# Why does the control effort increase if the bandwidth of the system is higher then the bandwidth of the plant?

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?

• Doesn't control effort have the plant function's response in the denominator, making it plausible that the effort rises when the plant can't follow? I vaguely remember that only, though: I don't have the actual formula. Mind adding the definition of control effort to your question? Feb 6, 2020 at 22:03
• Thanks for answering. I have added more details in my question.
– J.D.
Feb 6, 2020 at 22:32
• what exactly are you plotting for each color? what's the difference between "open loop" and "plant"? is "control effort" the magnitude of the closed-loop system response: $|Q(j\omega)|$? or is it $|L(j\omega)|$? Feb 6, 2020 at 22:51
• Thanks for answering. I have edited my question to make it clearer and also added a new plot with the colors specified better.
– J.D.
Feb 6, 2020 at 23:04
• I think you mixed up your equations above. At high frequencies, i.e. significantly above the bandwidth, $|L(j\omega)| <<1$ and thus $Q(j\omega) ~ C$. Feb 7, 2020 at 17:36

Intuitively (and I leave it as an exercise for you to work it out with math), it's because you have to push harder on the plant to get it to move fast.

Consider a bowl of honey and a spoon -- moving the spoon around in the bowl slowly takes little effort, but moving it fast requires a lot of effort, because honey is viscous.

It may be most informative to take a simple plant model, like a 1st-order low-pass filter, and without ever designing a closed-loop controller, design a signal for it to follow that has significant energy content above it's rolloff frequency. Then look at the relative amplitudes of the components of that drive signal compared to the resulting components of the motion of the plant.

Even in open loop your control effort is higher at higher frequencies, for reasons very much analogous to that spoon in honey.

Bandwidth in control system usually mean open-loop bandwidth and is usually defined as the frequency where the open loop transfer function gain crosses the 0 dB gain.

The open loop transfer function is defined as $$C(s)P(s)$$. Usually, but not always, processes have some kind of low-pass transfer function. Therefore, to increase the bandwidth your controller needs to add gain at frequencies higher than the bandwidth of your process.

Say you want to increase your bandwidth to 100 Hz and that your process gain is - 10 dB at 100 Hz, your controller needs a gain of 10 dB for the open-loop gain to be 0 dB.