Elsevier

European Journal of Control

Volume 61, September 2021, Pages 91-100
European Journal of Control

An intelligent parameter determination approach in iterative learning control

https://doi.org/10.1016/j.ejcon.2021.06.001Get rights and content

Highlights

  • An approach for determining parameters of learning function in iterative learning control without using the mathematical model.

  • Learning parameter for standard P- and D-type ILC is intelligently computed based on system’s tracking errors from the two previous consecutive trials.

  • Stability and convergence are guaranteed by using injective property and Jacobian matrix of the input-output mapping.

  • Existing pole placement methods can be also applied in combination with the proposed method for determining learning parameters.

  • An planar robot with two links was used for numerical simulation to verify and support the proposed method.

Abstract

This paper introduces an approach for determining parameters of learning function in iterative learning control without using the mathematical model of the controlled plant, which guarantees definitely the convergence requirement of the learning process. The learning parameter is intelligently computed based on system’s tracking errors from the two previous consecutive trials by minimizing a suitable cost function. Then, analysis of stability and convergence using injective property and Jacobian matrix of the input-output mapping is also given in this paper. Finally, some numerical simulations with actual system model of robot manipulators are performed to illustrate and support the proposed method.

Introduction

Iterative learning control (ILC) was well-known for plants repetitively working such as robots and manufacture processes in industry [21], [32], [34]. The definition of iterative learning was first coined by Uchiyama [21], [30] (in Japanese). The concept of ILC was later developed by Arimoto and his colleagues [2], [16]. Concurrent works [6], [10], [20] with similar concepts were proposed by independent authors. Most of the ILC controllers in the literature are P, I, and D types or their combinations such as PD, PI and PID types. In this paper, our interest falls in P and D-types ILC which is very simple and effective in practice.

There have been various methods for designing P-type ILC controllers. In [27], the weighting gain matrix and the learning gain matrix were determined such that the trace of the input error covariance matrix is minimized in the presence of full knowledge of the system’s dynamics and disturbances. Robust P-type and high-order P-type iterative learning controllers were proposed in [7], [9]. P-type controllers [11], [12] were developed for linear distributed parameter systems. A P-type ILC controller was proposed for one-sided Lipschitz nonlinear systems in [13]. Comparisons between P-type ILC controllers with other ILC ones have been made in [33]. An optimal approach for determining parameters of iterative learning PID controllers was proposed in [19] in which a quadratic cost function of error and control signal in two consecutive trials is minimized. The stability and convergence analysis of this method were only performed for linear discrete-time systems. Other optimization based methods can be also found in [26], [28]. There was a neural network based ILC controller for nonlinear systems [23]. An observer based ILC controller [8] was given to overcome non-repetitive uncertainties, random initial states and input constraints. An unified ILC for flexible systems with input constraints was proposed in [14] using the Lyapunov approach, and adaptive ILC for an Euler-Bernoulli Beam system with input constraint was also developed in [15]. Recently, several model free ILC control methods have been studied such as for SISO discrete-time nonlinear systems [1], [4], [18], SISO continuous-time linear systems with disturbance observer [17], second-order MIMO nonlinear systems [3] using state model.

ILC is known as an intelligent control method for repetitive processes. This method is simple but it brings high effectiveness, and the performance of tracking control is relatively robust to disturbances and model error if these disturbances and model error are also assumed to have the same cycle as the working cycle of the plant.

The common feature of both intelligent control and ILC is that no mathematical model of the plant is used for the controller design process, thus the application area of intelligent control methods is quite large, and applicable to both linear and nonlinear systems.

However, to obtain convergent parameters for the learning process based on the existing criteria, one needs to have an approximate model of the plant at least. This influences the meaning of “intelligence” of the method. In this work, we propose a method for convergent parameters calculation of the learning process without using any mathematical model of the plant.

The main contributions of this work are to (a) optimally compute the gain matrix for the P-type and D-type ILC controllers based on the tracking errors of the previous two consecutive trials without using any plant model, (b) prove the stability of the ILC controller for a general class of nonlinear systems represented with input-output mapping, in which the input-output mapping is smooth and the number of inputs is not smaller than the number of outputs, and (c) verify the proposed method using a SISO example and a robot manipulator with two links.

The rest of this paper is organized as follows. In Section 2, a problem formulation is given, and then optimal parameter calculations for P-type and D-type learning functions are proposed for single-input single-output (SISO) and multiple-input multiple output (MIMO) systems with stability and convergence analysis. Some numerical simulations are shown to illustrate the proposed method in Section 3. Final Section provides conclusions and future works.

Section snippets

Preliminaries and problem formulation

The nature purpose of input signal adjustment step by step through trials in ILC of a repetitively working process, which is described formally as{x̲˙=f̲(x̲,u̲,t)y̲=g̲(x̲,t),is that the process output y̲(t) will asymptotically track a desired repetitive reference signal r̲(t), r̲(t)=r̲(t+T), where T is the repetitive working interval of the system, u̲(t), x̲(t) and y̲(t) are m-dimension input vector, vector of process states and n-dimension output vector of the process (also called the system),

Illustrated examples

To verify the effectiveness of proposed formulas (18) and (20) for intelligent parameter computation of P- and D-type learning functions, the following two simulation examples will be carried out.

Conclusions

In this work, an approach for intelligent parameter computation of P- and D-type learning functions was proposed for repetitive systems based on minimizing an objective function, which depends only on the system’s tracking errors of the two previous consecutive trials. A noticeable distinction between the proposed method and the others in the literature is that the first one is able to optimally provide a gain matrix within the range [0;1] for P- and D-type learning functions after each trial.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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