1. Stabilization of TORA System: 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 TORA System
Adaptive Backstepping Sliding Mode Control
Hierarchical Sliding Mode Control
Control Law for TORA System
Simulation Results
Conclusions

3. A Few Words on TORA System
Degree of Freedom: 2
θ No of Control Input: 1
u
x
State Model of TORA System

q1 p1 q1=x

p1 k 3 q1
2
k 2 sin q 2 p1 k 2 cos q 2 u q2=θ

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 x, x2 
x, x3 , x4 
2
f1 X k 1 q1 k 2 sin q 2 p 2
b1 X k 2 cos q 2
f2 X 0
b2 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 s at s1 1 d
x 1 h1 s1 1
s at 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 s at s1 1 d
x 1 h1 s1 1
s at s1
1
u2 b2 X 1
e4 c 2 e3 2 2
f2 X d 2 M s at s 2 2 d
x 2 h2 s 2 2
s at 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 sat ( S ) K .S
 Coupling Law: u sw
1 1 2
b
1 1
X 2
b2 X
b X u1 b2 X u 2 sat ( S ) KS
 Composite Control Law: u
1 1 2
b
1 1
X 2
b2 X

8. Control Law for TORA System .
 Expression of Control Input for Translational Motion
k1 2

u1 1
e2 c1 e1 1 1
k 1 q1 1 d
x k 2 sin q 2 p 2
d1M 1 h1 s1 1
sat ( s1 )
cos q 2
 Expression of Control Input for Rotational Motion

u2 2
e4 c 2 e3 2 2
2 d
x d 2M 2 h2 s 2 2
sat ( s 2 )
 Coupling Control Law:
1
k 2 cos q 2 u 2 2
u1 sat ( S ) K .S
u sw
1
k 2 cos q 2 2
 Composite Control Law:
k cos q 2 u1
1 1 2
u2 sat ( S ) KS
u
1 1
k cos q 2 2

9. Simulation Results
 Initial Conditions: q1 0 .2 5 and p1 0
Cart Position Cart Velocity

10. Contd.
Phase Portrait of q1-p1

11. Contd.
 Initial Conditions: q2 pi / 3 and p2 0
Rotor Position Rotor 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