A presentation about the control theory for controlling a cart balancing an inverted pendulum.

1. 1 of 12
By Carlos Asmat and David López Sansò
ECSE 493 - Control and Robotics Laboratory. McGill University.

2. Introduction 2 of 12
Intro · Lead Controller · Full State Feedback Controller · LQR · Conclusion
● Inverted pendulum controller
– Lead controller
– Pole placement
– LQR y
● State-space equations α x
mp
– Energies & Lagrangian lp
F
mc
ECSE 493 - Control and Robotics Laboratory. McGill University.

3. Introduction 3 of 12
Intro · Lead Controller · Full State Feedback Controller · LQR · Conclusion
State Space Representation
[ ][ ]
0 0 1 0 0
0 0 0 1 x
[] []
x
˙ 0
2
 −m p g −K 1 K1
˙
= 0 0   V
x
¨ mc R m mc x
˙ R m mc

¨  m p m c  g K 1
2 
˙ −K 1
0 0
mc l p R m mc l p R m mc l p
ECSE 493 - Control and Robotics Laboratory. McGill University.

4. Lead Controller 4 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
● Sole objective: to control the pendulum angle.
 −C 1
Plant: = 2
V s − o 2

ℑ 
V

ℜ 
− 0 0 V
ECSE 493 - Control and Robotics Laboratory. McGill University.

5. Lead Controller 5 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
● The lead controller can “pull” on the RHP pole.
s  a
D  s =K
s  b
Root Locus
ECSE 493 - Control and Robotics Laboratory. McGill University.

6. Full State Feedback 6 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
u =−K x
x = A −B K  x
˙
● Specify pole locations
● Obtain gain values
ECSE 493 - Control and Robotics Laboratory. McGill University.

7. Full-State Feedback 7 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
Results
Poles Comments
P1 P2 P3 P4 Angle Response Position Response Overall
-0.5+i -0.5-i -10 -12 - - Doesn't work, too slow
-3+i -3-i -30 -35 Underdamped Almost critically damped Doesn't work
-3+3i -3-3i -30 -35 Very underdamped - Doesn't work
-3 -3 -30 -30 - - On the verge of working
-3 -3 -25 -35 Has large overshoots Overdamped Prone to stop working
-2 -4 -25 -35 Slightly underdamped Slightly overdamped Good behaviour
-1 -2 -25 -35 Critically damped Critically damped Good behaviour
-2 -4 -40 -50 - - Doesn't work
-2 -5 -25 -35 - Very violent movements Doesn't work
-1 -4 -25 -35 Slightly underdamped Slightly overdamped Best Response
ECSE 493 - Control and Robotics Laboratory. McGill University.

8. Full-State Feedback 8 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
Gain Variation
K1 (position) K2 (angle) K3 (velocity) K4 (angle velocity)
-8 -30 -30 -120 -12 -50 -3 -14
Low freq. Starts too High freq.
Responds violently.
Slow Fast On the verge High Amp. Turned off
Comments movement movement of instability
well to
by safety
Low Amp.
disturbances Large Turned off
by safety Cart shakes
overshoots
ECSE 493 - Control and Robotics Laboratory. McGill University.

9. Full-State Feedback 9 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
Disturbance Reaction
Angle (rad)
ECSE 493 - Control and Robotics Laboratory. McGill University.

10. LQR 10 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
● A more efficient way of finding the coefficients.
∞
J =∫  x T Q x u T R u  dt
0
[ ]
Q 11 0 0 0
0 Q 22 0 0
0 0 Q 33 0
0 0 0 Q 44
ECSE 493 - Control and Robotics Laboratory. McGill University.

11. LQR 11 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
● Angle and position performance
ECSE 493 - Control and Robotics Laboratory. McGill University.

12. Conclusion 12 of 12
Intro · Lead Controller · Full-State Feedback Controller · LQR · Conclusion
● Worst: lead controller
● FSF tolerant to
– Disturbances
– Parameter variation (model & gains)
● LQR optimal for pole placement
ECSE 493 - Control and Robotics Laboratory. McGill University.