Note that we are using"negsmall"as an abbreviation for"negative small in size"and so on for the other variables Such abbreviations help keep the linguistic descriptions short yet precise. For an even shorter description we could use Integers -2" to represent"neglarge I"to represent"possmall 2 to represent"poslarge This is a particularly appealing choice for the linguistic values since the descriptions are short and nicely represent that the variable we are concerned with has a numeric quality. We are not, for example, associating"-1"with any articular number of radians of error; the use of the numbers for linguistic descriptions simply quantifies the sign of the error (in the usual way)and indicates the size in relation to the other linguistic values. We shall find the use of this type of linguistic value quite convenient and hence will give it the special name, "linguistic-numeric value The linguistic variables and values provide a language for the expert to express her or his ideas about the control decision-making process in the context of the framework established by our choice of fuzzy controller inputs and outputs d Recall that for the inverted pendulum r=0 and e=r-y so thate=-y and- e 0 dt will study how we can quantify certain dynamic behaviors with linguistics. In the next subsection we will study how to quantify knowledge about how to control the pendulum using linguistic descriptions For the inverted pendulum each of the following statements quantifies a different configuration of the pendulum The statement"error is poslarge" can represent the situation where the pendulum is at a significant angle to the lefn The statement"error is negsmall"can represent the situation where the pendulum is just slightly to the right of the vertical, but not too close to the vertical to justify quantifying it as"zero"and not too far away to justify quantifying it as The statement"error is zero" can represent the situation where the pendulum is very near the vertical position(a linguistic quantification is not precise, hence we are willing to accept any value of the error around e( 0=0 as being quantified linguistically by"zero"since this can be considered a better quantification than"possmall"or"negsmall") The statement"error is poslarge and change-in-error is possmall"can represent the situation where the pendulum is to the left of the vertical and, since, the pendulum is moving away from the upright position(note that in this case the pendulum is moving counterclockwise) Overall, we see that to quantify the dynamics of the process we need to have a good understanding of the physics of the underlying process we are trying to control. While for the pendulum problem, the task of coming to a good understanding of the dynamics is relatively easy, this is not the case for many physical processes. Quantifying the cs is not al ways easy, and certainly a better understanding of the process dynamics generally leads to a better linguistic quantification. Often, this win naturally lead to a better fuzzy controller provided that you can adequately measure the system dynamics so that the fuzzy controller can make the right decisions at thez "negsmall" z "zero" z "possmall" z "poslarge" Note that we are using "negsmall" as an abbreviation for "negative small in size" and so on for the other variables. Such abbreviations help keep the linguistic descriptions short yet precise. For an even shorter description we could use integers: z "-2" to represent "neglarge" z "-1" to represent "negsmall" z "0" to represent "zero" z "1" to represent "possmall" . z "2" to represent "poslarge" This is a particularly appealing choice for the linguistic values since the descriptions are short and nicely represent that the variable we are concerned with has a numeric quality. We are not, for example, associating "-1" with any particular number of radians of error; the use of the numbers for linguistic descriptions simply quantifies the sign of the error (in the usual way) and indicates the size in relation to the other linguistic values. We shall find the use of this type of linguistic value quite convenient and hence will give it the special name, "linguistic-numeric value." The linguistic variables and values provide a language for the expert to express her or his ideas about the control decision-making process in the context of the framework established by our choice of fuzzy controller inputs and outputs. Recall that for the inverted pendulum r = 0 and e = r - y so thate y = − and d d e y dt dt = − since 0 d r dt = . First, we will study how we can quantify certain dynamic behaviors with linguistics. In the next subsection we will study how to quantify knowledge about how to control the pendulum using linguistic descriptions. For the inverted pendulum each of the following statements quantifies a different configuration of the pendulum: • The statement "error is poslarge" can represent the situation where the pendulum is at a significant angle to the left of the vertical. • The statement"error is negsmall" can represent the situation where the pendulum is just slightly to the right of the vertical, but not too close to the vertical to justify quantifying it as ''zero" and not too far away to justify quantifying it as "neglarge." • The statement "error is zero" can represent the situation where the pendulum is very near the vertical position (a linguistic quantification is not precise, hence we are willing to accept any value of the error around e(t) = 0 as being quantified linguistically by "zero" since this can be considered a better quantification than "possmall" or "negsmall“). • The statement "error is poslarge and change-in-error is possmall" can represent the situation where the pendulum is to the left of the vertical and, since , the pendulum is moving away from the upright position (note that in this case the pendulum is moving counterclockwise). Overall, we see that to quantify the dynamics of the process we need to have a good understanding of the physics of the underlying process we are trying to control. While for the pendulum problem, the task of coming to a good understanding of the dynamics is relatively easy, this is not the case for many physical processes. Quantifying the process dynamics with linguistics is not always easy, and certainly a better understanding of the process dynamics generally leads to a better linguistic quantification. Often, this win naturally lead to a better fuzzy controller provided that you can adequately measure the system dynamics so that the fuzzy controller can make the right decisions at the