The state-space model of a pendulum without damping can be described with the 4 state-space matrices.
$$
\begin{matrix}
A = \begin{bmatrix}0 & 1 \ \frac{g}{l} & 0\end{bmatrix} \\
B = \begin{bmatrix} 0 \ \frac{1}{ml^2}\end{bmatrix} \\
C = [1 \ 0] \\
D = 0
\end{matrix}$$
Where the states \(x_1\) and \(x_2\) are \(\theta\) and \(\dot{\theta}\) respectively and \(g\) is negative.
For a linearized pendulum model we get the following matrix equations
$$J\ddot{\theta}+b\dot{\theta}+mlg\theta=\tau$$
$$\begin{bmatrix}
\ddot{\theta} \\
\dot{\theta}
\end{bmatrix}=
\begin{bmatrix}
\frac{-b}{J} & \frac{-mgl}{J} \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
\dot{\theta}\\
\theta
\end{bmatrix}+
\begin{bmatrix}
1 \\ 0
\end{bmatrix}\tau$$
$$y=
\begin{bmatrix}
0&1
\end{bmatrix}
\begin{bmatrix}
\dot{\theta} \\
\theta
\end{bmatrix}$$
- UnderstandingKalmanFilterse
- bevlyMECH4420Lectureb
[[Pendulum]]
[[State-Space Model]]