Parametric identification of transfer functions in the frequencydomain-a survey

This paper gives a survey of frequency domain identification methods for rational transfer functions in the Laplace (s) or z-domain. The interrelations between the different approaches are highlighted through a study of the (equivalent) cost functions. The properties of the various estimators are discussed and illustrated by several examples     

Multidimensional BIBO stability and Jury’s conjecture

Twenty years ago E. I. Jury conjectured by analogy to the case of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function has the form H = 1/P where P is a very strict Hurwitz polynomial (VSHP). In more detail he conjectured that the impulse response of the filter is an absolutely integrable function. However, he did not specify the exact equations of these filters and did not prove the existence of the impulse response. In the present paper we generalise Jury’s conjecture to arbitrary proper transfer functions H = Q/P where P is a bivariate VSHP and prove this generalisation. In particular, we show the existence of a suitable impulse response or fundamental solution for any multivariate proper rational function. However, this impulse response is a measure and not a function. We have not succeeded to prove an analogue of Jury’s conjecture in higher dimensions than two yet, but we propose a new conjecture in context with the robustly stable multivariate polynomials investigated by Kharitonov et. al. For the discrete case we prove that the structurally stable rational functions after Bose, Lin et al. coincide with the stable rational functions discussed in context with the stabilisation of discrete input/output systems. These rational functions are BIBO stable, but the converse is not true as established by several authors. Financial support of M. Scheicher through the Austrian FWF-project P18974 is gratefully acknowledged.     

A minimal canonical form realization for a multivariable econometric system

A minimal realization theory for a multivariable system is developed based on the Hankel matrices, which is closely related to the method given by Silverman (1971). The procedure provided here yields a canonical form dynamical equation. The results obtained are applied to a multivariable econometric system. The proposition of realizability and existence of a minimal realization is demonstrated, which contains both proper and strictly proper systems. Thus the correspondence between minimal realization and canonical form is clarified.     

A z -domain transfer function solution to the non-minimum phase acoustic beamformer

The two input Griffiths–Jim acoustic beamformer is analysed in the frequency domain using a causal Wiener z-transfer function formulation. Unlike previous solutions the approach here examines the particular problem of non-minimum phase acoustic transfer functions which are encountered in many real filtering problems. The polynomial transfer function approach gives an elegant way of obtaining the frequency response of the beamformer. The effect of signal leakage is also considered and a general polynomial solution is given.     

WeightedH 2 approximation of transfer functions

The aim of this work is to generalize to the weighted case some results and algorithms concerningL ² approximation by analytic and rational functions which are useful when performing the identification of unknow transfer functions of a class of stable (linear causal time-invariant) systems from incomplete frequency data.     

离散事件系统广义分式方法

本文在Z-域上考虑一类离散事件系统,引入极大代数上的有理式作为传递函数的紧致表达式,将有理式展开成广义分式进行系统分析与设计,讨论了广义分式的约简与标准形;研究了有理传递函数的谱特征与周期响应,同时给出了最优控制等应用实例。     

Tangent-phase continued-fraction expansion for stable reduced models of linear discrete-time systems

The tangent-phase continued-fraction expansion for stable reduced models is based on the tangent-phase frequency response to the expansion and the factorization technique to obtain reduced models. In this paper, we propose a new procedure for deriving stable and minimum-phase reduced z-transfer functions by the tangent-phase continued-fraction expansion. The procedure is based on transforming the z-domain tangent-phase function to the p domain, where p = z + z^?1?2, and then expanding the p-domain tangent-phase function into a modified continued fraction. An example is given to illustrate the utility of the procedure.     

H∞ optimization with plant uncertainty and semidefinite programming

The fundamental H^∞ problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H^∞ problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H^∞ optimization problem of control can be naturally treated with modern primal–dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H^∞ optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H^∞ optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. © 1998 John Wiley & Sons, Ltd.     

Neural networks,rational functions,and realization theory

The problem of parametrizing single hidden layer scalar neural networks with continuous activation functions is investigated. A connection is drawn between realization theory for linear dynamical systems, rational functions, and neural networks that appears to be new. A result of this connection is a general parametrization of such neural networks in terms of strictly proper rational functions. Some existence and uniqueness results are derived. Jordan decompositions are developed, which show how the general form can be expressed in terms of a sum of canonical second order sections. The parametrization may be useful for studying learning algorithms.This work was supported by the Australian Research Council, the Australian Telecommunications and Electronics Research Board, and the Boeing Commencai Aircraft Company (thanks to John Moore).     

On robust asymptotic tracking: perturbations on coprime factors andparameterization of all solutions

The robust asymptotic tracking problem is analyzed in this paper relative to unstructured perturbations on each coprime factor of the transfer functions from the control input to the measured and controlled outputs. In each case, necessary and sufficient conditions for the existence of solutions are presented which are explicitly given in terms of problem data. Under such conditions, explicit parameterizations are given of all controllers which achieve robust asymptotic tracking, in terms of free, rational proper, and stable matrices     

Realization and generalized frequency response for non-linear input-output maps

A realization theory is developed and the ‘generalized frequency response’ is presented for an input-output map introduced in a previous work. Relations with the realization of finite Volterra series are outlined. A simple example is given to show that the new representations have improved convergence properties.     

State space realizations of rational interpolants with prescribed poles

We give a generic algorithm for computing rational interpolants with prescribed poles. The resulting rational function is expressed in the so-called Newton form. State space realizations for this expression of rational functions are given. Our main tool for finding state space realizations is Fuhrmann's shift realization theory from which we obtain concrete realizations by introducing suitable bases of the state space and expressing the abstract operators with respect to these bases in matrix form.     

On the structure of a class of nonlinear systems

Nonlinear systems composed of linear dynamic sub-systems in cascade with static power nonlinearities are considered. A structure theory is developed based on the representation of input/output behavior by rational functions in several variables. An algorithm is given which solves the minimal complete realization problem for two forms of the representation. Examples are worked in detail to illustrate the results.     

A time-domain approach to z-domain model identification

A time-domain parameter-optimization approach to discrete linear-model identification is presented, z-domain transfer functions are sought with outputs closely matching corresponding outputs of observed systems. Optimal transfer-function coefficients are determined through the minimization of mean-square output error. The general technique developed is applied to problems in model reduction and identification in the presence of measurement noise.     

AR and ARMA identification algorithms of Levinson type: An innovations approach

Fast recursive-in-time identification procedures for both AR and ARMA processes (e.g., Chandrasekhar, square root algorithms,... ) have been available for a few years. These algorithms realize the desired transfer functions in the classical polynomial or rational form. On the other hand, the synthesis of polynomial and rational transfer functions in lattice and ladder form has fostered great interest in network theory by virtue of its pleasant properties. This type of synthesis is strongly related to the theory of orthogonal polynomials on the unit circle. An identification procedure with the realization of the desired whitening filter in lattice form was available for AR processes. We give here a simple approach for obtaining such algorithms, investigating furthermore the connections between the so obtained algorithms and the classical ones (least squares procedure). In the same way, we obtain identification procedures with realization of the desired filter in lattice and ladder form for ARMA processes, together with the connection with the classical extended least squares procedure. The method is based upon a fairly general Levinson orthogonalization lemma in Hilbert space, involving innovation techniques. We extend the method to various other estimation problems. The algorithms we obtain are fast (even somewhat faster than the previous fast ones) and seem to be well conditioned.     

Characterization and computation of the solution to the optimal L∞ approximation problem

The characterization of the solution to the problem of approximating a given stable, proper, rational transfer function, in L _∞ norm sense, by a rational function with prescribed number of stable and unstable modes is developed. A simple state-space suboptimal computational algorithm for the solution is presented     

New results in realization theory for linear time-varying analytic systems

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.     

Barycentric coordinates for polytopes

In Wachspress (1975) [1], theory was developed for constructing rational basis functions for convex polygons and polyhedra. These barycentric coordinates were positive within the elements. Generalization to higher space dimensions is described here. The GADJ algorithm developed by Dasgupta (2003) [5] and in Dasgupta and Wachspress (2008) [6] is crucial for simple construction of rational barycentric basis functions.