## 2 Kalman Filter Design for Real-Time Estimation of Electric Power Characteristic Of Fuel Cells
### Simplified Model and Kalman Filter Design
In (10), the process noise \(\mathbf{w}_{k}\) is assumed to consist of three independent Gaussian noise processes with zero mean \(\boldsymbol{\mu}_{w,k}=\mathbf{0}\) with the constant disturbance input matrix \(\mathbf{E}_{k}=\mathbf{I}_{3\times 3}\) and the associateddiagonal covariance matrix \(\mathbf{C}_{w,k}=\text{diag}\left\{\left[\sigma_{w,1}^{2}\quad\sigma_{w,2}^{2} \quad\sigma_{w,3}^{2}\right]\right\}\) with the standard deviations \(\sigma_{w,i}\), \(i\in\{1,2,3\}\) of the state variability between two subsequent sampling steps. According to this integrator disturbance model, the predicted parameters of the current-power characteristic can be described by their expected value \(\mathrm{E}\left\{\mathbf{x}_{k+1}\right\}=\boldsymbol{\mu}_{x,k+1}^{\text{p}}\) as well as their covariance matrix \(\mathbf{C}_{x,k+1}^{\text{p}}\) according to  
\[\boldsymbol{\mu}_{x,k+1}^{\text{p}} =\boldsymbol{\mu}_{x,k}^{\text{e}} \tag{19}\] \[\mathbf{C}_{x,k+1}^{\text{p}} =\mathbf{C}_{x,k}^{\text{e}}+\mathbf{C}_{w,k}\enspace, \tag{20}\]  
where all variables denoted with the superscript e refer to the outcome of the previous innovation step. Accounting for the specific structure of the scalar power measurement in Equation (12), the innovation step at the time instant \(k\) can be expressed with the help of a zero-mean Gaussian approximation of the power measurement noise (since precise knowledge about the measurement noise variance is crucial for an effective Kalman filter design, it has been determined for the available test rig by means of experiments within the voltage and current measurement range considered in the following investigation) with the variance \(C_{v,k}\). This innovation step precedes the prediction in Equations (19) and (20) and is given by the expectation and covariance update  
\[\boldsymbol{\mu}_{x,k}^{\text{e}} =\boldsymbol{\mu}_{x,k}^{\text{p}}+\mathbf{L}_{k}\cdot\left( \hat{y}_{k}-\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2}\right]\cdot \boldsymbol{\mu}_{x,k}^{\text{p}}\right) \tag{21}\] \[\mathbf{C}_{x,k}^{\text{e}} =\mathbf{C}_{x,k}^{\text{p}}-\mathbf{L}_{k}\cdot\left[I_{k}^{0} \quad I_{k}^{1}\quad I_{k}^{2}\right]\cdot\mathbf{C}_{x,k}^{\text{p}} \tag{22}\]  
with the gain vector  
\[\mathbf{L}_{k}=\mathbf{C}_{x,k}^{\text{p}}\cdot \begin{bmatrix}I_{k}^{0}\\ I_{k}^{1}\\ I_{k}^{2}\end{bmatrix} \cdot\left( \begin{bmatrix}I_{k}^{0}&I_{k}^{1}&I_{k}^{2} \end{bmatrix} \cdot\mathbf{C}_{x,k}^{\text{p}}\cdot \begin{bmatrix}I_{k}^{0}\\ I_{k}^{1}\\ I_{k}^{2}\end{bmatrix} +C_{v,k}\right)^{-1}\in\mathbb{R}^{3}\enspace. \tag{23}\]  
A forecast of the uncertainty in the approximation of the current-power characteristic is based on an evaluation of the respective standard deviation given by  
\[\sigma_{\text{PLL},k} =\sqrt{\mathrm{E}\left\{\left(P_{\text{EL},k}-\mathrm{E}\left\{ P_{\text{EL},k}\right\}\right)^{2}\right\}} \tag{24}\] \[=\sqrt{\mathrm{E}\left\{\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k }^{2}\right]\cdot\left(\mathbf{x}_{k}-\boldsymbol{\mu}_{x,k}^{\text{e}} \right)\cdot\left(\mathbf{x}_{k}-\boldsymbol{\mu}_{x,k}^{\text{e}}\right)^{T} \cdot\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2}\right]^{T}\right\}}\] \[=\sqrt{\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2}\right]\cdot \mathrm{E}\left\{\left(\mathbf{x}_{k}-\boldsymbol{\mu}_{x,k}^{\text{e}} \right)\cdot\left(\mathbf{x}_{k}-\boldsymbol{\mu}_{x,k}^{\text{e}}\right)^{T} \right\}\cdot\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2}\right]^{T}}\] \[=\sqrt{\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2}\right]\cdot \mathbf{C}_{x,k}^{\text{e}}\cdot\left[I_{k}^{0}\quad I_{k}^{1}\quad I_{k}^{2} \right]^{T}}\enspace.\]  
This value is used in the following section together with the trace  
\[\tau_{k}:=\text{trace}\left\{\mathbf{C}_{x,k}^{\text{e}}\right\}\] (2588.674 Tm [(k)]TJ)  
**Remark 1**: _Note, thus far only the dependency of the electric power characteristic on the stack current \(I_{k}\) is explicitly included in the real-time identification model.