8.3 Genetic Algorithm Improves the System

The rendezvous controller of Chapter 4 was developed without the aid of a GA to locate membership functions and rules. Since a real-valued consequent approach was used there are an infinite number of rule sets from which to choose. Thus, there is good reason to believe that the controller presented in Chapter 4 can be improved. In this section, we will use a GA to locate rules and membership functions that improve the performance of the rendezvous controller. Again, the two issues to be addressed are the coding scheme and the fitness function.

8.3.1 Coding Scheme

The concatenated, mapped, unsigned binary coding employed in previous chapters is quite robust and can represent both rules and membership functions. It will also be used here to represent these parameters taken together.

With the linguistic terms selected to describe the condition variables in the rendezvous system (triangular membership functions were used as shown in Figure 4.2), there are ten triangle base points that must be defined for each condition variable, and thus there are 60 points that must be defined to completely describe the membership functions. With four bits each allotted to each base point, the substrings representing an entire set of membership functions is 240 bits long.

The real-valued consequents associated with the rules do not require any special attention; they are continuous variables existing on a fixed range. Thus, these values can also be represented using our familiar concatenated, mapped, unsigned binary coding. The substrings representing the individual rules are simply concatenated to the substring representing the membership functions. Recall that the rendezvous controller consists of 124 total rules (of the 124, 16 are coupled rules). Each rule was represented with a binary string of length three. The substring representing a complete rule set was therefore 372 bits in length. When combined, the substring representing a set of fuzzy membership functions and a substring representing a rule set were of length 612. This is an extremely long string length for the GA to work with. As an indication, consider that for this discretization, there are 2⁶¹² = 1.699 * 10¹⁸⁴ possible solutions to the search problem.

8.3.2 Fitness Function

The goal of the rendezvous controller is to drive the values of x, y, and z to setpoint or desired values (to complete the rendezvous) in as little time as possible. This goal is quite similar to those defined for the liquid level and cart-pole systems. Thus, it is natural to consider the fitness functions set forth in the previous chapters. A reasonable fitness function for the rendezvous system would seem to be:

However, upon closer inspection, such a fitness function is not adequate. The goal is to drive the relative distance between the chaser and target vehicles to zero. This distance, r, is defined by the equation

There is a constraint on the problem: r must be driven to zero gradually. This distance should not be zero while there are large relative velocities; this means the chaser vehicle crashes into the target. Certainly, this is a concern especially if one recalls the critically damped behavior exhibited by the liquid level and cart-pole systems. A constraint must be added into the fitness function to dissuade the GA from crashing the vehicles. The constraint is a limitation to the magnitudes of the velocities (, , and ) that can exist when the distance (r) is driven to zero. Goldberg (1989) suggests a penalty function for this type of constrained optimization. The penalty function used here is

where K is a penalty that is proportional to the degree to which the constraint is violated. In this case,

In this way, the magnitude of a fitness value is increased proportional to the magnitude of the relative velocities existing when the rendezvous is complete.

8.3.3 Results

A GA was used to select both the membership functions and the rule set employed by a rendezvous controller. The results are summarized in Figure 8.2. In this figure, the performance of the GA-FC is compared with the performance of the controller developed by the authors (AD-FC) and presented in Chapter 4. Notice that not only does the GA-FC complete the rendezvous in less time, but it does not allow the distance between the vehicles to go to zero when there are large velocities. In this case, the overdamping effect displayed by the GA-FC is necessary because of the constraints that restrict the system from allowing the distance r to go to zero if the chaser craft has a large velocity.

Figure 8.2  The GA-FC completes the rendezvous in less time than the AD-FC presented in Chapter 4. In addition, the GA-FC prevents the “crash” that the AD-FC causes at t = ** seconds.

8.4 Implication Operators

In our original presentation of the rendezvous controller in Chapter 4, we used a multiplication operator instead of the min-max implication operator we used in previous chapters. To gauge the effects of the two operators, we decided to compare the multiplication and min-max implication operators in the rendezvous problem. However, to ensure a fair comparison, we used a GA to design each controller. Figure 8.3 summarizes the results achieved. There is much discussion in the fuzzy community concerning the relative merit of implication operators. It is our experience that when a GA is used, the choice of implication operator is virtually inconsequential (Karr, Fleming, and Vann, 1994).

Figure 8.3  The choice of implication operator appears to be inconsequential when a controller is optimized using a GA.

To gain a better understanding of this observation, we did a thorough investigation using numerous implication operators, membership function forms, and rule forms. A complete discussion of the results of this study appear in the report by Karr, Fleming, and Vann (1994). Briefly, the selection of implication operator did not seem to affect the performance of the controller when a GA was used to design the controller.