On the computation and statistical properties of the MLE for the frequency response of NCFs

Abstract

This paper deals with computations and statistical properties, as well as transfer function selections involved in frequency response estimation for normalized coprime factors (NCF). A simple expression and the probability density function (PDF) are derived for the maximum likelihood estimate (MLE). It is shown that the statistical characteristics of the MLE are independent of the auxiliary plant. Based on this observation, all the coprime factors of the auxiliary transfer function have been derived which lead to an almost unbiased estimate when the noise level is low. Moreover, an analytic expression has been established for the involved free transfer functions which minimize both the bias and the variance of the estimate. The theoretical conclusions have been verified through numerical simulations.

Frequency Response Estimation for NCFs of an MIMO Plant From Closed-Loop Time-Domain Experimental Data

“…Compared with the frequency-domain experimental data based estimation problem discussed in [16], [17], there are no assumptions in the problem formulation on the statistical properties of the external disturbances and the measurement errors. It will become clear afterwards (Theorems 1 and 3) that a consistent estimate can be derived under some weak conditions.…”

“…Moreover, let denote a zero of with multiplicity , then, it is obvious that . 1 In this paper, the Z-transform of a sequence a j , denote it by a(z), is defined as a(z) = a z , which is different from that of [16], [17] but widely adopted in control theory. However, the parameterization of [17] is still valid after introducing a transformation z !…”

“…Although model sets with NCF perturbations (NCFPMS) are well utilized in robust control theory, it is not very easy to identify them from experimental data. While great advances have been made, respectively, in [3], [6], and [12] toward NCFPMS estimation, practical applications of the suggested algorithms can still be hardly regarded as very convenient [16]- [18]. An essential issue here seems to be the lack of mathematically appreciable relations between NCFs and physically measurable variables [9], [6], [3], [12].…”

“…Its detailed definition is therefore not included, but can be found, for example, in [5], [9], [6], and [16].…”

“…An essential issue here seems to be the lack of mathematically appreciable relations between NCFs and physically measurable variables [9], [6], [3], [12]. To solve this identification problem, it is suggested in [16]- [18] to divide it into two steps. At first, frequency response is estimated for NCFs.…”