基于矩阵半张量积方法的模糊系统分析与设计
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摘要
与传统控制技术相比,模糊控制技术具有两大不可比拟的优点：(1)在许多实际应用中可以有效且方便地实现人的控制策略和知识经验；(2)可以不依赖于被控对象的数学模型,便可达到良好的控制效果.近几十年来,广大学者对模糊逻辑和模糊逻辑系统进行了广泛深入的研究,并取得了丰硕的成果,推动了模糊系统理论与应用的极大发展.对于非线性系统,甚至非解析系统来说,模糊逻辑控制已被证明是一种有效的控制方法.
工业生产过程中的多数系统都具有多个输入输出变量,对于多变量模糊系统来说,控制规则的条数随着输入变量的增加呈指数递增.为解决这类问题,产生了很多处理方法,但多数方法不可避免地产生推理误差,降低了模糊推理的精度.随机性和模糊性是两种不同性质的不确定性,它们常常共存于现实世界中的许多物理过程和系统中,例如机器人控制系统、电力系统和信号处理过程等.所以研究带有随机性的模糊逻辑系统是一件很有价值和意义的工作.近年来,程代展教授提出了一种新的矩阵理论-矩阵半张量积理论.迄今为止,半张量积理论已在很多领域取得了丰富的成果.注意到,模糊逻辑其实是扩展的混合值逻辑.因此,半张量积理论也可以用于模糊控制器的研究.基于半张量积方法,复杂的模糊推理过程可以转换为一系列代数方程的求解问题,大大简化了模糊逻辑推理.我们希望通过本论文的研究为模糊系统的研究开辟一些新的途径,以进一步丰富模糊控制系统的理论和应用.
本论文针对多变量模糊系统、分层模糊系统、随机模糊系统的分析和控制设计等问题进行了研究,主要包括以下内容：
第一部分基于半张量积理论研究了多变量模糊系统的分析和控制器设计.通过用逻辑向量表示输入输出语言变量,得到了控制规则新的表示形式.基于该表示形式,通过构建结构矩阵,复杂的模糊推理转换成了代数等式的形式.利用半张量积理论建立了多变量模糊逻辑控制器的研究框架.仿真例子验证了本章提出的方法是有效的.通过对模糊逻辑控制器结构矩阵的分析,得到了最小入度模糊控制.而且,当控制器的控制规则不完备时,给出了计算最小入度模糊控制的算法.当控制规则不符合一致性时,给出了得到一致性控制规则的几条原则.最后,将此方法用到了并行混合电动汽车(PHEV)的能量管理和控制策略的模糊控制器设计上.
第二部分提出了混合值逻辑函数的半张量积分解方法,实现了混合值逻辑函数的串联分解、并联不相交分解和并联相交分解.给出了混合值逻辑函数可以进行半张量积分解的充分必要条件.如果已知其中一个分解函数,给出了另外一个分解函数的求解方法与其所有可能的取值.本方法同样适用于k值逻辑与布尔逻辑.利用多变量模糊控制系统的半张量积设计方法以及混合值逻辑函数的半张量积分解,建立了实现多输入单输出(MISO)模糊系统的串联分层、并联分层以及混合分层建模的新方法.利用这种方法可以很方便的得到中间层的结构矩阵与模糊规则.本方法得到的分层模糊系统与原来的多变量模糊系统相比较,分层前后模糊系统系统的输入输出模型是等价的,而且大大减少了控制规则的条数.
第三部分研究了随机模糊逻辑和随机模糊控制器设计.介绍了随机模糊逻辑的概念和相关性质,利用矩阵半张量积理论给出了随机模糊控制规则的向量表示形式.基于随机模糊规则的新的表示形式,构造了随机模糊控制器的结构矩阵和概率转移矩阵,得到了随机模糊推理的代数表达式.最后给出例子验证了所得结果的有效性.
本论文的主要创新点是：
·基于矩阵半张量积理论,得到了多变量模糊系统模糊推理的代数形式,当模糊规则不完备时,给出了最小入度模糊控制算法.当模糊控制规则不一致时给出了相应的处理方法.
·提出了混合值逻辑函数的半张量积分解方法,实现了混合值逻辑函数的串联分解、并联不相交分解和并联相交分解.基于混合值逻辑函数的半张量积分解,建立了实现多变量模糊系统的串联分层、并联分层和混合分层建模的新方法.
·利用矩阵半张量积理论研究了随机模糊逻辑和随机模糊系统的控制器设计.基于随机模糊规则的向量表示形式,构造了随机模糊控制器的结构矩阵和概率转移矩阵,得到了随机模糊推理的代数表达式.
Compared with conventional control systems, fuzzy control systems have the fol-lowing two unmatched advantages. First, it can realize easily and effectively human control strategies and experience in many real applications. Second, it can achieve bet-ter control performance independent of the mathematic model of the controlled system. During the past few decades, increasing attention has been devoted to fuzzy logic and fuzzy logic systems, and many nice results have been proposed for the analysis and synthesis of fuzzy control systems promoting the development of fuzzy system theory and its application. The fuzzy logic control has been proved to be a successful control approach to many complex nonlinear systems or even nonanalytic systems.
Many of industrial processes and systems have multiple input and output variables. Whereas, in a multi-variables fuzzy systems, the number of rules increases exponen-tially with the number of variable involved. To deal with the rule-explosion problem, a lot of processing methods have been generated. But inference error of many of these methods is inevitable. Moreover, there are two different kinds of uncertainty-the ran-domness and fuzziness often coexist in many physical processes and systems in the real-world, such as robot control systems, power systems and signal processing, etc. As probabilistic methods and fuzzy techniques are effective approaches to deal with the randomness and fuzziness respectively, it is a worthwhile and meaningful job to bridge the gap between fuzziness and probability. Recently, the semi-tensor product (STP) of matrices was proposed, and up to now, it has been widely applied in many fields and lots of fundamental results have been presented. It is noted that fuzzy logic can be consid-ered as an extended mix-valued logic, and by the STP method, the complex reasoning process can be converted into a problem of solving a set of algebraic equations, which greatly simplifies the process of logical reasoning. We hope that this thesis can provide some new ways to study fuzzy systems and further enrich the theory and application of fuzzy systems.
In this paper, we investigate the analysis and control design for multi-variable fuzzy systems, hierarchical fuzzy systems and stochastic fuzzy system. The main con-tents of this thesis are composed of the following parts:
The first part investigate the fuzzy logic controller(FLC) analysis and design for multi-input multi-output fuzzy systems based on the semi-tensor product of matrices. Firstly, we give a new expression for the fuzzy control rules via expressing the input and output variables with the sector form. Based on this form, we convert the fuzzy reasoning into an algebraic form by constructing structural matrices of the FLC. Then a new framework is established to study multi-variable FLC. According to the proposed approach, a simulation example is given to demonstrate its effectiveness. A set of rea-sonable least in-degree controls are obtained through the analysis of structural matrices. when the control rules are incomplete, the algorithm of least in-degree controls is given. Moreover, the consistency of fuzzy control rules is introduced, and some principles are proposed for dealing with the inconsistency of fuzzy controls. Finally, the proposed ap-proach is applied to design of fuzzy controllers for the energy management and control strategy of parallel hybrid vehicles (PHV).
The second part studies the semi-tensor product decomposition of mixed-valued logical functions and the modeling of hierarchical fuzzy systems. Firstly, the approach to the semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Then the sufficient and necessary condition is given for the semi-tensor product decomposition of mixed-valued logical functions, and the al-gorithm of the other decomposition function and all its possible solutions are devel-oped when one decomposition function is known. This method is also applicable to the decomposition of k-valued logic function and Boolean logic function. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierarchical structure and hybrid hierar-chical structure. The algorithm of this scheme is developed such that one can easily design the involved structural matrices and fuzzy rules in the muddle layers of the hi-erarchical structure. It is well worth pointing out that, using this method, the same input-output model can be get as the conventional layer fuzzy logic system, and the total number of the rules can be greatly reduced.
The third part considers the stochastic fuzzy logic and stochastic fuzzy systems based on one new method, that is, the semi-tensor product of matrices, and obtain some new results about stochastic fuzzy systems. Firstly, some concepts and properties on stochastic fuzzy logic are given. Then, the design of stochastic fuzzy controller is studied based on the semi-tensor product, and the algebraic equation of fuzzy reasoning is obtained. Moreover, the structural matrix and the probability transition matrix are obtained. Finally, a numerical example is provided to demonstrate our new results.
Innovations of the thesis mainly include the following aspects:
●The algebraic form of fuzzy reasoning of multi-variable fuzzy systems is ob-tained based on semi-tensor product of matrices. when the control rules are incomplete, the algorithm of least in-degree controls has been given. Some principles are proposed for dealing with the inconsistency of fuzzy controls.
●The semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierar-chical structure and hybrid hierarchical structure. It is noted that the results obtained in this paper not only are simple, but also have more advantages than the existing ones in some cases.
●The stochastic fuzzy logic and stochastic fuzzy systems are discussed based on the semi-tensor product of matrices. By vector representation of stochastic fuzzy rules, the structural matrix and the probability transition matrix are constructed. Then, the algebraic equation of stochastic fuzzy reasoning is obtained.
工业生产过程中的多数系统都具有多个输入输出变量,对于多变量模糊系统来说,控制规则的条数随着输入变量的增加呈指数递增.为解决这类问题,产生了很多处理方法,但多数方法不可避免地产生推理误差,降低了模糊推理的精度.随机性和模糊性是两种不同性质的不确定性,它们常常共存于现实世界中的许多物理过程和系统中,例如机器人控制系统、电力系统和信号处理过程等.所以研究带有随机性的模糊逻辑系统是一件很有价值和意义的工作.近年来,程代展教授提出了一种新的矩阵理论-矩阵半张量积理论.迄今为止,半张量积理论已在很多领域取得了丰富的成果.注意到,模糊逻辑其实是扩展的混合值逻辑.因此,半张量积理论也可以用于模糊控制器的研究.基于半张量积方法,复杂的模糊推理过程可以转换为一系列代数方程的求解问题,大大简化了模糊逻辑推理.我们希望通过本论文的研究为模糊系统的研究开辟一些新的途径,以进一步丰富模糊控制系统的理论和应用.
本论文针对多变量模糊系统、分层模糊系统、随机模糊系统的分析和控制设计等问题进行了研究,主要包括以下内容：
第一部分基于半张量积理论研究了多变量模糊系统的分析和控制器设计.通过用逻辑向量表示输入输出语言变量,得到了控制规则新的表示形式.基于该表示形式,通过构建结构矩阵,复杂的模糊推理转换成了代数等式的形式.利用半张量积理论建立了多变量模糊逻辑控制器的研究框架.仿真例子验证了本章提出的方法是有效的.通过对模糊逻辑控制器结构矩阵的分析,得到了最小入度模糊控制.而且,当控制器的控制规则不完备时,给出了计算最小入度模糊控制的算法.当控制规则不符合一致性时,给出了得到一致性控制规则的几条原则.最后,将此方法用到了并行混合电动汽车(PHEV)的能量管理和控制策略的模糊控制器设计上.
第二部分提出了混合值逻辑函数的半张量积分解方法,实现了混合值逻辑函数的串联分解、并联不相交分解和并联相交分解.给出了混合值逻辑函数可以进行半张量积分解的充分必要条件.如果已知其中一个分解函数,给出了另外一个分解函数的求解方法与其所有可能的取值.本方法同样适用于k值逻辑与布尔逻辑.利用多变量模糊控制系统的半张量积设计方法以及混合值逻辑函数的半张量积分解,建立了实现多输入单输出(MISO)模糊系统的串联分层、并联分层以及混合分层建模的新方法.利用这种方法可以很方便的得到中间层的结构矩阵与模糊规则.本方法得到的分层模糊系统与原来的多变量模糊系统相比较,分层前后模糊系统系统的输入输出模型是等价的,而且大大减少了控制规则的条数.
第三部分研究了随机模糊逻辑和随机模糊控制器设计.介绍了随机模糊逻辑的概念和相关性质,利用矩阵半张量积理论给出了随机模糊控制规则的向量表示形式.基于随机模糊规则的新的表示形式,构造了随机模糊控制器的结构矩阵和概率转移矩阵,得到了随机模糊推理的代数表达式.最后给出例子验证了所得结果的有效性.
本论文的主要创新点是：
·基于矩阵半张量积理论,得到了多变量模糊系统模糊推理的代数形式,当模糊规则不完备时,给出了最小入度模糊控制算法.当模糊控制规则不一致时给出了相应的处理方法.
·提出了混合值逻辑函数的半张量积分解方法,实现了混合值逻辑函数的串联分解、并联不相交分解和并联相交分解.基于混合值逻辑函数的半张量积分解,建立了实现多变量模糊系统的串联分层、并联分层和混合分层建模的新方法.
·利用矩阵半张量积理论研究了随机模糊逻辑和随机模糊系统的控制器设计.基于随机模糊规则的向量表示形式,构造了随机模糊控制器的结构矩阵和概率转移矩阵,得到了随机模糊推理的代数表达式.
Compared with conventional control systems, fuzzy control systems have the fol-lowing two unmatched advantages. First, it can realize easily and effectively human control strategies and experience in many real applications. Second, it can achieve bet-ter control performance independent of the mathematic model of the controlled system. During the past few decades, increasing attention has been devoted to fuzzy logic and fuzzy logic systems, and many nice results have been proposed for the analysis and synthesis of fuzzy control systems promoting the development of fuzzy system theory and its application. The fuzzy logic control has been proved to be a successful control approach to many complex nonlinear systems or even nonanalytic systems.
Many of industrial processes and systems have multiple input and output variables. Whereas, in a multi-variables fuzzy systems, the number of rules increases exponen-tially with the number of variable involved. To deal with the rule-explosion problem, a lot of processing methods have been generated. But inference error of many of these methods is inevitable. Moreover, there are two different kinds of uncertainty-the ran-domness and fuzziness often coexist in many physical processes and systems in the real-world, such as robot control systems, power systems and signal processing, etc. As probabilistic methods and fuzzy techniques are effective approaches to deal with the randomness and fuzziness respectively, it is a worthwhile and meaningful job to bridge the gap between fuzziness and probability. Recently, the semi-tensor product (STP) of matrices was proposed, and up to now, it has been widely applied in many fields and lots of fundamental results have been presented. It is noted that fuzzy logic can be consid-ered as an extended mix-valued logic, and by the STP method, the complex reasoning process can be converted into a problem of solving a set of algebraic equations, which greatly simplifies the process of logical reasoning. We hope that this thesis can provide some new ways to study fuzzy systems and further enrich the theory and application of fuzzy systems.
In this paper, we investigate the analysis and control design for multi-variable fuzzy systems, hierarchical fuzzy systems and stochastic fuzzy system. The main con-tents of this thesis are composed of the following parts:
The first part investigate the fuzzy logic controller(FLC) analysis and design for multi-input multi-output fuzzy systems based on the semi-tensor product of matrices. Firstly, we give a new expression for the fuzzy control rules via expressing the input and output variables with the sector form. Based on this form, we convert the fuzzy reasoning into an algebraic form by constructing structural matrices of the FLC. Then a new framework is established to study multi-variable FLC. According to the proposed approach, a simulation example is given to demonstrate its effectiveness. A set of rea-sonable least in-degree controls are obtained through the analysis of structural matrices. when the control rules are incomplete, the algorithm of least in-degree controls is given. Moreover, the consistency of fuzzy control rules is introduced, and some principles are proposed for dealing with the inconsistency of fuzzy controls. Finally, the proposed ap-proach is applied to design of fuzzy controllers for the energy management and control strategy of parallel hybrid vehicles (PHV).
The second part studies the semi-tensor product decomposition of mixed-valued logical functions and the modeling of hierarchical fuzzy systems. Firstly, the approach to the semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Then the sufficient and necessary condition is given for the semi-tensor product decomposition of mixed-valued logical functions, and the al-gorithm of the other decomposition function and all its possible solutions are devel-oped when one decomposition function is known. This method is also applicable to the decomposition of k-valued logic function and Boolean logic function. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierarchical structure and hybrid hierar-chical structure. The algorithm of this scheme is developed such that one can easily design the involved structural matrices and fuzzy rules in the muddle layers of the hi-erarchical structure. It is well worth pointing out that, using this method, the same input-output model can be get as the conventional layer fuzzy logic system, and the total number of the rules can be greatly reduced.
The third part considers the stochastic fuzzy logic and stochastic fuzzy systems based on one new method, that is, the semi-tensor product of matrices, and obtain some new results about stochastic fuzzy systems. Firstly, some concepts and properties on stochastic fuzzy logic are given. Then, the design of stochastic fuzzy controller is studied based on the semi-tensor product, and the algebraic equation of fuzzy reasoning is obtained. Moreover, the structural matrix and the probability transition matrix are obtained. Finally, a numerical example is provided to demonstrate our new results.
Innovations of the thesis mainly include the following aspects:
●The algebraic form of fuzzy reasoning of multi-variable fuzzy systems is ob-tained based on semi-tensor product of matrices. when the control rules are incomplete, the algorithm of least in-degree controls has been given. Some principles are proposed for dealing with the inconsistency of fuzzy controls.
●The semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierar-chical structure and hybrid hierarchical structure. It is noted that the results obtained in this paper not only are simple, but also have more advantages than the existing ones in some cases.
●The stochastic fuzzy logic and stochastic fuzzy systems are discussed based on the semi-tensor product of matrices. By vector representation of stochastic fuzzy rules, the structural matrix and the probability transition matrix are constructed. Then, the algebraic equation of stochastic fuzzy reasoning is obtained.
引文
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