This paper investigates the consistency between frequency domain closed-loop experimental data and a model set with linear fractional unstructured uncertainties.When the number of unstable poles of the transfer function matrices (TFM) in the model set are identical, a necessary and sufficient condition is derived for the reproduction of experimental data. This condition is expressed as the non-positiveness of a quadratic form of the external disturbances and measurement errors, together with a linear constraint.In this derivation, it is not required that all the TFMs in the model set are internally stabilizable by the controller.The condition is also not necessary for the closed-loop system consisting of the nominal model and the controller to be internally stable.It is believed that this condition is theoretically and computationally attractive in model set validation, no matter whether a deterministic or stochastic framework is adopted.

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. 1. Introd

This paper deals with probabilistic model set validation. It is assumed that the dynamics of a multi-input multi-output (MIMO) plant is described by a model set with unstructured uncertainties, and identi4cation experiments are performed in closed loop. A necessary and su6cient condition has been derived for the consistency of the model set with both the stabilizing controller and closed-loop frequency domain experimental data (FDED). In this condition, only the Euclidean norm of a complex vector is involved, and this complex vector depends linearly on both the disturbances and the measurement errors. Based on this condition, an analytic formula has been derived for the sample unfalsi4ed probability (SUP) of the model set. Some of the asymptotic statistical properties of the SUP have also been briefly discussed. A numerical example is included to illustrate the e6ciency of the suggested method in model set quality evaluation.

Quality assessment is investigated under a probabilistic framework in this note for a prescribed model set. The results on unfalsified probability estimation are extended from additive modeling errors to normalized coprime factor perturbations. An analytic formula has been derived for the sample unfalsified probability. It is shown that with increasing the data length, the sample unfalsified probability converges in probability to a number which is independent of experimental data. Numerical simulations show that the proposed sample unfalsified probability is appropriate in the evaluation of the quality of a model set.

The main objective of this technical note is to derive a simple necessary and sufficient condition for a linear fractional transformation (LFT) perturbed model set being consistent with frequency domain plant input-output data. Only discrete-time models and unstructured modeling errors are dealt with. Compared with the available results in which the eigenvalues of a matrix are involved, this condition is related only to the Euclidean norms of two vectors. Moreover, these vectors linearly depend on measurement errors. Some of its applications to model set validation have been briefly discussed. Based on this condition, an almost analytic solution has been established for model set validation under a deterministic framework when the measurement errors are energy bounded. Numerical simulations show that this consistency condition can lead to a significant computation cost reduction.