# Doubt on the relationship between the sensitivity and the complementary sensitivity function

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

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

## 1 Answer

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.