1. Robust Adaptive Integral Backstepping
Control and its Implementation on
Motion Control System
Presented By
Shubhobrata Rudra
Assistant Professor
Department of Electrical Engineering
Calcutta Institute of Engineering and Management

2. 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

3. State Model of the Motion Control System
 Differential Equations of the Motion Control System
dθ
=ω
dt
dω
J = Tq − TL
dt
 State model: state variables z1 = θ and z 2 = θ
z1 = z 2

J ( z 2 + h ) = Tq

TL
h=
J

4. State Model of the Motion Control System
Tq 1 ω θ
+
- J ∫ ∫
TL
State Model

5. Objective of the Control Design:
i) The primary objective of the motion control system
is 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.

6. Integral Backstepping Control Design.
 Equation of Control Input :
Tq = J ( z 2 + h)

 Definition of 1st error variable:
e1 = θ - θ ref ω is acting as a virtual
control input for the
 Modified Stabilizing Function
Stabilizing Function: first integrator
z refz ref c1 c1e1 + θref + λ 1 χ 1
= = e + θ ref
t
 Choice of 2nd error variable:
χ 1 = ∫ e1 ( t ) dt
e 2 = z ref - z 2
0
 Control Lyapunov Function:
1 2 1 2 1 2
V2 = λ1 χ 1 + e1 + e2
2 2 2

7. Contd.
 Derivative of the error variable e2:
de2 T
 + λ χ − q + h
= z ref − z 2 = c1e1 + θ ref
   1 1
dt J
(
= c1 ( − c1e1 + e2 − λ 1 χ 1 ) + θ ref )
 + λ e − q + h
1 1
T
J
 Derivative of Lyapunov Function:
 = λ χ χ + e e + e e = λ χ e + e ( − c e + e − λ χ ) + e  c ( − c e + e − λ χ ) + θ + λ e − q + h
( ) T
V2 1 1 1 1 1 2 2 1 1 1 1 1 1 2 1 1 2  1 1 1 2 1 1 ref 1 1 
 J 
2 2
( 

2
)  + h − q 
= − c1e1 − c2 e2 + e2  1 − c1 + λ 1 e1 + ( c1 + c2 ) e2 − c1λ 1 χ 1 + θ ref
T
J

Incomplete
Design
 
 Control Input:
ˆ 2
(( )
Tq = J 1 − c1 + λ 1 e1 + ( c1 + c2 ) e2 − c1λ 1 χ 1 + θref + h
 ˆ )

8. Adaptation Scheme
2 2 1 2 1 1 1 1
 Augmented Lyapunov Function:Va = λ1 χ 1 + e1 + e 2 + J2 + h2
2 2 2 2γ 1 J 2γ 2
 Derivative of the Lyapunov function:
ˆ
 = -c e 2 - c e 2 + J { e (( 1 − c 2 + λ )e − c λ χ + ( c + c )e + θ + h ) - 1 dJ } + h ( e - 1 dh )
Va  ˆ
1 1 2 2 2 1 1 1 1 1 1 1 2 2 ref 2
J γ 1 dt γ 2 dt
 Parameter Update Law:
ˆ
dJ 2  ˆ
= γ 1e2 (( 1 − c1 + λ1 )e1 − c1λ1 χ1 + ( c1 + c2 )e2 + θref + h )
dt
ˆ
dh
= γ 2e2
dt

9. 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 .

10. Contd.
 A continuous Switching function is use to implement the Robustification
Different type of switching
measures : techniques can be used to

J 1 2 (
ˆ = γ e ( 1 − c 2 + λ )e − c λ χ + ( c + c )Adaptiveˆ − γ σ J
1 1 1
prevent the abnormal
1 1 1 Robust e2 + θref + h
1

2 the rate of
variation of 1 Js
ˆ )

ˆ ˆ
h = γ 2 e 2 − γ 2σ sh h
Control!!!!!
adaptation
where
 0 ˆ
if J < J 0  ˆ
 0 if h < h0

  J −J  ˆ   h−h ˆ
  0
ˆ   0
σ Js = σ J 0  if J 0 ≤ J ≤ 2J 0 σ hs = σ h0  ˆ
if h 0 ≤ h ≤ 2h0
   J  ˆ    h  ˆ 
    
 σ J0 ˆ
if J ≥ 2J 0  σ h0 ˆ
if h ≥ 2h0
 

11. Simulation Results and Discussion
Reference Trajectory and Response of the System
t  t 
Reference Signal θ ref = 10 sin( pi ) sin pi 
2  50 

12. Simulation Results and Discussion
Estimation of Inertia Variation
Robust Adaptive Integral Backstepping Adaptive Integral Backstepping
Control Scheme Control Scheme

13. Simulation Results and Discussion
Estimation of Load Torque Variation
Robust Adaptive Integral Backstepping Adaptive Integral Backstepping
Control Scheme Control Scheme

14. 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.

15. Questions
Polygonia interrogationis known as Question Mark

16. 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.

17. 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.

20. 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