On-line RBFNN based identification of rapidly time-varying nonlinear systems with optimal structure-adaptation

Abstract

This paper presents an adaptive RBF network for the on-line identification and tracking of rapidly-changing time-varying nonlinear systems. The proposed algorithm is capable of maintaining the accuracy of learned patterns even when a large number of aged patterns are replaced by new ones through the adaptation process. Moreover, the algorithm exhibits a strong learning capacity with instant embodiment of new data which makes it suitable for tracking of fast-changing systems. However, the accuracy and speed in the adaptation is balanced by the computational cost which increases with the square of the number of the radial basis functions, resulting in a computational expensive, but still practically feasible, algorithm. The simulation results show the effectiveness (in terms of degradation of learned patterns and learning capacity) of this architecture for adaptive modeling.

Introduction

Following their broad learning and modeling capabilities, artificial neural networks are established as primary tools for optimization, pattern recognition and nonlinear system identification [1], [4], [6], [12] and control [5], [13], [14], [15]. Radial basis function (RBF) networks are three-layered neural networks composed of nodes whose output is proportional to the distance from the node center. Broomhead and Lowe [1] first studied the RBF networks as nonlinear function estimators and indicated their interpolation capabilities. Hartman et al. [2] and later Park and Sandberg [9], [10] proved that RBFs are capable of approximating any function with arbitrary accuracy. Since then, substantial efforts have been made to establish the learning efficiency and convergence rates of RBFs [4], [7], [16].

RBF networks have emerged as an alternative approach to multi-layer perceptrons (MLP) for pattern recognition and nonlinear systems modeling. This is due to the fast two-stage learning algorithms developed for RBFs in contrast to the slow convergence of the back-propagation algorithm which is widely used for the training of MLPs [6]. In the first phase of these popular RBF training methods, the centers and widths of the nodes of the network are selected. This could be done either randomly according to the training data or by employing an unsupervised procedure. The incentive behind this is that by placing the node centers following the training data density, near-optimum coverage of the input space can be expected. Various clustering algorithms for the choice of centers and widths have been applied successfully to many problem domains. In the second phase the output weights can be calculated using simple supervised least mean squares procedures since the network output is linear with respect to the weights. The decoupling could cause in some cases some loss of information although these algorithms are capable to compensate this loss by usually having significant smaller computational cost. Two-stage procedures often lead to better solutions given limited training data and computational time, although they can scarcely reach an optimal solution even if they go through intensive training.

Supervised training involving simultaneous adaptation of all the RBF network parameters attracted less research interest and not until recently have researchers started to study more systematically this domain [3]. The localized structure of RBF networks inevitably leads to inferior generalization performance compared to sigmoidal MLP neural networks. This constraint has up to now being tackled with the above mentioned methods of unsupervised selection of RBF centers and widths which are more capable of capturing the modeled function’s input space structure. This implies that in order for a supervised trained RBFNN to achieve similar to MLPNN interpolation capabilities, it needs significantly more computational resources.

The problem formulation is briefly discussed in Section 2. The proposed methodology is presented in Section 3. Section 4 discusses the implementation details and in Section 5 the mathematical formulation of the model is given. Some simulation results which show the model effectiveness are given in Section 6. Finally, Section 7 contains some concluding remarks.