1. Design of Nonlinear State Feedback Control Law for
Rotating Pendulum System: A Block Backstepping
Approach
by
Shubhobrata Rudra
Inspire Research Fellow
Electrical Engineering Department
Jadavpur University
Kolkata

2. Content
A Few Words on Rotating Pendulum
Block Backstepping 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 x 2

x2 f1 X g1 X u

x3 x4

x4 f2 X g2 X u
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. Block Backstepping Control
 Define new Control Variable z1 :
z1 q2 k q1 g 2 p1 g1 p 2
 Dynamics of z1:

z1 p2 k p1 g 2 f1 p1 dg 2 .P g1 f 2 p 2 dg 1 .P
 Stabilizing function
1
c1 z1 k p1 g 2 f1 p1 dg 2 .P g1 f 2 p 2 dg 1 .P
 Second error variable:
z2 p2 1

z1 z2 c1 z1
 Dynamics of z2:

z2 
p2 1
u c1 z 2 c1 z 1

6. Contd.
f1 2
f1 g2 g2
g2 k g1 g1 g 2 g 2
g 1 dg 2 . p p1 g1 p1 g2
p1 p2 q1 q2
2
f2 f2 g1 g1
g1 g1 g 2 g 2 dg 1 . p p2 g1 p2 g2
p1 p1 q1 q2
and
g2 g2 f1 f1 f1 f1
f2 k f1 2 f 1 dg 2 . p p1 f1 f2 g2 p1 p2 f1 f2
q1 q2 q1 q2 p1 p2
2 2 2
g2 2
g2 g2 2
g1 g1
p1 2
p 1
2 p1 p 2 2
p2 2 f 2 dg 1 . p p2 f1 f2
q 1
q1 q 2 q 2
q1 q2
2 2 2
f2 f2 f2 f2 g2 2
g2 g2 2
g1 p1 p2 f1 f2 p2 2
p 1
2 p1 p 2 2
p2
q1 q2 p1 p2 q1 q1 q 2 q2
Desired dynamics of z2 : 
z2 z1 c2 z 2
Required Control Input: u
1
1 c1
2
z1 c1 c2 z 2

7. Contd.
 Remark 1: The similar type of control algorithm can be obtained by defining
a new set of state variable as follows .
z1 q1 k q2 g1 p 2 g 2 p1
 Remark 2: The control law relies on the fact that ψ is invertible.
 Remark 3: The proposed control law asymptotically semi globally stabilizes
the equilibrium of the fourth order system.
 Remark 4: The proposed control law is more flexible than its predecessor
backstepping based control laws. The only requirement of the control law is
that at-least one entry of g-vector field is nonzero at configuration space,
which is generally satisfied by the common underactuated system used for
realistic applications.

8. Control Law for Rotating Pendulum System .
 Definition of z1:
z1 q2 k q1 p1 k1 sec q 2 p 2
 Dynamics of z1:
2 2

z1 p2 k p1 k 2 tan q 2 k 3 sin q 2 p1 k 1 sec q 2 tan q 2 p 2
 Stabilizing Function:
2 2
1
c1 z 1 k p1 k 2 tan q 2 k 3 sin q 2 p1 k 1 sec q 2 tan q 2 p 2
 Definition of z2 : z p2
2 1
 Control Input Required to realize the desired dynamics for z2:
1 2
u 1 c1 z1 c1 c2 z 2
where
1
1 1 k k 1 sec q 2 2 k 1 k 3 tan q 2 p1 2 k 1 p 2 tan q 2 sec q 2
2 2 2
k k 2 tan q 2 k 3 sin q 2 p1 k 2 sec q 2 p 2 k 3 cos q 2 p 2 p1
2 2 3 3
2 k 3 sin q 2 p1 k 2 tan q 2 k 3 sin q 2 p1 k 1 sec q 2 tan q 2 sec q 2 p 2

9. Simulation Results
 Initial Conditions: q1 pi / 6 and p1 0
Shaft Position Shaft Velocity

10. Contd.
Phase Portrait of q1-p1

11. Contd.
 Initial Conditions: q2 pi / 4 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.
 The proposed algorithm can be generalized for n-degree of freedom
system.
 The proposed algorithm is not adaptive. So research can be pursued in
future to develop an Adaptive Block Backstepping Control for
underactuated system.

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15. Thank You