EP2647001A2

EP2647001A2 - Active noise reducing filter apparatus, and a method of manufacturing such an apparatus

Info

Publication number: EP2647001A2
Application number: EP11797371.9A
Authority: EP
Inventors: Arthur Perry Berkhoff; Johan Wesselink
Original assignee: Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek TNO
Current assignee: Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek TNO
Priority date: 2010-12-01
Filing date: 2011-12-01
Publication date: 2013-10-09
Also published as: WO2012074403A3; WO2012074403A2

Abstract

The invention provides an active noise reducing filter apparatus for actively reducing noise d from at least one primary noise source reference signal x. The apparatus comprises a secondary source signal connector connected to a secondary source generating secondary noise y based on a secondary signal u for reducing said primary noise d. A sensor connector is connected to a sensor for measuring said primary and secondary noise as an error signal e. A reference signal connector receives the reference signal x, and a control filter W receives the reference signal x and calculates a control signal for providing the signal u. An adaptation circuit receives said error signal e and reference signal x and adapts the control filter W. The adaptation circuit adapts the filter W using a representation of said secondary source including a secondary source transfer function G of said secondary source and an additional damping for providing improved stability.

Description

Active noise reducing filter apparatus, and a method of manufacturing such an apparatus

Technical field

The present invention is directed to an active noise reducing filter apparatus, and a method of manufacturing such an apparatus, for actively reducing noise from a primary noise source, applying a filtered- error scheme.

Such a filter apparatus typically implements a so called secondary path wherein an actuator is fed with control signals to provide a secondary source that is added to the primary source providing noise to be reduced. The resultant sensed noise is measured by a microphone and fed back into the filter apparatus as an error signal. The filter apparatus comprises a control filter for providing a control signal based on an input reference signal and a time-reversed model of the secondary path formed as the open loop transfer path between the control signal and the sensed resultant error signal. The input reference signal is coherent with the primary noise, for example by providing a signal that is physically derived from the primary noise source, while other sources, in particular the secondary source have a relatively small contribution.

Accordingly, the conventional filter apparatus comprises a secondary source signal connector for connecting to at least one secondary source, such as a loudspeaker, wherein the secondary source generates secondary noise to reduce the primary noise. A sensor connector is provided for connecting to at least one sensor, such as a microphone, for measuring the primary and secondary noise as an error signal. The error signal is delayed and filtered by a time reversed secondary path filter, which is a time-reversed and transposed version of the secondary path as formed by the open loop transfer path between the control signal and the sensed resultant error signal. Accordingly a delayed filtered error signal is provided. An adaptation circuit is arranged to adapt the control filter based on a delayed reference signal and an error signal derived from the delayed filtered error signal. The adaptation circuit can be a least mean square circuit, known in the art.

A known problem of these type of systems is the reduced stability in response to sudden changes. For example, a noise control system for actively reducing noise in a room may operate stable and satisfactory, until suddenly a door in the room is opened changing the acoustic properties of the room.

Summary of the invention

It is an object of the present invention to provide an active noise reducing filter apparatus and method having improved stability.

The above mentioned objects of the invention are achieved in that there is provided an active noise reducing filter apparatus for actively reducing noise d from at least one primary noise source reference signal x, comprising:

- a secondary source signal connector for connecting to at least one secondary source, wherein said secondary source generates secondary noise y based on a secondary source signal u received through said secondary source signal connector, for reducing said primary noise d with said secondary noise y;

- a sensor connector for connecting to at least one sensor for measuring said primary and secondary noise as an error signal e;

- a reference signal connector for receiving said at least one reference signal x;

- a control filter W connected to said reference signal connector for receiving said reference signal x and for calculating a control signal from said reference signal for providing said secondary source signal u;

an adaptation circuit for adapting said control filter W, said adaptation circuit connected to said sensor connector for receiving said error signal e and to said reference signal connector for receiving said reference signal x, and said adaptation circuit being arranged for adapting said control filter W based on said error signal e and said reference signal x using a representation of said secondary source including a secondary source transfer function G of said secondary source and an additional damping.

According to a second aspect, there is provided a method of manufacturing an active noise reducing filter apparatus for actively reducing noise d from at least one primary noise source reference signal x, said apparatus comprising:

- a control filter W connected to said reference signal connector for receiving said reference signal x and for calculating a control signal from said reference signal for providing said secondary source signal u; and

- an adaptation circuit for adapting said control filter W, said adaptation circuit connected to said sensor connector for receiving said error signal e and to said reference signal connector for receiving said reference signal x, and said adaptation circuit being arranged for adapting said control filter W based on said error signal e and said reference signal x using a representation of said secondary source

wherein said method comprising a step of modeling said representation such as to include a secondary source transfer function G of said secondary source and an additional damping. Brief description of the drawings

The invention will further be elucidated by description of some specific embodiments thereof, making reference to the attached drawings, wherein:

Figure 1 illustrates a prior art filter apparatus implementing a prior art filtered- error adaptive control scheme;

Figure 2 illustrates a prior art filter apparatus implementing a postconditioned filtered-error adaptive control scheme;

Figure 3 illustrates a regularized modified filtered-error adaptive control scheme with internal model control (IMC);

Figure 4 is a graph of the magnitude of the transfer function between actuator 1 and sensor 1 for different conditions;

Figure 5 provides a graph of the phase of the transfer function between actuator 1 and sensor 1 for different conditions;

Figure 6 is a graph of the difference of the phase of the transfer function between actuator 1 and sensor 1 for different conditions as compared to the nominal condition;

Figure 7 is a graph of the minimum real eigenvalue of Eq. (11) for a secondary path G obtained in a room for the nominal situation and without model mismatch, i.e. G = G.

Figure 8: as figure 5, except that G is obtained one hour later;

Figure 9: as figure 5, except that G is obtained with the door open; Figure 10: as figure 5, except that G is obtained with the window open;

Figure 11: as figure 5, except that G is obtained with the door and the window open;

Figure 12: minimum real eigenvalue of Eq. (11) in which G is the original secondary path and in which G is the secondary path with damping obtained by LQR state feedback;

Figure 13: as figure 12, except that G was obtained one hour later;

Figure 14: as figure 12, except that G is obtained with the door open; Figure 15: as figure 12, except that G is obtained with the window open;

Figure 16: as figure 12, except that G is obtained with the door and the window open;

Figure 17: magnitude of G_reg as used for frequency dependent regularization;

Figure 18: Configuration of the high- authority/low -authority control architecture applied to a sandwich panel with piezoelectric patch actuators, piezoelectric patch sensors and accelerometers.

Figure 19: Dimensions of the active panel used in the experiments. A mount with a thread was used to attach extra weight to the panel (indicated by a circle).

Figure 20: Architecture of the FPGA (without the LAC-unit).

Figure 21: Simplified block diagram of the control system containing the low-authority controller and down-/upsampling for the high- authority controller.

Figure 22: Regularized modified filtered- error adaptive control scheme with IMC [3, 22].

Figure 23: The transfer functions for the collocated sensor-actuator pairs from 0 to 1 kHz; piezoelectric actuator to acceleration sensor, 5 pairs of the available 9 pairs were used.

Figure 24: As Fig. 23, except for the frequency range being 0-10 kHz.

Figure 25: Influence of low -authority controller on the transfer function of the plant, as measured from the piezoelectric actuator to the piezoelectric sensor, while the feedback loop was from the piezoelectric actuator to the acceleration sensor. The decimation filter was switched off; an interpolation filter with a stopband attenuation of 50 dB was used.

Figure 26: Convergence curves for different 's using a feedforward controller. The format in the legend is: Reduction in dB; step size a.

Figure 27: Performance measured on the sensors for a feedback controller using IMC.

Figure 28: Performance of the feedforward controller. Figure 29: Convergence curves for MIMO feedback with and without

LAC.

Figure 30: Convergence curves for MIMO feedforward control with and without LAC.

Figure 31: A plot of the phase difference between systems with and without additional weight. The first curve uses no LAC and the second one has LAC enabled. In this case a weight of 40.60 gram was added to the panel.

Detailed description

Part i

This document presents techniques for the improvement of the stability of an adaptive feedforward control system. An analysis is made of an active noise control system with model mismatch, i.e. a system in which a difference exists between the secondary path and the model as used in the controller. In particular, the eigenvalues that determine the stability of the system are analyzed. A modification of the feedforward control scheme is presented which improves stability and reduction of the error signals in case of model mismatch.

Methods

Adaptive feedforward controller

Improved stability is desirable in many implementations of adaptive control algorithms based on the filtered-reference least mean square (LMS) algorithm or the filtered-error LMS algorithm. Preferably, such robustness improvements do not lead to increases of the error signal. In this document, some techniques for improved robustness are presented. The methods are tested in combination with a particular version of a filtered-error type LMS algorithm [1] . A block diagram of the multiple-input multiple-output adaptive controller is shown in Fig. 3. A detailed description of this algorithm can be found in Refs. [2,3]. For the description of the multiple input multiple output (MIMO) controller, we assume that there are K reference signals, L error sensors and M actuators. The transfer function between the actuators and the error sensors is denoted by the L x Tridimensional secondary path G. Denoting n as the sample instant, the update rule for the controller coefficients is

Wi(n + 1) = Wi(n) - ae"(n)x'^T(n - ΐ), (1) in which the i-th filter coefficients of the control filters are represented by the M x K matrix Wi, where

with q the unit delay operator. Furthermore. e"(ri) is the x l vector of auxiliary error signals, x '(ri) is the K x 1 vector of delayed reference signals, and a is the convergence coefficient. In the actual implementation, a normalized LMS update rule was used, combined with 'leakage' of the control coefficients [4] . Although the algorithm differs in some aspects from the standard filtered-reference LMS of filtered- error LMS algorithms, especially with respect to the speed of convergence, the stability properties are governed by the same underlying equations. Therefore, for the remainder of this part, it is assumed that the analysis applies to general filtered-reference LMS or filtered- error LMS algorithms. Nevertheless, it will be seen that the algorithm of Fig. 3 is well suited for the

implementation of the techniques discussed in this document. In particular, the scheme facilitates the implementation of frequency dependent regularization techniques, which can be useful for robust control

approaches. Robust control approaches are described by, for example [5] . Probabilistic methods leading to frequency dependent regularization for optimum filtering are described by [6,7] . Methods for adaptive control are given in [4,8] . Such algorithms can be tuned for a particular application but require additional effort in the design stage and presume that sufficient a- priori knowledge is available about the uncertainty. An alternative approach is to use a high-authority and low-authority control (HAC/LAC) architecture [9] where the goal of the low-authority controller is to add active damping to the structure. Active damping can be implemented using different strategies. The use of a HAC/LAC architecture yields three major advantages [9] . Firstly, the active damping extends outside the bandwidth of the HAC control loop, which reduces the settling times outside the control bandwidth. Secondly, it is easier to gain- stabilize the modes outside the bandwidth of the outer loop. And thirdly, the large damping of the modes inside the controller bandwidth makes them more robust to parametric uncertainty. In this part it will be shown that an alternative approach is possible which does not require the complex hardware of a HAC/LAC scheme nor does it need detailed a-priori knowledge of the uncertainty.

A block diagram of a conventional filtered- error scheme can be found in Fig. 1. The parts of the diagram which constitute the controller are indicated by a dashed line. All signals are assumed to be stationary. In this scheme, x is the K x 1- dimensional reference signal and d is the L x 1- dimensional primary disturbance signal, which is obtained from the reference signal by the L x K dimensional transfer function P(z) . The goal of the algorithm is to add a secondary signal y to the primary disturbance signal d such that the total signal is smaller than d in some predefined sense. The signal y is generated by driving actuators with the M x 1- dimensional driving signal u. The transfer function between u and y is denoted as the L x M- dimensional transfer function G(z), the secondary path. The actuator driving signals u are generated by passing the reference signal x through an M x K- dimensional transfer function W (z) which is implemented by an M x K-dimensional matrix of Finite Impulse Response (FIR) control filters. The i-th coefficients of this FIR matrix are denoted as the M x K matrix Wi. The transfer function matrices Wi are tuned in such a way that the error signal e = d + y is minimum. This tuning is obtained with the least-mean square (LMS) algorithm, which in Fig. 1, is implemented by modifying the control filters Wi at each sample n according to the update rule Wi(n +1) = Wi(n) - a f n) x '^T (n - ΐ) (2) where T denotes matrix transpose and where x \n) is a delayed version of the reference signal such that x \z) = D_K(z)x(z) (3) in which DK (Z) is a K X K- dimensional matrix delay operator resulting in a delay of J samples:

D_K(z) = Z -J 7K (3) (4) and in which f n) is a filtered and delayed version of the error signal, such that i Xz) = G^* (z)D_L(z)e(z) (5)

In the above equation the filtering is done with the adjoint G^* (z), which is the time-reversed and transposed version of the secondary path G(z), i.e. G^*(z)= G^T (ζ^Λ). The adjoint G^*(z) is anti-causal and has dimension M x L. The delay for the error signal, and consequently also the delay for the reference signal, is necessary in order to ensure that the transfer function G^*{z) DL(Z) is predominantly causal. The convergence coefficient a controls the rate of convergence of the adaptation process, which is stable only if the convergence coefficient is smaller than a certain maximum value.

An advantage of the filtered-error algorithm as compared to the filtered-reference algorithm [13] is that computational complexity is smaller for multiple reference signals [14], i.e. if K> 1. A disadvantage of the filtered-error algorithm as compared to the filtered-reference algorithm is that the convergence speed is smaller due to the increased delay in the adaptation path, which requires the use of a lower value of the convergence coefficient a in order to maintain stability. One of the reasons for a possible reduced convergence rate of the algorithm of Fig. 1 is the frequency dependence of the secondary path G(z) as well as the interaction between the individual transfer functions in G(z). The convergence rate can be improved by incorporating an inverse of the secondary path between the control filter W (z) and the secondary path G(z) [15] . In order to ensure stability of such an inverse, only the minimum- phase part Go(z) of G(z) is to be inverted. The secondary path is written as

G(z)= Gi(z)G₀(z) (6) where the following properties hold:

G^* (z)G(z) = G^*o(z)Go(z) (7) Gf (z)Gi(z) = IM (8)

Assuming that the number of error signals is at least as large as the number of actuators, i.e. L > M, the transfer function Gi(z) has dimensions L x M and the transfer function G₀(z) has dimensions M x M. The extraction of the minimum-phase part and the all-pass part is performed with so-called inner-outer factorization [16] . A control scheme in which such an inverse G ¹o (z) is used can be found in Fig. 2. The update rule for the control filters Wi in Fig. 2 is Win +1) = Wi(n) - e \n) x '^T (n - ΐ) (9)

Indeed, if the magnitude of the frequency response of G(z) varies considerably and/or if there is strong interaction between the different channels of G(z) then the convergence rate of the scheme of Fig. 2 can be significantly better than that of Fig. 1. In Fig. 2, the filtered error signal is denoted with e '(ri) in order to emphasize that the frequency response magnitude of the filtered error signal has a close correspondence with the real error signal e(ri). It should be noted however that e(ri) is an L x 1 dimensional signal, while e \n) is an M x 1-dimensional signal.

Figure 3 illustrates a regularized modified filtered-error adaptive control scheme with internal model control (IMC).

Uncertainty

We assume that the uncertainty AG in the secondary path G is such that

G=G+AG (10) in which G is the model of the secondary path G. The condition for stability of the LMS update rule is determined by the minimum real part of any eigenvalue of a matrix determined by the secondary path and the model of the secondary path [4] :

Alternatively

Regularization

Regularization can be used to ensure that all real parts of the eigenvalues are positive. The regularization can be implemented by definin; an augmented plant G : in which the L x M- dimensional secondary path G(q) is augmented with an L ' x M- dimensional transfer function Greg (q). This augmented plant allows us to define a cost function J - E(e^re) (14)

in which

e

t-reg (15)

Hence

./ = E(e^re) + Ki ^-

(16)

The error signal e is defined as usual

e - Gu + d (17) whereas the regularizing error signal is obtained from

6reg— Gregll (18)

The requirement for stability now becomes

in which

The stability condition can be written as

The task is to determine a minimum Mx -dimensional matrix Λ =

^re reg f_{or eac}h _{ω suc}h that the selected condition holds. A diagonal matrix for A should be sufficient, but the elements on the diagonal are not necessarily identical. Spectral factorization of Λ then leads to Greg. In practical situations, considerable time and effort is required to obtain sufficient information about the different conditions such that a reliable estimate of Λ can be obtained. If Λ is to be determined from the transfer function deviations for all possible conditions that may occur during operation, then this approach may be too time-consuming for many applications since each individual installation may require such an a-priori procedure.

The allpass factor G_I and the minimum-phase factor G₀ are obtained from an inner-outer factorization such that G = G_IG_A. The adjoint G ^* is combined with a delay D of ND samples in order to ensure that DG ^* is predominantly causal. The transfer function Grp subtracts the contribution of the actuators on the reference signals, as required for internal model control (IMC) [10].

Damping by state feedback

Improved robustness of adaptive algorithms can be achieved by increasing the damping of G, which can be realized with separate highspeed control loops in a HAC/LAC strategy [11]. In the HAC/LAC strategy, LAC is active in the system identification phase but also during adaptive control. An observation in experiments was that increased robustness of the adaptive algorithms could be obtained even if damping was active in the identification phase only, i.e., active damping was not active during adaptive control. This implies that improved robustness can also be obtained if damping is applied numerically to the transfer function G. In this section we try to realize such a numerical damping.

Let x be the state vector and u the actuator driving signals. Let A, B, C, D be the state space system describing the system G. Feedback control can be implemented by using an LQR regulator [5] which determines the feedback gain K defined by

U ——K x (22) minimizing the cost function

J - [z^T(n)Qx(n) + u^T(n)R

n

subject to riii + 1 ) — Ax(ri) + Bu(n (24)

Additional damping can be realized if the feedback gain is not too high, otherwise additional resonances with low damping may be introduced. A relatively low feedback gain K is obtained by setting the weighting by R relatively high as compared to the weighting by Q. The new state-space system with added damping is obtained by setting:

A — A -BK

B

C ^ C - DK

D — D (25)

Alternatively, one could use feedback of the output y such that

./ = T [y^T(n)Q'y{n) + u^r(n)R'u{ n_)j

(26) in which y = Cx + Du. If G contains significant phase delays then the LQR regulator could be applied to the minimum-phase factor Go only.

Stability analysis based on measured transfer functions

This section presents a stability analysis for an active noise control system in a room. The active noise control system uses 3 loudspeakers and 4 sensors. The sensor signals are a pressure signal and 3 particle velocity signals. The 4 sensors are positioned very close to each other using a Microflown USP probe. Experiments were performed to obtain transfer functions under different conditions. The nominal situation consists of a room in which the door and the window are closed. The dimensions of the room are 5m x 3m x 2.6m. The loudspeakers are located in corners on the floor of the room, while the sensor is located near the longest wall at about lm from the wall and at a height of lm. An example of a transfer function, for actuator 1 and sensor 1, is given in Figs. 4 and 5. The difference of the phase as compared to nominal condition is shown in Fig. 6.

The minimum real part of the eigenvalue according to Eq. (11) for different conditions is given in Figs. 7-11. Fig. 7 shows the situation for which G = G. As a result all real parts of the eigenvalues are positive and the system is stable. Fig. 8 shows the situation for which the transfer functions are measured at two different instants with one hour in between but for which the conditions are the same. It can be seen that for the stability condition the minimum real part of the eigenvalues is positive for all frequencies, i.e., the system is expected to be stable. Fig. 9 shows the situation for the case that a door is fully opened. It can be seen that the minimum real part of the eigenvalues is negative for some frequencies, leading to possibly unstable behavior. Problematic frequencies according to the stability condition are 36 Hz and 69 Hz. Fig. 10 shows the results for the minimum real eigenvalue for the case that a window is opened. In this case stability problems are expected at very low frequencies from 0 Hz to 5 Hz according to the stability condition. Fig. 11 shows the minimum real eigenvalue for the case that both the door and the window are open.

According to the stability condition, problematic frequencies are the range of 0 Hz to 9 Hz, 36 Hz and 69 Hz.

The minimum real part of the eigenvalue according to Eq. (11) for different conditions using state feedback is given in Figs. 12 - 15. In this case the weighting matrices Q and R for the LQR regulator were Q = I and R = 10³J These values ensure that all real parts of the eigenvalues are positive and also that the resulting curve of the real part of the eigenvalue vs. frequency has approximately the same smoothness as in the nominal situation. It can be seen that, except for very low frequencies below 7 Hz, the stability condition is satisfied, i.e. the minimum real part of the eigenvalues is positive. For the very low frequencies for which still a negative real part of the eigenvalues exists, an alternative stabilization technique is required. Frequency dependent regularization is considered, which should ensure that the minimum real part of the eigenvalues becomes positive at the low frequencies while having a minimum influence at higher frequencies. Simulation results

Using the measured transfer functions as described in the previous section, simulations were performed to verify the robustness as predicted by the stability analysis for different control strategies. Furthermore, the final reduction of the error signals was determinded for a converged algorithm in a stationary situation. The results are shown in Table 1. The nominal condition denotes the situation in which all windows and doors are closed. The modified condition denotes the situation in which the windows and doors are opened. The nominal controller denotes a controller which uses the model obtained during the nominal condition and which uses effort weighting, i.e. frequency independent weighting. In the case that damping is used, damping is applied to the nominal model. Frequency dependent weighting is also used in combination with the nominal model, with or without damping. Frequency dependent weighting is based on frequency independent weighting for frequencies above 20 Hz with additional amplification for frequencies below 10 Hz using a 2nd-order filter, as shown in Fig. 17. This regularization technique emphasizes regularization at low- frequencies while being less conservative at higher frequencies. At low frequencies the gain of the frequency dependent regularization filter is 12 dB higher than at high frequencies. The regularization level for frequency dependent weighting as indicated in the table is the value at high

frequencies. The primary field was obtained by providing three independent white noise signals with a delay of 20 samples to the inputs of the nominal model or the modified model, depending on the condition that was used. The reference signals were the signals from the noise generators. The algorithm of Ref. [3] was used with affine projection order KA = 4, delay length Nd = 150, number of controller coefficients for each channel Wi = 250, leakage coefficient = γ = 10^~5 , affine projection regularization parameter δ = 0.25, convergence coefficient a— 0.025. A convergence coefficient higher than 0.025 led to somewhat faster convergence in case there was no model mismatch. However, such a higher convergence coefficient resulted in less robustness and higher error signals in case of model mismatch.

In Table 1, it can be seen that adding damping to the secondary path model has a positive effect on the reduction of the error signals that can be achieved. Damping also has a positive effect on the stability of the system in the sense that a lower value of regularization level is possible for stabilizing the system. For the modified condition, the highest reductions of the error signals are possible when a combination is used of added damping and frequency dependent regularization, leading to maximum MSE reductions of 21.6 dB to 26.4 dB (marginally stable). For the nominal controller the maximum reduction for the same condition is 12.0 dB, i.e. considerably less.

Frequency dependent regularization alone does not improve the noise reduction for this case. The addition of damping leads to an improvement, in this case 16.8 dB maximum reduction. Subsequent addition of frequency dependent regularization leads to a further possible improvement of the reduction of the error signals. These results are in agreement with the stability analysis of the previous section. Also the regularization levels that are needed for stabilization are in agreement with the results of the previous section. Remarkable is that damping also has a positive effect on the amount of reduction in the nominal situation. Additional simulations were performed with longer filter lengths, i.e. higher values of Nd for the realization of the delayed adjoint operator DG*. However, this did not result in higher noise reductions. A possible explanation could be that errors in the modeling of undamped poles is critical and that, in order to avoid computed gradients with large errors, it is better to use cautious gradients based on poles which are assumed to have more damping.

Concluding remarks for part I

The combined controller modifications of adding damping to the secondary path model and a frequency dependent regularization technique leads to a particularly interesting control scheme. The scheme improves the stability for the case that the secondary path model does not correspond anymore with the real secondary path. This also leads to higher possible reductions of the error signal. For the present configuration, a high- frequency regularization level of -10 dB yields good performance for the nominal situation (21.7 dB reduction) as well as for the situation with model mismatch (21.6 dB reduction). Even when the model of the secondary path corresponds to the real secondary path, the new scheme outperforms the nominal controller, which yields 20.7 dB reduction for the same

regularization level. Furthermore, the new scheme does not require detailed models from additional system identification cycles.

Table 1: Mean-square reduction of the error signals in dB after 1000 s for different physical situations, controller models and control strategies. A dash indicates an unstable system, an asterisk indicates marginal stability.

Part II

Recent implementations of multiple-input multiple -output adaptive controllers for reduction of broadband noise and vibrations provide considerably improved performance as compared to more traditional adaptive algorithms. The most significant performance improvements are in terms of speed of convergence, the amount of reduction, and stability of the algorithm. Nevertheless, if the error in the model of the relevant transfer functions becomes too large then the system may become unstable or lose performance. Online adaptation of the model leads to increased complexity and, for rapid changes in the model, necessitates a large amount of additional noise to be injected in the system. It has been known for decades that a combination of high-authority control (HAC) and low-authority control (LAC) could lead to improvements with respect to parametric uncertainties and unmodeled dynamics. In this part a full digital

implementation of such a control system is presented in which the HAC (adaptive MIMO control) is implemented on a CPU and in which the LAC (decentralized control) is implemented on a high-speed Field Programmable Gate Array. Experimental results are given in which it is demonstrated that the HAC/LAC combination leads to performance advantages in terms of stabilization under parametric uncertainties and reduction of the error signal.

Many algorithms used for broadband active noise control are based on the adaptive Least-Mean- Square (LMS) algorithm [17]. The low complexity and the relatively good robustness properties are the major advantages of the LMS algorithm. Recent algorithms solve many of the problems associated with the speed of convergence of the older algorithms. The basis for a particular class of such algorithms has been given by Elliott [18] as the preconditioned LMS algorithm. The version based on the filtered- error algorithm [1] is more efficient for multiple reference signals than the filtered-reference algorithm. A proper implementation of the filtered- error preconditioned LMS algorithm solves many of the problems associated with early implementations of the LMS algorithm, such as slow convergence due to frequency dependence of the secondary path and cross-coupling in the secondary path [3]. However, the controller is model-based and is therefore still sensitive for mismatch between the model and the plant.

This model mismatch reduces the overall performance of the controller. Model mismatch can be caused by variations in parameters such as temperature, boundary conditions etc. For some control schemes, on-line adaptation of the model is possible in principle but a large amount of additional noise has to be injected in the system for rapid changes in the model [19]. Furthermore, if the controller uses model-based preconditioning or factorization, then these time-consuming operations should be performed on-line as well. Robust control approaches are known [5] as well as probabilistic methods leading to frequency dependent regularization for optimum filtering [6, 7] and adaptive control [4, 8].

Such algorithms can be tuned for a particular application but require additional effort in the design stage and presume that sufficient a-priori knowledge is available about the uncertainty. An alternative approach is to use a high-authority and low-authority control (HAC/LAC) architecture [11] where the goal of the low-authority controller is to add active damping to the structure. Active damping can be implemented using different strategies. The use of a HAC/LAC architecture yields three major

advantages [11]. Firstly, the active damping extends outside the bandwidth of the HAC control loop, which reduces the settling times outside the control bandwidth. Secondly, it is easier to gain- stabilize the modes outside the bandwidth of the outer loop. And thirdly, the large damping of the modes inside the controller bandwidth makes them more robust to parametric uncertainty. In the paper by Herold et al. [20], a method using piezoelectric sensors and actuators and positive position feedback (PPF) was described. In the PPF-method, a second-order filter is used as the control filter which is combined with positive feedback. The control filter is then tuned to reduce one of the desired resonance peaks. In the present part, an approximately collocated and dual sensor- actuator pair is used, suitable for broadband damping as described by Elliott et al. [21]. If the actuator-sensor system is dual and collocated, a simple decentralized proportional feedback controller is sufficient to add damping due to the fact that the overall energy that is stored in the system will be reduced [11]. As such, less detailed a-priori information is required about the model uncertainty. Active damping is not very effective for frequencies that do not coincide with the poles and zeros. To gain further reductions for such frequency components a model-based controller is used such as the RMFeLMS algorithm as described in this part.

Methods

In this part a particular implementation a MIMO adaptive algorithm (HAC) is combined with a decentralized feedback controller (LAC). The implementation of the adaptive algorithm uses the inverse of the minimum- phase factor of the secondary path, combined with a double set of control filters to eliminate the negative effect of the delay in the adaptation loop [22] . The latter algorithm is combined with a regularization technique that preserves the factorization properties [22]. The secondary path is estimated using subspace identification techniques [23]. This enables the use of reliable numerical techniques for the minimum-phase/all-pass

decomposition using inner-outer factorization [24, 25] . The resulting algorithm, the so-called regularized modified filtered- error least mean square algorithm (RMFeLMS) has good convergence properties as compared to the standard filtered-reference and filtered- error algorithm [3].

The HAC/LAC architecture was tested on a piezoelectric panel for reduction of noise transmission, a cross-section of which can be found in Fig. 18. The dimensions of the panel and the positions of the actuators and sensors are given in Fig. 19. Nine piezoelectric patch actuators and nine piezoelectric patch sensors were attached to the panel [3], of which the middle pair and the four pairs in the corners were used. The panel was built from two Printed Circuit Boards (PCBs) with a honeycomb layer in between. One advantage of this approach is that electronics can be integrated. Another advantage is that the actuator and the sensor can be placed on different faces of the panel, which improves the control of the acoustically relevant out-of-plane vibrations because the in-plane coupling between the actuator and the sensor is reduced [26]. Five collocated accelerometers were used for active damping with decentralized control using the piezoelectric patch actuators. The control results were obtained with a perspex box on which the panel was mounted. Inside this perspex box, noise was created with a loudspeaker which led to vibrations of the panel [3].

Active damping is often realized using an analog controller [27, 28,

29]. One of the advantages of an analog controller is its low delay when compared to a digital controller. In this part, a digital controller with a high- sample rate is used, such that the analog and digital controllers have identical performance for frequencies within the control bandwidth. The dedicated analog interface board was developed containing ADDA

converters operating at a relatively high sample rate (in this case 100 kHz). The lower sample rate was derived from the high sample rate by

downsampling. This made it possible to run the LAC and HAC controller at different sample rates, i.e. 100 kHz and 2 kHz, respectively. The interface between the PCI- 104 system running the HAC algorithm and the ADDA unit was implemented in reconfigurable hardware, in this case a Field Programmable Gate Array (FPGA),(See Fig. 20). The FPGA incorporates the following functional units: the decimation filters, the interpolation filters, the glue logic for the PCI bus interface and the low-authority controller. The decimation filters and interpolation filters were designed in such a way that the desired compromise between group-delay, filter transition band characteristics and stopband attenuation was obtained.

A block diagram of the multiple-input multiple- output adaptive controller as used for the high-authority controller is shown in Fig. 22. A detailed description of this algorithm can be found in Refs. [3, 22]. Relevant for the experiments as described in this part are the definition of the update rule for the controller and the regularization of the secondary path. For the description of the MIMO controller, we assume that there are K reference signals, L error sensors and M actuators. Denoting n as the sample instant, the update rule for the controller coefficients is

Wiin + 1) = Wi (n) - e"{n)x^,T(n - i ) . ₍₂₇₎ in which the i-th filter coefficients of the control filters are represented by the unit delay operator. Furthermore, e"(n) is the M x l vector of auxiliary error signals, x'(ri) is the K x 1 vector of delayed reference signals, and a is the convergence coefficient. In the actual implementation, a normalized 1ms update rule was used, combined with 'leakage' of the control coefficients [4] . The regularization was implemented by defining an augmented plant G (q) : in which the L x M- dimensional secondary path G(q) is augmented with an L ' x -dimensional transfer function Greg(q). For the implementation as described in this part, the regularization was based on a simple weighting of the M x l vector of control signals u(ri), which also limits the inversion of zeros in G(q) that are close to the unit circle, resulting in more stable behavior of the M x -dimensional inverse G₀{qY^l . The regularizing transfer function Greg(q) was defined as: (29)

in which β is a scalar quantity and in which IM is an M x M identity matrix. The allpass factor G_I and the minimum-phase factor G₀ are obtained from an inner-outer factorization such that G = G_TG₀ . The adjoint G ^* is combined with a delay D of ND samples in order to ensure that DG ^* is predominantly causal. The transfer function G_rp subtracts the contribution of the actuators on the reference signals, as required for internal model control (IMC) [10] . Results

Influence of LAC on the secondary path

For the idealized low-authority controller having the purpose to add damping to the system the phase should be between -90 and +90 degrees for each collocated pair. Figs. 23 and 24 can be used to judge the practical setup involving the piezoelectric patch actuator and the accelerometer regarding this requirement. In Fig. 23, it can be seen that for frequencies up to 1 kHz the phase is between 0 degrees and +180 degrees. Thus, with an integrator the phase will be between -90 and +90 degrees. In Fig. 24, it can be seen that for higher frequencies the phase lag is larger. Based on these results, the decentralized controllers were configured with an integrator and a 1st- order roll-off above 1 kHz. The gain was adjusted for each collocated pair such that a gain margin of at least 6 dB was obtained. The influence of the local controllers on the measured transfer function of the high-authority controller is shown in Fig. 25. It can be seen that resonances and

antiresonances can be damped, particularly at low frequencies. For higher frequencies and for higher feedback gains some spillover can be observed. For this particular configuration also a controller configuration was tested in which the center actuator- sensor pair had a higher gain than the other pairs (see Fig. 25). Of the configurations studied, this configuration was found to give the best compromise between damping performance and spillover.

Selection of the convergence coefficient

In order to select a suitable value of the convergence coefficient a, the influence of a on the convergence speed and MSE was studied by measuring the error signals for a duration of 60 seconds. At the beginning of each measurement the controller was switched off for 5 seconds and then it was switched on for 55 seconds. The reduction was calculated by taking the first 4 seconds and the last 4 seconds, using ensemble averages based on 32 measurements. Table 2: Reduction of the error signals for a feedback controller for different step sizes a after 60 seconds. For the feedback controller the following parameters were used: N_w =

80. ^j\∑⁾ — 80, — 10 _; in which γ sets the amount of leakage [10] of the control coefficients. The regularization parameter β was set to -20 dB. In Table 2, the result for a number of different step sizes a is given. A plot of the different convergence curves is not included due to the fact that the curves were almost identical: only the MSE errors were different. From Table 2, it can be concluded that =— gives the best possible reduction

40

after 60 seconds.

For the feedforward controller, the following parameters were used: Nw = ^'D— SO. — 10 _ regularization parameter β was set to -30 dB. The procedure used to measure the results for feedforward control was the same as used in the feedback scenario described in the previous paragraph, this time using 16 measurements. The different MSE values after convergence for different values of a can be found in Table 3. The convergence curves for different values of a can be found in Figure 26. From these results, it was concluded that an a of— ^{was a} good trade-off between

40

convergence speed and steady-state MSE for HAC.

Table 3: Reduction of the error signals for a feedforward controller using different step sizes a after 60 seconds Influence of LAC on the steady state mean square error

Control results for a HAC/LAC architecture using an adaptive MIMO feedback algorithm are shown in Fig. 27. The convergence coefficient was set to _a of _L (see Subsection titled 'selection of the convergence coefficient' of

40

part II herein). It can be seen that the reduction of the error signals for

MIMO control (HAC) is higher than for the decentralized control (LAC). The combination of HAC and LAC leads to the largest reduction of the error signals. The average improvement by adding the low-authority controller to the high-authority controller is approximately 1.4 dB.

Control results for a HAC/LAC architecture using an adaptive MIMO feedforward algorithm are shown in Fig. 28. The convergence coefficent was set to _a of _L (see Subsection titled 'selection of the convergence coefficient'

40

of part II herein). It can be seen that the reduction of the error signals for MIMO control (HAC) is considerably higher than for the decentralized control (LAC). As with feedback control the combination of HAC and LAC leads to the largest reduction of the error signals. The average improvement by adding the low- authority controller to the high- authority controller is approximately 4.4 dB. Influence of LAC on the convergence speed

The objective was to find out how LAC influences the overall performance in terms of speed of convergence. Based on the subsection titled 'selection of the convergence coefficient' of part II herein it was decided to set a to a value of -L for the feedback controller as well as for feedforward

40

controller. The convergence under influence of LAC using a feedback HAC strategy can be found in Figure 29.

To show the impact of the LAC clearly, it was decided to leave the LAC switched off for the first 5 seconds and switched on for the remainder of the time. The two plots in the Figure 29 demonstrate the difference between a high- authority controller with and without LAC. The plots show that LAC does not significantly influence the speed of convergence.

However, the MSE improves with approximately 1.5 to 2 dB.

The same experiment was also carried out for the feedforward HAC algorithm. In this scenario, the reference signal was directly taken from the noise generator. The results of this measurement can be found in Fig. 30. In this scenario, the LAC unit was again switched off for the first 5 seconds and then switched on for the remainder of the duration. It can be concluded that the LAC unit does not influence the speed of convergence. However, it does improve the MSE with approximately 5 dB.

For both feedback HAC and feedforward HAC, LAC does not modify the speed of convergence. Apparently, the preconditioning part of the RMFeLMS algorithm, which is designed to remove the eigenvalue spread of the autocorrelation matrix of the filtered reference signal [4], works as expected since LAC has a significant influence on this eigenvalue spread.

Influence of LAC on the robustness

A subsequent set of tests was performed to study the influence of LAC on the robustness of the controller. The robustness was evaluated by adding different weights to the piezoelectric panel. The high-authority controller for these tests was based on a model that was obtained without the additional weight. Fig. 31 shows the phase difference between the situations with added mass and without added mass of the models as identified for the high- authority controller for the cases that the low-authority controller was switched on and for the case that the low-authority controller was switched off. In this figure, it can be seen that for low frequencies the system is less sensitive to the addition of mass when the low-authority controller is switched on. Therefore it was expected that the robustness of the present high- authority controller would benefit from the addition of the low- authority controller since the robustness of the adaptive high -authority controller is primarily determined by the phase of the secondary path [4] .

Indeed, the robustness of the adaptive controllers improved by the addition of the low-authority controller. The results for the robustness of the adaptive feedback controller are summarized in Table 4. This table contains the average reductions of the error signals provided the system was stable. The reductions are given for different values of added weight as well as different values of the regularization of the controller. It can be seen from the results in table 4 that, firstly, larger weights can be added if LAC is switched on, and therefore the robustness improves by the addition of LAC, secondly, that higher levels of the regularization parameter β also lead to increased robustness, and thirdly, for low values of β the improvement by LAC is marginal. Furthermore, the regularization by β does not seem to influence the robustness if LAC is switched off. Especially for regularization levels of β =-20 dB and β =-25 dB there is a significant improvement of the robustness by the addition of LAC. Similar results can be found for the feedforward controller, as shown in Table 5, although in this case the emphasis of the improvement is on the reduction of the mean square value of the error signal instead of the robustness.

The relative importance of the robustness and the reduction of the mean- square error is influenced by the value of the regularization level β, which was not equal for the feedforward controller and the feedback controller. The regularization level for the feedback controller was set to a somewhat higher level than for the feedforward controller. One reason is that, on the one-hand, it doesn't make sense to use an extremely small regularization level for feedback controllers since it will not reduce the mean-square error anymore. On the other hand, if the value of the regularization level is set too high for a feedforward controller then the performance gain of a feedforward controller over a feedback controller is relatively small. An interesting observation was that the robustness of the adaptive feedback controller also increased if the model was obtained with LAC switched on but for which, during control operation, the LAC was switched off.

The results of these tests can be found in Table 6. Apparently, the reduced phase in the model itself is beneficial for the robustness of the controller. This suggests that the addition of numerical damping to the a- priori determined transfer functions would lead to improved robustness. By doing so one could obtain more robust controllers without the additional effort of implementing the decentralized controllers. Nevertheless, it was found that the full HAC-LAC control strategy resulted in the best performance and robustness properties.

Conclusion for part II

In this part, real-time results were shown of a combination of fixed decentralized feedback control (low -authority control) with multiple-input multiple -output adaptive control (high- authority control). The system was applied to a panel with piezoelectric actuators, piezoelectric sensors, and acceleration sensors. The HAC/LAC architecture was realized as a highspeed decentralized controller on a field programmable gate array (FPGA) and a medium speed centralized controller on a central processing unit

(CPU).

Table 4: Influence of added weight on performance of feedback controller.

Reduction after 180sec (average MSE reduction (dB) over all 5 sensors). Step a = ._n . IMC and RMFeLMS algorithm identified without the additional weight.

Table 5: Influence of added weight on the performance of a feedforward controller.

The performance was measured after 180 seconds. A step size a =— was used. The

40

model for the RMFeLMS algorithm was identified without additional weight.

Table 6: Influence of added weight on the performance of a feedback controller. The performance was measured after 180 seconds. A step size a =— was used. In this

40

case the LAC unit was switched off but the model used LAC and was used for IMC and the RMFeLMS controller. This model was identified without additional weight but with LAC switched on. The regularization level β was set to -25 dB.

For the configurations that were studied, the increase in robustness was most noticeable for an adaptive feedback controller, whereas, for the adaptive feedforward controller, the improvement of performance was most noticeable with respect to the reduction of the mean- square value of the error signals.

It was found that low authority control has no significant influence on the speed of convergence of the high authority controller as used in this part, both for feedback and feedforward configurations.

It was also shown that secondary path models with added damping, as obtained with low-authority control, can lead to improved robustness even if high-authority feedback control is used without low-authority control. This suggests that artificially added damping, which could be added numerically to the models obtained from system identification, can lead to improved robustness properties.

Part III

Model errors in multiple-input multiple-output (MIMO) adaptive controllers for reduction of broadband noise and vibrations may lead to unstable systems or increased error signals. In this part a combination of high- authority control (HAC) and low- authority control (LAC) is considered for improved performance in case of such model errors. A digital

implementation of a control system is presented in which the HAC (adaptive MIMO control) is implemented on a CPU and in which the LAC (decentralized control) is implemented on a high-speed Field Programmable Gate Array. Experimental results are given which demonstrate that the HAC/LAC combination leads to performance advantages in terms of stabilization under parametric uncertainties and reduction of the error signal.

Many algorithms used for broadband active noise control are based on the adaptive Least-Mean- Square (LMS) algorithm [17]. The low complexity and the relatively good robustness properties are the major advantages of the LMS algorithm. Recent algorithms solve many of the problems associated with the speed of convergence of the older algorithms. The basis for a particular class of such algorithms has been given by Elliott [18] as the preconditioned LMS algorithm. The version based on the filtered- error algorithm [1] is more efficient for multiple reference signals than the filtered-reference algorithm. A proper implementation of the filtered- error preconditioned LMS algorithm solves many of the problems associated with early implementations of the LMS algorithm, such as slow convergence due to frequency dependence of the secondary path and cross-coupling in the secondary path [3]. However, the controller is model-based and is therefore still sensitive for mismatch between the model and the plant. This model mismatch reduces the overall performance of the controller. Model mismatch can be caused by variations in parameters such as temperature, boundary conditions etc. For some control schemes, on-line adaptation of the model is possible in principle but a large amount of additional noise has to be injected in the system for rapid changes in the model [19]. Furthermore, if the controller uses model-based preconditioning or factorization, then these time-consuming operations should be performed on-line as well.

Robust control approaches are known [5] as well as probabilistic methods leading to frequency dependent regularization for optimum filtering [6, 7] and adaptive control [4, 8]. Such algorithms can be tuned for a particular application but require additional effort in the design stage and presume that sufficient a-priori knowledge is available about the uncertainty. An alternative approach is to use a high- authority and low-authority control (HAC/LAC) architecture [11] where the goal of the low-authority controller is to add active damping to the structure. Active damping can be

implemented using different strategies. The use of a HAC/LAC architecture yields three major advantages [11]. Firstly, the active damping extends outside the bandwidth of the HAC control loop, which reduces the settling times outside the control bandwidth. Secondly, it is easier to gain- stabilize the modes outside the bandwidth of the outer loop. And thirdly, the large damping of the modes inside the controller bandwidth makes them more robust to parametric uncertainty. In the paper by Herold et al. [20], a method using piezoelectric sensors and actuators and positive position feedback (PPF) was described. In the PPF-method, a second-order filter is used as the control filter which is combined with positive feedback. The control filter is then tuned to reduce one of the desired resonance peaks. In the present part, an approximately collocated and dual sensor-actuator pair is used, suitable for broadband damping as described by Elliott et al. [21]. If the actuator-sensor system is dual and collocated, a simple decentralized proportional feedback controller is sufficient to add damping due to the fact that the overall energy that is stored in the system will be reduced [11]. As such, less detailed a-priori information is required about the model uncertainty. Active damping is not very effective for frequencies that do not coincide with the poles and zeros. To gain further reductions for such frequency components a model-based controller is used such as the

RMFeLMS algorithm as described in this part. Section 2 gives a description of the panel, the control architecture, the control hardware and the particular implementation of the adaptive MIMO control algorithm. Section 3 presents results on the design of the decentralized feedback loops and the combination of decentralized feedback with adaptive MIMO control. Results are given of the speed of convergence, reduction of the mean-squared error and the robustness of the system. Methods

The HAC/LAC architecture was tested on a panel with piezoelectric transducers for reduction of noise transmission, a cross-section of which can be found in Fig. 18. The dimensions of the panel and the positions of the actuators and sensors are given in Fig. 19. The height and width of the piezoelectric actuators and sensors are both 76 mm; the thickness is 0.5 mm. Nine piezoelectric patch actuators and nine piezoelectric patch sensors were attached to the panel [3], of which the middle pair and the four pairs in the corners were used. The panel was built from two Printed Circuit Boards (PCBs) with a honeycomb layer in between. One advantage of this approach is that electronics can be integrated. Another advantage is that the actuator and the sensor can be placed on different faces of the panel, which improves the control of the acoustically relevant out-of-plane vibrations because the in-plane coupling between the actuator and the sensor is reduced [26]. Five collocated accelerometers were used for active damping with decentralized control using the piezoelectric patch actuators. The control results were obtained with a perspex box on which the panel was mounted. Inside this perspex box, noise was created with a loudspeaker which led to vibrations of the panel [3] . It is noted that large reductions of the error signals may lead to pinning of the control locations and thus to formation of a new boundary condition yielding additional resonances at higher frequencies in the sound transmission spectrum [21]. These new resonances may cause an increase in the transmitted sound. Noise reductions for a similar panel can be found in Ref. [3].

Active damping is often realized using an analog controller [27, 28, 29]. One of the advantages of an analog controller is its low delay when compared to a digital controller. In this part, a digital controller with a high- sample rate is used, such that the analog and digital controllers have identical performance for frequencies within the control bandwidth. The dedicated analog interface board was developed containing ADDA

downsampling. This made it possible to run the LAC and HAC controller at different sample rates, i.e. 100 kHz and 2 kHz, respectively. The interface between the PCI- 104 system running the HAC algorithm and the ADDA unit was implemented in reconfigurable hardware, in this case a Field Programmable Gate Array (FPGA), (See Fig. 20). The FPGA incorporates the following functional units: the decimation filters, the interpolation filters, the glue logic for the PCI bus interface and the low-authority controller. The decimation filters and interpolation filters were designed in such a way that the desired compromise between group- delay, filter transition band characteristics and stopband attenuation was obtained.

U ^". ! // + 1) = U i // i - i n - ¾) , ₍₂₇₎ where the i-th set of coefficients of the control filters are represented by the M x matrix Wi, where ^{ = ^{)-^NW ^~ 1 , i.e.. W(q) =∑-J¾ ζΓ'Ή⁷* , where q is the unit delay operator. Furthermore, e"(n) is the M x 1 vector of auxiliary error signals, x'(ri) is the K x 1 vector of delayed reference signals, and a is the convergence coefficient. In the actual implementation, a normalized LMS update rule was used, combined with 'leakage' of the control coefficients [4] . The regularization was implemented by defining an augmented plant G (q) :

G(q) =

G_reg (q)

in which the L x M- dimensional secondary path G(q) is augmented with an L ' x -dimensional transfer function Greg(q). For the implementation as described in this part, the regularization was based on a simple weighting of the M x 1 vector of control signals u(n), which also limits the inversion of zeros in G(q) that are close to the unit circle, resulting in more stable behavior of the M x -dimensional inverse G₀ {q)^~l . The regularizing transfer function Greg was define in which β is a scalar quantity and in which IM is an M x M identity matrix. The allpass factor G_i and the minimum-phase factor G₀ are obtained from an inner-outer factorization such that G = G_tG₀ . The adjoint G_i is combined with a delay D of ND samples in order to ensure that DG_i is predominantly causal. The transfer function G_rp subtracts the contribution of the actuators on the reference signals, as required for internal model control (IMC) [10].

Results

Influence of LAC on the secondary path

For the idealized low-authority controller having the purpose to add damping to the system the phase should be between -90 and +90 degrees for each collocated pair. Figs. 23 and 24 can be used to judge the practical setup involving the piezoelectric patch actuator and the accelerometer regarding this requirement. In Fig. 23, it can be seen that for frequencies up to 1 kHz the phase is between 0 degrees and +180 degrees. Thus, with an integrator the phase will be between -90 and +90 degrees. In Fig. 24, it can be seen that for higher frequencies the phase lag is larger. Based on these results, the decentralized controllers were configured with an integrator and a 1st- order roll-off above 1 kHz. The gain was adjusted in such a way that a gain margin of at least 6 dB was obtained for each collocated pair, resulting in a maximum feedback gain of 64. For lower feedback gains the gain margin was higher. Significant cross-talk exists between the non-collocated pairs, which for some frequencies can be as large as the transfer function between the collocated pairs. The results confirm that if the actuator and sensor are located at the same place and are energetically conjugated, or

approximately as in this part, then the controller can be implemented in a decentralized manner [11, 27]. The influence of LAC on HAC is taken into account in the system identification for HAC. The influence of LAC on the measured transfer function for HAC is shown in Fig. 25. It can be seen that resonances and antiresonances can be damped, particularly at low frequencies. For higher frequencies and for higher feedback gains some spillover can be observed. For this particular configuration also a controller configuration was tested in which the center actuator-sensor pair had a higher gain than the other pairs (see Fig. 25). Among the configurations studied, this configuration was found to give the best compromise between damping performance and spillover. Selection of the convergence coefficient

In order to select a suitable value of the convergence coefficient a, the influence of a on the convergence speed and MSE was studied by measuring the error signals for a duration of 60 seconds. At the beginning of each measurement the controller was switched off for 5 seconds and then it was switched on for 55 seconds. The reduction was calculated by taking the first 4 seconds and the last 4 seconds, using ensemble averages based on 32 measurements. For the feedback controller the following parameters were used: "^ ¹ · ^~~ ^ ^{D ~} ' ^~ ^ , in which γ sets the amount of leakage [4] of the control coefficients. The regularization parameter β was set to -20 dB. In Table 1, the result for a number of different step sizes a is given. A plot of the different convergence curves is not included due to the fact that the curves were almost identical: only the MSE errors were different. From Table 7, it can be concluded that =— gives the best

40

possible reduction after 60 seconds.

Table 7: Reduction of the error signals for a feedback controller for different step sizes a after 60 seconds.

■a l j_ .1

5. Si ^■zo 4:0 1SS

MSE Mict on 7.8 dB 94 dB S,3 dB tu) dB 8,i

For the feedforward controller, the following parameters were used:

Nw = '3^0^, Λ ΐ⁾— b0. /— 10 . The regularization parameter _j5 was set to -30 dB. The procedure used to measure the results for feedforward control was the same as used in the feedback scenario described in the previous paragraph, this time using 16 measurements. The different MSE values after convergence for different values of a can be found in Table 8. Table 8: Reduction of the error signals for a feedforward controller different step sizes a after 60 seconds

1. 3.0

MSE r< :- A dB 22, 1 dE 22.4 dB 22,6 dB

The convergence curves for different values of a can be found in Figure 26. From these results, it was concluded that an ce of— was a good

40

trade-off between convergence speed and steady-state MSE for HAC.

Influence of LAC on the steady state mean square error

Control results for a HAC/LAC architecture using an adaptive MIMO feedback algorithm are shown in Fig. 27. The convergence coefficient was set to a ₀f— (see Subsection titled 'selection of the convergence coefficient' of

40

part III herein). It can be seen that the reduction of the error signals for MIMO control (HAC) is higher than for the decentralized control (LAC). The combination of HAC and LAC leads to the largest reduction of the error signals. The average improvement by adding the low-authority controller to the high-authority controller is approximately 1.4 dB.

Control results for a HAC/LAC architecture using an adaptive MIMO feedforward algorithm are shown in Fig. 28. The convergence coefficent was set to of— (see Subsection titled 'selection of the convergence coefficient'

40

of part III herein). It can be seen that the reduction of the error signals for MIMO control (HAC) is considerably higher than for the decentralized control (LAC). As with feedback control the combination of HAC and LAC leads to the largest reduction of the error signals. The average improvement by adding the low- authority controller to the high- authority controller is approximately 4.4 dB. Influence of LAC on the convergence speed

The objective was to find out how LAC influences the overall performance in terms of speed of convergence. Based on the subsection titled 'selection of the convergence coefficient' of part II herein it was decided to set a to a value of— for the feedback controller as well as for feedforward

40

controller. The convergence under influence of LAC using a feedback HAC strategy can be found in Figure 29. To show the impact of the LAC clearly, it was decided to leave the LAC switched off for the first 5 seconds and switched on for the remainder of the time. The two plots in the Figure 29 demonstrate the difference between a high-authority controller with and without LAC. The plots show that LAC does not significantly influence the speed of convergence. However, the MSE improves with approximately 1.5 to 2 dB.

The same experiment was also carried out for the feedforward HAC algorithm. In this scenario, the reference signal was directly taken from the noise generator. The results of this measurement can be found in Fig. 30. In this scenario, the LAC unit was again switched off for the first 5 seconds and then switched on for the remainder of the duration. It can be concluded that the LAC unit does not influence the speed of convergence. However, it does improve the MSE by approximately 5 dB.

Influence of LAC on the robustness

A subsequent set of tests was performed to study the influence of LAC on the robustness of the controller. The robustness was evaluated by adding different weights to the panel. The high-authority controller for these tests was based on a model that was obtained without the additional weight. Fig. 31 shows the phase difference between the situations with added mass and without added mass of the models as identified for the high-authority controller for the cases that the low-authority controller was switched on and for the case that the low-authority controller was switched off.

In this figure, it can be seen that for low frequencies the system is less sensitive to the addition of mass when the low-authority controller is switched on. Therefore it was expected that the robustness of the present high- authority controller would benefit from the addition of the low- authority controller since the robustness of the adaptive high -authority controller is primarily determined by the phase of the secondary path [4] .

Indeed, the robustness of the adaptive controllers improved by the addition of the low-authority controller. The results for the robustness of the adaptive feedback controller are summarized in Table 9. This table contains the average reductions of the error signals provided the system was stable. The reductions are given for different values of added weight as well as different values of the regularization of the controller. It can be seen from the results in table 9 that, firstly, larger weights can be added if LAC is switched on, and therefore the robustness improves by the addition of LAC, secondly, that higher levels of the regularization parameter β also lead to increased robustness, and thirdly, for low values of β the improvement by LAC is marginal.

Furthermore, the regularization by β does not seem to influence the robustness if LAC is switched off. Especially for regularization levels of β =- 20 dB and β =-25 dB there is a significant improvement of the robustness by the addition of LAC. Similar results can be found for the feedforward controller, as shown in Table 10, although in this case the emphasis of the improvement is on the reduction of the mean square value of the error signal instead of the robustness. The relative importance of the robustness and the reduction of the mean- square error is influenced by the value of the regularization level β, which was not equal for the feedforward controller and the feedback controller. The regularization level for the feedback controller was set to a somewhat higher level than for the feedforward controller. One reason is that, on the one-hand, it does not make sense to use an extremely small regularization level for feedback controllers since it will not reduce the mean-square error anymore. On the other hand, if the value of the regularization level is set too high for a feedforward controller then the performance gain of a feedforward controller over a feedback controller is relatively small. An interesting observation was that the robustness of the adaptive feedback controller also increased if the model was obtained with LAC switched on but for which, during control operation, the LAC was switched off.

The results of these tests can be found in Table 11. Apparently, the reduced phase in the model itself is beneficial for the robustness of the controller. This suggests that the addition of numerical damping to the a- priori determined transfer functions would lead to improved robustness. By doing so one could obtain more robust controllers without the additional effort of implementing the decentralized controllers. One technique to realize the numerical damping is to apply an LQR regulator [5] to the identified plant and set the weighting matrices in such a way that the desired amount of damping is obtained. Alternatively, if the plant contains significant phase delays, one could add damping to the minimum-phase factor of the plant. Nevertheless, it was found that the full HAC-LAC control strategy resulted in the best performance and robustness properties.

Conclusion for part III

In this part, real-time results were shown of a combination of fixed decentralized feedback control (low -authority control) with multiple-input multiple -output adaptive control (high- authority control). The system was applied to a panel with piezoelectric actuators, piezoelectric sensors, and acceleration sensors. The HAC/LAC architecture was realized as a high- speed decentralized controller on a field programmable gate array (FPGA) and a medium speed centralized controller on a central processing unit (CPU). For the configurations that were studied, the increase in robustness was most noticeable for an adaptive feedback controller, whereas, for the adaptive feedforward controller, the improvement of performance was most noticeable with respect to the reduction of the mean- square value of the error signals.

Table 9: Influence of added weight on the performance of a feedback controller. The reduction was measured after 180 seconds and was the average MSE reduction in dB over all 5 sensors. A step size a. = - ^Λ was used. The model for IMC and the RMFeLMS algorithm were identified without the additional weight.

Table 10: Influence of added weight on the performance of a feedforward controller.

The performance was measured after 180 seconds. A step size _a =— ^was used. The

40

model for the RMFeLMS algorithm was identified without additional weight.

Table 11: Influence of added weight on the performance of a feedback controller.

The performance was measured after 180 seconds. A step size Q: = _Lwas used. In

40

this case the LAC unit was switched off but the model used LAC and was used for IMC and the RMFeLMS controller. This model was identified without additional weight but with LAC switched on. The regularization level β was set to -25 dB.

Remarks for whole document

The present invention has been described in terms of some specific embodiments thereof. It will be appreciated that the embodiments shown in the drawings and described here and above are intended for illustrative purposes only, and are not by any manner or means intended to be restrictive on the invention. The context of the invention discussed here is merely restricted by the scope of the appended claims. List of reference signs figures 1, 2 and 3

Any units in figures 1, 2 and 3 performing similar functions have received equal reference signs, as listed below:

1. reference signal connector

2. control filter

3. secondary source

4. secondary source signal connector

5. sensor

6. sensor connector

7. first delay

8. time reversed secondary path filter

9. update control filter (for updating control filter 2)

10. least mean square module

11. second delay

13. third delay

15. internal model control feedback module

16. internal model control unit

17. primary source (transfer function model)

18. minimum-phase inverse filter (inverse minimum-phase or outer- factor)

23. regularization unit

In figures 1 and 2, the adaptation circuit includes units 7, 8, and 10.

In figure 3, the adaptation circuit includes units 7, 8, 9 and 10.

The figures are schematic and may include additional units in reality, which are left out in figures 1, 2 and 3 such as to reduce complexity for explanatory purpose. References to literature in text

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[11] A. Preumont. Vibration control of active structures. Kluwer Academic Publishers. Dordrecht. 1997.

[12] E. A. Wan, "Adjoint LMS: an efficient alternative to the filtered-X LMS and multiple error LMS algorithms," in Proc. Int. Conf. on Acoustics, Speech and Signal Processing ICASSP96 (IEEE, Atlanta, 1996), pp. 1842- 1845. [13] E. Bjarnason, "Analysis of the Filtered-X LMS algorithm," IEEE Transactions on Speech and Audio Processing 3, 504-514 (1995).

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Claims

1. An active noise reducing filter apparatus for actively reducing noise d from at least one primary noise source reference signal x,

comprising:

an adaptation circuit for adapting said control filter W, said adaptation circuit connected to said sensor connector for receiving said error signal e and to said reference signal connector for receiving said reference signal x, and said adaptation circuit being arranged for adapting said control filter W based on said error signal e and said reference signal x using a

representation of said secondary source including a secondary source transfer function G of said secondary source and an additional damping.

2. An active noise reducing filter apparatus in accordance with claim 1, wherein said representation of said secondary source is a numerical representation of said secondary source transfer function G of said secondary source, additionally including a numerical damping in addition to said secondary source transfer function G.

3. An active noise reducing filter apparatus in accordance with any of the previous claims, wherein said adaptation circuit comprises a secondary path filter comprising said representation of said secondary source.

4. An active noise reducing filter apparatus in accordance with claim 2 and 3, wherein said secondary path filter having a transfer function G* being a time reversed transposed version of said numerical

representation of said secondary source transfer function G including said additional numerical damping.

5. An active noise reducing filter apparatus in accordance with any of the previous claims, further comprising a regularization filter Greg such as to provide regularized secondary source transfer function G based on said numerical representation of said secondary source transfer function G including said additional numerical damping, for use by said adaptation circuit.

6. An active noise reducing filter apparatus in accordance with claim 5, further comprising an minimum-phase inverse filter G₀ ^_1 connected to said control filter for receiving said control signal and for calculating said secondary source signal u; wherein said minimum-phase inverse filter is obtained by computing the inverse of an minimum-phase factor Go obtained from an inner-outer factorization G = Gt G₀ of an open loop transfer path between said secondary source signal u and said error signal e.

7. An active noise reducing filter apparatus in accordance with claim 4 and 6, wherein said transfer function G* of said secondary path filter is provided by a time-reversed transpose of an allpass factor G, obtained from said inner-outer factorization G = Gi G₀ for yielding G* = G,^■ * .

8. An active noise reducing filter apparatus in accordance with any of the claims 5-7, wherein said regularization filter is a frequency dependent regularization filter.

9. An active noise reducing filter apparatus in accordance with claim 8, wherein said regularization filter is frequency dependent at least such that for any signal frequencies below a cut-off frequency, said regularization filter has different values than for any signal frequencies above said cut-off frequency, with a transition frequency range around said cut-off frequency.

10. Method of manufacturing an active noise reducing filter apparatus for actively reducing noise d from at least one primary noise source reference signal x, said apparatus comprising:

- an adaptation circuit for adapting said control filter W, said adaptation circuit connected to said sensor connector for receiving said error signal e and to said reference signal connector for receiving said reference signal x, and said adaptation circuit being arranged for adapting said control filter W based on said error signal e and said reference signal x using a representation of said secondary source wherein said method comprising a step of modeling said representation such as to include a secondary source transfer function G of said secondary source and an additional damping.

11. Method according to claim 10, wherein said representation of said secondary source is a numerical representation of said secondary source transfer function G of said secondary source, and wherein said step of modeling includes a step of iteratively optimizing a numerical damping included in said numerical representation in addition to said secondary source transfer function G.

12. Method according to claim 10 or 11, wherein said active noise reducing filter apparatus further comprises means for performing high authority control and low authority control for providing said secondary source signal u, wherein said step of modeling is performed by:

- performing a system initiation step including both a step of a high authority control and a step of low authority control such as to adapt said representation of said secondary source transfer function G to a transfer function including said low authority control;

- performing a normal operation step including high authority control in absence of low authority control, using said adapted

representation of said secondary source transfer function G established during said system initiation step.