WIP: Working on Implementation Section

60_The_MPC_Problem.tex

@@ -1,4 +1,48 @@
 \section{The MPC Problem}
 
+The Optimal Control Problem 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}
+
+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