Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA 2000) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April 2000
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1 Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA ) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April Experimental Approach of Mutation Probability Selection of Floating-point-based Genetic Algorithms (Case Study: Swing-up Feedforward Control of Rotary Inverted Pendulum) Son Kuswadi Electronic Engineering Polytechnic Institute of Surabaya Institut Teknologi Sepuluh Nopember Kampus ITS Keputih Sukolilo Surabaya, 6111, Indonesia Tel: ; Ext 16 Fax: sonk@eepis-itsac-idnet Abstract: The research achievement on the influence of mutation to the optimization result, which is one of standard operator of GA, using floating point representation will be presented The optimization case study to be chosen is the feed-forward swing up control system of rotary inverted pendulum This research is very important because there is no researchers have found the appropriate mutation of floating-point representation-based GA, in order to get the better optimization result In this research, the rotary inverted pendulum was used because this system is well known to be difficult to control, hence the research result could be guaranteed not intentionally happen The output of this research is rule-of-thumb on how to determine the appropriate mutation, to help the floating-point-based GA users on optimization problems KEYWORDS: soft-computing, genetic algorithms, floating point representation, mutation, rotary inverted pendulum, swing-up control 1 Introduction Genetic algorithms (GA) indeed the most popular optimization tools nowadays However, there are many algorithms that using evolution principle For example, Fogel [1] propose evolutionary programming, Rechenberg and Schwefel [] develop evolution strategies, and evolutionary dynamics was proposed by Conrad [3] For the survey on this algorithms, see [4][5][6] There are two GA s main operators namely cross over and mutation In the actual implementation, some extended operators needed to improve the optimization results Traditionally, mutation is seen as background operator [7] that responsible to introduce the lost genes, to avoid genetic destruction and to provide small elements of random search in population when it already on convergent state However, some example in nature shows that asexual reproduction process could produce better generation without crossover For example bdelloid rotifiers Therefore some biology expert said that mutation is main source of change of evolution Schaffer and its colleagues [8] conducting large scale experiment to determine optimum GA parameters They found that crossover have the smaller role comparing with the existing knowledge They propose naïve evolution that consist only selection and mutation and will produce similar result with hill climb search and better performance without crossover Indeed, in their experiments, it is shows that crossover has role to speed up evolution process compared with using mutation only However, they were show that mutation will produce better generation comparing with the process that using crossover only Davis shows the similar results [9] Although mutation operator generally using small, but it is important operator The exact value of this operator is more important than crossover [6] Son Kuswadi and his colleagues [1] use GA to design neuro-fuzzy controller with memory In the experiment they use binary representation of chromosome Due to the simple structure, its computation time is relatively short (about 5 hours using PC Pentium 75) The further study on floating point-chromosome [11] was conducted, on swing up feedforward control of rotary inverted pendulum [1] Crossover and mutation was experimentally tested in small scale and was using the similar reference as stated in common literature Since the result was not satisfied, in this research, the largescale experiment was conducted to obtain the
2 Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA ) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April suitable operator, especially its mutation The research was trying to determine experimentally the mutation suitable for floating point chromosome The problem to be solved was swing up feed forward control of rotary inverted pendulum, a difficult one, in order to avoid coincident Research Methodology (1) Plant modeling of rotary inverted pendulum, by using standard procedure: ie Euler- Lagrange method () Determine shape of input pattern that may use as swing up feed forward control signal, in which its parameters will be calculated using GA (3) Determine fitness function, including its weight parameters The maximum-search problem was considered (without loss its generality) for the above optimization Therefore, spinning roulette wheel method as suggested by Goldberg [7] is used (4) Determine initial setting of crossover and mutation It could be done by referring to the well-known literature on GA The of 6 and 1 were selected for crossover and mutation respectively (5) Run the designed GA and observe the evolution process Repeat this procedure by using different parameters (6) From the step (5), the trend and weakness of parameters could be observed Hence, based on the above step, the suitable parameters and operators could be selected Steps (3) to (6) were done repeatedly, and the suitable parameters candidate was selected 3 Rotary Inverted Pendulum Rotary inverted pendulum used in the research is shown in Figure 1 Mathematical model of rotary inverted pendulum was formulated as follows: J+ m1 L+ L1 1 m1ll ( sin θ ) 1cosθ1 θ mll 1 1cos 1 J1+ ml θ θ C + ml sinθ 1θ1 ml 1 L1 sinθ 1θ1 + ml sinθ 1θ + θ 1 ml 1 sinθ θ 1θ C1 Torque + (1) sin = 1 m L g θ where L is length of arm (15 m), L 1 is length of pendulum (18 m), C is arm s friction coefficient (3 Nms), C 1 is pendulum s friction coefficient ( Nms), J is arm s inertia (175 kgm), J 1 is inertia around center of gravity, m is arm s mass (1358 kg) and m 1 is pendulum mass (538 kg) Moreover, Torque is torque input that will be given to pendulum, and its outputs were θ (arm angle) and θ 1 (pendulum angle) 4 Genetic Algorithms 41 GA experimental parameters Experimental parameters of GA used in the experiment were: Crossover : 6 Mutation : 1; ; 3; 4; 5; 1; 1; 15; ; ; 4; 6; 8; 3; 35 Fitness function: linearly modified by factor 4 Feed forward signal Torque input as feed forward control that will be optimized by GA was designed intuitively in such a way that the signal could makes the pendulum will move from hanging position to upright position Such signal is shown in Figure A t 1 -A Fig t t 3 t 4 Feed forward signal Fig 1 Rotary inverted pendulum The signal parameters to be optimized were t 1, t, t 3, t 4 (in second) and its amplitude A (in Nm) 43 Fitness function
3 Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA ) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April The optimization objective of this problem is to make arm (θ ) and pendulum (θ 1 ) angles minimum In other words, pendulum angle is as close as possible to upright position and arm angle is return to its initial position (known as home-position) Since spinning of roulette wheel method was used, hence optimization means maximization; it is make sense to determine fitness function as follows: f()=a/(1+θ )+b/(1+θ 1 )+ c/(1+dθ /dt)+d/(1+dθ 1 /dt) () where a, b, c and d were constants that were selected intuitively based on the importance of each variable to be optimized In this experiment, the following variables were selected: a=55; b=1; c=d=1 It means that pendulum angle was considered as most important variable in the optimization 5 Experimental Results Figure 3 shows a typical example of implementation of optimization result into the rotary inverted pendulum In this implementation, the following feed forward parameters were used: A=91113 [Nm], t 1 = [s], t = [s], t 3 = 1369 [s] and t 4 = [s] Several experimental result of GA operation (in terms of fitness function vs number of generation) are shown in Figure 4 to 8 Meanwhile, Table 1 shows the performance of GA operation in terms of its fitness value for several values of mutation Table 1 Performance test result No Mutation Probability Fitness value Fig 4 Experimental result for 4% of mutation Fig 5 Experimental result for 5% of mutation Fig 3 Implementation of feed forward control to rotary inverted pendulum Fig 6 Experimental result for 1% of mutation
4 Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA ) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April fit for 5% mutation, whereas for others, it is need further study Fig 7 Experimental result for % of mutation Fig 8 Experimental result for 4% of mutation Fig 9 Fitness value as a function of mutation Figure 9 shows fitness value for several mutation values From the above figures and table, it is difficult to conclude the general tendency of the role of mutation However, the following fact could be drawn from the experimental results It is clear that 5% mutation seems better than others in terms of its stability, even though its fitness value (173833) was not the best Meanwhile for other value of mutation there were some individuals that have fitness too good (usually called as super-fit individual) This problem is sometimes happen in GA operation, but it was not scope of this research In this experiment, a linear scaling was used to preprocess the calculated fitness value by scale factor It seems that this value was only 6 Conclusions The role of mutation on GA operation by using floating point chromosome is presented The swing up feed forward control optimization of rotary inverted pendulum was selected as test bed The experimental result shows that only 5% mutation GA operation could avoid instability of fitness value, meanwhile for others seems could produce super-fit individuals and, therefore, could trapped into local minima and produce premature individuals This research should be elaborated for other mutation operators such as non-uniform and arithmetical crossover Also, the role of linear scaling value on GA operation subject to further study 7 Acknowledgement Part of this research was conducted at Department of Knowledge-based Information Engineering, Toyohashi University of Technology, Japan, under guidance of Prof Osami Saito and Dr Li Xu Thanks due to Japan International Cooperation Agency (JICA) for their support to this research, especially for Mr Tsuzuki of Institute for International Cooperation (IFIC) This research also was supported by URGE (University Research for Graduate Education) Project of World Bank Batch IV (1998/1999) References: [1] Fogel L, 'Autonomous automata', Industrial Research, Vol 4, 196, pp14-19 [] Schwefel, HP, 'Numerical optimization of computer models', John Wiley, Chichester, 1981 [3] Conrad M, 'Evolutionary learning circuit', J Theo Biol, Vol 46, 1974 pp [4] Fogel, 'Evolutionary computation: Toward a new phylosophy of machine intelligent', IEEE Press, New York, 1995 [5] D Beasley, DR Bull, RR Martin, 'An overview of genetic algorithms: Part 1, Fundamental', University Computing, 1993, 15() [6] D Beasley, DR Bull, RR Martin, 'An overview of genetic algorithms: Part, Research Topics', University Computing, 1993, 15(4) [7] Goldberg, DE, 'Genetic algorithms in search, optimization, and machine learning', Addison Wesley, Read MA, 1989 [8] Schaffer JD etal, 'A study of control parameters affecting online performance of genetic algorithms for
5 Proceeding, Seminar of Intelligent Technology and Its Applications (SITIA ) Graha Institut Teknologi Sepuluh Nopember, Surabaya, 19 April function optimization', in JDSchaffer (ed), Proceeding of the Third International Conference on Genetic Algorithms, Morgan Kauffman, 1989, pp 51-6 [9] Davis L ed, 'Handbook of genetic algorithms', Van Nostrand Reinhold, New York, 1991 [1] Son Kuswadi, Hyunrak Choi, Li Xu, Osami Saito, 'Memory neuron fuzzy network controller and its aplication to rotary inverted pendulum stabilization', Proceeding International Conference on Microelectronics 1997, October 1997, Bandung, Indonesia, pp 5-8 [11] Son Kuswadi, Munir, Mohammad NUH, Osami Saito, On performance test of floating-point-based genetic algorithms, Proceeding Seminar Nasional I Kecerdasan Komputasional, Fakultas Ilmu Komputer-UI, 6-8 Juli 1999, pp [1] Son Kuswadi, Li Xu, Osami Saito, 'A unified approach to swing-up and stabilization of rotary inverted pendulum by indirect adaptive fuzzy control', in preparation
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