## 2 ARNC Using an ESG
### Proposed MR-SISO ARNC System with an ESG  
An MR-SISO ARNC system was proposed in this study to comprise multiple reference signals, a single error signal, and a single secondary actuator, as illustrated in Figure 2. The control system utilizes an adaptive feedforward FxLMS control algorithm that aims to minimize the road noise measured at the driver's ear, that is, the disturbance signal. The reference signals are captured by \(L\) accelerometers that are mounted around the wheels of a car, and these signals are represented by the reference signal vector \(\mathbf{x}\). The error signal \(e\) is measured by a single microphone at the ear location, and a single ESG installed at the cowl  
Figure 1: Radiated anti-sound from the cowl panel excited by the ESG propagating to the error microphone.  
panel works as the secondary actuator to make canceling sound. The ESG is excited by the control signal \(u\) calculated by Equation (4) in the proposed ARNC algorithm.  
\[u=\mathbf{w}^{\mathsf{T}}\mathbf{x}, \tag{4}\]  
where \(\mathbf{w}\) is the MR-SISO control filter vector and \(\mathbf{x}\) is the reference signal vector. The length of \(\mathbf{w}\) is \(L\times I\), where \(L\) is the number of the control filter, which is the same as the number of reference signals, and \(I\) is the length of each control filter. Thus, the control filter vector can be written as \(\mathbf{w}=\left[\begin{array}{cccc}\mathbf{w}_{1}^{\mathsf{T}}&\mathbf{w}_ {2}^{\mathsf{T}}&\cdots&\mathbf{w}_{L}^{\mathsf{T}}\end{array}\right]^{\mathsf{T}}\), where \(\ell\)= 1, 2,.. \(L\) and \(\mathbf{w}_{\ell}=\left[\begin{array}{cccc}w_{\ell,0}&w_{\ell,1}&\cdots&w_{ \ell,I-1}\end{array}\right]^{\mathsf{T}}\). The reference signal vector is defined by \(\mathbf{x}=\left[\begin{array}{cccc}\mathbf{x}_{1}^{\mathsf{T}}&\mathbf{x}_ {2}^{\mathsf{T}}&\cdots&\mathbf{x}_{L}^{\mathsf{T}}\end{array}\right]^{ \mathsf{T}}\), where \(\mathbf{x}_{\ell}=\left[\begin{array}{cccc}x_{\ell,0}&x_{\ell,1}&\cdots&x_{ \ell,I-1}\end{array}\right]^{\mathsf{T}}\).  
In this system, the primary paths \(\mathbf{P}\) are defined as transfer functions between each reference signal and the disturbance signal. The secondary path \(S\) is a transfer function between the control signal and the error signal.  
In this system, a cost function is defined to consider the mean-square error and the sum of squared weighted control filter coefficients and is defined as [1]:  
\[J_{\ell}=E\Big{[}e^{2}\Big{]}+\beta_{\ell}\mathbf{w}_{\ell}^{\mathsf{T}} \mathbf{w}_{\ell} \tag{5}\]  
and its instantaneous gradient with respect to \(\mathbf{w}_{\ell}\) is given by, after some manipulation,  
\[\frac{\partial J_{\ell}}{\partial\mathbf{w}_{\ell}}=2\left(\overset{\mathsf{ A}}{\mathbf{x}}_{\ell}e+\beta_{\ell}\mathbf{w}_{\ell}\right) \tag{6}\]  
where \(\overset{\mathsf{A}}{\mathbf{x}}\) is the filtered reference signal vector and \(\beta\) is a positive weighting coefficient.  
Thus, the MR-SISO ARNC system tries to cancel unwanted road noise adaptively by updating the control filter \(\mathbf{w}\), and its update equation can be expressed as [1; 3]:  
\[\mathbf{w}_{\mathrm{new},\ell}=(1-\alpha_{\ell}\beta_{\ell})\mathbf{w}_{ \mathrm{old},\ell}-\alpha_{\ell}\overset{\mathsf{A}}{\mathbf{x}}_{\ell}e, \tag{7}\]  
Figure 2: Illustration of an MR-SISO ARNC system with an ESG attached to the cowl panel using an FxLMS strategy.  
where \(\alpha_{\ell}\) is the convergence coefficient for relevant adaptation. The coefficients \(\beta_{\ell}\) is included to prevent the control filter from diverging in case excessive reference signals are involved, such as a loud peaky noise when a tire passes a pothole. The coefficient \(\beta_{\ell}\) is known to be able to improve the robustness of the proposed control algorithm even though it is just small values [1].