Convergence Analysis of Wireless Remote Iterative Learning Control Systems with Dropout Compensation

Abstract

The wireless remote iterative learning control (ILC) system with random data dropouts is considered. The data dropout is viewed as a binary switching sequence which obeys the Bernoulli distribution. In order to eliminate the effect of data dropouts on the convergence property of output error, the signal at the same time with the lost one but in the last iteration is used to compensate the data dropout at the actuator. With the dropout compensation, the convergence property of output error is analyzed by studying the element values of system transition matrix. Finally, some simulation results are given to illustrate the validity of the proposed method.

1. Introduction

One feature of the wireless remote control system is that signals are transmitted in wireless network from the sensor to the controller and from the controller to the actuator, and then the controller is separated from the system platform [1–3]. Consequently, the system has such advantages as easy installation, reduced wiring, and maintenance. In wireless remote control systems, designing a remote controller to track the desired trajectory is not an easy task. Fortunately, when systems execute the same task periodically in a fixed time interval, iterative learning control (ILC) is an effective method [4]. This method uses information obtained from the previous operation to improve the control signal for the next trial. If some conditions are met, the output error could monotonically converge to zero from iteration to iteration.

However, the introduction of wireless network makes the tracking of desired trajectory more complicated than traditional point-to-point connection method due to the unreliability of wireless network. The main issue is that some signals would be lost during transmitting, which can be divided into two different types: control signal dropouts and measurement signal dropouts. The first one occurs when the signal is transmitted from the controller to the actuator after updating, which would disturb the learning process indirectly; the second one arises when the signal is transmitted from the sensor to the controller, which would be involved in the learning process directly. The convergence of output error cannot be guaranteed without considering data dropouts.

The research on networked control systems has attracted much attention. In [5–9], authors are concerned with the state estimation problem for networked control systems with various uncertainties of the communication channels. In [10, 11], the author discussed decentralized stabilization of networked control systems with nonlinear perturbations. In [12], the nonfragile state feedback 𝐻∞ control problem for networked control systems with quantized signals is studied. However, most researches related to the case with data dropouts are conducted only in sensor-to-controller side and did not consider the control scheme adopted by controller. In the very recent years, some researches on ILC systems considering the problem of data dropout appeared (see [13–21] and references therein). Liu et al. investigated the implementation of ILC in a remote control systems environment and specifically focused on compensation when both random data dropouts and delays occur at the communication network between the plant output and the controller [13]. In [14], the author proposed an averaging ILC algorithm to overcome the random transport delay and data dropout, which guarantees the convergence property of the ensemble average of the output tracking errors along the iteration axis. In [15], Bu and Hou discussed the stability of ILC with data dropouts via asynchronous system and offered the stability condition in the form of linear matrix inequalities. Bu et al. also studied the stability of first and high order ILC with data dropout when the plant is subject to measurement signal dropouts [16]. Ahn et al. presented a mathematical formulation of the problem of robust ILC design when the system is subject to measurement signal dropouts and used the Kalman filtering approach to design a learning gain such that the system eventually converges to a desired trajectory if there are not complete data dropouts [17], but the results were restricted to the case when the network from sensor to controller has dependency. Aiming at this problem, the author considered discrete-time intermittent iterative learning controller with independent data dropouts in his further study [18]. However, all the previously mentioned researches only considered the dropout of measurement signals. In [19], the author considered the problem of ILC for a class of nonlinear systems with control signal dropouts and measurement signal dropouts, but the convergence analysis needs controller and actuator to know the received signal whether lost or not. In [20], a sampled-data ILC approach was proposed for a class of nonlinear networked control systems to deal with the existence of time delays and packet losses in control signal and measurement signal transmissions. In [21], Pan et al. assumed that the controller and the actuator are all event driven and analyzed the effect of packet loss from controller to actuator side and from sensor to controller side on the convergence property of output error. But the convergence is asymptotic because the control error at the actuator cannot converge to zero. To the best of our knowledge, no one has studied the compensation for the data dropouts in both control signals and measurement signals to guarantee the convergence property of output error. This observation is the motivation of the present paper.

As depicted in the second paragraph, there are two different kinds of data dropouts in wireless remote ILC systems. For the sake of convenience, we only consider the control signal dropouts in this paper, but the results can be extended to the measurement signal dropouts. The data dropout would be described as a binary sequence which obeys a Bernoulli distribution taking the value of one and zero with certain probability. In order to eliminate the effect of data dropouts on the convergence property of output error, we consider using the signal at the same time with the lost one but in the last iteration to compensate the data dropout at the actuator. The convergence performance of the output error with dropout compensation would be analyzed by studying eigenvalues and other elements in the lower triangular of the system transition matrix.

2. Problem Formulations

The discrete-time linear system is considered by the following equation: 𝐱_𝑘(𝑡+1)=𝐀𝐱_𝑘(𝑡)+𝐁𝐮_𝑘(𝑡),𝐲_𝑘(𝑡)=𝐂𝐱_𝑘(𝑡), (1) where 𝐱_𝑘(𝑡), 𝐮_𝑘(𝑡), and 𝐲_𝑘(𝑡) are state, control, and output variables, respectively, and 𝐀, 𝐁, and 𝐂 are the system matrices. The subscript 𝑘=0,1,2,… is used to denote iteration and 𝑡∈[0,𝑇] is used to denote time. The system operates repeatedly in the iteration domain with the desired trajectory 𝐲_𝑑(𝑡), which can be of the form 𝐱_𝑑(𝑡+1)=𝐀𝐱_𝑑(𝑡)+𝐁𝐮_𝑑(𝑡),𝐲_𝑑(𝑡)=𝐂𝐱_𝑑(𝑡), (2) where 𝐮_𝑑(𝑡) and 𝐱_𝑑(𝑡) are desired control and state. In order to track 𝐲_𝑑(𝑡) precisely, adopting ILC method at the controller is a useful method and the first order ILC can be expressed as 𝐮_𝑘+1(𝑡)=𝐮_𝑘(𝑡)+Γ(𝑡)𝐞_𝑘(𝑡+1), (3) where Γ(𝑡) is learning gain, and output error 𝐞_𝑘(𝑡)=𝐲_𝑑(𝑡)−𝐲_𝑘(𝑡),  𝑡∈[0,𝑇−1]. The setup of wireless remote ILC systems is illustrated as in Figure 1 [22, 23]. During the measurement signal 𝐞_𝑘(𝑡) transmitting from the sensor to the controller and the control signal 𝐮_𝑘(𝑡) transmitting from the controller to the actuator, data dropouts occur due to the unreliability of wireless network. Taking the effect of data dropouts into account, the system controlled remotely by iterative learning controller can be represented as 𝐱_𝑘(𝑡+1)=𝐀𝐱_𝑘(𝑡)+𝐁𝐮_𝑘(𝑡),𝐲_𝑘(𝑡)=𝐂𝐱_𝑘(𝑡),𝐮_𝑘+1(𝑡)=𝐮_𝑘(𝑡)+Γ(𝑡)𝐞̃_𝑘(𝑡+1), (4)(5) where control signal 𝐮_𝑘(𝑡) received at the actuator and measurement signal 𝐞̃_𝑘(𝑡+1) received at the controller can be expressed as 𝐮_𝑘(𝑡)=𝜉_𝑘(𝑡)𝐮_𝑘(𝑡),𝐞̃_𝑘(𝑡+1)=𝜂_𝑘(𝑡)𝐞_𝑘(𝑡+1), (6) where 𝐮_𝑘(𝑡) and 𝐞_𝑘(𝑡+1) are signals at the controller and the sensor, respectively. The stochastic parameters 𝜉_𝑘(𝑡) and 𝜂_𝑘(𝑡) are two scalar Bernoulli distributed random variables taking value 0 or 1 (i.e., 𝜉_𝑘(𝑡),𝜂_𝑘(𝑡)∈{0,1},∀𝑘,𝑡). 𝜉_𝑘(𝑡) is uncorrelated with 𝜂_𝑘(𝑡). In this model, if the variable takes value 0, then the data is lost correspondingly; otherwise there is no data dropout.

Figure 1 

Block diagram of wireless remote ILC systems.

Random data dropouts could disturb the learning process of controller and have an effect on the convergence performance of output error. In order to remove the effect of data dropouts, a compensation method would be proposed in Section 3.

3. The Data Dropout Compensation Scheme

In this section, we will derive the relation between output error and control error at the actuator and the relation of control errors at the controller in different iterations first, and then we will propose a compensation scheme based on the relations and attribute of control error. In order to simplify the analysis, the following assumptions are made

Assumption 1. Consider  ‖Γ(𝑡)‖≤𝛽_Γ,‖𝐀‖≤𝛽_𝐴,‖𝐁‖≤𝛽_𝐵,‖𝐂‖≤𝛽_𝐶.

Assumption 2. Consider 𝐱_𝑘(0)=𝐱_𝑑(0),for all 𝑘.
The output error 𝐞_𝑘(𝑡+1) can be represented as 𝐞_𝑘(𝑡+1)=𝐲_𝑑(𝑡+1)−𝐲_𝑘(𝑡+1)=𝐂𝛿𝐱_𝑘(𝑡+1). (7) From (2) and (4), it is easy to find the nonrecursive solution to 𝛿𝐱_𝑘(𝑡+1) as 𝛿𝐱_𝑘(𝑡+1)=𝐀𝛿𝐱_𝑘(𝑡)+𝐁𝛿𝐮_𝑘(𝑡)=𝐀^𝑡+1𝛿𝐱_𝑘(0)+^𝑡_𝑖=0𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘(𝑖), (8) where 𝛿𝐮_𝑘(𝑖) denotes the control error at the actuator. If Assumption 2 is satisfied, substitute (8) into (7) to get 𝐞_𝑘(𝑡+1)=^𝑡_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘(𝑖). (9) Take norms on both sides of (9) to yield 𝐞_𝑘(𝑡+1)=^𝑡_𝑖=0𝛽_𝐶𝛽_𝐴^𝑡−𝑖𝛽_𝐵𝛿𝐮_𝑘(𝑖). (10)
From (10), the relation between output error 𝐞_𝑘(𝑖),𝑖∈[1,𝑇], and control error 𝛿𝐮_𝑘(𝑖),𝑖∈[0,𝑇−1], at the actuator can be expressed in the matrix form 𝐞_𝑘(1⋮𝐞_𝑘)(𝑇=𝜌)^′(0)𝜌⋯0⋮⋱⋮_′(𝑇−1)⋯𝜌^′(0)×𝛿𝐮_𝑘(0⋮𝛿𝐮_𝑘)(,𝑇−1) (11) where 𝜌^′(𝑖)=𝛽_𝐶𝛽^𝑖_𝐴𝛽_𝐵,𝑖∈[0,𝑇−1].
On the other hand, the relation of control errors at the controller in different iterations can be get from (5) as follows: 𝛿𝐮_𝑘+1(𝑡)=𝐮_𝑑(𝑡)−𝐮_𝑘+1(𝑡)=𝐮_𝑑(𝑡)−𝐮_𝑘(𝑡)−Γ(𝑡)𝐞̃_𝑘(𝑡+1)=𝛿𝐮_𝑘(𝑡)−Γ(𝑡)𝐞̃_𝑘(𝑡+1). (12)
If there are no signals happen data dropouts, 𝐞̃_𝑘(𝑡+1)=𝐞_𝑘(𝑡+1) and 𝐮_𝑘(𝑡)=𝐮_𝑘(𝑡). Substituting (9) into (12), we have 𝛿𝐮_𝑘+1(𝑡)=𝛿𝐮_𝑘(𝑡)−Γ(𝑡)^𝑡_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘(𝑖). (13) Taking norms of both sides of (13), the relation can be rewritten as 𝛿𝐮_𝑘+1(𝑡)≤𝐇_𝑘+1,𝑘(𝑡)𝐇_{𝑘,𝑘−1}(𝑡)𝝍_𝑘−1(𝑡),𝑡∈[0,𝑇−1], (14) where the vector of control error 𝝍_𝑘−1, 𝐇_𝑘+1,𝑘(𝑡), and transition matrix 𝐇_{𝑘,𝑘−1}(𝑡) can be represented as (15), (16), and (17), respectively 𝝍_𝑘−1(𝑡)=𝛿𝐮_𝑘−1(0),𝛿𝐮_𝑘−1(1),…,𝛿𝐮_𝑘−1(𝑡)^𝑇,𝐇_𝑘+1,𝑘(𝑡)=𝜌₁(𝑡)⋯𝜌₁(1)𝜌₀(𝑡)𝐇_{𝑘,𝑘−1},(𝑡)=𝜌₀(0)𝜌0⋯0₁(1)𝜌₀(1)𝜌⋱⋮⋮⋱⋱0₁(𝑡)⋯𝜌₁(1)𝜌₀(𝑡), (15)(16)(17) where 𝜌₁(𝑖)=𝛽_Γ𝛽_𝐶𝛽^𝑖_𝐴𝛽_𝐵 and 𝜌₀(𝑖)=‖𝐈−Γ(𝑖)𝐂𝐁‖,𝑖∈[0,𝑡].
If 𝐞̃_𝑘(𝑡+1)=𝐞_𝑘(𝑡+1) and 𝐮_𝑘(𝑡)=𝐮_𝑘(𝑡) are satisfied, the convergence of ‖𝐞_𝑘(𝑡+1)‖ is determined by ‖𝛿𝐮_𝑘(𝑖)‖,𝑖∈[0,𝑡], and the convergence of ‖𝛿𝐮_𝑘(𝑡)‖ is determined by the eigenvalues of transition matrix. From (14), we have ‖𝛿𝐮_𝑘+1(𝑡)‖≤𝐇_𝑘+1,𝑘(𝑡)∏^𝑘_𝑖=0𝐇_𝑖+1,𝑖(𝑡)𝝍₀(𝑡). Since 𝐇_𝑖+1,𝑖(𝑡) is a lower triangular matrix, all of its eigenvalues are the diagonal elements 𝜌₀(𝑖). Selecting the learning gain Γ(𝑡) such that all eigenvalues of 𝐇_𝑖+1,𝑖(𝑡) are within unit circle; that is, 𝜌₀(𝑖)=‖𝚰−Γ(𝑖)𝐂𝐁‖<1, lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑡)‖=0,𝑡∈[0,𝑇−1], and then lim_𝑘→∞‖𝐞_𝑘(𝑡)‖=0,for all 𝑡∈[1,𝑇], is guaranteed. However, with the introduction of random data dropouts in control signals, 𝐮_𝑘(𝑡)≠𝐮_𝑘(𝑡), the element values of 𝐇_𝑘+1,𝑘(𝑡) and 𝐇_𝑖+1,𝑖(𝑡) would be changed, so the convergence of ‖𝛿𝐮_𝑘(𝑡)‖ cannot be guaranteed and then they have an effect on the convergence performance of output error ‖𝐞_𝑘(𝑡+1)‖.
In order to remove the effect, we propose to use the control signal at same time with the lost one but in the last iteration to compensate the data dropout at the actuator based on the attribute that control signal converges in the iteration domain, which can be represented by 𝐮_𝑘(𝑡)=𝜉_𝑘(𝑡)𝐮_𝑘(𝑡)+1−𝜉_𝑘(𝑡)𝐮_𝑘−1(𝑡). (18) The convergence of output error with the proposed compensation scheme would be analyzed in the next section.

4. Convergence Analysis of Output Error with Data Dropout Compensation

In the following part, we start by assuming that there is one control signal 𝐮_𝑘(𝑡) loss during the 𝑘th iteration and that 𝐮_𝑘−1(𝑡) is applied to compensate the lost 𝐮_𝑘(𝑡) at the actuator, and then we extend it to the condition with multiple control signals loss. The convergence property of output error with data dropout compensation can be concluded in the following theorem.

Theorem 1. With the compensation method presented in (18), if the condition 0<𝜌(Γ(𝑡)𝐂𝐁)<1 is satisfied, then lim_𝑘→∞𝐞_𝑘(𝑡)=0,∀𝑡∈[1,𝑇]. (19)

Proof. If the control signal 𝐮_𝑘(𝑡) is lost, 𝐮_𝑘−1(𝑡) would be applied at the actuator to compensate the dropout of 𝐮_𝑘(𝑡). In this condition, output errors 𝐞_𝑘(𝑖),(0≤𝑖≤𝑡) are not affected but 𝐞_𝑘(𝑖), (𝑡+1≤𝑖≤𝑇) are all changed according to (11). Consequently, the proof can be divided into the following two parts due to changing 𝐞_𝑘(𝑡+1), 𝐞_𝑘(𝑡+2) and afterwards.
(1)  Convergence Analysis of ‖𝐞_𝑘(𝑡+1)‖  with Dropout Compensation.
According to 𝐞_𝑘(𝑡)=𝐲_𝑑(𝑡)−𝐲_𝑘(𝑡), the output error 𝐞_𝑘(𝑡+1) can be represented as 𝐞_𝑘(𝑡+1)=𝐲_𝑑(𝑡+1)−𝐲_𝑘(𝑡+1)=𝐂𝛿𝐱_𝑘(𝑡+1). (20) Using (2), (4), and Assumption 2, the relation between state error 𝛿𝐱_𝑘(𝑡) and control error 𝛿𝐮_𝑘(𝑡) can be expressed as 𝛿𝐱_𝑘(𝑡+1)=𝐱_𝑑(𝑡+1)−𝐱_𝑘(𝑡+1)=𝐀𝛿𝐱_𝑘(𝑡)+𝐁𝜉_𝑘(𝑡)𝛿𝐮_𝑘(𝑡)+1−𝜉_𝑘(𝑡)𝛿𝐮_𝑘−1(𝑡)=^𝑡−1_𝑖=0𝐀^𝑡−𝑖𝐁𝜉_𝑘(𝑖)𝛿𝐮_𝑘(𝑖)+1−𝜉_𝑘(𝑖)𝛿𝐮_𝑘−1(𝑖). (21) From (5), (20), (21), and 𝐞̃_𝑘(𝑡+1)=𝐞_𝑘(𝑡+1), the relation of control errors in different iterations can be represented as 𝛿𝐮_𝑘+1(𝑡)=𝐮_𝑑(𝑡)−𝐮_𝑘+1(𝑡)=𝐮_𝑑(𝑡)−𝐮_𝑘(𝑡)−Γ(𝑡)𝐞̃_𝑘(𝑡+1)=𝛿𝐮_𝑘(𝑡)−Γ(𝑡)𝐂𝛿𝐱_𝑘(𝑡+1)=𝛿𝐮_𝑘(𝑡)(𝑡)×−Γ_𝑡𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝜉_𝑘(𝑖)𝛿𝐮_𝑘(𝑖)+1−𝜉_𝑘(𝑖)𝛿𝐮_𝑘−1(𝑖). (22) If the control signal 𝐮_𝑘(𝑡) is lost, the random parameter 𝜉_𝑘(𝑡)=0 correspondingly, then (22) could be changed to 𝛿𝐮_𝑘+1(𝑡)=𝛿𝐮_𝑘(𝑡)−Γ(𝑡)𝐂𝐁𝛿𝐮_𝑘−1(𝑡)−Γ(𝑡)^𝑡−1_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘(𝑖). (23) We can show 𝛿𝐮_𝑘+1(𝑡) using control error in (𝑘−1)st iteration further, which is given by 𝛿𝐮_𝑘+1(𝑡)=(𝐈−2Γ(𝑡)𝐂𝐁)𝛿𝐮_𝑘−1(𝑡)−Γ(𝑡)^𝑡−1_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘−1(𝑖)−Γ(𝑡)^𝑡−1_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘−1(𝑖)−Γ(𝑖)^𝑖_𝑗=0𝐂𝐀^𝑖−𝑗𝐁𝛿𝐮_𝑘−1𝑗. (24) Furthermore, (24) could be rewritten in the form of transition matrix𝛿𝐮_𝑘+1(𝑡)=^{Γ(𝑡)𝐂𝐀𝑡𝐁⋮⋮}_{𝐈Γ(𝑡)𝐂𝐀𝐁}^𝑇(0)Γ(1)_Γ()^{𝐈−Γ𝐂𝐁0⋯⋯0𝐂𝐀𝐁⋱⋱⋮⋮⋮⋱⋱⋮}_{𝑡−1𝐂𝐀𝑡−1()()Γ(𝑡)𝐁⋯Γ𝑡−1𝐂𝐀𝐁𝐈−Γ𝑡−1𝐂𝐁0𝐂𝐀𝑡(𝑡)(𝑡)𝐁⋯⋯Γ𝐂𝐀𝐁𝐈−2Γ𝐂𝐁}^{𝛿𝐮𝑘−1(0)⋮⋮}_{⋮𝛿𝐮𝑘−1(𝑡)}. (25)Taking norms of both sides of (25), we have 𝛿𝐮_𝑘+1(𝑡)≤𝐇^′_𝑘+1,𝑘(𝑡)𝐇^′_{𝑘,𝑘−1}(𝑡)𝝍_𝑘−1(𝑡),𝐇^′_𝑘+1,𝑘(𝑡)=𝜌₁(𝑡)⋯𝜌₁(1)1,𝐇^′_{𝑘,𝑘−1}(𝑡)=𝜌₀(0)𝜌0⋯⋯0₁(1)𝜌⋱⋱⋮⋮⋮⋱⋱⋮₁(𝑡−1)⋯𝜌₁(1)𝜌₀()0𝜌𝑡−1₁(𝑡)⋯⋯𝜌₁(1)𝜌₂, (26) where 𝜌₂=‖𝐈−2Γ(𝑡)𝐂𝐁‖.
Clearly, 𝐇_𝑘+1,𝑘(𝑡) in (16) becomes 𝐇^′_𝑘+1,𝑘(𝑡) in which the element in (𝑡+1)st column is changed from 𝜌₀(𝑡) to 1, and what is more is that transition matrix 𝐇^′_{𝑘,𝑘−1}(𝑡) in (17) becomes 𝐇^′_{𝑘,𝑘−1}(𝑡) in which the eigenvalue in (𝑡+1)st row is changed from 𝜌₀(𝑡) to 𝜌₂. If learning gain Γ(𝑡) is selected to satisfy the condition 0<𝜌(Γ(𝑡)𝐂𝐁)<1, then 0<1−𝜌(Γ(𝑡)𝐂𝐁)<1 and −1<1−2𝜌(Γ(𝑡)𝐂𝐁)<0 are also satisfied, which indicate that ‖𝚰−Γ(𝑡)𝐂𝐁‖<1 and ‖𝚰−2Γ(𝑡)𝐂𝐁‖<1. Since 𝐇^′_{𝑘,𝑘−1}(𝑡) is a lower triangular matrix and all of its eigenvalues are the diagonal elements 𝜌₀(𝑖), 0≤𝑖≤𝑡−1 and 𝜌₂, we have 𝜆_𝑚[(∏^𝑘−2_𝑖=0𝐇_𝑖+1,𝑖(𝑡))𝐇^′_{𝑘,𝑘−1}(𝑡)(∏^∞_𝑖=𝑘𝐇_𝑖+1,𝑖(𝑡))]→0,𝑚=0,…,𝑡+1, which guarantees that lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑡)‖=0.
According to the previous observation, control signals lost at time 𝑡 in each iteration will result in more “𝜌₂” in the diagonal elements of transition matrix 𝐇^′_𝑖+1,𝑖(𝑡), 0≤𝑖≤∞. As the structure of 𝐇^′_𝑖+1,𝑖(𝑡) remains the lower triangular form, the structure of the serial product of transition matrix 𝐇^′_𝑖+1,𝑖(𝑡) stays the lower triangular form as well. As a result, we can conclude that 𝜆_𝑚[∏^∞_𝑖=0𝐇^′_𝑖+1,𝑖(𝑡)]→0, 𝑚=0,1,…𝑡, which indicates lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑡)‖=0.
Afterwards, we should highlight that if 𝐮_𝑘(𝑡) happens data dropout and 𝐮_𝑘−1(𝑡) is used the compensate the lost 𝐮_𝑘(𝑡), the control signal 𝐮_𝑘(𝑡) at the actuator could also converge as the convergence of control signal 𝐮_𝑘(𝑡); that is, lim_𝑘→∞‖𝛿𝐮_𝑘(𝑡)‖=lim_𝑘→∞‖𝐮_𝑑(𝑡)−𝐮_𝑘−1(𝑡)‖=0 because lim_𝑘→∞𝐮_𝑘(𝑡)=lim_𝑘→∞𝐮_𝑘−1(𝑡). In other words, the control signal not only at the controller but also at the actuator converge as iteration goes on. Furthermore, we know that the output error ‖𝐞_𝑘(𝑡+1)‖ is a function of control errors ‖𝛿𝐮_𝑘(𝑖)‖, 𝑖∈[0,𝑡] from (11), so the convergence of ‖𝛿𝐮_𝑘(𝑡)‖ indicates lim_𝑘→∞‖𝐞_𝑘(𝑡+1)‖=0.
(2)  Convergence Analysis of ‖𝐞_𝑘(𝑡+1+𝑖)‖, 1≤𝑖≤𝑇−𝑡−1  with Dropout Compensation.
In this part, we continue to analyze the convergence of ‖𝐞_𝑘(𝑡+1+𝑖)‖, 1≤𝑖≤𝑇−𝑡−1. Due to the similarity in the analysis process, we would prove the convergence of ‖𝐞_𝑘(𝑡+2)‖ as the previous part and prove the convergence of ‖𝐞_𝑘(𝑡+1+𝑖)‖, 2≤𝑖≤𝑇−𝑡−1 by analogy.
Similarly, the output error 𝐞_𝑘(𝑡+2) can be represented as 𝐞_𝑘(𝑡+2)=𝐲_𝑑(𝑡+2)−𝐲_𝑘(𝑡+2)=𝐂𝛿𝐱_𝑘(𝑡+2)=𝐂^𝑡+1_𝑖=0𝐀^{𝑡+1−𝑖}𝐁𝜉_𝑘(𝑖)𝛿𝐮_𝑘(𝑖)+1−𝜉_𝑘(𝑖)𝛿𝐮_𝑘−1(𝑖). (27) According to (5), (27), and 𝐞̃_𝑘(𝑡+2)=𝐞_𝑘(𝑡+2), 𝛿𝐮_𝑘+1(𝑡+1) can be expressed as 𝛿𝐮_𝑘+1(𝑡+1)=𝐮_𝑑(𝑡+1)−𝐮_𝑘+1(𝑡+1)=𝐮_𝑑(𝑡+1)−𝐮_𝑘(𝑡+1)−Γ(𝑡+1)𝐞̃_𝑘(𝑡+2)=𝛿𝐮_𝑘(𝑡)−Γ(𝑡+1)𝐂𝛿𝐱_𝑘(𝑡+2)=𝛿𝐮_𝑘()()×𝑡+1−Γ𝑡+1_𝑡+1𝑖=0𝐂𝐀^{𝑡+1−𝑖}𝐁𝜉_𝑘(𝑖)𝛿𝐮_𝑘(𝑖)+1−𝜉_𝑘(𝑖)𝛿𝐮_𝑘−1(𝑖). (28) If the control signal 𝐮_𝑘(𝑡) is lost, 𝜉_𝑘(𝑡)=0 correspondingly, then (28) can be rewritten as 𝛿𝐮_𝑘+1()=(())𝑡+1𝐈−Γ𝑡+1𝐂𝐁𝛿𝐮_𝑘()(𝑡)𝑡+1−Γ𝐂𝐀𝐁𝛿𝐮_𝑘−1(𝑡)−Γ(𝑡)^𝑡−1_𝑖=0𝐂𝐀^𝑡−𝑖𝐁𝛿𝐮_𝑘(𝑖). (29) Expressing 𝛿𝐮_𝑘+1(𝑡) by means of control error in (𝑘−1)st iteration, we have 𝛿𝐮_𝑘+1()=(())×𝑡+1𝐈−Γ𝑡+1𝐂𝐁𝛿𝐮_𝑘−1(𝑡+1)−Γ(𝑡+1)^𝑡+1_𝑖=0𝐂𝐀^{𝑡+1−𝑖}𝐁𝛿𝐮_𝑘−1(𝑖)−Γ(𝑡+1)𝐂𝐀𝐁𝛿𝐮_𝑘−1(𝑡)()×−Γ𝑡+1_{𝑡−1𝑖=0}𝐂𝐀^{𝑡+1−𝑖}𝐁𝛿𝐮_𝑘−1(𝑖)−Γ(𝑖)^𝑖_𝑗=0𝐂𝐀^𝑖−𝑗𝐁𝛿𝐮_𝑘−1𝑗. (30)
Furthermore, (30) could be rewritten in the form of transition matrix𝛿𝐮_𝑘+1(𝑡+1)=Γ(𝑡+1)𝐂𝐀^𝑡+1𝐁⋮⋮Γ(𝑡+1)𝐂𝐀²𝐁Γ(𝑡+1)𝐂𝐀𝐁𝐈−Γ(𝑡+1)𝐂𝐁^𝑇×^(0)Γ(1)_Γ()^{𝐈−Γ𝐂𝐁0⋯⋯⋯0𝐂𝐀𝐁⋱⋱⋮⋮⋱⋱⋮}_{𝑡−1𝐂𝐀𝑡−1()()Γ()𝐁⋯Γ𝑡−1𝐂𝐀𝐁𝐈−Γ𝑡−1𝐂𝐁⋱⋮0⋯⋯0𝐈0𝑡+1𝐂𝐀𝑡+1()()𝐁⋯⋯⋯Γ𝑡+1𝐂𝐀𝐁𝐈−Γ𝑡+1𝐂𝐁}^{𝛿𝐮𝑘−1(0)⋮⋮}_{⋮⋮𝛿𝐮𝑘−1()𝑡+1}. (31)
Taking norms of both sides of (31), we have 𝛿𝐮_𝑘+1(𝑡+1)≤𝐇^′_𝑘+1,𝑘(𝑡+1)𝐇^′_{𝑘,𝑘−1}(𝑡+1)𝝍_𝑘−1(),𝐇𝑡+1_{′𝑘+1,𝑘}()=𝜌𝑡+1₁(𝑡+1)⋯𝜌₁(1)𝜌₀(),𝐇𝑡+1_{′𝑘,𝑘−1}()=𝜌𝑡+1₀(0)𝜌0⋯⋯⋯0₁(1)𝜌⋱⋱⋮⋮⋱⋱⋮₁(𝑡−1)⋯𝜌₁(1)𝜌₀()𝜌𝑡−1⋱⋮0⋯⋯010₁(𝑡+1)⋯⋯⋯𝜌₁(1)𝜌₀().𝑡+1 (32)
The analysis shows that 𝐇^′_𝑘+1,𝑘(𝑡+1) is equal to 𝐇_𝑘+1,𝑘(𝑡+1) in (16). 𝐇_{𝑘,𝑘−1}(𝑡+1) in (17) becomes 𝐇^′_{𝑘,𝑘−1}(𝑡+1) in which the eigenvalue in (𝑡+1)st row is changed from 𝜌₀(𝑡) to 1, and elements in the left of the eigenvalue are all changed to 0. Since 𝐇^′_{𝑘,𝑘−1}(𝑡+1) is a lower triangular matrix, and all of its eigenvalues are the diagonal elements with one element being “1” at (𝑡+1,𝑡+1) and other elements 𝜌₀(𝑖)<1,  𝑖∈[0,𝑡+1)∪(𝑡+1,𝑡+2]. Similar the proof into the previous part, we can have 𝜆_𝑚[(∏^𝑘−2_𝑖=0𝐇_𝑖+1,𝑖(𝑡+1))𝐇^′_{𝑘,𝑘−1}(𝑡+1)(∏^∞_𝑖=𝑘𝐇_𝑖+1,𝑖(𝑡+1))]→0, 𝑚=0,1,…,𝑡+2, and then lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑡+1)‖=0 can be guaranteed.
With regard to ‖𝛿𝐮_𝑘+1(𝑡+𝑖)‖, 2≤𝑖≤𝑇−𝑡−1, according to the convergence analysis of ‖𝛿𝐮_𝑘+1(𝑡+1)‖, we may know that the eigenvalue in (𝑡+𝑖)th row of 𝐇_{𝑘,𝑘−1}(𝑡+𝑖) would be changed to 1, and elements in the left of the eigenvalue would be all changed to 0. For this condition, we can also have 𝜆_𝑚((∏^𝑘−2_𝑖=0𝐇_𝑖+1,𝑖(𝑗))𝐇^′_{𝑘,𝑘−1}(𝑗)(∏^∞_𝑖=𝑘𝐇_𝑖+1,𝑖(𝑗)))→0, 𝑚∈[0,𝑗], 𝑗∈[𝑡+2,𝑇−1], and lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑖)‖=0, 𝑖∈[𝑡+2,𝑇−1], can also be guaranteed.
By analogy, if multiple control signals are lost at time 𝑡 in each iteration, which will result in more “1” in the diagonal elements of transition matrix, we also have 𝜆_𝑚(∏^∞_𝑖=0𝐇_𝑖+1,𝑖(𝑗))→0, 𝑚∈[0,𝑗], 𝑗∈[𝑡+1,𝑇−1], and then lim_𝑘→∞‖𝛿𝐮_𝑘+1(𝑖)‖=0, 𝑖∈[𝑡+1,𝑇−1].
Similarly, we can conclude that lim_𝑘→∞‖𝛿𝐮_𝑘(𝑖)‖=lim_𝑘→∞‖𝐮_𝑑(𝑖)−𝐮_𝑘−1(𝑖)‖=0, 𝑖∈[𝑡+2,𝑇−1] because lim_𝑘→∞𝐮_𝑘(𝑖)=lim_𝑘→∞𝐮_𝑘−1(𝑖). Because the output error ‖𝐞_𝑘(𝑡+1+𝑖)‖, 𝑖∈[1,𝑇−𝑡−1] is a function of the control error ‖𝛿𝐮_𝑘(𝑗)‖, 𝑗∈[0,𝑡+𝑖], according to (11), the convergence of ‖𝛿𝐮_𝑘(𝑗)‖, 𝑗∈[0,𝑡+𝑖], indicates lim_𝑘→∞‖𝐞_𝑘(𝑡+1+𝑖)‖, 𝑖∈[1,𝑇−𝑡−1].

Up to now, by compensating the data dropout with the control signal at the same time with the lost one but in the last iteration, the convergence property of output error has been proved. The effect of data dropout was also discussed in [21] by assuming that the actuator is event driven. That is, when there is one signal loss at 𝑡 during the 𝑘th iteration, the signal sent at 𝑡−1 will continue to be applied in the system before the signal sent at 𝑡+1 arrives. If 𝐮_𝑘(𝑡) is lost, this method can guarantee the control error ‖𝛿𝐮_𝑘(𝑡)‖=0, 𝑡∈[0,𝑇−1], at the controller converge, but the control error 𝛿𝐮_𝑘(𝑡) at the actuator cannot converge as iteration goes on because lim_𝑘→∞‖𝛿𝐮_𝑘(𝑡)‖=lim_𝑘→∞‖𝐮_𝑑(𝑡)−𝐮_𝑘(𝑡−1)‖≠0. According to (11), the output error ‖𝐞_𝑘(𝑡)‖ cannot converge to zero, and the effect of random data dropouts increases with the increment of random data dropout probability.

5. Simulation Results

In this section, some simulation results are provided to illustrate the validity of the proposed method. Let us consider the following discrete-time system: 𝐱_𝑘(𝑡+1)=𝐱−0.50011.24−0.8700.870^𝑘(𝑡)+100𝐮_𝑘(𝑡),𝐲_𝑘(𝑡)=22.6−2.8𝐱_𝑘(𝑡). (33) The desired trajectory is 𝐲_𝑑(𝑡)8()=5sin𝑡−1𝑇. (34)

ILC method is described in (5). Initial state error 𝛿𝐱_𝑘(0) and initial control error 𝛿𝐮₀(𝑡) are 0, 𝑇=200. Γ(𝑡)=0.2, so 0<𝜌(Γ(𝑡)𝐂𝐁)=0.4<1 is satisfied. The mean of output errors ∑^𝑇_𝑡=1‖𝐞_𝑘(𝑡)‖ in each iteration is selected to show the convergence of output error. In each iteration, 𝜉_𝑘(𝑡) and 𝜂_𝑘(𝑡) take value 0 or 1 according to the data dropout rate correspondingly.

The mean of output errors without any compensation versus iterations is shown in Figure 2. From Figure 2, it is easy to find that the effect of control signal dropouts on the mean of output error increases as random data dropout rate increases. The mean of output errors compensated by the control signal in the same iteration with the lost one but at the last time is shown in Figure 3. The comparison of the simulation results in Figures 3 and 2 shows that although the effect of data dropouts is reduced obviously, the mean of output errors does not converge to zero because this method cannot guarantee control error at the actuator converge to zero. And what is more, the effect of data dropouts increases with the increment of dropout rate. The mean of output errors compensated by the control signal at the same time with the lost one but in the last iteration is shown in Figure 4. By comparing Figure 4 with Figures 2 and 3, it is clear that even though the dropout rate would affect the convergence speed of the output error mean, which can still converge to zero from iteration to iteration with different dropout rates.

Figure 2 

Without compensation.

Figure 3 

Compensated by the signal in the same iteration with the lost one but at the last time.

Figure 4 

Compensated by the signal at the same time with the lost one but in the last iteration.

6. Conclusions

In order to remove the effect of data dropouts on the wireless remote ILC systems, a compensation method using the signal at the same time with the lost one but in last iteration is proposed to guarantee the convergence property of output error. The method can not only guarantee the convergence of control error at the controller, but also guarantee the convergence of control error at the actuator. By modeling the random data dropouts as a Bernoulli distributed sequence, the convergence property of control error at the controller is proved through studying the element values of system transition matrix, and then the convergence of control error at the actuator and output error is analyzed. Finally, the method is supported by some simulation results.

In wireless remote ILC systems, channel noises and time delays are other two main issues. In the future, we should consider the effects of data dropouts, channel noises, and time delays simultaneously. Besides, in order to improve the tracking performance further, we may select the learning gain adaptively according to such optimization criterions as least mean square and so on.