Physics
PID controller applies correction to a control variable 
by evaluating the difference 
between the measured process variable (PV) and desired setpoint (SP). A combination of terms proportional (P), integral (I) and derivative (D) together is called as PID controller. These three control terms are varied together or in combination of two based on the application to get an accurate and optimal response. The typical workflow of PID controller is shown in the following figure.
by evaluating the difference between the measured process variable (PV) and desired setpoint (SP). A combination of terms proportional (P), integral (I) and derivative (D) together is called as PID controller. These three control terms are varied together or in combination of two based on the application to get an accurate and optimal response. The typical workflow of PID controller is shown in the following figure.Block diagram of PID controller
Source: Wikipedia
The function that governs the control variable exists in Parallel (Ideal) and Standard forms. The Parallel (Ideal) form of the control function is
where,
= Control variable = Error between the process variable (PV) and setpoint (SP)
= Coefficient for the proportional term (Proportional Gain)
= Coefficient for the integral term (Integral Gain)
= Coefficient for the derivative term (Derivative Gain)The standard form of control function is
where,
= Coefficient for the proportional term (Proportional Gain)
= Integral time, the time sample in which the I-controller tries to eliminate the error completely
= Derivative time, the time at which the derivative term tries to predict the future errorThe behavior of three control terms of PID are:
Proportional Term (P - Controller)
P-controller provides an output that is proportional to current value of error (SP - PV). The resulting error between the Setpoint (SP) and Process Variable (PV) is multiplied with a proportional constant () to get the output. The speed of the output response depends on the proportional gain (). Higher value of () results in large change in output for a given value of error and can make the system unstable. On contrast, smaller value of () makes the system less responsive for a given change in the error and for any fluctuations in the system. The P-controller always operates with a steady state error, as it is completely driven by non-zero value of error. The setpoint cannot be achieved with P-controller as the applied correction approaches zero with the error approaching to zero. In general, industrial practices suggest that the proportional term must contribute the bulk of the output response.
) to get the output. The speed of the output response depends on the proportional gain (). Higher value of () results in large change in output for a given value of error and can make the system unstable. On contrast, smaller value of () makes the system less responsive for a given change in the error and for any fluctuations in the system. The P-controller always operates with a steady state error, as it is completely driven by non-zero value of error. The setpoint cannot be achieved with P-controller as the applied correction approaches zero with the error approaching to zero. In general, industrial practices suggest that the proportional term must contribute the bulk of the output response.The response of a system to proportional gain () is shown in the following image. It can be seen with increase in (), the Process Variable overshoots the Setpoint and starts oscillating.
) is shown in the following image. It can be seen with increase in (), the Process Variable overshoots the Setpoint and starts oscillating.Response of PV to step change of SP with time. () and () are held constant
) and () are held constantSource: Wikipedia
Integral Term (I - Controller)
Integral term accounts for past values of error and integrates them over a period of time, until the error reaches zero. I-controller considers how long and how far process variable is away from setpoint, unlike the P-controller that only considers how far it is away from the setpoint. I-controller seeks to eliminate the residual error after the application of P-controller by adding a gain () to the cumulative value of error. I-Controller is primarily used to reduce steady state error in the system. For many applications, P-I controllers are combined and sufficient to get a good response, accelerating to the set point with P-controller and eliminating the steady state error with I- controller. However, in the process of bringing the cumulative error to zero, I-controller can sometimes overshoot the output response as shown in the following message.
) to the cumulative value of error. I-Controller is primarily used to reduce steady state error in the system. For many applications, P-I controllers are combined and sufficient to get a good response, accelerating to the set point with P-controller and eliminating the steady state error with I- controller. However, in the process of bringing the cumulative error to zero, I-controller can sometimes overshoot the output response as shown in the following message.Response of PV to step change of SP with time. () and () are held constant
) and () are held constantSource: Wikipedia
Derivative Term (D - Controller)
Derivative term determines the slope of the error over time and multiplies it with a derivative gain (). D-controller anticipates the future behaviour of the error based on the current rate of change and fastens the system output response. If the change is high or slope is varying continuously, high dampening effect is required to control the change. D-controller moves the control device in a direction to counteract the sudden change of the process variable. A pure D-controller cannot bring the error to zero, as it considers only the rate of change of error. It only tries to bring the rate of change to zero by damping and thereby reducing the overshoot of the output response, as shown in the following figure.
). D-controller anticipates the future behaviour of the error based on the current rate of change and fastens the system output response. If the change is high or slope is varying continuously, high dampening effect is required to control the change. D-controller moves the control device in a direction to counteract the sudden change of the process variable. A pure D-controller cannot bring the error to zero, as it considers only the rate of change of error. It only tries to bring the rate of change to zero by damping and thereby reducing the overshoot of the output response, as shown in the following figure.Response of PV to step change of SP with time. () and () are held constant
) and () are held constantSource: Wikipedia
