[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.