WIP: Working on Implementation Section
This commit is contained in:
parent
524074b2e2
commit
d33f0fd4d6
47 changed files with 1842 additions and 27 deletions
|
@ -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
|
||||
|
|
Loading…
Add table
Add a link
Reference in a new issue