Newton method based iterative learning control for nonlinear systems


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A novel iterative learning control algorithm for nonlinear systems

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Newton method based iterative learning control for nonlinear systems

  1. 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. 2. Outline <ul><li>Introduction of Iterative Learning Control (ILC) </li></ul><ul><li>Development </li></ul><ul><li>Problem Definition </li></ul><ul><li>ILC Strategy </li></ul><ul><li>Introduction of Newton-Method Based ILC </li></ul><ul><li>Previous Work </li></ul><ul><li>Merits </li></ul><ul><li>Idea </li></ul><ul><li>Interpretation of Newton-Method Based ILC </li></ul><ul><li>Algorithm </li></ul><ul><li>Convergence </li></ul><ul><li>Simulation </li></ul>
  3. 3. Iterative Learning Control <ul><li>Was originally proposed by Arimoto in 1984 </li></ul><ul><li>Considers systems REPETITIVELY tracking a reference signal over a fixed interval </li></ul><ul><li>Improves control input and increases the accuracy trial by trial </li></ul><ul><li>Has been extensively researched with plenty of results </li></ul><ul><li>Has been applied to many industries </li></ul>
  4. 4. ILC Problem <ul><li>Consider the discrete-time dynamical system defined over finite time interval, t ∈[0,1,..., N ] : </li></ul><ul><li>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. </li></ul><ul><li>SPECIAL feature of the system: </li></ul><ul><li>After t = N , x is reset back to x 0 and the system will follow y d ( t ) again. </li></ul>
  5. 5. ILC Strategy <ul><li>The repetitive nature of the problem leads to ILC strategy: </li></ul><ul><li>where </li></ul><ul><li>k is the number of trials. </li></ul><ul><li>A most common and simple ILC strategy is </li></ul>
  6. 6. ILC for Nonlinear Systems <ul><li>(Xu, 1997) investigates a class of discrete nonlinear systems with direct transmission from inputs to outputs; </li></ul><ul><li>(Avrachenkov, 1998) suggests ILC schemes based on Quasi-Newton Method for general nonlinear operators; </li></ul><ul><li>(Wang, 1998) investigates a class of discrete nonlinear systems with initial uncertainties and disturbances; </li></ul><ul><li>(Xu and Tan, 2002a) uses P-type ILC first to ensure global convergence then uses Newton-type ILC to speed up local convergence; </li></ul><ul><li>(Xu and Tan, 2002b) adopts robust ILC to resist system uncertainties and guarantee fast convergence at the same time. </li></ul>
  7. 7. ILC Based on Newton-Method <ul><li>MERITS of the Newton-method based ILC: </li></ul><ul><li>It decomposes a nonlinear ILC problem into a sequence of linear time-varying ILC problems </li></ul><ul><li>For the linear ILC problems derived, any suitable ILC algorithm can be applied </li></ul><ul><li>It needn’t calculate inverse systems </li></ul><ul><li>It converges semi-locally </li></ul>
  8. 8. The Idea <ul><li>Consider the nonlinear system as nonlinear equations </li></ul><ul><li>Apply Newton method for nonlinear equations to the ILC problem of the nonlinear system </li></ul><ul><li>Decompose the nonlinear ILC problem into linear ILC problems </li></ul><ul><li>Apply suitable linear ILC algorithms </li></ul>
  9. 9. Nonlinear System --- Nonlinear Equations <ul><li>Consider the discrete nonlinear system defined over finite time interval, t ∈[0,1,..., N ] : </li></ul><ul><li>Set </li></ul><ul><li>Since </li></ul><ul><li>Then </li></ul>
  10. 10. Apply Newton Method <ul><li>Apply Newton method to the nonlinear equation system </li></ul><ul><li>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. </li></ul>
  11. 11. Solve the Linear ILC <ul><li>The linearisation of the nonlinear system, i.e., , has the form of </li></ul><ul><li>with the desired output e k , this time-varying ILC problem can be solved by Norm-Optimal ILC (Amann et al., 1996) </li></ul><ul><li>while l is the trial number of the linear ILC problem. </li></ul>
  12. 12. Semi-Local Convergence <ul><li>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 . </li></ul><ul><li>The Newton-Kantrovich Theorem and its proof (Ortega and Rheinboldt, 1970) can be applied to the Newton-method based ILC after slight modification. </li></ul>
  13. 13. Simulation <ul><li>Consider the single link direct joint driven manipulator model (Bien and Xu, 1998) on t ∈[0,1] </li></ul><ul><li>The reference signal is </li></ul><ul><li>T s =0.01 sec is the sampling period. </li></ul><ul><li>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. </li></ul>
  14. 14. Simulation <ul><li>The result is shown by the following figure </li></ul>
  15. 15. Future Work <ul><li>Global convergence </li></ul><ul><li>Stability and robustness </li></ul><ul><li>Monotonic convergence </li></ul><ul><li>Extension to continuous-time case </li></ul>
  16. 16. References <ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>Bien, Z. and J. Xu (1998). Iterative learning control: analysis, design, integration and applications . Kluwer Academic Publishers. </li></ul><ul><li>Ortega, J.M. and W.C. Rheinboldt (1970). Iterative solution of a nonlinear equation in several variables . Academic Press. </li></ul><ul><li>Wang, D. (2002). Convergence properties of discrete-time nonlinear systems with iterative learning control. Automatica 34(11), 1445–1448. </li></ul><ul><li>Xu, J. (1997). Analysis of iterative learning control for a class of nonlinear discrete-time systems. Automatica 33(10), 1905–1907. </li></ul><ul><li>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. </li></ul><ul><li>Xu, J. and Y. Tan (2002 b ). Robust optimal design and convergence properties analysis of iterative learning control algorithms. Automatica 38, 1867–1880. </li></ul>