Torsten Söderström

Identification of processes in closed loop-identifiability and accuracy aspects

It is often necessary in practice to perform identification experiments on systems operating in closed loop. There has been some confusion about the possibilities of successful identification in such cases, evidently due to the fact that certain common methods then fail. A rapidly increasing literature on the problem is briefly surveyed in this paper, and an overview of a particular approach is given. It is shown that prediction error identification methods, applied in a direct fashion will given correct estimates in a number of feedback cases. Furthermore, the accuracy is not necessarily worse in the presence of feedback; in fact optimal inputs may very well require feedback terms. Some practical applications are also described.

Instrumental variable methods for system identification

This paper gives a tutorial overview of instrumental variable methods. Comparisons are made to the least-squares method. An analysis including consistency and asymptotic distribution of the parameter estimates is included.

Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements

The problem of estimating the frequencies, phases, and amplitudes of sinusoidal signals is considered. A simplified maximum-likelihood Gauss-Newton algorithm which provides asymptotically efficient estimates of these parameters is proposed. Initial estimates for this algorithm are obtained by a variation of the overdetermined Yule-Walker method and periodogram-based procedure. Use of the maximum-likelihood Gauss-Newton algorithm is not, however, limited to this particular initialization method. Some other possibilities to get suitable initial estimates are briefly discussed. An analytical and numerical study of the shape of the likelihood function associated with the sinusoids-in-noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure. Some numerical examples are presented to illustrate the performance of the proposed estimation procedure. Comparison to the performance corresponding to the Cramer-Rao lower bound is also presented, using a simple expression for the asymptotic Cramer-Rao bound covariance matrix derived in the paper. >

Survey paper: Errors-in-variables methods in system identification

The paper gives a survey of errors-in-variables methods in system identification. Background and motivation are given, and examples illustrate why the identification problem can be difficult. Under general weak assumptions, the systems are not identifiable, but can be parameterized using one degree-of-freedom. Examples where identifiability is achieved under additional assumptions are also provided. A number of approaches for parameter estimation of errors-in-variables models are presented. The underlying assumptions and principles for each approach are highlighted.

A theoretical analysis of recursive identification methods

In this paper five different recursive identification methods will be analyzed and compared, namely recursive versions of the least squares method, the instrumental variable method, the generalized least squares method, the extended least squares method and the maximum likelihood method. They are shown to be similar in structure and need of computer storage and time. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, these methods are examined from a theoretical viewpoint. Possible convergence points and their global and local convergence properties are studied. The theoretical analysis is illustrated and supplemented by simulations.

Papers: Identification of stochastic linear systems in presence of input noise

Most identification methods rely on the assumption that the input is known exactly. However, when collecting data under an identification experiment it may not be possible to avoid noise when measuring the input signal. In the paper some different ways to identify systems from noisy data are discussed. Sufficient conditions for identifiability are given. Also accuracy properties and the computational requirements are discussed. A promising approach is to treat the measured input and output signals as outputs of a multivariable stochastic system. If a prediction error method is applied using this approach the system will be identifiable under mild conditions.

Uniqueness of the maximum likelihood estimates of the parameters of an ARMA model

Estimation of the parameters in a mixed autoregressive moving average process leads to a nonlinear optimization problem. The negative logarithm of the likelihood function, suitably normalized, converges to a deterministic function as the sample length increases. The local and global extrema of this function are investigated. Conditions for the existence of a unique global and local minimum are given.

Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies

Consideration is given to the analysis of the large-sample second-order properties of multiple signal classification (MUSIC) and subspace rotation (SUR) methods, such as ESPRIT, for sinusoidal frequency estimation. Explicit expressions for the covariance elements of the estimation errors associated with either method are derived. These expressions of covariances are then used to analyze and compare the statistical performances of the MUSIC and SUR estimation (SURE) methods. Both MUSIC and SURE are based on the eigendecomposition of a sample data covariance matrix. The expressions for the estimation error variances derived are used to study the dependence of MUSIC and SURE performances on the dimension of the data covariance matrix used. >

Comparison of some instrumental variable methods-Consistency and accuracy aspects

A general analysis of various instrumental variable methods is given. The investigation includes methods for open loop operation as well as methods designed for closed loop operation. The instrumental variables are formed as different combinations of inputs, delayed inputs, delayed outputs, filtered inputs and external setpoint variations. Necessary conditions as well as sufficient conditions for consistency are explicitly derived and discussed. The paper also includes some simple counter-examples to general consistency for some common instrumental variable methods. Also asymptotic accuracy properties are reviewed. It is e.g. studied how the use of filters will influence the covariance matrix of the parameter estimates.

On model structure testing in system identification

In system identification it is assumed in many cases that the order of the system is known. Thus it is important to perform tests for determining the correct model order. For a single-input, single-output system the order is the only structural parameter, but for multivariable systems there are several structural parameters. It is in such cases not enough to test only for the order but to investigate if a model structure is appropriate or not. The paper contains a brief review of some methods for model structure testing. Two methods proposed by Akaike and the F-test are compared and it is shown that they are asymptotically equivalent. The methods are analysed theoretically through analytical calculations. The results are also illustrated with simulations.