Now, the rule set must be applied in its entirety. Each rule recommends a potential action, and the recommended actions are not necessarily the same from one rule to another. Thus, the conflict provided by the rules must be reconciled so that a single crisp value of F_x is computed. This procedure has, to this point, been presented as a graphical calculation in which a COA is computed. However, this is not really necessary in the real-valued consequent approach. Actually, the weighted average of the applicable rules is taken — each rule is weighted with the product of the μ values associated with the condition portion of the rule. (If the minimum operator has been employed, then the rule would be weighted with the minimum μ value associated with the condition portion of the rule.) The mathematical formula is:

where a_i represents the real-value of the action in the i^th rule and n is the number of rules. The fuzzy controller prescribes a value of 0.40 for the thrust in the x direction for the conditions given. Similar calculations are performed to compute values for the thrusts in the y and z directions.

A computer program was written to implement the fuzzy control system for the satellite rendezvous problem for a specific set of initial conditions. Figure 4.5 is a plot of the relative position of the spacecraft and the target. Notice that after approximately 600 seconds the rendezvous is complete. These results demonstrate that controllers that solve difficult, real-world control problems can be developed using the step-by-step procedure presented in Chapter 2. By the way, it is important to note that the authors are not experts in the art of spacecraft rendezvous. Thus, difficult problems can be solved by people who have only a crude feel for the system being considered.

Figure 4.5  The FLC successfully completes the rendezvous using rules and membership functions developed by the authors.

4.5 Review/Preview

In this chapter we have applied the step-by-step procedure set forth in Chapter 2 to a spacecraft rendezvous problem. We have used this procedure with the help of an expert to develop a controller for a difficult problem about which we actually knew little. In developing the rendezvous controller, we have introduced three new ideas.

The first new idea involves the defuzzification of the rules. In previous chapters, the minimum operator was used in which the final action set forth by the controller was determined using the minimum confidence associated with individual rules. In the current chapter, the multiplication operator is used in which individual confidences associated with a rule are multiplied together. Basically, this approach did not provide any better results (Karr, Fleming, and Vann, 1994); it is just an alternative method the reader should be familiar with since it is used in some fuzzy control systems.

The second new idea was the real-valued consequent approach. In this approach the rules have their actions described by real values instead of by linguistic terms. This rule form provides the user with greater flexibility in choosing the rule set, and allows for control systems that are very precise.

The third new idea involves the use of two different rule sets. In previous chapters, all of the rules had identical forms. However, because of the physics of the rendezvous system, it was necessary to write two types of rules. Specifically, rules were written for each of the coordinate directions individually, then the coupling of the system was addressed using a set of specific coupling rules. In adopting this approach, the system required only 124 rules instead of the 66656 rules required without this approach. We recommend exploiting information about the problem when possible.

The most noteworthy feature introduced in this chapter is the real-valued consequent approach. When we discovered this approach during the course of the rendezvous project, we were fascinated at the possibilities now available to us. However, we were faced with a major dilemma: how were we to select the floating point values appearing on the right-hand side of the rules? In fact, the rules for the rendezvous were extremely difficult to write even using the “old approach” in which fuzzy terms were used for the actions. We went through a difficult trial-and-error process in which we would write some rules, test the controller, refine the rules, test the controller, etc. Incidentally, this is the procedure that developers of fuzzy systems use routinely to get their systems to achieve the goals set forth for their controllers. Although the basic mechanics of the procedure are relatively straightforward, defining all of the rules and membership functions so that they work well in concert with one another is not an easy task.

The problem of defining the rules is very much related to the problem of defining the membership functions. There are an unlimited number of choices for both the membership functions and the rules, and they must work well together and neither set should be too large. Well, it is true that both of these ideas came to the forefront one day in our little research lab via a very intense philosophical discussion. The solution to the problem represents the authors’ major contribution to the field of fuzzy logic, and also represents the end of Part I of this book. The next part of this book is the beginning of our journey into adaptive control; it is about the genetic algorithm and its role in rule and membership function selection. Using the GA we can produce a relatively small, co-adapted set of rules and membership functions.

References

Burden, R. L., Faires, J. D., and Reynolds, A. C. (1978). Numerical analysis. Boston, MA: Prindle, Weber, & Schmidt.

Clohessy, W. H., and Wiltshire, R. S. (1960). Terminal guidance system for satellite rendezvous. Journal of the Aerospace Sciences, Sept, 653–674.

Kaplan, M. H. (1976). Modern spacecraft dynamics & control. New York: John Wiley & Sons.

Natenbruk, P., and Ragnittk, D. (1983). Control aspects as elaborated in space rendezvous simulations. Conference Proceedings on Guidance and Control Techniques for Advanced Space Vehicles, Florence, Italy.