Page 137 - Mechatronics with Experiments

CLOSED LOOP CONTROL 123
the time delay is known in advance accurately. Digital implementation of the time delay
logic in the control loop (Figure 2.56c) requires N = t ∕T sample number of sampling time
d
d
delays in the processed signal, where T sample is the sampling period of the digital controller,
that is

e −t d s ≈ z −N d (2.199)

that is, t = 2.0, T sample = 0.1, then N = 20. In order to implement this time delay in a
d
d
digital Otto-Smith regular controller, we would have to keep track of the past 20 samples
of the control signals.

Example: PID Control of a Motion System with Friction Many mechan-
ical motion systems have friction opposing the motion. When two surfaces move against
each other, the resistance force (or torque) before the relative motion starts is larger than the
resistance force once the motion has started. This friction resists motion. For the relative
motion to start, the applied external force must be larger than the friction force. Further-
more, that initial friction force, so called “stiction” (Coulomb) friction, is not constant
and varies for the same system due to the condition of the surfaces and lubrication levels.
The dynamic model of such a basic motion system is a mass-force system with friction as
follows,

m ̈ x(t) = f control (t) − f friction (t) (2.200)

where f friction (t) represents the friction in the system. If the mass is at rest, the only way
for the motion to start is for the control force to be larger than the friction force. Once the
motion starts, the net force (difference between control force and friction force) determines
the acceleration and deceleration of the mass. Again, once it stops, the motion can start
again only if the control force is larger than the friction force.
Without the integral control action, that is the PD controller, a closed loop position
control system which has stiction friction will result in a finite steady-state positioning
error. The reason is that when the control force, determined by the PD controller, is less
than the friction force, the motion of the mass will start to decelerate and come to zero
velocity. At that point, if the actual position is a finite value (the likelihood of the actual
position being exactly the desired position is a random possibility) the control signal from
the PD controller will be only due to the proportional controller since the mass speed is
zero, and that control signal value would be smaller than friction force. As a result, the
mass cannot move and will be stuck at that position with the finite steady-state position
error. The following equations describe this condition,

f control (t) = K e(t) + K ̇ e(t) (2.201)
d
p
f control (t) < f friction (t) ⟶ ̈ x(t) < 0.0 (2.202)
⟶ ̇ x(t) = 0.0 eventually (2.203)
f control (t) = K e(t) when ̇ x(t) = 0.0 (2.204)
p
f control (t) < f friction (t) ⟶ ̈ x(t) = 0.0 and ̇ x(t) = 0.0 no motion. (2.205)

With integral control action, that is a PID controller, a closed loop position system will
tend to oscillate about the target position which is referred to as limit cycle oscillations.This
is a fundamental condition that is common in many closed loop motion control systems.