Design and Validation of Improved Modal Parameter Estimators

Abstract

Despite several modal parameter estimation techniques had been proposed in the literature, it was found that there is a trade-off between several factors: the computational time, accuracy of the estimated parameters and the user friendliness that is quit important for the non-expert users. So, a major challenge when developing a new modal parameter estimation technique is how to compromise between those three factors. A very popular modal parameter estimator is the polyreference Least-Squares Complex Frequency-domain (pLSCF) estimator, commercially known as PolyMAX estimator. This estimator is computationally fast and always produces very clear stabilization diagrams, even when numerous modes are to be identified with high noise level. Regardless of its positive evaluations, it was found that the accuracy of its estimates, the damping estimates in particular, deteriorates for high noise levels. In addition, this estimator uses a discrete-time model that could introduce some modeling errors. In this dissertation, the research work focuses on designing and validating improved modal parameter estimators, which keep the benefits of the pLSCF (PolyMAX) estimator while give more accurate modal parameter estimates together with their confidence bounds. This dissertation met this research aim by suggesting improvements for the pLSCF (PolyMAX) estimator in three distinct ways. The first part concerns the use of a continuous-time frequency-domain model rather than discrete-time frequency-domain model. This entailed the use of matrix orthogonal polynomials (e.g. Forsythe polynomials) to cure the ill-conditioning problem of the normal equations matrix that happens when identifying continuous-time models with high model orders or wide frequency bands. Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-condition if not handled properly. The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this part, a new approach is proposed to compute the poles and the participation factors directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems. This approach generalized the results (i.e. in the literature) for scalar orthogonal polynomials to multivariable (matrix) orthogonal polynomials that can be used in case of multiple inputs multiple output (MIMO) system. The proposed approach has been compared with two other classical transformation methods, which are based on the back-transformation of the orthogonal polynomial coefficients to the power polynomial basis. The proposed approach outperformed the classical transformation methods in terms of the accuracy of the estimated modal parameters. The outcome of this part is a linear least-squares frequency-domain estimator like PolyMAX estimator but it uses a continuous-time frequency-domain model rather than discrete-time model. The name of this estimator has been abbreviated as pLSF-Orth. In the second part, an improved modal parameter estimator has been proposed. This estimator is a combination between the maximum likelihood estimator (MLE), based on a common denominator rational fraction polynomial model (MLE-CDM), and the pLSCF estimator. This estimator has been called PolyMAX Plus. The PolyMAX Plus approach tries to compromise between the benefits of the maximum likelihood estimator (MLE-CDM), which is slower, yields unclear stabilization diagram but is consistent and efficient, and the pLSCF estimator, which is fast, yields clear stabilization diagram but, in general, is not consistent. This estimator consists of two steps. In the first step, the MLE-CDM is applied to the measured data where the uncertainty on the measurements can be used as weighting functions to improve the quality of the fitting process. Then, the pLSCF estimator is applied to the MLE-CDM synthesized data in a second step to construct a stabilization chart in a fast way. The Main idea behind this new proposed estimator is that: the MLE-CDM can be considered as kind of smoothing of data after which it is expected that pLSCF performs very well, since noise has been removed. In the third part, another improved modal estimation method is proposed, which estimates the modal parameters by identifying the modal model directly instead of identifying a rational fraction polynomial model. The proposed estimator belongs to the class of the Maximum Likelihood Estimators, and the name of this estimator has been abbreviated as ML-MM. Like the PolyMAX Plus estimator, the ML-MM estimator tries to combine the benefits of both the pLSCF estimator (i.e. deterministic approach) and the MLE (i.e. stochastic approach). Unlike the PolyMAX Plus estimator, the ML-MM estimator uses a continuous-time model that could help to decrease the modeling errors. In addition, it delivers the uncertainty bounds on all the estimated modal parameters (i.e. poles, mode shapes, participation factors and lower and upper residual elements) in a direct way without the need to use many complex linearization formulas which are normally used in case of identifying a rational fraction polynomial model. Despite the ML-MM estimator uses a continuous-time model, there was no any remarkable numerical conditioning problem since the used parameterization (i.e. the modal model) is a reduced-order model. In this part of the dissertation, a new model has been proposed to express the effects of the out-of-band modes (i.e. the residual terms). Indeed, the new proposed residual model has been compared with the classical upper and lower residual terms. The results showed that the proposed residual model highly improves the performance of the ML-MM estimator compared to the classical lower and upper residual terms. The application of the proposed modal parameter estimation methods to real-life and simulations case studies gives the following conclusions. The pLSF-Orth is quite slow compared to the pLSCF estimator, and it can achieve improvement over the pLSCF estimator only at a low noise level. When the noise level increase (e.g. > 10% of the measured signal), the pLSF-Orth and the pLSCF estimators have a very comparable performance since the pLSF-Orth is still linear least-squares estimator. The proposed PolyMAX Plus and the ML-MM estimators perform very well in terms of the accuracy of the estimated modal parameters compared with the pLSCF (PolyMAX), polyreference Least-Squares Complex Exponential (pLSCE) estimators and MLE-CDM.

Date of Award	May 2013
Original language	English
Supervisor	Patrick Guillaume (Promotor) & Tim De Troyer (Promotor)