Identification Of Nonstationary Systems Research Articles

On “cheap smoothing” opportunities in identification of time-varying systems

In certain applications of nonstationary system identification the model-based decisions can be postponed, i.e. executed with a delay. This allows one to incorporate into the identification process not only the currently available information, but also a number of “future” data points. The resulting estimation schemes, which involve smoothing, are not causal. Despite the possible performance improvements, the existing smoothing algorithms are seldom used in practice, mainly because of their high computational requirements. We show that the computationally attractive smoothing procedures can be obtained by means of compensating estimation delays that arise in the standard exponentially weighted least squares, least mean squares and Kalman filter-based parameter trackers.

Optimal and Suboptimal Smoothing Algorithms for Identification of Time-varying Systems with Randomly Drifting Parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér-Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least squares algorithm is often only slightly inferior to that of the Kalman filter based smoother.

On the lower smoothing bound in identification of time-varying systems

In certain applications of nonstationary system identification the model-based decisions can be postponed, i.e. executed with a delay. This allows one to incorporate in the identification process not only the currently available information, but also a number of “future” data points. The resulting estimation schemes, which involve smoothing, are not causal. Assuming that the infinite observation history is available, the paper establishes the lower steady-state estimation bound for any noncausal estimator applied to a linear system with randomly drifting coefficients (under Gaussian assumptions). This lower bound complements the currently available one, which is restricted to causal estimators.

Blind identification of non-stationary MA systems

A new adaptive algorithm for blind identification of time-varying MA channels is derived. This algorithm proposes the use of a novel system of equations derived by combining the third- and fourth-order statistics of the output signals of MA models. This overdetermined system of equations has the important property that it can be solved adaptively because of their symmetries via an overdetermined recursive instrumental variable-type algorithm. This algorithm shows good behaviour in arbitrary noisy environments and good performance in tracking time-varying systems.

Lyapunov-theory-based radial basis function networks for adaptive filtering

Two important convergence properties of Lyapunov-theory-based adaptive filtering (LAF) adaptive filters are first explored. The LAF finite impulse response and infinite impulse response adaptive filters are then realized using the radial basis function (RBF) neural networks (NNs). The proposed adaptive RBF neural filtering system possesses the distinctive properties of RBF NN and the LAF filtering system. Unlike many adaptive filtering schemes using gradient search techniques, a Lyapunov function of the error between the desired signal and the RBF NN output is first defined. By properly choosing the weights update law in the Lyapunov sense, the RBF filter output can asymptotically converge to the desired signal. The design is independent of the stochastic properties of the input disturbances and the stability is guaranteed by the Lyapunov stability theory. Simulation examples for nonlinear adaptive prediction of nonstationary signal and system identification are performed.

Editorial

Editorial

Simulation and Identification of Nonstationary Systems Using Linear Time-Frequency Methods

This work investigates synthesis and identification methods for nonstationary systems. Use of the time or frequency domains individually has limitations in many applications. Analysis in two domains jointly—that is, time frequency (TF)—gives us the potential to describe complex system behavior. A localized transfer function in time and frequency can describe time-variant filters. Operations between the TF expansion of an input signal and the two-dimensional transfer function of a system are equivalent to filtering in the TF plane. Simple analytic examples explain the basic methods and are applied to simulated and real data. Models of evolving systems are realized parametrically and nonparametrically in the TF domain.

Combined LMS/F algorithm

A new adaptive filter algorithm has been developed that combines the benefits of the least mean square (LMS) and least mean fourth (LMF) methods. This algorithm, called LMS/F, outperforms the standard LMS algorithm judging either constant convergence rate or constant misadjustment. While LMF outperforms LMS for certain noise profiles, its stability cannot be guaranteed for known input signals even For very small step sizes. However, both LMS and LMS/F have good stability properties and LMS/F only adds a few more computations per iteration compared to LMS. Simulations of a non-stationary system identification problem demonstrate the performance benefits of the LMS/F algorithm.

Identification of nonstationary stochastic systems using parallel estimation schemes

The parallel (multiple-model) schemes for identification of nonstationary stochastic systems are considered. First, the form of the optimal-local Bayesian predictor is derived under the assumptions that system coefficients vary according to the random walk model and that the Kalman-filter-based algorithms are used for identification purposes. A rational extension of this strategy, which can be applied to identification algorithms of any form, is discussed. Specific suggestions are made concerning the possible choice of adaptation gains of the competitive adaptive filters. Computer simulation results, confirming the good estimation robustness properties of the parallel identification schemes, are presented. It is shown that the proposed scheme can significantly decrease sensitivity of the identification algorithm to the rate of nonstationarity of the analyzed system or (alternatively) to the choice of design parameters such as adaptation gains and forgetting factors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Identification of non-stationary continuous systems using modulating functions

The identification of non-stationary multi-input multi-output continuous systems by the use of trigonometric modulating functions is considered. The time variable matrices to be identified are developed in polynomial form, and the fact that the product of a modulating function with these series also gives a modulating function is discussed. The proposed method avoids the necessity of knowledge and estimation of the initial conditions and signal derivatives.

Application of multiple Fourier series to identification of multivariable non-stationary systems

An algorithm for the on-line identification of multivariable non-stalionary systems is presented. The procedure is based on the first arithmetic means (C, 1) of multiple Fourier series. Convergence in the probability and with probability one is proved. Systems with multiplicative non-stalionarity are discussed in details.

Functional series modeling approach to identification of nonstationary stochastic systems

The problem of identification of a nonstationary stochastic system is considered, and an estimation method based on the functional series modeling (FSM) of the system parameter trajectory is proposed for its solution. It is shown that the parameter-matching properties of FSM estimators can be described in terms of the appropriately defined (time-varying) impulse and frequency responses. It is suggested and verified by means of computer simulation that the averaged frequency characteristics associated with FSM estimators can yield useful information about their parameter-matching abilities. >

On tracking characteristics of weighted least squares estimators applied to nonstationary system identification

The parameter-tracking abilities of weighted-least-squares (WLS) estimators are characterized in terms of a suitably defined variability matrix and associated frequency characteristics. Using these concepts, the conflict arising between the estimation variance and estimation bandwidth is easily explained and quantified. >

First-order tracking properties of weighted least squares estimators

The first-order tracking properties (evolution of the mean tracking error) of weighted-least-squares (WLS) estimators applied to nonstationary system identification are investigated. The expected path of the parameter estimates is evaluated. The concept of the impulse response associated with the WLS estimator is introduced. >

Recursive Identification of Nonstationary Random Processes and Systems

Identification of time-varying random processes and systems by adaptive filtering methods is considered. A method for the synthesis of a stochastic difference equation modeling the dynamics of unknown parameters is proposed. A priori information required for the construction of this equation consists of the choice of the basis of functions approximating time evolution of the parameters. A recursive algorithm for the model structure selection and for adaptive parameter estimation is presented. An example of parametric spectral analysis of nonstationary autoregressive process is given.

Real-time identification of time-varying systems by non-parametric algorithms based on Parzen kernels

Non-parametric algorithms, based on Parzen kernels, for the real-time identification of non-stationary systems are presented. Weak and strong pointwise convergence of identification procedures is s...

On nonparametric identification with prediction of time-varying systems

Nonparametric procedures-based on the orthogonal series approach-for the real-time identification and prediction of nonstationary systems are presented. It is shown that algorithms are convergent in the mean square error sense and with probability 1.

Analysis of Structural Nonstationary Behaviour and Dynamic System Identification Based on the Auto-Regressive Model Fitting

Analysis of Structural Nonstationary Behaviour and Dynamic System Identification Based on the Auto-Regressive Model Fitting

On-line identification of time-varying systems by nonparametric techniques

A nonparametric algorithm, based on orthogonal series, is proposed for real-time identification of nonstationary systems. It is proved that the algorithm converges in the mean-square sense. The convergence conditions are closely related to those in Dupac [2].

Practicable identification of nonstationary nonlinear systems

Practicable identification of nonstationary nonlinear systems