Linear And Nonlinear Systems Identification Based On Higher-Order Statistics

This dissertation considers the identification of linear ARMA models as well as truncated nonlinear Volterra systems- quadratic nonlinear systems based on higher-order statistics and artificial neural networks. Very promising results are drawn as follows.Firstly, third-order cumulant RLS algorithm for nonGaussian nonminimum phase ARMA models (CRLS) is suggested in this thesis. A cost function based on the third-order cumulants and cross-cumulants is defined for the development of the CRLS identification algorithm. While dealing with ARMA models, the orders of AR and MA sub-models p, q are assumed known a priori. The construction of a residual time series (RLS) is required for the MA parameters estimation. Due to the third-order cumulant properties, the CRLS algorithm can suppress Gaussian noise and is capable of providing an consistent estimate under a noisy environment. Theoretical analyses and numerical simulations show that the proposed approach has strong consistency and convergency.When both inputs and outputs of ARMA models are contaminated by additive Gaussian noises of unknown power spectral density, the above CRLS method is not suitable any more. A novel order-recursive methodology for identifying ARMA models is then introduced herein. The major objective of the proposed algorithm, in which a priori knowledge of the model orders are not required, is to determine the model orders together with the corresponding parameters simultaneously. It is performed order-recursively until the updated order is equal to or greater than the true AR submodel's order, where the norm of output error squares (NES) reaches its minimum. Both ARMA orders and parameters are then estimated correctly. Numerical simulations testify the convergency of the proposed approach.In many practical cases, however, the inputs of the identified systems are not measurable and the identification must be performed blindly, that simply based on some statistical properties of the output signals. This thesis develops a novel identification methodology for nonminimum phase ARMA models of which the models' orders are not given. It is based on the third-order statistics of given noisy output observations and assumed input random sequences. Semiblind identification approach is thereby named. At each updated order the MA parameters are estimated without computing the residual time series with the result of decreasing the computational complexity and memory consumption. Effects of the AR estimation error on the MA parameters estimation are reduced also. The procedure begins with a first-order AR and MA submodel and updates the model order with an increment of one. At each test order, the corresponding parameters are estimated by using the least squares algorithm to minimize the well-defined cost function. When the test order reaches its true one, the magnitude of the cost function is minimum and may hold stable even when the test order further increases. we can conclude that the estimates of ARMA orders are just at the last turning point of the cost function characteristic curves, where the corresponding parameters can be estimated. Theoretical statements and simulation results illustrate that the method provides accurate estimates of unknown ARMA models despite the output measurements are corrupted by arbitrary Gaussian noises of unknown p.d.f..In view of unblind feature of the above MA submodels identification, a blind identification approach is then presented in this study. As per the above AR scheme, the developed method performs order-recursively to estimate MA orders and parameters.Because of wide application of nonlinear Volterra systems, the blind identifiability of quadratic nonlinear systems is considered in this thesis.Blind identification of quadratic nonlinear systems based on constrained neural networks and autocorrelations of the system outputs is addressed firstly. The driven sequence of the system is assumed an unobservable i.i.d. random variable. Different from the conventional neural network, the proposed network is of a three-layer constrained feedforward topology, in which activation functions hold constant and weights of summing junctions represent the system kernels. The inputs of the network consist of pseudo-random binary signals which are generated in terms of the monitored system's order. Conventional back-propagation policy is used to train the constrained network such that the outputs of the network arrive at the desired second-order statistics of the system output when the weights equal their corresponding exact kernels. Very promising results on identifying nonlinear systems are obtained and discussed through numerical simulations.The blind identification methods for ARMA models and quadratic nonlinear systems mentioned above are also utilized to model the real vibration signals measured from a running railway carriage. Based on the above development, blind identifiability of quadratic nonlinear systems in higher-order statistics domain is presented in this study. Some nonlinear analytic relations between the third-order cumulants of the outputs and the quadratic kernels are characterized through theoretical statements and simulations. It provides a useful starting point for implementation of truncated Volterra nonlinear system identification using conventional techniques or neural networks methodologies.Lastly, an input-output model based on feedforward neural networks for a generic nonlinear dynamic system identification is considered. In the developed neural networks based nonlinear system identification model, the input space of the network consists of input-output past observations of the identified system and the size of the input space is therefore directly related to the system order, i.e., the number of the input space is at least double the system order. The neural network uses a higher-order cumulants based cost function together with an effective training algorithm, the cumulant-based Weights Decoupled Extended Kalman Filter (CWDEKF) strategy. By monitoring the identification error characteristic curve, the system order and subsequently an appropriate network structure for systems identification can be determined. The obtained results are promising which indicate that generic nonlinear systems can be identified by the proposed neural network models.