Robust adaptive integral backstepping control and its implementation on
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The document presents a robust adaptive integral backstepping control scheme for motion control systems. It describes the state model of the system and control objective of tracking a reference signal. An integral backstepping control design is developed using error variables and a control Lyapunov function. An adaptation scheme estimates parameter variations using a parameter update law. To improve robustness, a continuous switching function is used to prevent abnormal variation of the adaptation rates. Simulation results show the proposed robust adaptive controller has better tracking performance and reduces parameter estimation error compared to a standard adaptive controller.
Robust adaptive integral backstepping control and its implementation on
- 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 + θref + 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 + θref + 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