## 2 Problem Statement  
Consider the following diffusion-convection-reaction system with Danckwert's boundary conditions  
\[\partial_{t}x(s,t) =D\partial_{s}^{2}x(s,t)-v\partial_{s}x(s,t)+r(s)x(s,t),\quad\quad s \in(0,1),t>0 \tag{1a}\] \[D\partial_{s}x(0,t)-vx(0,t) =-vu(t), t>0\] (1b) \[\partial_{s}x(1,t) =0, t>0\] (1c) \[x(s,0) =x_{0}(s), s\in[0,1]\] (1d) \[y(t) =x(s_{m},t), t\geq 0 \tag{1e}\]  
with the diffusion coefficient \(D>0\), (superficial) flow velocity \(v>0\), reaction coefficient function \(r\in L^{2}(0,1)\), state \(x(\cdot,t)\in L^{2}(0,1)\) at time \(t\geq 0\) with initial value \(x_{0}\in L^{2}(0,1)\), input \(u(t)\in\mathbb{R}\), and output \(y(t)\in\mathbb{R}\) corresponding to a pointwise measurement of \(x\) at the position \(s_{m}\in[0,1]\). The structure of the model is motivated by chemical engineering applications, in particular tubular reactor models (with length \(L=1\)) [7; 24; 25; 26; 27; 28; 29] linearized along equilibrium solutions (see Appendix A) (Note that in such applications the flow velocity \(v\) might be varying over the spatial extension of what is assumed to be neglectable here.)  
In the sequel, the developments are illustrated and tested for a case study with  
\[D=0.1,v=0.5,r(s)=-k_{0}s/(1+k_{1}s^{2}),\quad k_{0}=1,k_{1}=5 \tag{2}\]  
with a reaction rate maximum at \(s=0.2\), and measurement locations  
\[s_{m}\in\{0.5,1\} \tag{3}\]  
motivated by related studies on approximate observability of the pde model [29] suggesting \(s_{m}=1\) and recent studies on the pointwise measurement injection observer [6; 7; 8; 9; 10] suggesting an in-domain measurement (here chosen as \(s_{m}=0.5\)). Giving the transport-reaction character of the considered system model, the associated system specific time scales can best be captured using the Peclet (\(P_{\ell}\)) and Damkohler (\(D_{a}\)) numbers [24; 28]  
\[P_{e}=\frac{Lv}{D}=\frac{0.5}{0.1}=5,\quad D_{a}=\frac{k_{0}L}{v}=\frac{1}{0. 5}=2\]  
with \(P_{e}\) giving the relation between convective and diffusive characteristic times and \(D_{a}\) the relation between reaction and convection characteristic times. Given a moderate pair \((P_{e},D_{a})\) the solution behavior is expected to be relatively smooth without steep spatial gradients, so that different numerical solution schemes can be implemented. The pde model is solved numerically using a finite-difference approximation with \(N=100\) collocation points, implying a spatial step of \(\mathrm{d}z=0.01\) yielding a finite-dimensional approximation  
\[\dot{x}(t) =Fx(t)+gu(t), t>0,\quad x(0)=x_{0} \tag{4a}\] \[y(t) =\mathbf{h}^{\mathrm{T}}x(t), t\geq 0 \tag{4b}\]  
with state vector \(\mathbf{x}(t)\in\mathbb{R}^{N}\) at time \(t\geq 0\), and \(F\in\mathbb{R}^{N\times N},\mathbf{g},\mathbf{h}\in\mathbb{R}^{N}\). For this purpose, central differences (\(\mathcal{O}^{2}(\mathrm{d}z)\)) are used for the advection and the diffusion parts. The finite-dimensional ode model (4) is subsequently solved using the standard Euler forward solver (\(\mathcal{O}^{2}(\mathrm{d}t)\)) in Matlab. Note that for solving the ode system (4) in Matlab a discrete-time system approximation of the form  
\[\mathbf{x}(t+1)=\Phi\mathbf{x}(t)+\gamma u(t),\quad t\geq 0,\mathbf{x}(0)=\mathbf{x}_{0} \tag{5}\]with \(\Phi\in\mathbb{R}^{N\times N}\) and \(\gamma\in\mathbb{R}^{N}\) is used. In summary, the approximation error is \(\mathcal{O}^{2}(\mathrm{d}t,\mathrm{d}z)\). To achieve an accurate simulation result, the step size in time was set to \(\mathrm{d}t=2\times 10^{-5}\).  
The solution of the resulting equations for the input and initial value given by  
\[u(t)=3\sigma(t)+7\cos(3t)(\sigma(t-5)-2\sigma(t-8)+\sigma(t-12)\sin(3t))\]  
with \(\sigma\) denoting the Heaviside function, and initial condition \(x_{0}(s)=0.2\sin(\pi/2s)^{2}\) is shown in Figure 1.