The summation point may be a motoneuron; the system may be a muscle and limb and the load it encounters; the feedback circuit may be due to spindle afferents, or from the slip detectors of motion on the retina.
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§ 1 / 1 The varieties of feedback experience
Our look at velocity storage in optokinesis revealed the role of feedback to control eye position. As engineers we want to move beyond a hand-waving verbal account of feedback, to a mathematical description which will allow us to model and predict various actions of the motor system. This chapter introduces the neurological context of feedback and tells you about the virtues of feedback in any system. You'll learn how feedback can stabilize, speed up and regulate a system. Part of what you'll learn is terminology: feedback vs feedforward, negative vs positive feedback, discrete vs continuous and linear vs non-linear feedback.
A SIMULINK demo may be given for a non-linear feedback system. Use Mathworks program SIMULINK to build dynamic systems with feedback. Consider dynamics of delay and integration. Simulink is like LabVIEW: A graphic-based
Some knowledge of Laplace transforms is helpful. You have seen Laplace transforms in AM33 and EN52.
Feedback means sending a copy of an output signal back to an input summation part of
the system, where it can influence the system components which helped form it in the first
place. If this definition sounds circular, it is! It may remind you of recursion in programming.
The copy of output sent as feedback may be attenuated or amplified; it may even be
reversed in sign; the feedback may be filtered for a frequency range. In any physical
system the feedback path is inevitably associated with some delay. The dynamics of that
delay can create some of the most useful or destabilizing effects. In the diagram below a
summation unit adds external input to output transformed through a feedback circuit. A
plus sign is shown on the summation path from the feedback circuit but depending on what
the feedback circuit does, the effect of feedback may be positive or negative.
The summation point may be a motoneuron; the system may be a muscle and limb and the load it encounters; the feedback circuit may be due to spindle afferents, or from the slip detectors of motion on the retina.
Negative feedback is useful in various situations where an output should be maintained at a desired level in spite of "load" disturbances. In physiology, where negative feedback is called homeostasis, it helps regulate blood pressure, blood sugar level, etc. The stretch reflex which keeps muscle length constant is an example of negative feedback.
§ 1 / 2 Closed loop gain calculation
Shown below are feed-forward and feedback configurations, with negative comparisons
at the summation units.
The feedforward path shows an example of feedforward inhibition, for making the OUTput
a selective version of input. (What if F(IN) responds best to large INPUTS?)
Now we develop the basic feedback equation, using algebra. Assume a system output y(t)
is intended to match a goal x(t), shown in the figure below. Without feedback y(t) = g(x(t)),
where g is called gain and is a function-anything from multiplication by a constant to a
differential equation. With negative feedback an error = x-f' is amplified by gain g until the
negative feedback reduces error to an acceptable level. As an example, the feedback
signal is formed by the product of y·f, where f is the scalar feedback gain and f* is the
feedback signal presented to the summation amplifier on the left.
Notice that the feedback and the input must be the same "units", usually volts.
It's negative feedback because of the subtraction of f from goal x.
Note that closed-loop gain is lower than open loop gain. In fact if f=1, closed loop gain is
always less than 1! Reduction in gain is normally not something to be desired, so there
better be other "positive" effects of negative feedback!
§ 1 / 3 Virtues of negative feedback
§ 1 / 3 / 1 Reduce sensitivity to internal parameter changes
Suppose g = 100 in the system above. Then the close loop gain is 
Now suppose g changes by a factor of 2, to 200. The close loop gain is still about 0.99.
Negative feedback reduces sensitivity of the system to changes in internal parameters. In
this case the only internal parameter we have is open-loop gain. Think of reduced
sensitivity to fatigue.
§ 1 / 3 / 2 Reduce sensitivity to external load changes
Let a variable load affect the output, as shown below:
Now 
Let F = 1.
Isolating y on one side of the equation, 
If L = 0 then the same G/(1+G) factor results. Even if L 0, the effect of load on the output is
reduced by 1/(1+G) !! Thus negative feedback systems can have reduced sensitivity to
external load changes. As an example of external load, consider a beaker held up by the
neuromuscular system of the arm; the aim is to hold the beaker at a constant position in
spite of liquid being poured in. What is the gain G of such a system?
§ 1 / 3 / 3 Increase speed of response
So far we've thought of the gain and feedback paths as algebraic or attenuation terms. Now
to dynamics. Consider the negative feedback system shown below. Plant and feedback
are represented by Laplace transforms, so multiplication of transfer functions can take the
place of time-domain convolution integrals. Let G(s) be 
, a first-order plant.
What time function is this a transform of? What differential equation does the Laplace
transform represent?
Let F(s)=k, 0 k , be a feedback factor.
If F(s) = 1 then the open loop response of the system is Y(s) = In(s) · G(s).
If In(s)=1 (impulse function) then 
a decaying exponential with time constant 1/a. The larger a is, the faster the system
decays. And decay to zero is what we want, when the input is an impulse. The impulse is a
brief disturbance, and we want the system to return to its zero state as soon as possible.
After t=0 input is zero, and we want y(t) to track the input. Now consider the closed loop.
Substitute G(s) in the formula G/(1+G), 
which we can do because we are in the frequency domain. If we were in the time domain,
convolution would be called for.
The formula simplifies to 
which has inverse Laplace transform 
Compared to the open loop form the time constant 1/(a+1) is smaller, so the system
decays to zero faster. If F>1 then the time constant will be even smaller than for F=1.
Bottom line: Negative feedback can speed up the response of a system.
§ 1 / 3 / 4 Improve stability
Now consider an unstable dynamic system:
Give the open loop block a Laplace transform with a negative time constant and set
feedback F = 0. The inverse Laplace transform is 
The exponential is positive. The system is unstable. It's impulse response grows
exponentially. Now consider what happens if negative feedback is put in place.
Let F=1.
Substituting 
in the formula we obtain
Let IN(s) be 1, for the input to be an impulse function.
now 
which has a time domain form of 
A stable form with a decaying exponential. What's the time constant?
Bottom line here: Negative feedback can stabilize a system.
Add that to the list of negative feedback virtues.
Notice that in the dynamic examples given, we haven't been worried about large open loop
gain, but certainly large open loop gain can be a feature of dynamic systems.
The next example has unity open loop gain.
tests for stability: poles in right half plane are unstable. Wolovich, page 174: Routh-Hurwitz criteria
§ 1 / 4 Two more uses of feedback: Inverse systems and positive feedback
§ 1 / 4 / 1 Inverse systems in feedback
Place the dynamic element in the feedback path,
while maintaining a high open loop gain A :
Let 
where p(s), as you may recall from EN52, is a polynomial for the poles of G(s).
Applying the feedback formula 
to this arrangement, while letting
IN(s) be 1, for the impulse input, we find,
As usual, let's assume A is a large number,
Voila: The output is the inverse of the feedback path. Sometimes we seek the inverse of a
system in order to "cancel" some effect or other.
§ 1 / 4 / 2 Increasing the time constant with positive feedback
You saw this in the lecture on time constant of the VOR.
Look at a dynamic system with positive feedback k less than 1:
Apply the feedback formula to find
Let 
and we have
OK, let's say k < a. If so, then a-k is a positive number smaller than a.
And the time constant of the system, 
is correspondingly larger.
Positive feedback can be used to "plug up" a leaky integrator, a theme for §26 later in the
book.
§ 1 / 5 Positive feedback producing an oscillator
For now consider the the effect of delay in the one-inverter-with-feedback:
OUT becomes IN almost instantly, in "rise time" s, but IN must wait a longer transport time D before it starts to change to the new value of IN. Assume D >> s. The result is the following "timing diagram" for input and output waveforms. Thinking about what causes what can send you around in circles, so start by considering that the circuit has just been turned on, and IN is zero. As soon as power is applied, OUT goes (with rise-time s) to HI, but a change of IN to HI must wait for delay D to expire.
An oscillation, with period 2·(D+s), results.
[If s > D, then the metastable condition can last "a long time"; such may be the case with various IC inverters.]
(To be more careful in this analysis of feedback we must specify the thresholds
qH and qL
at which the inverter snaps from one binary value to another.
Assume for now that
OUT will snap up if IN < 1 volt = qL, and
OUT will snap down if IN > 4 volts = qH .
§ 1 / 6 Delay in feedback
The Laplace transform of the pure delay operation is computed from the definition of the
one sided LT: 
where t0 is the delay of output from input.
The signal going up from a spindle to a motoneuron in a feedback loop is not just slowed
down by an integration process, it is absolutely delayed by finite (and slow) conduction
speed. In engineering terms such delay is called transport delay. You can find it in the
"nonlinear" menu of SIMULINK. It's commonly seen is chemical engineering systems
where fluid to be measured is transported down a pipe until it arrives at a sensor. How can
we handle pure delay in terms of Laplace transforms? The Laplace transform of pure delay
is 
where t0 is the duration of the transport delay. The exponential can be approximated
by the Maclaurin series,
or by the Pade approx: 
which I will try to demonstrate with SIMULINK!
[can also use the Z transform to study delay in feedback systems]
§ 1 / 7 Proportional, integral, derivative feedback
Control engineers call it PID. So far we have considered mainly proportional feedback, where the feedback signal is some proportional version of the output. Since the primary afferent sends a rate-of-change-of-stretch signal back to the motoneuron, it is derivative feedback. We will look briefly at the effects of PID. For example, integral feedback is capable of driving the error of a system to zero, not just to some small value.
§ 1 / 8 Other Reading
Kuo, B.C. Automatic Control Systems, 6th Edition, Prentice-Hall (1991)
chapter 1, "Introduction," pages 2-14.
Wolovich, W.A., Automatic Control Systems, Saunders (1994); book for EN166
Part 1: Dynamic systems representations (EN52)
Part 2: Performance Goals and Tests
General control and nominal stability
See Fig 5.1, page 166
The controller is to be designed
Loop goals
stability; gain margin, phase margin
disturbance rejection and noise attenuation
Response goals
transient tracking response
Part 3: Compensation
The controller design: input is regulation signal and feedback + noise
STABILITY: All the poles of the transfer function lie in the left half complex plane
Routh-Hurwitz criteria examine sign changes in P(s)
Nyquist criteria look at pattern of root locus(k)
Some knowledge of Laplace transforms is helpful.
You have seen Laplace transforms in AM33 and EN52.
While you read, keep in mind the feedback reflex circuits in the spinal cord.
§ 1 / 9 Further Reading
Clare D. McGillem & G.R. Cooper, chapter 5, §15, "Feedback systems,"
pages 270-283, in Continuous and Discrete Signal and System Analysis, 3rd Edition,
Holt-Rinehart and Winston (1991).
V. S. Vaidhyanathan, Regulation and Control Mechanisms in Biological Systems,
Prentice-Hall (1993).
Oppenheim & Willsky, chapter 11 (first 2 §'s) pages 685-701 in Signals and Systems,
Prentice-Hall (1983). O&W was the EN157 book a few years ago.
§ 1 / 10 Summary
* G/(1+G) NOTE LOWERED GAIN
* G/(1+FG) If FG>>1 then an inverse system F-1 is created.
* Virtues of negative feedback
insensitivity to parameter or load changes
increased speed
better stability
* Frequency response: What spindle input does in the cord
* Delay & stability & non-linear/time varying
* PID = proportional/derivative/differential controllers