Proportional-Integral-Derivative (PID) controllers are widely used in various control systems to enhance stability and performance. In a thermostat, it adjusts heating or cooling based on the temperature difference between the actual and desired levels. They are often used in automotive speed systems, effectively managing sudden speed changes while maintaining a constant speed under varying conditions. On the other hand, PI controllers, commonly employed in voltage regulation, enhance stability and reduce steady-state error but increase the time to achieve the desired voltage.

A PID controller combines the features of both PD and PI controllers, balancing their respective advantages and addressing their limitations. When designing a PID controller, it is initially treated as a PI part connected in series with a PD part. The proportional constant of the PD section is set to unity since the PID controller requires only three parameters.

First, only the PD component is activated. The derivative gain is adjusted to achieve the desired stability, which is evaluated by observing the maximum overshoot in the time domain and phase margin measurements in the frequency domain. This step ensures that the controller responds swiftly to changes while maintaining adequate stability.

Next, the integral and proportional gains for the PI section are selected to fulfill the overall stability requirements. The integral gain helps eliminate steady-state errors, while the proportional gain adjusts the system response, ensuring that the controller meets the relative stability criteria.

By combining these components, a PID controller effectively manages both transient and steady-state behavior, offering a more comprehensive control solution. The proportional, integral, and derivative elements work together to provide a balanced response, mitigating the drawbacks of using PD or PI controllers alone. This integrated approach is essential in applications requiring precise and stable control, such as in thermostats and various industrial systems