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I have two subsystems which are shown in the picture below.

One of them is a system of stable and controlled differential equations of a mechanical system.

Second is a block that implements a function $f(\theta)$ from a signal entering its input.

$\theta,\Omega,\omega$ - variables, $u(t)$ - control signal.

enter image description here

I need to combine the two systems in such a way that $\omega=\Omega$. What will be the structure of such a system and how to generate a control signal $u(t)$? As control laws, it is allowed to use simple ones, including $P-$,$PI-$ and $PID-$ones.

I would be grateful to the community for advice and help.

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    $\begingroup$ Draw it out. Connect the second block to the first, so you have $u$ going in, and $\omega$, $\Omega$ going out. Then subtract them. That difference is the output of the loop to be formed. You want to drive the difference to zero? So draw a loop now, and put in a controller block that receives $(\Omega - \omega)$, and produces $u$, closing the loop. No way to tell what form (eg PI etc) is appropriate. That depends on what's in the "boxes". I have a feeling the is something missing here, but it should become evident with a fuller diagram. $\endgroup$
    – Pete W
    Nov 18 '21 at 21:44
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    $\begingroup$ Is $\theta$ a function of time? Is it controlled by the user? $\endgroup$
    – AJN
    Nov 19 '21 at 0:35
  • $\begingroup$ @AJN $θ$ - function of time. Perhaps I should have made an additional clarification. Well, no big deal ... In general ... A block with equations of motion is just some state-space, and $\theta$ and $\Omega$ are its variables, which are interconnected. We accept a simple dependency to begin with, i.e. $\theta = k_1 \Omega$, where $k_1$ - positive constant. As for the control of this variable. Not directly, but it is related to the $\Omega$, which is controllable and therefore indirectly regulated and $\theta$. And yet, $\theta$ is measurable. $\endgroup$
    – dtn
    Nov 19 '21 at 4:12
  • $\begingroup$ @PeteW A block with equations of motion is just some state-space, and $\theta$ and $\Omega$ are its variables, which are interconnected. I wrote about their interconnection in the comment above. The second block contains the function $f(\theta)$, and this dependence can be either a linear function, for example, $\omega=f(\theta) = k_2 \theta$, where $k_2$ - positive constant, or nonlinear, for example, $\omega=f(\theta) = \sin(\theta)$. $\endgroup$
    – dtn
    Nov 19 '21 at 4:15
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    $\begingroup$ @PeteW it turned out as follows ibb.co/ySLpc41 $\endgroup$
    – dtn
    Nov 19 '21 at 4:39

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