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equation_depot.tex
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%
% Equations used for the Inkscape diagrams
%
$\Omega_k$
$\theta_{1,k}$
$\theta_{2,k}$
$\theta_{3,k}$
$V_k$
\fbox{$\bar{\Omega}_k = f_2\left(\zeta_{\Omega},\omega_{\Omega}; \bar{\Omega}_{k-1},\bar{\Omega}_{k-2}, \Omega_{k}, \Omega_{k-1}, \Omega_{k-2} \right)$}
\fbox{$\bar e_{\Omega,k} = f_n \left( \zeta_{n,1}, \zeta_{n,2}, \omega_{n}; \bar e_{\Omega,k-1}, \bar e_{\Omega,k-2}, e_{\Omega,k}, e_{\Omega,k-1}, e_{\Omega,k-2} \right)$}
\fbox{$\bar e_{P,k} = f_n \left( \zeta_{n,1}, \zeta_{n,2}, \omega_{n}; \bar e_{P,k-1}, \bar e_{P,k-2}, e_{P,k}, e_{P,k-1}, e_{P,k-2} \right)$}
\fbox{$\Omega_{d,k} = f_p \left(\zeta_{d},\omega_{n}; \Omega_{d,k-1}, \Omega_{d,k-2}, \Omega_k, \Omega_{k-1}, \Omega_{k-2} \right)$}
$\Omega_{d,k}$
\fbox{$\theta_{m,k}=\frac13 \sum_{i=1}^3 \theta_{i,k}$}
\fbox{$\bar \theta_{m,k} = f_1 \left(\tau_{\theta}; \bar \theta_{m,k-1} , \theta_{m,k} , \theta_{m,k-1} \right)$}
$\bar \theta_{m,k}$
\fbox{$\bar V_k = f_1 \left(\tau_{V}; \bar V_{k-1}, V_k, V_{k-1}\right)$}
$\bar V_k$
\fbox{$Q_{g,ref,k}^{full} = P_0/\Omega_k \;\;\; \mbox{or} \;\;\; Q_{g,ref,k}^{full} = Q_0$}
\fbox{\parbox{60mm}{\vspace{-\abovedisplayskip} \begin{align*}
Q_{g,ref,k}^{full} &= P_0/\Omega_k \;\;\; \mbox{Constant power control}\\
Q_{g,ref,k}^{full} &= P_0/\Omega_0 \;\;\; \mbox{Constant torque control}
\end{align*}\vspace{-1.5\belowdisplayskip}}}
\fbox{$\theta_{min,k} = \theta_{min}\left(\bar V_k \right)$}
\fbox{$\sigma_{\theta,k} = f_1 \left(\frac{2\pi}{\Omega_0}; \sigma_{\theta,k-1} , \sigma\left(\theta_{f_0}, \theta_{f_1}; \theta_{m,k} \right) , \sigma\left(\theta_{f_0}, \theta_{f_1}; \theta_{m,k-1} \right)\right)$}
\fbox{\parbox{50mm}{\vspace{-\abovedisplayskip} \begin{align*}
\sigma_{min,k} &= \sigma \left(\Omega_{min_1}, \Omega_{min_2}; \bar \Omega_k \right)\\
\sigma_{max,k} &= \sigma \left(\Omega_{max_1}, \Omega_{max_2}; \bar \Omega_k \right)
\end{align*}\vspace{-1.5\belowdisplayskip}}}
\fbox{\parbox{80mm}{\vspace{-\abovedisplayskip} \begin{align*}
Q_{g,min,k}^{part}&=\min{\bigl(\sigma_{min,k}(K\bar\Omega_k^2-K_{dot}\dot{\bar{\Omega}}),K\Omega^2_{max_1}\bigr)}\\
Q_{g,max,k}^{part}&=\max{\bigl((1-\sigma_{max,k})(K\bar\Omega_k^2-K_{dot}\dot{\bar{\Omega}})+\sigma_{max,k}Q_{g,ref,k}^{full},K\Omega^2_{min_2}\bigr)}
\end{align*}\vspace{-1.5\belowdisplayskip}}}
\fbox{\parbox{50mm}{\vspace{-\abovedisplayskip} \begin{align*}
Q_{g,min,k} &= \left(1-\sigma_{\theta,k}\right) Q_{g,min,k}^{part} + \sigma_{\theta,k} \, Q_{g,ref,k}^{full}\\
Q_{g,max,k} &= \left(1-\sigma_{\theta,k}\right) Q_{g,max,k}^{part} + \sigma_{\theta,k} \, Q_{g,ref,k}^{full}
\end{align*}\vspace{-1.5\belowdisplayskip}}}
\fbox{$e_{Q,k} = \bar \Omega_k - \Omega_{set,k}$}
\fbox{$Q_{I,k} = Q_{I,k-1} + k_{I}^g \frac{\Delta t}2 \left( e_{Q,k} + e_{Q,k-1} \right) $}
\fbox{$Q_{P,k} = k_{P}^g \frac12 \left( e_{Q,k} + e_{Q,k-1} \right) $}
\fbox{$Q_{D,k} = k_{D}^g \left( e_{Q,k} - e_{Q,k-1} \right)/\Delta t $}
\fbox{$Q_{PID,k} = \max\left( Q_{g,min,k}, \min\left( Q_{g,max,k}, Q_{P,k} + Q_{I,k} + Q_{D,k} \right) \right)$}
\fbox{$Q_{PID,k} = \max\left( Q_{g,min,k}^{part}, \min\left( Q_{g,max,k}^{part}, Q_{P,k} + Q_{I,k} + Q_{D,k} \right) \right)$}
\fbox{$Q_{PID,k} = Q_{g,ref,k}^{full}$}
\fbox{$Q_{I,k-1} = Q_{PID,k-1} - Q_{P,k-1} - Q_{D,k-1}$}
\fbox{$Q_{dmp,k} = k_{dmp} \Omega_{d,k}$}
\fbox{$Q_{dmp, k} = f_{dmp}(k_{dtd}, \omega_{dt}, \omega_{n, dtd}, \zeta_{bp, dtd}, \zeta_{n, dtd}, t_{d, dtd}; \Omega_k, \Omega_{k-1},\Omega_{k-2})$}
\fbox{$\theta_{dmp, k} = f(k_{td}, \omega_{bp, td}, \omega_{n, td}, \zeta_{bp, td}, \zeta_{n, td}, t_{d, td}; a_k, a_{k-1}, a_{k-2})$}
\fbox{$\Omega_{set,k} = \left\{ \begin{array}{l} \Omega_{min} \;\;\mbox{for} \;\; \Omega_k < \frac12(\Omega_{min} + \Omega_0) \\ \Omega_0 \;\;\;\;\;\; \mbox{otherwise} \end{array} \right.$}
\fbox{$Q_{ref,k} = Q_{PID,k} + Q_{dmp,k}$}
$Q_{ref,k}$
\fbox{$P_{ref,k} = Q_{PID,k} \Omega_k$}
\fbox{$e_{\Omega,k} = \bar \Omega_k - \Omega_0$}
\fbox{$e_{P,k} = P_{ref,k} - P_0$}
\fbox{$\theta_{P,k} = \eta_k \left( k_{P} \frac12 \left(\bar e_{\Omega,k} + \bar e_{\Omega,k-1}\right) + k_P^P \frac12 \left(\bar e_{P,k} + \bar e_{P,k-1}\right) \right)$}
\fbox{$\theta_{I,k} = \theta_{I,k-1} + \eta_k \left( k_I \frac{\Delta t}2\left(\bar e_{\Omega,k} + \bar e_{\Omega,k-1}\right) + k_I^P \frac{\Delta t}2\left(\bar e_{P,k} + \bar e_{P,k-1}\right) \right) $}
\fbox{$\theta_{D,k} = \eta_k k_{D} \left(\bar e_{\Omega,k} - \bar e_{\Omega,k-1}\right)/\Delta t $}
\fbox{$\theta_{ref,k} = \max\left( \theta_{min,k}, \theta_{P,k} + \theta_{I,k} + \theta_{D,k}\right)$}
\fbox{$\theta_{I,k-1} = \theta_{ref,k-1} - \theta_{P,k-1} - \theta_{D,k-1} $}
\fbox{\parbox{50mm}{\vspace{-\abovedisplayskip} \begin{align*}
\eta_{A,k}&=\left( 1 +\bar \theta_{m,k}/K_{1} + \bar \theta_{m,k}^2/K_{2} \right)^{-1} \\
\eta_{\Omega,k} &= 1+\bar \theta_{m,k}/K_{\Omega 1}+\bar \theta_{m,k}^2/K_{\Omega 2}
\end{align*}\vspace{-1.5\belowdisplayskip}}}
\fbox{$\eta_{nl,k}=1+\frac{e_{\Omega,k}^2}{\left(\Omega_2-\Omega_0\right)^2}$}
\fbox{$\eta_k=\eta_{A,k} \eta_{nl,k}$}
$\theta_{ref,k}$
\fbox{$k_{P} = k_{P0} +k_{P0,\Omega} \eta_{\Omega,k}$}
\fbox{$y_{k} = f(\zeta,\omega;y_{k-1},y_{k-2},u_{k}, u_{k-1}, u_{k-2})$}
\fbox{$y_{k} = f(\zeta_1,\zeta_2,\omega;y_{k-1},y_{k-2},u_{k}, u_{k-1}, u_{k-2})$}
\fbox{$y_{k} = f(t_d; u_{k}, u_{k-1}, u_{k-2},... u_n)$}
\fbox{$\theta_{ref, k} = \theta_{PID, k} + \theta_{dmp, k}$}
% Exclusion zone
\fbox{$Q_{g,min,k}, Q_{g,max,k},e_{Q,k} = f(Q_{g,min,k}, Q_{g,max,k},e_{Q,k},Q_{g,min,k}^{part},Q_{g,max,k}^{part})$}