ComputersElect. Engng Vol. 19, No. 4, pp. 333-341,1993 0045-7906/93 $6.00 + 0.00 Printedin GreatBritain.All rightsreserved Copyright © 1993 Pergamon Press Ltd THE RECURSIVE NEURAL NETWORK AND ITS APPLICATIONS IN CONTROL THEORY DON HUSH, CHAOUKI ABDALLAH and BILL HORNE Department of Electrical Engineeringand Computer Engineering, Universityof New Mexico, Albuquerque, NM 87131, U.S.A. (Received 1 September 1990; accepted in final revisedform 20 November 1991) Abstract--This paper introducesa new dynamicalneural network and a corresponding learningalgorithm based on a gradient search. A model-following controller using the network is presented and is shown to be useful in the identificationand control of discretenonlinear systems.A discussionof the advantages and limitations of the new network is also included. 1. INTRODUCTION In general the control of nonlinear systems is a difficult task where intuition and experience are the guiding approaches. Even when a controller is found, its implementation is far from simple. Nonlinear controllers are basically nonlinear dynamical mappings which may not have a simple analytical form. On the other hand, some neural networks have been shown to be able to implement arbitrarily complex static mappings [1]. In order to use neural networks in control one must make them dynamic and describe the control objective in a neural network language. This paper introduces a new dynamical neural network and a corresponding learning algorithm based on a gradient search. The network is shown in Fig. 1 and may be used for modeling or controlling discrete-time nonlinear systems. It is a single-input single-output nonlinear dynamical system. The output of the network is the sum of the outputs of two subnets, a nonrecursive and a recursive subnet, both of which are multilayer perceptrons (MLPs). The inputs to the nonrecursive subnet are current and delayed (previous) versions of the input sequence, and the inputs to the recursive subnet are delayed versions of the network output. Because of its resemblance to a recursive digital filter we refer to it as the recursive neural net, RNN. In fact if the MLPs are replaced by ADALINES [2] the RNN reduces to a linear recursive digital filter (IIR filter) and the learning algorithm reduces to the well-known recursive LMS (RLMS) algorithm [2]. The work in this paper is closely related to the work of Narendra in Ref. [3]. Narendra describes, in diagrammatic fashion, a variety of nonlinear dynamical models which make use of the multilayer perceptron without providing a detailed description of the corresponding update algorithms. This paper presents such a description along with some properties of the resulting network. The RNN lends itself to a variety of uses such as nonlinear prediction, forward or inverse modeling for a nonlinear system, or as a controller for a nonlinear plant. A few of these applications are illustrated in Fig. 2. The remainder of this paper is organized as follows. Section 2 describes the operation of the RNN and introduces the notation that will be used. Section 3 derives a learning algorithm for the RNN based on a gradient search. Section 4 presents an application of the RNN in modeling a nonlinear plant and Section 5 contains a summary and discussion. 2. THE RECURSIVE NEURAL NET The output of the recursive neural netwok at time k as shown in Fig. 1, is the sum of the outputs of two subnets, the nonrecursive and recursive subnet: u(/O = v~(/~) + v~,(/c). (l) Each of these networks is a multilayer perceptron. The superscripts n and r are used to distinguish between parameters in the nonrecursive and recursive subnets respectively. Let L and M represent the number of layers in the nonrecursive and recursive subnets, respectively. If we let ¢t(k) represent 333