Feedforward model calculations

[From Bill Powers (931109.1330 MST)]

Martin Taylor (931108.1915) --

Added thought on the L&H system.

I believe there's a quicker way through your diagram. Using your
notation and a linear system:

···

r

                     -----------------
                    > >
                    B A
                    > >
                    V + + V +
                  [comp]--->G------->[ ]
                    ^ - |
                    > >
                    s |
                    > >
=================== P =============== | ============
                    > >
                    c ----- E <-------

First, the optimum values of A and B:

For zero disturbance, we want s = rAEP = r, so

   (a) A = 1/EP

When s = r, we want Br - AEPr = 0 (zero error)

   but A = 1/EP, so Br - r = 0, or

   B = 1.

Now we can solve the system equations in general (without
dynamics) and then substitute the optimum A and B values.

s = PE(Ar + GBr - Gs), or

       PEAr + PEGBr
s = -----------------------
       1 + PEG

For optimum A (1/PE) and B (1), we have

        r(1+ PEG)
s = --------------------- = r, exactly.
        1 + PEG

In the canonical diagram, A = 0, giving

s (canonical) = [PEG/(1+PEG)]r

So the feedforward addition brings s to the reference value while
without the feedforward it brings s to PEG/(1+PEG) of the
reference value. The difference is r/PEG, which approaches zero
as PEG approaches infinity.

The real advantage that is visualized for this arrangement only
shows up when the environmental feedback has a dynamic lag. Then
the factor A can add a correction equal to the dynamical inverse
of E, just making up for the lag in the closed loop path.

This salubrious result, however, is trivial. Simply by moving A,
we can get the same dynamical improvement in a canonical design:

                             >r
                     --------
                    >
                    B = 1
                    >
                    V +
                  [comp]--->G-------->
                    ^ - |
                    > >
                    s A = inv(E)
                    > >
=================== P =============== | ============
                    > >
                    c ----- E <-------

The only improvement that is lost is the exact steady-state match
of s to r in the absence of disturbances of c. If A has an
integrating component, that exact match will also be achieved
with the canonical design. Even without the integrator, setting

B = 1 + 1/PEG

will achieve the exact match with zero disturbance, if that is
important. Of course that calibration will remain correct only as
long as P, E and G do not change. In the original, the
calibration requires that P and E not change.

This is the biggest difference between the canonical and the
feedforward cases. If E doubles, for example, the value of A is
then twice as large as it should be, requiring a large correction
from the closed-loop portion. Without A, doubling E will simply
halve the error. Since that error can be small to begin with,
halving it is equivalent to essentially no effect on the
controlled variable.

A properly designed system with pure feedback can do just as well
as a feedforward system, and its operation is not materially
affected by reasonable changes in its own output parameters or of
the external feedback loop. A pure feedforward system requires
that these parameters remain unchanged. All that saves the Lang
and Ham system from being totally susceptible to changes in
environmental parameters is the negative feedback loop.

Note that the L&H system, like the canonical system, will not
work properly if the feedback connection is lost.

So far I have seen no advantages of feedforward over feedback,
and there seem to be a number of obvious drawbacks.
---------------------------------------------------------------
Best,

Bill P.