Electrical – How to tackle noise with PI controller

controlpid controller

Updated: I am not sure if the title fits my inquiry here but I will try to explain.

Fig1.

I have this system show in Fig1. It is a simple plant \$\frac{1}{as}\$ (expressed in the frequency domain). Then I use a simple controller such a a proportional controller (Contr) to follow a reference \$x_{ref}\$ (constant). The controller I use is a proportional controller, lets call it \$k\$ (the parameter of my controller could be easily computed by \$k=a/\tau\$, where \$\tau\$ is the time constant of design).

Then I have a modification of the previous system show in the Fig2.

Fig2

The addition is a delay with a signal adding and subtracting at the beginning and at the end of it, respectively (it is drew in black in the figure). Since \$b\$ is relatively small (in the range of 0.1-0.001) this addition could be neglected, having the first figure as a result. However, I notice that the addition produce high frequency noises in the output \$x\$. Although those noises are not big and therefore cause no much trouble. I would like to know if it would be possible to mitigate completely or reduce the noise by using a PI controller instead of a proportional only?

So far, I have reduce the system, and got a reduced system show in Fig. 3.

Fig3

Building that system in Matlab and using PID tuner tool, it provides the gains of the PI controller so that the noise in the output \$x\$ is reduced considerebly. But I dont know how to proceed analitically. I would appreciate any help or suggestion you can provide me. Thanks!

Ps: In case it is necessary some values of \$a,b,c\$ are a=27e-6,b=0.002,c=50, and the gain of the PID tool are kp=0.0479, ki=0.5339.

Best Answer

Bode plot will show lack of phase margin at unit gain. So the solution is to add phase lead compensation (s/T + 1) near the unity gain T=1/ωo for ωo being the unity loop gain frequency and adding 45 deg phase margin, so that the 2nd order system becomes a 1st order slope during the unity gain region. The frequency response should then have have more gain margin, less overshoot to a step input and less peaking in LPF breakpoint.

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