### Improve 'Battery operated systems and Energy Harvesting'

 ... ... @@ -123,36 +123,44 @@ Battery powered device rely on energy harvesting and storing power. \\ A simple structur to control the battery:\\ \includegraphics[width=1\linewidth]{images/power-model.png} \\ \textbf{Parameters:}\\ \textbf{Parameters:} \begin{itemize} \item $p(t)$: Energy that is harvested. \item $\Tilde{p}(\tau)$:Estimation of future harvested energy. \item $\Tilde{p}(\tau)$: Estimation of future harvested energy. \item $b(t)$: Battery Level \item $B$: Battery Capacity \item $u(t)$: Used energy \end{itemize} \end{itemize} How should we chose the energy consumption $u(t)$ such that: \begin{itemize} \item Device continuously operates: \begin{center} $b(t) + p(t) - u(t) \geq 0 \ \forall t \in (0,T)$ \end{center} \item Battery has same charge after interval: \item Battery has same charge after interval: \begin{center} $b(0) = b(T)$ $b(0) = b(T)$ \end{center} \item u(t) maximizes to minimal used Energy. \\ \texttt{"}Every other use function provides \textbf{less} energy at some time" \item $u(t)$ maximizes to minimal used Energy. Every other use function provides \textbf{less} energy at some time. \begin{center} $\displaystyle \forall \hat{u}: \min(u(t)) \geq \min(\hat{u}(t))$ \end{center} \end{itemize} The following holds for the optimal energy consumption $u(t)$: \begin{itemize} \item If the battery is neither full nor empty, $u(t)$ stays constant. \item If the battery is \textbf{empty}, $u(t)$ \textbf{increases} \item If the battery is \textbf{full}, $u(t)$ \textbf{decreases} \end{itemize} \bigskip If the following relations hold for all $\tau \in (t,T)$ \begin{align} u^*(\tau -1) < u^*(\tau) \quad & \Longrightarrow \quad b^*(\tau) = 0 \qquad (\textnormal{empy battery})\\ u^*(\tau -1) > u^*(\tau) \quad & \Longrightarrow \quad b^*(\tau) = B \qquad (\textnormal{full battery)} \end{align} then $u^*(t)$ is optimal with respect to maximizing the minimal used energy among all use functions and maximizes the utility $U(\tau, T)$. \bigskip Since the perfect use function $u(t)$ was chosen with knowledge of all harvested energy for all times, the battery level will drop to 0\% only at times, when the harvesting functions starts delivering more energy than we currently use. \\ The energy consumption will then be chosen, such that the battery is full, when the harvesting function is decreasing, thus after the battery is fully charged, the use function should decrease. \\ ... ...