PID and Phase Compensators

[jonnybgood]

What are the main differences between PID controller and lead/lag compensators? When and were are they implemented?


I am working on position control of a DC motor connected to a pulley and getting position feedback by means of a potentiometer. The motor is given a square wave for obtaining a step response. I deduced the transfer function and designed a PID using the Manual Method and ZN. Now I need to design a phase lead and phase lag compensator. thanks

 

[LvW]

I suppose the transfer function of your dc motor can be approximated by an I-T2 characteristic

H(s)=Phi(s)/V(s)=A/[s*(1+sT1)*(1+sT2)] in rad/V

The system will be stable - however, the response time will be rather large.
Therefore, you can use a phase-lead controller (PD-T1) to enhance magnitude and phase in the region of the first pole (pole-zero cancellation)., thereby improving the step response considerably.
Regarding the mentioned phase-lag element I cannot see any good reason to use such a controller.

 

[jonnybgood]

I assume the final bandwidth of the system cascaded with lead/lag compensator is the whole CLTF not the OLTF.. is that correct? Do you have any idea of the matlab commands required?

 

[LvW]

I assume the final bandwidth of the system cascaded with lead/lag compensator is the whole CLTF not the OLTF.. is that correct? Do you have any idea of the matlab commands required?

The bandwidth of the closed-loop is approx. identical to the cross-over frequency (0 dB line) of the open-loop TF (that means: Loop gain).
Sorry, I don't use MATLAB.

 

[jonnybgood]

Just to clarify; I deduced the values of alpha, z and p for my lead compensator for a Phase Margin of 50 degrees steady state error of 0.05. I managed to get the time response graphs attached for uncompensated, lag and lead(png can be zoomed).

In order to verify my modification of phase for stability safety (margins) etc, I will need a bode and phase plot of each CLTF.

 

[LvW]

Jonny, problem solved?

 

[jonnybgood]

Actually I need to verify my results (the ones I attached). And I am still not sure about how to represent the bandwidth of the final systems(2nd & 3rd) if i should use the CLTF or OLTF.

 

[FvM]

I have difficulties to recognize a systematical controller design approach behind the transfer functions and related step responses shown in post #5.

I suppose, that the second order transfer function (PT2) is meaned to represent the control system. So the first case in post #5 shows a pure P controller. Instead of connecting arbitrary lead-lag or lag-leads elements, you could e.g. adjust the P gain for suitable closed loop behaviour and analyze the results. I didn't yet hear a motivation for the lead-lag design.

Regarding the expectable control system transfer function, I would rather expect something like the said I-T2 characteristic.

 

[jonnybgood]

This pdf attached should show all my working. I was asked to correct and improve the response of the system (DC motor + flywheel) using manual pid, ZN PID, phase lag and a phase lead compensator and after argue about each benefits and disadvantages. I would appreciate any hints.

 

[LvW]

Hi Jonny,

your pdf document contains the line

In order to check the achieved Phase Margin, an Open Loop bode plot is considered;

However, where is such a plot?
For my opinion, it is the classical and most logical way to determine parameters of the controller (gain and time constants) based on the systems open-loop response (of course, without controller).

 

[jonnybgood]

I am uploading a new version pdf. I did the system identification experiment myself and sometimes I think that either there were too much sources of error or I did something wrong.

 

[LvW]

I did the system identification experiment myself and sometimes I think that either there were too much sources of error or I did something wrong.

Jonny, I do not understand this comment. Are you satisfied with the result - yes or no?

Let me summarize:
I think, your target should be a closed-loop with a step response that exhibits (1) a short rise time (in comparison to the uncompensated case) and (2) no or only a negligible ringing.

For this purpose, the open-loop frequency response is needed (without any controller, cannot be found in your pdf-paper !).
From this plot you can see which gain and time constants the controller should have in order to (1) shift the cross-over frequency to larger frequencies (bandwidth of the closed-loop)
and (2) establish a good phase margin (50...70 deg) - magnitude slope of approx. -20 dB/dec at the cross-over.

Remark: It is hard to evaluate the time domain plots because of missing units.

 

[FvM]

O.K., there are the open loop bode plots.

I also don't see a table of PID parameters that has been calculated according to Ziegler-Nichols parameter optimization method.

More generally, controller design and optimization is based on performance criteria that may be quite different for different applications. Besides overshoot percent (which could be roughly related to phase margin) we also have settling time to a certain error band or error "area" ∫|e|dt as possible candidates.

 

[LvW]

Jonny,

another question/remark: Is it really true that you derive the phase margin from the closed-loop response? (page 6 and last page). Indeed, this would be wrong.