Abstract
Neural network (NN) approaches have been widely applied for modeling and identification of nonlinear multiple-input multiple-output (MIMO) systems. This paper proposes a stochastic analysis of a class of these NN algorithms. The class of MIMO systems considered in this paper is composed of a set of single-input nonlinearities followed by a linear combiner. The NN model consists of a set of single-input memoryless NN blocks followed by a linear combiner. A gradient descent algorithm is used for the learning process. Here we give analytical expressions for the mean squared error (MSE), explore the stationary points of the algorithm, evaluate the misadjustment error due to weight fluctuations, and derive recursions for the mean weight transient behavior during the learning process. The paper shows that in the case of independent inputs, the adaptive linear combiner identifies the linear combining matrix of the MIMO system (to within a scaling diagonal matrix) and that each NN block identifies the corresponding unknown nonlinearity to within a scale factor. The paper also investigates the particular case of linear identification of the nonlinear MIMO system. It is shown in this case that, for independent inputs, the adaptive linear combiner identifies a scaled version of the unknown linear combining matrix. The paper is supported with computer simulations which confirm the theoretical results.
Highlights
Neural network [1] approaches have been extensively used in the past few years for nonlinear multiple-input multiple-output (MIMO) system modeling, identification and control where they have shown very good performances compared to classical techniques [2,3,4,5,6]
This paper deals with a typical class of nonlinear MIMO systems (Figure 1) which is composed of M inputs, M memoryless nonlinearities, a linear combiner, and L outputs
The purpose of this paper is to provide a stochastic analysis of Neural network (NN) modeling of this class of MIMO systems
Summary
Neural network [1] approaches have been extensively used in the past few years for nonlinear MIMO system modeling, identification and control where they have shown very good performances compared to classical techniques [2,3,4,5,6]. This paper deals with a typical class of nonlinear MIMO systems (Figure 1) which is composed of M inputs, M memoryless nonlinearities, a linear combiner, and L outputs. This corresponds, for example, to MIMO channels used in wireless terrestrial communications. Linear adaptive system This section studies the linear adaptive system that tries to model the nonlinear MIMO system (Figure 4): Mean weight behavior and Wiener solution. Μ λ max ð16Þ where λmax is the largest eigenvalue of the covariance matrix RXX In this case, the linear adaptation allows the identification of matrix W to within a scaling matrix, which depends on the nonlinearities and the input signals.
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