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Sections/60_The_MPC_Problem.tex

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+\section{The MPC Problem}\label{sec:mpc_problem}
+
+The \acrlong{ocp} to be solved was chosen in such a way as to make the
+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, $\mathcal{U}$ is the
+set of allowed inputs, $\mathbf{x}_{t}$ 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~\cite{sia180:2014ProtectionThermiqueProtection2014}}
+    \label{fig:sia_temperature_norm}
+\end{figure}
+
+\clearpage