In chapter 10 we saw that GAs can be effective parameter tuners when the basic equation of the model is known. Thus, a simple GA has been used here to tune the parameters. At this point, an analysis element has been forged in which a GA, as described earlier in the book, is used to compute the model parameters necessary to accurately predict the response of the laboratory pH system. A binary coding consistent with that described in Part II of this book was used to form 200-bit strings representing the appropriate model parameters. The first 40 bits of the strings were used to represent the concentration of the acid on the control input stream, the second 40 bits were used to represent the concentration of the base on the control input stream, the third 40 bits were used to represent the flow rate of the acid of the external streams, and the final 80 bits were used to represent the flow rates of the buffer and the base of the external streams, respectively.

Figure 13.6  The analysis element tracks the performance of the real-world environment and compares it with a computer simulation. When the two performances differ by a substantial amount, then the real-world system is considered to have changed.

An effective fitness function to select model parameters that mimic the response of the laboratory pH system is:

With this definition of the fitness function, the problem becomes a minimization problem: the GA must minimize f, which as it has been defined, represents the difference between the response predicted by the model and the response of the laboratory system.

Figure 13.7 illustrates the ability of a GA to locate the parameters needed by the computer model to calculate the response of the physical pH system. This figure includes information concerning the performance of the GA in locating these parameters. The GA was able to locate the correct parameters after only 3000 function evaluations. Locating the correct parameters took approximately 5 minutes on a 386 personal computer. The physical system often mandates that a control action be taken in less than 5 minutes. In this case, the time the GA is allotted to update the model parameters can be restricted. In such situations, the model simply must operate with inaccurate parameters until the analysis element is again employed. However, the magnitude of this problem will be diminished as computers become ever faster. And, the execution speed is adequate for the pH control system.

Figure 13.7  The GA is capable of solving the curve-fitting problem that arises in the operation of the analysis element.

The purpose of the analysis element is to recognize changes in the parameters associated with the problem environment that are not accounted for by the control element, and to compute the new values of these parameters. Once new parameters (and thus the new response characteristics of the problem environment) have been determined, the adaptive controller must alter the control element.

13.6 Adaptive Element

The adaptive element is responsible for altering the control element in response to changes in the problem environment. Recall that the relevant changes occurring in the pH system include: (1) changes in the concentrations of the acid and base of the control input stream, (2) random additions of acid, base, and buffer from the external streams, and (3) changes in the system setpoint. As set forth in a previous section, none of the parameters associated with the above changes are included in the rule set of the FC that serves as the control element. Therefore, the only way to account for these conditions (outside of completely revamping the system) is to alter the membership functions employed by the FC. The approach we have developed and implemented for using a GA to alter the membership functions associated with an FC has been well documented in previous chapters of this book. Thus, only the details for the pH system implementation are provided here.

A GA used bit strings of length 224 to represent the 32 parameters associated with the search problem. The 32 parameters represent the fuzzy membership functions shown in Figure 13.3 (symmetry considerations allowed for a reduction in the number of parameters that had to be represented). The fitness function must reflect the goal(s) of the control system. In the pH system, the objective is to drive the system pH to a desired setpoint in the shortest time possible, and to keep it there. The fitness function used in this application is:

where the summation is performed over a 100-second time period as simulated using the mathematical model of the system has been updated by the analysis element. This simulation is initiated from the current state of the laboratory system, i.e., the current values of pH, Q_ACID, and Q_BASE.