# Control Systems/Gain

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## What is Gain?

Gain is a proportional value that shows the relationship between the magnitude of the input to the magnitude of the output signal at steady state. Many systems contain a method by which the gain can be altered, providing more or less "power" to the system. However, increasing gain or decreasing gain beyond a particular safety zone can cause the system to become unstable.

Consider the given second-order system:

$T(s) = \frac{1}{s^2 + 2s + 1}$

We can include an arbitrary gain term, K in this system that will represent an amplification, or a power increase:

$T(s) = K\frac{1}{s^2 + 2s + 1}$

In a state-space system, the gain term k can be inserted as follows:

$x'(t) = Ax(t) + kBu(t)$
$y(t) = Cx(t) + kDu(t)$

The gain term can also be inserted into other places in the system, and in those cases the equations will be slightly different.

### Example: Gain

Here are some good examples of arbitrary gain values being used in physical systems:

Volume Knob
On your stereo there is a volume knob that controls the gain of your amplifier circuit. Higher levels of volume (turning the volume "up") corresponds to higher amplification of the sound signal.
Gas Pedal
The gas pedal in your car is an example of gain. Pressing harder on the gas pedal causes the engine to receive more gas, and causes the engine to output higher RPMs.
Brightness Buttons
Most computer monitors come with brightness buttons that control how bright the screen image is. More brightness causes more power to be outputed to the screen.
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## Responses to Gain

As the gain to a system increases, generally the rise-time decreases, the percent overshoot increases, and the settling time increases. However, these relationships are not always the same. A critically damped system, for example, may decrease in rise time while not experiencing any effects of percent overshoot or settling time.

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## Gain and Stability

If the gain increases to a high enough extent, some systems can become unstable. We will examine this effect in the chapter on Root Locus.

### Conditional Stability

Systems that are stable for some gain values, and unstable for other values are called conditionally stable systems. The stability is conditional upon the value of the gain, and oftentimes the threshold where the system becomes unstable is important to find.

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