Kautz Models Research Articles

Efficient Gram-matrix computation for irrational resonant systems using Kautz models

We present a new method for evaluating the Gram matrix containing the inner products of repeated integrals and derivatives of irrational transfer functions via a Kautz model. The technique is particularly interesting for describing resonant systems and it has an immediate application in model order reduction.

Least-squares estimation of a class of frequency functions: A finite sample variance expression

A new expression for the variance of scalar frequency functions estimated using the least-squares method is presented. The expression is valid for finite sample size and for a class of model structures, which includes finite impulse response, Laguerre and Kautz models, when the number of estimated parameters coincides with the number of excitation frequencies of the input. The expression gives direct insight into how excitation frequencies and amplitudes affect the accuracy of frequency function estimates. With the help of this expression, a severe sensitivity of the accuracy with respect to the excitation frequencies is exposed. The relevance of the expression when more excitation frequencies are used is also discussed.

Optimal Pole Conditions for Laguerre and Two-Parameter Kautz Models: A Survey of Known Results

In this paper we present the stationarity conditions for Laguerre and two-parameter Kautz models under a plethora of different conditions, viz., any combination ofi) continuous-time or discrete-time modelsii) impulsive (or white-noise) input signals or arbitrary input signalsiii) a time- or frequency-domain performance criterion based on an Lp, lp or an Hp norm, 1 < p < ∞iv) existence, or not, of interpolation constraints in the response of the model to known signals, in the time- and/or frequency domainsv) utilization, or not, of a predefined set of fixed poles in the model.For Laguerre models, in all possible cases the optimality conditions take the same form, namely, either the last optimal weight of the model vanishes or the last optimal weight of the model of the next higher order vanishes. For two-parameter Kautz models similar conditions hold, with the sole difference that the last two weights of the model vanish, instead of only a single weight. Unfortunately, these conditions are also satisfied in other stationarity points of the performance criterion.

Stable worst-case iterative identification and control redesign via unfalsification

This work is a further development of a recently introduced iterative identification and control redesign scheme by Veres which is based on the concept of model unfalsification. In this scheme convergence of the iterations and attainment of nearly optimal worst-case control performance is ensured under the a priori given set of controller structures. This paper makes two contributions: proposes the use of Laguerre and Kautz models and introduces caution into the iterative steps of this scheme similarly as it was done by Anderson et al. An advantage will be that structure selection of the models will be based on control related criteria, hence model structure selection will be intrinsic to the iterative scheme. Copyright © 1999 John Wiley & Sons Ltd.

Stationarity conditions for the L2 error surface of the generalized orthonormal basis functions lattice filter

Recently, a novel model to perform system identification, based on the so-called generalized orthonormal basis functions, appeared in the automatic control literature. This model generalizes the Laguerre and “two-parameter” Kautz models (which are IIR models with a restricted structure), and has some remarkable properties. Among them is the fact that under ideal conditions the correlation matrix of its internal signals has a block Toeplitz structure. In this paper we explore this property of the correlation matrix, with the result that a lattice version of the aforementioned model is uncovered. This lattice model is then used to determine under which conditions the model's mean-squared-error has a stationary point with respect to the position of its poles, assuming that there is no external feedback between the model's output and input. The results of this study generalize known similar results for Laguerre and “two-parameter” Kautz filters. As a by-product of this study the Stationarity conditions for the error surface of ARMA( m, m − 1) filters with respect to their pole positions, in an output error configuration and for an arbitrary input signal, are obtained.

Globally convergent adaptive IIR filters based on fixed pole locations

A new class of adaptive filters, dubbed fixed pole adaptive filters (FPAF's), is introduced. These adaptive filters have infinite impulse responses, yet their adaptation exhibits provable global convergence. Good filter performance with a relatively small number of adapted parameters is permitted by the new filter structure, thus reducing the computational burden needed to implement adaptive filters. The implementation and computational complexity of the FPAF is described, and its modeling capabilities are determined. Excitation conditions on the filter input are established that guarantee global convergence of a standard set of adaptive algorithms. Some methods are described for selecting the fixed pole locations based on a priori information regarding the operating environment of the adaptive filter. The FPAF is tailored to applications by such a procedure, enabling improved performance. In examples, the FPAF is shown to achieve a smaller minimum mean square error, given an equal number of adapted parameters, in comparison with adaptive FIR filters and adaptive filters based on Laguerre and Kautz models.

Robust identification in the disc algebra using rational wavelets and orthonormal basis functions

Worst-case identification using corrupted frequency response measurements is considered in a framework which allows one to consider a variety of rational basis functions; these include Laguerre models, Kautz models and rational wavelets. Conditions are given for model sets to be complete in the disc algebra and examples are given. Explicit error bounds are provided for approximate modelling using specific basis functions.

On approximation of stable linear dynamical systems using Laguerre and Kautz functions

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the “two-parameter” Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformai mapping techniques the “two-parameter” Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on þ 2 and þ ∞ approximation using Kautz models are given. Furthermore, the weighted L 2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation.

Simultaneous estimation of parameters for different orders of discrete orthonormal function models

The augmented UD identification (AUDI) method is used to simultaneously estimate parameters of all 1 to Nth-order discrete orthonormal function models in one computational step. This method is tested on different types of orthonormal functions such as the Laguerre, Kautz, FIR, and Markov-Laguerre models.

Applications of kautz models in system identification : P. Lindskog, B. Wahlberg, pp 41–44

Applications of kautz models in system identification : P. Lindskog, B. Wahlberg, pp 41–44

Selecting Pole Locations for System Identifiers with Fixed Poles

Model sets consisting of rational transfer functions with fixed pole locations are considered for system identification. Such model sets are attractive because the identifier is linear in the parameters, while the flexibility in selecting pole locations provides modeling capabilities better than those of finite impulse response, Laguerre, or related models. Procedures are presented for selecting the fixed pole locations, using a priori information about the system to be identified. Some examples show that this identifier achieves more accurate identification, given an equal number of adjusted parameters, in comparison with finite impulse response, Laguerre, and Kautz models.

Identification of Lightly Damped Systems using Frequency Domain Techniques

In this paper we present recent work on a sequential algorithm which combines the two stage nonlinear algorithm of Gu and Khargonekar (1992a) with Kautz models (Wahlberg, 1991).

System identification using Kautz models

In this paper, the problem of approximating a linear time-invariant stable system by a finite weighted sum of given exponentials is considered. System identification schemes using Laguerre models are extended and generalized to Kautz models, which correspond to representations using several different possible complex exponentials. In particular, linear regression methods to estimate this sort of model from measured data are analyzed. The advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters. The subsequent analysis is based on the result that the corresponding linear regression normal equations have a block Toeplitz structure. Several results on transfer function estimation are extended to discrete Kautz models, for example, asymptotic frequency domain variance expressions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Applications of Kautz Models in System Identification

FIR, ARX or AR model structures can be used to describe many industrial processes. Simple linear regression techniques can be applied to estimate such models from experimental data. However, for low signal to noise ratios in combination with transfer function poles and noise model zeros close to the unit circle, a large number of model parameters are needed to generate adequate models. The Kautz model structure generalizes FIR, ARX and AR models. By using a priori knowledge about the dominating time constants and damping factors of the system, the model complexity is reduced, and the linear regression structure is retained. The objective of this contribution is to study an industrial example, where Kautz models have distinct advantages. The data investigated corresponds to aircraft flight flutter, which is a state when an aircraft component starts to oscillate.

Estimation of Noise Models By Means of Discrete Laguerre/Kautz Filters

Autoregressive (AR) modelling is generalized by replacing the delay operator by discrete Laguerre or Kautz filters. The motivation is to reduce the number of parameters needed to obtain useful approximate models of stochastic processes, without increasing the computational complexity. The key observation is that Tooplitz strucure of covariance matrix of the regression vector (normal equations) for AR estimation also holds for Laguerre/ Kautz models. This result enables several AR estimation results to be generalized to the Laguerre/Kautz case. The potential of Kautz models for modelling of narrow-band signal is discussed. The Laguerre/Kautz estimation technique is illustrated by simple examples.