Comments on “Robust optimal design and convergence properties analysis of iterative learning control approaches” and “On the P-type and Newton-type ILC schemes for dynamic systems with non-affine input factors”

doi:10.1016/j.automatica.2007.02.007

Automatica

Volume 43, Issue 9, September 2007, Pages 1666-1669

https://doi.org/10.1016/j.automatica.2007.02.007 Get rights and content

Abstract

The papers by Xu and Tan [Robust optimal design and convergence properties analysis of iterative learning control approaches, Automatica 38 (2002) 1867–1880], and Xu and Tan [On the P-type and Newton-type ILC schemes for dynamic systems with non-affine input factors, Automatica 38 (2002) 1237–1242], give a convergence analysis for several iterative learning control approaches. Unfortunately, these papers contains several mathematical errors that render the proofs of the claimed results invalid. As there are no obvious ways to correct these errors, the results presented in these papers are questionable.

Introduction

The analysis of the convergence speed of the iterative learning control (ILC) schemes discussed in Xu and Tan, 2002a, Xu and Tan, 2002b is based upon a comparison of their quotient convergence factors ( $Q$ -factor). Unfortunately, the analysis used to calculate the $Q$ -factors of the ILC schemes is erroneous and hence the comparison of their convergence speeds must be regarded as unproven. In this paper we will describe these technical shortcomings and discuss what would be required to overcome then.

Section snippets

Robust optimal design

In Section 3 of Xu and Tan (2002a), the authors introduce a first order ILC scheme $I_{1}$ and use its Q-factor $Q (I_{1}, 0)$ as a measure of the convergence speed. In Eq. (9), the authors establish the inequality $| Δ u_{i + 1} |_{λ} ⩽ [γ_{1} + δ_{1}] | Δ u_{i} |_{λ}$ for any convergent sequence ${u_{i}} \in I_{1}$ , and then claim that $Q (I_{1}, 0) = γ_{1} + δ_{1} .$

The error is to take the upper bound $γ_{1} + δ_{1}$ for the ratio $| Δ u_{i + 1} |_{λ} / | Δ u_{i} |_{λ}$ established in (9) as the least upper bound. Numerous upper bounds have been applied to establish the upper bound $γ_{1} + δ_{1}$ . In

Convergence speed analysis of higher order ILC

In Section 5 of Xu and Tan (2002a), the authors provide a comparison of the convergence speeds of first order ILC schemes with higher order schemes, by comparing their Q-factors. Again, the calculation of the Q-factors contains several mathematical errors, invalidating the comparison.

In Section 5.1 the authors consider a second order ILC scheme $I_{2}$ and obtain (38) for any sequence ${u_{i}} \in I_{2}$ : $| Δ u_{i + 1} |_{λ} ⩽ [γ_{1} + q_{1} c (λ)] | Δ u_{i} |_{λ} + [γ_{2} + q_{2} c (λ)] | Δ u_{i - 1} |_{λ} .$

Dividing through (38) by $| Δ u_{i} |_{λ}$ and taking the ${\lim \sup}_{i \to \infty}$ ,

Convergence speed analysis of P-type and Newton-type ILC

In 4 Convergence speed analysis of P-type and Newton-type ILC, 5 Analogies with other control theories of Xu and Tan (2002b), the authors introduce P-type and Newton-type ILC schemes $I_{P}$ and $I_{N}$ and proceed to obtain their Q-factors $Q (I_{P}, 0)$ and $Q (I_{N}, 0)$ as a measure of their convergence speed.

This authors repeat the same errors as in Xu and Tan (2002a). For the P-type scheme, the error is to take the upper bound from (10) and treat it as the least upper bound in (11) and similarly for the

Analogies with other control theories

Additional insights into the difficulties in the Xu and Tan papers, and what would be required to overcome them, may be gained by considering an analogy with the problem of finding the robust stability margin of a given controller K for a plant P subject to unstructured perturbations $Δ$ . The robust stability margin $r (P, K)$ of the controller is defined to be the largest value r such that the closed loop system will be stable for all perturbations $Δ$ with $∥ Δ ∥ < r$ . The small gain theorem gives a

Book review

The analysis provided in these papers was subsequently published in Chapters 2, 3 and 5 of Xu and Tan (2003). This book was reviewed in Kuc and Won (2004), and the reviewers remarked that

In general, it is expected that if more control information from the previous trials is used in ILC, a better performance of ILC is likely to be obtained. The authors show in Chapter 3 that this intuition is not always true as far as the convergence speed of higher order ILC is concerned ... the simplest ILC

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There are more references available in the full text version of this article.

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^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Wei Kang under the direction of Editor André Tits.

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CorrespondenceComments on “Robust optimal design and convergence properties analysis of iterative learning control approaches” and “On the P-type and Newton-type ILC schemes for dynamic systems with non-affine input factors”☆

Abstract

Introduction

Section snippets

Robust optimal design

Convergence speed analysis of higher order ILC

Convergence speed analysis of P-type and Newton-type ILC

Analogies with other control theories

Book review

Automatica

Automatica

Correspondence
Comments on “Robust optimal design and convergence properties analysis of iterative learning control approaches” and “On the P-type and Newton-type ILC schemes for dynamic systems with non-affine input factors”☆