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Multiple Estimation Models for Discrete-time Adaptive Iterative Learning Control

Ram Padmanabhan, Rajini Makam, Koshy George

TL;DR

The paper tackles discrete-time Adaptive ILC by introducing a multiple estimation model framework (MMST) that updates parameters using the identification error rather than tracking error. It derives a single-model solution with a projection-based, identification-error-driven update and proves convergence using composite energy functions and the properties of $\ell_2$ sequences, avoiding the Key Technical Lemma. The MMST extension constructs multiple candidate models and offers two switching schemes (end-of-iteration and per-sample) with unified convergence guarantees for all schemes. Simulation results across linear and nonlinear, with and without disturbances, corroborate that multiple models accelerate convergence, with Case 2 (per-sample switching) typically providing the fastest tracking and identification error decay. The framework lays a foundation for robust, fast adaptive ILC applicable to time-varying references and disturbances, with notes on computational complexity and avenues for complexity-reducing variants such as MM-SLA.

Abstract

This article focuses on making discrete-time Adaptive Iterative Learning Control (ILC) more effective using multiple estimation models. Existing strategies use the tracking error to adjust the parametric estimates. Our strategy uses the last component of the identification error to tune these estimates of the model parameters. We prove that this strategy results in bounded estimates of the parameters, and bounded and convergent identification and tracking errors. We emphasize that the proof does not use the key technical lemma. Rather, it uses the properties of square-summable sequences. We extend this strategy to include multiple estimation models and show that all the signals are bounded, and the errors converge. It is also shown that this works whether we switch between the models at every instant and every iteration or at the end of every iteration. Simulation results demonstrate the efficacy of the proposed method with a faster convergence using multiple estimation models.

Multiple Estimation Models for Discrete-time Adaptive Iterative Learning Control

TL;DR

The paper tackles discrete-time Adaptive ILC by introducing a multiple estimation model framework (MMST) that updates parameters using the identification error rather than tracking error. It derives a single-model solution with a projection-based, identification-error-driven update and proves convergence using composite energy functions and the properties of sequences, avoiding the Key Technical Lemma. The MMST extension constructs multiple candidate models and offers two switching schemes (end-of-iteration and per-sample) with unified convergence guarantees for all schemes. Simulation results across linear and nonlinear, with and without disturbances, corroborate that multiple models accelerate convergence, with Case 2 (per-sample switching) typically providing the fastest tracking and identification error decay. The framework lays a foundation for robust, fast adaptive ILC applicable to time-varying references and disturbances, with notes on computational complexity and avenues for complexity-reducing variants such as MM-SLA.

Abstract

This article focuses on making discrete-time Adaptive Iterative Learning Control (ILC) more effective using multiple estimation models. Existing strategies use the tracking error to adjust the parametric estimates. Our strategy uses the last component of the identification error to tune these estimates of the model parameters. We prove that this strategy results in bounded estimates of the parameters, and bounded and convergent identification and tracking errors. We emphasize that the proof does not use the key technical lemma. Rather, it uses the properties of square-summable sequences. We extend this strategy to include multiple estimation models and show that all the signals are bounded, and the errors converge. It is also shown that this works whether we switch between the models at every instant and every iteration or at the end of every iteration. Simulation results demonstrate the efficacy of the proposed method with a faster convergence using multiple estimation models.
Paper Structure (17 sections, 2 theorems, 63 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 2 theorems, 63 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. A New Solution for Discrete-time Adaptive ILC
  4. Control and Identification Laws
  5. Convergence Analysis
  6. Multiple Estimation Models for Adaptive ILC
  7. Control and Identification Laws
  8. Case 1
  9. Case 2
  10. Convergence Analysis
  11. Simulation Examples
  12. Example 1: LTI System without Disturbances
  13. Example 2: LTV System with Disturbances
  14. Example 3: Nonlinear System without Disturbances
  15. Example 4: Nonlinear System with Disturbances
  16. ...and 2 more sections

Key Result

Theorem 1

For the system eq:Plant with the objective of tracking the reference model eq:RefModel, the control law eq:ControlSM along with the adaptive law eq:ParEstSM guarantees the following:

Figures (5)

  • Figure 1: Example 1: Error profiles over iterations for an LTI system without disturbances.
  • Figure 2: Example 2: Error profiles over iterations for an LTV system with disturbances.
  • Figure 3: Example 3: Error profiles over iterations for a nonlinear system without disturbances.
  • Figure 4: Example 4: Error profiles over iterations for a nonlinear system with disturbances.
  • Figure 5: Example 5: Error profiles over iterations for a nonlinear system with disturbances, with an iteration-varying reference trajectory.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • proof
  • Remark 6
  • Theorem 2
  • proof