Output-only parameter identification of a colored-noise-driven Van-der-Pol oscillator: Thermoacoustic instabilities as an example

The problem of output-only parameter identification for nonlinear oscillators forced by colored noise is considered. In this context, it is often assumed that the forcing noise is white, since its actual spectral content is unknown. The impact of this white-noise forcing assumption upon parameter identification is quantitatively analyzed. First, a Van-der-Pol oscillator forced by an Ornstein-Uhlenbeck process is considered. Second, the practical case of thermoacoustic limit cycles in combustion chambers with turbulence-induced forcing is investigated. It is shown that in both cases, the system parameters are accurately identified if time signals are appropriately band-pass-filtered around the oscillator eigenfrequency.

Figure 1

Summary and context of the paper. Center: stochastically forced dynamic system. To perform model-based output-only SI, models for the stochastic input and for the dynamic system (being in the present study a Van-der-Pol oscillator) are required. Left: effects of noise color on oscillator dynamics and statistics, with $x\left(t\right)$ being the system state and $A\left(t\right)$ its envelope (the energy is $\propto A^2$). Right: filtering the data to isolate the dynamics of interest can be needed. The corresponding filter bandwidth affects the statistics and dynamics of the data, which has to be accounted for in the parameter identification procedure.

Figure 2

Comparison between white noise (gray) and OU noise (red) power spectra, normalized by the white-noise intensity $\mathrm{\Gamma }$, for different isopower semibandwidths $\mathrm{\Delta }\mathrm{\Omega }$. The power provided by the two types of noise is equal in the considered band (same area under the curve: note the linear scale). Note that $S_{\xi \xi }\left({\omega }_0\right)={\mathrm{\Gamma }}_{\text{e}}/2\pi \ne \mathrm{\Gamma }/2\pi$.

Figure 3

Power spectra of the input $\xi$ and of the output $x$ for three different values of correlation time ${\tau }_{\xi }$, normalized by $T_0=1/f_0$.

Figure 4

Time scales involved in the stochastically forced oscillator: ${\tau }_{\xi }$ is the correlation time of the noise source $\xi , T_0=1/f_0$ is the oscillation period of $x$, and ${\tau }_A=\pi /\left|\nu \right|$ is the envelope amplitude $A$ characteristic time scale. Note the two different time scales adopted for the two halves of the plot.

Figure 5

Map of the coefficient $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\text{e}}=(1+{\omega }_0^2{\tau }_{\xi }^2)/\gamma$. The closer this is to 1, the closer are the analytical expressions for $P_{\mathrm{OU}}$ and $P_{\mathrm{w}}$.

Figure 6

Probability density function for two different linear growth rates $\nu$, two different isopower semibandwidths $\mathrm{\Delta }\mathrm{\Omega }$, and five different adimensional correlation times ${\tau }_{\xi }/T_0$ of the driving noise (where $T_0=2\pi /{\omega }_0$ is the oscillation period). Shaded area and solid lines, respectively, correspond to the PDFs of the VDP driven by white noise $[P_{\mathrm{w}}$ given in Eq. (11)] and to the VDP driven by the OU noise $[P_{\mathrm{OU}}$ given in Eq. (10)] for the same parameters $\nu , \kappa , {\omega }_0$, and $\mathrm{\Gamma }$. The amplitude $A$ is normalized by $A_{\mathrm{m}}$, which is the amplitude of the maximum of $P_{\mathrm{w}}$.

Figure 7

Hellinger distance (12), quantifying the difference between the PDFs of OU noise and white-noise-driven VDP oscillators. (a) Different maps in the space $(\mathrm{\Delta }\mathrm{\Omega },{\tau }_{\xi },\nu )$. (b) Detail of one linearly stable and one linearly unstable case. (c) PDF comparison of two points in the map. The correlation time of the noise is normalized on the correlation time of the pressure amplitude ${\tau }_A=\pi /\left|\nu \right|$. The linear growth rate $\nu$ is normalized on the oscillator's angular frequency ${\omega }_0$.

Figure 8

(a) Identified growth rate as a function of the filter bandwidth. (b) Power spectrum of the signal (gray) and two filter windows (color highlight) for the VDP driven by OU noise. The thin black line is the spectrum of a VDP driven by a white noise of intensity ${\mathrm{\Gamma }}_{\text{e}}$. (c) Amplitude time traces (red and green) obtained from the two proposed filters, superimposed to the unfiltered signal (gray). In panel (a) the corresponding points are highlighted.

Figure 9

Map of OU noise-driven oscillator identification, for different noise correlation time ${\tau }_{\xi }$ and isopower semibandwidth $\mathrm{\Delta }\mathrm{\Omega }$. (a) Identification using the unfiltered data. (b) Identification using signals filtered in the band $\left[f_0(1\pm 0.5)\right]$. The identification result is given as relative error: $\varepsilon =|{\nu }_{\text{t}}-{\nu }_{\text{id}}|/{\nu }_{\text{t}}$.

Figure 10

Example of power spectra ($S_{qq}$ and $S_{pp}$) of turbulence-induced heat release rate fluctuations ${\widehat{q}}_{\mathrm{n}}$ and combustion noise of a flame radiating sound in the free field (adapted from Ref. [71]). $S_{pp}$ can be approximated with a bandpass model (-$◯$-), plotted also in the inset, in linear scale.

Figure 11

Block diagram for the sound field generated by turbulent flames in open and closed environments. The open-loop configuration (a) corresponds to open flames radiating noise in the free field. When the flame is enclosed (b), a thermoacoustic feedback operates and the acoustic block is fed by the total heat release rate fluctuations $\widehat{q}={\widehat{q}}_{\mathrm{a}}+{\widehat{q}}_{\mathrm{n}}$, where ${\widehat{q}}_{\mathrm{a}}$ and ${\widehat{q}}_{\mathrm{n}}$, respectively, stand for the acoustically and turbulence-induced components.

Figure 12

(a) Example of normalized combustion noise spectra measured for different open flame configurations (adapted from [71]). In the inset, the frequency of the spectrum maximum $f_{\max }$ is given as a function of the flow characteristics for a large set of operating conditions (see the main text for definitions). (b) Typical acoustic pressure spectrum recorded in a combustion chamber.

Figure 13

Comparison between white-noise (gray) and colored-noise (red) power spectra, normalized by the white-noise intensity, for different isopower bandwidths $\mathrm{\Delta }\mathrm{\Omega }$. The power provided by the two types of noise is equal in the considered band (same area under the curve: note the linear scale). Note that $S_{\xi \xi }\left({\omega }_0\right)={\mathrm{\Gamma }}_{\text{e}}/2\pi \ne \mathrm{\Gamma }/2\pi$.

Figure 14

Mutual position of noise and pressure spectrum maxima for three different adimensional noise correlation times $2\pi {\tau }_{\xi }/T_0$. Depending on the correlation time of the forcing noise, its spectrum maximum changes position accordingly [Eq. (21)]. This modifies the response of the system, as can be observed on the output spectra.

Figure 15

Map of the coefficient $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\text{e}}={(1+{\omega }_0^2{\tau }^2)}^2/\gamma {\omega }_0^2{\tau }^4$. The closer this is to 1, the closer are the analytical expressions for $P_{\mathrm{c}}$ and $P_{\mathrm{w}}$. Note that the coefficient can be either greater than 1 (red scale) or less than 1 (blue scale).

Figure 16

Probability density function for two different linear growth rates $\nu$, two different isopower semibandwidths $\mathrm{\Delta }\mathrm{\Omega }$, and three different adimensional correlation times $2\pi {\tau }_{\xi }/T_0$ of the driving noise (where $T_0=2\pi /{\omega }_0$ is the acoustic period). Solid lines are the PDFs for colored noise VDP [Eq. (27)]; shaded area is the white-noise driven PDF [Eq. (11)] for the same parameters. The amplitude $A$ is given as relative to $A_{\mathrm{m}}$, the amplitude where $P_{\mathrm{w}}\left(A\right)$ is maximum.

Figure 17

Hellinger distance (12), quantifying the difference between the PDFs of colored- and white-noise-driven VDP oscillators. (a) Different maps in the space $(\mathrm{\Delta }\mathrm{\Omega },{\tau }_{\xi },\nu )$. (b) Detail of a linearly unstable case. (c) PDFs for two linearly unstable points.

Figure 18

Map of band-pass colored-noise-driven oscillator identification, for different noise correlation time ${\tau }_{\xi }$ and isopower semibandwidth $\mathrm{\Delta }\mathrm{\Omega }$. (a) Identification using the unfiltered data. (b) Identification using signals filtered in the band $\left[f_0(1\pm 0.5)\right]$. The identification result is given as relative error: $\varepsilon =|{\nu }_{\text{t}}-{\nu }_{\text{id}}|/{\nu }_{\text{t}}$.

Figure 19

Effects of three different filter bandwidths on the analysis of combustor pressure experimental data. (a) Acoustic pressure spectrum and filter bands. (b) Time traces resulting from the three different filtering. (c) Detail of the envelopes over a time span of two ${\tau }_A$.

Figure 20

Double-oscillator simulation results. The model is made of two nonlinear oscillators linearly coupled, and is fed by colored noise. Oscillator no. 1 is linearly unstable, oscillator no. 2 is stable. Values adopted for the parameters [refer to Eq. (28)]: ${\omega }_2=1.3{\omega }_1, {\alpha }_1/{\omega }_1=0.02, {\beta }_1/{\omega }_1=0.03, {\alpha }_2/{\omega }_1=0.03, {\beta }_2/{\omega }_1=-0.02$, and ${\kappa }_1/{\omega }_1={\kappa }_2/{\omega }_1=0.015$. The colored noise parameters [refer to Eq. (20)] are $f_{\text{max}}/f_1=0.2, \mathrm{\Delta }\mathrm{\Omega }/{\omega }_1=0.5$, and $\mathrm{\Gamma }/4{\omega }_1^2=1$. (a) Overview of the total pressure and forcing noise spectra. (b) The spectral SPL of total output, single oscillator outputs $p_1$ and $p_2$, and forcing noise, in the frequency band that encloses the two oscillators' natural frequencies. (c) The poles of the linearized coupled system move due to the feedback action, which can either decrease or increase the stability margin of each mode, or even fully destabilize a mode.

Figure 21

Identified growth rate against the filter semibandwidth. The source signal is obtained via a Simulink® simulation of a double VDP oscillator, of known parameter (e.g., ${\nu }_{\text{t}}$ is the true growth rate). The identification is performed on mode no. 1, of eigenfrequency $f_1$, while another mode of eigenfrequency $f_2$ is in place. Three identification methods of [56] are used, respectively, based on the power spectral density of the amplitude (-$△$-), the probability density function of the amplitude (-$◯$-), and the coefficients of the Fokker-Planck equation (-$\square$-).