# 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. – TimWescott Dec 3 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 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 at 8:49