分析和设计。通过构造模糊Lyapunov函数,基于并行分布补偿原理和线性矩阵不等式,推导了T-S模糊系统的状态观测器、最优控制器的稳定性充分条件和参数化设计方法,并提出了一种带有补偿量的并行分布补偿控制器的设计方法。在此基础上,基于典型模糊控制器的解析结构,将模糊Lyapunov方法应用于模糊系统的稳定性分析,得出了一个新的判断Mamdani模糊系统H∞稳定性的充分条件。
(2)提出了一种关于输入、输出隶属函数的系统化设计方法。系统分析了输入、输出隶属函数各个设计要素对控制系统性能的影响,为隶属函数的设计提供了理论基础。
(3)在总结工程中常用的三角形和梯形隶属函数的基础上,提出了一种将这两种隶属函数作为特例的广义线性隶属函数。基于模糊控制器的解析结构,分析了广义线性隶属函数的结构特性,证明了输入为广义线性隶属函数的模糊控制器的解析结构是一个带有控制补偿量的全局多值继电器和局部PD(PI)控制器之和,并推导了其极限特性和非线性特点。
(4)研究了具有广义线性输入隶属函数的Mamdani模糊控制器的通用逼近性,证明了该类模糊控制器在论域范围内能以任意精度逼近任意连续实函数,并提出了该类模糊控制器作为通用逼近器的一个充分条件。从模糊控制器通用逼近性的角度,分析和验证了所设计隶属函数的有效性。
(5)设计了一种具有加权因子的PID型模糊控制器并提出了基于普通PID控制器参数整定模糊控制器参数的方法。首先,推导了模糊控制器的解析结构,揭示了具有加权因子PID型模糊控制器的控制本质,并基于模糊Lyapunov方法分析了具有加权因子模糊控制系统的闭环稳定性,从稳定性分析的角度证明了所设计模糊控制器的控制性能优于常规PID控制器。在此基础上,针对大惯性、大滞后等被控对象易出现积分饱和的现象,提出了一种变积分PID型模糊控制器。最后,将所设计两种模糊控制器用于过程控制实验系统的温度控制和液位控制,仿真和实验结果表明了所设计的模糊控制器设计简单、易于实现、具有较强的鲁棒性和稳定性。
Since American professor Zadeh firstly proposed the concept of fuzzy set in 1965 and English engineer Mamdani applied fuzzy theory in steamer control successfully in 1975, fuzzy control has been widely used in many fields such as chemical engineering, metallurgy, aerospace engineering, communication and transportation etc. Compared with the traditional control, it is proved that fuzzy control has a better control effect on those systems which has the features of nonlinear, big time delay, and non-mathematical model etc. It has become an important branch of intelligence control.
Compared with the traditional control, fuzzy control lacks enough research on the systematic analysis and design, due to its special control theory and complicated structure, for example, the dividing of fuzzy universe, the design of membership function, and acquisition of fuzzy rules etc., specially, the stability analysis of fuzzy system which is relative to the feasibility of system. These problems make fuzzy controller be a“black-box”. So it is hard to mathematically analyze the characters of fuzzy systems and to explain the better nature of fuzzy control. As a result, these above shortages make the development of fuzzy control theory fall behind its application. Thus, how to make a research on fuzzy theory and systematic design method of fuzzy controller, has become an important research issue in fuzzy control field. Specially, how to establish a kind of design method for the general, applicable and engineering application of fuzzy controller, is an challengable job for designers.
The analytical structure of fuzzy controller builds the explicit expression between input and output of fuzzy controller, and establishes the relationship between fuzzy controller and traditional PID controller. So with the analytical structure of fuzzy controller, it is possible to reveal the essence of fuzzy control. The analytical structure of fuzzy controller provides a useful theoretical tool to design and analyze fuzzy controller. At present, many researchers have proposed many kinds of analytical structures of fuzzy controllers at different perspectives and attain a lot of important results. It is valuable to deeply study on the analytical structures of the different fuzzy controllers, and specially, on the systematic analysis and design of fuzzy controllers with their analytical structures.
In order to develop analytical theory, simplify design process and make full use of engineering application of fuzzy controller, some key issues on the design and stability analysis of fuzzy controller are researched in detail in the paper, based on the derivation of analytical structure of fuzzy controllers. Firstly, the fuzzy Lyapunov function is used in stability analysis of T-S fuzzy control systems and H∞stability of Mamdani fuzzy control systems by the analytical structure of a typical fuzzy controller. Secondly, with the discussion of input and output membership function, a kind of general linear membership function is put forward and its universal approximation is discussed. Finally, a new method for the design of fuzzy controller is presented, whose parameters are defined according to the traditional PID control technology. Simulation and experiment results manifest the effect of the researches in the paper. As a conclusion, the main contents of the paper are as follows:
(1) Apply fuzzy Lyapunov function in the stability analysis and design of T-S and Mamdani fuzzy system. For T-S fuzzy control system, the sufficient conditions and parameters design method, for the state observer, optimal controller and a parallel distributed compensation controller with a compensator, are derived by formulating the fuzzy Lyapunov function of the fuzzy system, the parallel distributed compensation strategy and linear matrix inequality. Based on the above researches, the fuzzy Lyapunov function is adopted in the H∞stability analysis of Mamdani fuzzy system, and a new sufficient condition is derived on the basis of the analytical structure of the fuzzy controller.
(2) A kind of systematic design method is put forward about the input and output membership function. The design factors of membership function are analyzed in detail on the basis of analytical structure of a typical fuzzy controller. And the research provides theoretical foundation for the design of fuzzy controller.
(3) Based on the summary of the characteristic of triangle and trapezoid membership function, a kind of general linear membership function is put forward, and triangle and trapezoid membership function can be look as its special cases. It is proved that the controller with general linear membership function is the sum of a global multi-delay controller with an offset and a local PD(or PI) controller. Meanwhile, the limiting structure and nonlinear character are derived.
(4) The universal approximation and sufficient condition of Mamdani fuzzy controller with general linear membership function is derived and proved that the controller can approximate any continuous real function with arbitrary precision in discourse. The effect of the designed membership function is verified by the research of universal approximation of fuzzy controller. (5) A kind of PID fuzzy controller with a weighted factor is designed and a systematic design method is proposed to set parameters of fuzzy controller according to the ordinary PID controller. On the basis of analytical structure of fuzzy controller, the control essence of PID fuzzy controller is revealed and the stability of fuzzy system is researched with the fuzzy Lyapunov method. These results manifest that the performance of the designed fuzzy controller is better than that of ordinary PID controller. Secondly, a kind of PID fuzzy controller with a changeable integral parameter is designed to control these systems with big inertial and delay, etc. Finally, the designed fuzzy controllers are used in temperature and level control of the process control experiment system. Simulation and experiment results show that the designed fuzzy controllers are simple and easy to be realized, and meanwhile, they have some advantages of robustness and stability.
引文
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