Modelling and Simulation in Engineering

Volume 2018, Article ID 4195938, 9 pages

https://doi.org/10.1155/2018/4195938

## New Iterative Learning Control Algorithm Using Learning Gain Based on *σ* Inversion for Nonsquare Multi-Input Multi-Output Systems

^{1}Laboratoire Analyse, Conception et, Commande des Systèmes, École Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, Tunis LR11ES20, Tunisia^{2}Laboratoire Analyse, Conception et, Commande des Systèmes, Institut Préparatoire aux, Etudes d’Ingénieurs d’El Manar, Université de Tunis El Manar, Tunis LR11ES20, Tunisia

Correspondence should be addressed to Leila Noueili; nt.unr.tine@ilieuon.aliel

Received 6 November 2017; Revised 8 March 2018; Accepted 25 March 2018; Published 7 June 2018

Academic Editor: Azah Mohamed

Copyright © 2018 Leila Noueili et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Model inversion Iterative Learning Control (ILC) for a class of nonsquare linear time variant/invariant multi-input multi-output (MIMO) systems is considered in this paper. A new ILC algorithm is developed based on -right inversion of nonsquare learning gain matrices to resolve the matrix inversion problems appeared in the direct model inversion of nonsquare MIMO systems. Furthermore, a sufficient and necessary monotonic convergence condition is established. With rigorous analysis, the proposed ILC scheme guarantees the convergence of the tracking error. To prove the effectiveness and to illustrate the performance of the proposed approach for linear time-invariant (LTI) and time-varying nonsquare systems, two illustrative examples are simulated.

#### 1. Introduction

Iterative Learning Control is an intelligent control strategy to deal with repetitive processes. The aim of the ILC is to control systems which execute the same task over a finite duration, such as the industrial robot manipulator [1]. The basic principle behind ILC is that the data generated from previous trial are used to adapt the control input for current trial. The control input in each trial is adjusted by using the tracking error obtained from the previous trial. As the iterations continue, the control system eventually learns the task and follows the desired trajectory with minimized tracking errors. The concept has been well developed in terms of both the fundamental theory and experimental applications that were accomplished in [2–6]. Thus, ILC research has considerably coped with mechanical systems such as multijoint hand-arm robots [7], the station stop control of train [8], and wafer stage motion systems [9]. The most of ILC schemes in the literature focus on proportional type learning law [10] and Optimal Iterative Learning Control (OILC) [11–14] where in [13] the learning approach has been applied to a rapid thermal processing. However, the majority of proposed algorithms were based on the notion of direct model inversion where the learning gain is obtained by Markov matrix inverse which represents the input-output map of the system [15]. However, this kind of ILC cannot be developed for general nonsquare (rectangular) MIMO systems that are systems in which the numbers of inputs and outputs are unequal. Indeed, a problem of the matrix inversion is encountered.

Many industrial processes require nonsquare MIMO mapping, especially chemical plants such as crude distillation process [16], mixing tank process [17], three-tank systems [18], distillation column [19], and chemical mechanical planarization process [20]. Therefore, the need of control schemes treating this type of systems is of major interest. The model inverse design and its related applications in control design have been widely studied in [21–24].

In the literature, there are control works that treated the nonsquare MIMO systems such as a Proportional Integrator Derivator (PID) controller cited in [25] which has been applied to a voltage model of Proton Exchange Membrane Fuel Cell (PEMFC); Model Predictive Control (MPC) was studied in [26] and has been applied to a Shell Heavy Oil Fractionator (SHOF); Minimum Variance Control (MVC) which is an inverse model control was studied in [27] and other examples of control strategies applied to nonsquare systems [28]. In the above cited works [25, 26], the control schemes were achieved without a need to matrix inversion. However, in [27] the synthesis of the inverse model in the control procedure required a matrix inversion based on left and right inverses which are developed in [29–31].

However, all the above mentioned methods could not deal with repetitive systems that require a learning strategy based on inversion model to achieve the control of repetitive nonsquare MIMO systems. Therefore, developing an ILC procedure for nonsquare systems remains an open problem.

Based on nonsquare polynomial matrix inversions [32], we propose the design and the analysis of an ILC scheme based on the model inversion called -ILC for linear time-invariant and varying-time nonsquare MIMO systems. The main contribution of this work is to prove the monotonic convergence of the proposed scheme where the tracking error trial-to-trial will converge to zero even though the system has the initial resetting state. Through the simulation results, we prove the effectiveness of the proposed method.

The rest of this paper is organized as follows: In Section 2, the problem formulation is presented. The proposed inverse learning gain is developed in Section 3. A sufficient and necessary monotonic convergence of the -ILC is established in Section 4. Further, the proposed -ILC law is extended to time-varying systems in Section 5. Simulation results are illustrated in Section 6 to prove the effectiveness of the scheme for nonsquare MIMO systems. Finally, conclusions and an outlook on future work are given in Section 7.

#### 2. Problem Formulation

The state-space representation of an LTI discrete-time system is given by (8):where denotes the iteration numbers and is the total number of trials. represents the number of the discrete-time sampling steps and is the total number of discrete-time steps at each trial. is the iteration vector of system states, is the iteration vector of the system inputs which will be recursively generated by an iterative learning algorithm, is the iteration of the system outputs and is a fixed initial state. Further, , , and are constant matrices with appropriate dimensions, and . Therefore, this state-space system can be described as follows:where is a Markov matrix of rank and whose terms are Markov parameters of the plant as cited in [33] and

The control objective is to find a control sequence with the ability to reduce tracking error for the whole trajectory based on the past tracking experiences until , the system tracking error limit iswhere is the infinite norm.

The main formulation of the ILC design problem is to achieve an update mechanism for the control input trajectory of a new cycle based on the information from previous cycles, so that the output trajectory converges asymptotically to the reference trajectory [34–36]. This idea is depicted in the block diagram form in Figure 1, which shows the next trial’s control input to be calculated from the previous trials control input and transient output error .