**Feedback**

**The Feedback Equation**

The amplifier A produces and output that is A times its input, y = Ax. The feedback network B produces an output that is B times its input.

Thus in the system above y = A(x + By)

And y = Ax/(1-AB)

Hence the gain of the system with feedback is

**Negative feedback**

The original use of feedback was to improve the quality of amplifiers. If A
is very large it is easy to show that the gain becomes -1/*B*. Thus if *B*
is a negative number, the gain of the overall system is independent of the
properties of the amplifying device itself. This was the saving grace for the
transistor, which could be manufactured with but not well controlled high gain.
Applied to a whole amplifier it made Hi Fi possible. In general negative feedback
is regarded as a stabilising process, but there are hazards.

Negative feedback is used in biological systems to maintain stability. This process (homeostasis) keeps conditions in, for example, the healthy human body constant, regardless of a wide variation of magnitudes in the external stimuli. Disregarding such processes is used to create various health scares (salt being an important case in point).

**Positive feedback**

We can easily see that something strange happens when the product *AB*
(known as the loop gain) is equal to one. The gain of the system become
infinity. The result of this is that the system becomes unstable. The
type of instability that occurs depends upon the frequency response of *AB*.
The system can become an oscillator or a switch, the latter being a basic
component of computers. The public address system becomes an oscillator when the
amateur orator waves his microphone in front of the loudspeaker. He has turned
an **open loop** into a **closed loop**.

The attraction to computer modellers, however, is that, by judicious choice
of *B*, the gain of the system can be made any number they like, between *A*
and infinity. A famous example is with the putative contribution of carbon
dioxide to the Greenhouse Effect. This is basically negligible, but by invoking
a feedback mechanism involving the dominant greenhouse gas (water vapour) it can
be, and is, made to look substantial and scary.

Another important example of positive feedback is in disease. When Dr John
Snow removed the handle of the Broad Street pump and saved hundreds of people
from cholera, he had surmised that the cause was a minute entity capable of
reproducing in the human body. However, the exponential growth within the whole
human population only occurred when there was a **closed loop** via the cess-pits
and pump. Similarly feeding cattle on their own kind probably caused the BSE and
CJD crises.

**Feedback and numbers**

The concept of feedback can also be applied to sequences of numbers. An example of a process that produces number is the Fibonacci sequence.

y

_{i }= y_{i-1}+ y_{i-2}

Each number is the sum of the two numbers preceding it. It has important
analogues in the living world. It is also an example of a **recursive process**,
i.e. one that uses the previous outputs of the process to form each new number.
The general process that uses both inputs and outputs is

y

_{i}= ax_{i}+bx_{i-1}+c_{xi-2}+........ +B_{yi-1}+C_{yi-2}+.....

It is relatively easy to predict the properties of any such sequence from the coefficients, using a device known as the z transform, but as this requires the use of complex numbers it would not be appropriate here.

Because a recursive process involves feedback, it is capable of instability. Here is a simple example from Sorry, wrong number!

y

_{i}= x_{i}_{ }+ y_{i-1}- 1.2 y_{i-2}

Applying the simplest possible input test sequence (1,0,0,0,0,.....), we find that the output is a rising sinusoid, just as with the orator's PA system. Such a system is unstable, as the infinitely rising output occurs regardless of any input. Recursion, however, is a potentially powerful process, giving rise to an important branch of signal processing known as recursive digital filtering.

**Feedback and computer models**

We have seen that, even in these simple examples, feedback is a powerful but
dangerous concept. Computer models are also
powerful and even more dangerous. If they include one or more feedback loops
they are capable not only of instability but also several other problems.
Furthermore, they are not amenable to analysis as are the simple systems
outlined above. For a start, as in the simple amplifier we began with, small
changes in the feedback ratio can produce large changes in behaviour. This can
be used in misleading ways; as observed above, a notorious example being the
exaggeration of the negligible contribution of carbon dioxide to the greenhouse
effect by linking it in a feedback loop to the far more important water
vapour. Models with multiple feedback loops are also likely to be **chaotic**
(very small changes in coefficients produce very large changes of behaviour).
This puts great temptation in the path of those prone to self-deception (or,
indeed, out and out liars). Feedback can also exaggerate the effects of
computational errors (noise) and systems can become capable of operating on the
noise alone regardless of any inputs. Most computer models are non-linear
systems, for which there is no comprehensive stability analysis, and they can go
from stability to instability for different combinations of inputs.

Feedback is one of the main reasons that computer models that are not rigorously tested against reality are not worth the magnetic oxide they are written on. They are incapable of analysis.

The combination of feedback and delay can be deadly. Even with the supposedly stabilizing negative feedback, the introduction of delay can produce instability. Finance ministers look at (inaccurate) numbers three months old, compare them with numbers six months old, then introduce what they think is stabilising negative feedback, usually by way of taxation or interest rates. They are surprised when the result is recession (or worse) and blame it on everyone and everything except themselves. Examples are legion.