Although the laboratory pH system is complex and nonlinear, an effective pH FC can be written that contains only 14 rules. The 14 rules are necessary because there are seven fuzzy terms describing the pH and two fuzzy terms describing ΔpH (7*2=14 rules to describe all possible combinations that could exist in the pH system as described by the fuzzy terms represented by the membership functions shown). Each rule is of the form:

Thus, each of the 14 rules prescribe two actions. The entire rule set for the pH FC is shown in Figure 13.4. In this figure, the left table shows the actions for Q_ACID while the right table shows the actions for Q_BASE.

Figure 13.3  The membership functions shown above represent the developer’s rough conception of the linguistic terms used in the control element.

Figure 13.4  The 14 rules shown above prove to be adequate to control the pH system when all of the environmental changes are disregarded.

Now, the only aspect of the initial control element design that is left is the technique for determining one value for the flow rates of the input acid and base streams. The center-of-area (COA) method (Larkin, 1985) discussed in the early chapters of this book is used again. This method provides a convenient way to compute a weighted average of the different control actions prescribed by the rules that are eligible to fire. The COA method results in the selection of a single control action to be taken on the problem environment.

One detail associated with the pH system being considered warrants special mention. There is a limit on the allowable change in the flow rates of the input streams, i.e., the flow rates cannot change by more than 0.5 mL/s/s. But, the membership functions describing the action variables used in the COA method (shown in Figure 13.3) allow for values of Q_ACID and Q_BASE to range between 0.0 and 2.5 mL/s. The constraint is imposed by computing the value of the flow rates using the COA method. If the computed value is such that it exceeds the constrained flow rate, then the flow rate is changed by the maximum allowable value of 0.5 mL/s.

Figure 13.5 shows the effectiveness of a fuzzy controller in the pH problem. The control element is able to drive the pH to a setpoint of 7 as long as the pH system remains unbuffered. However, when the system is buffered, the control element is not effective because the pH system has undergone substantial changes and the rules in the control element need to be modified to account for such changes.

Figure 13.5  The fuzzy controller is capable of neutralizing the pH in the unbuffered system.

13.5 Analysis Element

The analysis element must recognize changes in parameters associated with the problem environment that are not taken into account by the rules of the control element. These parameters include: (1) the concentrations of the acid and base of the input control streams, (2) the flow rates of the acid, the base, and the buffer that are altered by an external agent, and (3) the system setpoint. Changes to any of these parameters can dramatically alter the way in which the system pH reacts to additions of acid or base. Recall that the FC used for the control element presented includes none of these parameters in its 14 rules. Therefore, some mechanism for altering the prescribed actions must be included in the control system. But before the control element can be altered, the control system must recognize that the problem environment has been changed, and compute the nature and magnitude of the changes.

Fortunately, the dynamics of the pH system are well understood for buffered reactions, and can be modeled using a single cubic equation (Hand and Blewitt, 1986) that is solved for [H₃O⁺] ion concentrations to directly yield the pH of the solution:

and bracketed terms ( [ ] ) represent molar concentrations. Further details of the computer model appear in a paper by Karr and Gentry (1992). In physical systems for which the physics are not as clearly understood, alternative modeling techniques such as those discussed in Part III of this book can be employed.

Figure 13.6 shows a schematic of an analysis element. In the approach represented by the schematic, a computer model mirrors the actual problem environment. The response of the physical pH system is compared with the response as predicted by the computer model. When these responses differ by some threshold amount over a substantial period of time, the parameters of the pH system have changed and the model must be updated. Certainly, the threshold and the “substantial” period of time depend on the problem environment. For the pH system considered, when the predicted pH differed from the actual pH of the physical system by 1 unit of measure or more for a period of 5 seconds, the pH system parameters are considered to have changed and new parameters are required. The problem of computing the new system parameters is a curve-fitting problem. The parameters must be recalculated so that the response of the model matches the response of the actual problem environment.