proper time Rules Next, we will use the above linguistic quantification to specify a set of rules(a rule-base) that captures the expert's knowledge about how to control the plant. In particular, for the inverted pendulum in the three positions shown in Figure 2.5, we have the following rules(notice that we drop the quotes since the whole rule is linguistic 1. If error is neglarge and change-in-error is neglarge Then force is poslarge This rule quantifies the situation in Figure 2.5(a) where the pendulum has a large positive angle and is moving clockwise; hence it is clear that we should apply a strong positive force(to the right) so that we can try to start the pendulum moving in the proper dire Figure 2.5 Inverted m in various positi 2. If error is zero and change-in-error is possmall Then force is negsmall This rule quantifies the situation in Figure 2.5(b) where the pendulum has nearly a zero angle with the vertical(a guistic quantification of zero does not imply that e(t)=0 exactly) and is moving counterclockwise, hence we should apply a small negative force(to the left)to counteract the movement so that it moves toward zero(a positive force could result in the pendulum overshooting the desired position) 3. If error is poslarge and change-in-error is negsmall Then force is negsmall This rule quantifies the situation in Figure 2.5(c)where the pendulum is far to the left of the vertical and is moving clockwise; hence we should apply a small negative forceto the left) to assist the movement, but not a big one since the pendulum is already moving in the proper direction. Each of the three rules listed above is a"linguistic rule"since it is formed solely from linguistic variables and values Since linguistic values are not precise representations of the underlying quantities that they describe, linguistic rules are not precise either. They are simply abstract ideas about how to achieve good control that could mean somewhat different things to different people. They are, however, at a level of abstraction that humans are often comfortable with in terms specifying how to control a process The general form of the linguistic rules listed abov If premise Then consequent As you can see from the three rules listed above, the premises(which are sometimes called"antecedents")are associated with the fuzzy controller inputs and are on the left-hand-side of the rules. The consequents(sometimes called"actions") are associated with the fuzzy controller outputs and are on the right-hand-side of the rules Notice that each premise(or consequent) can be composed of the conjunction of several"terms"(e.g, in rule 3 above"error is poslarge and change-in-error is negsmall"is a premise that is the Conjunction of two terms ). The numberproper time. Rules： Next, we will use the above linguistic quantification to specify a set of rules (a rule-base) that captures the expert's knowledge about how to control the plant. In particular, for the inverted pendulum in the three positions shown in Figure 2.5, we have the following rules (notice that we drop the quotes since the whole rule is linguistic): 1. If error is neglarge and change-in-error is neglarge Then force is poslarge This rule quantifies the situation in Figure 2.5(a) where the pendulum has a large positive angle and is moving clockwise; hence it is clear that we should apply a strong positive force (to the right) so that we can try to start the pendulum moving in the proper direction. (a) (b) (c) Figure 2.5 Inverted pendulum in various positions 2. If error is zero and change-in-error is possmall Then force is negsmall This rule quantifies the situation in Figure 2.5(b) where the pendulum has nearly a zero angle with the vertical (a linguistic quantification of zero does not imply that e(t) = 0 exactly) and is moving counterclockwise; hence we should apply a small negative force (to the left) to counteract the movement so that it moves toward zero (a positive force could result in the pendulum overshooting the desired position). 3. If error is poslarge and change-in-error is negsmall Then force is negsmall This rule quantifies the situation in Figure 2.5(c) where the pendulum is far to the left of the vertical and is moving clockwise; hence we should apply a small negative force (to the left) to assist the movement, but not a big one since the pendulum is already moving in the proper direction. Each of the three rules listed above is a "linguistic rule" since it is formed solely from linguistic variables and values. Since linguistic values are not precise representations of the underlying quantities that they describe, linguistic rules are not precise either. They are simply abstract ideas about how to achieve good control that could mean somewhat different things to different people. They are, however, at a level of abstraction that humans are often comfortable with in terms of specifying how to control a process. The general form of the linguistic rules listed above is If premise Then consequent As you can see from the three rules listed above, the premises (which are sometimes called "antecedents") are associated with the fuzzy controller inputs and are on the left-hand-side of the rules. The consequents (sometimes called "actions") are associated with the fuzzy controller outputs and are on the right-hand-side of the rules. Notice that each premise (or consequent) can be composed of the conjunction of several "terms" (e.g., in rule 3 above "error is poslarge and change-in-error is negsmall" is a premise that is the Conjunction of two terms). The number