A covariance matching approach for identifying errors-in-variables systems

The errors-in-variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables models. In this paper a covariance matching approach is proposed to solve the identification problem. It applies for general types of input signals. The method utilizes a small set of covariances of the measured input–output data. This property applies also for some other methods, such as the Frisch scheme and the bias-eliminating least squares method. Algorithmic details for the proposed method are provided. User choices, for example specification of which input–output covariances to utilize, are discussed in some detail. The method is evaluated by using numerical examples, and is shown to have competitive properties as compared to alternative methods.     

A generalized instrumental variable estimation method for errors-in-variables identification problems

The errors-in-variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables models. In this paper it is shown how a number of common methods for errors-in-variables methods can be put into a general framework, resulting into a Generalized Instrumental Variable Estimator (GIVE). Various computational aspects of GIVE are presented, and the asymptotic distribution of the parameter estimates is derived.     

On the accuracy of a covariance matching method for continuous-time errors-in-variables identification

An analysis of a covariance matching method for continuous-time errors-in-variables system identification from discrete-time data is made. In the covariance matching method, the noise-free input signal is not explicitly modeled and only assumed to be a stationary process. The asymptotic normalized covariance matrix, valid for a large number of data and a small sampling interval, is derived. This involves the evaluation of a covariance matrix of estimated covariance elements and estimated derivatives of such elements, and large parts of the paper are devoted to this task. The latter covariance matrix consists of two parts, where the first part contains integrals that are approximations of Riemann sums, and the second part depends on the measurement noise variances.     

An improved bias-compensation approach for errors-in-variables model identification

Parametric estimation of the dynamic errors-in-variables models is considered in this paper. In particular, a bias compensation approach is examined in a generalized framework. Sufficient conditions for uniqueness of the identified model are presented. Subsequently, a statistical accuracy analysis of the estimation algorithm is carried out. The asymptotic covariance matrix of the system parameter estimates depends on a user chosen filter and a certain weighting matrix. It is shown how these can be tuned to boost the estimation performance. The numerical simulation results suggest that the covariance matrix of the estimated parameter vector is very close to the Cramér-Rao lower bound for the estimation problem.     

Accuracy analysis of bias-eliminating least squares estimates for errors-in-variables systems

The bias-eliminating least squares (BELS) method is one of the consistent estimators for identifying dynamic errors-in-variables systems. In this paper, we investigate the accuracy properties of the BELS estimates. An explicit expression for the normalized asymptotic covariance matrix of the estimated parameters is derived and supported by some numerical examples.     

Identification of continuous-time errors-in-variables models

A novel direct approach for identifying continuous-time linear dynamic errors-in-variables models is presented in this paper. The effects of the noise on the state-variable filter outputs are analyzed. Subsequently, a few algorithms to obtain consistent continuous-time parameter estimates in the errors-in-variables framework are derived. It is also possible to design search-free algorithms within our framework. The algorithms can be used for non-uniformly sampled data. The asymptotic distributions of the estimates are derived. The performances of the proposed algorithms are illustrated with some numerical simulation examples.     

Third-order cumulants based methods for continuous-time errors-in-variables model identification

In this paper, the problem of identifying stochastic linear continuous-time systems from noisy input/output data is addressed. The input of the system is assumed to have a skewed probability density function, whereas the noises contaminating the data are assumed to be symmetrically distributed. The third-order cumulants of the input/output data are then (asymptotically) insensitive to the noises, that can be coloured and/or mutually correlated. Using this noise-cancellation property two computationally simple estimators are proposed. The usefulness of the proposed algorithms is assessed through a numerical simulation.     

L-two-optimal identification of errors-in-variables models: a frequency-domain approach

This paper proposes an L-two-optimal identification approach to cope with errors-in-variables model (EIVM) identification. With normalized coprime factor model (NCFM) representations, L-two-optimal approximate models are derived from the framework of an EIVM according to the kernel and image representations of related signals. Based on the optimal approximate models, the v-gap metric is employed as a minimization criterion to optimize the parameters of a system model, and thus the resulting optimization problem can be solved by linear matrix inequalities (LMIs). In terms of the optimized system model, the noise model (NM) can be readily obtained by right multiplication of an inner. Compared with other EIVM identification methods, the proposed one has a wider scope of applications because the statistical properties of disturbing noises are not demanded. It is also capable of giving identifiability. Finally, a numerical simulation is used to verify the effectiveness of the proposed method.     

Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification

For identifying errors-in-variables models, the time domain maximum likelihood (TML) method and the sample maximum likelihood (SML) method are two approaches. Both methods give optimal estimation accuracy but under different assumptions. In the TML method, an important assumption is that the noise-free input signal is modelled as a stationary process with rational spectrum. For SML, the noise-free input needs to be periodic. It is interesting to know which of these assumptions contain more information to boost the estimation performance. In this paper, the estimation accuracy of the two methods is analyzed statistically for both errors-in-variables (EIV) and output error models (OEM). Numerical comparisons between these two estimates are also done under different signal-to-noise ratios (SNRs). The results suggest that TML and SML have similar estimation accuracy at moderate or high SNR for EIV. For OEM identification, these two methods have the same accuracy at any SNR.     

Relations between Bias-Eliminating Least Squares, the Frisch scheme and Extended Compensated Least Squares methods for identifying errors-in-variables systems

There are many methods for identifying errors-in-variables systems. Among them Bias-Eliminating Least Squares (BELS), the Frisch scheme and Extended Compensated Least Squares (ECLS) methods are attractive approaches because of their simplicity and good estimation accuracy. These three methods are all based on a Bias-Compensated Least-Squares (BCLS) principle. In this paper, the relationships between them are considered. In particular, the set of nonlinear equations utilized in these three methods are proved to be equivalent under different noise conditions also for finite samples. It is shown that BELS, Frisch and ECLS methods have the same asymptotic estimation accuracy providing the same extended vector is used.     

Unifying some higher-order statistic-based methods for errors-in-variables model identification

In this paper, the problem of identifying linear discrete-time systems from noisy input and output data is addressed. Several existing methods based on higher-order statistics are presented. It is shown that they stem from the same set of equations and can thus be united from the viewpoint of extended instrumental variable methods. A numerical example is presented which confirms the theoretical results. Some possible extensions of the methods are then given.     

State-space analysis and identification for a class of hysteretic systems

In this paper we present results on the twin subjects of system analysis and system identification for a class of state-space realizable dynamic systems under the influence of hysteresis. The class of systems in question consists of models in the form of a linear time-invariant dynamic system in series with a differential model of hysteresis. It will be demonstrated that under fairly light constraints on the differential model of hysteresis, it is possible to design a series of experiments leading towards the identification of the full state-space realization. The approach is tested successfully on a high-precision mechanical translation system affected by hysteresis.     

Checking the adequacy for a distortion errors-in-variables parametric regression model

This paper studies tools for checking the validity of a parametric regression model, when both response and predictors are unobserved and distorted in a multiplicative fashion by an observed confounding variable. A residual based empirical process test statistic marked by proper functions of the regressors is proposed. We derive asymptotic distribution of the proposed empirical process test statistic: a centered Gaussian process under the null hypothesis and a non-centered one under local alternatives converging to the null hypothesis at parametric rates. We also suggest a bootstrap procedure to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed test statistic and real examples are analyzed for illustrations.     

Parameter estimation from noisy measurements

This article considers the problem of estimating linear model parameters from noisy measurements. The starting point is the classical approach by Koopmans for linear regression analysis. It is known that concerning the direct application of those early results for process identification, neither the original Koopmans algorithm nor its updated forms called Koopmans–Levin algorithms exhibit maximum-likelihood (ML) parameter estimation. In this article, a new, numerically advanced method is developed to ensure ML property for the parameter estimation, assuming noisy inputs and outputs, respectively.     

Fast approximate identification of nonlinear systems

In this paper, a method is presented to extend the classical identification methods for linear systems towards nonlinear modelling of linear systems that suffer from nonlinear distortions. A well chosen, general nonlinear model structure is proposed that is identified in a two-step procedure. First, a best linear approximation is identified using the classical linear identification methods. In the second step, the nonlinear extensions are identified with a linear least-squares method. The proposed model not only includes Wiener and Hammerstein systems, it is also suitable to model nonlinear feedback systems. The stability of the nonlinear model can be easily verified. The method is illustrated on experimental data.     

时滞系统稳定性分析:一种积分等式方法

提出一种积分等式方法研究时滞系统稳定性问题.该方法是已有积分不等式方法的改进.结合自由权矩阵构建积分等式,且自由权矩阵的选取由Lyapunov-Krasovskii泛函的导数确定,从而在稳定条件的推导过程中,不引入任何模型转换及限界方法.基于该方法,可得到具有更低保守性的稳定条件.仿真例子说明了所提方法的有效性.     

Identification of dynamic errors-in-variables models: Approaches based on two-dimensional ARMA modeling of the data

In this paper we propose a parametric and a non-parametric identification algorithm for dynamic errors-in-variables model. We show that the two-dimensional process composed of the input-output data admits a finite order ARMA representation. The non-parametric method uses the ARMA structure to compute a consistent estimate of the joint spectrum of the input and the output. A Frisch scheme is then employed to extract an estimate of the joint spectrum of the noise free input-output data, which in turn is used to estimate the transfer function of the system. The parametric method exploits the ARMA structure to give estimates of the system parameters. The performances of the algorithms are illustrated using the results obtained from a numerical simulation study.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.