14.3.3 Learning Element

The analysis element notifies the learning element of the nature and extent of the changes occurring in the hexamine system. Based on this information, the learning element employs a GA to alter either the membership functions or the rules used in the FC of the control element. To do this successfully, the two issues of coding and fitness function must be addressed.

The parameters that have to be coded in the quest for an efficient set of fuzzy membership functions and a viable rule set are points that define all of the membership functions (the linguistic terms) and the real-valued consequents associated with the rules. The tasks of both tuning the membership functions and selecting the rules have been addressed in earlier chapters of the book. As before, concatenated, mapped, unsigned binary coding was used in this application. In this coding, each portion of the bit strings represented either one of the four points needed to define a trapezoidal fuzzy membership function, or a real-value consequent associated with one of the rules. Since there are 31 trapezoidal membership functions needed to characterize all of the variables considered in the FC of the control element, there are 124 parameters that must be represented to completely define the membership functions. However, due to the symmetry present in the problem, only 62 parameters had to be defined to completely describe a membership function set. Since there were 300 rules in the FC of the control element, there were an additional 300 parameters to represent. The 362 parameters were each allotted four bits, resulting in bit-strings of length 1448.

Now that an appropriate coding has been determined, a second issue must be addressed: how to evaluate the merit of each string (each membership function set). This task of defining a fitness function is always application specific; it always comes down to accurately describing the goal of the controller. In this case, the objective of the controller is to drive the hexamine to the desired temperature setpoint in the shortest time possible and to maintain that desired temperature. Thus, it would appear that the fitness function of the pH system described in Chapter 14 is again applicable. However, we would also like for the hexamine FC to maximize the amount of hexamine being produced without wasting reactants. After all, this is the way the company makes a profit and stays in business. The ability of the adaptive system to achieve these objectives can be represented by a fitness function that, first and foremost, specifies how well the controller reduces the difference between the current temperature and the desired temperature over some finite time period. Additionally, the fitness function must indicate the amount of hexamine produced. Finally, to ensure that a robust FC is obtained, the fitness function should reflect the controller’s ability to reach the setpoint from a number of initial condition cases. Mathematically, this fitness function is expressed as:

where f is the cost function, m_hex is the total moles of hexamine leaving the tank, w₁ = 3.0, w₂ = 1000.0, and w₃ = 100.0 are weighting function values. The weighting function values are selected so that all three products in the objective function are of the same order of magnitude. In this way, the GA places the same emphasis on all three objectives. The objective function was evaluated over two different forcing functions so that a general purpose adaptive controller would be produced.

It is important to note here that the learning element took approximately two hours to locate an effective control strategy to be used by the control element. The lengthy computational time was due to the amount of information that was being processed — recall that the learning element was searching for effective rules and membership functions encapsulated in strings that were 1448 bits long. However, in the hexamine system, the process dynamics generally do not change quickly. In fact, it is not uncommon for the process dynamics to change only on the order of days. Thus, the computational time required by the learning element is acceptable. However, in other systems (for instance the chaotic ball system of Chapter 17), the time necessary to locate effective rules in real time is simply not available, and other alternatives must be considered.

14.4 Results

An adaptive hexamine GA-FC was produced for the hexamine system. Each time a change was detected in the hexamine system by the analysis element, a GA was used by the learning element to locate a new, more efficient rule set and membership functions. The performance of this adaptive hexamine GA-FC is demonstrated for a situation in which the concentration of one of the reactants is changing.

Consider the situation where the concentration of a reactant is changing due to an external agent. The adaptive hexamine GA-FC alters the rules and membership functions it uses to enact its production rules (which do not change) because the process dynamics are altered when the concentration changes. Figure 14.3 compares the performance of the adaptive GA-FC with the non-adaptive FC of the control element that has been disconnected from the analysis and learning elements. According to Figure 14.3(b), both the non-adaptive and the adaptive controllers do a reasonably good job of maintaining the temperature in the reactor. However, the adaptive GA-FC produces more hexamine (the adaptive GA-FL produces 0.81 moles of hexamine per mole of ammonia while the non-adaptive FC produces only 0.65 moles of hexamine per mole of ammonia). The adaptive controller outperforms the non-adaptive FC because the adaptive controller is flexible enough to accommodate the changing process dynamics.

Figure 14.3  (a) The external agent alters the concentration of a reactant. (b) The adaptive GA-FC was able to drive the system temperature to the setpoint while producing more hexamine than the non-adaptive FC.

The results presented in this section demonstrate much of the power of the adaptive FCs possible through the implementation of the general purpose software architecture defined in Chapter 14. The adaptive controller is able to maintain a high degree of control over the hexamine system despite drastic changes in the system characteristics. Perhaps the most intriguing aspect of this application is the fact that the system parameters that were altered do not appear in the rule set the FC employs. All changes in FC performance are due to alterations in the membership functions. Based on the results presented, adaptive GA-FCs will allow the volatile characteristics of many industrial chemical systems to be controlled via on-line changes to the membership functions.