Good day,
I have two linear and coupled differential equations:
$J_{11}\ddot{\theta_1}-n(J_{11}-J_{22}+J_{33})\dot{\theta_3}+n^2(J_{22}-J_{33})\theta_1=T_{c_1}+T_d \\ J_{33}\ddot{\theta_3}+n(J_{11}-J_{22}+J_{33})\dot{\theta_1}-n^2(J_{11}-J_{22})\theta_3=T_{c_3}+T_d$
and I have specified:
$T_{c_1}(s) = -K_{d_1}e_1(s)s-K_{p_1}e_1(s) \\ T_{c_3}(s) = -K_{d_3}e_3(s)s-K_{p_3}e_3(s)$
Where:
$e_1(s) = \theta_1(s)-\theta_{1_{ref}} \\ e_3(s) = \theta_3(s)-\theta_{3_{ref}}$
$J_{11}$, $J_{22}$, $J_{33}$, $n$, $K_{p_1}$, $K_{d_1}$, $K_{p_3}$, $K_{d_3}$ are simply constants.
Now I would like to calculate the steady-state error when I apply a known step input to $\theta_{1_{ref}}$, $\theta_{3_{ref}}$ and $T_d$.
My approach, I write:
$\begin{bmatrix} s^2+n^2\frac{J_{22}-J_{33}}{J_{11}} & -n\frac{J_{11}-J_{22}+J_{33}}{J_{11}}s\\ n\frac{J_{11}-J_{22}+J_{33}}{J_{33}}s & s^2-n^2\frac{J_{11}-J_{22}}{J_{33}} \end{bmatrix}\begin{bmatrix} \theta_1(s)\\ \theta_3(s) \end{bmatrix}=\begin{bmatrix} \frac{1}{J_{11}}\left (T_{c_1}(s)+T_{d}(s) \right )\\ \frac{1}{J_{33}}\left (T_{c_3}(s)+T_{d}(s) \right ) \end{bmatrix}$
Substituting $T_{c_1}$ and $T_{c_3}$:
\begin{bmatrix} s^2+n^2\frac{J_{22}-J_{33}}{J_{11}} & -n\frac{J_{11}-J_{22}+J_{33}}{J_{11}}s\\ n\frac{J_{11}-J_{22}+J_{33}}{J_{33}}s & s^2-n^2\frac{J_{11}-J_{22}}{J_{33}} \end{bmatrix}\begin{bmatrix} e_1(s)+\theta_{1_{ref}}\\ e_3(s)+\theta_{3_{ref}} \end{bmatrix}=\begin{bmatrix} \frac{1}{J_{11}}\left (-K_{d_1}e_1(s)s-K_{p_1}e_1(s)+T_{d}(s) \right )\\ \frac{1}{J_{33}}\left (-K_{d_3}e_3(s)s-K_{p_3}e_3(s)+T_{d}(s) \right ) \end{bmatrix}
Simplifying the notation by introducing some new (constant) parameters:
$\begin{bmatrix} s^2+a & -bs\\ cs & s^2+d \end{bmatrix}\begin{bmatrix} e_1(s)+\theta_{1_{ref}}\\ e_3(s)+\theta_{3_{ref}} \end{bmatrix}=\begin{bmatrix} \frac{1}{J_{11}}\left (-K_{d_1}e_1(s)s-K_{p_1}e_1(s)+T_{d}(s) \right )\\ \frac{1}{J_{33}}\left (-K_{d_3}e_3(s)s-K_{p_3}e_3(s)+T_{d}(s) \right ) \end{bmatrix}$
Rewriting:
$\begin{bmatrix} s^2+a-\frac{K_{d_1}}{J_{11}}s-\frac{K_{p_1}}{J_{11}} & -bs\\ cs & s^2+d-\frac{K_{d_3}}{J_{33}}s-\frac{K_{p_3}}{J_{33}} \end{bmatrix}\begin{bmatrix} e_1(s)\\ e_3(s) \end{bmatrix}=\begin{bmatrix} (s^2+a)\theta_{1_{ref}}-bs\theta_{3_{ref}}+\frac{T_{d}(s)}{J_{11}} \\ (s^2+d)\theta_{3_{ref}}-cs\theta_{1_{ref}}+\frac{T_{d}(s)}{J_{33}} \end{bmatrix}$
Now I think I should take the inverse of the equation on the left hand side, in order to have only the errors on the left hand side of the equation. Then I can take the limit of s to 0 to find the steady-state error with respect to step inputs on $\theta_{1_{ref}}$, $\theta_{3_{ref}}$ and $T_d$. Is that indeed correct?
If so, what is the easiest way to find $e_1(\infty)$ and $e_3(\infty)$ using Matlab? Because the equations are getting quite complex now.