【入門】状態空間モデルをPID制御【数値計算】

\(
u(t)=\int\{K_p \dot{e}(t) + K_i e(t)dt + K_d \ddot{e}(t)\}\dots(変形式)
\)
\(\dot{e}(t)\)と\(\ddot{e}(t)\)をオイラー法で解決する。

\(
\eqalign{
\dot{e}(t)&=&\frac{e(t)-e(t-\Delta T)}{\Delta T[Nsec]}\cr
&=&\frac{e(t)-e(t-\Delta T)}{\Delta T/T_1}\cr
&=&\frac{\{e(t)-e(t-\Delta T)\}T_1}{\Delta T}\cr
}
\)

上記の\(e(t)\)に\(\dot{e}(t)\)を代入することで2階微分となるため、
\(
\eqalign{
\ddot{e}(t)&=&\frac{\{\dot{e}(t)-\dot{e}(t-\Delta T)\}T_1}{\Delta T}\cr
&=&\frac{\displaystyle\frac{\{e(t)-e(t-\Delta T)\}T_1}{\Delta T}-\frac{\{e(t-\Delta T)-e(t-\Delta T-\Delta T)\}T_1}{\Delta T}}{\Delta T}\cr
&=&\frac{T_1}{\Delta T}\{\frac{T_1}{\Delta T}\{e(t)-e(t-\Delta T)\}-\frac{T_1}{\Delta T}\{e(t-\Delta T)-e(t-2\Delta T)\}\}
}
\)

積分に関しては以下で近似

\(
\displaystyle\int x(t)dt=\sum x(t-\Delta T)+x(t)\frac{\Delta T}{T_1}
\)

\(
\eqalign{
u(t)&=&\int\{K_p \dot{e}(t) + K_i e(t)dt + K_d \ddot{e}(t)\}dt\cr
&=&\int\{\color{red}{\{K_p\frac{T_1}{\Delta T}\{e(t)-e(t-\Delta T)\}}\color{black}+K_ie(t)\cr
&+&K_d\frac{T_1}{\Delta T}\{\color{blue}{\{\frac{T_1}{\Delta T}\{e(t)-e(t-\Delta T)\}}\cr
&-&\color{blue}{\frac{T_1}{\Delta T}\{e(t-\Delta T)-e(t-2\Delta T)\}}\color{black}\}\}dt
}
\)