The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalised Fourier series and the asymptotic properties of Toeplitz-like matrices in such a way that the new variance expressions presented here encompass pre-existing ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, or...

This paper is concerned with the frequency domain quantification of noise induced errors in dynamic system estimates. Preceding seminal work on this problem provides general expressions that are approximations whose accuracy increases with observed data length and model order. In the interests of improved accuracy, this paper provides new expressions whose accuracy depends only on data length.

In this paper we consider a three-step procedure for identification of timeseries, based on covariance extension and modelreduction, and we present a complete analysis of its statistical convergence properties. A partial covariance sequence is estimated from statistical data. Then a high-order maximum-entropy model is determined, which is finally approximated by a lower-order model by stochastically balanced model reduction. Such procedures have been studied before, in various combinations, but an overall convergence analysis comprising all three steps has been lacking. Supposing the data is generated from a true finitedimensional system which is minimumphase, it is shown that the transfer function of the estimated system tends in H # tothe true transfer function as the data length tends to infinity, if the covariance extension and the model reduction is done properly. The proposed identification procedure, and some variations ofit, are evaluated by simulations. 1.

A method is considered for identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain. To this end use is made of the empirical transfer function estimate (ETFE), identified from time-domain data. As a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions, being generalizations of the classical pulse, Laguerre and Kautz types of bases. An asymptotic analysis of the estimated models is provided and conditions for consistency are formulated. Explicit and transparent bias and variance expressions are established, the latter ones also valid in a situation of undermodelling.

Statistical model validation is treated for a class of parametric uncertainty models and also for a more general class of nonparametric uncertainty models. We show that, in many cases of interest, this problem reduces to computing relative weighted volumes of convex sets in R N (where N is the number of uncertain parameters) for parametric uncertainty models, and to computing the limit of a sequence (Vk ) 1 1 of relative weighted volumes of convex sets in R k for nonparametric uncertainty models. We then present and discuss a randomized algorithm based on gas kinetics for probable approximate computation of these volumes. We also review the existing Hit-and-Run family of algorithms for this purpose. Finally, we introduce the notion of testability to describe uncertainty models that can be statistically validated with arbitrary reliability using input-output data records of sufficient (finite) length. It is then shown that some common nonparametric uncertainty models, such as thos...

In this paper we formulate a particular statistical model validation problem in which we wish to determine the probability that a certain hypothesized parametric uncertainty model is consistent with a given input-output data record. Using a Bayesian approach and ideas from the field of hypothesis testing, we show that in many cases of interest this problem reduces to computing relative weighted volumes of convex sets in R N (where N is the number of uncertain parameters). We also present and discuss a randomized algorithm based on gas kinetics, as well as the existing Hit-and-Run family of algorithms, for probable approximate computation of these volumes. 1 Introduction Motivated by the desire to produce identified models that are compatible with modern robust control design methodologies, many researchers have recently been working in the area of control-oriented system identification (see for example [5, 6, 7, 11, 12, 16, 17, 19, 23, 24, 25, 27, 28] and the references cited therein...

. Statistical model validation is treated for a general class of nonparametric uncertainty models. This problem is shown to reduce, in many cases of interest, to computing the limit of a sequence (V k ) 1 1 of relative weighted volumes of convex sets in R k . An associated decision problem is shown to reduce to a pair of likelihood ratio tests. The notion of testability is introduced to describe uncertainty models that can be statistically validated with arbitrary reliability using input-output data records of sufficient (finite) length. It is then shown that some common uncertainty models, such as those involving ` 1 or H1 norms, do not possess this property. Keywords. Decision theory; hypothesis testing; model validation; probability; robust control; system identification. 1 INTRODUCTION Motivated by the desire to produce identified models that are compatible with modern robust control design methodologies, many researchers have recently been doing work in the area of control-o...

This paper investigates the accuracy of the linear component that forms part of an overall Hammerstein model-structure estimate, and a key finding is that the process of estimating the non-linear element can have a strong effect on the associated estimate of the linear dynamics. Furthermore, this effect is not explained simply by way of considering how the input spectrum is changed by the non-linearity. Instead, it arises that the linear model-estimate variability may be dominated by a term that depends on the frequency response of the linear system itself. Amongst other things, the main results derived here have experiment design implications for Hammerstein system estimation. Technical Report EE9933, Department of Electrical and Computer Engineering, University of Newcastle, AUSTRALIA 1

This paper establishes that when using a least squares criterion to estimate an output error type model structure, then the measurement noise induced variability of the frequency response estimate depends on the estimated (and hence also on the true) pole positions. This dependence on pole position is perhaps counter to prevailing wisdom that for any `shift invariant' model structure, the variability depends only on model order, data length, and input and noise spectral densities. That is, it is counter to the belief that variance error is model-structure independent.

from Z N to a space of models: GN = A(Z N ) (4) This algorithm is then said to be robustly convergent (Helmicki et al., 1991) if lim ffi!0 lim N!1 jj GN \Gamma G 0 jj = 0 (5) jv(t)j ffi 8t (6) for all G 0 2 G. The algorithm is "untuned" if it does not use knowledge of either G or ffi. In (5) jj \Delta jj denotes a suitable norm, for linear systems, it is typically chosen as the H1 - norm (sup ! j G t (e i! ) \Gamma G 0 (e i!