If error is positive-large and change-in-error is negative-small Then plant-input is positive-big (in this case e="error",E="positive-large", etc. ) We use a standard choice for all the membership functions on all the input universes of discourse, such as the ones shown in Figure 4.4. Hence, we would simply use some membership functions similar to those in Figure 4.4, but with a scaled horizontal axis, for the c(kT)input kT) Figure 4. 4 Membership functions for input universe of discourse We will use all possible combinations of rules for the rule-base. For example, we could choose to have ll membership functions on each of the twe out universes at course, in which ca3we would avQ212=121 rules n the rule-base. At first glance it would appear that the complexity of the controller could make implementation prohibitive for applications where it is necessary to have many inputs to the fuzzy controller. However, we must remind the reader of the results in Section 2.6 where we explain how implementation tricks can be used to significantly reduce computation time when there are input membership functions of the form shown in Figure 4.4 Rule-Base initialization The input membership functions are defined to characterize the premises of the rules that define the various situations in which rules should be applied. The input-e 8 ship funa 0s ge left consta 0) are not tun 0:2e FMRLC. The membership functions on the output universe of discourse are ed to be unknown. They are what the FMRLC will automatically synthesize or tune. Hence, the Fmrlc tries to fill in what actions ought to be taken for the various situations that are characterized by the premises We must choose initial values for each of the output membership functions. For example, for an output universe of discourse[-1, I we could choose triangular-shaped membership functions with base widths of 0. 4 and centers at zero This choice represents that the fuzzy controller initially knows nothing about how to control the plant so it inputs u=0 to the plant initially(well, really it does know something since we specify the remainder of the fuzzy controller a priori)Of course, one can often make a reasonable best guess at how to specify a fuzzy controller that is"more knowledgeable than simply placing the output membership function centers at zero. For example, we could pick the initial fuzzy controller to be the best one that we can design for the nominal plant. Notice, however, that this choice is not always the best one. Really, what you often want to choose is the fuzzy controller that is best for the operating condition that the plant will begin in(this may not be the nominal condition). Unfortunately, it is not always possible to pick such aIf error is positive-large and change-in-error is negative-small Then plant-input is positive-big (in this case e = "error", 4 E = "positive-large", etc.). We use a standard choice for all the membership functions on all the input universes of discourse, such as the ones shown in Figure 4.4. Hence, we would simply use some membership functions similar to those in Figure 4.4, but with a scaled horizontal axis, for the c(kT) input. e kT ( ) Figure 4.4 Membership functions for input universe of discourse We will use all possible combinations of rules for the rule-base. For example, we could choose to have 11 membership functions on each of the two input universes of discourse, in which case we would have 112 = 121 rules in the rule-base. At first glance it would appear that the complexity of the controller could make implementation prohibitive for applications where it is necessary to have many inputs to the fuzzy controller. However, we must remind the reader of the results in Section 2.6 where we explain how implementation tricks can be used to significantly reduce computation time when there are input membership functions of the form shown in Figure 4.4. Rule-Base Initialization The input membership functions are defined to characterize the premises of the rules that define the various situations in which rules should be applied. The input membership functions are left constant and are not tuned by the FMRLC. The membership functions on the output universe of discourse are assumed to be unknown. They are what the FMRLC will automatically synthesize or tune. Hence, the FMRLC tries to fill in what actions ought to be taken for the various situations that are characterized by the premises. We must choose initial values for each of the output membership functions. For example, for an output universe of discourse [-1, 1] we could choose triangular-shaped membership functions with base widths of 0.4 and centers at zero. This choice represents that the fuzzy controller initially knows nothing about how to control the plant so it inputs u = 0 to the plant initially (well, really it does know something since we specify the remainder of the fuzzy controller a priori). Of course, one can often make a reasonable best guess at how to specify a fuzzy controller that is "more knowledgeable" than simply placing the output membership function centers at zero. For example, we could pick the initial fuzzy controller to be the best one that we can design for the nominal plant. Notice, however, that this choice is not always the best one. Really, what you often want to choose is the fuzzy controller that is best for the operating condition that the plant will begin in (this may not be the nominal condition). Unfortunately, it is not always possible to pick such a