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