Stabilization of Furuta Pendulum: A Backstepping Based Hierarchical Sliding Mode Approach with Disturbance Estimation
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:
V2
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
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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