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

Question

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 ?

Answer 1

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:

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