Structure Identification of NN-ANARX Model by Genetic Algorithm with Combined Cross-correlation-test Based Evaluation Function Sven N˜ omm, Kristina Vassiljeva, Juri Belikov and Eduard Petlenkov Abstract— Application of genetic algorithm to determine structure of Neural Networks based Additive Nonlinear eX- ogenous (NN-ANARX) model and if possible to simplify the architecture of corresponding neural network constitutes sub- ject of present paper. In this paper, we construct a specific fitness function, which depends on mean square error, certain cross correlation coefficients and an order of the model. I. INTRODUCTION Model structure belong to the set of most important prerequisites for obtaining an accurate model. Choosing the model structure one should be guided not only by the factors affecting the quality of identification but also take into account conditions imposed by the further usage of identified model. Those conditions can be imposed by the necessity to apply certain methods for model analysis and control syntheses or by necessity to limit computational complexity. In spite of all advantages made in developing effective training algorithms neural networks (here and after NN) are not exception of this rule and model structure (reflected by topology of the corresponding NN) still plays important role. The class of NN based Additive Nonlinear eXogenous (NN-ANARX) models [4] has proven itself to be wide and flexible enough to model input-output behavior of a wide range of nonlinear processes. From such an academic examples as system of interconnected water tanks [2] to real-life applications like modeling backwards motion of a truck trailer [1] and movements of surgeon’s right hand [7]. ANARX class has three key advantages over some more general model classes • ANARX models are always realizable in the classical state-space form [4], which makes this class very at- tractive for model analysis and control design; • ANARX models are always linearizable via dynamic output feedback [8], again important from the control point of view; • additive structure of ANARX implemented in the form of neural network has fewer number of connections and S. Nomm and J. Belikov are with Control Systems Department, institute of cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia sven@cc.ioc.com jbelikov@cc.ioc.ee This work was supported by the Estonian Science Foundation through the state funding project SF0140018s08 and research grand ETF 8365. K. Vassiljeva and E. Petlenkov are with the Department of Computer Control, Tallinn University of Technology, Ehitajate tee 5,19086 Tallinn, Estonia kristina.vassiljeva@dcc.ttu.ee eduard.petlenkov@dcc.ttu.ee This work was partially supported by the Governmental funding project no. SF0140113As08, the Estonian Science Foundation Grant no. 8738 and Supported by the Estonian Doctoral School in ICT. weights and therefore requires less computations for training and simulation. In spite of all advantages mentioned above, one still have to determine correct order of the model. While usually mean square error is considered to provide enough information about quality of identified model. Procedure to validate identified neural network on the basis of correlation test was described in [13] and later adopted to compare quality of NN-NARX and NN-ANARX models of the same system in [6]. The idea is based on a more general results [14] and [12] where it was formulated for validation of nonlinear models. In [5] it was noted that even if model order is correct it can be possible to simplify the structure of the model by eliminating certain connections between the neurons. In [9] and [10] genetic algorithm (GA) was applied to improve quality of general model class by eliminating certain redundant interconnections and weights. Present contribution is devoted to the application of ge- netic algorithm to determine structure of the model (of course remaining within ANARX class) whereas fitness function of the GA will depend both on mean square error and results of correlation based test. Such process can be seen as trade off between ”playing” with model order and removing certain interconnections and weights. The paper is organized as follows. NN-ANARX class and necessary mathematical tools together with brief explanation of correlation test based procedure presented in Section II. Model selection algorithm is described in detail in Section III. Experimental results and their discussion are presented in Section IV. Concluding remarks are drawn in the last section. II. PROBLEM STATEMENT AND MATHEMATICAL TOOLS For the given unknown nonlinear plant, described by its input-output behavior, find NN-ANARX model satisfying given limitations: • mean square error should not exceed certain preset value; • model should be valid (no correlations between residu- als and delayed residuals, delayed outputs and delayed inputs). In other words a model quality criterion is based on both mean square error of the model and results of correlation based test. As usually it is desirable to find a model such that its order is optimal and if possible all redundant connections between layers are eliminated.