7.4 Introduction of Error

Up to this point in the book we have developed fuzzy systems for controlling environments simulated on a computer. This is nice for us because we have been working with the proverbial “perfect world.” As shown in Figure 7.5, the computer simulations have been used to provide feedback information to the controller concerning the state of the problem environment. In this situation, the simulation is used both to model the problem of interest (the liquid level, cart-pole, and rendezvous systems) and to provide sensory information to the controller. Thus, the sensory information has been free of error (except roundoff error). The sensory information provided in the simulated environments we have worked with to date is not representative of the sensory information available in real-world systems for a number of reasons. First, the models are considered to be perfect when real-world models are not always perfect. Second, there are no mistakes in the simulated sensory information; no outliers. Third, the only error is roundoff error which is very small.

Even though these simulated systems have provided us an effective vehicle for presenting our ideas concerning fuzzy systems and GAs, we need to address the issue of error in sensory information if we are to be prepared for the real-world systems addressed later in this book. Thus, section 7.4 is meant to examine the effectiveness of GA-FCs in the presence of error.

Figure 7.5  The simulated environments addressed to this point take advantage of the availability of perfect sensory information when describing the status of the state variables to the controller.

The presence of sensor error is easily simulated in the cart-pole system we are considering. The values of x, , θ, and are simply altered with the addition of various amounts of Gaussian noise. Then, a GA is used to select optimum rules for systems with various amounts of sensory error. Figure 7.6 shows the performance of a GA-FC for the cart-pole system in the presence of 5 and 10 percent noise. Notice that the performance of the optimized controller degrades as the level of noise is increased. This is exactly the effect one would expect with the introduction of noise. The point here is that fuzzy controllers can tolerate substantial amounts of noise.

Figure 7.6  The performance of a fuzzy controller degrades as the magnitude of the sensory error is increased. Fuzzy controllers cannot completely overcome the presence of noise even when used in conjunction with a GA.

7.5 Review/Preview

This chapter has demonstrated the effectiveness of using GAs for selecting optimum rule sets employed by a fuzzy cart-pole controller. Two specific cases were considered. In the first, a GA was used to select the optimum values for a seven-action fuzzy controller. In this instance, the GA located the optimum rule set determined by the authors in a four-month trial-and-error process. The GA was able to find the rule set overnight. Second, a GA was used to determine an expanded rule set in which the developer was free to choose one of nine actions for each of 108 rules in the cart-pole controller (as opposed to the seven choices available in the controller presented in Chapter 3). In this instance a GA defined a rule set that allowed more-effective control of the cart-pole system. Finally in this chapter, the concept of error in sensory information was introduced and accounted for in a simulated environment.

The next chapter addresses the use of a GA to improve the performance of our satellite rendezvous controller introduced in Chapter 4. We will address the issue of using a GA to select real-valued consequent values, and we will also see that a GA can be used to simultaneously locate rules and membership functions. Additionally, we will discuss the ramifications of using GAs to optimize controllers to fuzzy implication operators.

References

Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison-Wesley.