ASE Filtering AFM2007

Noise Cancellation Technology Applied to Aeroservoelastic Adaptive Filtering AIAA AFM 2007 Steve Rogers Northrop Grumman 304.367.8540 Steven.Rogers@ngc.com

Abstract Adaptive filtering is used in many applications in telecommunications and aerospace, including system identification and noise cancellation. Generally, the coefficients of an FIR (all numerator) filter are modified according to the function of an error signal. In the case of system identification the behavior of an unknown system is modeled by accessing its input and output. An adaptive FIR filter can be used to adapt to the output of the unknown system based on the same input. When the deviation of the adapted FIR filter reduces to an acceptable value the adapted filter successfully models the unknown system.

The design of a noise cancellation system is achieved similar to the system identification described above123. The objective of a noise cancellation system is to remove or attenuate an undesired signal component from a given signal. If an additional signal is available that has only the undesired signal components it is possible to use it to remove the undesired components from the given signal. This process is active/adaptive noise control and uses feedback control principles. This process has been used for several decades in telecommunication systems in echo cancellation. A system for adaptive noise cancellation has two inputs, a noise-corrupted signal and a noise source. The figure below illustrates such a system. A desired signal is corrupted by noise v 1[n] which originates from a noise source signal v0[n].

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The Adaptive Filter is designed to minimize the output signal e[n]. In the process of this minimization any common components are attenuated according to performance criteria.

If both signals contain a disturbance component, the effects of this component will be minimized in the input signal s[n].

The paper will develop the general concept of Adaptive Noise Control and show an application to ASE Adaptive Filtering. The signal s[n] will be pqr (feedback control) channels and the additional signal v0[n] will come from an array of strain gage equivalent sensors.

Keywords: Adaptive Noise Control, ASE Filters

Introduction It is instructive to consider system identification prior to a discussion of adaptive noise control because of their similarities. System identification is the experimental approach to the modeling of a process or a plant. Generally, the following steps are involved: 1) experiment planning, 2) model structure selection, 3) model parameter estimation, and 4) performance assessment. In the case of adaptive filtering, there is less iteration permissible on selection of model structures and the performance assessment is an on-line error minimization. The concept is shown in Figure 1 below.

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Figure 1

System Identification Schematic

The error term, e[n], which is the difference between the unknown system output d[n] and the adaptive FIR filter y[n] output, is used to update the coefficients of the adaptive filter. One of the most common approaches is the least mean square (LMS) algorithm 123.

The update equation is: hk [ n ] = hk −1 [ n] + δe[ n]x[ n − k ] The h’s denote the unit sample response of FIR filter coefficients, and δ is the adaptation coefficient. The adaptation coefficient causes the output y[n] to approach the desired value d[n] at an appropriate convergence rate. A small δ causes a slow adaptation; however, a large δ may lead to skipping over the solution.

The design of a noise cancellation system 4,5 is achieved similar to the system identification described above123. A system for adaptive noise cancellation has two inputs, a noise-corrupted signal and a noise source. Figure illustrates such a system. A desired signal is corrupted by noise v1[n] which originates from a noise source signal v0[n]. Keep in mind that the noise signal corrupting s[n] is different from the reference noise signal due to environmental effects. Another example is the dual acoustic sensor approach to measurement of a leak. The acoustic sensors are directional, so theoretically, only one will pick up the leak signal. But, both will pick up ambient background noise. The background noise will be picked up differently by each acoustic sensor; consequently, the background noise can’t be simply subtracted out. The adaptive filter will then be used to estimate the signal v1[n].

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Figure 2 Noise Cancellation Schematic

The weights of the filter are adjusted in the same manner stated previously. The error term of this system is given by2,3: e[ n] = s[ n ] + v1 [ n] − y[ n ]

The error e[n] approaches the signal s[n] as the filter output adapts to the noise component of the input v1[n].

The FIR adaptive filter matlab function is in the Appendix.

Results An adaptive FIR filter is designed to adapt to the response of this system. A simulink based experiment was designed for a proof of concept. A simple four-state jet transport yaw/roll model6 was used.

A = [-0.0558 -0.9968 0.0802 0.0415 0.598 -0.115 -0.0318 0 -3.05 0.388 -0.465 0 0 0.0805 1 0]; B = [0.0729 0.0001 -4.75 1.23 1.53 10.63 0 0]; C = [0 1 0 0 0 0 0 1]; D = [0 0 0 0];

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0.36 s2 +0.048s+0.36

sgage

MATLAB Function AdFiltyaw

sgageyaw noise

0.36

meas

yaw

s2 +0.048s+0.36 tfyaw

myaw

err

Gp jet_tran

Clock 0.36 s2 +0.048s+0.36 tfroll

noise1

0.36

mroll MATLAB Function AdFiltroll

roll AF

s2 +0.048s+0.36 sgageroll

Figure 2 Simulink block diagram of System Identification experiment

The adaptive filter is contained in the matlab functions AdFiltyaw and AdFiltroll in Figure 2 above.

Figure 3 Yaw Channel Results

Figure 2 above is the block diagram of the experiment. Since a simple FIR filter structure is used the only tuning required is to determine the number of lags and the adaptation

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coefficient. The tuning was by using matlab’s fminsearch, which is a Nelder-Mead simplex unconstrained optimization method.

Figure 3 above shows the results of the yaw channel experiment with an adaptive FIR filter. The top plot contains the two measurement signals: yaw rate and strain gage equivalent. The strain gage equivalent has a different sinusoid gain, noise, and the sinusoid has a phase lag of 3*pi/8. The middle plot has the ‘truth’ signal the adapted signal and the yaw rate measurement. As can be seen the adapted signal converges to the ‘truth’ signal. The error or difference between the ‘truth’ and the adapted signal estimate is shown in the bottom plot. Figure 4 below gives the results at a slightly higher disturbance signal. Note that the convergence is quicker. Slower disturbance signals require more FIR adaptive lags for cancellation in order to capture the disturbance model.

If the common disturbance sinusoid is too low in frequency, it will not cancel and will require additional lags to converge to the adapted signal. Thus, under certain circumstances, convergence will be too slow to use for the ‘truth’ signal.

Figure 4 Yaw rate results

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Figure 5 Example of poor tracking on the Yaw rate channel At very low disturbance signal frequencies, the adapted signal fails to track well. An example is shown in Figure 5. Since the disturbance frequency is at such a low frequency, more lags may be needed. An alternative is to explore using IIR adaptive filters. These can model a much broader bandwidth with an equivalent number of lags as compared to FIR filters.

Normally, harmonics of the principal flexible mode are present. In Figure 6 below, the results of the principal mode plus three harmonics is shown. Reasonable convergence is shown.

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Figure 6 Yaw rate with harmonics

Conclusion A simple adaptive FIR filter approach to attenuation of flexible structural modes has been demonstrated. The approach is based on active noise cancellation or feedback control principles. Tracking of a truth signal has been shown to be feasible when an additional array of signals is available, such as from strain gage equivalents. It was assumed that the same disturbance signal frequency with a phase delay and harmonics is present in the additional signal. Common frequencies in both the signal of interest and the additional signal(s) are attenuated which produces an adaptive notch filter. Although it tracks it requires tuning and hundreds of lags, thus, computer throughput may be a limitation of the method. Future work would be to test it further on a series of high fidelity simulations, apply different filter structures, and insert known dynamic models into the filter structure.

References 1. Kehtarnavaz, N. & Kim, N., Digital Signal Processing System-Level Design Using Labview, 2005, Elsevier, ISBN 0-7506-7914-X. 2. Haykin, S., Adaptive Filter Theory, 3rd Ed., 1996, Prentice Hall, ISBN 0-13322760-X.

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3. Manolakis, D., etal, Statistical and Adaptive Signal Processing, 2000, McGrawHill, ISBN 0-07-11660-2.

4. Tokhi, M. & Leitch, R., Active Noise Control, 1992, Clarendon Press, ISBN 0-19856243-8.

5. Kuo, S. & Morgan, D., Active Noise Control Systems, 1996, Wiley Interscience, ISBN 0-471-13424-4.

6. Control System Toolbox User’s Guide, version 4, 1996, Mathworks, Inc.

Appendix The FIR adaptive filter function is written in matlab and is given below. % AdaptFilt.m function yout = AdaptFilt(u); % global mu mu1 n n1 T x = u(1); r = u(2); t = u(3); persistent X if 1 if isempty(X)||t == 0 X.x = x*ones(1,n); X.a = ones(1,n)/n; end y = X.x*X.a'; err = r - y; X.a = X.a + mu*X.x*err; % X.a = X.a/sum(abs(X.a)); X.x = [x,X.x(1:end-1)]; end if 0 if isempty(X)||t == 0 X.x = x*ones(1,n); X.y = X.x; X.a = ones(1,n)/n/3; X.b = X.a; end y = X.x*X.a' - X.y*X.b'; err = r - y; X.a = X.a + mu*X.x*err; X.b = X.b + mu*X.y*err; X.b = X.b/sum(abs(X.a)+abs(X.b))*1.98; X.x = [x,X.x(1:end-1)]; X.y = [y,X.y(1:end-1)]; end yout = err;

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