[From Bill Powers (951001.0850 MDT)]
Hans Blom (9509xx) --
I seem to be coming up on a slight bit of understanding about the
difference between the "modern control theory" approach and the PCT
approach. I realized this morning that all this stuff about statistics
and uncertainties is just confusing me, and conceals the structure of
the calculations. It will be much easier for me to understand in terms
of ordinary algebraic variables. Once the basic method is clear for a
simple example, we can always generalize and introduce statistical
methods for doing the same thing in the presence of random noise
components.
Don't expect too much, as I'm working this out as I go.
Start with your world-model and the real system outside:
r
>
------------> f(y,r)-------->---
> >
> >
> >
> world model |
y <--------- K*u + d------------ u
>
x <--------- Ko*u + Do<--------- u
real system
The world-model equations are
(1) y = K*u + D
(2) u = f(y,r)
The "modern control theory" approach, as I understand it, uses the first
of these equations to deduce what is necessary if y is to equal r. :
If y = r, then r = K*u + D, and
(3) u = (r - D)/K
If the model has certain values of D and K, the value of u that will
make its output y match r can be calculated from equation (3). This
value of u can be used as input to the real-world plant, and the output
x of the plant can be observed. In general, K and D will not be the same
as Ko and Do, so the value of x will be different from y. This
difference, however, can be used as the basis for adjusting K and D to
bring x and y closer together. If the procedure converges, we will
arrive at K and D values which make x = y = r. These values may or may
not be the same as Ko and Do.
The basic procedure is
1. Start with assumed values of K and D
2. Compute u = (r - D)/K.
3. calculate x from Ko*u + Do (i.e., get the response of the plant)
4. Calculate slopes dK/dt and dD/dt based on the sensitivity of each to
dy/dt. Calculate a delta-K and a delta-D from the values of dK/dt,
dD/dt, and the error, x - y.
5. Add delta-K to K and delta-D to D, and repeat from step 2.
So far so good.
Now I have to figure out how step 4 is done. I think this may be a good
time to stop and ask you for a hand.
···
----------------------------------------------------------------------
Best,
Bill P.