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Research Papers

Stochastic Averaging for Identification of Feedback Nonlinearities in Thermoacoustic Systems

[+] Author and Article Information
Gregory Hagen

 Systems Department, United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108, USA e-mail: HagenGS@utrc.utc.com

J. Dyn. Sys., Meas., Control 133(6), 061017 (Nov 21, 2011) (10 pages) doi:10.1115/1.4003799 History: Received December 02, 2009; Revised January 14, 2011; Published November 21, 2011; Online November 21, 2011

We present algorithms based on stochastic averaging for estimating nonlinear feedback parameters obtained from time series data with application to noise-driven nonlinear vibration systems, with particular emphasis on limit-cycling thermo-acoustic systems. The harmonic and Gaussian components of relevant signals are estimated from the probability density function (pdf) of an output signal from a single experiment. The respective feedback gains, along with a phase-shifting element are fit to a nominal (given) linear oscillator model from which the parameters of a nonlinearity are fit. When input-output data are available from multiple experiments, the feedback nonlinearity can be estimated point-wise via an iterative algorithm, applicable when the appropriate input signals have a constant (Gaussian) variance. The estimation procedures are demonstrated on a benchmark thermo-acoustic model and applied to time-series data obtained from a limit-cycling combustor rig experiment. In the latter case, relations between the feedback parameters and the fuel to air ratio are briefly discussed.

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Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Block diagram of a noise-driven system with a feedback nonlinearity

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Figure 2

Top: noise-driven stable system response PSD estimate (a) and histogram (b). Bottom: noisy limit cycle system response PSD estimate (c) and histogram (d).

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Figure 3

For example 2.1. Left: h∧A1(ϖ) and h∧Ad(ϖ) with the scaling of Eq. 7 shown. Right: Their corresponding histograms.

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Figure 4

Block diagram of a noise-driven system with a feedback nonlinearity

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Figure 5

Ratio appearing in Eq. 30 as a function of δ for different values of A/σ. The plots were all computed with a saturation nonlinearity with unity slope and saturation value.

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Figure 6

Left: characteristic of data, estimates of the harmonic and gaussian characteristic components, and estimate of the total characteristic. Right: data pdf, estimates of the harmonic and gaussian pdf components, and the resulting estimate of the total pdf.

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Figure 7

Estimates resulting from Algorithms 2.2 and 3.1 from 10 simulation runs (blue dash) compared with the actual saturation nonlinearity (black solid) and 10% error ranges. The plots are normalized to show the relative errors.

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Figure 8

Actual and estimated pdfs from single nozzle rig data corresponding to a range a fuel to air ratios

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Figure 9

Estimated variances and limit cycle amplitudes from rig data

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Figure 10

Estimated oscillation frequency and linear phase-shift elements from rig data

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Figure 11

Estimated slope (blue diamonds) and saturation values from rig data

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Figure 12

The nonlinear component in the feedback model has an input composed of a harmonic plus a Gaussian

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Figure 13

Plot of the saturation function and its estimate from the iterative procedure of Algorithm 4.6

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