Hi Martin,
Suppose you have a system G(s) and you have designed a controller C(s). Assuming the low frequency disturbances and high frequency noise and deriving the equations will give the sensitivity (S= 1/(1+GC)) and complementary sensitivity (T=GC/(1+GC)) function. It's desirable to have a system insensitive to disturbance and high frequency noise (Robust system with good performance) which implies both S and T to be zero. However, if you look carefully at S and T, you will see that S+T=1/(1+GC)+GC/(1+GC))=(1+GC)/(1+GC)=1. It is clear that S+T=1 and we can never have both zero and it means if you want to have the system to be robust in presence of low frequency disturbances, your system will be sensitive to high frequency noises. This is a trade-off between S and T. Anyhow, now you can imagine the waterbed effect, since S+T=1 always holds, therefore, if you try to push the S further down at low frequencies in order to make the system robust against low frequency disturbances, then the price to pay is that, S will pop up in higher frequencies which makes the system perform very pool at those frequencies. A wise man once said: "Every stupid can design a controller, but only a very intelligent person can design a very good controller".
Regards,
I introduce a book that can help you.skogestad multivariable feedback control
Hi Martin,
Suppose you have a system G(s) and you have designed a controller C(s). Assuming the low frequency disturbances and high frequency noise and deriving the equations will give the sensitivity (S= 1/(1+GC)) and complementary sensitivity (T=GC/(1+GC)) function. It's desirable to have a system insensitive to disturbance and high frequency noise (Robust system with good performance) which implies both S and T to be zero. However, if you look carefully at S and T, you will see that S+T=1/(1+GC)+GC/(1+GC))=(1+GC)/(1+GC)=1. It is clear that S+T=1 and we can never have both zero and it means if you want to have the system to be robust in presence of low frequency disturbances, your system will be sensitive to high frequency noises. This is a trade-off between S and T. Anyhow, now you can imagine the waterbed effect, since S+T=1 always holds, therefore, if you try to push the S further down at low frequencies in order to make the system robust against low frequency disturbances, then the price to pay is that, S will pop up in higher frequencies which makes the system perform very pool at those frequencies. A wise man once said: "Every stupid can design a controller, but only a very intelligent person can design a very good controller".
Regards,
Hi Martin,
Control loops are usually low pass filters- as they work on signals and not noise. Therefore, control loops should allow low frequencies and attenueate high frequency signals (noise). We are not worried what the system does after some frequency in general. Closed loop transfer function is called complimentary sensitivity function and has high magnitude at low frequency. While the sensitivity function has higher values at higher frequencies. This being the case, as Noshadi explained S+T=1, therefore if you want to increase sensitivity at low frequency, then you cannot do it without affecting T, and vice-versa at high frequency. One can understand that now bumping T down, S increases, and if you bump S down T increases. This leads to what is called water-bed effect, because you can imagine changing S and T works like this. A good reading would be skogestad multivariable feedback control, and paper titled "respect the unstable" by Gunter Stein.
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