13.3 Control Element

In this section, an FC is used as the control element for the pH system. The details concerning the development of the FC are consistent with the process for FC design presented earlier in the book, and they are presented in such a way as to make them easily extensible to other problem environments.

As in conventional expert systems, FCs use production rules to arrive at appropriate control operations. The left-hand-side of the rules (the condition side) consists of combinations of the controlled variables; the right-hand side of the rules (the action side) consists of combinations of the manipulated variables. Unlike conventional expert systems, FCs use fuzzy linguistic terms like those appearing in human rules of thumb. For example, a rule used to manipulate the pH system is:

This rule says in words that if the solution is very acidic and the pH is not changing rapidly, the flow rate of the base should be made large and the flow rate of the acid should be made zero.

The fact that FCs use fuzzy terms gives rise to another fundamental difference between them and conventional expert systems: a condition variable can be described by more than one fuzzy term. For instance, a system pH of 5 can be described as “very acidic” and “mildly acidic.” FCs use fuzzy membership functions to allow the condition variables in several rules to be described, to some degree, by each of the fuzzy terms. Therefore, more than one rule is qualified, or eligible, to enact its action at any given time, i.e., to “fire.”

Typically, since more than one rule can have its condition met at any given time, FCs must include a mechanism for determining a single control action. For this purpose a weighted average of the actions prescribed by the appropriate rules is calculated. The emphasis placed on each rule’s action is based on the confidence that exists in its condition (the degree to which a concrete value is described by a fuzzy term). The weights used in this averaging technique come directly from the fuzzy membership functions that provide the fuzzy terms with meaning.

Figure 13.2 shows a schematic of the structure of a control element composed of an FC. The control element receives definite values of the condition variables, uses fuzzy membership functions to characterize the definite values with fuzzy terms (it “fuzzifies” the variable values), employs a rule set, and computes one definitive action to be taken on the problem environment by calculating a weighted average (it “defuzzifies” the prescribed actions). This process is clarified in the following paragraphs as the membership functions and the rule set used in the pH system are set forth.

Figure 13.2  The control element must prescribe effective control actions based on the present condition of the pH environment.

The initial phase of FC development is to determine the appropriate condition and action variables. After a period of experimentation (an inevitable requirement for the development of a quality FC), two condition variables were selected: the current value of pH in the beaker and the absolute value of the current time-rate-of-change of the pH in the tank (ΔpH). Therefore, the two action variables were the flow rates for the acid (Q_ACID) and the base (Q_BASE), respectively, of the input streams.

Next, fuzzy linguistic variables are selected to represent the condition and action variables. After experimentation, seven terms were selected to describe pH, two terms were selected to describe ΔpH, and four terms each were selected to describe Q_ACID and Q_BASE. The specific linguistic terms used to describe the pertinent variables in the pH system follow:

These fuzzy linguistic terms are subjective, but the developer of the pH FC understands what they mean in the context of the physical system to be controlled.

The developer’s understanding of the linguistic terms is defined by the membership functions. The initial membership functions used in the FC appear in Figure 13.3. Figure 13.3a depicts the membership functions for pH, Figure 13.3b shows the membership functions for ΔpH, and Figure 13.3c depicts the membership functions for Q_ACID and Q_BASE. These membership functions are later altered by the adaptive element, via a GA, in response to changes occurring in the pH system. As will be seen, alterations in these functions can dramatically change the performance of the FC.