Proportional plus derivative PD controller

The minimum controller configuration containing derivative action is the combination of proportional plus derivative action shown in Equation 4.18. This combination is not used very often and is primarily apphed in batch pH control loops. However, it will help in the definition of derivative time T.  

For a ramp input it takes a period of time for the proportional action to reach the same level as the derivative action. This period of time is called the derivative time and is measured in minutes. Increasing the derivative time 7d increases MVd, or the contribution of the derivative action to the movement of the final control element. 

In Equation 4.18, for the PD controller the derivative action acts on the error. Since e = SP - CV for I/D action, de/df is a function of both the derivative of the set point dSP/df, and the derivative of the controlled variable dCV/dt  

In other words, there is no derivative action on a set-point change, only proportional action. On a load upset, both proportional and derivative actions are enabled. (Note also that, in some controller implementations, the proportional action is also decoupled from set-point changes, as the kick from a set-point change is also considered to be too aggressive.) 

In a PI controller, in order to minimize the integral action, Tj was made a large number. This makes the integral gain approach zero, and the controller then behaves essentially like a P-only controller. However, in the PD controller, even by setting Ti to a very small value, there is still the possibility of a sizeable derivative contribution if there is a noisy input, i.e. if dCV/dr is large. 

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We have just described proportional (P) control and proportional plus derivative (PD) control. Integration can be added to a controller, which not only gives it reset action, but also can exacerbate instability. There are proportional plus integral (PI) and proportional-integral-derivative (PID) controllers. These classical types are used where the system dynamics (the Plant) are well defined. 

In order to demonstrate the effect of derivative action we will formulate a proportional plus derivative (PD) controller. This probably has no practical application but including integral action would make the trends very difficult to interpret. Combining Equations (3.5)... 

This control mode is called proportional plus rate (PD) control because the derivative section responds to the rate of change of the error signal. 

Controller modes—settings and functions that include proportional (P), proportional plus integral (PI), proportional plus derivative (PD), and proportbnal-integral-derivative (PID). Proportional control is primarily used to provide gain where little or no load change typically occurs in the process. Proportional plus integral is used to eliminate offset between the setpoint and process variables PI works best where... 

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