What is the denominator of the transfer function when PI-regulator is connected?

Question

A system is described by the transfer function

$$G_p(s)=\frac{s+2}{(s+1)(s+3)}.$$

A PI-regulator is connected to the system making it a closed loop system.

So the transfer function for the PI-regulator is $$G_r(s)=K\left(1+\frac{1}{T_is}\right).$$

So the transfer function for the system becomes $$G(s)=\frac{G_pG_r}{1+G_pG_r}$$ where $$G_pG_r=\frac{K(sT_i+1)(s+2)}{sT_i(s+1)(s+3)}$$ and $$1+G_pG_r=\frac{sT_i(s+1)(s+3)+K(sT_i+1)(s+2)}{sT_i(s+1)(s+3)}$$ thus the denominator of $G$ is $$sT_i(s+1)(s+3)+K(sT_i+1)(s+2).$$ But the answer is $$s^3 + (4 + K)s^2 + \left( \frac{1}{T_i}+ 2K + 4\right)s + 2K.$$

Answer 1

If I put your systems into Matlab:

% Q90151

syms s K Ti

Gp = (s+2)/((s+1)*(s+3));
Gr = K*(1 + 1/(Ti*s));

G = Gp*Gr/(1+Gp*Gr);

[num, den] = numden(G);
sympref('PolynomialDisplayStyle','ascend');
num
den

Then I get

num = K*(1 + Ti*s)*(2 + s) 
den = 2*K + K*s + 3*Ti*s + 2*K*Ti*s + 4*Ti*s^2 + K*Ti*s^2 + Ti*s^3

as the output.

Looking at "the answer" this sort of matches except that

• The Ti is cancelled out
• The constant coefficient should then be 2*K/Ti ($\color{red} \times$).
• The coefficient of $s$ seems to be (K/Ti + 2K + 3) ($\color{red} \times$).
• The coefficient of $s^2$ is (4 + K) ($\color{green} ✔$).
• The coefficient of $s^3$ is 1 ($\color{green} ✔$).

The bottom line is that we also need to know what your numerator is!