Linearizing a nonlinear model - simple water tank

[Peter Nachtwey]

Rytko, can you help me sort this out?

I don't have Matlab so I can't play along. However, it is easy enough to see that there should be no difference in the steady state levels where flow in=flow out and h is constant.

 

[Pandiani]

rytko,
can you also have a look at the following thread:
http://plctalk.net/qanda/showthread.php?t=38696&page=4&pp=15 ?
Don't be affraid of large number of posts.

 

[coolblind]

The two should be the same. Something is not right. What are constant 1 and constant 3. If they are inflow then why are they different? You should always sanity check your work.

Peter, sorry for the late reply, in fact you can find where the constant1 and constant 3 come from in my "TankerNonlinear.pdf" file in post #2.

Even the inflow is the same (here I just use 0.5 according to Pandiani's assumption), but as long as you linearize it, it cannot be 0.5 anymore because of the none initial condition. You can also find this in rytko's "Linearization.pdf" in Page 2 just above equation (9), where he defines u=F-Fi. F is 0.5 in this case, and Fi is changed with different hi (please refer how rytko defines the Fi in equation 7, the left part of equation 7 is set to be ZERO in order to get Fi).

 

[morrido]

Hi! I'm developing a fuzzy controler for a series of two water tanks with a 5 second delay. The outflow of tank1 is the inflow of tank2.
I was asked to linearize the system, but im having problems doing so since it's a series of two tanks, instead of just a simple water tank.

In steady state, inflow = outflow in both tanks, since this is true, can I use the method you are using for the simple tank simply ignoring tank1?

Tanks parameter are as follow:
Tank1
Height: 0.5 m
Diameter: 0.4 m
Outflow Diameter: 0.04 m

Tank2
Height: 1 m
Diameter: 0.5 m
Outflow Diameter: 0.03 m

(I want to linearize for a tank2 height of 0.5m)

Any help would be appreciated. Thnx

 

[freeze]

can you please tell why you added constant 1.2 to the linearized transfer function' output.

 

[shooter]

you will not get an answer as the post is 4 years old.

 

[Peter Nachtwey]

I don't have Matlab but I do have Mathcad. I generated this for a student AFTER he handed in his home work just so that he would know how to solve the two tank level control problem correctly instead of just tweaking gains, hoping that it works and being satisfied with good enough.

http://deltamotion.com/peter/Mathcad/TwoTanks/Mathcad - t0p2 p pi Alin's two tanks Cascade.pdf

I didn't bother to 'linearize' anything. Instead I solved the non-linear equations and since I know how to calculate the controller gains I could make the controller gains adapt as the level changes and not suffer from the problems that Pandiani had with initial conditions or the set point changing.

The correct way to start is to write the differential equations. Only then can one understand how to attack the problem the correct way. There were a lot of clueless engineers that wanted to tweak gains that basically were saying to the student don't bother to learn how to do this right, just good enough. While it may be OK for a tank level control to be good enough that isn't the point.

Notice that only a P and PI controller are required.

BTW, I talked to Pandiani yesterday via skype. He is alive and doing well and is busy.

 

[saharul]

linearization

hello,

i am newbie to control system, my question related to level system why we try to avoid to start from 0?

 

[Peter Nachtwey]

hello,

i am newbie to control system, my question related to level system why we try to avoid to start from 0?

There is no reason the initial level can't be anything from empty to full.

Here is an example where there are two tanks but both tanks start with non-zero levels.
http://deltamotion.com/peter/Mathcad/TwoTanks/Mathcad - t0p2 pi tc - Alin's two tanks OL.pdf