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Newton method based iterative learning control for nonlinear systems
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Newton method based iterative learning control for nonlinear systems

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A novel iterative learning control algorithm for nonlinear systems

A novel iterative learning control algorithm for nonlinear systems


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  • 1. NEWTON-METHOD BASED ITERATIVE LEARNING CONTROL METHOD FOR NONLINEAR SYSTEMS T. Lin, D. H. Owens, J. Hätönen Department of Automatic Control and Systems Engineering University of Sheffield, UK
  • 2. Outline
    • Introduction of Iterative Learning Control (ILC)
    • Development
    • Problem Definition
    • ILC Strategy
    • Introduction of Newton-Method Based ILC
    • Previous Work
    • Merits
    • Idea
    • Interpretation of Newton-Method Based ILC
    • Algorithm
    • Convergence
    • Simulation
  • 3. Iterative Learning Control
    • Was originally proposed by Arimoto in 1984
    • Considers systems REPETITIVELY tracking a reference signal over a fixed interval
    • Improves control input and increases the accuracy trial by trial
    • Has been extensively researched with plenty of results
    • Has been applied to many industries
  • 4. ILC Problem
    • Consider the discrete-time dynamical system defined over finite time interval, t ∈[0,1,..., N ] :
    • OBJECTIVE: to find the ideal control input u d ( t ) for the system to track accurately the reference signal y d ( t ) over the same time interval.
    • SPECIAL feature of the system:
    • After t = N , x is reset back to x 0 and the system will follow y d ( t ) again.
  • 5. ILC Strategy
    • The repetitive nature of the problem leads to ILC strategy:
    • where
    • k is the number of trials.
    • A most common and simple ILC strategy is
  • 6. ILC for Nonlinear Systems
    • (Xu, 1997) investigates a class of discrete nonlinear systems with direct transmission from inputs to outputs;
    • (Avrachenkov, 1998) suggests ILC schemes based on Quasi-Newton Method for general nonlinear operators;
    • (Wang, 1998) investigates a class of discrete nonlinear systems with initial uncertainties and disturbances;
    • (Xu and Tan, 2002a) uses P-type ILC first to ensure global convergence then uses Newton-type ILC to speed up local convergence;
    • (Xu and Tan, 2002b) adopts robust ILC to resist system uncertainties and guarantee fast convergence at the same time.
  • 7. ILC Based on Newton-Method
    • MERITS of the Newton-method based ILC:
    • It decomposes a nonlinear ILC problem into a sequence of linear time-varying ILC problems
    • For the linear ILC problems derived, any suitable ILC algorithm can be applied
    • It needn’t calculate inverse systems
    • It converges semi-locally
  • 8. The Idea
    • Consider the nonlinear system as nonlinear equations
    • Apply Newton method for nonlinear equations to the ILC problem of the nonlinear system
    • Decompose the nonlinear ILC problem into linear ILC problems
    • Apply suitable linear ILC algorithms
  • 9. Nonlinear System --- Nonlinear Equations
    • Consider the discrete nonlinear system defined over finite time interval, t ∈[0,1,..., N ] :
    • Set
    • Since
    • Then
  • 10. Apply Newton Method
    • Apply Newton method to the nonlinear equation system
    • Note that is just the linearisation of the nonlinear system at u k , with z k +1 as the control input and e k as the desired output. Then z k +1 can be acquired by solving the linear ILC problem.
  • 11. Solve the Linear ILC
    • The linearisation of the nonlinear system, i.e., , has the form of
    • with the desired output e k , this time-varying ILC problem can be solved by Norm-Optimal ILC (Amann et al., 1996)
    • while l is the trial number of the linear ILC problem.
  • 12. Semi-Local Convergence
    • The Newton-method based ILC algorithm is equivalent to the Newton method for nonlinear equations, therefore it should inherit the Newton method’s convergence properties .
    • The Newton-Kantrovich Theorem and its proof (Ortega and Rheinboldt, 1970) can be applied to the Newton-method based ILC after slight modification.
  • 13. Simulation
    • Consider the single link direct joint driven manipulator model (Bien and Xu, 1998) on t ∈[0,1]
    • The reference signal is
    • T s =0.01 sec is the sampling period.
    • The initial control input is u 0 =0 . The terminal condition is E =sup [0,1] ( θ d - θ )<1 ˚. The Norm-Optimal ILC algorithm (Amann et al., 1996) is adopted for solving linear ILC problems.
  • 14. Simulation
    • The result is shown by the following figure
  • 15. Future Work
    • Global convergence
    • Stability and robustness
    • Monotonic convergence
    • Extension to continuous-time case
  • 16. References
    • Amann, N., D.H. Owens and E. Rogers (1996). Iterative learning control using optimal feedback and feedforward actions. International Journal of Control 65(2), 277–293.
    • Avrachenkov, K.E. (1998). Iterative learning control based on quasi-Newton methods. In: Proceedings of 37th IEEE Conference on Decision and Control . Tampa, FL, USA.
    • Bien, Z. and J. Xu (1998). Iterative learning control: analysis, design, integration and applications . Kluwer Academic Publishers.
    • Ortega, J.M. and W.C. Rheinboldt (1970). Iterative solution of a nonlinear equation in several variables . Academic Press.
    • Wang, D. (2002). Convergence properties of discrete-time nonlinear systems with iterative learning control. Automatica 34(11), 1445–1448.
    • Xu, J. (1997). Analysis of iterative learning control for a class of nonlinear discrete-time systems. Automatica 33(10), 1905–1907.
    • Xu, J. and Y. Tan (2002 a ). On the P-type and Newton-type ILC scheme for dynamic systems with non-affine-in-input factors. Automatica 38, 1237–1242.
    • Xu, J. and Y. Tan (2002 b ). Robust optimal design and convergence properties analysis of iterative learning control algorithms. Automatica 38, 1867–1880.