How to obtain transfer function of control diagram with Internal Model Control?

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

• Diadikasia. I love Greek :-) Mar 27 '20 at 11:09
• Yes, I like Greek too 😀 But what about the transfer function ? Any ideas ? :-) Mar 27 '20 at 12:41
• 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. Mar 27 '20 at 12:48
• 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. Mar 27 '20 at 12:49
• 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)? Mar 27 '20 at 13:43

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:

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:

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