Process Control for 3 variables: Target and 2 Disturbances

Peter Nachtwey

"Peter Nachtwey" schrieb im Newsbeitrag news:yPCdnRS1_ZRmvdLbnZ2dnUVZ snipped-for-privacy@comcast.com...

Basically I fed back a PI controller for handling the disturbances. I'd like to call it optimal feedforward control with PI feedback.

The integral feedback I was discussing before I called 'correcting'. Test showed that PI works better.

See appropriate diagrams:

F2(s) Feedforward controller with PI

------------------------------------

Error e = u - v1

v2 = u + B1 * u' + B2 * u'' + B3 * u''' + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Recognize PD3!

• K * (e + 30 * Integrator(e)) + z1 ^^^^^^^^^^^^^^^^^^^^^^ ^^ PI! K = 1 Integration[0...t] e dt + C C=0 Forced disturbance z1 z1=0.1 at t=0.3

F1(s) Process transfer function

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A3*v1''' + A2*v1'' + A1*v1' + v1 + z2 = v2 ^^ Forced disturbance z2 z2=-0.1 at t=0.5

F3(s) Target filter

-------------------

C3*u''' + C2*u'' + C1*u' + u = w (w = scheme)

If you combine all DEs mentioned you will get the result I showed in

The set point is changed and the simulated disturbances are controlled to zero. It works as you can see. That's all about. It is MATHEMATICS. No doubt!

Here you also find the start values.

Looking at the graphics shows that the 'benchmark data are met' despite the disturbances I forced for testing.

Data used:

A1 = 0.0371717 A2 = 0.0004699841 A3 = 0.000002771629

B1 = A1 B2 = A2 B3 = A3

C1 = A1 C2 = A2 C3 = A3

These simplified example data make tests easier understandable.

General note:

F2(s) became an integro differential equation. It sounds complicated but it is just an integrator to add.

This is more realistic. : : See appropriate diagrams: :

: : F2(s) Feedforward controller with PI : ------------------------------------ : : Error e = u - v1 : : v2 = u + B1 * u' + B2 * u'' + B3 * u''' + : ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ You need a B0. What you have may be OK for a type 0 system. B0 would be 0 for a type 1 system like a position control system. In a position control system you don't have a feed forward gain for the position.

: Recognize PD3! Yes, but these coefficients are used as feed forards, not as closed loop gains. There is a difference. There are problems trying to compute the second and third derivatives from the feedback. It is easy to calculate the second and third derivative from the target generator. : : F3(s) Target filter : ------------------- : : C3*u''' + C2*u'' + C1*u' + u = w (w = scheme) : Now you need a better target generator so one can enter that maximum velocity and accelation etc. : : If you combine all DEs mentioned you will get the result I showed in :

: : F2(s) became an integro differential equation. It sounds complicated but it : is just an integrator to add. : This makes sense. I am a big believer in feed forwards. You have shown the ideal case where the system is known so the feed foward gains can generate exactly correct control outpuit.. In reality there are disturbances, as you show, and the system model is not exact. In this case the PI controller must make of the difference in the control output. If the feed forwards are accurate within 5% then the PI controller only needs to make of the last 5% which means the errors are reduced by roughly 95%. I have never understood those that said gains must be tuned and dismissed the use of models. Any decent estimate will be better than none.

Notice the overshoot of the PI's response to the disturbances. What are you going to do about that? You didn't show what you used for the PI gains.

Stick with the feed forwards with closed loop control and give up on the infinite gain controller.

Peter Nachtwey