if you already have a reference (given by the estimation)  fuzzy PI is an option as Aly said,  or you can try neural networks,  your problem now is how to tune the PID,  try sn error coul take ages. 

If you have nonlinear model as a family of linear model around operating points ,  then design PID controller for local linear model and use some fuzzy gain scheduling mechanism or multi model approach or weighed some of the output from local PID controllers.  (indirect adaptive control).

 or

You can use "Enhanced Nonlinear PID controller" which has two differentiator for measuring non linear gain change of process. The PID output is function of this gain.              

http://downloads.deusm.com/designnews/1477-Elsevier06.pdf

Thank you for the answers, but the system is non-linear and time varying, so I think that the Fuzzy Adaptive Control is the best way. I would like to underline that the reference signal is a constant. If you have other suggestions I will be very grateful.

To me, your question seems too general, sort of looking for "the general solution for the general problem."

If you did some successful estimation, then your plant is not too wild and maybe even stable or close to be stable. Is this so? Or, can you just run it,. Even before looking for good performance, is it stable, or, at least, do you have any idea how you could stabilize it?If it is, then we could start from there.

Mr Barkana, 

thank you for your answer. I will try to be more precise this time.

I have to do the control of a heat exchanger that receives hot air that has to be cooled by chill water. Air and water temperature in input oscillate as a sort of sine due to day/night temperature variations. Moreover the system is controlled by the mass-flow-rate of water injected in the heat exchanger in a problem that is non-linear. The system is influenced by physical characteristic of heat exchanger, piping, valves, pressure losses, temperature variations that are not possible to model and also oscillations of the air mass-flow rate in input in the system. The only available measure of the process are the inlet temperatures of air and water and the outlet ones. 

So I have:

xdot = A(mdotwater)*x(t) + B(mdotwater)*u(t)

y = C*x(t)

where clearly y is the outlet air temperature of the process, that must be as close as possible to a constant reference temperature. The state x is represented by the air and water temperatures in the heat exchanger.

The water mass-flow-rate changes at every instant the A and B matrices. 

I would like to implement a PID control that can adjust itself its Ki,Kp,Kd parameters according to the real system behaviour having identified the system (online parameter estimation), and acting only on the water mass-flow-rate as input of the system. 

The system with conventional PID considering constant values of temperatures and air input mass-flowrate is stable and converges to the reference signal. 

I don't know neither the desired behaviour of the system, nor a model reference to follow. I only have a constraint on the output that must converge to the constant reference signal. 

(For some reason, my answer did not go. I hope it will no suddenly show twice)

First of all (except for  strictly official occasions), I am just Itzhak for all, no need for Prof., Dr., Mr., or Sir.  :-)

Now, we start talking. Still, in the same spirit of "no general solution for the general problem," it will take a while to get what you want.

BTW "what you want," you write:

>I don't know either the desired behavior of the system, or a model reference to follow. I only have a constraint on the output that must converge to the constant reference signal.

If you want to control something you must know what you want it to do. I don't mean the behavior of all its state variables, or anything like that. However, if the output now converges to what you want, should it do it in 1 msec, 1 minute, 1 hour? If you do know, then build a simple model, say a good 2nd order system which does what you want in response to the step input. Then, supply the step input to the model and supply the model output as the new input to the plant.

Unlike the standard MRAC which may require a pretty complex adaptive controller for any other than very low order systems, the techniques that i called Simple Adaptive Control (SAC) does not assume that the Plant is sooo unknown that you cannot do anything. You start with something that somewhat "works" and then adding the simple adaptive control make it better and also maintains performance in spite of parameter and other plant changes.

If you say that your nominal Plant is stable and converges, then first of all see how it behaves in response to the new input. Don't be disappointed by the new delay in its response. we will take care of it.

I saw you started following me on RG. Maybe you try to get "Classical and Simple Adaptive Control..." It pretty much describes how to deal in a similar case.

In any case, start and see how it works and we will talk again.

Salute,

Itzhak

I recently wrote about a simple way of deriving parameters (with integrated low-pass filtering) for a digital PID controller, given a low-order (approximate) plant model. The parameters are derived by minimizing the white-noise gain, subject to a specified phase-margin constraint (e.g. 60 deg) for reasonable robustness and a nice transient response. It may be suitable for your problem.

https://www.researchgate.net/publication/299569076

Conference Paper Optimal Design of Digital Anticipatory Servomechanisms with ...

Please refer to this article:

https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design

Article On the Improved Nonlinear Tracking Differentiator based Nonl...