Now, we are at that same point we arrived at in Chapter 1. Specifically, we have a means for converting a crisp set of measurements in the liquid level system to a set of fuzzy conditions, and a set of fuzzy production rules prescribing a fuzzy action associated with a particular set of fuzzy conditions have been developed. Unlike conventional expert systems where only one rule is applicable for any given set of conditions, all of the rules in an FLC take effect to some degree at every time step (albeit that we have zero confidence that some of the rules are appropriate as reflected by membership function values of μ=0). Therefore, there still remains the task of converting the 20 fuzzy actions prescribed by the fuzzy production rules into a single, crisp action to be taken on the liquid level system.

To review the approach we introduced in Chapter 1, Larkin (1985) found that a center of area (COA) method (sometimes called the centroid method) provides an efficient means for determining this crisp action. In the COA method, the fuzzy membership functions shown in Figure 2.4 are used in a weighted summing procedure to find one crisp action. These manipulated variable membership functions actually serve only to define the base points (those located on the membership function value axis) of the triangles because the heights of the membership functions are varied according to the membership function values associated with the individual rules as applied to a particular situation. Each production rule has its action (as described by a fuzzy membership function) scaled to the height equal to the minimum likelihood associated with the condition portion of the rule, i.e., the minimum μ value associated with the conditions under which the rule is to be applied.

For example, consider the sample rule provided above as applied when the conditions in the system are E=5 and ΔE=0.521. Since μ_PB(E=5)=0.3 and μ_PB(ΔE=0.521)=0.7, the membership function for ΔQ_net=BN is “chopped off” at a height of 0.3. This is because the minimum membership function value for the variables on the condition portion of the rule is 0.3. When all 20 conditions have been compared with the rules, 20 trapezoids of varying height and width will have been plotted. Figure 2.6 shows a plot for the aforementioned condition existing in the environment: E=5 and ΔE=0.521. The two trapezoids occur as a consequence of the two rules that have their conditions met to some degree (the other 18 rules have a portion of their conditions met to the degree of 0.0 and are thus accounted for by trapezoids with a height of zero). The crisp action is determined simply by finding the center of area of the total area comprising the two trapezoids; in this case the crisp action is ΔQ_net=-12.0 percent. The COA is computed according to the following which is repeated from Chapter 1 for convenience:

where A_i is the area of the triangle i, x_i is its COA, and n is the number of rules. The center of area of the composite shapes (they will be trapezoids and possibly triangles — if any situation exists in which μ = 1) is a straightforward calculation easily performed on a computer. This value defines the single crisp value of percent change in Q_net applied at the next time step. The rules that create triangles with large areas, those arising from the conditions that are most applicable, have the greatest effect on the action, as well they should.

Figure 2.6  The center of area method (sometimes called the centroid method) is a commonly used technique for defuzzification. In the above figure the COA of the two trapezoids occurs at ΔQ_net = -12.0.

The step-by-step procedure described above for developing an FLC is summarized below:

1)  Determine the condition variables to be considered. These are the variables that are measured when a change to the system is considered.

2)  Determine the action variables. These are the variables that are manipulated to elicit a change in the system state.

3)  Describe the fuzzy sets for both the condition and action variables.

4)  Establish fuzzy production rules that cover all of the possible conditions that exist in the problem environment.

5)  Define the fuzzy membership functions.

6)  Determine all condition membership values for each rule.

7)  Apply the fuzzy production rules by taking a weighted average of the actions prescribed by all of the rules. The fuzzy production rules are generally written in matrix form as shown in Figure 2.5 and the essence of the defuzzification process is depicted in Figure 2.6.

Certainly, this procedure is easily put in the form of a computer program.

To someone familiar with traditional control methods, this may seem like a strange approach to process control. To someone who has worked extensively with FLCs, this approach seems powerful, logical, and natural. At any rate, to help the reader become more comfortable with the effectiveness of this fuzzy approach to process control, consider the results from a computer program implementing the FLC described above that manipulates a mathematical model of the liquid level system. Figure 2.7 shows the liquid level height plotted as a function of time for one particular initial condition in the problem environment. The FLC uses only the set of 20 fuzzy production rules and the associated membership functions to govern its selection of actions. The FLC is able to drive the height to the setpoint of 25 m in approximately 70 sec at which time a small oscillation about the setpoint begins.

Figure 2.7  The author-developed FLC successfully drives the liquid level to the setpoint.