be understood that the choice of the controller inputs and outputs undamentally important part of the control design This is a very simple and academic nonlinear control problem, and many good techniques already exist for its solution. Indeed, for this standard configuration, a simple PId controller works well even in implementation In the remainder of this section, we will use the inverted pendulum as a convenient problem to illustrate the design and basic mechanics of the operation of a fuzzy control system. We will also use this problem in Section 2.4 to discuss much more general issues in fuzzy control system design that the reader will find useful for more challenging applications(e.g, the ones in the next chapter) 2.2.2 Putting Control Knowledge into Rule-Bases How do we put control knowledge in to rule-bases? Suppose that the human expert shown in Figure 2.3 provides a description of how best to control the plant in some natural language(e.g, English). We seek to take this"linguistic"description and load it into the fuzzy controller, indicated by the arrow in Figure 2.4 a Linguistic description Linguistic Descriptions The linguistic description provided by the expert can generally be broken into several parts. There will be"linguistic variables"that describe each of the time-varying fuzzy controller inputs and outputs. For the inverted pendulum, error"describes e(t) d change-in-error"describes(t) cribes u(t) Note that we use quotes to emphasize that certain words or phrases are linguistic descriptions, and that we have d added the time index to, for example, e(0), e(t) to emphasize that generally e varies with time. There are many possible choices for the linguistic descriptions for variables. Some designers like to choose them so that they are quite descriptive for documentation purposes However, this can sometimes lead to long descriptions. Others seek to keep the linguistic descriptions as short possible(e.g, using"e(t"as the linguistic variable for e(t )) yet accurate enough so that they adequately represent the variables, Regardless, the choice of the linguistic variable has no impact on the way that the fuzzy controller operates; it is simply a notation that helps to facilitate the construction of the fuzzy controller via fuzzy logic Just as e(l) takes on a value of, for examples 0 I at /=2(e(2)=0.1), linguistic variables assume"linguistic value That is, the values that linguistic variables take on overtime change dynamically. Suppose for the pendulum example that error, change-in-error, "and"force"take on the following valuesbe understood that the choice of the controller inputs and outputs is a fundamentally important part of the control design process. This is a very simple and academic nonlinear control problem, and many good techniques already exist for its solution. Indeed, for this standard configuration, a simple PID controller works well even in implementation. In the remainder of this section, we will use the inverted pendulum as a convenient problem to illustrate the design and basic mechanics of the operation of a fuzzy control system. We will also use this problem in Section 2.4 to discuss much more general issues in fuzzy control system design that the reader will find useful for more challenging applications (e.g., the ones in the next chapter). 2.2.2 Putting Control Knowledge into Rule-Bases How do we put control knowledge in to rule-bases? Suppose that the human expert shown in Figure 2.3 provides a description of how best to control the plant in some natural language (e.g., English). We seek to take this "linguistic" description and load it into the fuzzy controller, as indicated by the arrow in Figure 2.4. „ Linguistic description „ Rules „ Rule-bases Linguistic Descriptions The linguistic description provided by the expert can generally be broken into several parts. There will be "linguistic variables" that describe each of the time-varying fuzzy controller inputs and outputs. For the inverted pendulum, z "error" describes e(t) z "change-in-error" describes ( ) d e t dt z "force" describes u(t) Note that we use quotes to emphasize that certain words or phrases are linguistic descriptions, and that we have added the time index to, for example, e(t), ( ) d e t dt to emphasize that generally e varies with time. There are many possible choices for the linguistic descriptions for variables. Some designers like to choose them so that they are quite descriptive for documentation purposes. However, this can sometimes lead to long descriptions. Others seek to keep the linguistic descriptions as short as possible (e.g., using "e(t)" as the linguistic variable for e(t )), yet accurate enough so that they adequately represent the variables, Regardless, the choice of the linguistic variable has no impact on the way that the fuzzy controller operates; it is simply a notation that helps to facilitate the construction of the fuzzy controller via fuzzy logic. Just as e(t) takes on a value of, for examples 0.1 at t = 2 (e(2) = 0.1), linguistic variables assume "linguistic values" . That is, the values that linguistic variables take on overtime change dynamically. Suppose for the pendulum example that "error ," "change-in-error , " and "force" take on the following values: z "neglarge