Evolutionary Algorithms Based Speed Optimization of Servo Motor in Optical Disc Systems

Evolutionary Algorithms Based Speed Optimization of Servo Motor in Optical Disc Systems Radha Thangaraj1, Millie Pant1 and Ajith Abraham2 1Indian Institute of Technology Roorkee, India 2Norwegian University of Science and Technology, Norway [email protected], [email protected], [email protected] Abstract a wide range of engineering optimization problems [10] - [14] etc. In this study we investigate the Evolutionary Algorithms are inspired by biological performance of PSO and DE for optimizing the and sociological motivations and can take care of average bit rate of an optical disc servo system. optimality on rough, discontinuous and multimodal Servo motor is one of the most sophisticated motion surfaces. During the last few decades, these algorithms control devices in electric motors. In CD-ROM or have been successfully applied for solving numerical DVD ROM, the objective function mainly consists of bench mark problems and real life problems. This maximizing the average bit transfer rate subject to paper presents the application of two popular various constraints due to servo motor, control circuits Evolutionary Algorithms (EA); namely Particle Swarm decoding electronics etc. In the present article we have Optimization (PSO) and Differential Evolution (DE) taken a popular yet complex problem of CD/ DVD for optimizing the average bit rate of an optical disc ROM systems where the objective is to optimize the servo system. Two optimization models are considered speed of the servo motor. We considered two cases of in the present study subject to the various constraints optimization (i) average bit rate in seeking and (ii) due to servo motor. The results obtained by PSO and average bit rate of zoned CLV. The mathematical DE are compared with the experimental and the design models are taken from Jung and Sheu [15]. A results given in the literature. Simulation results preliminary version of this study was already presented clearly show the superior performance of PSO and DE by Pant et al [16], but in the present study more algorithms. elaborated analysis is given. The structure of the paper is as follows; Section 2 1. Introduction briefly describes the mathematical models of the optimization problems. Section 3, gives a general The Electrical Engineering community has shown a introduction to PSO and DE algorithms. In Section 4, significant interest in optimization for many years [1] – the penalty approach for handling constraints is [4]. In particular, there has been a focus on global discussed. Section 5 gives the parameter settings and optimization of numerical, real- valued problems for numerical results; finally this paper concludes with which exact and analytical methods do not apply. section 6. During the last few decades, many general-purpose . optimization algorithms have been proposed for finding optimal solutions, some of which are; 2. Mathematical Model of the Problem Evolution strategies [5], evolutionary programming [6], Genetic algorithms (GA) [7], Particle Swarm With the boost in CD/ DVD ROM market, the Optimization (PSO) [8] and Differential Evolution demand of smaller access times and higher data rates (DE) [9]. These algorithms are also known as are also increasing. The design of a CD servo system Evolutionary Algorithms (EAs) or Nature Inspired employs a constant linear velocity (CLV) strategy in Algorithms because they follow simple rules of nature. which the disc is rotated at a varying rotation speed to These algorithms have also become popular because of maintain a synchronized velocity between the pickup their advantages over the traditional optimization head and track across the disc radius [17], [18]. techniques (decent method, quadratic programming Several adaptive speed algorithms for CD ROM approach, etc). They have been successfully applied to systems are proposed in [19]. The objective of the present paper is to observe the effectiveness of PSO 2.2. Zoned Constant Linear Velocity Control and DE algorithms for solving two different (CLV) optimization models to maximize the average bit rate, subject to various constraints imposed by the servo In zoned CLV control, the disc area is partitioned into motor, control circuits, decoding electronics and other m zones R1 = Rin, R2, R3… Rm. Zoned CLV assumes limitation factors. The model of the design of the speed that within each zone the linear velocity or over speed profile has been adopted from Jung and Sheu [15]. factor is fixed. If Ni denotes the over speed factor in the ith zone, then the rotation speed is given as 2.1. Speed Profile in Seeking N v ω(r) = i 0 for R ≤ r ≤ R (6) One advantage of adaptive speed control in r i i+1 optical disc servo system is reduction of seek time and The objective function for maximizing the average bit hence access time during seek motion. Typically there rate is given as are two simultaneous control activities involved in 2πB m seek motion. The spindle motor is controlled to adjust 0 Ri+1 2 Bavg = ∑ ∫R ω(r)r dr the speed and the sledge motor (as well as the voice Lpv0 i=1 i coil) is commanded to move the optical head to the m πB0 2 2 desired position. If the speed profile is well designed = ∑ N (R − R ) (7) Lp i i+1 i the system is able to read out data for processing once i=1 the pickup head is in its target position. Otherwise, the Subject to the constraints access time will be increased. N v ≤ ω R ∀i (8) Maximize i 0 max i 2πB0 n −1 B = . N min ≤ Ni ≤ N max ∀i (9) avg Lpv 12(R − R ) o out in 4 3 4 {ω1[3R1 − 4R1 R2 + 3R2 ]+ N v ⎡ N v ⎤ K i 0 −α i 0 + β ≤ m u ∀i (10) n−1 r ⎢ 2 ⎥ R max 4 3 4 3 4 ⎣ r ⎦ α ∑ωi[Ri+1 − 4Ri+1Ri + 6Ri − 4Ri−1Ri + Ri−1 ] i=2 rj Nmin ≤ N j + ( Ni − N j )e − (β / Jvh ) | ri − rj | 4 3 4 ri + ωn[Rn−1 − 4Rn Rn−1 + 3Rn ]} (1) ≤ N max (11) Where Table 1 Design Parameters π (R 2 − R 2 ) L = out in (2) Parameter Value Unit p Rin 25 mm Rout 58 mm Ri = Rin + ((Rout − Rin ) /(n −1))(i −1) (3) B0 150 KB/sec The objective function is a linear function with p 1.6 µm unknown vector ω. v0 1.3 m/sec Subject to the constraints: vh 12 mm/sec Km 0.0062 Nm/A ωi ≤ ωmax i =1,2,....,n (4) Kb 0.0062 Nm/A −(β|R −R |) / Jv j i h Ra 5 Ω Nminvo ≤ R jωie D 5.0 x 10-6 Kg m2 −(β|R j −Ri|)/ Jvh + R jω j [1− e ] ≤ Nmaxvo (5) Nmin 12 - Nmax 24 - ωmax 7500 rpm 3. Particle Swarm Optimization and part represents the cooperation among particles and is Differential Evolution therefore named as the social component [20]. Acceleration constants c1, c2 [21] and inertia weight ω PSO and DE algorithms may be termed as general [22] are predefined by the user and r1, r2 are the purpose algorithms for solving optimization problems. uniformly generated random numbers in the range of Both of these methods are assisted with special [0, 1]. operators that are based on some natural phenomenon. These algorithms are iterative in nature and in each 3.2. Differential Evolution iteration the operators are invoked to reach to optimal (or near optimal) solution. Pseudo codes of all the Differential evolution (DE) is an Evolutionary algorithms used in this study are given in Appendix A. Algorithm (EA) proposed by Storn and Price in 1995 A brief description of PSO and DE are given in the [9]. DE is similar to other EAs particularly Genetic following subsections: Algorithms (GA) [23] in the sense that it uses the same evolutionary operators like selection recombination 3.1. Particle Swarm Optimization and mutation like that of GA. However the significant difference is that DE uses distance and direction PSO was proposed by Kennedy and Eberhart in information from the current population to guide the 1995 [8]. It is inspired by the complex socio search process. The performance of DE depends on the cooperative behavior displayed by various species like manipulation of target vector and difference vector in flocks of birds and shoals of fish. In PSO, the members order to obtain a trial vector. of the swarm or the particles are placed in the A general DE variant may be denoted as parameter space of a particular problem, and each DE/X/Y/Z, where X denotes the vector to be mutated, particle evaluates the fitness at its current location. The Y specifies the number of difference vectors used and movement of each particle in space is determined by Z specifies the crossover scheme which may be the history of its own fitness and also by the fitness of binomial or exponential. For the more details the its neighbors. It then moves through the parameter interested reader may please refer to [24]. In this study, space with a velocity determined by the locations and the mutation strategy DE/rand/1/bin [9] is considered. processed fitness values of those other members, along It is also known as the classical version of DE and is with some random perturbations. The members of the perhaps the most frequently used version of DE. DE swarm that a particle can interact with are called its works as follows: First, all individuals are initialized social neighborhood. Together the social with uniformly distributed random numbers and neighborhoods of all particles form a social network of evaluated using the fitness function provided. Then the PSO. following will be executed until maximum number of For a D-dimensional search space the position of generation has been reached or an optimum solution is th the i particle is represented as Xi = (xi1,xi2,..xiD).

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