Presented ByShubhobrata RudraAssistant Professor

Department of Electrical EngineeringCalcutta Institute of Engineering and Management

Robust Adaptive Integral Backstepping Control and its Implementation on

Motion Control System

Content

State Model of the Motion Control System

Control Objective

Integral Backstepping Control Design

Adaptation Scheme

Robustification of the Adaptive Design

Simulation Results and Discussion

Conclusion

State Model of the Motion Control System

Differential Equations of the Motion Control System

State model: state variables and

dt

d

Lq TTdt

dJ

1z 2z

21 zz

qThzJ 2

J

Th L

∫J

1∫+-

Tq

TL

ω θ

State Model

State Model of the Motion Control System

Objective of the Control Design:

i) The primary objective of the motion control systemis to track a continuous bounded reference signal θref.

ii) Design a suitable parameter adaptation algorithm to estimate the variation of Inertia and Load Toque.

iii)Explore the concept of continuous switching function to design a robust adaptive law for parameter adaptation.

Integral Backstepping Control Design.

Equation of Control Input :

Definition of 1st error variable:

Stabilizing Function:

Choice of 2nd error variable:

Control Lyapunov Function:

hzJTq 2

refe -1

refref ecz 11

22 - zze ref

22

21

2112 2

1

2

1

2

1eeV

ω is acting as a virtual control input for the

first integrator Modified Stabilizing Function

1111 refref ecz

t

dtte0

11

Derivative of the error variable e2:

Derivative of Lyapunov Function:

Control Input:

J

Thcecceceecec

hJ

TeeecceeeceeeeeeV

qref

qref

11122111212

222

211

11112111211211111122111112

1

Contd.

hceccecJT refq 11122111

211

Incomplete Design

hJ

Teeecc

hJ

Teczz

dt

de

qref

qrefref

11112111

111122

hceccecJT refq 11122111

211

Augmented Lyapunov Function:

Derivative of the Lyapunov function:

Parameter Update Law:

Adaptation Scheme

)dt

dhe(h}

dt

Jd)he)cc(ce)c((e{

J

JececVa

22

122111111

212

222

211

111

- --- ref

)he)cc(ce)c((edt

Jd ref

221111112121 1

22

ˆe

dt

hd

2

2

2

1

22

21

211 2

1

2

1

2

1

2

1

2

1hJ

JeeVa

Robust Adaptive Backstepping

Difficulties for the designer of Adaptive Control

Mathematical Models are not free from Un modeled Dynamics

Parameter Drift may occur in the time of real world implementation

Noises are unavoidable in real time application.

Bounded disturbances may cause a high rate of adaptation which leads to an unstable/undesirable system performance.

Contd.

Robust Adaptive Control!!!!!

Different type of switching techniques can be used to

prevent the abnormal variation of the rate of

adaptation

A continuous Switching function is use to implement the Robustification measures :

where

0J0

00

0

0

0

2J if

2J if ˆ

ˆ

J if 0

J

JJJ

JJ

J

JJs

Jheccce)c(eJ Jsref 1221111112121 1

heh shˆˆ

222

0h0

00

0

0

0

2h if

2ˆh if ˆ

ˆ

h if 0

h

hhh

hh

h

hhs

Simulation Results and Discussion

Reference Trajectory and Response of the System

50sin)

2sin(10

tpi

tpirefReference Signal

Simulation Results and Discussion

Estimation of Inertia Variation

Robust Adaptive Integral Backstepping Control Scheme

Adaptive Integral Backstepping Control Scheme

Simulation Results and Discussion

Estimation of Load Torque Variation

Robust Adaptive Integral Backstepping Control Scheme

Adaptive Integral Backstepping Control Scheme

Conclusion

The system response always closely follows the given reference signal, while the maximum tracking error is less than 0.1rad.

Robust Adaptive Controller reduce the parameter estimation error.

This robust adaptive controller offers a smart estimation of the parameters variation. Sudden variations in parameter is not able to affect the estimation of the other parameter.

Questions

Polygonia interrogationis known as Question Mark

References

Y.Tan, J. Hu, J.Chang, H. Tan,”Adaptive Integral Backstepping Motion Control and Experiment Implementation”, IEEE Conference on Industry Applications, pp 1081-1088, vol-2, 2000.

M. Krstic, I. Kanellakopoulos, and P.V. Kokotovic, Nonlinear and Adaptive Control Design, New York : Wiley Interscience, 1995.

J. Jhou and C. Wen, Adaptive Backstepping Control of Uncertain System, Springer-verlag, Berlin, Heidelbarg 2008.

•

References

H.Tan and J. Chang, “Adaptive Position Control of Induction Motor Systems under Mechanical Uncertainties”, Proceedings of the IEEE 1999 International Conference on Power Electronics and Drive Systems, pp 597-602 Hong Kong, July 1999.

Y.Tan, J.Chang and H.Tan,” Adaptive backstepping control and friction compensation for AC servo with inertia and load uncertainties,” IEEE Transaction on Industrial Electronics, vol-50, pp944-952, 2003.

Ioannou PA and Sun J, Robust Adaptive Control. Prentice Hall, Englewood Cliff, 1996.

List of Design Parameters

Name of the Parameters Parameters Value

C1 4

C2 4

Λ1 1.25

γ1 6

γ2 6

J0 0.9

h0 0.25