Parameter identification of distributed parameter systems in the presence of measurement noise

A new approach to recursive parameter identification of second-order distributed parameter systems in the presence of measurement noise under unknown initial and boundary conditions is proposed. A two-dimensional low-pass filter is introduced to pre-filter the observed data corrupted by measurement noise. The low-pass filter is designed in the continuous time-space domain and discretized by bilinear transformation. Thus a discrete estimation model of the system under study is easily constructed with filtered input-output data for recursive identification algorithms. The recursive least squares method is still efficient in the presence of low measurement noise if the filter parameters are designed so that the noise effects are reduced sufficiently. Using filtered input data as instrumental variables, a recursive instrumental variable method is also presented to obtain consistent estimates when the digital low-pass filters are not designed successfully or when the output data is corrupted by high measurement noise. Illustrative examples are given to demonstrate the applicability of the proposed methods.     

Self-tuning weighted measurement fusion white noise deconvolution estimator and its convergence analysis

White noise deconvolution or input white noise estimation has a wide range of applications including oil seismic exploration, communication, signal processing, and state estimation. For the multisensor linear discrete time-invariant stochastic systems with correlated measurement noises, and with unknown ARMA model parameters and noise statistics, the on-line AR model parameter estimator based on the Recursive Instrumental Variable (RIV) algorithm, the on-line MA model parameter estimator based on Gevers–Wouters algorithm and the on-line noise statistic estimator by using the correlation method are presented. Using the Kalman filtering method, a self-tuning weighted measurement fusion white noise deconvolution estimator is presented based on the self-tuning Riccati equation. It is proved that the self-tuning fusion white noise deconvolution estimator converges to the optimal fusion steady-state white noise deconvolution estimator in a realization by using the dynamic error system analysis (DESA) method, so that it has the asymptotic global optimality. The simulation example for a 3-sensor system with the Bernoulli–Gaussian input white noise shows its effectiveness.     

A cumulant based algorithm for the identification of input-output quadratic systems

The identification of a special class of polynomial models is pursued in this paper. In particular a parameter estimation algorithm is developed for the identification of an input-output quadratic model excited by a zero mean white Gaussian input and with the output corrupted by additive measurement noise. Input-output crosscumulants up to the fifth order are employed and the identification problem of the unknown model parameters is reduced to the solution of successive triangular linear systems of equations that are solved at each step of the algorithm. Simulation studies are carried out and the proposed methodology is compared with two least squares type identification algorithms, the output error method and a combination of the instrumental variables and the output error approach. The proposed cumulant based algorithm and the output error method are tested with real data produced by a robotic manipulator.     

Hammerstein system identification by non-parametric instrumental variables

A mixed, parametric–non-parametric routine for Hammerstein system identification is presented. Parameters of a non-linear characteristic and of ARMA linear dynamical part of Hammerstein system are estimated by least squares and instrumental variables assuming poor a priori knowledge about the random input and random noise. Both subsystems are identified separately, thanks to the fact that the unmeasurable interaction inputs and suitable instrumental variables are estimated in a preliminary step by the use of a non-parametric regression function estimation method. A wide class of non-linear characteristics including functions which are not linear in the parameters is admitted. It is shown that the resulting estimates of system parameters are consistent for both white and coloured noise. The problem of generating optimal instruments is discussed and proper non-parametric method of computing the best instrumental variables is proposed. The analytical findings are validated using numerical simulation results.     

On sparsity-inducing methods in system identification and state estimation

The purpose of this article is to survey some sparsity-inducing methods in system identification and state estimation. Such methods can be divided into two main categories: methods inducing sparsity in the parameters and those sparsifying the prediction error. In the last class we discuss in particular the least absolute deviation estimator and its robustness properties with respect to sparse noise in both cases of univariate and multivariate measurements. We also discuss the application of sparsity-inducing methods to switched system identification and to state estimation for linear systems in the presence sparse and dense measurement noises. While the presentation focuses essentially on bridging some existing results, some technical refinements, and new features are also provided.     

A recursive maximum a posteriori estimator

Kalman Filter (KF) is the optimal state estimator for linear dynamical systems in the presence of zero mean white Gaussian noise. It is a minimum mean square error (MMSE) estimator. In the present work a recursive maximum a posteriori estimator (RMAPE) has been developed from basic principles of estimation. This estimator may be used for realtime state estimation of linear dynamical systems in presence of zero mean white Gaussian noise. It is further shown here that the KF can be derived from this RMAPE algorithm, i.e. this work shows an alternative method way to derive the KF. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Instrumental variable methods for closed-loop system identification

In this paper, several instrumental variable (IV) and instrumental variable-related methods for closed-loop system identification are considered and set in an extended IV framework. Extended IV methods require the appropriate choice of particular design variables, as the number and type of instrumental signals, data prefiltering and the choice of an appropriate norm of the extended IV-criterion. The optimal IV estimator achieves minimum variance, but requires the exact knowledge of the noise model. For the closed-loop situation several IV methods are put in an extended IV framework and characterized by different choices of design variables. Their variance properties are considered and illustrated with a simulation example.     

Simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise

In the present paper, the identification and estimation problem of a single-input–single-output (SISO) fractional order state-space system will be addressed. A SISO state-space model is considered in which parameters and also state variables should be estimated. The canonical fractional order state-space system will be transformed into a regression equation by using a linear transformation and a shift operator that are appropriate for identification. The identification method provided in this paper is based on a recursive identification algorithm that has the capability of identifying the parameters of fractional order state-space system recursively. Another subject that will be addressed in this paper is a novel fractional order Kalman filter suitable for the systems with coloured measurement noise. The promising performance of the proposed methods is verified using two stable fractional order systems.     

A new information fusion white noise deconvolution estimator

The white noise deconvolution or input white noise estimation problem has important applications in oil seismic exploration, communication and signal processing. By the modern time series analysis method, based on the autoregressive moving average （ARMA） innovation model, a new information fusion white noise deconvolution estimator is presented for the general multisensor systems with different local dynamic models and correlated noises. It can handle the input white noise fused filtering, prediction and smoothing problems, and it is applicable to systems with colored measurement noises. It is locally optimal, and is globally suboptimal. The accuracy of the fuser is higher than that of each local white noise estimator. In order to compute the optimal weights, the formula computing the local estimation error cross-covariances is given. A Monte Carlo simulation example for the system with Bernoulli-Gaussian input white noise shows the effectiveness and performances.     

基于子空间方法的精馏塔系统辨识建模研究

子空间辨识方法作为一种有效的针对多输入-多输出系统(MIMO)的辨识建模方法近年来受到广泛的重视.目前主要采用的子空间辨识算法只能适用于白噪声环境,而实际的工业现场数据很多是受到较大有色噪声干扰的.针对问题采用了一种新的子空间辨识算法,利用马尔可夫参数用于处理随机性部分,同时引入辅助变量用以去除噪声的干扰,能够适用于存在较大有色噪声干扰情况下的辨识建模,并可得到对象的无偏模型,建模的精度优于通常所采用的子空间辨识算法.通过对精馏塔仿真模型的辨识结果证明了该方法的可行性和有效性,以及在实际工业过程对象建模中良好的应用前景.     

自校正多传感器观测融合Kalman估值器及其收敛性分析

对于带未知噪声方差的多传感器系统,应用加权最小二乘(WLS)法得到了一个加权融合观测方程,且它与状态方程构成一个等价的观测融合系统.应用现代时间序列分析方法,基于观测融合系统的滑动平均(MA)新息模型参数的在线辨识,可在线估计未知噪声方差,进而提出了一种加权观测融合自校正Kalman估值器,可统一处理自校正融合滤波、预报和平滑问题,并用动态误差系统分析方法证明了它的收敛性,即若MA新息模型参数估计是一致的,则它按实现或按概率1收敛到全局最优加权观测融合Kalman估值器,因而具有渐近全局最优性.一个带3传感器跟踪系统的仿真例子说明了其有效性.     

带白色观测噪声的ARMA模型参数的无偏估计

本文研究了如何利用受白色噪声污染的观测数据辨识ＡＲＭＡ（ｐ，ｑ）模型参数数据的问题，提出了一种递推辅助变量法．利用这种方法首先辨识出ＡＲ（ｐ）部分的参数及观测噪声的方差，然后根据所得的估计利用常用的Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ方法确定ＭＡ（ｑ）部分的参数。     

On identifying stochastic closed-loop systems

The identification of closed-loop feedback systems having a white or colored stochastic input and a feedback structure is discussed for cases with and without measurement noise. It is shown that a consistent identification scheme can be obtained for the above cases.     

Identification of delayed system using instrumental variable method

System identification uses system inputs and outputs to raise mathematical models.Various techniques of system identification exist that offer a nominal model and an uncertainty bound.Many practical systems such as thermal processes & chemical processes have inbuilt time delay.If the time delay used in the system model for controller design does not concur with the actual process time delay,a closed-loop system may be unstable or demonstrate unacceptable transient response characteristics so here the time delay is assumed to be time-invariant. This paper proposes on-line identification of delayed complex/uncertain systems using instrumental variable(Ⅳ) method.Parametric uncertainty has been considered which may be represented by variations of certain system parameters over some possible range.This method allows consistent estimation when the system parameters are associated with the noise terms,as the IV methods(IVM’s)usually make no assumption on the noise correlation configuration.The faster convergence of the parameters including noise terms has been proved in this paper.Iterative prefiltering(IP)method has also been used for the identification of the delayed uncertain system and the graphical results given in this paper demonstrate that the convergence results are inferior to the instrumental variable method.     

Guest Editorial

The identification of continuous time models from non-uniformly sampled data records is investigated and a new identification algorithm based on the state variable filter approach is derived. It is shown that the orthogonal least squares estimator can be adapted for the identification of continuous time models from non-uniformly sampled data records and instrumental variables are introduced to reduce the bias in stochastic system identification. Multiplying the filtered variables obtained from the state variable filter, with higher powers of the noise free output signal prior to the estimation, is shown to enhance the parameter estimates. Simulated examples are included to illustrate the models.     

The relationship between a continuous-time identification algorithm based on the deterministic filter and least-squares methods

It is shown that an identification technique recently derived from the continuous-time Kalman filter may also be deduced from recursive algorithms for least-squares and minimum-variance methods of parameter estimation. Preliminary repeated integration, as a method for the identification of continuous systems, is enhanced by its incorporation into the procedure. The investigation permits a greater appreciation of certain features of the least-squares and minimum-variance methods, including the inference that the recursive minimum-variance algorithm can only exist when the measurement noise is an uncorrelated sequence. A study of well-known starting procedures relate these techniques to a deterministic state estimator derived from stability considerations.     

An alternative approach to state estimation in linear time-invariant systems

An alternative approach to state estimation problem in linear, time-invariant dynamic systems is presented in this paper. The approach developed first identifies the initial state of the system by using a proportional plus integral parameter identification method. The Lyapunov design technique is used to guarantee the asymptotic convergence of the initial state identifier. A state estimator is then constructed to operate in series with the initial state identifier. The estimator generates an estimate of the unobserved part of the system state. Simulation studies have shown that satisfactory state estimation can be achieved in the presence of measurement or disturbance noise. An example problem is considered to demonstrate the response characteristics of the estimator-identifier combination.     

Strongly consistent parameter estimation by the introduction of strong instrumental variables

This paper introduces the concept of strong instrumental variables and strong instrumental matrix sequences for the estimation of the transfer function parameters of discrete-time, time-invariant models of linear systems. It is shown that the strong instrumental variable estimators are strongly consistent and a sufficient condition for the estimator to be asymtotically unbiased is given. Moreover, it is shown that with a persistently exciting signal of appropriate order for an input, "virtually" any discrete-time, time-invariant, linear system model of appropriate order can be used to generate strong instrumental variables.     

Blind maximum likelihood identification of Hammerstein systems

This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér–Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss–Newton based optimization scheme. The paper includes the illustration of the method on a simulation example. A theoretical analysis demonstrates that additive output measurement noise introduces a bias that is proportional to the variance of that additive, unmodeled noise source. The simulations support this result, and show that this bias is insignificant beyond a certain Signal-to-Noise Ratio (40 dB in the example).     

Parameter identification of linear structural dynamic systems

The parameter estimations of linear multi-degree-of-freedom structural dynamic systems are carried out in time domain. Methods for estimating the system parameters and the modal parameters are presented. The equation of motion is transformed into the state space equation of the observable canonical form, and then into the auto-regressive and moving average model with auxiliary stochastic input (ARMAX) model to process the measurement data contaminated by the system noise as well as the output noise. The parameters of the ARMAX model are estimated by using the sequential prediction error method. Then, the parameters of equation of motion are recovered thereafter. In order to verify the accuracy of the estimation method, analytical simulation studies are performed for a model with two degrees of freedom on the basis of simulated data under various noise conditions. It is shown that the presented methods yield good estimates even under large noise conditions. The method is also applied to the identification of the modal parameters of a building model based on the experimental data.