Newton method based iterative learning control for nonlinear systems
7 likes1,327 views
The document proposes a Newton-method based iterative learning control (ILC) method for nonlinear systems. It decomposes the nonlinear ILC problem into a sequence of linear time-varying ILC problems that can be solved using existing linear ILC algorithms. This approach converges semi-locally and avoids calculating inverse systems. The algorithm is demonstrated in a simulation of a single link manipulator model that successfully tracks the reference signal across trials. Future work aims to improve global convergence, stability, and monotonic convergence properties.
![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.](https://image.slidesharecdn.com/nolcos2004101-1303170365835-phpapp01/85/Newton-method-based-iterative-learning-control-for-nonlinear-systems-4-320.jpg)
![Nonlinear System --- Nonlinear Equations Consider the discrete nonlinear system defined over finite time interval, t ∈[0,1,..., N ] : Set Since Then](https://image.slidesharecdn.com/nolcos2004101-1303170365835-phpapp01/85/Newton-method-based-iterative-learning-control-for-nonlinear-systems-9-320.jpg)
![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.](https://image.slidesharecdn.com/nolcos2004101-1303170365835-phpapp01/85/Newton-method-based-iterative-learning-control-for-nonlinear-systems-13-320.jpg)
Newton method based iterative learning control for nonlinear systems
- 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.