Non-linear Time-Varying System Modeling Using Time-Varying High-Order Neural Networks

The non-linear time-varying system is difficult to be modeled due to its complexdynamics. Artificial neural network, a black box model, which presented as a powerfulapproximation tool, is widely used in identification and modeling of non-linear systems.However, the ability of conventional neural networks with its training algorithm is limited,especially when it is applied to the identification of time-varying nonlinear systems withunknown noise, the accuracy of this approach is always cannot reach a satisfactory expec-tation. Iterative learning identification is a method derived from iterative learning controltheory, its identification procedure mainly focus on the repetitive learning of input/outputmapping relationship in each sample time point, which makes it can implements a fullyidentification of system parameters and more appropriate to a high accuracy modeling ofhighly time-varying non-linear system. The iterative neural identification approach is dis-cussed with the architecture of time-varying high-order neural network. Our contributionsmainly lie in the following aspects:1. A time-varying radial basis function neural network is proposed on the basis ofits conventional counterpart(ie, weight-constant neural network) which is created througha subtractive clustering method. Based on this architecture, an iterative learning algorithmwith dead-zone is derived to train this neural model. The training error can converge toa zero-centered unit circle with the assumption of that we have posterior knowledge ofmodeling accuracy or upper noise bound. On the other hand, we introduce time factor t toinput space of conventional RBF network with the expectation to track system dynamics.To make a contrast, a similar training method called dead zone RLS is applied to train thisweight-constant neural model. The simulation cases illustrated the efficiency of this scheme on high accuracy modeling of time-varying non-linear system.2. On the consideration of redundant hidden neurons existed in RBF identificationmodel, a time-varying volterra polynomial network is introduced to solve the same identi-fication problem by using iterative learning least squares with dead-zone. In order to avoid"Dimension Explosion", which is a natural defect of this network architecture,we use or-thogonal least squares to select the most significant regressors to eliminate the redundanthidden neurons.finally, the time-invariant network is used to make a contrast with its time-varying counterpart.3. To address the problem of huge computation burden of iterative learning algo-rithm and with the consideration of ill-posed information matrix collected from learningiterations,givens transformation is utilized to introduce a more computational effective or-thogonal iterative learning least squares algorithm, it has better numerical performance andless computation complexity. A periodical time-varying neural network with this time-varying network architecture is also proposed for identification of non-linear periodicaltime-varying system. Both on-line and off-line learning strategy of iterative learning ap-proach are given with numerical cases illustrated its efficiency.