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I want to design a robust control system using the internal model design specifications. The block diagram is the one shown below:

enter image description here

I am trying to obtain the transfer function $\ \frac{Y(s)}{R(s)} $ in order to acquire the characteristic polynomial of the closed loop system but for some reason I am stuck. This is what I have done so far:

$$ E_{a}(s) = R(s) - Y(s) $$

$$ U(s) = E_{a}(s)G_{c}(s) - KX(s) $$ ( $\ U(s) $: control input to the process $\ G(s) $)

$$ Y(s) = U(s)G(s) = E_{a}(s)G(s)G_{c}(s)-KX(s)G(s) $$

$\ $

$$ Y(s) = R(s)G(s)G_{c}(s) - Y(s)G(s)G_{c}(s) - KX(s)G(s) $$ $\ $

$$ Y(s) = \frac{G(s)[R(s)G_{c}(s)-KX(s)]}{1+G(s)G_{c}(s)} $$

From this point, I really don't know how to continue. Obviously the term that confuses me is $\ X(s) $. So, now how should I proceed and obtain the overall transfer function ?

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  • $\begingroup$ Diadikasia. I love Greek :-) $\endgroup$ – peterh - Reinstate Monica Mar 27 '20 at 11:09
  • $\begingroup$ Yes, I like Greek too 😀 But what about the transfer function ? Any ideas ? :-) $\endgroup$ – Teo Protoulis Mar 27 '20 at 12:41
  • $\begingroup$ There was a deleted answer: "I think the problem is that you are not considering that X(s) = Y(s)." It was deleted on obvious reasons, but maybe it helps a little bit. $\endgroup$ – peterh - Reinstate Monica Mar 27 '20 at 12:48
  • $\begingroup$ Well $X(s)$ consists of the states of the system (position,velocity) and $Y(s)$ is just the output of the system which is position. I don’t think this is a correct approach. $\endgroup$ – Teo Protoulis Mar 27 '20 at 12:49
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    $\begingroup$ But that would be the transfer function of an output feedback, we have a state feedback. I think you need to write the transfer function in some state space fashion, involving A, B, C and D, to describe how x and y differ. y = Cx, so your approach only works if C is filled with 1s. Is there no more info on the internals of G(s)? $\endgroup$ – OpticalResonator Mar 27 '20 at 13:43
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Well, after a while this is what I finally came up with. I have tested the results in simulation environment as well as on the real system and the produced control behaviour was what I expected. Firs of all, I rewrote the block diagram in the form shown below:

                                        enter image description here

The whole procedure of the transfer function is shown below:

The state variables of the plant process are $\ x_1, \ x_2 $ and they are equal to:

$$ x_1 = y \rightarrow X_1(s) = Y(s) $$ $$ x_2 = \dot{y} \rightarrow X_2(s) = sY(s) - y(0) $$

Considering zero initial conditions, the transfer function of the inner loop is (simple feedback system):

$$ H(s) = \frac{G_p(s)}{1+k_1G_p(s)+sk_2G_p(s)} $$

And now it all comes down to the outer loop being again a simple feedback loop:

$$ T(s) = \frac{G_c(s)H(s)}{1+G_c(s)H(s)} $$

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In recent comments, you have attached some important info that you missed to put in your question, primarily that there are two states: position $y=x_1$, and velocity $x_2$ and the output is $x_1$. In that sense $X(s)\neq Y(s)$. An image of your block diagram can be represented as

enter image description here

That means that the matrix $C$ is different from the Identity matrix. Remember that $\dot{x_1}=x_2$ and that you can consider your plant as $\dot{x}=Ax+Bu$.

You are probably trying to obtain the state-space equation. I'd say that your problem also has important info about Gc(s).

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