ME 4710 Motion and Control
Example: Phase Lead Control of a Weighted Sled
o Consider position control of a weighted sled system using a continuous phase-lead
compensator
Phase Lead
X d ( s) 

 155  s  26 
51


 26  s  155 
Valve
16641
s  178.7 s  15773
2
Weighted Sled
13.227
s ( s  20.7)
X (s)
Closed-Loop Position Control of a Weighted Sled System
o In the Simulink model below, we model closed loop control of the weighted sled using
both continuous and digital control. The embedded MATLAB function allows us to
implement the compensator directly as a difference equation.
o The coefficients of the difference equation depend on the sampling time T . The
difference equation used here is the same equation that would be coded into the digital
controlling software. (e.g. LabVIEW code)
o The effects of saturation and dead-band are included.
Kamman – ME 4710: page 1/3
o To run the Simulink file, values for the gain (Gain), desired position (DesiredPosition),
sample time (T), and the maximum time (tmax) for the simulation must be entered in
the MATLAB workspace.
o The file “DigitalControlDemoMaster.m” can be used to plot command and position
responses for continuous control and digital control (with different sample times).
o The difference equation coded in the MATLAB embedded function was developed
from the continuous compensator using Tustin’s approximation. Using   155/ 26
 2  z  1


26


T
 z  1 
U (s)
 s  26 
Gc ( s ) 
 
 

E (s)
 s  155 
 2  z  1   155 
 T  z  1 

After some algebraic manipulation, this equation can be reduced to
 26T  2 
 26T  2  1
155T  2  1
U ( z)   
E
(
z
)


z
E
(
z
)

155T  2 
155T  2  z U ( z )
155T  2 
This is equivalent to the difference equation
 26T  2 
 26T  2 
155T  2 
u (k )   
e
(
k
)


e
(
k

1)

155T  2 
155T  2  u (k  1)
155T  2 
o Here, e(k ) and u (k ) represent the signal values at the present time, and e(k  1) and
u (k  1) represent the signal values at the previous time step.
o The figures below show results for the compensator command and the sled position
response for continuous and series of discrete compensators with
Variable
Value
Gain
51
Desired Position
1 (in)
Sample Times (0.001, 0.005, 0.01, 0.03) (sec)
o The Bode diagram of the closed loop system is also shown, indicating a closed loop
bandwidth of approximately 60 (rad/s)  9.55 (Hz) . For good results, we should
sample at approximately f  20(9.55)  191 (Hz) . This corresponds to a sample time of
T  0.005 (sec) .
Kamman – ME 4710: page 2/3
Compensator Commands
Sled Position Response
Closed Loop
Frequency Response
Kamman – ME 4710: page 3/3