Understanding PID

By watching this video, you can understand PID in just 4 minutes. More explanation is given below on this website.
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There are four different types of controllers and these are P, PI, PD and PID explanined below.


P Controller:


 P controller Block Diagram

>>In Proportional Only mode, the controller simply multiplies the Error by the Proportional Gain (Kp) to get the controller output.

>>Small proportional gain (Kp) is the safest way to get to setpoint, but your controller performance will be slow. If the Kp is increased, Overshoot in the signal will be present.

In the next figure we show an example of a P controller with different Proportional Gains.


 P controller Block Diagram - Proportional gains Kp of 100, 500, and 2000. As the gain is increased the system time response is faster and less damped. 

Comments on result: Clearly, it is not possible to achieve low steady state error and good transient response using only proportional control. As the gain is increased, the response becomes faster, but it has a lower phase margin. To remove the steady-state error and have better response, integral and/or derivative  terms must be included in the controller.

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PI Controller:


 PI controller Block Diagram

In Proportional Integral mode, the controller make the following: 
Multiplies the Error by the Proportional Gain (Kp) and Added to the Integral error multiplied by Ki, to get the controller output. In the next figure we show an example of a P controller with different Proportional Gains.

The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual  steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). 

In the next figure we show an example of a PI controller with different Proportional and Integral Gains.


 PI controller Block Diagram - This Figure shows step responses of the closed loop system for a = 3 and proportional gains of Kp = 25, 50, and 75. Where a = Ki / Kp. 

Comments on result: Integral control has removed the steady-state error and improved the transient response, but it has also increased the system settling time. Settling times can be lowered by increasing the gain. This will increase the system bandwidth, but it will also decrease the stability margin. 




PD Controller:


 PD controller Block Diagrambr />
In Proportional Derivative mode, the controller make the following:
Multiplies the Error by the Proportional Gain (Kp) and Added to the Derivative error multiplied by Kd, to get the controller output. In the next figure we show an example of a P controller with different Proportional Gains.

The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the stability. However, differentiation of a signal amplifies noise and thus this is highly sensitive to noise in the error term, and can cause a process to become unstable. 

IIn the next figure we show an example of a PD controller with different Proportional andDerivative Gains.


 This Figure shows step responses of the closed loop system for a =10 and derivativegains of KD = 10, 27, 50, and 75. Where a = Kp / Kd. br /> 
Comments on result: The PD controller has decreased the system settling time considerably; however, to control the steady-state error, the derivative gain KD must be high. This will decrease the response times and increase the bandwidth of the system and may make it susceptible to noise. 

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PID Controller:


 PID controller Block Diagrambr />
In Proportional Derivative Integral mode, the controller make the following:
Multiplies the Error by the Proportional Gain (Kp), Added to the Derivative error multiplied by Kd and Added to the Integral error multiplied by Ki, to get the controller output. In the next figure we show an example of a P controller with different Proportional Gains.

IIn the next figure we show an example of a PID controller with different Proportional, Integral and Derivative Gains.


 This Figure shows shows step responses of the closed-loop system for a =15, b = 50, and derivative gains of KD = 5, 10, and 15. Where a = Kp / Kd and b = Ki / Kd br /> 
Comments on result: Using both integral and derivative control (PID) has removed steady-state error and decreased system settling times while maintaining a reasonable transient response. 



PI, PD, PID Summery Characteristics:

PD
>> Compensator is anticipatory; it responds to the error and its derivative.
>> Phase lead is provided starting one decade below the zero.
>> Generally, increases damping and reduces %OS.
>> Generally, reduces rise and settling times.
>> Increases bandwidth.
>> Increases phase and gain margins.
>> May render a system susceptible to high frequency noise.
>> Acts as a high-pass filter.

PI
>> Compensator increases the system type by one, which helps with error control.
>> Increases phase-lag at low frequencies.
>> Generally, increases damping, rise times, and settling times and reduces overshoot.
>> Decreases bandwidth.
>> Not sensitive to high frequency noise.
>> Acts as a low-pass filter.

PID
>> Combined effects of PI and PD compensation.
>> Cascade of a PI and PD compensator.


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