modifications to the fuzzy controller. On the other hand if ye(kn) is big, the desired performance is not achieved and the learning mechanism must adjust the fuzzy controller. Next, we describe the operation of the learning mechanism 4.2.3 The learning mechanism The learning mechanism tunes the rule-base of the direct fuzzy controller so that the closed-loop system behaves like the reference model. These rule-base modifications are made by observing data from the controlled process, the reference model, and the fuzzy controller. The learning mechanism consists of two parts: a"fuzzy inverse model"and a knowledge-base modifier". The fuzzy inverse model performs the function of mapping ye(kn)(representing the deviation from the desired behavior ) to changes in the process inputs p(kn) that are necessary to force ye (kn)to zero The knowledge-base modifier performs the function of modifying the fuzzy controller's rule-base to affect the needed changes in the process inputs. We explain each of these components in detail next Fuzzy Inverse model Using the fact that most often a control engineer will know how to roughly characterize the inverse model of the plant (examples of how to do this will be given in several examples in this chapter), we use a fuzzy system to map ye (kT), and sibly functions of y.knsuch as y(k7)=(v(kT)-y(kT (or any other closed-loop system data),to the necessary changes in the process inputs. This fuzzy system is sometimes called the"fuzzy inverse model"since information about the plant inverse dynamics is used in its specification. Some, however, avoid this terminology and simply view the fuzzy system in the adaptation loop in Figure 4.3 to be a controller that tries to pick p(kn) to reduce the error ye( kn). This is the view taken for some of the design and implementation case studies in the next section Note that similar to the fuzzy controller, the fuzzy inverse model shown in Figure 4.3 contains scaling gains, but now we denote them with g,, 8, and g,. We will explain how to choose these scaling gains below. Given that gy, ye and 8y, y are inputs to the fuzzy inverse model, the rule-base for the fuzzy inverse model contains rules of the IFj。 is y, and j。 is y: Then p is P Where Y and y! denote linguistic values and Pm denotes the linguistic value associated with the mth output In this book we often utilize membership functions for the input universes of discourse as shown in Figure 4.4, symmetric triangular-shaped membership functions for the output universes of discourse, minimum to represent the premise and implication, and CoG defuzzification. Other choices can work equally well. For instance, we could make the same choices, except use singleton output membership functions and center-average defuzzification Knowledge-Base modifiermodifications to the fuzzy controller. On the other hand if ye(kT) is big, the desired performance is not achieved and the learning mechanism must adjust the fuzzy controller. Next, we describe the operation of the learning mechanism. 4.2.3 The Learning Mechanism The learning mechanism tunes the rule-base of the direct fuzzy controller so that the closed-loop system behaves like the reference model. These rule-base modifications are made by observing data from the controlled process, the reference model, and the fuzzy controller. The learning mechanism consists of two parts: a "fuzzy inverse model" and a "knowledge-base modifier". The fuzzy inverse model performs the function of mapping ye(kT) (representing the deviation from the desired behavior), to changes in the process inputs p(kT) that are necessary to force ye(kT) to zero. The knowledge-base modifier performs the function of modifying the fuzzy controller's rule-base to affect the needed changes in the process inputs. We explain each of these components in detail next. Fuzzy Inverse Model Using the fact that most often a control engineer will know how to roughly characterize the inverse model of the plant (examples of how to do this will be given in several examples in this chapter), we use a fuzzy system to map ye(kT) , and possibly functions of ye(kT)such as y kT y kT y kT T T c ee ( ) = −− ( ( ) ( )) (or any other closed-loop system data), to the necessary changes in the process inputs. This fuzzy system is sometimes called the "fuzzy inverse model" since information about the plant inverse dynamics is used in its specification. Some, however, avoid this terminology and simply view the fuzzy system in the adaptation loop in Figure 4.3 to be a controller that tries to pick p(kT) to reduce the error ye(kT) . This is the view taken for some of the design and implementation case studies in the next section. Note that similar to the fuzzy controller, the fuzzy inverse model shown in Figure 4.3 contains scaling gains, but now we denote them with e y g , c y g and . We will explain how to choose these scaling gains below. Given that p g e y e g y and c y c g y are inputs to the fuzzy inverse model, the rule-base for the fuzzy inverse model contains rules of the form is is is j l m ee c c IF and Then y Y y Y pP      Where j Ye  and denote linguistic values and denotes the linguistic value associated with the l Y c  m P mth output fuzzy set. In this book we often utilize membership functions for the input universes of discourse as shown in Figure 4.4, symmetric triangular-shaped membership functions for the output universes of discourse, minimum to represent the premise and implication, and COG defuzzification. Other choices can work equally well. For instance, we could make the same choices, except use singleton output membership functions and center-average defuzzification. Knowledge-Base Modifier