#### 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.

The paper addresses the output convergence of the system as the number of iterative models tends to infinity under the assumptions that the actuator can miss data from the remote controller while the controller can miss data from the output sensors. The output convergence requires the compensation of both types of data dropouts. The control scheme takes into account the following * features*.(a)The actuator associated to the th iterative model receives its corresponding control signals vector as well as those corresponding to its precedent iterative models, namely, for , from the remote controller at the current sampling instant. Each one of these vectors is transmitted through a different communication channel from the controller to the actuator. Such an actuator applies the control vector to such an iterative model if it has been transmitted without failure. Otherwise, the control vector corresponding to the closest precedent iterative model to the current one which has been received without transmission failure is used to replace the lost one.(b)The remote controller requires the output measurements vector of each iterative model in order to synthesize the control signals vector of the subsequent iterative model at the current sampling instant. Namely, the controller needs the output measurements vector of the th iterative model to calculate the control vector corresponding to the th one. In this context, the controller can miss data due to transmission failures from the sensors providing such output measurements vectors. In such a situation, the replacement of the missed output data with alternative ones is required. In that case, the controller replaces the lost output measurements vector, namely, , by that corresponding to the closest precedent iterative model whose output measurements vector was appropriately transmitted, namely, , if it has been transmitted from its sensor to the controller without failure, otherwise, and so on.(c)All the iterative models potentially suffer from transmission failures in their communication channels from the controller to the actuator and/or from the sensor to the controller. Such failures are distributed within the time interval for the execution of a task. In this context, the scalar variables and , for , , and , taking values 0 or 1, which are mutually uncorrelated, are used to denote these failures. The value 1 means that the corresponding signal has been transmitted without failure while 0 implies the contrary.(d)The data are missed or not from the sensors to the controller and from the controller to the actuators independently of each other.

##### 2.2. Design of the Remote Controller for Compensating the Presence of Dropouts

An ILC law is implemented in the remote controller in order to compensate the data dropouts. Such a control law is given by the following equations:where is the number of iterative models, , for all , is a free-design control signals vector to the first iterative model in the ILC scheme, is a learning gain matrix, andwithwhere and , for , is the th order identity matrix, and , for , are the scalar variables taking values 0 or 1, as it was described in feature (c) of the previous subsection, which indicate whether a transmission failure between the sensor of the th iterative model and the controller has occurred or not at the sampling instant . If the value of is 1 then the first block of is while the other ones are zero. It means that the measurement signals vector of the th iterative model, namely, , has been appropriately transmitted from the sensor of such a model to the controller and then , which is the error signals vector, is available to be used in the calculation of the control signals vector of the subsequent th iterative model. Otherwise, such a datum has not been received by the controller and then the controller has to replace such a lost datum by another one corresponding to one of the precedent iterative models without transmission failure between its sensor and the controller. From the construction of , such an alternative measurements vector corresponds to the closest precedent iterative model to the current one without transmission failure. Note that, at most, only one of the blocks of can be equal to at each sampling instant while the other ones are zero. Also, if all of such blocks are zero then the controller uses the error signals vector of the th iterative model at the previous sampling instant, that is, , for generating the control signals vector .

The actuator associated with the th iterative model can receive several control vectors from the controller, each one through a different transmission channel. They can be compactly written as for . The actuator applies the control given bywithwhere , for , is the th order identity matrix, and , for , are scalar variables taking values 0 or 1 as those described in feature (c) of the previous subsection. If the value of is 1 it means that the control vector has been appropriately transmitted from the controller to the actuator of the th iterative model at the current sampling instant. From the construction of the control vector is used as input for the th iterative model if it has been appropriately transmitted from the controller to the actuator, that is, if . Otherwise, such a datum has not been received by the actuator and then the actuator has to apply to the th iterative model another input vector. From the construction of , such an input vector is the control vector of the closest precedent iterative model to the current one without transmission failure through the channel which carries such a control vector from the controller to the actuator of the th iterative model.

Note that if the control signals vector corresponding to the th iterative model, namely, , is transmitted without failure from the controller to the th actuator and the measurements vector of the th iterative model, namely, , is transmitted without failure from the th sensor to the controller at the sampling instant , that is, , then one gets from (3)–(7) that the control vector applied to the th iterative model at such a sampling instant is . If the control signal corresponding to the th iterative model fails in the transmission from the controller to the th actuator but that corresponding to the th one is well transmitted at the sampling instant , that is, and , then provided that ; that is, the measurements vector corresponding to the th iterative model is transmitted without failure from the th sensor to the controller. Otherwise, the output measurement vector corresponding to the th iterative model will be used if it is transmitted without failure from the th sensor to the controller, that is, if and so on. Finally, if all the transmissions from the controller to the th actuator fail simultaneously at the sampling instant then ; that is, the actuator maintains the control vector of the previous sampling instant.

*Remark 1. *From a practical viewpoint, the scheme of Figure 1 can be simplified such that the actuator corresponding to the th iterative model potentially receives a set of control signals vectors, namely, , instead of if . Otherwise, it potentially receives the set of control signals vector as it has been explained in the main text. Then, the number of communication channels between the controller and all the actuators of the ILC system is while there are channels for the transmissions from the sensors to the controller. Thus, the total* number of transmissions* in the system within the time period is .

#### 3. Convergence Analysis of the Output Errors

The following result is concerned with the convergence of the output errors to zero as tends to infinity for all .

Theorem 2. *Assume the following.*(a)* for all .*(b)*At least one of the scalar variables and at least one of those , for any , are 1 at each sampling time instant for with being a large enough integer number.*(c)*Each th iterative model, for , suffers at most one dropout in its actuator at the sampling instant because of and another one in its sensor at ; that is, .**Then, the following properties hold.*(i)* as provided that is chosen such that and .*(ii)* as for all .*

*Proof. *(i) From (1) and (2), one obtains thatwhere and with . Moreover, the fact that if and if has been used. In this sense, note that only takes two values, namely, if all the transmissions from the controller to the actuator of the th iterative model fail at the sampling instant and otherwise; that is, if at least one of such transmissions is successful then . By proceeding recursively from (8),where the fact that has been used. By using (9), one obtainsFrom (3), (4), and (10), it follows thatwhere . From (10) and (11), one obtains thatfor any , where the facts that , for all , and in view of assumption (b), and alsofor all have been applied since and in view of such an assumption. Moreover, it follows thatwhere the facts that and have been used. Such an expression holds for all so that one can obtain by taking norms in (14)where for andwithNote that in the ideal case of absence of dropouts for all the iterative models and for all one obtains, from (16), thatsincefor all , and all the matrices are zero for all . In such a situation, if is chosen such that for all , then as since the spectral radius of is smaller than 1. The following facts have to be taken into account when there are dropouts distributed during the time interval .*Fact 1*. When a dropout occurs in the actuator of the th iterative model at the sampling instant because of , then the entries in the column of are zero except that in the main diagonal which is 1. Furthermore, the entries in the column of one, and only one, of the matrices , for , namely, , are the components of the vector:The rest of the columns of are zero and the rest of the matrices with are zero.*Fact 2*. When there is a dropout in the sensor of the th iterative model at the sampling instant because of , then the entries in the row of are zero except that in the main diagonal which is 1. Furthermore, the entries in the row of one, and only one, of the matrices , for , namely, , are the components of the vector:The rest of the rows of are zero and the rest of the matrices with are zero.*Fact 3*. If there are dropouts in the th iterative model at several sampling instants, then each one of them implies a change in one of the matrix following the previous Facts 1 and 2 depending on the occurrence of the failure in the actuator or in the sensor.

If there is a dropout in the actuator of the th iterative model at the sampling instant because of and a dropout in its sensor at the sampling instant , that is, , then it follows from (16) thatfor one and one , where , and are built as it has been pointed out in the above Facts 1 and 2. In the case that , the matrix has the column and the row as it is described in Facts 1 and 2, respectively, and the rest of its entries are zero. By applying again the recursion (16) in (23), one obtains thatfor one and one , if the actuator of the th iterative model fails at the sampling instant because of and its sensor fails at the sampling instant ; that is, . By analyzing the entries of the main diagonal of the matrices in the five terms of the right hand side of (24), it follows thatwhere the fact that is such that has been taken into account. Moreover, has been used with , that is, the set of sampling instants at which the th iterative model suffers a dropout in its actuator or in its sensor. Note that with . Now, the recursion (16) in (24) is used. Four situations can occur, namely, (s1) and , (s2) and , (s3) but , and (s4) . In the situation (s1), one obtains thatif the actuator of the th iterative model fails at the sampling instant because of and its sensor fails at the sampling instant ; that is, . If or or or , then the entries of the main diagonal of the matrices in the five terms of the right hand side of (26) fulfill the fact thatwhere , , and has been used. Note also that in this situation (s1) if , , , and , thenwhere , , , , have been used. In the situation (s2) one obtains, by applying again the recursion (16) in (24), thatif the actuator of the th iterative model fails at the sampling instant because of and its sensor fails at the sampling instant ; that is, . If or or or , then the entries of the main diagonal of the matrices in the new terms of the right hand side of (29) fulfill the fact thatNote that in this situation (s2) the expressions in (30) are also valid if , , , and . Similar results to that of (29) and (30) can be obtained in the other situations (s3) and (s4). In the following the situation (s1) is analyzed since the results can be easily extended to the other cases. By applying the recursion (16) in (26) a finite number of times, namely, , one obtains thatBy iterating this process one can express the right hand side of (31) as a sum of terms where each of them is the product of a matrix and a vector where with defined byOne obtains, from (31) and (32), thatwhere is a sum of terms of the form , , , or plus the last two terms, namely, and . The vector is defined asThe entries of the main diagonal of the terms in have the following properties.

*Property 1. * where with being the number of iterative models included in the set which suffer a dropout at the sampling instant because of and/or for and . If means that there is at least one sampling instant within , namely, , at which all the iterative models included in the set fail in its actuator because of and/or its sensor; that is, , with at such a sampling instant. If , it means that there are not dropouts at any sampling instant in the iterative models included in . When the failures are distributed within the time interval and among the set of iterative models in , if a large enough integer is considered, then by taking into account that a small number of different iterative models included in present failures at the same sampling instant.

*Property 2. *All the terms of the form , , or have zeros in their main diagonal except those related to dropouts in the actuator because for all at the same sampling instant and those related to dropouts in the sensor because for all at the same sampling instant . Again, all of them can be made zero if a large enough integer is considered since a small number of different iterative models included in present failures at the same sampling instant.

*Property 3. *Finally, from (25), and .

By applying recursively (33), it follows thatwhere with and so on until with . Note that each , for , is an upper-triangular matrix since all its nonzero terms are upper-triangular. Moreover, its spectral radius, which is the maximum of its diagonal entries, can be made smaller than 1 if is large enough so that and if where and are defined, respectively, as and were defined for describing the features of the matrix . Then, as is deduced from (35) since when and from the fact that has been chosen so that . Thus, as