Proportional plus derivative controllers

The amplifier network provides signal conversion and suitable static and dynamic compensation for good positioner performance. Control from this block usually reduces to a form of proportional or proportional plus derivative control. The output from this block in the case of a pneumatic positioner is a single connection to the spring and diaphragm actuator or two connections for push/pull operation of a... 

With the exception of derivative action any of these control modes may be used alone in certain applications. Integral and derivative actions are most usually combined with proportional control to give proportional plus integral control (proportional control with automatic reset) proportional plus derivative control or three-mode control, which is proportional plus integral plus derivative. 

Proportional-Plus-Derivative Control. Here, derivative action is added to proportional controllers for processes with large capacitance and appreciable dead-time. Control action is now proportional to the rate of change (the time derivative) of the error signal. The response equation is written as... 

Derivative-action time is defined as the amount of lead, in seconds, that the derivative action advances the effect of pure proportional action. Figure 24.7 illustrates a proportional-plus-derivative controller. 

Figure 24.7. The proportional-plus-derivative controller. Derivative action is accomplished by a shunt capacitor C across Rf. When deviation from the setpoint is rapid, the low reactance of the capacitor causes less negative feedback—hence, greater amplifier gain. The derivative time resistor Ra allows adjustment of the magnitude of derivative control action to a given rate of change of the error signal. Courtesy of the Foxboro Company.

If the derivative time of the controller is set equal to t2, the above expression can be substituted into the proportional-plus-derivative controller equation, with the result... 

In the two-capacity process, then, setting D = n will not completely cancel tj, but will replace it with a lag equal to T2/IO. The effect is considerable, however, in that the characteristics of Ihe same process under proportional control are improved tenfold. For pi amplitude damping with proportional-plus-derivative control. 

Construct a vector diagram for the proportional-plus-derivative controller described by Eq. (1.26). Indicate the magnitude and phase angle of each vector. 

A certain process consists of a 1-min dead time and a 30-min lag. Estimate the period and settings for J-i-amplitude damping under proportional-plus-derivative control. Repeat for a proportional-plus-reset controller, assuming 45 phase lag in the controller. 

Use pulse width and amplitude modulation of a proportional plus derivative controller output to mimic lab titration for batch pH control. 

The time difference by which the output of a proportional-plus-derivative controller leads the controller input when the input changes linearly with time. 

---