how to design controller based on approximate close-loop model

You have two unknowns and one known. That isn't enough. Normally one has a plant model and a desired response and then calculates the controller gains.

BTW, what makes you so sure that the controller needs to be a PI controller if you don't know the plant model.

Peter Nachtwey

He should have one known for each pole location, which makes one known point for each plant state plus one for the integrator. If he knows the PI gains he should, in theory at least, be able to derive the system polynomial for the closed-loop system and from there get to the system polynomial for the open-loop system.

It's terribly round about, though.

Any method that you would use to get the pole locations should be able to give you a more direct measurement of the plant transfer function.

Either ARMA methods (say with a step disturbance input) or frequency sweep methods (which I prefer, and have detailed both on my website and in my book*) can give you a plant model, even when you're in closed-loop. Why, then, do you want to mess around with such indirect data?

The article is

The book is "Applied Control Theory for Embedded Systems", by me, published by Elsevier. will buy you a book, will get you the publisher's description.

• I would have liked to include a section on determining plant transfer functions using ARMA methods, but I was running out of time to have the book published this year -- I hope to have an article at some point; should the book go to a 2nd edition the article will make it's way into the measurement chapter.

Yes, but you are assuming that forest knows that. I am not making that assumption. He could have told us what the transfer function was if he knew. How often have you see people use the wrong number of gains? Too often. Many people people don't understand what you just said and why it is so.

thanks for your good ideas! I don't know plant model, so I can't design other controller. Therefore I only use PI controller. I know closed-loop dominant poles. It is equivalent that I get the reduced-order model of closed-loop. I think it's wrong that derived plant model from reduced close-loop model and controller, so I can't archive plant model. This is not a practical problem. I only do a research. So I don't identify plant model.