I am studying control systems and I am focusing on the control effort.
I have seen that if I place a pole at higher and higher frequencies, if the bandwidth of the system goes above the bandwidth of the plant, the control effort increases at high frequencies. In particular, at higher frequencies than the bandwidth of the plant, the control effort behaves as the controller, while at lower frequencies it behaves as the plant, and the frequency response of the control effort increases until the frequency of the open loop bandwidth.
The definition of control sensitivity function is:
$Q(j\omega)=\frac{C(j\omega)}{1+C(j\omega)P(j\omega)}$
by looking at this definition, we can make some approximations, so if $|L(j\omega)|>>1$ then $Q(j\omega)=\frac{1}{|P(j\omega)|}$ and if $|L(j\omega)|<<1$ then $Q(j\omega)=|C(j\omega)|$ where $L(j\omega)=C(j\omega)P(\omega)$. From the plot below, this means that for high frequencies, the control effort behaves like the controller, while for low frequencies it behaves like the plant.
This is the plot I have created using Matlab.
where by control effort I meant the frequency response of the control sensitivity function and the open loop is $L(j\omega) = C(j\omega)P(j\omega)$. Moreover, I have plotted the frequency response of the plant and of the controller. Just to be more clear as possible, the code for plotting this plot is:
s = tf('s');
P = 1/[(1+s)*(1+0.05*s)^2];
C = (s+1)/s;
tau_3 = 0.001;
CF_3 = [(1+s)*(1+0.05*s)^2]/((1+tau_3*s)^3);
Q3 = (C+CF_3)/(1+P*C);
C3 = (C+CF_3);
L3 = C3*P;
figure;
bodemag(Q3,'g',C3,'r',P,'b',L3,'y'),grid
legend('control effort Q(jw)','controller C(jw)','plant P(jw)','open loop L(jw)')
title('behaviour of the control sensitivity function')
In this case I am considering a two degrees of freedom controller, which is why I have used C+CF_3
in my code.
Why does this happen?