## 3 Lettuce Greenhouse Model  
This work uses a validated greenhouse model by [PERSON] [7; 8] to describe climate dynamics and lettuce growth. An overview of the model is presented in Figure 1. The model includes four state variables: lettuce dry weight (biomass), indoor CO\({}_{2}\) concentration, air temperature, and humidity. The control inputs are CO\({}_{2}\) injection rate, ventilation opening, and heating power. In Figure 1, labels and connections distinguish the control inputs, system measurements (derived from state variables), and external disturbances, with interactions represented by dashed lines. The discrete-time state-space model is formulated as:  
\[x(k+1) =f(x(k),u(k),d(k),p) \tag{1}\] \[y(k) =g(x(k),p)\]  
Figure 1: The employed lettuce greenhouse model [7]. Here, \(k\in Z^{+}\) denotes the discrete time, \(x(k)\in R^{4}\) represents the state variables, \(y(k)\in R^{4}\) is the output, \(u(k)\in R^{3}\) denotes the control inputs, and \(d(k)\in R^{4}\) represents the (weather) disturbances. Table 1 provides the meaning of all signals. The parameter vector \(p\in R^{28}\), along with the nonlinear functions \(f(\cdot)\) and \(g(\cdot)\), is detailed in (2) and (4), respectively. The dynamic model of the lettuce greenhouse is detailed with the nonlinear functions \(f(\cdot)\) as:  
\[\left\{ \begin{array}{ll}\frac{dx_{1}(t)}{dt}=p_{1}\phi_{phot,c}(t)-p_{2}x_{ 1}(t)2^{x_{3}(t)/10-5/2}\\ \frac{dx_{2}(t)}{dt}=\frac{1}{p_{0}}\Big{(}-\phi_{phot,c}(t)+p_{10}x_{1}(t)2^{ x_{3}(t)/10-5/2}+10^{-6}u_{1}(t)-\phi_{\text{vent},c}(t)\Big{)}\\ \frac{dx_{3}(t)}{dt}=\frac{1}{p_{16}}\big{(}u_{3}(t)-\big{(}10^{-3}p_{17}u_{2 }(t)+p_{18}\big{)}(x_{3}(t)-d_{3}(t))+p_{19}d_{1}(t)\big{)}\\ \frac{dx_{4}(t)}{dt}=\frac{1}{p_{20}}\Big{(}\phi_{\text{trans},h}(t)-\phi_{ \text{vent},h}(t)\Big{)}\end{array} \right. \tag{2}\]  
with the flux terms are defined as:  
\[\phi_{phot,c}(t) = \Big{(}1-e^{-p_{3}x_{1}(t)}\Big{)}\frac{p_{4}d_{4}(t)\big{(}-p_{ 5}x_{3}^{2}(t)+p_{6}x_{3}(t)-p_{7}\big{)}(x_{2}(t)-p_{8})}{p_{4}d_{1}(t)+ \big{(}-p_{5}x_{3}^{2}(t)+p_{6}x_{3}(t)-p_{7}\big{)}(x_{2}(t)-p_{8})}\] \[\phi_{\text{trans},h}(t) = p_{21}\Big{(}1-e^{-p_{3}x_{1}(t)}\Big{)}\bigg{(}\frac{p_{22}}{p_ {23}(x_{3}(t)+p_{24})}e^{\frac{p_{23}(t)}{2(t)+p_{26}}}-x_{4}(t)\Big{)} \tag{3}\] \[\phi_{\text{vent},c}(t) = \big{(}10^{-3}u_{2}(t)+p_{11}\big{)}(x_{2}(t)-d_{2}(t))\] \[\phi_{\text{vent},h}(t) = \big{(}10^{-3}u_{2}(t)+p_{11}\big{)}(x_{4}(t)-d_{4}(t))\]  
Here, \(t\in\mathbb{R}\) represents continuous time. The terms \(\phi_{phot,c}(t)\), \(\phi_{vent,c}(t)\), \(\phi_{transp,h}(t)\), and \(\phi_{\text{vent},h}(t)\) correspond to the gross photosynthesis rate, CO\({}_{2}\) mass exchange through ventilation, canopy transpiration, and H\({}_{2}\)O exchange through ventilation, respectively. The output equations \(g(\cdot)\) are defined as: \begin{table}
\begin{tabular}{c c c c} \hline
**Category** & **Symbol** & **Description** & **Unit** \\ \hline \multirow{4}{*}{State variables} & \(x_{1}\) & Lettuce dry weight & kg/m\({}^{2}\) \\  & \(x_{2}\) & Indoor CO\({}_{2}\) concentration & kg/m\({}^{3}\) \\  & \(x_{3}\) & Indoor Air temperature & \({}^{\circ}\)C \\  & \(x_{4}\) & Indoor Humidity & kg/m\({}^{3}\) \\ \hline \multirow{4}{*}{Measurement variables} & \(y_{1}\) & Lettuce dry weight & g/m\({}^{2}\) \\  & \(y_{2}\) & Indoor CO\({}_{2}\) concentration & ppm \\  & \(y_{3}\) & Indoor Air temperature & \({}^{\circ}\)C \\  & \(y_{4}\) & Indoor Humidity & \({}^{\circ}\)\(\%\) \\ \hline \multirow{4}{*}{Disturbances} & \(d_{1}\) & Solar radiation & W/m\({}^{2}\) \\  & \(d_{2}\) & Outdoor CO\({}_{2}\) & kg/m\({}^{3}\) \\  & \(d_{3}\) & Outdoor Air temperature & \({}^{\circ}\)C \\  & \(d_{4}\) & Outdoor Humidity & kg/m\({}^{3}\) \\ \hline \multirow{4}{*}{Control inputs} & \(u_{1}\) & CO\({}_{2}\) supply & mg/m\({}^{2}\)/s \\  & \(u_{2}\) & Ventilation & mm/s \\ \cline{1-1}  & \(u_{3}\) & Heating & W/m\({}^{2}\) \\ \hline \end{tabular}
\end{table}