Parameter estimation in mathematical models using the real coded genetic algorithms Nedim Tutkun * Department of Electrical and Electronic Engineering, University of Zonguldak, University Park, 67100 Zonguldak, Turkey Abstract In this study, parameter estimation in mathematical models using the real coded genetic algorithms (RCGA) approach is presented. Although the RCGA is similar with the binary coded genetic algorithms (BCGA) in terms of genetic process, it has few advantages such as high precision, non-existence of Hamming’s cliff etc., over the BCGA. In this approach, creating initial population and selection pro- cedure are almost the same with the BCGA, but crossover and mutation operations. The proposed approach is implemented on the sec- ond order ordinary differential equations modeling the enzyme effusion problem and it is compared with previous approaches. The results indicate that the proposed approach produced better estimated results with respect to previous findings. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Real coded genetic algorithms; Binary coded genetic algorithms; Nonlinear curve fitting; Parameter estimation; Ordinary differential equat- ions; Dynamic system identification 1. Introduction Mathematical models based on a system of ordinary dif- ferential equations (ODEs) are widely used in many appli- cations. The solutions of such models can not be expressed by elementary functions in most cases and contain unknown parameters which are usually estimated by exper- imental data obtained from well-defined standard condi- tions. This type of problem is usually called parameter estimation in the literature and is often solved by determin- istic optimization methods such as Nelder–Mead, Leven- berg–Marquardt, Gauss–Newton etc., (Abbasi, 2006; Bayram & Yildiz, 1999; Yildirim, 2003). Unfortunately, the solution to the problem using these methods is usually around the local minima if there are more than one mini- mum available. However, better solutions may be found by one of sto- chastic optimization methods to obtain global minima if they exist in the given search space. The genetic algorithms (GA) method, the best known stochastic optimization tech- nique, imitates the natural evolution process (Coley, 2003; Gen & Cheng, 2000; Goldberg, 2003). One of the main advantages of this method is that it require no gradient of the objective function and additional information. Recently there has been an increasing trend to the real coded GA (RCGA) method for parameter estimation in various applications due to the number of drawbacks of the binary coded GA (BCGA) method. In this study, the RCGA approach is employed for the problem under con- sideration. The results are considerably improved with respect to previous findings in many aspects such as accu- racy, computation time etc. 2. Parameter estimation for dynamic systems ODEs have been a useful tool for describing the behav- ior of wide variety of dynamic physical systems since they were numerically solved by using few methods (Chapra & Canale, 2002; Schiling & Harris, 2000). A system of the first order ODEs can be stated by 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.01.060 * Tel.: +90 (372) 2578394; fax: +90 (372) 2574023. E-mail addresses: nedimtutkun@yahoo.com, tutkun@karaelmas. edu.tr www.elsevier.com/locate/eswa Available online at www.sciencedirect.com Expert Systems with Applications 36 (2009) 3342–3345 Expert Systems with Applications