Re: control law modification

OK Tim,

I'm going to snip a bit here as its getting hard to read....

y" + 4.2y' + 9y = C.

But your original equation doesn't involve C (at least as you've
presented it here). It looks like you took the specified natural
frequency and damping factor and 'derived' your end equation by looking
in a handbook.


# No. My failure to communicate well. The y"+4.2y'+9y was/is based on our
last homework. He added the C, as a wind disturbnace input for this
problem. The equation came from class notes/ application of the
formula/equations:

y" + 2zw y' + w^2 y = 0.

given zeta,z, = 0.7 and w,omega, = 3, we get the equation above.


Really, the problem should be stated as something like

y" - k1*y = k2 * B[] + (some expression involving C),
then when you find your gains you can work out where C fits in to the
output. You're being asked to make a generic PI controller, which should
reject any steady-state disturbance from C, so the question is fair as
long as they are telling you where C comes into the dynamics of your
system.

Argh! No, you're not! It's early enough in the morning here in Oregon
that the caffeine hasn't taken effect!

# I can understand that!


You're being asked to make a _PD_ controller (one gain times proportional,
another times differential). Sorry for any confusion.

#Do I have the steady-state part correct?

How does one make a generic PI controller? Trying to read between the
lines here. Are you suggesting that
B[] is modified by: (integrating?)

B = g1Y(s)/s + g2 sY(s)/s = g1 Y(s)/s + g2 Y(s) ?

Actually, if you implement this as it implies (integrate everything, then
differentiate) you'll get a problem because of the pole-zero cancellation.
In the s domain you want something like

B = k_p Y(s) + k_i Y(s)/s + k_d Y(s)s

But isn't this a PID controller? I think I see where the instructor is
going. Our last homework was a PD with no forcing function and now he wants
us to see (?) that by adding the forcing function we'll need a PID??? If
so, then if I take my original B[] = g1*y + g2*y' and integrate it I'll get:

new B[] = g1 * y/s + g2* y + Ci where Ci is an integration constant?

I think I kind of gave away the show. Oh well.

I don't think so...I'm as confused as ever!

Am I on the right track?
Aside from the pole-zero cancellation, yes.

You lost me there. I'm going to re-read this chapter because I'm missing
something.

Another unrelated question:

What is the physical meaning of 'residue' when computing the residues of a
G(s)? The book really doesn't explain that very well.



It seems to me that not integrating, but differentiating the equation is
the only way to get rid of C in


y" - k1*y = k2 * B[] + C

Ah! You can differentiate y, and get rid of C -- but that just tells you
that y will settle to some steady value. You want (presumably) y to
settle to 0 (or zero steady-state error). For that you need an integrator
in the feedback.

Yes I do want steady-state=0. So I want a PID--not a PI or PD only. Is that
right?

You'll go through all the math, exhaustively, soon. But for a preview:

When you integrate your error the output of the integrator is guaranteed
to follow the average error*. So _any_ non-zero average error will make
your integrator value build up. A well-behaved plant** will respond to
this by driving toward the set point, which will drive the average error
to zero, which is what you want.

So integrators are good. But integrators in your loop will reduce
stability.
So integrators are bad.

They're kind of like managers (or girl friends) that way.

Sounds like a control theory paradigm applied to women... ha ha

* That's _average_ error. So in the absence of any clever schemes your
error may have to go to the other side of zero for a while.

** I.e. a linear plant, or one who's response is strictly monotonic. If
you're where I think you are in your course of study I just spouted
gibberish -- just wait until you first try to close a loop around a sticky
motor driving a loose gearbox, then you'll understand.

Sounds complex.... I'm only 3 weeks into the class as a Grad student, but I
had Controls 20+ yrs ago as an undergrad, so I needed the
refresher..apparently more than I realized.

Thanks,

Bo


.

Relevant Pages

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