# Control Systems/Gain

## 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.

![](Gain_Block.svg "Gain_Block.svg"){width="400"}

### Example: Gain

## 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.

## 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**.
But it will decrease the steady state error.

### 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 often the threshold where
the system becomes unstable is important to find.

