Online neural identification of multi-input multi-output systems A. Bazaei and M. Moallem Abstract: A feedforward neural network tuning algorithm is developed, which is suitable for identification of multi-input multi-output nonlinear functions, by utilising the learning method of a conventional neuro-adaptive control technique. Using Lyapunov functions, it is shown that not only the approximation error converges to values that have arbitrarily reducible upper bounds, but also the weights of the neural network remain bounded. The effectiveness of the identification method and its application in force-control of an uncertain robot interacting with an unknown flexible environment are investigated as an application example. 1 Introduction Feedforward neural networks (FFNNs) have been widely used for function approximation [1]. Most of the existing FFNN learning algorithms for function approximation, such as the popular back-propagation algorithm and its variants [2–5], steepest descent and other gradient-based methods [6–11], layer-by-layer learning [12], iterative methods [13], hybrid learning algorithms [14], and methods used in optimal filtering [15, 16], are either offline batch learning algorithms or discrete-time methods. Online continuous-time training methods for FFNNs have been developed by Lewis et al. [17], Ge and Wang [18] and Hovakimyan et al. [19] in which the training algorithms are designed such that the tracking error of a feedback control system is reduced. In this paper, we address the online nonlinear function approximation problem of multi- input multi-output systems. In this respect, we develop online continuous-time FFNN training methods that can be used for function approximation in identification problems. The proposed algorithm can reduce the neural network (NN) approximation error below an upper bound, which can be made arbitrarily small by increasing a constant design parameter. A proof showing uniform ultimate boundedness the weights of the NN is also presented. The effectiveness of the algorithm is investigated by simulation studies. The application of the proposed algorithm for force control of a robot manipulator constrained by a flexible environment is studied in one of the simulation studies. 2 Problem statement Consider the problem of approximating a nonlinear multi- input multi-output (MIMO) function via a single-hidden- layer neural network (NN). We assume that the function that is to be approximated is in the form y ¼ f ðxÞ ð1Þ where x [ < m21 is the input vector containing the indepen- dent variables of function f ( . ), and y [ < l is the output vector containing the dependent variables of the function. NNs are suitable tools for estimating functions and have been known as universal approximators [20]. The NN that we consider has n neurons in its hidden layer with its output described by the following equation ^ y ¼ ^ M T ^ s a ð2Þ where ^ M is an (n þ 1)  l matrix containing the weights and biases of the output layer. The outputs of the hidden layer neurons have been included in the augmented vector s ˆ a described by ^ s a ¼ s ð ˆ N T x a Þ 1   ð3Þ where the m  n matrix ^ N contains the weights and biases of the input layer. Vector s ( ^ N T x a ) is a function of independent variables x and can be represented by s ð ^ N T x a Þ¼ ^ z 1 . . . ^ z n 2 4 3 5 ð4Þ where ^ z i ¼ s i ð ^ n T i x a Þ; i ¼ 1; ... ; n ð5Þ in which s i ( . ) is the activation function of the ith hidden neuron, ^ n i is the ith column in matrix ^ N , and the m  1 vector x a contains the independent variables given by x a ¼ x 1   ð6Þ This type of NN has been successfully used in the work of Lewis et al. [17] and Hovakimyan et al. [19] for adaptive control, in which the tuning algorithm tries to improve the control performance. In this work, we would like to address how NN structures can be used for online identifi- cation of unknown nonlinear and/or time-varying MIMO # The Institution of Engineering and Technology 2006 doi:10.1049/iet-cta:20050259 Paper first received 14th July 2005 and in revised form 31st January 2006 The authors are with the Department of Electrical and Computer Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9 E-mail: mmoallem@uwo.ca IET Control Theory Appl., Vol. 1, No. 1, January 2007 44