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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