control path with differnent rise and fall characteristics

There is a unit delay function, which for a specified time delay, say d,
would allow you to write something like this:

U = Uo * (1-exp(t/T1) - (Uo(t-d))*exp(t/T2)  ...... where Uo(t-d) is
directly transformable.

How does this relate to your application?

Dave

richard bruckner wrote:

I haven't seen the 'PT1' notation; I'm not sure where it's from.

This cannot be modeled exactly as a linear time invariant system, so the
question you need to ask isn't "how do I model this", but "can do I
model this _well enough_ to match _my environment_, and how".  This
means that you have to make a realistic assessment of your needs and
your environment, because you can kill yourself both by being too
optimistic and too pessimistic.

You speak of separate rise and fall times -- is this system actuated by
an on/off actuator?  If your actuator isn't proportional then the system
cannot, by definition, be truly LTI.  If it's on/off and you're using
PWM to linearize it then you can close one eye and pretend it's a linear
system, except then you have that 2x difference in apparent time
constants to deal with.

Tell me more if I'm off base, but I'm going to assume it's something
akin to your PWM'd on/off system with two distinctly different time
constants, and rattle out some approaches:

The linearized (with PWM) system will have an effective time constant
that ranges between 70ms and 150ms depending on the duty cycle.  You
could just model the system as having a time constant that ranges
between these two values, and design a controller that's robust to the
changes.  If you're using classical control techniques then you should
be OK with Bode and Nyquist plots of both extremes, possibly backed up
with some root-locus plots of the final system as the time constant
varies.  I'd cling to this approach for a long time before I tried another.

You can use describing function analysis.  In a sense this is what
you've done already with step functions, and you've gotten two different
answers.  So if you do it with sine waves, i.e. stuff a sine wave into
the input of your plant and look at the amplitude and phase of the
fundamental of the output, then you may get something that's more
representative.

You may be able to linearize the system by identifying the root cause of
the time constant and controlling to that.  I have built virtual torque
drives for motors that measured the motor velocity, calculated the
motor's back-EMF, and controlled the motor's voltage to get a certain
current, all without actually measuring the motor current.  This was in
a system that could stand substantial inaccuracy, but it worked well
enough*.

In your case you would look at what your actuator actually pushes on,
and what actually pushes back to generate the two time constants.  Then
you'd work backward to whatever internal force is acting on the
integrating element, and have a nonlinear feedforward to control the
drive to that integrating element.  This kind of thing can be
astoundingly sensitive to plant variation though, so I wouldn't try it
unless you know your system can stand the variation as the plant
changes, or you _know_ you can avoid it in the first place.

* I think "it works well enough" should be the catchphrase for all
control systems designers, by the way -- you could kill off entire
companies if everyone were chasing "optimum" instead of "good enough".
I choose to think of it as "optimized for profit".

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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