第3章 Fuzzy Identification and estimation 教学内容 这一章主要讲述用模糊进行评估和辨识。模糊辨识设计的最主要的问题是用已知的 离散数据构建一个模糊系统。首先介绍最基本的函数近似问题,然后介绍传统的辨识方 法:最小二乘法,即怎样用批量最小二乘法和递归最小二乘法来辨识一个系统以匹配输 入输出数据。最后讲述用这两种方法直接训练模糊系统 教学重点 重点是最小二乘算法。模糊辨识的最小二乘算法包括成批最小二乘算法和递推最小 乘算法。 教学难点 对最小二乘算法的准确把握和理解,关键是应用模糊最小二乘法实现模糊辨识和估 计 教学要求 要求掌握模糊辨识和估计的基本概念和知识,主要掌握最小二乘算法的推导和应 用 3.1 Overview While up to this point we have focused on control, in this chapter we will examine how to use fuzzy systems for estimation and identification The basic problem to be studied here is how to construct a fuzzy system from numerical data. This is in contrast to our discussion in Chapters 2 and 3, where we used linguistics as the starting point to specify a fuzzy system. If the numerical data is plant input-output data obtained from ar experiment, we may identify a fuzzy system model of the plant. This may be useful for simulation purposes and sometimes for use in a controller On the other hand, the data may come from other sources, and a fuzzy
第3章 Fuzzy Identification and Estimation 教学内容 这一章主要讲述用模糊进行评估和辨识。模糊辨识设计的最主要的问题是用已知的 离散数据构建一个模糊系统。首先介绍最基本的函数近似问题，然后介绍传统的辨识方 法：最小二乘法，即怎样用批量最小二乘法和递归最小二乘法来辨识一个系统以匹配输 入输出数据。最后讲述用这两种方法直接训练模糊系统。 教学重点 重点是最小二乘算法。模糊辨识的最小二乘算法包括成批最小二乘算法和递推最小 二乘算法。 教学难点 对最小二乘算法的准确把握和理解,关键是应用模糊最小二乘法实现模糊辨识和估 计。 教学要求 要求掌握模糊辨识和估计的基本概念和知识，主要掌握最小二乘算法的推导和应 用。 3.1 Overview While up to this point we have focused on control, in this chapter we will examine how to use fuzzy systems for estimation and identification. The basic problem to be studied here is how to construct a fuzzy system from numerical data. This is in contrast to our discussion in Chapters 2 and 3, where we used linguistics as the starting point to specify a fuzzy system. If the numerical data is plant input-output data obtained from an experiment, we may identify a fuzzy system model of the plant. This may be useful for simulation purposes and sometimes for use in a controller. On the other hand, the data may come from other sources, and a fuzzy
system may be used to provide for a parameterized nonlinear function that fits the data by using its basic interpolation capabilities. For instance suppose that we have a human expert who controls some process and we observe how she or he does this by observing what numerical plant input the expert picks for the given numerical data that she or he observes Suppose further that we have many such associations between decision-making data "The methods in this chapter will show how to construct rules for a fuzzy controller from this data(i.e, identify a controller from the human-generated decision-making data) and in this sense they provide another method to design controllers Yet another problem that can be solved with the methods in this hapter is that of how to construct a fuzzy system that will serve as a parameter estimator. To do this, we need data that shows, roughly how the input-output mapping of the estimator should behave (i.e, how it should estimate). One way to generate this data is to begin by establishing a simulation test bed for the plant for which parameter estimation must be performed. Then a set of simulations can be conducted, each with a different value for the parameter to be estimated by coupling the test conditions and simulation-generated data with the parameter values, you can gather appropriate data pairs that allow for the construction of a fuzz estimator, For some plants it may be possible to perform this procedure with actual experimental data(by physically adjusting the parameter to be
system may be used to provide for a parameterized nonlinear function that fits the data by using its basic interpolation capabilities. For instance, suppose that we have a human expert who controls some process and we observe how she or he does this by observing what numerical plant input the expert picks for the given numerical data that she or he observes. Suppose further that we have many such associations between "decision-making data." The methods in this chapter will show how to construct rules for a fuzzy controller from this data (i.e., identify a controller from the human-generated decision-making data), and in this sense they provide another method to design controllers. Yet another problem that can be solved with the methods in this chapter is that of how to construct a fuzzy system that will serve as a parameter estimator. To do this, we need data that shows, roughly how the input-output mapping of the estimator should behave (i.e., how it should estimate). One way to generate this data is to begin by establishing a simulation test bed for the plant for which parameter estimation must be performed. Then a set of simulations can be conducted, each with a different value for the parameter to be estimated .by coupling the test conditions and simulation-generated data with the parameter values, you can gather appropriate data pairs that allow for the construction of a fuzzy estimator, Forsome plants it may be possible to perform this procedure with actual experimental data (by physically adjusting the parameter to be
estimated). In a similar way, you could construct fuzzy predictors using the approaches developed in this chapter We begin this chapter by setting up the basic function approximation problem in Section 3. 2, where we provide an overview of some of the fundamental issues in how to fit a' function to input-output data, including how to incorporate linguistic information into the function that we are trying to force to match the data. We explain how to measure how well a function fits data and provide an example of how to choose a data set for an engine failure estimation problem(a type of parameter estimation problem in which when estimates of the parameters take on certain values, we say that a failure has occurred) In Section 3, 3 we introduce conventional least squares methods for identification, explain how they can be used to tune fuzzy systems provide a simple example, and offer examples of how they can be used to train fuzzy systems-Next, in Section 3, 4 we show how gradient methods can be used to train a standard and Takagi-Sugeno fuzzy system, These methods are quite similar to the ones used to train neural networks(e. g the"back-propagation technique"). We provide examples for standard and Takagi-Sugeno fuzzy systems. We highlight the fact that via either the recursive least squares method for fuzzy systems or the gradient method we can perform on-line parameter estimation. We will see in Chapter 6 that these methods can be combined with a controller
estimated). In a similar way, you could construct' fuzzy predictors using the approaches developed in this chapter We begin this chapter by setting up the basic function approximation problem in Section 3.2, where we provide an overview of some of the fundamental issues in how to fit a' function to input-output data, including how to incorporate linguistic information into the function that we are trying to force to match the data. We explain how to measure how well a function fits data and provide an example of how to choose a data set for an engine failure estimation problem (a type of parameter estimation problem in which when estimates of the parameters take on certain values, we say that a failure has occurred). In Section 3,3 we introduce conventional least squares methods for identification, explain how they can be used to tune fuzzy systems, provide a simple example, and offer examples of how they can be used to train fuzzy systems- Next, in Section 3,4 we show how gradient methods can be used to train a standard and Takagi-Sugeno fuzzy system, These methods are quite similar to the ones used to train neural networks (e.g., the "back-propagation technique"). We provide examples for standard and Takagi-Sugeno fuzzy systems. We highlight the fact that via either the recursive least squares method for fuzzy systems or the gradient method we can perform on-line parameter estimation. We will see in Chapter 6 that these methods can be combined with a controller
construction procedure to provide a method for adaptive fuzzy control In Section 3.5 we introduce two techniques for training fuzzy systems based on clustering. The first uses"c-means clustering"and least squares to train the premises and consequents, respectively, of the Takagi-Sugeno fuzzy system; while the second uses a nearest neighborhood technique to train standard fuzzy systems. In Section 3.6 we present two"learning from examples"(LFE) methods for constructing rules for fuzzy systems from input-output data Compared to the previous methods, these do not use optimization to construct the fuzzy system parameters. Instead, the LFE methods are based on simple procedures to extract rules directly from the data In Section 3. 7 we show how hybrid methods for training fuzzy systems can be developed by combining the methods described in this chapter Finally, in Section 3.8, we provide a design and implementation case study for parameter estimation in an internal combustion engine Overall, the objective of this chapter is to show how to construct fuzzy systems from numerical data. This will provide the reader with another general approach for fuzzy system design that may augment or extend the approach described in Chapters 2 and 3, where we start from linguistic information. With a good understanding of Chapter 2, the reader can complete this chapter without having read Chapters 3 and 4 The section on indirect adaptive control in Chapter 6 relies on the
construction procedure to provide a method for adaptive fuzzy control. In Section 3.5 we introduce two techniques for training fuzzy systems based on clustering. The first uses "c-means clustering" and least squares to train the premises and consequents, respectively, of the Takagi-Sugeno fuzzy system; while the second uses a nearest neighborhood technique to train standard fuzzy systems. In Section 3.6 we present two "learning from examples" (LFE) methods for constructing rules for fuzzy systems from input-output data. Compared to the previous methods, these do not use optimization to construct the fuzzy system parameters. Instead, the LFE methods are based on simple procedures to extract rules directly from the data. In Section 3.7 we show how hybrid methods for training fuzzy systems can be developed by combining the methods described in this chapter. Finally, in Section 3.8, we provide .a design and implementation case study for parameter estimation in an internal combustion engine. Overall, the objective of this chapter is to show how to construct fuzzy systems from numerical data. This will provide the reader with another general approach for fuzzy system design that may augment or extend the approach described in Chapters 2 and 3, where we start from linguistic information. With a good understanding of Chapter 2, the reader can complete this chapter without having read Chapters 3 and 4- The section on indirect adaptive control in Chapter 6 relies on the
gradient and least squares methods discussed in this chapter, and a portion of the section on gain schedule construction in Chapter 7 relies on the reader knowing at least one method from this chapter. In other words this chapter is important since many adaptive control techniques depend on the use of an estimator moreover. the sections on neural networks and genetic algorithms in Chapter 8 depend on this chapter in the sense that if you understand this chapter and those sections, you will see how those techniques relate to the ones discussed here. Otherwise the remainder of the book can be completed without this chapter; however, this chapter will provide for a deeper understanding of many of the concepts to be presented in Chapters 6 and 7. For example, the learning mechanism for the fuzzy model reference learning controller(FMrLC)described in Chapter 6 can be viewed as an identification algorithm that is used to tune a fuzzy controller 3.2 Fitting functions to data We begin this section by precisely defining the function approximation problem, in which you seek to synthesize a function to approximate another function that is inherently represented via a finite number of input-output associations (i.e, we only know how the function maps a finite number of points in its domain to its range). Next, we show how the problem of how to construct nonlinear system identifiers and nonlinear estimators is a special case of the problem of how to perform
gradient and least squares methods discussed in this chapter, and a portion of the section on gain schedule construction in Chapter 7 relies on the reader knowing at least one method from this chapter. In other words, this chapter is important since many adaptive control techniques depend on the use of an estimator. Moreover, the sections on neural networks and genetic algorithms in Chapter 8 depend on this chapter in the sense that if you understand this chapter and those sections, you will see how those techniques relate to the ones discussed here. Otherwise, the remainder of the book can be completed without this chapter; however, this chapter will provide for a deeper understanding of many of the concepts to be presented in Chapters 6 and 7. For example, the learning mechanism for the fuzzy model reference learning controller (FMRLC) described in Chapter 6 can be viewed as an identification algorithm that is used to tune a fuzzy controller. 3.2 Fitting Functions to Data We begin this section by precisely defining the function approximation problem, in which you seek to synthesize a function to approximate another function that is inherently represented via a finite number of input-output associations (i.e., we only know how the function maps a finite number of points in its domain to its range). Next, we show how the problem of how to construct nonlinear system identifiers and nonlinear estimators is a special case of the problem of how to perform