Linear Feedback Analysis

[From Rick Marken (2005.10.03.1120)]

Bjorn Simonsen (2005.10.04,11:55 EST)

Back to my question. At the pages 84 and 85 we read: 1.) e = r � p �2.) p =
k*e� 3.) p = k*r/(1+k).

Page 87: ���. The act so as to maintain the perceptual signal at nearly the
same magnitude as the reference signal. (this is OK, -my words) This result
remains essentially true despite a variety of changes which without the
feedback to the subtractor neuron, would drastically alter the sensory signal.

I think I understand the last sentence, but I need a verification/denial. I
read it as: If the sensory signal did not go to the subtractor neuron, and we
did not have a feedback, the sensory signal would not change.

I think it means that without feedback the sensory signal _would_ change
drastically.

If k changes from 10 to 20, there is a drop in the error signal from r/11 to
r/21.

Then the muscle contraction should not be so intense, but the muscle is more
sensitive (20). Therefore it will contract more than if k = 10.

And now, my uncertainty. The two last sentences in the section.
Am I correct if I say. Think upon k = 10. The reference signal and the
perceptual signal are almost in balance. That is e is near zero.� If k becomes
20, p = 20* e.

That's what p would be if there were no feedback. With feedback, p =
(20/21)*r, making it even closer to r than it was when k = 10.

This is a rise in the perceptual signal. But if e was near
zero and p becomes greater, than e becomes negative. And that doesn�t� work.

I know I am wrong, but I don�t know where.

I think your problem comes from computing p based on the open loop equation
for p, p = ke. This equation doesn't take into account the feedback effect
of p on e. The feedback effect is taken into account when you replace e
with r-p so that p = k (r-p) which shows that p is a function of itself to
some extent; that's the feedback effect. Solving p = k (r-p) for p we get
the feedback equation for p, p = k*r/(1-k).

Best

Rick

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Richard S. Marken