**5.1 Linguistic description**

An expert uses linguistic variables to describe the time-varying inputs and outputs of the fuzzy controller. Thus, for our temperature system, we might have:


We used the quotes to emphasize how certain words or phrases. Though there are many possible ways to describe the variables linguistically, choosing one or another has no effect on how the fuzzy controller works, it only simplifies the task of constructing the controller using fuzzy logic.

Just as e(t) takes on a value, for example, 0.1 at t=2 (e(2)=0.1), so do linguistic variables take on "linguistic values", that is, the values of the linguistic variables change over time. For example, to control the temperature, we can have the "error", "error-variation" and "increase-energy-supplied" take on the following values:

Control Application Using Fuzzy Logic: Design of a Fuzzy Temperature Controller 387

Let us now consider how we can describe the system's dynamics based on the linguistic variables and the values they assume. In the case of the temperature controller, each of the

1. The error is Large Negative, indicating that the temperature of the liquid is much

2. The error is Small Negative and the error-variation is Small Positive, indicating that the temperature of the liquid is somewhat higher than the setpoint and dropping to the

3. The error is Zero and the error-variation is Small Negative, indicating that the temperature of the liquid is more or less at the setpoint but rising, Figure 6.c. 4. The error is Zero and the error-variation is Small Positive, indicating that the temperature of the liquid is more or less at the setpoint but falling, Figure 6.d. 5. The error is Small Positive and the error-variation is Small Positive, indicating that the temperature of the liquid is below the setpoint and dropping further, Figure 6.e. 6. The error is Large Positive and the error-variation is Large Negative, indicating that the temperature of the liquid is well below the setpoint but increasing, Figure 6.f.

Next we will use the linguistic quantifiers defined earlier to craft a rule set that captures the expert's knowledge regarding how to control the system. Specifically, we have the following

This rule quantifies the situation in which the liquid's temperature is above that desired,

This rule quantifies the situation in which the liquid's temperature is far below the setpoint

This rule quantifies the situation in which the liquid's temperature is close to the desired temperature but decreasing slightly, meaning that heat must be supplied to correct the error. Each of the three rules above is a "linguistic rule", since it uses linguistic variables and values. Since these linguistic values are not precise representations of the magnitudes they describe, then neither are the linguistic rules. They are merely abstract ideas on how to achieve proper control, and may represent different things to different people. And yet,

Using rules of the type described above, we can define every possible temperature control situation. Since we used a finite number of linguistic variables and values, there is a finite number of possible rules. For the temperature control problem, given two inputs and seven linguistic variables, there are 72=49 possible rules (every possible combination of the values

2. If the error is LP and the error-variation is SP, then increase-energy-supplied is LP.

3. If the error is ZE and the error-variation is SP, then increase-energy-supplied is SP.

1. If the error is LN, MN or SN, then increase-energy-supplied is LN.

(undesired situation) and decreasing, requiring a substantial heat input.

experts very often use linguistic rules to control systems.

following phrases represents different system states:

higher than desired, Figure 6.a.

desired value, Figure 6.b.

rules to control the temperature:

meaning heat must not be supplied.

**5.2 Rules** 

**5.3 Rule base** 

of the linguistic variables).


Fig. 6. Temperature system in different states.

Let us now consider how we can describe the system's dynamics based on the linguistic variables and the values they assume. In the case of the temperature controller, each of the following phrases represents different system states:

