# A Feasibility Study of the Use of PZT Actuators for Active Control to Enrich Engine Sound
## 2 Active Sound Enrichment Using PZT Actuators
### Novel ASE System Using a PZT-Plate Actuator and ASE Algorithm
In such a case, \(A(z)\) can be realized by the inverse modelling approach [20] using a modelling delay \(\Delta\), as shown in Figure 4a,b. The arrangement in Figure 4a shows a feedforward control system where \(\hat{g}(n)\) follows the desired signal \(d^{\prime}(n)\) which is equal to \(c(n-\Delta)\). \(\hat{g}(n)\) and \(\hat{d}(n)\) are the estimates of the output and disturbance signals, respectively, and \(\hat{S}(z)\) is the secondary path model. Assuming that \(A(z)\) and \(\hat{S}(z)\) are linear and time-invariant, they can be rearranged as shown in Figure 4b. Then, the optimal FIR filter coefficient vector of the FIR device \(A(z)\), \(\mathbf{a}_{opt}\), to minimize the expectation of the squared \(e(n)\) can be calculated by the Wiener filter design process and given as [21]:  
\[\mathbf{a}_{opt}=[\mathbf{R}+\beta\mathbf{I}]^{-1}\mathbf{p}, \tag{6}\]  
where \(\beta\) is a regulation parameter to prevent excessively large values of \(\mathbf{a}_{opt}\) and \(\mathbf{I}\) is the identity matrix. \(\mathbf{R}\) is the auto-correlation matrix of the estimate of the filtered target signal vector \(\hat{\mathbf{r}}(n)\) and can be written as:  
\[\mathbf{R}=E\Big{[}\hat{\mathbf{r}}(n)\hat{\mathbf{r}}^{\text{T}}(n)\Big{]}, \tag{7}\]  
where \(\hat{\mathbf{r}}(n)= \begin{bmatrix}\hat{r}(n)&\hat{r}(n-1)&\cdots&\hat{r}(n-I+ 1)\end{bmatrix} ^{\text{T}}\), \(I\) is the length of \(\mathbf{a}_{opt}\), and the superscript T means transpose. \(\mathbf{p}\) is cross-correlation vector of \(\hat{\mathbf{r}}(n)\) and \(d^{\prime}(n)\) and can be given as:  
\[\mathbf{p}=E\big{[}\hat{\mathbf{r}}(n)d^{\prime}(n)\big{]}. \tag{8}\]  
However, such implementation of \(A(z)\) as \(\mathbf{a}_{opt}\) can be computationally expensive for the convolution operation of \(c(n)\) and \(\mathbf{a}_{opt}\) and also lead to a time lag of \(u(n)\) due to the modelling delay. To overcome these drawbacks, in the proposed ASE algorithm, \(A(z)\) is redefined to compensate for only the magnitude response of the secondary path, and so it is given as [22]:  
\[A(z)=\Big{|}\hat{S}^{-1}(z)\Big{|}. \tag{9}\]  
Figure 4: (**a**) Block diagram of a feedforward control for following the delayed target sound using a modelling delay. (**b**) Rearrangement of the block diagram for the design of the inverse control filter \(A(z)\). This can be a reasonable implementation of \(A(z)\) if the purpose of ASE is to enrich the amplitude of the engine order sound to that of the target sound regardless of their phase. This approach has two advantages. The first is that the computational efficiency can be significantly improved since \(A(z)\) in Equation (9) can be simply implemented as a gain varied according to the frequency of \(c(n)\). Thus, \(u(n)\) can be produced through just one scalar multiplication operation without any convolution operation. The second is that the time lag due to the modelling delay is eliminated, so the supplemented engine sound \(y(n)\) can immediately correspond to a sudden frequency change in the actual engine sound \(d(n)\). It should be noted that even if \(A(z)\) is designed to perfectly cancel the magnitude of the secondary path, there may be a difference between the amplitudes of \(c(n)\) and \(e(n)\) when \(d(n)\) has a large value which is not negligible compared with that of \(c(n)\), and the magnitude of the difference depends on both the amplitude and phase of \(d(n)\). In fact, the arrangement of the ASE algorithm in Figure 3 can be considered a special case of a feedback control algorithm with the internal model control (IMC) structure [21] illustrated in Figure 5 when the actual secondary path \(S(z)\) and its model \(\hat{S}(z)\) is perfectly identical and \(d(n)\) is ignored.