Negative feedback is created when the output of a system is subtracted from its input (or command) to form the error signal. The error signal is normally fed into a controller, and the resulting signal is applied to the process to be controlled (the plant). Most controllers are either of the P (proportional), PI (proportional-integrator) or PID (proportional-integrator-derivator) kind. Once a negative feedback loop is in action in a servo system, it is said to be in closed loop mode, by opposition to the open loop mode, where the command is fed directly to the controller without any negative feedback loop.

As opposed to positive feedback.

A system with negative feedback will tend towards a certain stable state. The most famous example of this effect is the Watt steam governor, a device for regulating the flow of steam in a steam engine. The steam governor will decrease the flow if the machine is going too fast, and increase the flow if it is going too slow. This keeps the engine running at the desired speed.

So a system with negative feedback will tend towards a stable state, like a steam engine with a Watt steam governor, which was designed to keep steam engines from breaking from too much strain, from destabilising.

Example of a steam engine calibrated to run at 100 rotations per minute (10% negative feedback per time unit)

```
time    machine 1   machine 2
0       150         50
1       145         55
2       140.5       59.5
3       136.45      63.55
t=inf.     100      100
```

Machine 1 is going too fast, so the RPM decrease over time until it reaches 100. Machine 2 is going too slow, so the RPM increase.

Negative feedback is generally used to regulate processes. Body temperature, as well as practically all bodily functions, is regulated by negative feedback.

Animal populations usually tend towards an optimum due to negative feedback. If there are too many animals of a species, there will not be enough food to feed them all, so more of them will die than usual. Deaths will outweigh the births, and population will decrease.

Conversely, if there are not enough animals, food will be so plentiful that almost all animals will survive. So the births will outweigh the deaths, and the population will increase.

Negative feedback keeps the world stable and alive.

Definition of Negative Feedback
```               ______
+      |      |
I -----0--->|  A   |-------+----> O
+^    |______|       |
|     ______        |
|    |      |       |
+----|  B   |<------+
|______|

Open Loop Gain = A
Closed Loop Gain = O = Forward  =   A
I   1 - Loop   1 - AB

```
Thus the effect of feedback can be defined as closed loop gain divided by open loop gain:
```  _      _
|   A    |  /
|_1 - AB_| /  A

=    1
1 - AB

Negative Feedback is when:
1    < 1
1 - AB

and positive feedback is when:
1    > 1
1 - AB

```

On a nyquist diagram the region of positive feedback is the unit circle around the point (-1, j0); the rest is negative feedback.

Effects of Negative Feedback

1. It reduces errors in the system.
2. It reduces the effect of disturbances.
3. It reduces the effect of changes in A.
1. Reduced Errors
What we normally want for the system above is for O = -I/B (for B = -1, O = I).
Error can be defined: Output Error = Desired Output - Actual Output
This is absolute error. A more useful measure would be error ratio:
Error Ratio = Output Error / Desired Output
It can be shown from the definition of closed loop gain or open loop gain (actual output) and the desired system gain (desired output) that the error ratio can be shown to be:
```For Closed Loop:
ER = (-I/B) - AI/(1-AB) =   1
(-I/B)          (1-AB)

For Open Loop:
ER = (-I/B) - AI = 1 + AB
(-I/B)
```
Therefore error has been reduced if the error with feedback (closed loop) is less than the error without feedback (open loop). If this is plotted on a nyquist diagram the result looks a little like this:
```                  Im
__    |    __ <------ 0.5
e-- _/  \_  |  _/  \_  e--
/      \ | /      \
/  e++   \|/   e++  \
--|----------|----------|----Re
\        /|\        /^
\_    _/ | \_    _/ |
\__/   |   \__/   |
|          |
|         sqrt(2)
```
Where the line is where error is unchanged, inside error is increased and outside error is decreased.

2. Reduced Effect of Disturbances
Disturbances are introduced into the system in the following way:
```                          D
|
______     |
+      |      |    v
I -----0--->|  A   |----0--+----> O
+^    |______|       |
|     ______        |
|    |      |       |
+----|  B   |<------+
|______|

For simplicity's sake assume that I = 0

Open Loop, O = D
Closed Loop, O =   D
(1-AB)

Thus disturbance is reduced if:
D >   D    =>   1    < 1
(1-AB)    (1-AB)

```

This is the definition of negative feedback as above. Thus negative feedback reduces disturbances.

3. Reduced Effect of Changes in A
Closed Loop Gain = G. Thus change in G from changing A, assuming B is constant, is:
```dG = (1-AB)*(dA/dA) - A * d(1-AB)/da =   1
dA              (1-AB)2                (1-AB)2
```
This is not useful in itself but relative change dG/G is
```dG = dA/(1-AB)2 = dA   1
G =   A/(1-AB)    A  (1-AB)
```
dA/A is the proportional change in A.
The other term you may recognise as the effect of feedback term which will be less than one for negative feedback. Therefore if feedback is negative then the effect of the change in A will be reduced.

This information was gained while revising for my finals from the lecture notes provided by R.J.Mitchell, Dept. Cybernetics, Reading University
In relation to the biological system, negative feedback has many positive features. To take a very simplified model:

```                |---------|
+             |    K    |
--->O---------->| ------  |------------------------->
I   ^           |  sT+1   |           |     O
-|           |---------|           |
|                                 |
|---------------------------------|

```

Aside from being very accurate, the negative feedback system also has a very unintuitive feature. For example:

Our transfer function H = K/(sT+1), a simple low pass filter. The transfer function for the loop would be H = H/(1+H) which would if we subbed in the equations be:

{K/sT+1} / {1+K/(sT+1)}

which would simplify further to get

k/sT+1+k

The steady state gain (the gain at an unvarying input or s=0) would be come k/1+k.

Now, lets say in a normal person the gain of the filter is 9. Then the gain of the entire loop would be 9/1+9 which would be 0.9. Now lets say you had a lesion in your brain which killed off about 4 neurons in this simplified case. Your gain would drop to 5. However, the loop gain would be 5/1+5 which is 0.83. Therefore, a loss of 44% of the number of neurons involved in a pathway would only leave you with a loss of 7.8% of system gain. Very useful!

Of course this is all very simplified. There are indeed no biological system in which you have a negative feedback with a single low pass filter. But it does give you a nice example of how they help the body resist damage.

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