Mathematical Problems in Engineering

Volume 2015, Article ID 429892, 16 pages

http://dx.doi.org/10.1155/2015/429892

## A Data Dropout Compensation Algorithm Based on the Iterative Learning Control Methodology for Discrete-Time Systems

^{1}Department of Electricity and Electronics, University of the Basque Country, UPV/EHU, Campus of Leioa, 48940 Leioa, Bizkaia, Spain^{2}Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Barcelona, Spain

Received 6 June 2014; Accepted 28 August 2014

Academic Editor: Minrui Fei

Copyright © 2015 S. Alonso-Quesada 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

This paper deals with the convergence of a remote iterative learning control system subject to data dropouts. The system is composed by a set of discrete-time multiple input-multiple output linear models, each one with its corresponding actuator device and its sensor. Each actuator applies the input signals vector to its corresponding model at the sampling instants and the sensor measures the output signals vector. The iterative learning law is processed in a controller located far away of the models so the control signals vector has to be transmitted from the controller to the actuators through transmission channels. Such a law uses the measurements of each model to generate the input vector to be applied to its subsequent model so the measurements of the models have to be transmitted from the sensors to the controller. All transmissions are subject to failures which are described as a binary sequence taking value 1 or 0. A compensation dropout technique is used to replace the lost data in the transmission processes. The convergence to zero of the errors between the output signals vector and a reference one is achieved as the number of models tends to infinity.

#### 1. Introduction

Iterative learning control (ILC) strategies have been broadly used in many industrial applications, for instance in manufacturing, robotics, or chemical processes, to improve the performance of systems which executes the same task multiple times [1–5]. In the past years, such control strategies have been also applied to improve the performance of closed-loop discrete-time control systems where the controller device is sited far away from the set composed by the process to be controlled, its actuator, and its sensor. On the one hand, these systems require the transmission of the control signals from the controller to the actuator in order to apply the control action to the process at each sampling time. On the other hand, the output measurements have to be transmitted at each sampling time from the sensor to the controller for synthesizing the control signals by using the information of the output of the process. Unfortunately, such transmissions are susceptible to suffering eventual failures due to several causes as punctual disconnections or intermittent data dropouts and delays appearing as a consequence of the limited bandwidth of the communication channels or by the presence of uncertainties and noises in such channels [6, 7]. These failures cause the deterioration of the performance in the control system dynamics and potential instability. Such data dropouts and delays happen mainly when the transmission channels are used by several control systems working simultaneously and, also, in cases of large interconnected systems which need to have coupled information for control purposes [8, 9]. In such a context, a possibility to circumvent such difficulties is to consider several copies of the set composed by the process, actuator, and sensor for applying the control action several times at each sampling instant. Each one of these copies refers to an iterative model. This alternative is interesting for instance when the process to be controlled is implemented in a computer, so it is available to dispose several copies of it, and the controller is developed either in another computer or in an analog device sited far away from the process location [10].

All the iterative models are running simultaneously during the finite time interval for executing a certain task. However, the control signals vector is applied to each iterative model in a sequential way within each intersampling period. Namely, the control vector corresponding to the th iterative model is synthesized after the controller has received the measurements vector from the sensor corresponding to the precedent th iterative model. Later, such a control vector is transmitted to the actuator of the th iterative model to be applied to this model and finally its sensor sends the output signals vector to the controller in order to synthesize the control signals vector for the subsequent th iterative model. Later such a cycle is repeated for the th iterative model and so on. In this context, ILC strategies are a good choice to synthesize the control signals vector to be applied to the set of iterative models of the system since they use the information about the output errors vector of each iteration model to modify the input signals vector to be applied to the subsequent one. In such a way, the accuracy of the system can be improved if a reference model tracking is required. Obviously, the number of iterative models has to be appropriately chosen such that all the iterative models are run within each intersampling time period. Furthermore, the actuator of the th iterative model cannot update the control signals vector applied to such model in a realistic situation, with presence of failures in the transmissions at certain sampling instants, when there is a data dropout in the communication channel from the controller to such an actuator, which implies a deterioration in their performance. The same undesirable result in the performance of the th iterative model occurs when there is a data dropout in the transmission channel which links the sensor of the th iterative model with the controller. In both situations, the controller cannot use the measurements vector of the th iterative model to synthesize the control signals vector to be applied to the th iterative model. The performance deterioration caused by such transmission failures can be compensated by replacing the lost datum corresponding to the th iterative model, that is, the control signals vector of th iterative model or the measurements vector of the -iterative model, by that corresponding to the precedent th iterative model, that is, the control vector of th iterative model or the measurements vector of the -iterative model, respectively [11–13].

This paper studies the output error convergence of an ILC system, composed of a set of discrete linear and time-invariant models with their actuators and sensors and a remote control device, under potential data missing caused by transmission failures. Such failures, those from the sensors to the controller as well as those from the controller to the actuators, are distributed as among the iterative models as during the time interval which lasts a complete execution of a task by the system. The study proposes a new data dropouts compensation algorithm to guarantee the convergence to zero of the tracking error as the number of iterative models tends to infinity. Such an algorithm is an extension of those proposed in [11, 12].* This paper studies the presence of data dropouts in the transmission from the controller to the actuators as well as from the sensors to the controller*. In this sense, the algorithm proposed in [11] only considers transmission failures from the controller to the actuators while that in [12] only considers failures in the transmissions from the sensors to the controller. Moreover, the dropout compensation technique proposed in such papers replaces the lost datum of the th iterative model by that corresponding to the precedent iterative model at the same sampling instant. As a consequence, such algorithms can give place to a defective behaviour if two consecutive iterative models fail at the same sampling time. In this sense,* this paper considers replacing the lost datum with that corresponding to one of the precedent models*. Concretely, the algorithm takes the datum corresponding to the closest iterative model to the current one without failure at such a sampling instant. In this way, the behaviour of the system can be improved in the eventual case of several consecutive transmission failures in a set of consecutive iterative models.* The main aim of the paper is the proof of convergence of the ILC system with such a dropout compensation algorithm*. Also, a simulation example illustrates the behaviour of the system with such an algorithm and a comparison with the algorithm proposed in [11] is provided.

#### 2. Problem Statement

##### 2.1. System Scheme with the Set of Iterative Models

Consider an ILC system composed by a set of discrete-time linear time-invariant models described bywhere , , and are the state, control, and output vectors of respective dimensions , , and , with , of each th model. The matrices , , , and are of orders being compatible with the respective dimensions of the above vectors. The subscript and the discrete argument with for some integer (defining the horizon size) run, respectively, for the set of models and for the set of sampling instants (i.e., for the discrete time). Each th model is equipped with an actuator and a sensor. The actuator receives from the remote controller a set of control signals vectors at each sampling time instant, each one through a different communication channel, and it chooses one of them, namely, , to be applied to its corresponding model. This redundancy augments the probability that each actuator receives an actualized control signals vector at each sampling time when all of the transmissions are subject to data dropouts due to the unreliability of the communications. Each sensor measures the outputs vector of its corresponding model and sends such measurements to the controller which generates the control signals vectors to be transmitted to the actuators. Also, such transmissions are subject to failures.

The potential presence of data dropouts in the transmission of signals, from the controller to the actuators as well as from the sensors to the controller, motivates the use of a platform of several identical models described by (1) instead of an only one. Although all of such models possess the same dynamics, their time evolution will be different as a consequence of the presence of dropouts and the algorithm to compensate it. In this sense the ILC law calculates the input signal vector to be applied to the th iterative model by modifying that corresponding to the precedent th iterative model with an additional term being proportional to the tracking errors vector associated to the later iterative model. Such tracking errors are computed for each iterative model by comparison of the iterative model outputs vector with a reference signals vector being obtained from a discrete-time dynamic model defined bywhere , , and are the reference model state, input, and output vectors of respective dimensions , , and . In order that the model is BIBO-stable, it is assumed that the reference inputs vector is bounded for all time and that is a stable matrix; that is, all its eigenvalues are inside the open unit complex circle. The scheme of the control system is displayed in Figure 1.