# Dynamics of the harmonic acoustic pneumatic source for active control of time-varying tonal noise

Alexandre Schiavini, Philippe Micheau, Pierre Grandjean, Gwénaël Gabard, Thomas Humbert

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Alexandre Schiavini, Philippe Micheau, Pierre Grandjean, Gwénaël Gabard, Thomas Humbert. Dynamics of the harmonic acoustic pneumatic source for active control of time-varying tonal noise. Applied Acoustics, 2026, 245, pp.111219. 10.1016/j.apacoust.2025.111219. hal-05441126

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# Dynamics of the harmonic acoustic pneumatic source for active control of time-varying tonal noise☆

Alexandre Schiavini<sup>a, b, c,*</sup> ◐, Philippe Micheau<sup>a, b</sup>, Pierre Grandjean<sup>a, b</sup>, Gwénaël Gabard<sup>c ◐</sup>, Thomas Humbert<sup>c, d</sup>

## HIGHLIGHTS

* Detailed modelling and analysis of the dynamic response of a Harmonic Acoustic Pneumatic Source (HAPS) and its associated control system.

* The convergence rate of the control system can be significantly improved, within the limitations of the electromechanical components.

* The HAPS can quickly attenuate tonal noises that vary slowly in amplitude and phase.

* The HAPS can be used as a narrow-band anti-noise source rather than a purely tonal source.

## ARTICLE INFO

**Keywords:**

Acoustic pneumatic source

Modulator

Active noise control

Near-field controller

Complex envelope controller

## ABSTRACT

The Harmonic Acoustic Pneumatic Source (HAPS), a mechanical airflow modulator, is a promising alternative to loudspeakers as secondary sources for the active control of harmonic noise, thanks to its ability to generate high sound pressure levels and to precisely control signal frequency, phase, and amplitude. However, this source requires a specialized control strategy, as it operates through mechanical modulation at a predetermined frequency. Hence, conventional adaptive control techniques are not directly applicable. Furthermore, when dealing with time-varying harmonic noise, the dynamic performance of the control system, which is influenced by the dynamic properties of the HAPS, is challenged. The originality of this study lies in considering this source, not as a tonal source, but as a narrow-band mechanical modulator that generates a narrow-band anti-noise by driving its complex envelope, allowing the active control of time-varying primary noise. A model of the dynamic response of the Harmonic Acoustic Pneumatic Source and of its controller is presented, with an analysis of the theoretical dynamic performances and limitations of this control system. The controller uses either a far-field error microphone or a near-field error microphone with a previously established compensation strategy. The closed-loop model is validated with experimental results. Simulations and experiments of far-field and near-field active noise control on slow time-varying primary noise are conducted to characterize the dynamic performances and limitations of the controller. The simulations are in good agreement with the experimental results.

## 1. Introduction

The noise generated by aircraft engines is an important cause of noise pollution near airports. Notably, the fan is one of the dominant sources of noise in a turbofan engine during takeoff [1,2]. It generates a harmonic

noise whose fundamental frequency is related to the number of blades of the fan and its speed of rotation. The common method to reduce the radiated sound uses passive treatments, like honeycomb structures placed inside the nacelle, which are efficient at reducing fan noise as long as

<footnote><sup>a</sup> Mechanical Engineering Dept., Engineering Fac., Université de Sherbrooke, Sherbrooke, J1K 2R1, Québec, Canada</footnote>

<footnote><sup>b</sup> Centre of Research Acoustics-Signal-Human of Université de Sherbrooke, Sherbrooke, J1K 2R1, Québec, Canada</footnote>

<footnote><sup>c</sup> Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, 72000, France</footnote>

<footnote><sup>d</sup> Univ. Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia, UMR 8520 - IEMN, Lille, 59000, France</footnote>

<footnote>* This document is the results of the research project funded by Safran Nacelles and the Agence Nationale de la Recherche (France) through the Industrial Chair ASTRAL (grant ANR-20-CHIN-00050-01).</footnote>

<footnote>* Corresponding author at: Mechanical Engineering Dept., Engineering Fac., Université de Sherbrooke, Sherbrooke, J1K 2R1, Québec, Canada.
Email address: alexandre.schiavini@usherbrooke.ca (A. Schiavini).</footnote>
enough surface is treated [3,4]. However, for high bypass ratio engines, the trend is to shorten the nacelle, thus reducing the area available for acoustic treatments, and to increase the fan diameter, which implies a reduced rotation speed and thus a lower blade passing frequency [5]. Conventional passive absorbers are no longer optimal at low frequencies because they need to be thickened beyond acceptable integration constraints.

Therefore, active noise control has been studied as a solution for tonal fan noise reduction [6,7]. The high noise level required for this application is a limiting factor when using electrodynamic loudspeakers as secondary sources. This suggests that pneumatic sources could be good candidates for secondary sources. Indeed, they are known for their ability to produce high sound levels, as they are for example used to provide acoustic power up to 150–200 kW in reverberant rooms [8]. In 2021, Allard et al. designed a source that can generate a high tonal sound level (177 dB) under a significant mean flow (M > 0.1) while keeping a low harmonic distortion [9]. A few studies have investigated active noise control using pneumatic sources. In 1990, Blondel and Elliott designed a subsonic pneumatic source for active control, allowing 25 dB of attenuation with a harmonic controller [10,11]. Among the electroacoustic sources, the Harmonic Acoustic Pneumatic Source (HAPS) has been designed to be used as a secondary source for tonal active noise control in turbojet engines [12], although it has a broader range of applications such as in ultrasound [13]. It is an electromechanical device that generates a high intensity harmonic sound from airflow modulation that can be precisely controlled in phase and amplitude. It relies on the rotation of a perforated cage in front of a mouth, generating a pulsed jet by chopping the airflow, and a flow control valve placed upstream. The feasibility of active control of stationary primary noises using a HAPS was demonstrated, first on a plane wave in a duct, allowing 20 dB of attenuation between 500 and 800 Hz [14]. Then, a ring of 6 HAPS was used to perform multimodal noise control in a duct, achieving more than 20 dB of attenuation, and a control strategy was elaborated in order to obtain attenuation in the acoustic far-field while using error microphones placed in the near-field of the HAPS [15]. Indeed, for turbofan noise control, error sensors would typically be placed close to the sources, and a direct minimization of the error signal at the near-field microphones generally leads to a suboptimal attenuation in the far-field. A compensation strategy is thus required, which can be related to virtual sensing techniques [16].

However, in practice, fan noise can vary in amplitude, phase and frequency due to changes in the engine operating conditions. Therefore, the dynamics of the control system have to be sufficiently fast to follow the evolution of the primary noise. Several studies have investigated harmonic control, but they are usually based on filtered-x least mean square (Fx-LMS) algorithms [17–19], and the secondary sources are usually electrodynamic loudspeakers [18–21]. Contrary to loudspeakers, the HAPS mechanically synthesizes the acoustic signal from slow time-varying amplitude and phase commands that modulate the noise at a given frequency. Therefore, its dynamic response, which depends on its mechanical components, i.e., a valve and a rotating flow chopper with a dedicated phase controller, differs from that of a loudspeaker. This motivates the need for a model of the HAPS’s dynamics capable of capturing both mechanical actuation and signal evolution, diverging from traditional Fx-LMS implementations, and enabling the use of HAPS as a secondary actuator for active noise control.

This work aims to provide a comprehensive model of the HAPS response and its integration into an adaptive control scheme for suppressing time-varying tonal noise, along with an experimental validation under realistic conditions. The originality of this work lies in a model of the HAPS’s dynamics, as well as a model of the control system, including the near-field compensation controller. Simulations of the proposed model are provided, along with experimental results of the active control of unsteady noises using both far-field and near-field error microphone strategies, enabling a thorough assessment of control performance under realistic, fluctuating conditions. The proposed framework enhances the

applicability of mechanical actuators in real-time adaptive noise control and contributes to the development of signal-driven strategies within complex aeroacoustic systems.

Section 2 presents the HAPS, the considered physical problem, the model of the dynamics and the controller design. The experimental setup is then described in Section 3. Results of the active noise control experiments and simulations are shown in Section 4 and discussed in Section 5.

## 2. Model

### 2.1. Description of the HAPS

The HAPS, illustrated in the dashed box in Fig. 1, is composed of a flow control valve and a chopper, which is a perforated cage rotating in front of a mouth and driven by an electric motor. The airflow between a plenum at high pressure $p_0$ and the exhaust at the atmospheric pressure is regulated by the time-varying throat orifice formed by the chopper and the valve. These two mechanical components of the HAPS can be modelled as a single equivalent fluctuating orifice of area $A(t)$ followed by a short duct of section $\sigma$ [22]. The rotation of the cage in front of the mouth creates a harmonic fluctuation of the area $A(t)$, whose fundamental frequency $f_0$ is related to the speed of rotation of the cage. The Fourier decomposition of $A(t)$ is

$$A(t) = \operatorname{Re} [A_0 + A_1 e^{j\omega_0 t} + A_2 e^{j2\omega_0 t} + \dots] \quad (1)$$

with $\omega_0 = 2\pi f_0$. The complex envelope of the first harmonic, corresponding to $\omega_0$, is $A_1$. The valve opening impacts the amplitude of $A_1$, while the angular position of the cage affects its phase. As described by the pneumatic speaker theory, the periodic fluctuations of the throat area generates a harmonic pressure signal [23]. For a choked flow at the throat under a quasi-static hypothesis, the first harmonic $Q$ of the acoustic mass flow at the outlet of the HAPS is related to $A_1$ [23]

$$Q = \rho \sigma \left( \frac{\alpha}{Z_t + Z_i} \right) A_1 \quad (2)$$

where $\rho$ is the air density, $Z_i$ is the internal acoustic impedance of the HAPS, $Z_t$ is the impedance at the throat and

$$\alpha = \beta / \bar{u}_t, \quad \beta = \left(\frac{5}{6}\right)^3 \frac{p_0 c_0}{\sigma} \quad (3)$$

with $\bar{u}_t$ the mean velocity at the throat and $c_0$ the speed of sound. Eq. (2) still holds if $A_1$ is time dependent, as long as its variations are slow compared to the frequency $f_0$. Therefore, amplitude and phase variations of the first harmonic of the acoustic signal is made possible by varying the first harmonic of the equivalent orifice area. The HAPS is controlled by two slow time varying real signals: $u_A(t)$ controlling the valve opening and $u_\phi(t)$ controlling the angle of the chopper. In order to control the angular position of the chopper, a closed-loop phase controller based on a Phase Locked Loop (PLL) algorithm is used. A proportional-integral (PI) regulator provides the appropriate motor speed command to synchronize the instantaneous angle $\theta(t)$ of the chopper, measured through a resolver, with the reference phase $u_\phi(t)$ [14]. Thus, the HAPS performs a mechanical modulation of the fluctuating area first harmonic which is described by the time-dependent complex envelope $A_1(t)$. According to (1)–(3) and the hypothesis of quasi-stationarity, a slow time-varying $A_1(t)$ implies a first harmonic acoustic mass flow time modulated both in amplitude and phase and defined as

$$q(t) = \operatorname{Re} [Q(t) e^{j\omega_0 t}] \quad (4)$$

where $Q(t) = q_A(t) e^{j\phi_\phi(t)}$ is the complex envelope of the first harmonic with $q_A(t)$ its amplitude and $\phi_\phi(t)$ its phase. The two time-varying command signals $u_A(t)$ and $u_\phi(t)$ that modulate the amplitude and the phase of the acoustic mass flow, respectively, are gathered into a single complex command signal $U(t) = u_A(t) e^{j\phi_\phi(t)}$.
<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic of the active noise control situation.</figcaption></figure>

## 2.2. Active noise control

The problem of active control of a tonal noise propagating in a duct, illustrated in Fig. 1, is considered. The nominal instantaneous frequency of the primary noise $f_0$ is assumed to be known. The primary wave propagates inside the duct and reaches the error microphone where it induces the signal $d(t)$, referred to below as the primary noise. The addressed problem of this paper is to attenuate a disturbance described as a narrow bandpass signal, expressed without loss of generality as $d(t) = \text{Re}[D(t)e^{j\omega_0 t}]$ with the complex envelope $D(t) = d_A(t)e^{j\phi(t)}$ defined by its time-dependent amplitude $d_A(t)$ and phase $d_\phi(t)$. The active control is performed using a HAPS as an original parametric acoustic source emitting anti-noise in this same narrow band centred on the primary frequency $f_0$. The transfer between the acoustic mass flow $q(t)$ generated by the HAPS and the anti-noise at the error microphone $s(t)$ is described by the impulse response $h(t)$. When both the primary source and the HAPS are turned on, the error signal is

$$y(t) = d(t) + s(t) = d(t) + h(t) * q(t) \quad (5)$$

where * is the convolution operator. Because signals $d(t)$ and $s(t)$, expressed in (5), are in the same narrow frequency band centred at the carrier frequency $f_0$, it is possible to translate the problem at 0 Hz with the complex envelope approach. In other words, for the considered narrow frequency band, it exists a complex envelope $Y(t)$ such that $y(t) = \text{Re}[Y(t)e^{j\omega_0 t}]$. To recover the complex envelope $Y(t)$ from the signal $y(t)$, a complex demodulation at the frequency $\omega_0$ is performed [24]:

$$Y(t) = 2f(t) * [y(t)e^{-j\omega_0 t}] \quad (6)$$

where $f(t)$ is the impulse response of a low-pass filter with a cutoff frequency $\omega_c \ll \omega_0$. Demodulating the error signal when only the primary source is activated, and the frequency support of the spectrum of $D(t)$ is included in the frequency band of the low-pass filter, gives the complex envelop of the primary noise $D(t) = d_A(t)e^{j\phi(t)}$. The acoustic transfer between the HAPS and the error microphone being defined by the impulse response $h(t)$, its frequency response is $h(f)$. Thus, the complex gain corresponding to the acoustic transfer evaluated at the frequency $f_0$ is $h(f_0)$. Considering the demodulation principle (6), the system presented in (5), and that the signals are narrow band centred at the frequency $f_0$, the equivalent low-pass expression of the error's complex envelope is approximated as $Y(t) = D(t) + h(f_0)Q(t)$. After heterodyning the transfer function $h(f)$ to shift the carrier frequency $f_0$ to base-band, $h(f_0)$ gives the complex gain at 0 Hz of the transfer between

the complex envelops $Q(t)$ and $Y(t)$. Since the mass flow is not experimentally measured in this work, this complex gain is rather expressed with the unit of the valve command, i.e., in pascals per volt, as

$$H = h(f_0)C_A \quad (7)$$

where $C_A$ is the gain in grams per second per volt between the valve electric command $u_A(t)$ and the amplitude $q_A(t)$ of the acoustic mass flow. The complex envelop of the error signal is then

$$Y(t) = D(t) + \frac{H}{C_A} Q(t). \quad (8)$$

The complex envelope $Q(t)$ of the mass flow is described as the output of a state-space model with inputs $u_A(t)$ and $u_\phi(t)$, which represents the HAPS dynamics and is detailed in the next section.

## 2.3. Model of the HAPS dynamics

The model considered in the present study is presented in Fig. 2, with separate dynamic models for the amplitude and the phase of the oscillating mass flow generated by the HAPS, since they are controlled by distinct electromechanical parts. A first-order model is used to represent the dynamics of the upstream servovalve

$$\frac{dq_A(t)}{dt} = -\frac{1}{\tau_A} q_A(t) + \frac{C_A}{\tau_A} u_A(t) \quad (9)$$

with its time constant $\tau_A$.

The phase of the complex command $u_\phi(t)$ is equal to $n$ times the motor phase command $\theta_r(t)$, with $n$ being the number of holes on the cage. The phase dynamics are related to both the properties of the motor used to drive the chopper and the tuning of the PLL feedback coefficients $K_p$ and $K_i$, the proportional and integral feedback gains, respectively. A first-order model with a time constant $\tau$ is chosen to represent the dynamics of the motor speed. A phase error is computed by subtracting the measured instantaneous angle of the chopper $\theta(t)$ from the motor phase command $\theta_r(t)$. The phase of the mass flow $q_\phi(t)$ is equal to $n$ times the motor phase $\theta(t)$. Introducing the proportional integral feedback between the phase error and the motor speed command gives a third-order dynamics model for the phase

$$\begin{pmatrix} \dot{q}_{\phi}(t) \\ \dot{x}_{2}(t) \\ \dot{x}_{3}(t) \end{pmatrix} = \begin{pmatrix} 0 & n & 0 \\ -\frac{K_p}{n\tau} & -\frac{1}{\tau} & \frac{1}{\tau} \\ -\frac{K_i}{n} & 0 & 0 \end{pmatrix} \begin{pmatrix} q_{\phi}(t) \\ x_{2}(t) \\ x_{3}(t) \end{pmatrix} + \begin{pmatrix} 0 \\ \frac{K_p}{n\tau} \\ \frac{K_i}{n} \end{pmatrix} u_{\phi}(t) \quad (10)$$

where $x_2(t)$ and $x_3(t)$ are state variables corresponding to the motor speed and the PLL integral feedback output, respectively. A saturation
<figure><img src="image_5.png" /><figcaption>Fig. 2. Block diagram of the model of the HAPS and the active control dynamics. A different controller is used depending on the location of the error microphone, either in the far-field of the HAPS or in the near-field.</figcaption></figure>

is introduced on $\dot{x}_2(t)$ to represent the limitation on the acceleration of the motor. Except during the initial acceleration required to reach the reference frequency, the acceleration limiter is not activated, thus the linear dynamic model (10) can be considered.

## 2.4. Controller design

In this work, two different controllers are studied, one uses an error microphone placed in the acoustic far-field of the HAPS, and the other uses an error microphone placed in its near-field, with the established compensation method developed in [15]. Therefore, a configuration with two microphones is considered. The signals corresponding to the near-field microphone are denoted with subscript NF, while those corresponding to the far-field microphone are denoted with subscript FF.

### 2.4.1. Far-field controller

The error signal in the far-field is $Y_{FF}(t)$ and is related to the HAPS command $U(t)$ by the complex gain $H_{FF}$ at 0 Hz. The controller equation for the configuration with the far-field microphone is based on an iterative algorithm to reach the theoretically optimal solution minimizing the error signal denoted in (8), known as the Gauss-Newton algorithm [25]

$$ \frac{dU(t)}{dt} = -\mu K Y_{FF}(t) \quad (11) $$

where $\mu > 0$ is the tunable real gain of the controller and $K = \hat{H}_{FF}^{-1}$ is a complex gain with $\hat{H}_{FF}$ an estimation of $H_{FF}$.

### 2.4.2. Near-field controller

When the error sensor is placed in the acoustic near-field of the HAPS, the minimization of the error signal is not equivalent to minimizing the radiated acoustic power in the far-field. An adaptation of the control strategy with near-field compensation is therefore required [15]. Fig. 3 illustrates this situation. The objective is to find a compensation term $M$ that will be used to correct the error signal in order to compensate for the effect of the evanescent modes in the near-field. As it will be seen, the value of the compensation term is independent of the primary noise. Hence, its expression can be obtained with stationary signals and without considering the dynamics in the system's transfers. The value thus obtained remains relevant for a varying primary noise as long as it varies in a sufficiently narrow bandwidth around $f_0$ for the transfers to stay valid. Therefore, all the transfers are here considered as static gains. The primary disturbances $D_{NF}$ in the near-field and $D_{FF}$ in the

<figure><img src="image_6.png" /><figcaption>Fig. 3. Block diagram of the near-field compensation configuration. Rectangle boxes correspond to the static gains of the transfers, and $P$ is constant. Adapted from [15].</figcaption></figure>

far-field are related to the stationary primary source command $P$ by the complex gains $G_{NF}$ and $G_{FF}$, respectively. The objective of this method is to design a controller that converges toward the optimal command $U_{opt}$ that produces the minimized far-field signal $Y_{FF, opt} = D_{FF} + S_{FF, opt}$ from the measurement of the near-field signal. When the optimal command is provided, the near-field signal is $Y_{NF, opt} = D_{NF} + S_{NF, opt}$. The optimal anti-noises $S_{NF, opt}$ in the near-field and $S_{FF, opt}$ in the far-field are related to the optimal HAPS command $U_{opt}$ by the complex gains $H_{NF}$ and $H_{FF}$, respectively. A corrected error signal $E(t) = Y_{NF}(t) - MU(t)$ is built from the near-field signal $Y_{NF}(t)$ in order to subtract its components that are due to the evanescent modes. The compensation term $M$ can be expressed as a function of the transfers of the system [15]

$$ M = H_{NF} - \frac{G_{NF}}{G_{FF}} H_{FF}. \quad (12) $$

Practically, since the system's primary transfer gains $G_{FF}$ and $G_{NF}$ are not necessarily known, the compensation term $M$ is learned experimentally
using a configuration with a microphone in the near-field and another in the far-field of the HAPS. A first learning step is required, in which the active control is performed on a stationary primary noise with the far-field error microphone, in order to learn the compensation term M. The residual signal Y_NF, opt in the near-field is measured when the optimal command U_opt minimizing the far-field error signal is provided to the HAPS. The compensation term is then computed as

$$M = \frac{Y_{\text{NF, opt}}}{U_{\text{opt}}}. \qquad (13)$$

Now going back to time-dependent complex envelops, the controller equation when using the near-field error microphone, ensuring that the command will converge to the optimal command, is

$$\frac{dU(t)}{dt} = -\mu K(Y_{\text{NF}}(t) - MU(t)) \qquad (14)$$

where K = ($\hat{H}_{\text{NF}} - M$)$^{-1}$ is a complex gain chosen such that the controller is stable [15], and $\hat{H}_{\text{NF}}$ is the estimation of $H_{\text{NF}}$. Therefore, the control dynamics is impacted by the controller gain K, which depends on the compensation term M, which itself depends on the acoustic transfers of the system. In Laplace notation, the controller equation (14) leads to the controller gain:

$$C(s) = \frac{U(s)}{Y_{\text{NF}}(s)} = \frac{-\mu K}{s - \mu K M} \qquad (15)$$

If the error microphone is placed in the far-field, the controller is equivalent to the far-field controller (11), the compensation term M is zero and the controller has a pole in s = 0. If the error microphone is placed in the near-field, C(s) has a pole in s = $\mu K M \neq 0$, as the residual signal in the near-field Y_NF, opt is generally not zero, and the static gain of the controller is C(j0) = C_opt = 1/M.

## 2.5. Closed loop analysis

The controller performs a demodulation of the error signal y(t) and provides the complex command U(t). Practically, the real and imaginary parts of the right-hand side of (11) and (14) are integrated. Since there is a distinct dynamic for the amplitude and the phase, expressing the dynamics with the complex numbers Q(t) = q_R(t) + jq_I(t) and U(t) = u_R(t) + ju_I(t) introduces nonlinearities due to polar to Cartesian coordinates transformation

$$\dot{q}_{\text{R}}(t) = \dot{q}_{\text{A}}(t) \cos(q_{\phi}(t)) - \dot{q}_{\phi}(t) q_{\text{A}}(t) \sin(q_{\phi}(t)) \qquad (16)$$

$$\dot{q}_{\text{I}}(t) = \dot{q}_{\text{A}}(t) \sin(q_{\phi}(t)) + \dot{q}_{\phi}(t) q_{\text{A}}(t) \cos(q_{\phi}(t)) \qquad (17)$$

with the relations between polar and Cartesian coordinates being $q_A(t) = \sqrt{q_R^2(t) + q_I^2(t)}$ and $q_\phi(t) = \arctan(q_I(t)/q_R(t))$. In the particular case of a constant phase, i.e., under the hypothesis that $\dot{q}_\phi = 0$ and $q_\phi = u_\phi$ is constant, the dynamics can be reduced to the valve dynamics, which can be expressed linearly in complex numbers as

$$\frac{dQ(t)}{dt} = -\frac{1}{\tau_A}Q(t) + \frac{C_A}{\tau_A}U(t). \qquad (18)$$

Applying (8) to the far-field error microphone case gives

$$Y_{\text{FF}}(t) = \frac{H_{\text{FF}}}{C_A} Q(t) + D_{\text{FF}}(t). \qquad (19)$$

Combining (11), (18) and (19) leads to the expression in Laplace domain of the closed loop transfer function between the error signal Y_{FF}(s) and the primary disturbance D_{FF}(s) when the far-field controller is used

$$\frac{Y_{\text{FF}}(s)}{D_{\text{FF}}(s)} = \frac{\tau_A s^2 + s}{\tau_A s^2 + s + \mu K H_{\text{FF}}} \qquad (20)$$

where s is the Laplace variable.

Applying (8) to the near-field error microphone case gives

$$Y_{\text{NF}}(t) = \frac{H_{\text{NF}}}{C_A} Q(t) + D_{\text{NF}}(t). \qquad (21)$$

Similarly, the expression in Laplace domain of the closed loop transfer function for the near-field controller is obtained by combining (14), (18) and (21):

$$\frac{Y_{\text{NF}}(s)}{D_{\text{NF}}(s)} = \frac{\tau_A s^2 + s(1 - \mu K M \tau_A) - \mu K M}{\tau_A s^2 + s(1 - \mu K M \tau_A) + \mu K(H_{\text{NF}} - M)} \qquad (22)$$

The closed loop transfer function in the far-field when the near-field controller is used can then be obtained knowing the values of the complex gains of the system

$$\frac{Y_{\text{FF}}(s)}{D_{\text{FF}}(s)} = \frac{H_{\text{FF}}G_{\text{NF}}}{H_{\text{NF}}G_{\text{FF}}} \frac{Y_{\text{NF}}(s)}{D_{\text{NF}}(s)} + 1. \qquad (23)$$

## 3. Experimental setup

The experimental setup is presented in Figs. 4 and 5. Measurements are performed in a semi-anechoic chamber, in which a semi open rectangular duct of length 147 cm, width 28 cm and height 15 cm is placed. A loudspeaker, placed on the upper wall of the duct, at 12 cm from its closed end, generates the primary noise at a frequency $f_0$. A HAPS is mounted on the side of the duct at a distance of 29 cm from the loudspeaker. A pressure tank maintained at $p_0 = p_{\text{atm}} + 15$ psi (103421 Pa), where $p_{\text{atm}}$ is the atmospheric pressure, provides compressed air to the HAPS. The number of holes on the chopper is $n = 8$. A microphone is placed in the acoustic near-field of the HAPS, at an axial distance of 9 cm from it, and another in the far-field at an axial distance of 80 cm. Both microphones are placed inside the duct, carried on a stand. The exact positions of the sources and the microphones are presented in Table 1. The controller and the simulation model are implemented using Simulink. A Speedgoat Performance real-time target machine is used to perform real time acquisition and to control the servodrive that drives the HAPS motor. The values used in the simulations for the time constants of the valve dynamic model and the motor speed first order model are $\tau_A = 0.1$ s and $\tau = 0.01$ s, respectively. The proportional and integral gains of the PLL are $K_p = 10$ and $K_i = 5$, respectively.

<figure><img src="image_7.png" /><figcaption>Fig. 4. Diagram of the experimental setup.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Pictures of the experimental setup. 1. Primary loudspeaker, 2. HAPS, 3. Semi-open duct, 4. In-duct microphone stand. 5. Plenum. 6. Servodrive. 7. Real-time computer.</figcaption></figure>

<table><caption>**Table 1**
Positions of the sources and the microphones in the duct.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Position</th></tr><tr><th>x (cm)</th><th>y (cm)</th><th>z (cm)</th></tr></thead><tbody><tr><td>Primary loudspeaker</td><td>12</td><td>14</td><td>15</td></tr><tr><td>HAPS</td><td>41</td><td>28</td><td>10</td></tr><tr><td>Near-field microphone</td><td>50</td><td>26</td><td>11.5</td></tr><tr><td>Far-field microphone</td><td>121</td><td>26</td><td>11.5</td></tr></tbody></table>

## 4. Results

For the experiments, a frequency $f_0 = 440$ Hz is chosen, for which only the plane wave propagates in the duct. Before performing active noise control, preliminary measurements are carried out to obtain the values of the estimated complex gains of the system. The values of $\hat{H}_{FF}$ and $\hat{H}_{NF}$ used in the controller are based on the measured outputs $Y_{FF}$ and $Y_{NF}$ for constant valve and phase commands ($U = 5$ V), and without primary noise ($P = 0$ V). The values are: $\hat{H}_{FF} = 4 - 15j$ Pa/V for the far-field microphone and $\hat{H}_{NF} = 3 - 19j$ Pa/V for the near-field microphone. In the same way, the primary complex gains $G_{FF}$ and $G_{NF}$ are obtained by measuring the signals $Y_{FF}$ and $Y_{NF}$ with a primary command $P = 1$ V and a secondary command $U = 0$ V. The values measured on the experimental setup are: $G_{FF} = -32 + 15j$ Pa/V and $G_{NF} = -7 + 117j$ Pa/V. The value of the compensation term $M$ is obtained from a first active control measurement on a stationary primary noise using the far-field microphone, and is computed with (13): $M = 70 + 13j$ Pa/V. The experimental results are compared to simulations obtained with a Simulink implementation of the dynamic model presented in Fig. 2. The values of

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Gain of the closed loop transfer function in the far-field $Y_{FF}(s)/D_{FF}(s)$, representing the ratio between the error signal and the primary disturbance at the far-field microphone, as a function of the frequency of the primary noise amplitude modulation. (a) Far-field controller. (b) Near-field controller.</figcaption></figure>

the estimated complex gains of the system are used in the simulations to represent the acoustic transfers at $f_0$.

### 4.1. Closed loop model

Before discussing the experimental results, an analysis of the closed loop transfers (20) and (22) obtained from the model for the considered experimental configuration is presented. Fig. 6 shows the amplitude of the closed loop transfer function for the far-field controller (20) and for the near-field controller (23), for different tunings of the feedback gain $\mu$, considering a perfect estimation of the system's complex gains, i.e., $\hat{H}_{FF} = H_{FF}$ and $\hat{H}_{NF} = H_{NF}$. The horizontal axis corresponds to the frequency of the amplitude modulation of the primary noise. The gain in decibels is thus the opposite of the attenuation of the fluctuating part of the primary noise. The value of $M$ used to compute the near-field closed-loop transfer (22) is obtained using (12). For both controllers, the transfer is a high-pass filter whose cutoff frequency increases with the gain $\mu$. Close to the zero frequency, the gain $Y_{FF}(s)/D_{FF}(s)$ of the closed loop theoretically approaches zero (i.e., high negative gain in dB) since the amplitude of the anti-noise tends to that of the primary noise. At high frequencies, the gain tends to one (i.e., 0 dB), because the amplitude of the anti-noise does not follow the fast amplitude variations of the primary noise. For $\mu = 1$, the controller attenuates the disturbance up to 0.4 Hz. Since the time constant of the valve dynamics model is $\tau_A = 0.1$ s, the performance is theoretically limited to a modulation frequency of $1/(2\pi\tau_A) = 1.6$ Hz. When increasing the value of $\mu$, a bump appears with positive gain, for example around 2 Hz with $\mu = 10$, showing a reduced robustness. The performance may then decrease in case of uncertainties in the estimation of the complex gains $\hat{H}_{FF}$ and $\hat{H}_{NF}$. The curves are similar for both controllers, although the bump that appears with high gain values like $\mu = 10$ is less pronounced with the near-field controller.

### 4.2. Active control with unsteady primary noise

Several experiments on the active control of stationary primary noises with different values of the controller gain $\mu$ have been performed. For values of $\mu$ above 0.8, oscillations in the valve command are observed. In the following active noise control results, a controller gain $\mu = 0.65$ is used. With this value, according to Fig. 6 and empirical testing, a modulation frequency of 0.1 Hz–0.2 Hz represents an upper
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Simulations and experimental results of active control of tonal noise at 440 Hz with primary amplitude modulation of 10% with frequency 0.2 Hz. The control is enabled at $t = 6$ s. (a) Far-field controller. (b) Near-field controller. Top: Noise level at the far-field microphone. The horizontal dashed-dot line indicates 15 dB below the mean primary noise level. Middle: Amplitude of the HAPS command. Bottom: Phase of the HAPS command.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Simulations and experimental results of active control of tonal noise at 440 Hz with primary phase modulation of 11.25° with frequency 0.1 Hz. The control is enabled at $t = 6$ s. (a) Far-field controller. (b) Near-field controller. Top: Noise level at the far-field microphone. The horizontal dashed-dot line indicates 15 dB below the mean primary noise level. Middle: Amplitude of the HAPS command. Bottom: Phase of the HAPS command.</figcaption></figure>

limit beyond which attenuation becomes ineffective. Experiments on the active control of primary noises modulated in amplitude or phase have been performed for several amplitudes and frequencies of modulation. However, only results corresponding to this upper modulation frequency limit are presented here. Simulations have also been performed using the model of the dynamics presented in Section 2.3 and the values of the system's complex gains given in Section 4.

### 4.2.1. Amplitude modulation

Fig. 7 shows the simulated and measured noise levels for the case of a varying primary noise amplitude for both the far-field controller (Fig. 7a) and the near-field controller (Fig. 7b). The primary noise is modulated in amplitude with sinusoidal variations of 10% at a frequency of 0.2 Hz. This modulation frequency corresponds to a limiting case where the predicted rejection of the amplitude fluctuation is almost zero (Fig. 6). For both controllers, a 15 dB attenuation is achieved 1.5 s after the controller is switched on. This attenuation is maintained despite

the variations in the primary noise. The measured noise level matches the simulation. The amplitude of the command fluctuates with the same frequency as the primary noise amplitude for both simulation and measurement. In contrast, an oscillation of about 10° in the command phase at the same frequency as the amplitude modulation is observed in the experiment but not in the simulation. The command phase is about 10° higher in the measurements than in the simulations.

### 4.2.2. Phase modulation

Fig. 8 shows the simulated and measured noise levels for the case of a varying primary noise phase for both the far-field controller (Fig. 8a) and the near-field controller (Fig. 8b). The phase of the primary noise undergoes a peak-to-peak sinusoidal variation of 11.25° at a frequency of 0.1 Hz. Again, a 15 dB attenuation is achieved after 1.5 s and is maintained despite the variations in the primary noise phase. The attenuation predicted by the simulation matches the measurement. The phase of the command fluctuates with the same frequency as the primary noise
phase fluctuations, both in the measurement and simulation. The simulation predicts that the amplitude of the command remains constant after 10 s. However, with the far-field controller (Fig. 8a), a small oscillating pattern at a higher frequency is observed in the command amplitude during the experiment, which could be related to some quantization of the valve opening. The command phase is again about 10° higher in the measurement than in the simulation.

## 5. Discussion

Experimental validation shows that the dynamics of the HAPS can be described with a first-order model for the valve and a first-order model for the chopper rotation speed coupled with the PLL feedback. The closed-loop analysis shows that the controller is not limited to purely tonal signals, since it can compensate low frequency variations in the amplitude of the primary noise, theoretically up to 1.6 Hz, depending on the tuning of the controller gain $\mu$. However, practical limits have been observed, as measurements with high gain values, i.e. $\mu > 0.8$, lead to oscillations in the command amplitude, which can be related to robustness issues or unmodeled dynamics such as valve quantization and hysteresis. According to the first-order model, a controller gain of $\mu = 0.65$, used in the experiments, corresponds to a cutoff frequency of approximately 0.2 Hz. Therefore, attenuation is only expected as long as the amplitude or phase modulation remains below this frequency limit. In experiments, the controller based on the Gauss-Newton algorithm, equivalent to an integral controller, is able to maintain a 15 dB mean attenuation of primary noises with slowly varying amplitude or phase, e.g., amplitude modulation of 10% at 0.2 Hz and phase modulation of 10° at 0.1 Hz. This can be achieved with both a far-field error microphone and a near-field error microphone using the appropriate compensation strategy.

A future step is to use the model of the dynamics presented here to study the impact of the compensation term $M$ used in the near-field control strategy on the performance, stability and robustness in the general case with simultaneous amplitude and phase variations. The 10° dephasing between the simulated and measured command phase can be explained by nonlinearities in the secondary transfer that have not been completely compensated, especially a coupling between the valve opening and the phase of the radiated sound. Also some hysteresis and quantization have been experimentally observed on the HAPS valve, which are not included in the model, and have limited the performance obtained in the experiments. Upgrading the valve of the HAPS could then be required to further increase the attenuation band. Another improvement could be the compensation of the nonlinearities in the secondary transfer, based on a preliminary identification. To improve the dynamics of the phase, it is possible to replace the PI regulation of the PLL with an advanced command (state feedback), or a vector control of the synchronous motor [26].

## 6. Conclusion

Rather than functioning as a purely tonal source, the Harmonic Acoustic Pneumatic Source (HAPS) acts as a mechanical modulator generating anti-noise within a narrow frequency band. This work introduces a model of the system dynamics of this source and its controller for active noise control, and provides a thorough assessment of its performance and limitations under unsteady primary noise conditions for both far-field and near-field control. The two mechanical components of the HAPS are modelled separately: a first-order model for the valve controlling the amplitude of the acoustic signal, and a first-order model combined with proportional-integral feedback for the chopper regulating its phase. Experimental results confirm that the proposed model accurately captures the dynamic behaviour of the HAPS. The dynamics imposed by its mechanical components and its specific control strategy influence the dynamic response of the control system and therefore limit the attenuation performance on time-varying primary disturbances. Experiments are carried out to investigate the limitations

of the control system. Results show that a harmonic controller using either a far-field or a near-field error microphone with a compensation strategy, enables effective attenuation of slowly varying primary noises. Attenuation levels of at least 15 dB are achieved under amplitude modulation of 10% at 0.2 Hz and phase modulation of 10° at 0.1 Hz, which theoretically represent the upper limits of modulation frequency below which attenuation of the fluctuation remains feasible with the presented closed loop model and tuning of the controller gain. The dynamic model proposed here allows anticipation of the dynamic behaviour of the HAPS. The attenuation band can be improved by increasing the controller gain. This control system can cope with slow fluctuations of the amplitude or phase of the primary disturbance. Therefore, the HAPS does not reduce to a purely tonal noise source, but can be seen as a narrowband source, and hence its application scope is enlarged. The proposed framework enhances the applicability of mechanically actuated sources in real-time adaptive noise control and contributes to the development of signal-driven strategies within complex aeroacoustic systems. The performance limits are related to the mechanical system and could theoretically be pushed with a faster HAPS. Future work should include compensation for the nonlinearities of the secondary transfer, and a new electromechanical version of the HAPS in order to increase both the HAPS' reaction speed and its narrowband width, thereby enabling its use in a wider range of applications.

## CRediT authorship contribution statement

**Alexandre Schiavini:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. **Philippe Micheau:** Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Pierre Grandjean:** Writing – review & editing, Resources, Investigation, Data curation. **Gwénaël Gabard:** Writing – review & editing, Conceptualization. **Thomas Humbert:** Writing – review & editing, Resources.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships that may be considered as potential competing interests:

Gwenael Gabard reports that financial support was provided by French National Research Agency. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors acknowledge the financial support of Safran Nacelles and the Agence Nationale de la Recherche (France) through the Industrial Chair ASTRAL (grant ANR-20-CHIN-00050-01). We also gratefully acknowledge PIMUS (Plateforme d'Instrumentation Mécanique de l'Université de Sherbrooke) for providing access to equipment and infrastructure, which made these measurement results possible.

## Data availability

Data will be made available on request.

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