How do you calculate the steady-state error in a control system?

Step Input:

For a control system with a step input, which is a sudden change in the reference signal from one constant value to another, you can calculate the steady-state error as follows:

Calculate the difference between the desired output value (steady-state value) and the actual output value at a sufficiently large time after the system has stabilized.

The steady-state error is the absolute value of this difference.

Ramp Input:

In a control system with a ramp input, where the reference signal changes linearly over time, the steady-state error can be calculated using the following steps:

Find the slope of the reference signal.

Calculate the difference between the desired output value and the actual output value at a sufficiently large time after the system has stabilized.

Divide this difference by the slope of the reference signal.

The steady-state error is the absolute value of the result.

Other Input Signals:

For other types of input signals, such as sinusoidal or arbitrary signals, it is common to use frequency domain analysis or the concept of transfer functions to calculate the steady-state error. In these cases, you typically determine the error by evaluating the system's frequency response or the transfer function's gain and phase at the frequency of interest.

It's worth noting that the specific equations and calculations can vary depending on the control system's characteristics, such as the type of controller (proportional, integral, derivative, or combination), the plant dynamics, and any additional components or feedback loops in the system. The above methods provide a general overview, but it's recommended to consult the specific control system theory or documentation for detailed calculations in your particular case.