The transfer function,
G(s) , of the output
\Phi(s), pendulum position, with respect to
F, the driving force is:
G(s) = \frac{\Phi(s)}{F(s)} = \frac{\frac{ml}{q}s}{s^3 + \frac{b(I+ml^2)}{q}s^2-\frac{(M+m)mgl}{q}s - \frac{bmgl}{q}} \begin{bmatrix}\underline{rad}
\\N
\end{bmatrix}
Where:
q = (M+m)(I+ml^2)-(ml)^2
The transfer function,
H(s), of the output
X(s), cart position, with respect to
F, the driving force is:
H(s) = \frac{X(s)}{F(s)} = \frac{\frac{I+ml^2}{q}s^2-gml}{s^4 + \frac{b(I+ml^2)}{q}s^3-\frac{(M+m)mgl}{q}s^2 - \frac{bmgl}{q}s} \begin{bmatrix}\underline{m}
\\N
\end{bmatrix}
The system can be represented by its state space model:
\begin{bmatrix}
\dot{X} \\
\ddot{X} \\
\dot{\Phi} \\
\ddot{\Phi}
\end{bmatrix} =
\begin{bmatrix}
0 & 1 & 0 & 0 \\
0& \frac{-(I+ml^2)b}{I(M+m)+Mml^2} & \frac{ml^2gl^2}{I(M+m)+Mml^2} & 0 \\
0& 0 & 0 & 1 \\
0& \frac{-mlb}{I(M+m)+Mml^2}& \frac{mgl(M+m)}{I(M+m)+Mml^2} & 0 \\
\end{bmatrix} \begin{bmatrix}
X \\
\dot{X} \\
\Phi \\
\dot{\Phi}
\end{bmatrix} +
\begin{bmatrix}
0 \\
\frac{I+ml^2}{I(M+m)+Mml^2} \\
0 \\
\frac{ml}{I(M+m)+Mml^2}
\end{bmatrix} u
y = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0& 0 & 1 & 0
\end{bmatrix} \begin{bmatrix}
X \\
\dot{X} \\
\Phi \\
\dot{\Phi}
\end{bmatrix} + \begin{bmatrix}
0\\
0
\end{bmatrix} u
Enter the values of the parameters:
