Proportional-Integral-Derivative (Pid) Control; Enhanced Proportional-Integral-Derivative (Epid) Control - Honeywell AUTOMATIC CONTROL Engineering Manual

PROPORTIONAL-INTEGRAL-DERIVATIVE (PID)
CONTROL
Proportional-integral-derivative (PID) control adds the
derivative function to PI control. The derivative function
opposes any change and is proportional to the rate of change.
The more quickly the control point changes, the more corrective
action the derivative function provides.
If the control point moves away from the setpoint, the
derivative function outputs a corrective action to bring the
control point back more quickly than through integral action
alone. If the control point moves toward the setpoint, the
derivative function reduces the corrective action to slow down
the approach to setpoint, which reduces the possibility of
overshoot.
The rate time setting determines the effect of derivative
action. The proper setting depends on the time constants of
the system being controlled.
The derivative portion of PID control is expressed in the
following formula. Note that only a change in the magnitude
of the deviation can affect the output signal.
dE
V = KT
D dt
Where:
V = output signal
K = proportionality constant (gain)
T = rate time (time interval by which the
D
derivative advances the effect of
proportional action)
KT = rate gain constant
D
dE/dt = derivative of the deviation with respect to
time (error signal rate of change)
The complete mathematical expression for PID control
becomes:
V = KE +
K
T
1
Proportional Integral Derivative
Where:
V = output signal
K = proportionality constant (gain)
E = deviation (control point - setpoint)
T = reset time
1
K/T = reset gain
1
dt = differential of time (increment in time)
T = rate time (time interval by which the
D
derivative advances the effect of
proportional action)
KT = rate gain constant
D
dE/dt = derivative of the deviation with respect to
time (error signal rate of change)
M = value of the output when the deviation
is zero
∫Edt + KT + M
D dE
dt
The graphs in Figures 38, 39, and 40 show the effects of all
three modes on the controlled variable at system start-up. With
proportional control (Fig. 38), the output is a function of the
deviation of the controlled variable from the setpoint. As the
control point stabilizes, offset occurs. With the addition of
integral control (Fig. 39), the control point returns to setpoint
over a period of time with some degree of overshoot. The
significant difference is the elimination of offset after the
system has stabilized. Figure 40 shows that adding the
derivative element reduces overshoot and decreases response
time.
SETPOINT
T1
Fig. 38. Proportional Control.
SETPOINT
T1 T2
Fig. 39. Proportional-Integral Control.
SETPOINT
T1
Fig. 40. Proportional-Integral-Derivative Control.
ENHANCED PROPORTIONAL-INTEGRAL-
DERIVATIVE (EPID) CONTROL
The startup overshoot, or undershoot in some applications,
noted in Figures 38, 39, and 40 is attributable to the very
large error often present at system startup. Microprocessor-
based PID startup performance may be greatly enhanced by
exterior error management appendages available with

enhanced proportional-integral-derivative (EPID) control.

Two basic EPID functions are start value and error ramp time.
25 ENGINEERING MANUAL OF AUTOMATIC CONTROL
CONTROL FUNDAMENTALS
CONTROL
OFFSET
POINT
T2 T3 T4 T5 T6
TIME
C2099
CONTROL
OFFSET
POINT
T3 T4 T5 T6
TIME
C2100
OFFSET
T2 T3 T4 T5 T6
TIME
C2501