Dynamic Control of Genetic Algorithms using Fuzzy Logic Techniques

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1 Dynamic Control of Genetic Algorithms using Fuzzy Logic Techniques Michael A. LEE Computer Science Department University of California Davis, CA Hideyuki TAKAGI Computer Science Division University of California Berkeley, CA FAX (510) Abstract This paper proposes using fuzzy logic techniques to dynamically control parameter settings of genetic algorithms (s). We describe the Dynamic Parametric : a that uses a fuzzy knowledge-based system to control parameters. We then introduce a technique for automatically designing and tuning the fuzzy knowledgebase system using s. Results from initial experiments show a performance improvement over a simple static. One Dynamic Parametric system designed by our automatic method demonstrated improvement on an application not included in the design phase, which may indicate the general applicabilityof the Dynamic Parametric to a wide range of applications. 1 INTRODUCTION Selecting parameter settings for genetic algorithms (s), such as population size and crossover rates, is often left to the user. Empirical and theoretical results about the effect of control parameter settings on performance and how set them to improve performance have emerged in the literature [1-8][12, 13]. Because the interaction of control parameters with performance is known to be complex, we assert that control is a prime target for a fuzzy system approach. In this paper we propose the ; a that uses a fuzzy knowledge-based system to dynamically control parameters, such as population size, crossover rates, and mutation rates in a. Obtaining the fuzzy knowledge-based system used in the can be accomplished in several Visiting Industrial Fellow from Central Research Laboratories, Matsushita Electric Industrial Co., Ltd., Moriguchi, 570 Japan 0 This paper appered in the Proceeding of 5th Int l Conf. on Genetic Algorithms (IC 93), Urbana-Champaign, IL, pp.76-83, July 17-21, ways; design can be done by an expert on control, by an automatic fuzzy design technique, or by both (providing the automatic technique can incorporate the expert s knowledge). Using an automatic technique can uncover new high performance control strategies. The choice of a fuzzy knowledge-based system to represent control strategies was made not only because it is easy to inject expert knowledge into them, but because new knowledge obtained by an automatic technique is equally accessible. This new knowledge may ultimately lead to a better understanding of the complex relationship between the control parameters and the performance. The goal of this work is three-fold: to explore approaches for fusing fuzzy control techniques with s, to better understand the relationship between control parameters and their effects on performance, and to improve performance. In the following sections, we first provide a short introduction to fuzzy knowledge-based systems. In section 3, we outline the approach; a fuzzy-knowledge based system is used to to dynamically change control parameters of a. (see Figure 1). In section 4, we discuss how to design the Dynamic Parametric ; the main point is the design of fuzzy knowledge-based system. We present and demonstrate a technique for optimizing the fuzzy knowledge-based systems using s (see Figure 2). Section 4 evaluates a designed in the previous section on the task of designing a fuzzy controller for inverted pendulum control and compares performance results with results from a simple static. 2 FUZZY KNOWLEDGE-BASED SYSTEMS Fuzzy knowledge-based systems are rule-based systems, which are built on fuzzy logic and fuzzy set theory. They are parametric models used to realize an input-output mapping that can specified using linguistic values. Like other rule based-systems, fuzzy knowledge-based systems attack system complexity by breaking the problem into smaller localized sub-problems. For example, under condition A, action Z should be taken and under condition B, action Y is

2 called for. The difference between fuzzy and non fuzzy systems is how they partition the space and resolve conflicts. In a fuzzy system, the partitionsmay can overlap and a point in the input space may belong to more than one partition. In a non-fuzzy system, a point either belongs to one region or another. When an action has to be taken, in a non-fuzzy system, normally this action is based solely on the partition that the input point belongs to. In a fuzzy system, each partition that a point belongs to contributes to the decision to the extent that it belongs to each partition. This mixing mechanism gives fuzzy rule-based systems their smooth interpolative behavior. Fuzzy sets are the building blocks of fuzzy knowledgebased systems [17]. Generally fuzzy sets serve to partition single dimensions. Membership of a point in a fuzzy set is allowed to be anywhere in the range of [,] and is determined according to a membership function. Common representations of membership functions are Gaussian or triangular shaped functions. Multidimensional partitions are formed by combining fuzzy sets through fuzzy IF-THEN rules. The action associated with each partition is contained in the THEN part. In our system, both IF and THEN parts of rules are represented as a fuzzy sets. The firing strength of a fuzzy IF-THEN rule is based on the membership values of fuzzy sets in the IF part. The output of the system is determined by the reasoning method, which combines the results of all fired rules. In control applications such as the Dynamic Parametric, the output of the fuzzy knowledge-based system needs to be a single crisp value. Defuzzification is the process of converting a fuzzy output variable into a crisp output variable. The system used in our experiments used triangular membership functions, the min intersection operator and correlation-product inference procedure. Defuzzification of the outputs was performed using the fuzzy centroid method [9] A fuzzy knowledge-based system is an ideal way to represent a control strategy because the knowledge is in a form which permits both computers and humans to efficiently share it. Fuzzy knowledge-based systems were intended to allow humans and machines to communicate using linguistic terms. 3 DYNAMIC PARAMETRIC BASED ON A FUZZY RULE- BASED SYSTEM This section proposes the ; a that uses a fuzzy knowledge-based system to control parameters dynamically. Figure 1 shows the proposed system. Engine Inputs to the fuzzy knowledge-based system for control can be any combination of performance measures or current control settings and outputs can be any of the control parameters, such as the six mentioned in [7]. Examfuzzy rule base Figure 1: application task ple inputs might be (average fitness)/(best fitness), current population size, or current mutation rate. Rules in the fuzzy knowledge-based system can reason about these measures and prescribe some control action. DeJong and Goldberg have been researching the effects of population size on performance. We can encode the rules or intuitive notions presented by them as fuzzy rules in a fuzzy knowledge-based system. For example, as the current population grows, the sensitivity to mutation rate decreases and the best mutation rate to use also decreases. The interpolative properties of fuzzy systems give the fuzzy knowledge-base system the ability to provide control values in novel situations and in the event of conflicting rules. For example, a typical fuzzy control scheme may include the following relations involving population size: IF (average fitness)/(best fitness) is big THEN population size should increase. IF (worst fitness)/(average fitness) is small THEN population size should decrease. IF mutation is small AND population is small THEN population size should increase. It is easy to apply classical non-linear optimization methods to dynamically change the learning rate or momentum coefficient of neural networks [15], but not to control parameters. Other approaches to dynamically control parameters have also been presented [13]. The framework of the provides an method to unify some of the notions resulting from this previous work. Because much of the knowledge of control parameters, how to control them, and their effects on performance is mainly qualitative in nature, the fuzzy knowledge-based approach is well suited to representing it. In addition, there are numerous automatic fuzzy design techniques, which can make use of a priori application knowledge and further improve the system [16, 11]. 4 AUTOMATIC DESIGN OF THE FUZZY KNOWLEDGE- BASED SYSTEM FOR DYNAMIC PARAMETRIC In the previous section, we proposed the framework to control parameters dynamically. The next question is how to design the fuzzy knowledge-based system, which plays an essential role in the. While it is possible to manually design a fuzzy knowledge-based

3 Engine µ(θ) center fuzzy rule base DeJong s toy tasks Figure 2: approach to auto-design the fuzzy knowledgebased system in the system for control, it can be rather difficult. Although much literature on the subject of control has appeared, our initial attempts at using this information to manually construct a fuzzy system for genetic control were unfruitful. For this reason, we decided to turn to an automatic technique, which has been demonstrated to be able to design fuzzy systems for other control applications [10]. In this section, we introduce the method for automatically designing fuzzy systems that was previously mentioned. We then show how to use this technique to design an optimal fuzzy knowledge-based system for the Dynamic Parametric. Figure 2 shows the diagram. The final part of this section presents performance results of the Dynamic Parametric design using this automatic technique. By using an automatic technique, relevant relations and membership functions can be automatically determined and may offer insight to understanding the complex interaction between control parameters and performance. Before we continue with the next chapter, some discussion about the added complexity is warranted. Some may argue that adding this extra step of fuzzy inference before calculating the next generation can be expensive computationally. It is important to keep in mind, however, that the fuzzy inference happens only between generations and in practice, the fuzzy inference is implemented by table lookup methods. For some applications with relatively uncomplex fitness functions, as the population size gets smaller, the proportion of time spent doing the reasoning does increase. However, with fitness functions which may have a high time cost associated with it, for example designing dynamic controllers, the cost amortized over the time spent evaluating an entire generation becomes insignificant. 4.1 AUTOMATIC FUZZY SYSTEM DESIGN USING s To incorporate s into fuzzy system design, we must first decide on a fuzzy system parameterization. Next we must find a suitable genetic coding and then determine a method to evaluate its fitness. The fitness function will be discussed in more detail in a later section. Figure 3: Fully overlapped membership functions The fuzzy system parameterization used for control had explicit parameters for membership function shape and rule consequent parts, and implicit parameters for rule number. There are several different methods for parameterizing the shape of membership functions. For our fuzzy system, we restricted adjacent membership functions to fully overlap; that is the center of the previous membership function serves as the left base point of the next. We also constrained one membership functions to have its center resting at the lower boundaries of the input range. By using this coding, only n? 1 membership function centers need to be specified, where n is the maximum number of partitions for a given dimension (see Figure 3 and Figure 4(a)). The maximum number of rules in the system is equal to the number of possible combinations of input sets. Each rule has the possibility of generating a single rule for each output variable. For example, a system with three inputs which are all composed of three fuzzy sets and three output variables could have up to or 81 rules. Each of the possible rules in the system could have as its consequent part, one output fuzzy set per output dimension. For example, if there are three output variables, each of the three rules produced by a given combination of input variables could be assigned exactly one or none of the output fuzzy sets from each of the output variables. This is specified by a rule identification number (rule ID). For example, an output variable which is composed of three fuzzy sets could lead to four possible rules for a given set of input variables; either set one, two, three, or none of the sets could be assigned to a rule. In the case where no rule is assigned, the rule is discarded from the system. To address the coding problem, we first define a composite chromosome as a set of parameters that represent a higher level entity, such as a membership function s shape parameters, i.e. center in Figure 3, and output rule ID ( respectively defined as MFC and ORC in Figure 4(a)). These composite chromosomes are concatenated together to form the entire fuzzy system representation (see Figure 4(b)). 4.2 PERFORMANCE INDICES The next step is to find an appropriate fitness function for the task. The objective of the automatic fuzzy system design algorithm is to improve the performance of the it θ

4 center center membership function chromosome (MFC) output rule ID output rule ID chromosome (ORC) (a) Composite chromosomes fuzzy input variable fuzzy output variable output rule ID MFC 1 MFC 2 MFC 3 MFC 1 MFC 2 MFC 3 ORC 1 ORC 81 (b) Gene Map Figure 4: Gene representation of the fuzzy system: two chromosome types, (a), construct gene map, (b). controls. In [4], DeJong assembled a set of five functions to study the behavior and effectiveness of s as function optimizers. The five functions were carefully selected to represent a wide cross section of function families. He designed two measures to quantify the performance of the s: online performance to measure ongoing performance and offline performance to measure convergence. Online performance is the running average of all evaluations performed up to a given time (see Figure 5 Eq.(1)). Offline performance is the running average of the best performance value up to a given time (see Figure 5 Eq.(2)). Figure 5: x online (s) = 1 T x offline (s) = 1 T TX TX t=1 t=1 f e (t) (1) f e (t) (2) Eq.(1) online and Eq.(2) offline performance measures: s is the search strategy, e is the environment, f e (t) is the objective function value at time t, and f e (t) is the best function value obtained up to time t. 4.3 RELATED WORK From DeJong s experiments, the following parameter settings for s were given as settings which give good online and offline performance on the test suite and were subsequently used as common settings: population size=50-100, crossover rate=0.6, mutation rate=01. Later, Grefenstette used DeJong s test suite and applied a meta-level to optimize the control parameters. He gave parameter settings for best online performance (population size=20-30, crossover rate= , mutation rate=005-1) and best offline performance (population size=80, crossover rate=0.45, mutation rate=1) [7]. This was accomplished by measuring the online performance (offline in the case of finding best offline settings) of a parameter setting on each of the five DeJong functions. The five scores were equally weighted and summed to yield a fitness value for a given parameter setting. In this work, Grefenstette also optimized window size, generation gap, and selection strategy parameters. Several other researchers have proposed methods for dynamically controlling parameters. Bagley proposed encoding the meta-level knowledge, i.e. crossover and mutation probabilities, into the genetic bit string itself [2]. Many other schemes have been proposed that centralize the meta-level knowledge used to control the. Any technique which uses information not encoded into the genes is an example of this approach [13, 14]. Although the centralized approach to meta-level knowledge may not follow nature s lead, results obtained from studying them may provide useful concepts which can be reformulated and represented in a distributed manner. 4.4 OPTIMIZING THE FUZZY KNOWLEDGE-BASED SYSTEM FOR THE DYNAMIC PARAMETRIC As an initial demonstration of our method, we propose using a fuzzy system which takes three input variables and produces three output variables; (average fitness)/(best fitness), (worst fitness)/(average fitness), and change in fitness since last control action are the inputs and population size change, crossover rate change, and mutation rate change are the outputs. All input and output variables are divided into three fuzzy sets. The total number of parameters in our fuzzy system coding is 12 (two membership function center parameters for each input and output axis) + 81 (rule ID parameters). In addition, initial values for population size, crossover, and mutation rates are included in the coding. The ranges of the outputs were set such that the population size change could not change by more than half of the current population size and could not go below 10 or exceed 160. The crossover and mutation parameters were also restricted to change at most by half of their current value and were bounded by [0.2,] and [001,1,0] respectively. For the genetic representation, the membership parameters and initial conditions were encoded as 8 bit numbers and the each output rule ID required two bits (only four possible sets assignment per rule) for a total of 258 bits (see Figure 4). To judge the fitness of the, we used a similar procedure to [7] (see Figure 2). The set of five DeJong functions were used on a population of s controlled by fuzzy systems and the online and offline measures were used to provide fitness measures. The scores from each DeJong function was normalized with respect to the random search score and then averaged. The simple static, or meta-level, that was used to design the

5 F1 online performance Simple Static Evaluations Figure 6: Comparison of F1 online performance. F1 offline performance Simple Static Evaluations Figure 7: Comparison of F1 offline performance. fuzzy systems had fixed parameters of population size=50, crossover rate=0.6, mutation rate=1, and used an elitist selection strategy. Window sizes and generation gaps were fixed at 7 (infinite window size) and respectively. The meta-level was allowed to evaluate 1000 the Dynamic Parametric s. After the meta-level completed, we selected the best fuzzy system. Results for online and offline comparisons of the simple static (static DeJong parameter settings) and the are given in 6 and 7. The resulting online optimized controller had an initial population size of 13, crossover and mutation rates of 0.9 and 8 respectively. The rule base included 51 rules. The membership functions and rules are given in Figure 8 and Table 1. The system used in our experiments used triangular membership functions, the min intersection operator and correlation-product inference procedure. Defuzzification of the outputs was performed using the fuzzy centroid method [9] For example, the fuzzy rule #15 of changing population size means: IF x 0 is big(a 0b ) and x 1 is medium(a 1m ) and x 2 is small(a 2s ), THEN change of population size is big(b 0b ), where each membership function and each input variable, x 0, x 1, x 2 in () are defined in Figure 8. The current three parameters are multiplied by the final change of three parameters that are obtained as the result of fuzzy reasoning are multiplied by the current parameters. More analysis A 0 A 1 A x 0 x 1 x 2 B 0 B 1 B change of population size change of crossover rate change of mutation rate Figure 8: Obtained membership functions for optimized online fuzzy rule-base. x 0 is (average fitness)/(best fitness), x 1 is (worst fitness/average fitness), and x 2 is the change in best fitness since the last control action input variables. A 0, A 1, and A 2 are the membership functions for the above three input valuables, and B 0, B 1, and B 2 are ones of output change for consequent part, which should be multiplied with the current population size, crossover rate, and mutation rate, respectively. Each fuzzy set within an input or output variable is identified from left to right as (s)mall, (m)edium, or (b)ig (See Table 1 for rules.) of the firing behavior may show that some of the rules are unnecessary. The optimized offline controller had an initial population of 10 and initial crossover and mutation rates of 0.7 and 0.99 respectively. The number of rules in its rule base was also EVALUATION OF DYNAMIC PARAMETRIC To validate our findings we compared the simple static with our fuzzy controlled on the inverted pendulum control task [10]. Figure 9 shows the experiments evaluated. In this section we present results that demonstrate the feasibility of designing fuzzy controllers using the Dynamic Parametric s. The fuzzy system given in 8 and 1 is used in the is the system obtained from optimizing the DeJong functions for online performance. The results show an improvement in online performance and may indicate that the rules contained in it may be universally applicable to other applications. 5.1 INVERTED PENDULUM The inverted pendulum represents a classic non-linear control problem where the task is to find a control strategy or controller that can balance a pole on a movable cart. The bottom of the pole is attached to a cart that travels along an

6 Table 1: Obtained rules for optimized online fuzzy rulebase. Each entry in the table corresponds to a rule. Column entries are the corresponding fuzzy set associated with the variable denoted at the top. The THEN column correspond to output fuzzy sets where the second subscript identifies the set. (see Figure 8 for membership functions) fuzzy rules for changing population size rule# x 0 x 1 x 2 THEN 1 A 0s A1s A2b B0b 2 A 0s A1m A2s B0b 3 A 0s A1m A2m B0s 4 A 0s A1m A2b B0b 5 A 0s A1b A2s B0s 6 A 0s A1b A2b B0m 7 A 0m A1s A2s B0s 8 A 0m A1s A2m B0m 9 A 0m A1s A2b B0s 10 A 0m A1m A2s B0s 11 A 0m A1m A2m B0m 12 A 0m A1m A2b B0b 13 A 0b A1s A2s B0m 14 A 0b A1s A2b B0s 15 A 0b A1m A2s B0b 16 A 0b A1m A2m B0b 17 A 0b A1m A2b B0b fuzzy rules for changing crossover rate rule# x 0 x 1 x 2 THEN 1 A 0s A1s A2s B1s 2 A 0s A1s A2m B1m 3 A 0s A1s A2b B1s 4 A 0s A1m A2m B1b 5 A 0s A1b A2s B1m 6 A 0s A1b A2m B1s 7 A 0s A1b A2b B1s 8 A 0m A1s A2s B1m 9 A 0m A1s A2m B1m 10 A 0m A1m A2m B1b 11 A 0m A1m A2b B1s 12 A 0m A1b A2s B1s 13 A 0b A1s A2m B1s 14 A 0b A1s A2b B1b 15 A 0b A1m A2m B1s 16 A 0b A1m A2b B1s 17 A 0b A1b A2s B1m 18 A 0b A1b A2b B1m fuzzy rules for changing mutation rate rule# x 0 x 1 x 2 THEN 1 A 0s A1s A2s B2s 2 A 0s A1s A2b B2m 3 A 0s A1m A2m B2m 4 A 0s A1m A2b B2b 5 A 0s A1b A2s B2m 6 A 0m A1s A2s B2m 7 A 0m A1s A2m B2b 8 A 0m A1s A2b B2m 9 A 0m A1m A2s B2m 10 A 0m A1b A2s B2b 11 A 0m A1b A2m B2m 12 A 0m A1b A2b B2m 13 A 0b A1s A2s B2s 14 A 0b A1s A2b B2b 15 A 0b A1m A2b B2b 16 A 0b A1b A2b B2s Engine fuzzy rule base vs. Simple Static application task fuzzy controller fuzzy controller Figure 9: Evaluation of the and a simple static infinite track. In this simulation, the movement of both the pole and the cart is restricted to the vertical plane and the cart is allowed to move infinitely in either direction along the track. The state of the system is described by the pole s angle and angular velocity. The controller is allowed to exert only a constant force on the cart in either the left or right direction. The objective of the controller is to balance the pole as quickly as possible. Other criteria, such as limiting the allowable force and minimizing overshoot, are often added. 5.2 EVALUATION The target fuzzy system for this application is similar to the system presented in section 4, except it had 360 parameters and a bit encoding of 2880 bits. For this application, we maximize an evaluation function which is computed by presenting it with several initial conditions and measuring the time until failure or success of the pendulum system. Success is determined by whether the system was able to keep the pole in an upright position for two time steps. In this case, the controller was rewarded and the trial was able to terminate before a maximum time setting had elapsed. A comparison of online performance between a simple static and our is given in Figure 10. In Figure 10, the online performance of the Dynamic Parametric starts in a different position because the initial population sizes are different and population members are initialized randomly. The peculiar dip in the front of the curve appeared for many different initial conditions and warrants further investigated. Because the score of a controller indirectly reflects its ability to terminate trials early, the number of inverted pendulum simulator evaluations is most likely reduced significantly. Figure 11 compares the offline performance of the and the simple static. It is interesting to note, that as in Grefenstette s work, the improvement in offline performance is not as significant as the online improvement. The reason for this may be that inputs selected for the fuzzy knowledgebase were not optimal, although the fuzzy IF-THEN rules based on these inputs were optimally tuned in section 4.

7 online performance Simple Static Evaluations Figure 10: Comparison of online performance of the Dynamic Parametric. offline performance Simple Static Evaluations Figure 11: Comparison of offline performance of the Dynamic Parametric. We also would like to point out that using the meta-level to find the optimized fuzzy system for the Dynamic Parameterized did consume many hours of computation time. However, because the performance of the final system transferred to another task without additional tuning may suggest that if a good knowledge-base were found, then optimizing the fuzzy knowledge-based system for the Dynamic Parameterized need not be performed very often. Research on including different combinations of rules, eliminating useless rules, and determining relevant input variables should be explored and more analysis needs to be done on the resulting systems. In addition, other performance optimization criteria, other than the online and offline performance indices presented, should be investigated. Fusion of fuzzy logic to s is still in its infancy. Other techniques to combine and mutually strengthen fuzzy systems and s are anxiously waiting to be discovered. Acknowledgments This research is supported in part by NASA Grant NCC-2-275, MICRO State Program Award No , and EPRI Agreement RP We also would like to thank Prof. David Wessel and the Center for New Music and Audio Technologies at UC Berkeley for use of computing resources. 6 DISCUSSION AND CONCLUSIONS We have proposed a method for controlling s using fuzzy logic techniques. This framework allows one to qualitatively express control strategies based on experience or intuition. To make s accessible to all, we have also presented an automatic fuzzy design technique, which is based on s. This technique was in turn used to design an optimal fuzzy system for control. The result was a controlled by a fuzzy system that exhibited better performance than a simple static. This was then evaluated on a different task and without additional optimization, out performed a simple static. This indicates that the rules found by the automatic design technique may be universally applicable to control s in other optimization tasks. In addition, because the representation of control knowledge is based on a fuzzy rule base, knowledge about control can be easily extracted after optimization. Analyzing this acquired knowledge may lead to better insights on the relationship of control parameters to performance. We also want to emphasized that the system inputs and outputs used in our experiments is one of many such sets to explore. Persons with better knowledge of control may know of other measures of performance and parameters that may prove to be useful or more appropriate than those used in our experiments. References [1] Back, T., "Self-adaptation in genetic algorithms," Proceedings of the First European Conference on Artificial Life, Paris, France, 1991, pp [2] Bagley, J. D., "The behavior of adaptive systems which employ genetic and correlation algorithms," Doctoral Dissertation, University of Michigan, University Microfilms No , [3] Baker, J. E., "Adaptive selection method for genetic algorithms," Proceedings of an Int l Conf. on Genetic Algorithms, 1985, pp [4] DeJong, K. A., "An analysis of the behavior of a class of genetic adaptive systems," Doctoral Dissertation, University of Michigan, University Microfilms , [5] DeJong, K. A. and Spears, W. M., "An analysis of interacting roles of population size and crossover in genetic algorithms," Parallel Problem Solving from Nature. 1st Workshop, PPSN 1 Proceedings, Dortmund, West Germany, 1990, pp [6] Goldberg, D. E., Deb, K. and Clark, J. H., "Genetic algorithms, noise, and the sizing of populations," Complex Systems, Vol.6, No.4, 1992, pp [7] Grefenstette, J. J., "Optimization of control parameters for genetic algorithms," IEEE Trans. on Systems, Man, and Cybernetics, Vol.16, No.1, 1986, pp

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