and ramps. We definitely want to control the transient response and steady-state response of the ball motion. The
transient response is the ball motion that occurs before steady-state is reached. The controller gains must be adjusted
until the transient response is satisfactory—this is the design process. Note that a stable system can have a non-zero
steady state error—this is different from marginal stability. With the PID controller, the ball will return to the setpoint
with zero steady-state error after an impulsive external disturbance.
Design of the PID controller
The steady-state error of the closed-loop system is determined by the structure of the controller. The integral term of
the PID controller increases the system type of the loop transfer function, represented by L(s)=G(s)Gc(s)H(s), by one
which leads to improved steady-state error response. In fact, the PID controller is a particularly important structure
in practice. The transient performance of the closed-loop system is determined by the PID controller gains. The PID
controller has three gains: the proportional gain, denoted by Kp, the derivative gain, denoted by KD, and the integral
gain, denoted by KI. The effect of each of these gains on the transient response to an external disturbance can be
described conceptually as follows:
Increasing Kp leads to an increase in the percent overshoot, has minimal impact on the settling time, and reduces the
steady-state errors;
Increasing KI leads to an increase in the percent overshoot and the settling time, but reduces the steady-state errors,
and
Increasing KD leads to a decrease in the percent overshoot and the settling time.
Note: Control concepts described in detail in Modern Control Systems, by R. C. Dorf & R. H. Bishop, 13th Ed., Pearson
Education, Inc., 2017.
Figure 3 Closed-loop feedback control to a set-point.
Y(s)
Plant
Controller
G
c
(s)
G(s)
Actuator
Balancing Arm
T
d
(s)
R(s)=0
Sensor
H(s)
-
N(s
E
a
(s)
Distance from
set-point
Balancing Arm Assembly 121
Balancing Arm Assembly 121
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