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

Disturbance Estimation

byShubhobrata Rudra

Inspire Research FellowElectrical Engineering Department

Jadavpur UniversityKolkata

Content

A Few Words on Rotating Pendulum

Adaptive Backstepping Sliding Mode Control

Hierarchical Sliding Mode Control

Control Law for Rotating Pendulum

Simulation Results

Conclusions

Degree of Freedom: 2

φ

θ

A Few Words on Rotating Pendulum

State Model of Rotating Pendulum System

q1=θq2=φ

u

No of Control Input: 1

1 1

21 2 2 3 2 1 1 2

2 2

2

tan sin / cos

q p

p k q k q p k u q

q p

p u

Contd.

Standard State Model of Underactuated System

1 2

2 1 1 1

3 4

4 2 2 2

x x

x f X b X u d t

x x

x f X b X u d t

1 2 3 4, , ,x x x x

21 2 2 3 2 1

2

1 1 2

2

tan sin

0

sec

1

f X k q k q p

f X

g X k q

g X

Adaptive Backstepping Sliding Mode Control Define 1st Error variable & its dynamic as:

&

Stabilizing Function:

Control Lyapunov Function (CLF) and its derivative

Define 2nd error variable e2 and its derivative as:

and

Define first-layer sliding surface s1 and new CLF as

and

1 1 1de x x 1 2 1de x x

1 1 1 1 1c e

2 21 1 1

1 1

2 2V e

2 2 1 1de x x 2 1 1 1 1 1de f X b X u d t x

1 1 1 2s e e 22 1 1

1

2V V s

Contd. Derivative of CLF:

Control Input:

Augmented Lyapunov Function:

Adaptation Law:

22 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1dV e e c e s e c e f X b X u d t x

11 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1tanh tanhM du b X e c e f X d s x h s s

23 2 1 1 1 1

1

1 and

2 M M M MV V d d d d

1 1 1Md s

Hierarchical Sliding Mode Control Control Inputs:

Adaptation Laws: and

Composite control law:

Define 2nd Layer sliding surface:

Coupling Law:

Composite Control Law:

11 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1tanh tanhM du b X e c e f X d s x h s s

12 2 1 4 2 3 2 2 2 2 2 2 2 2 2 2 2tanh tanhM du b X e c e f X d s x h s s

1 1 1Md s

2 2 2Md s

1 1 2 2S s s

1 2 swu u u u

1 1 2 2 2 1

1 1 2 2

tanh( ) .sw

b X u b X u S K Su

b X b X

1 1 1 2 2 2

1 1 2 2

tanh( )b X u b X u S KSu

b X b X

Expression of Control Input for Translational Motion

Expression of Control Input for Rotational Motion

Coupling Control Law:

Composite Control Law:

11 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1tanhM du b X e c e f X d x h s s

12 2 2 4 2 3 2 2 2 2 3 2 2 2 2 2tanhM du b X e c e f X d x h s s

1 2 2 1

1 2

tanh( ) .sw

b X u b X u S K Su

b X b X

11 2

2

1

2

tanh( ) .cos

1cos

ku u S K S

qu

k

q

Control Law for Rotating Pendulum System .

Simulation Results

Initial Conditions: and

1 / 3q pi 1 0p

Shaft Position Shaft Velocity

Contd.Phase Portrait of q1-p1

Contd.

Initial Conditions: and

2 / 6q pi 2 0p

Pendulum Position Pendulum Velocity

Contd.Phase Portrait of q2-p2

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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Reference

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