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I have some governing equations of the form:

$$\begin{align} \ddot \theta(t) &= \frac{MgL + mgl}{J} \theta(t) + \frac B J \dot x(t) - \frac \alpha J V + \frac {mg}{J} d - \frac{c_1}{J} \dot \theta(t) \\ \ddot x &= \frac{\alpha}{R(M+m)} V - \frac{\beta/R + c_2}{M+m} \dot x(t) \end{align} \tag 1$$

And I would like to transform them to the state space. I'd like them final equation to take the form of $\dot z(t) = Az(t)+Bu(t)+T$, where $z(t) = (\theta, \dot \theta, \dot x)$ is the state vector. And I'd like for $U$ to contain $V$ and for $T$ to be a vector containing the disturbance torque $mgd/J$.

How would I go about this? I'm having trouble conceptualizing the internal relationships between the states. How would I write the governing equations in state space form?

And for the measurements, Id like to measure all three states.

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  • $\begingroup$ It's not as painful as one might imagine, but I really wish that every poster uses $\LaTeX$ math markup instead of posting pictures of equations. There is even a Stack Exchange for it. $\endgroup$ Jun 20, 2023 at 2:32
  • $\begingroup$ Sometimes if you're a newbie, someone will come along and do it for you -- once. $\endgroup$
    – TimWescott
    Jun 20, 2023 at 3:01
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    $\begingroup$ I believe you want $z(t) = \begin{bmatrix} \dot \theta & \theta & \dot x \end{bmatrix}^T$. If so, where you want that to appear, edit your question with $z(t) = \begin{bmatrix} \dot \theta & \theta & \dot x \end{bmatrix}^T$. $\endgroup$
    – TimWescott
    Jun 20, 2023 at 3:06
  • $\begingroup$ @TimWescott I have done that for a couple of newbies. But this was a much bigger job and I didn't wanna research how to do single or double dot. $\endgroup$ Jun 20, 2023 at 3:34
  • $\begingroup$ $\ddot \theta$ = $\ddot \theta$. $\endgroup$
    – TimWescott
    Jun 20, 2023 at 14:34

1 Answer 1

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Start with $z(t) = \begin{bmatrix} \dot \theta & \theta & \dot x \end{bmatrix}^T$. Take it's derivative: $$ \frac d {dt} z(t) = \begin{bmatrix}\dot \theta \\ \theta \\ \dot x\end{bmatrix} = \begin{bmatrix}\ddot \theta \\ \dot \theta \\ \ddot x\end{bmatrix} $$

Now add the $\frac{d}{dt} \theta$ term to your (1): $$\begin{align} \ddot \theta(t) &= \frac{MgL + mgl}{J} \theta(t) + \frac B J \dot x(t) - \frac \alpha J V + \frac {mg}{J} d - \frac{c_1}{J} \dot \theta(t) \\ \dot \theta &= \dot \theta \\ \ddot x &= \frac{\alpha}{R(M+m)} V - \frac{\beta/R + c_2}{M+m} \dot x(t) \end{align}$$

You should be able to pick out $\mathbf A$ by inspection:

$$ \frac{d}{dt} \begin{bmatrix}\dot \theta \\ \theta \\ \dot x\end{bmatrix} = \begin{bmatrix} - \frac{c_1}{J} & \frac{MgL + mgl}{J} & \frac B J \\ 1 & 0 & 0 \\ 0 & 0 & - \frac{\beta/R + c_2}{M+m} \end{bmatrix} \begin{bmatrix}\dot \theta \\ \theta \\ \dot x\end{bmatrix} + \begin{bmatrix} - \frac \alpha J V + \frac {mg}{J} d \\ 0 \\ \frac{\alpha}{R(M+m)} V \end{bmatrix}$$

I think you should see at this point how to separate out the rest.

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  • $\begingroup$ Thank you so much! And sorry about the format once again $\endgroup$
    – skylinev2
    Jun 21, 2023 at 18:38

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