James R. McCusker

Parameter Estimation by Parameter Signature Isolation in the Time-Scale Domain

It is shown that output sensitivities of dynamic models can be better delineated in the time-scale domain. This enhanced delineation provides the capacity to isolate regions of the time-scale plane, coined as parameter signatures, wherein individual output sensitivities dominate the others. Due to this dominance, the prediction error can be attributed to the error of a single parameter at each parameter signature so as to enable estimation of each model parameter error separately. As a test of fidelity, the estimated parameter errors are evaluated in iterative parameter estimation in this paper. The proposed parameter signature isolation method (PARSIM) that uses the parameter error estimates for parameter estimation is shown to have an estimation precision comparable to that of the Gauss―Newton method. The transparency afforded by the parameter signatures, however, extends PARSIMs features beyond rudimentary parameter estimation. One such potential feature is noise suppression by discounting the parameter error estimates obtained in the finer-scale (higher-frequency) regions of the time-scale plane. Another is the capacity to assess the observability of each output through the quality of parameter signatures it provides.

Improved parameter estimation by noise compensation in the time-scale domain

It was shown recently that parameter estimation can be performed directly in the time-scale domain by isolating regions wherein the prediction error can be attributed to the error of individual dynamic model parameters [1]. Based on these single-parameter equations of the prediction error, individual model parameters error can be estimated for iterative parameter estimation. An added benefit of this parameter estimation method, besides its unique convergence characteristics, is the added capacity for direct noise compensation in the time-scale domain. This paper explores this benefit by introducing a noise compensation method that estimates the distortion by noise of the prediction error in the time-scale domain and incorporates that as a confidence factor to bias the estimation of individual parameters error. This method is shown to improve the precision of the estimated parameters when the confidence factors accurately represent the noise distortion of the prediction error.

Measurement Selection for Engine Transients by Parameter Signatures

A method of measurement selection is introduced that relies on parameter signatures to assess the identifiability of dynamic model parameters by different outputs. A parameter signature is a region in the time-scale plane wherein the sensitivity of the output with respect to one model parameter is much larger than the rest of the output sensitivities. Since a parameter signature can be extracted when the corresponding output sensitivity is independent of the others, the ability to extract parameter signatures is indicative of parameter identifiability by the output and used here for output/measurement selection. The purpose of this paper is to introduce a strategy for measurement selection by parameter signatures and to demonstrate its applicability to the transient decks of turbojet engines. The validity of the selected outputs in providing observability to all the engine model parameters is independently verified by successful estimation of parameters by nonlinear least-squares estimation.

Improved Parameter Estimation by Noise Compensation in the Time-Scale Domain

It was shown recently that parameter estimation can be performed directly in the time-scale domain by isolating regions wherein the prediction error can be attributed to the error of individual dynamic model parameters [1]. Based on these single-parameter attributions of the prediction error, individual parameter errors can be estimated for iterative parameter estimation. A benefit of relying entirely on the time-scale domain for parameter estimation is the added capacity for noise suppression. This paper explores this benefit by introducing a noise compensation method that estimates the distortion by noise of the prediction error in the time-scale domain and incorporates it as a confidence factor when estimating individual parameter errors. This method is shown to further improve the estimated parameters beyond the time-filtering and denoising techniques developed for time-based estimation.Copyright

Integrating Parameter Estimation Solutions From the Time and Time-Scale Domains

James R. McCusker , K our osh Danai Depar tment of Mechanical and Industr ial Engineer ingUniv ersity of Massachusetts AmherstAmherst, Massachusetts 01003ABSTRA CTA method of parameter estimation w as recently intro-duced that separately estimates each parameter of the dynamicmodel [1]. In this method, regions coined as par ameter signa-tur es ,are identied in the time-scale domain wherein the predic-tion error can be attrib uted to the error of a single model param-eter . Based on these single-parameter associations, indi vidualmodel parameters can then be estimated for iterative estimation.Relati ve to nonlinear least squares, the proposed P ar ameter Sig-natur e Isolation Method (P ARSIM) has tw o distinct attrib utes.One attrib ute of PARSIM is to lea ve the estimation of a parame-ter dormant when a parameter signature cannot be extracted forit. Another attrib ute is independence from the contour of theprediction error . The rst attrib ute could cause erroneous param-eter estimates, when the parameters are not adapted continually .The second attrib ute, on the other hand, can pro vide a safe guardagainst local minima entrapments. These attrib utes moti vate in-tegrating PARSIM with a method, like nonlinear least-squares,that is less prone to dormanc y of parameter estimates. The pa-per demonstrates the merit of the proposed inte grated approachin application to a difcult estimation problem.1 INTR ODUCTIONMost natural systems exhibit dynamic characteristics.Therefore, dynamic models are the natural frame w ork for rep-resenting most systems. Dynamic models which generally com-prise a set of ordinary or partial differential equations are oftenconstructed according to the rst-principles or empirical kno wl-edge of the system. Once the qualitati ve delity of the model hasbeen ensured, the model parameters are estimated to impro ve the

Iterative Controller Tuning in the Time-Scale Domain

A novel method of controller tuning is introduced to achieve a desired closed-loop response. It uses the same strategy as Iterative Feedback Tuning (IFT), but instead of relying on a scalar cost function of the performance error between the desired response and system response, it utilizes the expanded version of the performance error in the time-scale domain for estimating the suitable controller parameters. The proposed method relies on the enhanced delineation of output sensitivities in the time-scale domain to identify regions in the time-scale domain wherein the performance error can be attributed to individual controller parameters [1]. It then relies on the error association in each region to estimate the corresponding controller parameter. It is shown that given a realistic desired response for the closed-loop system, the proposed method can lead to satisfactory controller parameters. It is also shown that the results from this method can be integrated with those from IFT to represent the best of the two solutions from the time and time-scale domains.Copyright

Selection of Outputs for Gas-Turbine Engines by Parameter Signatures

A method of measurement selection is introduced that relies on parameter signatures to assess the identifiability of dynamic model parameters by different outputs. A parameter signature is a region in the time-scale plane wherein the sensitivity of the output with respect to one model parameter is much larger than the rest of output sensitivities. Since the existence of a parameter signature is synonymous with the uniqueness of the corresponding output sensitivity, parameter signatures are directly linked to parameter identifiability by outputs and, hence, can be used for output/measurement selection. The purpose of this paper is to introduce a strategy for measurement selection by parameter signatures and to demonstrate its applicability to the transient decks of turbo-jet engines. The validity of the selected outputs in providing observability to the engine model parameters is independently verified by successful estimation of parameters by nonlinear least-squares.Copyright

Validation of Dynamic Models by Image Distances in the Time-Scale Domain

Model validation is the procedure whereby the fidelity of a model is evaluated. The traditional approaches to dynamic model validation either rely on the magnitude of the prediction error between the process observations and model outputs or consider the observations and model outputs as time series and use their similarity to assess the closeness of the model to the process. Here, we propose transforming these time series into the time-scale domain, to enhance their delineation, and using image distances between these transformed time series to assess the closeness of the model to the process. It is shown that the image distances provide a more consistent measure of model closeness than available from the magnitude of the prediction error.Copyright

Mutual Identifiability Analysis for Parameter Signature Isolation in the Time-Scale Plane

It was shown recently that regions in the time-scale plane can be isolated wherein the prediction error can be attributed to the error of an individual model parameter. A necessary condition for this isolation capacity is the mutual (pairwise) identifiability of the model parameters. This paper presents conditions for mutual identifiability of parameters of linear models and refines these conditions for models that exhibit rank-1 dependency on the parameters.Copyright

Parameter Adaptation by Parameter Signature Isolation in the Time-Scale Domain

It is shown that delineation of output sensitivities with respect to model parameters in dynamic models can be enhanced in the time-scale domain. This enhanced differentiation of output sensitivities then provides the capacity to isolate regions of the time-scale plane wherein a single output sensitivity dominates the others. Due to this dominance, the prediction error can be attributed to the error of a single parameter at these regions so as to estimate each model parameter error separately. The proposed Parameter Signature Isolation Method (PARSIM) that uses these parameter error estimates for parameter adaptation has been found to have an adaptation precision comparable to that of the Gauss-Newton method for noise-free cases. PARSIM, however, appears to be less sensitive to input conditions, while offering the promise of more effective noise suppression by the capabilities available in the time-scale domain.Copyright