Bias-compensated Recursive Least-squares Research Articles

A New Variable Forgetting Factor-Based Bias-Compensation Algorithm for Recursive Identification of Time-Varying Multi-Input Single-Output Systems With Measurement Noise

This article proposes a new variable forgetting factor (VFF) bias-compensated recursive least-squares (BCRLS) algorithm for the recursive identification of complex time-varying multi-input single-output (MISO) systems with measurement noise. It extends a previously developed real-valued BCRLS algorithm to complex signals and introduces new self-calibrated VFF and noise variance estimation schemes for tracking time-varying systems. The proposed VFF scheme offers faster tracking speed, especially for sudden system changes, while achieving a low steady-state (SS) mean square error (MSE) in a stationary environment. Moreover, the mean and mean square deviation of the complex RLS algorithm under zero-mean white Gaussian output additive noise are performed, from which the variance of the additive noise can be estimated. To mitigate the effect of finite-sample number, a self-calibration scheme is proposed to refine the FF at the SS and hence MSE. Simulations show that the proposed self-calibrated VFF-BCRLS algorithm offers improved tracking speed in sudden system changes and offers smaller MSE over the conventional BCRLS algorithm. Applications to real-world data for pH value prediction of a pH neutralization process and temperature prediction of a glass furnace also demonstrate the effectiveness of the proposed algorithm. The good performance and efficient implementation make it an attractive alternative to other conventional methods for system identification in control and optimization processes and other possible applications.

Adaptive EIV‐FIR filtering against coloured output noise by using linear prediction technique

The problem of finite impulse response (FIR) filtering in errors-in-variables (EIV) system is studied. Due to the input noise, traditional recursive least-squares (RLS) algorithms are biased in EIV system. Most existing bias-compensated approaches are proposed in the case that both the input-output noises are white Gaussian random processes. However, taking account of the situation where the output is corrupted by coloured noise, there are rare existing algorithms work well. Two bias-compensated RLS algorithms with acceptable computational complexity are proposed, which can obtain unbiased real-time filtering in non-stationary system when the input noise is white while the output noise is coloured. Under the assumption that the input signal is a coloured process, linear prediction technique is used to estimate the sample of the input signal. Exploiting the statistical properties of the cross-correlation function between the least-squares error and the forward/backward prediction error, the input noise variance can be estimated and the bias can be compensated. Simulation results illustrate the good performance of the proposed algorithms.

Distributed incremental bias-compensated RLS estimation over multi-agent networks

In this paper, we study the problem of distributed bias-compensated recursive least-squares (BC-RLS) estimation over multi-agent networks, where the agents collaborate to estimate a common parameter of interest. We consider the situation where both input and output of each agent are corrupted by unknown additive noise. Under this condition, traditional recursive least-squares (RLS) estimator is biased, and the bias is induced by the input noise variance. When the input noise variance is available, the effect of the noise-induced bias can be removed at the expense of an increase in estimation variance. Fortunately, it has been illustrated that distributed collaboration between agents can effectively reduce the variance and can improve the stability of the estimator. Therefore, a distributed incremental BC-RLS algorithm and its simplified version are proposed in this paper. The proposed algorithms can collaboratively obtain the estimates of the unknown input noise variance and remove the effect of the noise-induced bias. Then consistent estimation of the unknown parameter can be achieved in an incremental fashion. Simulation results show that the incremental BC-RLS solutions outperform existing solutions in some enlightening ways.

Diffusion Bias-Compensated RLS Estimation with Quantized Measurements over Adaptive Networks

We study the problem of distributed bias-compensated recursive least-squares (BCRLS) estimation with quantized measurements, which means that the nodes in adaptive networks collaborate to estimate a common deterministic parameter using quantized data. Traditional recursive least-squares (RLS) algorithms are biased when both the regressor and the output response are corrupted by stationary additive noise. And considering the limited bandwidth resources and the growing complexity in communication, the measurements transmitted in adaptive networks are always quantized in practical cases, which also contributes to the bias in estimation. Therefore, a distributed bias-compensated RLS algorithm is proposed in this paper to estimate the bias caused by both the background noise and the quantization noise and compensate for it cooperatively. Meanwhile, considering the advantages of diffusion strategies, the algorithm is developed in a diffusion fashion. Simulation results illustrate the good performance of the proposed algorithm.

Adaptive frequency estimation of three-phase power systems

The frequency of a three-phase power system can be estimated by identifying the parameter of a second-order autoregressive (AR2) linear predictive model for the complex-valued αβ signal of the system. Since, in practice, both input and output of the AR2 model are observed with noise, the recursive least-squares (RLS) estimate of the system frequency using this model is biased. We show that the estimation bias can be evaluated and subtracted from the RLS estimate to yield a bias-compensated RLS (BCRLS) estimate if the variance of the noise is known a priori. Moreover, in order to simultaneously compensate for the noise on both input and output of the AR2 model, we utilize the concept of total least-square (TLS) estimation and calculate a recursive TLS (RTLS) estimate of the system frequency by employing the inverse power method. Unlike the BCRLS algorithm, the RTLS algorithm does not require the prior knowledge of the noise variance. We prove mean convergence and asymptotic unbiasedness of the BCRLS and RTLS algorithms. Simulation results show that the RTLS algorithm outperforms the RLS and BCRLS algorithms as well as a recently-proposed widely-linear TLS-based algorithm in estimating the frequency of both balanced and unbalanced three-phase power systems.

Unbiased Recursive Least-Squares Estimation Utilizing Dichotomous Coordinate-Descent Iterations

We propose an unbiased recursive least-squares algorithm for errors-in-variables system identification. The proposed algorithm, called URLS, removes the noise-induced bias when both input and output are contaminated with noise and the input noise is colored and correlated with the output noise. To develop the algorithm, we define an exponentially-weighted least-squares optimization problem that yields an unbiased estimate. Then, we solve the system of linear equations of the associated normal equations utilizing the dichotomous coordinate-descent iterations. The URLS algorithm features significantly reduced computational complexity as well as improved numerical stability compared with a previously proposed bias-compensated recursive least-squares algorithm while having similar estimation performance. We show that the URLS algorithm is asymptotically unbiased and convergent in the mean-square sense. We also calculate its steady-state mean-square deviation. Simulation results corroborate the efficacy of the URLS algorithm and the accuracy of the theoretical findings.

Self-tuning Fusion Kalman Filter with Unknown Parameters and Its Convergence

For the multisensor systems with unknown model parameters and noise variances,a self-tuning decoupled fused Kalman filter is presented based on the optimal fusion rule weighted by scalars for components.Its convergence is proved by using the dynamic error system analysis(DESA) method.As an application to signal processing,the multidimensional and multiple bias compensated recursive least-squares(BCRLS) algorithms for estimating the AR parameters are presented for the multisensor multidimensional autoregressive(AR) signal with white and colored measurement noises.The equivalence between the two BCRLS algorithms is proved.The convergence of the two BCRLS algorithms is proved by DESA method.Further more,a self-tuning fused Kalman filter for the AR signal is presented,which has asymptotic optimality.A simulation example shows the effectiveness.

Self-Tuning Weighted Measurement Fusion Kalman Filter for Multi-Channel AR Signals and Its Convergence

For the multisensor multi-channel autoregressive (AR) signals with colored measurement noises, when the model parameters and noise variances are partially unknown, the consistent estimators of unknown AR model parameters and noise variance are obtained by the bias compensated recursive least-squares (BCRLS) algorithm. Then, substituting these estimators into the optimal weighted measurement fusion signal Kalman filter, a self-tuning weighted measurement fusion signal Kalman filter is presented. Further, applying the dynamic error system analysis (DESA) method, it is rigorously proved that the proposed self-tuning fusion signal Kalman filter converges to the optimal fuser in a realization, so that it has asymptotic global optimality. A simulation example shows their effectiveness.

Diffusion Bias-Compensated RLS Estimation Over Adaptive Networks

We study the problem of distributed least-squares estimation over ad hoc adaptive networks, where the nodes have a common objective to estimate and track a parameter vector. We consider the case where there is stationary additive colored noise on both the regressors and the output response, which results in biased local least-squares estimators. Assuming that the noise covariance can be estimated (or is known a priori), we first propose a bias-compensated recursive least-squares algorithm (BC-RLS). However, this bias compensation increases the variance or the mean-square deviation (MSD) of the local estimators, and errors in the noise covariance estimates may still result in residual bias. We demonstrate that the MSD and residual bias can then be significantly reduced by applying diffusion adaptation, i.e., by letting nodes combine their local estimates with those of their neighbors. We derive a necessary and sufficient condition for mean-square stability of the algorithm, under some mild assumptions. Furthermore, we derive closed-form expressions for its steady-state mean and mean-square performance. Simulation results are provided, which agree well with the theoretical results. We also consider some special cases where the mean-square stability improvement of diffusion BC-RLS over BC-RLS can be mathematically verified.

On Bias Compensated Recursive Least-Squares Algorithm for FIR Adaptive Filtering

It is known that recursive least squares (RLS) algorithm or least mean square (LMS) algorithm leads to biased FIR filter coefficients in the presence of input and output noise. In this paper, a new type of bias compensated recursive least squares (BCRLS) algorithm is proposed to produce consistent results for adaptive FIR filtering in the input and output noise case. The proposed algorithm introduces an auxiliary estimator to estimate unknown input noise variance. Owing to this, the bias of the RLS solution due to the input noise can be compensated to yield the consistent filter coefficients. Computational simulations indicate that the proposed algorithm is very efficient