A n example of a control loop is taken in which adjustment of hot and cold valves is done to maintain the water at a desired temperature. This involves the mixing of two process , the hot and cold water. The sensor senses its temperature. Based on this feedback a control action is perform to adjust the hot and cold water valves until the process temperature attains the desired value.
Proportional controller
The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain.
The proportional term is given by:
A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable. In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.
This can be understood as after measuring the temperature , and then calculating the difference between desired value and present value , the controller decides when to change the tap position . When the controller first turns the valve on, it may turn the hot valve only slightly if warm water is desired, or it may open the valve all the way if very hot water is desired.
Integral controller
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller is the sum of the instantaneous error over time and also compensate the previous errors. The net error is then multiplied by the integral gain (Ki) and added to the controller output.
The integral term is given by:
The integral term accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the setpoint value i.e. if there will be a large error then system will respond more quickly.
This can be understood as when hot water does not arrive quickly, the controller may try to speed-up the process by opening up the hot water valve more-and-more as time goes by.
Derivative controller
The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain Kd. The magnitude of the derivative term to the overall control action is termed the derivative gain, Kd.
The derivative term is given by:
The derivative term slows the rate of change of the controller output. Derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, the derivative term slows the transient response of the controller. This means that it actually controls the rate at which integral controller works.