 It can be mathematically proven that the working frequency of the controller is increased and the time constant of the object is reduced by using cascaded PID controllers..

For example, a temperature-controlled circulating bath has two PID controllers in cascade, each with its own thermocouple temperature sensor.The outer controller controls the temperature of the water using a thermocouple located far from the heater, where it accurately reads the temperature of the bulk of the water.The error term of this PID controller is the difference between the desired bath temperature and measured temperature.