Master-Thesis/60_The_MPC_Problem.tex
2021-06-25 11:27:25 +02:00

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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
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}
\label{fig:sia_temperature_norm}
\end{figure}
\clearpage