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Journal of Dynamic Systems, Measurement, and Control | Volume 133 | Issue 6 | Research Paper
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
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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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+Block diagram of a noise-driven system with a feedback nonlinearity
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Block diagram of a noise-driven system with a feedback nonlinearity
Figure 1

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

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+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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Top: noise-driven stable system response PSD estimate (a) and histogram (b). Bottom: noisy limit cycle system response PSD estimate (c) and histogram (d).
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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+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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For example 2.1. Left: h∧A1(ϖ) and h∧Ad(ϖ) with the scaling of Eq. 7 shown. Right: Their corresponding histograms.
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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+Block diagram of a noise-driven system with a feedback nonlinearity
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Block diagram of a noise-driven system with a feedback nonlinearity
Figure 4

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

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+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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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.
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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+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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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.
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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+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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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.
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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+Actual and estimated pdfs from single nozzle rig data corresponding to a range a fuel to air ratios
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Actual and estimated pdfs from single nozzle rig data corresponding to a range a fuel to air ratios
Figure 8

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

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+Estimated variances and limit cycle amplitudes from rig data
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Estimated variances and limit cycle amplitudes from rig data
Figure 9

Estimated variances and limit cycle amplitudes from rig data

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+Estimated oscillation frequency and linear phase-shift elements from rig data
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Estimated oscillation frequency and linear phase-shift elements from rig data
Figure 10

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

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+Estimated slope (blue diamonds) and saturation values from rig data
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Estimated slope (blue diamonds) and saturation values from rig data
Figure 11

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

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+The nonlinear component in the feedback model has an input composed of a harmonic plus a Gaussian
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The nonlinear component in the feedback model has an input composed of a harmonic plus a Gaussian
Figure 12

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

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+Plot of the saturation function and its estimate from the iterative procedure of Algorithm 4.6
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Plot of the saturation function and its estimate from the iterative procedure of Algorithm 4.6
Figure 13

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

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