Monotonic convergence of iterative learning control systems with variable pass length

ABSTRACT

A growing number of researchers consider iterative learning control (ILC) a promising tool for numerous control problems in biomedical application systems. We will briefly discuss why classical ILC theory is technically too restrictive for some of these applications. Subsequently, we will extend the classical ILC design in the lifted systems framework to the class of repetitive trajectory tracking tasks with variable pass length. We will analyse the closed-loop dynamics for two standard learning laws, and we will discuss in which sense the tracking error can be reduced by which controller design strategies. Necessary and sufficient conditions for monotonic convergence will be derived. We then summarise all results in a set of practical controller design guidelines. Finally, a simulation study is presented, which demonstrates the usefulness of these guidelines and illustrates the special dynamics that occur in variable pass length learning.

KEYWORDS:

• Iterative learning control
• non-uniform trial duration
• lifted system framework
• monotonic convergence
• biomedical applications

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For example, the frequency-domain representations of the system dynamics and the controller dynamics commute, which is in general not the case for their lifted-system representations.

2. By definition, ${\mathbf{H}}_{n_j}a={⟨{⟨a⟩}_{n_j}⟩}_{\bar{n}}\forall a\in {\mathbb{R}}^{\bar{n}}$.

3. ${\left|\right|\mathbf{x}\left|\right|}_p:=({\sum }_{i=1}^n{\left|\mathbf{x}\left(i\right)\right|}^p{)}^{(1/p)}\forall \mathbf{x}\in {\mathbb{R}}^n,p\ge 1$, with the limit case ${\left|\right|\mathbf{x}\left|\right|}_{\infty }={max}_i\left|\mathbf{x}\left(i\right)\right|\forall \mathbf{x}\in {\mathbb{R}}^n$.

4. i.e. ||A||_p ≔ max {||Ax||_p, ||x||_p = 1}.

5. i.e. ||ABx||_p ⩽ ||A||_p||B||_p||x||_p ∀A, B, x.

6. more precisely: ${\left|\right|\mathbf{A}\left|\right|}_{\le f_0,p}={max\left\{\right|\left|\mathbf{Ax}\right||}_p,\mathbf{x}={\mathbf{Q}}_{f_0}\widehat{\mathbf{x}},\left|\right|\widehat{\mathbf{x}}{\left|\right|}_p=1\}$.

7. in the sense of Corollary 3.1, i.e. $\left|\right|{\bar{\mathbf{e}}}_{j+1}{\left|\right|}_1<\left|\right|{\bar{\mathbf{e}}}_j{\left|\right|}_1$ for all $n_j\in \{\underset{\_}{n},...,\bar{n}\}$ unless e_j is the zero vector.

Additional information

Funding

This work was supported by the German Federal Ministry of Education and Research [FKZ 16SV7069K].