# Limitations on the control effort

I am studying the mixed sensitivity design for control systems, and I have seen that the control effort has limitations given by the bandwidth of the process and by the bandwidth of the sensitivity function. In particular I have seen that if we plot the control sensitivity function, we have that if I am above the bandwith of the process I have an increase of the control effort, while if I am below it decreases. And also I have a lower limitation given by the bandwidth of the sensitivity.

But I don't understand why. Also, does the controller bandwidth impose some limitations? Can please someone explain this to me? Thanks in advance.

[EDIT] I recognize that my question was to shallow, so I will try to do more reasonings and add more argumentations as I search for answers.

I will use an approch based on the mixed sensitivity for my discussion.

Suppose I have the following plant and controller:

s = tf('s');
G = 10/((s+10)*(s+1));
K1 = 0.5/s;


Now I define the following mixed sensitivity controller, where I used a weight for the sensitivity in order to achieve my desired specifications, which are a 60 dB attenuation at high frequency, a cross over frequency of 0.66353 and a maximum resonance peak of 2 dB. I also inroduce a weight for the control effort, which I use to limit the amount of effort the control has to do. In particular, I use a constant weight, so by doing so I am hopefully trying to limit the control effort to increase too much:

W_bs2 = 0.66353;   %same cross over frequency as the previuos point
M = 2;           %peak of the sensitivity
A = 0.001;      %attenuation
Ws2 = (s/M + W_bs2)/(s+W_bs2*A);%sensitivity weight
%bodemag(1/Ws),grid;
Wu = tf(1);
[K2,CL,GAM2] = mixsyn(G,Ws2,Wu,[]);  %define the controller with the mixed
sensitivity
display(GAM2);
K2 = minreal(K2);  %define a minimal order controller


some of the comments may not make sense because they refer to some previous code.

Now, I define the control effort:

Q1 = K2/(1+G*K2); %control effort


and I plot the control effort and the weight of the control effort, which should impose a limitation: in this case I used a control effort weight equal to 1, but probably an higher value would have been better.

Now, the bandwith of the controller is 0.9879, so if I go above it the control effor shoulb be increasing instead of decreasing. Let's try:

I define a second sensitivity weight which gives an higher cross over frequency to the control effort when defining the controller:

W_bs3 = 2;   %same cross over frequency as the previuos point
M = 2;           %peak of the sensitivity
A = 0.001;      %attenuation
Ws3 = (s/M + W_bs3)/(s+W_bs3*A);%sensitivity weight
%bodemag(1/Ws),grid;
Wu = tf(1);
[K3,CL,GAM3] = mixsyn(G,Ws3,Wu,[]);  %define the controller with the mixed
sensitivity
display(GAM2);
K3 = minreal(K3);  %define a minimal order controller


and if I plot the control effort of it and confron it with the previous result: the second mixed sensitivity controller gives me the yellow line, while the first gives me the red line.

Now, I am not sure I am going to the right direction, but hopefully I am getting closer.

Can someone help me? Thanks again.

[EDIT 2] So,after a lot of searching and thinking I think the point is that I haven't really understood the concept of bandwisth and how it influences the system. I am in particular looking at sensitivity bandwidth and complementary sensitivity bandwidth.

At this point, my idea is that the sensitivity bandwidth an the complementary sensitivity bandwidth should be limited by the fact that $$S+T=1$$. But still not really clear what happens. So it should be that if S is large, T cannot be too large. But is the bandwidth related to this?

• Can you point to a reference for your comment about the bandwidth of the sensitivity imposing a lower limit on the effort? That doesn't sound like there's enough detail there. Dec 3, 2019 at 0:23
• @TimWescott ,Thanks for answering. You are right, I have edited my question with more detail and reasonings. Regarding the sensitivity, I don' t really have a reference, because it was something that was said, but I think the fact is that the sensitivity bandwith is a lower bound to the bandwith of the system, so my idea is that is I go lower than it the control effort increases. Is it correct? Thanks again.
– J.D.
Dec 3, 2019 at 8:13
• but with the mixed sensitivity approch I should be able to regualate the sensitivity bandwith as I want in order to have desired specifications ,right? So following this reasoning I am the one imposing a lower limit to my system. s that correct? Thanks again.
– J.D.
Dec 3, 2019 at 8:49

The reason the closed-loop bandwidth is related to control effort is pretty clear: once you're outside the bandwidth of the plant, the higher the frequency of the command, the higher the controller command amplitude (i.e., controller effort) needs to be to get the same result. So it should just flow from intuition that if you're pushing the system loop closure above the plant's bandwidth that the controller effort will go up.

I'm not getting your comment about the sensitivity bandwidth being linked to control effort -- there's more to it than that, because sensitivity is at least in part a function of the plant itself.

Also, does the controller bandwidth impose some limitations?

Not in theory-land. There's an implicit assumption that you're going to define a controller of arbitrary bandwidth, then give it to some poor slob (possibly me) to implement; and that your job is then done. While real-world designs sometimes do feature drivers or sensors of limited bandwidth, it's more common that there are real-world limitations on how hard you can push the physical devices (or how much you can ask of power supplies); that limits the available control effort.

• Thanks for the answer. The bandwidth of the sensitivity function should put a lower bound to the bandwidth of the system, because in this way we can filter the disturbances. Below that bandwidth the system tracking error increases because it would be subject to disturbances. So it is alower bound. But how it is related to the control effort? Maybe the controller makes more effort to track the signal because there are also disturbances? Thanks. (sorry if I am commenting a lot)
– J.D.
Dec 3, 2019 at 9:47
• Also I have found that it is impossible to minimize both S and KS over the whole spectrum, but I don't find why. I have only found that it is usually not required because we minimize S a low frequencies and KS at high frequencies. So I got that it is not required, but why it can't be done?
– J.D.
Dec 3, 2019 at 11:40