\section{The MPC Problem}\label{sec:mpc_problem} The \acrlong{ocp} to be solved was chosen in such a way as to make analysis of the models' performances more straightforward. The objective is tracking a defined reference temperature as close as possible, while ensuring the heat input stays within the HVAC capacity. The \textit{zero-variance} method is used for multi-step prediction when using an existing \acrshort{gp}. This does ignore the propagation of uncertainty for multi step ahead prediction, but even with its simplicity, this method has been proven to work ~\cite{kocijanModellingControlDynamic2016,jainLearningControlUsing2018, pleweSupervisoryModelPredictive2020}. The optimization problem is therefore defined as follows: \begin{subequations}\label{eq:optimal_control_problem} \begin{align} & \text{minimize} & & \sum_{i=0}^{N-1} (\bar{y}_{t+i} - y_{ref, t})^2 \\ & \text{subject to} & & \bar{y}_{t+i} = K_*K^{-1}\mathbf{x}_{t+i-1} \\ &&& \mathbf{x}_{t+i-1} = \left[\mathbf{w}_{t+i-1},\quad \mathbf{u}_{t+i-1},\quad \mathbf{y}_{t+i-1}\right]^T \\ \label{eq:components} &&& u_{t+i} \in \mathcal{U} \end{align} \end{subequations} where $y_{ref, t}$ is the reference temperature at time t, $\mathbf{x}_{t}$ is the GP input vector at time t, composed of the exogenous autoregressive inputs $\mathbf{w}_{t}$, the autoregressive controlled inputs $\mathbf{u}_{t}$ and the autoregressive outputs $\mathbf{y}_{t}$. \subsection{Temperature reference}\label{sec:reference_temperature} The temperature reference for the controller has been taken as the mean value of the SIA~180:2014~\cite{sia180:2014ProtectionThermiqueProtection2014} temperature norm. It imposes a range of temperatures that are allowed for residential buildings based on the rolling 48h average outside temperature. \begin{figure}[ht] \centering \includegraphics[width = \textwidth]{Images/sia_180_2014.png} \caption{The SIA 180:2014 norm for residential building temperatures} \label{fig:sia_temperature_norm} \end{figure} \clearpage