The document presents a novel control approach for stabilizing the underactuated Furuta pendulum using adaptive backstepping and hierarchical sliding mode control techniques. It details the control laws, simulation results, and introduces methods to reduce chattering and extend applicability to other systems. The findings suggest an effective solution for two-degree of freedom underactuated mechanical systems, proposing further research for broader application.

1. Stabilization of Furuta Pendulum: A Backstepping
Based Hierarchical Sliding Mode Approach with
Disturbance Estimation
by
Shubhobrata Rudra
Inspire Research Fellow
Electrical Engineering Department
Jadavpur University
Kolkata

2. Content
A Few Words on Rotating Pendulum
Adaptive Backstepping Sliding Mode Control
Hierarchical Sliding Mode Control
Control Law for Rotating Pendulum
Simulation Results
Conclusions

3. A Few Words on Rotating Pendulum
φ
u
Degree of Freedom: 2
θ No of Control Input: 1
State Model of Rotating Pendulum System

q1 p1
q1=θ
2

p1 k 2 tan q 2 k 3 sin q 2 p 1
k 1 u / cos q 2 q2=φ

q2 p2

p2 u

4. Contd.
 Standard State Model of Underactuated System

x1 x2

x2 f1 X b1 X u d1 t

x3 x4

x4 f2 X b2 X u d2 t
x1 , x2 , x , x4 
3
2
f1 X k 2 tan q 2 k 3 sin q 2 p1
f2 X 0
g1 X k 1 sec q 2
g2 X 1

5. Adaptive Backstepping Sliding Mode Control
 Define 1st Error variable & its dynamic as:
e1 x1 x1d & 
e1 x 2 
x1d
 Stabilizing Function:
1
c1 e1 1 1
 Control Lyapunov Function (CLF) and its derivative
1 2 1 2
V1 1
e1
2 2
 Define 2nd error variable e2 and its derivative as:
e2 x2 
x1 d and 
e2 f1 X b1 X u d1 t 1 d
x 1
1
 Define first-layer sliding surface s1 and new CLF as
1
s1 e
1 1
e2 and V2 V1
2
s1
2

6. Contd.
 Derivative of CLF:
V2
2
e1 e 2 c1 e1 s1 1
e2 c1 e1 1 1
f1 X b1 X u d1 t 1 d
x 1
 Control Input:
1
u1 b1 X 1
e2 c1 e1 1 1
f1 X d 1 M tanh s1 1 d
x 1 h1 s1 1
tanh s1
 Augmented Lyapunov Function:
1 2

V3 V2 d 1M
and d 1 M d1M d 1M
2 1
 Adaptation Law:


d1M s
1 1

7. Hierarchical Sliding Mode Control
 Control Inputs:
1
u1 b1 X 1
e2 c1 e1 1 1
f1 X d 1 M tanh s1 1 d
x 1 h1 s1 1
tanh s1
1
u2 b2 X 1
e4 c 2 e3 2 2
f2 X d 2 M tanh s 2 2 d
x 2 h2 s 2 2
tanh s 2

 
 Adaptation Laws: d1M s
1 1 and d 2M 2
s2
 Composite control law: u u1 u2 u sw
 Define 2nd Layer sliding surface: S s
1 1 2
s2
b X u2 b2 X u1 tanh( S ) K .S
 Coupling Law: u sw
1 1 2
b
1 1
X 2
b2 X
b X u1 b2 X u 2 tanh( S ) KS
 Composite Control Law: u
1 1 2
b
1 1
X 2
b2 X

8. Control Law for Rotating Pendulum System .
 Expression of Control Input for Translational Motion
1

u1 b1 X 1
e2 c1 e1 1 1
f1 X d 1M 1 d
x 1 h1 s1 1
tanh s1
 Expression of Control Input for Rotational Motion
1

u2 b2 X 2
e4 c 2 e3 2 2
f2 X d 2M 3 d
x 2 h2 s 2 2
tanh s 2
 Coupling Control Law:
b1 X u 2 b2 X u1 tanh( S ) K .S
u sw
b1 X b2 X
 Composite Control Law: k1
u1 u2 tanh( S ) K .S
cos q 2
u
k1
1
cos q 2

9. Simulation Results
 Initial Conditions: q1 pi / 3 and p1 0
Shaft Position Shaft Velocity

10. Contd.
Phase Portrait of q1-p1

11. Contd.
 Initial Conditions: q2 pi / 6 and p2 0
Pendulum Position Pendulum Velocity

12. Contd.
Phase Portrait of q2-p2

13. Conclusions
 Another new method of addressing the stabilization problem for
underactuated system.
 Can easily be extended to address the stabilization problem of other two
degree of freedom underactuated mechanical systems.
 Chattering problem can be reduced with the introduction of second
order sliding mode control.
 Proposed algorithm is applicable for only two-degree of freedom single
input systems, research can be pursued to make the control algorithm
more generalized such that it will able to address the control problem of
any arbitrary underactuated system.

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15. Thank You