## 2 Spatially Fixed Virtual Sensing Algorithms
### The Kalman filtering virtual sensing method
The Kalman gain matrix and the virtual innovation gain matrix are found by  
\[\mathbf{K}_{ps} =(\mathbf{A}\mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{ \mathbf{S}}_{ps})\mathbf{R}_{pv}^{-1}, \tag{32}\] \[\mathbf{M}_{vs} =(\mathbf{C}_{v}\mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{ \mathbf{R}}_{pv}^{-1})\mathbf{R}_{pv}^{-1}, \tag{33}\]  
with \(\mathbf{P}_{ps}=\mathbf{P}_{ps}^{\mathrm{T}}\), the unique solution to the discrete algebraic Riccati equation given by  
\[\mathbf{P}_{ps}=\mathbf{A}\mathbf{P}_{ps}\mathbf{A}^{\mathrm{T}}-(\mathbf{A} \mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{\mathbf{S}}_{ps})(\mathbf{C}_{ p}\mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{\mathbf{R}}_{p})^{-1}( \mathbf{A}\mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{\mathbf{S}}_{ps})^{ \mathrm{T}}+\bar{\mathbf{Q}}_{s}, \tag{34}\]  
where \(\mathbf{R}_{pv}\) is the covariance matrix of the innovation signals \(\epsilon_{p}(n)=\mathbf{e}_{p}(n)-\tilde{\mathbf{e}}_{p}(n|n-1)\) given by  
\[\mathbf{R}_{pv}=\left[\epsilon_{p}(n)\epsilon_{p}(n)^{\mathrm{T}}\right]= \mathbf{C}_{p}\mathbf{P}_{ps}\mathbf{C}_{p}^{\mathrm{T}}+\bar{\mathbf{R}}_{p}. \tag{35}\]To implement the Kalman filtering virtual sensing method, the state space matrices \(\mathbf{A}\), \(\mathbf{B}_{u}\), \(\mathbf{C}_{p}\), \(\mathbf{C}_{v}\), \(\mathbf{D}_{pu}\) and \(\mathbf{D}_{vu}\) of the state space model in Eq. (23) and the covariance matrices \(\bar{\mathbf{Q}}_{s}\), \(\bar{\mathbf{S}}_{ps}\), \(\bar{\mathbf{R}}_{p}\) and \(\bar{\mathbf{R}}_{pv}\) need to be known [12]. Together, the state space model in Eq. (23) and covariance matrices describe the behaviour of the active noise control system and the covariance properties of the input signals. In practice, the behaviour of the active noise control system can be estimated in a preliminary system identification stage using subspace identification techniques [54]. Subspace identification techniques estimate a model of the active noise control system in an innovations form [54]. Therefore, the Kalman filtering virtual sensing method needs to be reformulated for practical implementation with an innovations model of the active noise control system. The steps to practical implementation of the Kalman filtering virtual sensing method using an innovations model of the active noise control system are as follows [12]  
1. Temporarily locate physical sensors at the spatially fixed virtual locations and measure an input-output data-set \[\{\mathbf{u}(n),\left[ \begin{array}{c}\mathbf{e}_{p}(n)\\ \mathbf{e}_{v}(n)\end{array} \right]\}_{n=1}^{Ns}.\] (36)
2. Use subspace identification techniques [54] to estimate an innovations model of the physical and virtual error signals \[\tilde{\mathbf{z}}(n+1|n)=\tilde{\mathbf{A}}\tilde{\mathbf{z}}(n| n-1)+\tilde{\mathbf{B}}_{u}\mathbf{u}(n)+\tilde{\mathbf{K}}_{s}\left[ \begin{array}{c}\epsilon_{p}(n)^{\mathrm{T}}\epsilon_{v}(n)^{\mathrm{T}} \end{array} \right]^{\mathrm{T}}\] \[\mathbf{e}_{p}(n)=\tilde{\mathbf{C}}_{p}\tilde{\mathbf{z}}(n|n-1) +\tilde{\mathbf{D}}_{pu}\mathbf{u}(n)+\epsilon_{p}(n)\] (37) \[\mathbf{e}_{v}(n)=\tilde{\mathbf{C}}_{v}\tilde{\mathbf{z}}(n|n-1) +\tilde{\mathbf{D}}_{vu}\mathbf{u}(n)+\epsilon_{v}(n),\] and estimate the covariance matrix of the white innovation signals \[\tilde{\mathbf{R}}_{\epsilon}=\left[ \begin{array}{cc}\tilde{\bar{\mathbf{R }}}_{pe}&\tilde{\bar{\mathbf{R}}}_{pv\epsilon}\\ \tilde{\bar{\mathbf{R}}}_{pv\epsilon}&\tilde{\bar{\mathbf{R}}}_{v\epsilon} \end{array} \right].\] (38)
3. Implement the Kalman filtering virtual sensing method as \[\left[\frac{\tilde{\mathbf{z}}(n+1|n)}{\tilde{\mathbf{e}}_{v}(n|n)}\right]= \left[ \begin{array}{cc}\tilde{\mathbf{A}}-\tilde{\mathbf{K}}_{ps}\tilde{ \mathbf{C}}_{p}&\tilde{\mathbf{B}}_{u}-\tilde{\mathbf{K}}_{ps}\tilde{\mathbf{ D}}_{pu}&\tilde{\mathbf{K}}_{ps}\\ \hline\tilde{\mathbf{C}}_{v}-\bar{\mathbf{M}}_{vs}\tilde{\mathbf{C}}_{p}& \tilde{\mathbf{D}}_{vu}-\bar{\mathbf{M}}_{vs}\tilde{\mathbf{D}}_{pu}&\bar{ \mathbf{M}}_{vs}\end{array} \right]\left[