TUNING ROUGH CONTROLLERS BY GENETIC ALGORITHMS

Transcription

1 TUNING ROUGH CONTROLLERS BY GENETIC ALGORITHMS Teresa Chiu Department of Mathematics and Computer Science Sun Jose State University San Jose, California T.Y. Lin* Department of Electrical and Computer Science University of California Berkeley, California Engineering A design process of intelligent control systems, called rough logic govemment, consists of a sequence of transformations of mathematical models of control systems. It is a modification of the design process of fuzzy controllers using rough sets, rough logic, and evolutionary computing. Rough logic govemment starts with a symbolic model, called rough linguistic model, and concludes with tedious experiments. The final mathematical structure is called rough verification and validation model. In this paper, genetic algorithm is used to partially automate the final step, and experiments from various angles are conducted and analyzed. 1. Introduction Rough logic government is a formal model for a design process of intelligent controllers [ 11. It is a variant of classical fuzzy controller that integrates rough sets, fuzzy logic, genetic algorithm and differential geometric view of non-linear dynamic systems together. The most obvious additions are (1) at the front, we add rough set methodology to extract linguistic rules from a set of training data, (2) at the very end, we use genetic algorithm to tune the final model. Others additions are intrinsically viewpoints and formalism; individually, both have been studied by others. In our formalism, fuzzy logic is viewed as a methodology of constructing functions by a grand scale interpolation guided by qualitative information. Intrinsically interpolations are guessing the missing values. In this paper, we use genetic algorithm to sharpen these guesses; this is the main focal point of this paper. 3. Control and Rough Logic Government A control system can be expressed by (e.g.,[4]): dx/dt = F(t,U, X) Y = H(t, U, X) Then from the solutions of (1), one obtains the controzfunction * This research is partially supported by Electric Power Research Institute, Palo Alto, California. ** T.Y. Lin is currently on leave from SJSU, San Jose, Ca (tylin@cs.sjsu.edu) / IEEE 326

2 Y= K(t, U) (3) where X, U, and Y represent respectively the state, the input, and the observable output variables in vector forms. In linear control theory, the function K can be explicitly expressed by classical functions. In intelligent control and modern differential geometric approaches, we focus on nonlinear systems, where the system functions F and H, and control functions K are analytically unavailable. In differential geometric approach, we study the qualitative theory of non-linear equations (1)-(3). In intelligent control, we focus on constructing K directly. Rough logic government is such a process. It consists of five steps [ 11: Step 0. Extracting Control Rules: We use the rough set methodology to extract linguistic control rules from such a set of "training data" [3]. Step 1. A Rough Linguistic Model: A set of linguistic rules (obtained from Step 0 or recommended by domain experts) is expressed as a set of proper axioms in rough logic [2]. Formally, it is a theory of rough logic, called a rough linguistic model. In practices, it is a set of statements including the linguistic rules and qualitative properties of the system. Step 2. A Rough Fuzzy Model (A Family of Possible Fuzzy Worlds)}: Formally, a theory is interpreted by membership functions. In other words, all linguistic constant are replaced by membership functions Step 3 A Rough Candidate Model (A Family of Candidates of Control Functions): For each possible fuzzy world, an inference method determines a function. This function is a candidate of "solutions" of the un-constructed system equations. Step 4: A Rough W&V Model (A Family of Control Functions): The family in Step 3 is a set of potential control functions. To show that indeed they are, we need to verify and validate them by experiments. Step 5 Evolutionary Tuning: Rough logic government is a complex mathematical structure. So the amount of tuning is astronomical, they are tedious and time consuming. So evolutionary computing is adopted to conduct such tuning to select a "best answers." 3. GA Experiments 3.1. The problem: Rough logic government is a process to capture the control function K from observable data. We will illustrate the idea by a simple example. Suppose the control function, just happen to be, say Y= sin(x); however designers does not know that. From Step 0, we obtain the following: Let Ai represent some "quantity or range" in X-domain, Let Bi for Y-domain. The rough linguistic model is: If X is A1 then Y is B2 327

3 If X is A2 If X is A3 If X is A4 then Y is B3 then Y is B2 then Y is B1 In Step 2 & 3, the fuzzy interpretations and inference method define a function, which is a potential control function. When Bj are constants, TVFI inference method produces a function that is equivalent to Y(X)=(B2*Al +B3*A2+B2*A3+B 1*A4 + B2*A5)/(B2+B3+B2+B 1+B2) In step 4 & 5, we use Genetic Algorithm to search proper membership functions that represent Ai and Bj, and to verify and validate that indeed Y(X) and sin(x) is approximately equal Experiments The experimental runs reported here are performed by using a genetic algorithm with a population size of 500. Each individual in the population consists of the concatenation of a representation of each of the 5 sets. In this research, we use 72 numbers to represent a member function, which results in individual strings of length 360. To emphasize the strength of the genetic algorithms, we have run our implementation on populations that are both seeded and unseeded. Seeding in genetic algorithms refers to the manual construction of the initial population, thus allowing the GA to evolve from some restricted area(s) in the entire search space in the succeeding generations. In the following we present and discuss our several experiments using a powerful GA software, LibGA. Throughout this project, we employed the uniform crossover operator and a crossover rate of 0.6. The simple mutation operator is adopted; however, the selection of mutation rate is a crucial element in the experiments. Intuitively, the fuzzy sets used in an expert system are functions that can be depicted by smooth curves. Therefore, we would like to see the end result possessing this characteristic as well. This naturally suggests the choice of a small mutation rate, such as the commonly used lln, where n is the length of the strings. Using this technique in this research has the advantage of better preserving the shapes of the original curves. On the other hand, a small mutation rate may often fail to effectively explore the search space, resulting in insignificant improvement and premature convergence among the strings Random Population To observe the behavior of the genetic algorithm without the instruction of handinput functions, we started by running the GA without seeding. An unseeded initial population is simply a random initial population. Our initial 500 individuals 328

4 are produced by repeatedly generating 360 random floating numbers between 0 and 1 inclusively. Since the search by the GA is random in this case, we prefer a large mutation rate in order to efficiently locate the feasible regions. We multiplied the reciprocal of the string length, 1/360, by 10 and used this number as the mutation rate. As expected, GA is capable of manipulating its way to get very close to the goal curve. This is achieved by ignoring any pattern or restriction imposed on the fuzzy sets. Strictly speaking, therefore, the results we found have demonstrated the power of GA instead of its applicability on fuzzy sets Guided Evolution Now we would like to see, with the influence of the input functions, how much improvement or what difference the GA makes. Using the seeding strategy, we first generated the initial population by slightly altering each of the 5 original sets 500 times. More specifically, we took the original curves and bent (outwardly or inwardly) each of them by a slightly larger degree every time until we had 500 new sets to start the genetic algorithm. We used the same initial population and two different mutation rates to inspect its impact on the final results. The first mutation rate we adopted is the reciprocal of the string length, namely, 1/360. Illustrations of the resulting curves are listed in Figures l(a) through l(e). Comparing the graphs with the original ones shown in the previous section, we notice substantial preservation of information in the resulting curves. This is due to the lack of capability to further investigate the search space, leading to premature convergence of GA. We obtained unsatisfactory results as a consequence - See Figure Guided but Free Adaptive Evolution To remedy the problem described in the previous paragraph, we again used a large mutation rate, U360 multiplied by 10. Figure 2(a) through 2(e) show an example of the results given by the GA. From Figure 2(f), the result after applying the inference rule, we see improvement over the result yielded by using small mutation rate as illustrated in Figure 2(f). However, at the same time, we see heavy fluctuations in the 5 functions as shown in Figure 2(a) through 2(e) because a large mutation rate has led the search beyond the scopes of the original functions. Thus it is suggested that the tradeoffs between different mutation rates be taken into account when dealing with problems of this nature Discrete Evolution As an alternative, we have also used step functions as input to the GA. Instead of a curve, each of the five functions is represented by an arbitrarily choseh constant between 0 and 1. It is to our interest to observe that the results not only converge in a few generations but also converge to the immediately adjacent regions in the 329

5 search space. An illustration of the results using step functions is shown in Figure 3(a) through 3(e). For pure rough set theorists, they will choose such step functions as the membership functions. This is advisable when one use hand tuning. It converges very quickly and approximate roughly. A3 Figure 1 References 1. Lin, T. Y., Rough ic Formalism For Fuzzy Controllers-A Hard and Soft Computing Approach, 9 ournal of Approximate Reasoning, to appear. 2. Lin, T.Y. and Liu, Q, First Order Rough Logic: Approximate Reasoning via Rough Sets, Fundementa Informaticae, to appear. 3. Z. Pawlak, Rough Sets, Theoretical Aspects of Reasoning about Data, Kluwer Academic Pub., Vidyasagar, M., Nonlinear Systems Analysis, 2nd ed., Prentice Hall.,

6 2 o.:ml o.;f=q I Figure 2 o.:fl AI A2 o.:mj A3 o.;e OO A5 1, I Figure