A wavelet-based modal-parameter-extraction procedure has been developed to augment wavelet filtering and thereby produce more-realistic, robust aeroservoelastic-stability margins. The procedure is intended for use in processing data from aircraft flight tests.
Some background information is prerequisite to an explanation of this development. Deterministic nonstationary input test signals are often essential for extracting aeroelastic-stability trends from noisy measurements. The analysis of flight data is improved by discrimination among areas of low signal-to-noise ratio, unmodeled dynamics, and external disturbances. Wavelet signal processing has shown promise for identification of the conceptual structures, operators, and parameters of mathematical models (hereafter called "system identification") of aeroservoelastic systems for these purposes.
Nonparametric wavelet filtering removes aspects of signal responses detrimental to linear system-identification methods to improve stability tracking. Wavelet transforms are also used to directly supply information on time-dependent modal decay rates and phases for estimation of parameters of mathematical models of time-varying systems. Without any approximation of the range of parameters of a system, natural frequencies and damping ratios are extracted from the response of the system. Damping and frequency trends are useful for noting changes in system dynamics as functions of flight conditions.
Model validation is a critical procedure in the computation of robust stability margins. The margins are adversely affected by poor characterizations of uncertainty size and structure, which are determined by the magnitudes of perturbations, locations of perturbations within the system, and the types (real or complex) of perturbations. This completes the background information.
In the present wavelet-based modal-parameter-extraction procedure, both complex, nonparametric and real, parametric perturbations are decreased to generate reduced-norm uncertainty sets, which result in models with less conservatism. The models are used in a robust stability-boundary-prediction method called the "µ method" because it is based on a structured singular value called "µ." [This method was described in "Characterizing Worst-Case Flutter Margins From Flight Data" (DRC-97-03), NASA Tech Briefs, Vol. 21, No. 4 (April 1997), page 62.]
Within the µ conceptual framework, a system is represented as an operator, F(P,Δ), which, in turn, represents a feedback interconnection of a plant P and uncertainty Δ. Flight data can be incorporated into the µ method by formulating a description of uncertainty that accounts for observed variations and errors. A model-validation analysis is performed on the plant model to ensure that the range of dynamics admitted by the uncertainty is sufficient to cover the range of dynamics observed with the flight data.
The µ method can be coupled with wavelet processing for parametric and nonparametric estimation. This coupling is achieved by introducing, into the basic process, several time-frequency operations based on wavelet filtering (see figure). Wavelet transform operations are introduced to process time-domain data, x(t), before computation of a frequency-domain representation, x̂(ω). These operations map the time-domain data into the time-frequency domain via a wavelet transform, then map them back to the time domain via an inverse wavelet transform. A time-frequency filtering operation is performed between the wavelet transform and the inverse wavelet transform to remove unwanted features before the inverse wavelet transform yields a time-domain signal, x̂(t).
A modal-parameter-estimation algorithm is introduced by use of the wavelet algorithm. The estimated parameters are used to update the elements of a nominal plant model, P, and a new plant model, p̂ , is used to represent the dynamics of the aeroservoelastic system.
The final operations of the µ method are traditional robust-stability operations on frequency-domain data. The effect of the wavelet filtering is to introduce filtered versions of the data and the plant model for model validation. Thus, a new uncertainty operator, Δ̂, is associated with the parameter-updated plant, p̂, to account for errors observed from the filtered data, x̂(t). There is computed a robust stability margin, Γ, that describes the largest change in dynamic pressure for which p̂ is robustly stable to the errors, Δ̂ .
Nominal stability margins are computed for the plant model by use of the original theoretical modal parameters and are computed for the updated model by use of parameters estimated from wavelet filtering. These margins are computed from a µ analysis with respect to variation in dynamic pressure, q̄, but ignoring the modal and complex uncertainty operators.
This work was done by Martin J. Brenner of Dryden Flight Research Center and Rick Lind of NRC. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com under the Information Sciences category, or circle no. 175 on the TSP Order Card in this issue to receive a copy by mail ($5 charge). DRC-98-26