Page 138 - Mechatronics with Experiments

124 MECHATRONICS
This behavior can be explained by simply adding the effect of the integral control action
to the PD control behavior above. When the position of the inertia stops at a finite error,
integral control will eventually build the control signal to be larger than the friction and
will be able to start the motion again.

f control (t) = K e(t) + K ̇ e(t) + ∫ e( )d (2.206)
d
p
But, in the process of doing so, it will result in back and forth overshoot behavior, and
hence result in oscillations about the desired position.
The remedy is either to design and maintain the system to minimize the stiction
friction such that the resulting position error is acceptably small, or to predict it and
compensate for it in the control algorithm explicitly based on the friction prediction.
For a given stiction friction level, the positioning error cannot be guaranteed to be
zero by any PID type controller unless there is a way to eliminate the source of the
stiction friction. Friction prediction and compensation in real-time to achieve zero
positioning error is not practically realistic due to the random and highly varying nature of
friction physics.
The simulation results and physical explanations are shown in Figure 2.57.

PID controller
[u] [xd]
s Goto
[xd] Kd From 2
0.01s+1
Goto 2 Gain 1 Transfer Fon Mass-force with stiction friction [u] Scope
+ – Kp + + + + 1 1 From
– s s [ui]
Step Gain Integrator 1 Integrator From 1
1
Ki s
Gain 2 Integrator 2 [uI]
Coulomb &
Goto 1 Viscous friction = 50
Commanded and actual position Commanded and actual position
1 1.5
1
0.5
0.5
0 0
Control: force from PD controller Control: force from PID controller
1000 1000
500 500
0 0
–500 –500
Integral control contribution Integral control contribution
1 100
0 0
–1 –100
0 5 10 15 20 25 30 0 5 10 15 20 25 30
PD controller: Kp = 39.47, Kd = 8.79 Ki = 0.0 PID controller: Kp = 39.47, Kd = 8.79 Ki = 394.7
FIGURE 2.57: Open loop system which has stiction friction and a PID closed loop controller.
Due to the stiction friction, the integral action of the PID controller causes a cycle of oscillation,
as seen on the simulation results for the PID controller on the right. Without the integral
control, there will be a finite steady-state error, as seen on the simulation results for the PD
controller on the left.