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.



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.