## 2 Kalman Filter Design for Real-Time Estimation of Electric Power Characteristic Of Fuel Cells
### Simplified Model and Kalman Filter Design  
To implement an instationary Kalman filter [24; 25; 26] with the general discrete-time dynamic system model  
\[\mathbf{x}_{k+1}=\mathbf{A}_{k}\mathbf{x}_{k}+\mathbf{E}_{k}\mathbf{w}_{k} \tag{10}\]  
and the measurement equation  
\[\mathbf{y}_{k}=\mathbf{C}_{k}\mathbf{x}_{k}+\mathbf{v}_{k}\enspace, \tag{11}\]  
the electric power characteristic of the SOFC is first approximated by a polynomial of order two according to  
\[P_{\text{EL},k}\approx \begin{bmatrix}I_{k}^{0}&I_{k}^{1}&I_{k}^{2}\end{bmatrix}  \cdot\mathbf{x}_{k}\quad\text{with}\quad\mathbf{x}_{k}= \begin{bmatrix}x_{1,k}& x_{2,k}&x_{3,k}\end{bmatrix} ^{T}\in\mathbb{R}^{3} \tag{12}\]  
at each time instant \(t_{k}\). The vector \(\mathbf{x}_{k}\) (serving as a simplification of the previously introduced parameter vector \(\mathbf{a}\)) is treated in the following as the state vector of the Kalman filter to adapt the locally valid quadratic approximation of the current-power characteristic in terms of the available measurements and to account for dependencies of the power curve that are caused by effects other than the current \(I_{k}\). In what follows, the power \(P_{\text{EL},k}\) serves as a scalar realization of the general output vector \(\mathbf{y}_{k}\). In general, the Kalman filter for the system model in Equation (10) with Equation (11) consists of the prediction step  
\[\boldsymbol{\mu}_{x,k+1}^{\text{P}} =\mathbf{A}_{k}\boldsymbol{\mu}_{x,k}^{\text{e}}+\mathbf{E}_{k} \boldsymbol{\mu}_{w,k} \tag{13}\] \[\mathbf{C}_{x,k+1}^{\text{P}} =\mathbf{A}_{k}\mathbf{C}_{x,k}^{\text{e}}\mathbf{A}_{k}^{T}+ \mathbf{E}_{k}\mathbf{C}_{w,k}\mathbf{E}_{k}^{T} \tag{14}\]  
as well as the measurement-based innovation  
\[\boldsymbol{\mu}_{x,k}^{\text{e}} =\boldsymbol{\mu}_{x,k}^{\text{P}}+\mathbf{L}_{k}\cdot\left( \boldsymbol{\hat{y}}_{k}-\mathbf{C}_{k}\boldsymbol{\mu}_{x,k}^{\text{P}}- \boldsymbol{\mu}_{v,k}\right) \tag{15}\] \[\mathbf{C}_{x,k}^{\text{e}} =\mathbf{C}_{x,k}^{\text{P}}-\mathbf{L}_{k}\mathbf{C}_{k} \mathbf{C}_{x,k}^{\text{P}} \tag{16}\]  
with the time-varying Kalman gain  
\[\mathbf{L}_{k}=\mathbf{C}_{x,k}^{\text{P}}\mathbf{C}_{k}^{T}\left(\mathbf{C} _{k}\mathbf{C}_{x,k}^{\text{P}}\mathbf{C}_{k}^{T}+\mathbf{C}_{v,k}\right)^{- 1}\enspace. \tag{17}\]  
Here, the noise processes \(\mathbf{w}_{k}\) and \(\mathbf{v}_{k}\) are assumed to be given by the stochastically independent normal distributions \(\mathcal{N}\left(\boldsymbol{\xi},\boldsymbol{\mu}_{\xi},\mathbf{C}_{\xi}\right)\), \(\boldsymbol{\xi}\in\{\mathbf{w}_{k},\mathbf{v}_{k}\},\boldsymbol{\xi}\in \mathbb{R}^{n_{\xi}}\) with  
\[\mathcal{N}\left(\boldsymbol{\xi},\boldsymbol{\mu}_{\xi},\mathbf{C}_{\xi} \right)=\frac{1}{\sqrt{(2\pi)^{n_{\xi}}\cdot\det\left(\mathbf{C}_{\xi}\right)} }\cdot\exp\left(-\frac{1}{2}(\boldsymbol{\xi}-\boldsymbol{\mu}_{\xi})^{T} \mathbf{C}_{\xi}^{-1}(\boldsymbol{\xi}-\boldsymbol{\mu}_{\xi})\right) \tag{18}\]  
and the corresponding mean vectors and covariances \(\boldsymbol{\mu}_{\xi}\) and \(\mathbf{C}_{\xi}\), respectively. Because the vector \(\mathbf{x}_{k}\) describing the current-power characteristic of the SOFC is typically slowly varying, the system matrix \(\mathbf{A}_{k}\) in Equation (10) is set to the identity matrix \(\mathbf{I}_{3\times 3}\in\mathbb{R}^{3\times 3}\) in order to represent the neglected system dynamics in terms of a discrete-time integrator disturbance model. In this context, discrete-time integrator disturbance models according to \(\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{w}_{k}\), in which the zero-mean process noise \(\mathbf{w}_{k}\) is included in additive form, provide an effective means to represent temporally slow variations of the quantities \(\mathbf{x}_{k}\), where their typical variability (without any a priori knowledge concerning a preferred direction of change) is expressed by the process noise covariance of \(\mathbf{w}_{k}\).