Hi everybody, I want to identify a system that has an integrator. My system, is a mechanical system, for example a proportional valve that is connected to a flow sensor and this collection are connected to a container. Because of existence of container, system has integration property. The Container has an input for entrance of liquid and hasn't any drain for exhaust, therefor control actions can only be positive (because of input is flow and flow always is positive). Is there any solution ? The aim is control of fluid flow in this system. Before, I must to know the mathematical model of the system, so i would to identify the system. I am looking for methods to identify this system. Identification is linear. How can i do this ? what is the appropriate input for identification of this type of system? are there any ref.s about this ?
Can you stabilise the system, e.g. with proportional feedback? integrator plus one real pole type systems can be easily stabilised. Once it is stable, you should be able to do many different things based on the bandwidth of your sensor: perhaps apply chirp signal, take Fourier transform of input and output and find empirical transfer function estimate (ETFE - see Lennart Ljung's book)? Then you can fit a transfer function to this ETFE. Please let us know how you get on.
If you are dealing with Mechanical Systems, I should recommend you Worden's book
https://www.sheffield.ac.uk/drg/people/keith
It could be also useful Nayfeh's book
http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471593567.html
and this survey by Kerschen et al.
http://www.sciencedirect.com/science/article/pii/S0888327005000828
First. stabilize your system with a simple controller, i.e. a proportional one if possible.
Second, identify the resulting closed loop system.
Third, obtain your open loop transfer function from the identified closed loop system plus the knowledge of the stabilizing controller.
Fo a system that contains a pure integrator, you can use an input consisting of a (limited) fourier series input, where the coefficients are tuned such that your system does not saturate. If drift is a problem, due to input noise, add a small propotional feedback, just enough to avoid the drift.
However, the example system that you describe,has another property that makes it more difficult. The valve can only add liquid, not remove it, so the control actions can only be positive? Can you specify your system more clearly, such that we can know whether this is really the case?
Thanks to all for your attentions.
Dear Dr Aertbeliën, you exactly guessed it right. My system has an input to entrance of liquid and hasn't any drain for exhaust. Also as you say, control actions can only be positive (because of input is flow and flow always is positive). Is there any solution ?
Thanks.
Dear Davood Shaghaghi,
after You find a simple and stabile controller - any methods, try close loop indentification using PBR signal. Here You can study Landau (Ioan Doree) close loop dentification methods and algorithms based on RLS (recursive least square). One of presented method adress to contained integrator system.
You can try a triangular wave with a large time period and with a offset signal. In such a way you could identify some parameters of the system: time delay, steady-state error, and also you could measure continuous gain.
Thanks Albert. Thank you Ciprian but i emphasize that my system has an liquid input and hasn't any drain for exhaust and control actions can only be positive (because of input is flow and flow always is positive).
Thank you Pablo,
I read that paper. there was not any discussion about identification of integrator system with my application conditions.
Most suggestions assume a close-loop experiment. That is just fine, if you use some closed-loop identification procedure.
Here are two practical tips that, I suppose, will work well for you should you desire to perform the identification in open loop. Use as an input a pulse, lets call it u. This should be suficiently short for the system not to leave the linear operating range and should be smooth (avoid switching). Call the area under the pulse A. The output, y, should look like a step response for a system without integration. Call y_0 the response signal after removing the value it had previous to the test, such that y_0 starts from 0. Tip 1) Assume the input was a step function with amplitue A. Call this hypothetical input u_A. Use any method you like to estimate a model from the data Z=(y, u_A). After obtaining that model, add a pure integration, that is, add a pole at s=0 if the identified model is continuous or one at z=1 if it is disrete. Tip 2) Remove the offset (opearting point) from u, call it u_0. Take the first difference of the output, call it y* (notice that *now* both signals are absolutely summable and therefore are Fourier transformable). Use the data Z=(y*, u_0) to obtain a linear model and add to it a pure integration (as indicated above).
These tips are possible only because you know that the system has an integrator. Hence you can use that as prior information. An elegant way to solve the problem would be to use an estimator like the constrained-LS algorithm and constrain the final model to have a pole at z=1.
Good luck!
Your system seems to be analogous to a rigid wall lung simulator. That system "inhales" and "exhales" through a single opening according to what is known as the equation of motion: P(t) = E x V(t) + R x V'(t)
where P is the pressure difference across the system (Pout minus Pin), t is time, E is elastance (slope of P-V curve), V is (compressed) volume change, R is resistance (slope of P-V' curve, and V' is flow (positive flow is in, negative flow is out).
This is also an RC low pass filter and can be used to integrate an oscillating pressure signal to get average pressure (see attached article)
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