User:WDCCHEN/sandbox

\begin{frame}{State-Space Specification for Temporal Constraints} \begin{equation*} \left[\begin{array}{c} Q^{f, no}_{3(\tau-1)+1} \\ Q^{f, no}_{3(\tau-1)+2} \\ Q^{f, o}_{3(\tau-1)+3}\end{array} \right] =\begin{bmatrix} 1& 0 & 0 \\ 1& 1 & 0 \\  1& 1 & 1 \end{bmatrix} \left[\begin{array}{c} Q^{no}_{3(\tau-1)+1}\\ Q^{no}_{3(\tau-1)+2}\\ Q^{no}_{3(\tau-1)+3} \end{array} \right] \end{equation*}\pause

\begin{equation*} Q^{f,no}_{3(\tau-1)+r} =\sum^{r}_{s=1} Q^{no}_{3(\tau-1)+s}. \end{equation*}, and obtain \begin{align*} Q^{f,no}_{3(\tau-1)+1} & = A_{Q_{3(\tau-1)+1}}X_{3(\tau-1)+1} + \upsilon_{Q_{3(\tau-1)+1}} \\ Q^{f,no}_{3(\tau-1)+2} &= Q^{f,no}_{3(\tau-1)+1} + A_{Q_{3(\tau-1)+2}}X_{3(\tau-1)+2} + \upsilon_{Q_{3(\tau-1)+2}}, \\ Q^{f,o}_{3(\tau-1)+3} &=Q^{f,no}_{3(\tau-1)+2} + A_{Q_{3(\tau-1)+3}}X_{3(\tau-1)+3} + \upsilon_{Q_{3(\tau-1)+3}}, \end{align*} \pause

\end{frame}

\begin{frame}{State-Space Specification for Temporal Constraints}

\begin{alertblock}{A Time-Dependent Temporal Constraint Function} \begin{equation*} Q^{f,no}_{t} = \psi_{t}Q^{f,no}_{t-1} + A_{Q_{t}} \varPhi_{t} X_{t-1} + A_{Q_{t}}\varTheta_{t}\omega_{X_{t}} +\upsilon_{Q_{t}}. \end{equation*} \end{alertblock}

$\psi_{t} $ is an time-dependent flow variable indicator for temporal aggregation process. \begin{equation*} \psi_{t} = \begin{cases} 0 & \mbox{if } t=3(\tau-1)+1 \\ 1 & \mbox{if } t=3(\tau-1)+2 \quad \mbox{and}\quad 3(\tau-1)+3 , \end{cases} \end{equation*} where $ \tau = 1,2,\cdots, T/3.$ Only at $t=3(\tau-1)+3$, temporal cumulated variable $ Q^{f,o}_{t} $ is observed but its aggregation process ends.

%When $t=3(\tau-1)+3$, it is binding point for observed $ Q^{f,o}_{t} $ cumulated variable, state variables and the equality of constraint.

\end{frame}