Re: Determining Plant Bandwidth
- From: Tim Wescott <tim@xxxxxxxxxxxxxxxx>
- Date: Tue, 21 Mar 2006 22:02:03 -0800
Colby wrote:
Hello everyone, I'll do my best to be as thorough as possible in posingFirst, I question your assertion that your plant behavior is really a straight 20dB/decade line. Your servo valve will have its own dynamics, with a low-pass characteristic, that will affect the plant behavior. Your measurement will also have a low-pass characteristic.
my questions:
My plant is a single hydraulic actuator that is controlled using a
servovalve and Temposonic position feedback. I've identified the open
loop characteristics by using the method of least squares to fit
response data to a z-domain model that has second order dynamics, one
zero, and eight extra time delays (8ms delay). The data was sampled at
1kHz.
I'm designing a pole placement controller, with a 1kHz control loop
frequency, using Karl Astom's method presented in his "Computer
Controlled Systems" book. The desired system model I'm using keeps the
same process zero and order of time delay that was identified in the
open loop model, and includes second order dynamics for defining the
desired response. I'm finding that as I push the natural frequency of
the desired system model higher, I begin to create unstable poles in my
closed loop system. (To a lesser extent, I see this same effect occur
when I increase the desired model's damping ratio). I'm not 100% sure,
but I speculate that the instability is a result of trying to create a
closed loop system with a bandwidth that is higher than that of the
open loop system. Any input with respect to this proposition would be
appreciated.
Assuming that my speculation is true, I am left to determine the
bandwidth of the open loop system in order to design the closed loop
system with a bandwidth that will give me an acceptable margin of
stability. (Does that make sense?) My difficulty here is due to the
fact that the open loop system takes a velocity input (servovalve
position = flow), and gives a position output, so I'm not really sure
how the bandwidth is defined. The magnitude of the frequency response
plot is pretty much just a straight line. If I differentiate the
output to get velocity, does the bandwidth of the velocity transfer
function still correlate in some way to the achievable bandwidth of the
closed position loop?
Pole placement design only works to the extent that you know your plant characteristics. As such it is probably the best design method for leading you down the garden path. Unfortunately it is one of the worst methods that I know of for designing robust control systems. One is reduced to setting poles "where you know they're going to work" and iteratively cutting and fitting a control system 'design' to these theoretical pole positions.
I think the question you want to ask isn't "what is my plant bandwidth", but "to what degree can I trust my plant model at what frequencies". If you know the highest frequency at which you can trust your plant model then you can use this information to direct many different design methods. Unless it bumped up against some other requirement I would take a 10- or 20-degree phase shift due to the delay (or about 3.5 - 7Hz) as my highest frequency. The more you push against this limit the more you'll be sweating.
As you may have gathered, I'm not a fan of pole placement. For general system design I tend to favor designing with a Bode plot and a Nyquist plot -- I keep my gain and phase margins looking good on the Bode, and make sure that the trace on the Nyquist plot doesn't look like a pretzel. If I'm designing for a system that I know is going to vary I will use robust design techniques, but I find that my intuition is good enough for most design tasks, and faster.
I assume that you are working with the real system, with a design that should work yet is unstable in real life. Are you sure that you aren't running into any nonlinearities that are causing instability? Sticktion, square-law flow characteristics and actuator saturation all come to mind as nonlinearities that could cause significant variation between model and reality, and would create difficulties with any aggressively designed controller that assumes a linear plant.
Even if you aren't being tripped up by nonlinearities if your plant model diverges from the real system to any great degree around your loop closure frequency you're going to have trouble. To date I have always preferred using swept-sine frequency response measurements to time-domain ARMA fitting. This is partially because I prefer Bode plot design, but it is also because I feel that a good swept-sine measurement can get a lot of fine detail, as well as an indication of reliability, that is not apparent with curve fitting.
If it were me I would redo the measurements in swept sine, or at least with a really long random input. I would take the measurements at at least two different amplitudes to check for nonlinearities. Then I would design my system taking nonlinearities and expected plant variation into account.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
Posting from Google? See http://cfaj.freeshell.org/google/
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