Once the condition variables have been chosen, the action variable(s) must be identified. In this liquid level system, identifying the action variables is a straightforward task, for there are only two things the controller can adjust to alter the state of the liquid level system: either of the input or output flow rates (Q_i or Q_o) can be increased or decreased by adjusting a valve. To further simplify the problem, Q_i and Q_o can be considered together as a net rate of flow into the tank, Q_net, where:

In this problem, since Q_net was limited, the action variable was taken to be the percent change in Q_net (referenced using the variable ΔQ_net). In more complex systems, determining the important action variables is not always a straightforward task for there are often a large number of potentially important parameters that have complex relationships with the desired control result. Likewise, selecting the appropriate condition variables can be a daunting task because there are often a number of changes that, in the final analysis, really are of little or no consequence.

Once the important condition and action variables have been identified, the linguistic terms used to describe these variables must be defined (the fuzzy sets). In a classic computational intelligence approach to FLC development, fuzzy sets are written to describe the condition variables, E and ΔE. It must be understood that a greater number of linguistic terms can be expected to give more precise control. However, when more linguistic terms are used to describe the condition variables, more rules are needed by the FLC. (Recall, for instance, the expert system description of f(x) presented in Chapter 1 in which we improved the representation of f(x) by adding more rules.) Therefore, choosing the number of linguistic terms used to describe each variable is not an arbitrary decision; it is based on knowledge gained through working with the physical system.

For the liquid level system, based on the authors’ experience with the liquid level problem, four fuzzy sets were chosen to characterize E: Negative-Big (NB), Negative-Small (NS), Positive-Small (PS), and Positive-Big (PB).

Five fuzzy sets were used to characterize ΔE: Negative-Big (NB), Negative-Small (NS), Near-Zero (NZ), Positive-Small (PS), and Positive-Big (PB). Five fuzzy sets were used to characterize ΔQ_net: Big-Positive (BP), Small-Positive (SP), No-Change (NC), Small-Negative (SN), and Big-Negative (BN). All fuzzy sets were chosen because they are similar to the descriptive terms a human operator might use to control the liquid level system.

This number of linguistic terms for each variable allowed for an FLC of reasonable size while providing adequate control of the liquid level system. With this choice of fuzzy sets 20 different production rules are required to describe all of the possible conditions that could exist in the liquid level system when the rules are of the form:

where A, B, and C are linguistic terms represented by fuzzy sets.

The method for giving precise meaning to the fuzzy, linguistic variables involves the use of fuzzy membership functions. The fuzzy membership functions used in the liquid level FLC are shown in Figures 2.2, 2.3, and 2.4. Fuzzy membership functions allow the crisp values of E and ΔE to be transformed into a fuzzy membership value.

Figure 2.2  Four fuzzy sets were used to describe error where NB is Negative-Big, NS is Negative-Small, PS is Positive-Small, and PB is Positive-Big.

Figure 2.3  Five fuzzy sets were used to describe time rate of change of error where NB is Negative-Big, NS is Negative-Small, NZ is Near-Zero, PS is Positive-Small, and PB is Positive-Big.

Figure 2.4  Five fuzzy sets were used to describe percent change in flow rate where BN is Big-Negative, SN is Small-Negative, NC is No-Change, SP is Small-Positive, and BP is Big-Positive.

Actually, the fuzzy membership functions can be thought of as determining the degree to which a discrete value of a variable is described by a particular fuzzy, linguistic term.

For example, using the membership functions shown in Figure 2.2, a value of E=5 would produce the following membership values where μ represents a membership function value: μ_NB=0.0, μ_NS=0.0, μ_PS=0.8, and μ_PB=0.3. When a fuzzy membership function has a value of μ=1.0, the crisp value is accurately described by the fuzzy term. On the other hand, when μ=0.0, the crisp value is definitely not described by the fuzzy term. It is important to realize that for each crisp control value, each fuzzy set has a membership function value, even though some of the values are zero.

The crisp conditions (definite values of E and ΔE) existing in the liquid level system at any given time can be accounted for using production rules that include fuzzy linguistic terms. Now, a process for determining a crisp action to take on the liquid level system must be developed. In other words, the crisp conditions have been fuzzified (fuzzification). Now, they must be deffuzzified to yield a single crisp action. The set of fuzzy production rules provides a fuzzy action for any condition that could possibly exist in the problem environment. With the four fuzzy sets for E and the five fuzzy sets for ΔE, there are 4*5=20 possible conditions in the liquid level system, and a human expert provides a desirable action for each condition based on prior experience. An example of a fuzzy production rule used in the liquid level system follows:

This rule simply says that if the liquid level is well above the setpoint and rising rapidly, the net flow into the tank should be made negative big. The complete rule set for the liquid level FLC is depicted in Figure 2.5. This rule matrix is used by locating the descriptive term for E along the top of the matrix, locating the descriptive term for ΔE along the left side of the matrix, and then extracting the appropriate value for ΔQ_net for the given conditions.

Figure 2.5  The rule set included 20 rules that are summarized in the matrix above. The sample rule in the previous discussion — IF [E is PB and ΔE is PB] THEN [ΔQ_net is BN] — appears in the bottom right corner of the matrix.