I am doing loop shaping with the following transfer functions:

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

where G is the process and K1 is the controller, and If I increase the badwidth of the seisitivity function in this way:

enter image description here

I have that also the bandwidth of the complementary sensitivity function increases:

enter image description here

And this is what I don't understand. Infact, as far as I know, the relationship between the sensitivity and the complementary sensitivity function is $S+T=1$ in the case of SISO systems, so how is it possible the if I increase the sensitivity, also the complementary sensitivity increases?

Can someone help me?

[EDIT] I konow that if we increase the bandwidth of the sensitivity function, the bandwidth of the complementary sensitivity function should decrease. Am I wrong?

  • 1
    $\begingroup$ If you increase sensitivity at 1 Hz , complementary sensitivity at 1 Hz will decrease. $\endgroup$
    – Ben
    Dec 21 '19 at 20:39

You already have the answer $S+T = 1$, the relation must hold true at all frequencies.

Therefore if your sensitivity function $S$ has a high-pass characterisitic, the complementary sensitivity function $T$ will have low-pass characteristic.

As for bandwidth, your curves make sense. The red sensitivity function has a larger bandwidth than the green sensitivity function. Therefore, the red complementary sensivitity function has a smaller bandwidth than the green complementary sensitivity function.


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