## 3 Moving Virtual Sensing Algorithms
### The Kalman filtering moving virtual sensing method  
The Kalman filtering moving virtual sensing method [12] uses Kalman filtering theory to obtain estimates of the virtual error signals at the moving virtual locations. The Kalman filtering virtual microphone method as described in Section 2.6 is first used to obtain estimates of the virtual error signals, \(\tilde{\bar{\mathbf{e}}}_{v}(n)\), at the spatially fixed virtual locations, \(\bar{\mathbf{x}}_{v}\). A state space realisation of the Kalman filtering virtual sensing algorithm that estimates the virtual error signals \(\tilde{\bar{\mathbf{e}}}_{v}(n|n)\), given measurements of the physical error  
Figure 11: Block diagram of the adaptive LMS moving virtual microphone technique.  
signals \(\mathbf{e}_{p}(i)\) up to \(i=n\), is as follows [12]  
\[\left[\begin{array}{c}\tilde{\mathbf{z}}(n+1|n)\\ \tilde{\tilde{\mathbf{e}}}_{v}(n|n)\end{array}\right]=\left[\begin{array}{ ccc}\mathbf{A}-\mathbf{K}_{ps}\mathbf{C}_{p}&\mathbf{B}_{u}-\mathbf{K}_{ps} \mathbf{D}_{pu}&\mathbf{K}_{ps}\\ \tilde{\mathbf{C}}_{v}-\mathbf{M}_{vs}\mathbf{C}_{p}&\bar{\mathbf{D}}_{vu}- \bar{\mathbf{M}}_{vs}\mathbf{D}_{pu}&\bar{\mathbf{M}}_{vs}\end{array}\right] \left[\begin{array}{c}\tilde{\mathbf{z}}(n|n-1)\\ \mathbf{u}(n)\\ \mathbf{e}_{p}(n)\end{array}\right], \tag{69}\]  
where \(\bar{\mathbf{C}}_{v}\) and \(\bar{\mathbf{D}}_{vu}\) are the state space matrices of the virtual secondary transfer path matrix \(\tilde{\tilde{\mathbf{G}}}_{vu}\) at the spatially fixed virtual locations \(\bar{\mathbf{x}}_{v}\). The Kalman gain matrix \(\mathbf{K}_{ps}\) can be found using equation Eq. (32) and the virtual innovation gain matrix \(\bar{\mathbf{M}}_{vs}\), of size \(\bar{M}_{v}\times M_{p}\), is given by  
\[\bar{\mathbf{M}}_{vs}=(\bar{\mathbf{C}}_{v}\mathbf{P}_{ps}\mathbf{C}_{p}^{ \mathrm{T}}+\bar{\mathbf{R}}_{pv}^{-1})\mathbf{R}_{p\epsilon}^{-1}, \tag{70}\]  
with \(\mathbf{P}_{ps}=\mathbf{P}_{ps}^{\mathrm{T}}\), the unique stabilising solution to the discrete algebraic Riccati equation given in Eq. (34). The covariance matrix between the auxiliary measurement noises on the physical sensors and virtual sensors spatially fixed at \(\bar{\mathbf{x}}_{v}\), \(\bar{\mathbf{R}}_{pv}\), is defined as in Eq. (29).  
Estimates, \(\tilde{\mathbf{e}}_{v}(n)\), of the virtual error signals at the moving virtual locations, \(\mathbf{x}_{v}(n)\), are now obtained by spatially interpolating the virtual error signals, \(\tilde{\mathbf{e}}_{v}(n)\), at the spatially fixed virtual locations, \(\bar{\mathbf{x}}_{v}\).  
The performance of the Kalman filtering moving virtual sensing method has also been experimentally investigated in an acoustic duct at an acoustic resonance [12]. Again, the virtual microphone moved sinusoidally between a virtual distance of \(v=0.02\) m and 0.12 m with a period of 10 s. Experimental results demonstrated that minimising the moving virtual error signal using a feedforward control approach achieves an additional 14 dB of attenuation at the moving virtual location compared to minimising the error signal at a fixed physical microphone at \(v=0\) m or a fixed virtual microphone at \(v=0.02\) m. While this moving virtual sensing algorithm achieves significant attenuation at the moving virtual location, it is limited to use in systems of relatively low order such as an acoustic duct system.