An iterative learning control theory for a class of nonlinear dynamic systems

doi:10.1016/0005-1098(92)90063-L

Automatica

Volume 28, Issue 6, November 1992, Pages 1215-1221

https://doi.org/10.1016/0005-1098(92)90063-L Get rights and content

Abstract

An iterative learning control scheme is presented for a class of nonlinear dynamic systems which includes holonomic systems as its subset. The control scheme is composed of two types of control methodology: a linear feedback mechanism and a feedforward learning strategy. At each iteration, the linear feedback provides stability of the system and keeps its state errors within uniform bounds. The iterative learning rule, on the other hand, tracks the entire span of a reference input over a sequence of iterations. The proposed learning control scheme takes into account the dominant system dynamics in its update algorithm in the form of scaled feedback errors. In contrast to many other learning control techniques, the proposed learning algorithm neither uses derivative terms of feedback errors nor assumes external input perturbations as a prerequisite. The convergence proof of the proposed learning scheme is given under minor conditions on the system parameters.

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This paper discusses about Iterative Learning Control (ILC) which learns an appropriate feedforward input from iterative experiments. In particular, this paper focuses on systems which include integrator, and considers tracking to a desired output which converges to a constant. In this case, the feedforward input should converges to zero, however classical ILC does not take this condition into account, and interfere with the feedback controller. This paper proposes a new design of ILC whose resulting input converges to zero exponentially. The effectiveness of the proposed method is demonstrated by a practical experiment with a flexible arm. Since a flexible arm is known to nonlinear, the proposed method is expected to be effective for nonlinear systems.
Contraction analysis of nonlinear noncausal iterative learning control
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Citation Excerpt :
For convergence analysis of nonlinear ILC, there are two main approaches: the contraction mapping theorem (CMT) approach and the composite energy function (CEF) method. Both CMT [16,17] and CEF [7,18] methods have successfully been used to analyze causal, nonlinear ILC. However, for the noncausal nonlinear case, results are less comprehensive.
Iterative learning control (ILC) is a method for learning input signals for repetitive control tasks. In this paper, we provide a new method based on convex optimization for certifying convergence and estimating convergence rate in ILC schemes involving a nonlinear plant and a noncausal update law, which are common in practice. Using sum-of-squares (SOS) optimization, we compute the convergence rate of an example nonlinear, noncausal ILC system and verify its accuracy in experiment.
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Citation Excerpt :
It finds applications in many engineering systems where the system performs the same task over iterations with a high tracking precision requirement. Examples include chemical batch processes (Lee, Bang, & Chang, 1994; Mezghani et al., 2002), robotic manufacturing systems (Kawamura, Miyazaki, & Arimoto, 1988; Kuc, Lee, & Nam, 1992), precision motion control systems (Altin & Barton, 2014; Barton & Alleyne, 2008), traffic flow control (Hou, Xu, & Yan, 2008), control of wind turbine rotors (Tutty, Blackwell, Rogers, & Sandberg, 2014), modelling of human motor learning (Zhou et al., 2016), and robotic rehabilitation (Freeman, 2017; Xu, Chu, & Rogers, 2014). More applications can be found in the survey papers (Bristow, Tharayil, & Alleyne, 2006; Moore, 1999) and references therein.
The use of feedback-based iterative learning control (ILC) has been widely reported in the literature to reduce large transient errors in the iteration-domain. Using both feedback and feed-forward (ILC) would result in large control input. However, due to the existence of hardware limitation, input constraints exist. Such a hard constraint might lead to unacceptable performances such as unstable trajectories in the time-domain and steady-state error in the iteration-domain. In this paper, a feedback-based ILC design is proposed for a class of linear-time-invariant system in the presence of input constraints. The proposed control scheme consists of two parts: one (feedback) deals with the performance in time domain while the other (ILC) ensures ensure the perfect tracking performance. With the help of composite energy function, it is shown that the proposed algorithm can ensure the perfect tracking performance in the presence of hard input constraints under appropriate assumptions. Simulation results support the theoretical findings.
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Iterative learning control (ILC) is widely used as a simple method for precise tracking of systems under repetitive conditions. ILC operates by “learning” from the previous iteration’s errors, correcting them over a number of iterations. However, the question of whether or not a nonlinear ILC system converges is still in general an open one. Assuming a state-space formulation, we use contraction analysis to formulate a convergence condition for ILC system as a linear matrix inequality (LMI). Finally, we compute a convergence certificate for a simple example involving “anticogging” a permanent-magnet synchronous motor driving a pendulum in simulation.
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Iterative learning control (ILC) is a simple and efficient solution to improve tracking accuracy for systems that execute repetitively the same tracing operation. For engineering applications of ILC, the main concern is the monotonic decay of tracking errors, in the sense of infinity norm or peak error, along the trials. Low cost in implementation and robustness in performance are also critical factors. To achieve these important but sometimes contradicting goals, several multirate ILC schemes have been developed, in which different data sampling rates are used for feedback online loop and feedforward ILC offline loop. That is, multirate ILC uses a different (often lower) rate from the sampling rate of a feedback system to update input. Before the input signal is applied to the system for the next trial, it is upsampled to reach the original sampling rate. Since downsampling will cause distortion of frequency spectra, anti-aliasing and anti-imaging filters and signal extension are used together with downsampling and upsampling operations. In this paper, these technologies are integrated with three different multirate ILC schemes, pseudo-downsampled ILC, two-mode ILC, and cyclic pseudo-downsampled ILC, to achieve better performance. A series of experimental results on an industrial robot are presented to demonstrate the efficiency of multirate ILC schemes and compare the performance. The results demonstrate that multirate ILC schemes are able to achieve not only monotonic learning transient, but also much better tracking accuracy than conventional one-step-ahead ILC schemes.

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^☆: The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. J. McAvoy under the direction of Editor P. C. Parks.

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Brief paperAn iterative learning control theory for a class of nonlinear dynamic systems☆

Abstract

Bettering operation of dynamic system by learning: a new control theory for servomechanism or mechatronics system

On the iterative learning control theory for robotic manipulators

IEEE J. Robotics and Automation

A learning procedure for the control of movements of robotic manipulators

Adaptive control of manipulators through repeated trials

Brief paper
An iterative learning control theory for a class of nonlinear dynamic systems☆