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

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.

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.

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.

In this paper, the results on the parameterization of all the unfalsified models for single-input/single-output systems are extended to multi-input/multi-output systems. In this arameterization, the assumed a priori plant and noise information are upper bounds of the Hx-norm of the plant matrix-valued transfer function and the noise magnitude. The parameterization is on the basis of several series of plant time-domain identification experiment data. It is shown that for multi-input/multi-output systems, all the unfalsified plant matrix-valued transfer functions can still be expressed by the linear fractional transformation of a fixed matrix-valued transfer function and an Hx-norm bounded structure fixed, but uncertain matrix-valued transfer function. However, the high dimensions of the matrix-valued transfer functions involved in the parameterization hamper the direct application of these results to robust controller design. To overcome these difficulties, further efforts are needed.

This paper investigates estimating the frequency response of the normalized coprime factors of a possibly unstable transfer function from closed-loop frequency domain experimental data. A stochastic framework has been adopted. The dual Youla

In this paper, we investigate the structure of the transfer function set which includes all eliminate the transfer functions deduced from the plant available information. It is shown that when an upper bound of the plant transfer function's H∞-norm has been supplied, and the noise contaminating the time domain identification experiment data is not too significant, such a transfer function set can be parameterized by a linear fractional transformation of two transfer function matrices. One of them is a fixed transfer function matrix which is completely determined by the plant available information and the noise magnitude. The other is a norm bounded, structure fixed, free transfer function matrix. Moreover, it is shown that the problem of analytically obtaining the fixed complexity nominal model that best approximates this transfer function set is as difficult as the μ-synthesis problem.

The problem of parameterizing all unfalsified plant models from closed-loop time-domain dentification experimental data is investigated in this paper. The assumed a priori plant nformation is an upper bound of the Lx norm and the number of the unstable poles of its transfer function. Moreover, it is assumed that magnitude bounds of the disturbances that perturb the closed-loop system have been supplied. Under the condition that the stabilizing controller is known, it is shown that all the unfalsified plant transfer functions can be parameterized by a linear fractional transformation of a fixed transfer-function matrix and an Lx-norm-bounded, structure-fixed but uncertain transfer-function matrix. These results are very similar to those when plant open-loop time-domain identification experimental data are supplied. However, further efforts are still needed in order to apply these results to robust controller design. This results mainly from the computational omplexity and the high dimensions of the transfer-function matrices involved in the arameterization.

In this paper, an algorithm is proposed for the identification of fractal signals with wavelet-based models from corrupted data. This algorithm is based on curve fitting and wavelet transformation. It is proved that whenever is greater than 0, the algorithm provides an almost consistent estimation. oreover, the estimated parameters are asymptotically Gaussian distributed. A mean-square asymptotic convergence rate of the estimated parameters has also been established. Simulation results verify the efficiency of the proposed algorithm.