Robust ILC design with application to stroke rehabilitation

Whatever the hell this is. You'll have to ask your doctor how this can be applied to getting you to 100% recovery. That is the only criteria to evaluate stroke research. You'll have to look at section 2 so you could work out the math yourself since it is way above my head.
http://www.sciencedirect.com/science/article/pii/S000510981730198X

• Christopher Thomas Freeman

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http://doi.org/10.1016/j.automatica.2017.04.016

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Open Access funded by Engineering and Physical Sciences Research Council

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  Open Access

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Abstract

Iterative learning control (ILC) is a design technique which can achieve accurate tracking by learning over repeated task attempts. However, long-term stability remains a critical limitation to widespread application, and to-date robustness analysis has overwhelmingly considered structured uncertainties. This paper substantially expands the scope of existing ILC robustness analysis by addressing unstructured uncertainties, a widely used ILC update class, the presence of a feedback controller, and a general task description that incorporates the most recent expansions in the ILC tracking objective. Gap metric based analysis is applied to ILC by reformulating the finite horizon trial-to-trial feedforward dynamics into an equivalent along-the-trial feedback system, as well as deriving relationships to link their respective gap metric values. The results are used to generate a comprehensive design framework for robust control design of the interacting feedback and ILC loops. This is illustrated via application to rehabilitation engineering, an area where they meet an urgent need for high performance in the presence of significant modeling uncertainty.

Keywords

• Iterative learning control;
• Robustness;
• Electrical stimulation;
• Rehabilitation engineering

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1. Introduction

The iterative learning control (ILC) paradigm addresses tracking of a fixed reference trajectory over a finite time interval of T$T$ seconds. Each attempt is termed a ‘trial’, and the system is reset between trials to the same starting position. The tracking error is recorded during each trial, and in the reset period is used to update the control signal with the aim of reducing the error during the subsequent trial. ILC was originally developed to enable precision control of industrial robotics, but now covers a rich theoretical framework and broad range of applications, see e.g.  Ahn, Chen, and Moore (2007) and Bristow, Tharayil, and Alleyne (2006). While impressive tracking performance is achievable on nominal systems and satisfactory performance has been achieved in practical applications, robustness remains a serious issue. In practice it has been found that long term instabilities degrade the performance and convergence of the standard algorithms.

ILC long term stability is not well understood, and a variety of methods (e.g. quantization, filtering, suspension of learning) have been proposed to address the commonly encountered problem of convergence, followed by rapid divergence. These often lack theoretical basis and there remains debate on the cause of this phenomenon. Previous robustness results relate to multiplicative and additive uncertainty descriptions (De Roover and Bosgra, 2000; Donkers et al., 2008; Harte et al., 2005; Moon et al., 1998; Tayebi and Zeremba, 2003 ;  van de Wijdeven and Bosgra, 2007), or to parametric uncertainty (Ahn, Moore, & Chen, 2006). Unstructured uncertainties were addressed in French (2008) where it was shown that there exists a non-zero stability margin for a class of adaptive ILC algorithms. However, the analysis was not extended to more general ILC update classes. It is hence desirable for a general framework to quantify the effect of realistic model mismatch, thereby informing practical design. Furthermore, there is also a need to incorporate recent expansion in the ILC framework in which the tracking objective is generalized to permit tracking only at isolated time-points or over intervals in [0,T]$[0,T]$ (Janssens et al., 2013; Owens et al., 2015 ;  Son et al., 2013). This expanded class meets the needs of a wide range of industrial processes, such as robotic pick-and-place tasks, welding, and coordinated motion. However, the only robustness results for this expanded task framework relate to multiplicative uncertainty (Owens, Freeman, & Chu, 2014).

This paper substantially expands the scope of existing ILC robustness analysis by addressing for the first time: (1) unstructured uncertainties, (2) a general ILC update class, and (3) a full generalization of the task descriptions that have so far been considered in ILC. To maximize impact, we also consider inclusion of a feedback controller. Analysis is based on the nonlinear gap metric of Georgiou and Smith (1997), which is applied to ILC by reformulating the within-in trial feedforward action as trial to trial feedback action. The resulting gap on the trial to trial dynamics is then translated back to the original plant.

This paper is arranged as follows: a general problem description is defined in Section  2, and robust performance analysis is undertaken in Section  3 with proofs contained in the appendix. To illustrate the power of the framework, results are presented in Section  4 from an application to stroke rehabilitation. Section  5 contains conclusions and topics of future work.