Previous results on estimating errors or error bounds on identified transfer functions have relied upon prior assumptions about the noise and the unmodelled dynamics. This prior information took the form of parameterized bounding functions or parameterized probability density functions, in the time or frequency domain, with known parameters. Here we show that the parameters that quantify this prior information can themselves be estimated from the data using a Maximum Likelihood technique. This significantly reduces the prior information required to estimate transfer function error bounds. We illustrate the usefulness of the method with a number of simulation examples. The paper concludes by showing how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and undermodelling. Another simulation study compares our method to Akaike's well known FPE and AIC criteria.

The problem of assessing the quality of a given, or estimated model is a central issue in system identification. Various new techniques for estimating bias and variance contributions to the model error have been suggested in the recent literature. In this contribution, classical model validation procedures are placed at the focus of our attention. We discuss the principles by which we reach confidence in a model through such validation techniques, and also how the distance to a "true" description can be estimated this way. In particular, we stress how the typical model validation procedure gives a direct measure of the model error of the model test, without referring to its ensemble properties. Several model error bounds are developed for various assumptions about the disturbances entering the system.

A new approach to robust filtering, prediction and smoothing of discretetime signal vectors is presented. Linear time-invariant filters are designed to be insensitive to spectral uncertainty in signal models. The goal is to obtain a simple design method, leading to filters which are not overly conservative. Modelling errors are described by sets of models, parameterized by random variables with known covariances. These covariances could either be estimated from data, or be used as robustness "tuning knobs". A robust design is obtained by minimizing the H 2 -norm or, equivalently, the mean square estimation error, averaged with respect to the assumed model errors. A polynomial solution, based on an averaged spectral factorization and a unilateral Diophantine equation, is derived. The robust estimator is referred to as a cautious Wiener filter. It turns out to be only slightly more complicated to design than an ordinary Wiener filter. The methodology can be applied to any open loop filte...

A new approach to robust estimation of signals, prediction of time--series and robust feedforward control is considered. Signal and system parameter deviations are represented as random variables, with known covariances. A robust design is obtained by minimizing the squared estimation error, averaged both with respect to model errors and the noise. A polynomial equations approach, based on averaged spectral factorizations and averaged Diophantine equations, is derived. Mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators. The robust design turns out to be no more complicated than the design of an ordinary Wiener filter or LQG regulator. The proposed approach avoids two drawbacks of robust minimax design. First, probabilistic descriptions of model uncertainties may have soft bounds. These are more readily obtainable in a noisy environment than the hard bounds required for minimax design. Furthermore, not only the range of uncertainties...

This paper presents a consistent framework for the quantification of noise and undermodelling errors in transfer function model estimation. We use the, so-called, "stochastic embedding" approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodeling as a multiplicative error characterised by random walk processes in the frequency domain. The benefit of the present formulation is that it significantly simplifies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantification. An example illustrates how the random walk effectively captures typical cases of undermodelling found in practice.

Design equations are presented for robust and realizable decision feedback equalizers, for IIR channels with coloured noise. Given a probabilistic measure of model uncertainty, the mean MSE, averaged over the whole class of possible models, is minimized. A robustification parameter, which trades off error propagation against theoretical performance, is also introduced. The resulting design equations define a large class of equalizers, with DFE's and linear equalizers based on nominal models being special cases. If data sequences fd(n)g are transmitted in the presence of intersymbol interference, they have to be reconstructed from the received sequences fy(n)g. Equalizers compute estimates ¯ d(n) on a symbol by symbol basis. Their main advantage, compared to the MLSE Viterbi detector, is a low computational complexity. If channels are slowly time-- varying, filter coefficients can be adjusted during known training sequences, and held fixed until the next training. For fast time--variat...

Model Validation is at the heart of the System Identification process. Recently, much renewed interest has been expressed in so called "identification for control". This means that the design variables associated with the identification process are tailored to achieve models that are well suited for control design purposes. A separate, but closely related issue is to devise validation tests that give information about the model's quality and suitability for control design. This paper shows and discusses how a basic and classical residual test gives such information. ------------------------------------------------------------------------------- + Department of Electrical Engineering, Linkoping University, Linkoping, S-58183, Sweden. Email: Ljung@isy.liu.se, Fax: (+46)13 282622. ++ Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100080, P.R.China. Email: Lguo@iss03.iss.ac.cn, Fax: (86-10)2587343. 1 Introduction "Identification for Control" has since long been of m...

This paper addresses the problem of providing bounds on estimated plant frequency responses in a form suitable for robust control design. Our approach is to consider the undermodelling as a particular realisation of a random variable and to derive bounds based on averages over all possible noise realisations and over all possible undermodelling realisations. We critically examine the performance of these bounds relative to those that would be obtained by fitting a high order model to the data and then truncating to a low order model. We also show that the parameters in the distribution for the undermodelling can be estimated from the data analagously to the way measurement noise variance is estimated from prediction errors. We propose several new estimators and examine their finite data and asymptotic properties. 1 Introduction It is usually assumed in robust control design that a nominal model is available with appropriate uncertainty bounds. This paper addresses the issue of how th...

In this paper we consider situations in which model based analytical redundancy is to be used to detect faults via a Neyman-Pearson approach to hypothesis testing. In [3] we examined how the statistical power of the resultant test statistic could be improved via the use of reduced order models. Here we extend the work to cover the effect of static non-linearities by using the stochastic embedding approach of [8], [7], [14], [16] and [5]. We complete the paper by showing how the proposed algorithm can be implemented recursively for on line applications and present some simulations examples to illustrate the superiority of the new algorithm over more conventional techniques. 1 Introduction The detection of changes in the dynamics of systems from noisy measurements has been well studied and is comprehensively documented in the survey chapters and books [19],[10], [1] and [2]. Two main approaches that have sprung from this work are Kalman Filter based approaches introduced by Mehra and Pe...

A new approach to robust estimation of signals and prediction of time-- series is considered. Possible modelling errors are described by sets of systems, parametrized by random variables, with known covariances. A robust design is obtained by minimizing the squared estimation error, averaged both with respect to model errors and noise. A polynomial solution, based on averaged spectral factorizations and averaged Diophantine equations, is derived. The robust estimator is called a cautious Wiener filter. It turns out to be no more complicated to design than an ordinary Wiener filter. The methodology can be applied to any open loop filtering or control problem. Systems and Control Group, Uppsala University, P. O. Box 27 S-751 03 Uppsala, Sweden. The work has been partially supported by the Swedish Board for Technical Developement, under contract 8701573 and by the Swedish Institute. It was carried out while Anders Ahl'en was on leave at The Department of Electrical Engineering and Compu...