In edxu96/MatrixTSA: Tidy Multivariate (Non)Linear Dynamic Systems

equation \ref{eq-1} is

$$ \begin{align} \max_{\mathbf{E}(t), \mathbf{F}(t), \mathbf{U}(t)} \quad & \mathcal{M}t \left[\mathbf{Y}(t), \mathbf{X}(t), \mathbf{B}^{\mathrm{p}}(t), \mathbf{A}^{\mathrm{p}}(t) \right] \ & = - \mathbf{H}(t)^{\top} \mathbf{Q} - \left[\mathbf{G}^{+}(t) + \mathbf{G}^{-}(t) \right]^{\top} \mathbf{O} \ & - \mathbf{U}(t)^{\top} \mathbf{R} + l \left[ \mathbf{E}^{+}(t)^{\top} \mathbf{B}^{\mathrm{p}}(t) - \mathbf{E}^{-}(t)^{\top} \mathbf{A}^{\mathrm{p}}(t) \right] \ & - [\mathbf{F}^{+}(t) + \mathbf{F}^{-}(t)]^{\top} \mathbf{P} \ \text{s.t.} \quad & \mathbf{w}{i+k+1}(t) = \mathbf{A} \mathbf{w}{i+k}(t) + \mathbf{B} \mathbf{u}{i+k}(t) \quad \forall k \in K \ & \mathbf{v}{i+k}(t) = \mathbf{C} \mathbf{w}{i+k}(t) + \mathbf{D} \mathbf{u}{i+k}(t) \quad \forall k \in K \ & \mathbf{v}{\text{min}} \leq \mathbf{v}{i+k}(t) \leq \mathbf{v}{\text{max}} \quad \forall k \in K \ & \mathbf{u}{\text{min}} \leq \mathbf{u}{i+k}(t) \leq \mathbf{u}{\text{max}} \quad \forall k \in K \ & - \mathbf{u}{\Delta} \leq \mathbf{u}{i+k+1}(t) - \mathbf{u}{i+k}(t) \leq \mathbf{u}{\Delta} \quad \forall k \in K{-} \ & \mathbf{E}^{+}(t) \geq 0 \quad \ & \mathbf{E}^{+}(t) \geq \mathbf{E}(t) \ & \mathbf{E}^{-}(t) \geq 0 \ & \mathbf{E}^{-}(t) \geq - \mathbf{E}(t) \ & \mathbf{H}(t) \geq 0 \ & \mathbf{H}(t) \geq \mathbf{Y}(t) + \mathbf{E}(t) - \mathbf{U}(t) \tag{1} \ & \mathbf{G}^{+}(t) \geq 0 \ & \mathbf{G}^{+}(t) \geq \mathbf{V}(t) - \mathbf{X}(t) - \mathbf{F}(t) \tag{2} \ & \mathbf{G}^{-}(t) \geq 0 \ & \mathbf{G}^{-}(t) \geq \mathbf{X}(t) + \mathbf{F}(t) - \mathbf{V}(t) \tag{3} \ & \mathbf{F}^{+}(t) \geq 0 \quad \ & \mathbf{F}^{+}(t) \geq \mathbf{F}(t) \ & \mathbf{F}^{-}(t) \geq 0 \ & \mathbf{F}^{-}(t) \geq - \mathbf{F}(t) \ & \mathbf{E}(t) \in \mathbb{R}^{\kappa} \ & \mathbf{F}(t) \in \mathbb{R}^{\kappa} \ & \mathbf{U}(t) \in \mathbb{R}^{\kappa}_{+} \end{align} $$