Study of the periodic time-varying nonlinear iterative learning control ECE 6330: Nonlinear and Adaptive Control FISP Hyo-Sung Ahn Dept of Electrical and Computer Engineering Utah State University

Backgrounds of Iterative Learning Control (ILC) Leading researchers in ILC: Leading research groups around world: Dr. Arimoto, Dr. Moore and Dr. Chen’s group, Dr. Rogers and Dr. Owen’s group, Dr. Longman, Dr. Xu, Dr. Bien, Dr. Amann, etc. Categories: Linear system ILC, Nonlinear system linear ILC, Nonlinear system nonlinear ILC, Super-vector ILC.

Nonlinear System Iterative Learning Control Major assumption: and are globally Lipschitz continuous Nonlinear system: Type 1 Learning controller (high order ILC)

Stability condition (asymptotically convergence): Learning controller (if only typical gains are used, r =0) Stability condition: Stability Condition and Controller

Nonlinear system (SISO): Type 2 Learning controller: Stability condition:

Nonlinear system (MIMO): Type 3 Learning controller: Stability condition:

Iterative Learning Controller Design KLM and YQC: High order ILC in time domain, High order ILC in iteration domain, PI, PD type in iteration domain optimal design, Feedback controller (2002, ASCC), and Super-vector ILC. Owens, et al.: Optimal algorithm (2003 IJC). Hatonen, et al.: Time-variant ILC control laws (2004 IJC). Amann et al.: Optimization method (?). Jian-Xin Xu: Nonlinear ILC and convergence speed, time varying periodic parameter. LQ method(?): James A. Frueh, IJC Longman: Frequency domain analysis

Observer Based Time Varying Iterative Learning Control: Problem Definition There is periodically time dependent parameter uncertainty States are not measured directly, so observer is needed Periodically time dependent parameter is adapted States are estimated Lyapunov analysis is indispensable

Consider following system [Jina-Xin Xu and Jing Xu, IEEE TAC, Vol. 49, No. 2, Feb, 2004]: where is unknown periodically time varying parameter, is known system dynamics (assumed as Lipschitz continuous), and z could be X and Y. In this report, we assume that z = X. Because is periodically time varying and A, B, C are known, we can apply iterative learning control. Also, it is assumed that state X are not directly measured. So, observer is used in this method. Systems

Observer [Jina-Xin Xu and Jing Xu, IEEE TAC, Vol. 49, No. 2, Feb, 2004]: L is design parameter

Controller [Jina-Xin Xu and Jing Xu, IEEE TAC, Vol. 49, No. 2, Feb, 2004]: and are time dependent positive diagonal matrices

Theorem The control law, the algebraic learning law, and the adaptation law ensure the convergence of the state estimation and the output tracking in norm. Proof: [Jina-Xin Xu and Jing Xu, IEEE TAC, Vol. 49, No. 2, Feb, 2004]

Example

Target Trajectory, Input Disturbance, Design Parameters So,

Signal tracking errors

Estimated state errors

True periodically time varying parameter and adaptive parameter

Conclusions Good Results Two new approaches in observer based time varying nonlinear ILC - Observer design: Parameters are estimated in adaptive way. - Periodically time varying parameter: ILC tries to minimize the reference tracking signal error. Possible future research works - Can we find new adaptation dynamics based on Lyapunov analysis on new system model? - Can we apply above theorem to super-vector ILC, which has parameter uncertainties? - What does thing happen, when there is a periodic uncertain parameter in measured output? - With non-periodic uncertain parameter, the stochastic ILC (Kalman filter)? - What’s the relationship?