Applied Soft Computing 3 (2003) 343–352 The evolutionary learning rule for system identification Oscar Montiel a,∗ , Oscar Castillo b , Patricia Melin b , Roberto Sepulveda a a CITEDI, National Polytechnic Institute, Tijuana, Mexico b Deptartment of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Received 1 April 2003; accepted 30 May 2003 Abstract In this paper, we are proposing an approach for integrating evolutionary computation applied to the problem of system identification in the well-known statistical signal processing theory. Here, some mathematical expressions are developed in order to justify the learning rule in the adaptive process when a breeder genetic algorithm (BGA) is used as the optimization technique. In this work, we are including an analysis of errors, energy measures, and stability. © 2003 Elsevier B.V. All rights reserved. Keywords: Learning rule; Mathematical model; Breeder genetic algorithm; System identification; Breeder genetic algorithm; ARMA 1. Introduction The problem of determining a mathematical model for an unknown system by observing its input–output data pairs is known as system identification, and it is an important step when we wish to design a control law for a specific system. Real systems are non-linear and have time variations, hence, the best control laws that we can obtain are those based using real time data from continuous-time stochastic processes [1]. Traditionally, system identification has been per- formed in two ways: 1. Using analytic models, i.e. obtaining mathemati- cally the transfer function. 2. Using experimental input–output data. In this way, the identification can be achieved in two forms: non-parametric and parametric. In this paper, we are interested in parametric models. As we have mentioned, there are several ∗ Corresponding author. E-mail address: oross@citedi.mx (O. Montiel). well-known techniques to perform the system iden- tification process, most of the parametric techniques are gradient guided and are limited in highly multi- dimensional search spaces. The system identification process generally involves two top–down steps, and these are: structure identification, and parameter iden- tification. In the first step, we need to apply a priori knowledge about the target system for determining a class of model within the search for the most suitable model is going to be conducted [2,3]. Here, we are using an evolutionary algorithm known as breeder genetic algorithm (BGA) that lays somehow in between genetic algorithms and evolu- tionary strategies (ESs). Both methods usually start with a randomly generated population of individuals, which evolves over the time in a quest to get bet- ter solutions for a specific problem. GAs are coded in binary forming strings called chromosomes, they produce offsprings by sexual reproduction. Sexual reproduction is achieved when two strings (i.e. par- ents) are recombined (i.e. crossover), generally the parents are selected stochastically, the search process is mainly driven by the recombination operation, and 1568-4946/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2003.05.005