Study of the periodic time-varying nonlinear iterative learning control

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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

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Backgrounds of Iterative Learning Control (ILC)

• ? Leading researchers in ILC Leading research
  groups around world Dr. Arimoto, Dr. Moore and
  Dr. Chens group, Dr. Rogers and Dr. Owens
  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.

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Nonlinear System Iterative Learning Control

• ? Major assumption
• and are globally
  Lipschitz continuous

? Nonlinear system Type 1
? Learning controller (high order ILC)
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Stability Condition and Controller
? Stability condition (asymptotically
convergence)
? Learning controller (if only typical gains are
used, r 0) Stability condition
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Nonlinear system (SISO) Type 2
? Learning controller
? Stability condition
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Nonlinear system (MIMO) Type 3
? Learning controller
? Stability condition
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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 2000.
• Longman Frequency domain analysis

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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

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Systems
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.
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Observer Jina-Xin Xu and Jing Xu, IEEE TAC, Vol.
49, No. 2, Feb, 2004
L is design parameter
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Controller Jina-Xin Xu and Jing Xu, IEEE TAC,
Vol. 49, No. 2, Feb, 2004
and are time dependent positive diagonal
matrices
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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

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Example
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Target Trajectory, Input Disturbance, Design
Parameters
So,
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Signal tracking errors
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Estimated state errors
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True periodically time varying parameter and
adaptive parameter
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True periodically time varying parameter and
adaptive parameter
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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)?
• - Whats the relationship?