What is an integral controller (I-controller)?

The purpose of an integral controller is to eliminate the steady-state error in a system. Steady-state error refers to the difference between the desired output and the actual output of a system after it has reached a stable state. In many systems, such as those used in industrial processes or robotics, it is crucial to minimize or eliminate this error.

The integral controller achieves this by continuously integrating the error signal over time and adding it to the control signal. The error signal is the difference between the desired output and the actual output of the system. By integrating the error, the integral controller accumulates the historical error and uses it to adjust the control signal.

Mathematically, the output of an integral controller at any given time t is given by:

Output(t) = Kₐ ∫[0,t] (Error(τ) dτ

Where:

Kₐ is the gain or tuning parameter of the integral controller.

Error(τ) is the error signal at time τ.

∫[0,t] represents the integral operation from 0 to t.

The integration of the error signal allows the integral controller to gradually reduce the steady-state error by applying corrective actions over time. The longer the system experiences an error, the larger the correction applied by the integral controller. This process continues until the error is minimized or eliminated.

It's important to note that while integral controllers are effective at reducing steady-state error, they can introduce additional challenges. For instance, they can lead to overshoot or instability if not properly tuned or if the system has inherent limitations. Therefore, a balanced approach involving all three types of controllers (P, I, and D) is often used in practical control systems to achieve the desired performance. This combination is known as a PID controller, where P refers to proportional, I refers to integral, and D refers to derivative.