AN1213/D
MOTOROLA
3
TRANSFER FUNCTION TERMS
Each of the transfer function terms affects system performance.
The P Term
The
Proportional
P term amplifies the error signal by a constant amount. However, P is not in series, but in parallel with I and
D, which implies that P cannot be used to scale the transfer function amplitude at all frequencies. Instead,
the P term interacts with the I and D terms to determine the placement of the zeros in the controller open-
loop transfer function.
Figure 2
shows a root locus solution to the numerator of Equation 1 as P is varied
with respect to I and D.
term is the most subtle and perhaps most misunderstood term in the PID algorithm. The
Figure 2 Effect of Varying P on Zeros
The I Term
The
Integral
integrated error mixed to the output of the filter, any I value other than zero implies that
can be tolerated by the servo loop. In other words, given sufficient time, a PID control loop will eventually
servo the output to the exact value of the commanded input.
term gives the servo loop that inflexible, stubborn feel. Since the I term adjusts the amount of
no
steady state error
In the frequency domain, the I term also affects placement of the zeros, as shown in
one of the zeros is at s = –P/D, and the other zero is at s = 0, which means that it will cancel the integrator
pole at s = 0. This makes sense intuitively since the integrator is turned off if I equals zero. As I increases,
the servo loop becomes “snappier”, i.e., it responds more quickly to steady state error.
Figure 3
. For I = 0,
It appears that adding an integrator to the servo loop would be a panacea for motion control headaches.
However, while adding an integrator does address steady state error, it can also have a negative impact on
system dynamics. The effect is most easily seen in the time domain. Consider a linear PID system that per-
forms servo control. Initially, the controlled motor is at rest, with zero position error. Torque is applied to the
motor shaft, changing its position and holding it in the new position. The control system senses a steady-
state error and tries to return the shaft to the commanded position. Since the example system is linear, con-
trol voltage continues to increase as a result of integrated error. While the increasing control voltage could
cause the motor to overheat, this is not the only detrimental effect. If the applied torque is suddenly removed
while integrator output is large, the motor shaft will spin past the desired shaft position while control voltage
is “dumped”. Eventually, a zero steady-state condition is achieved, but in an underdamped (and potentially
unacceptable) manner. Because this situation is similar to winding up a spring and then letting it go, the term
“wind-up” is used to describe it. Techniques to mitigate wind-up are discussed later in this note.
DELTA P ON AN1213
P = 0
P =
∞
P = 0
(I
÷
D)
P =
∞
P =
(4DI)
– j
(I
÷
D)
j
(I
÷
D)
S PLANE