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
11 ◼ 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 inputoutput 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 design and implementation case study for'a chapter. Finally, in Section 3.8, we provide parameter estimation in an internal combustion engine 12
12 ◼ 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 impor- tant since many adaptive control techniques depend on the use of an estimator. Moreover, the sections on neural networks and genetic understand this chapter and those sections, you will see how those algorithms in Chapter depend on this chapter in the sense that if you 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 the fuzzy model reference learning controller (FMRLC) described i Or presented in Chapters 6 and 7. For example, the learning mechanism Chapter 6 can be viewed as an identification algorithm that is used to tune a fuzzy controller 13
13 ◼ 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 a 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 function approximation. Finally, we discuss issues in the choice of the data that we use to construct the approximators, discuss the incorporation of linguistic information, and provide an example of how to construct a data set for a parameter estimation problem 14
14 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 function approximation. Finally, we discuss issues in the choice of the data that we use to construct the approximators , discuss the incorporation of linguistic information, and provide an example of how to construct a data set for a parameter estimation problem
3.2.1 The Function Approximation Problem a Given some function 8: x>y a where xCs and y c we wish to construct a fuzzy system f:X→Y where XCx and r c yare some domain and range of interest, by choosing a parameter vector 6(which may include membership function centers, widths, etc. )so that 9(x)=f(x0)+c(x)( for all x=lx, x2,,x, I EX where the approximation error e(x is as small as possible. If we want to refer to the input at time k, we will use x(h) for the vector and x, (k) for its jth component 15
15 3.2.1 The Function Approximation Problem ◼ Given some function ◼ where and , we wish to construct a fuzzy system ◼ where and are some domain and range of interest, by choosing a parameter vector (which may include membership function centers, widths, etc.) so that ◼ (3.1) ◼ for all where the approximation error e(x) is as small as possible. If we want to refer to the input at time k, we will use x(k) for the vector and for its jth component. g x y : → n x y f X Y : → X x Y y g x f x e x ( ) ( ) ( ) = + 1 2 [ , ,..., ]T n x x x x X = ( ) j x k