Iterative Learning Control for Remote Control Systems with Communication Delay and Data Dropout

Abstract

Iterative learning control (ILC) is applied to remote control systems in which communication channels from the plant to the controller are subject to random data dropout and communication delay. Through analysis, it is shown that ILC can achieve asymptotical convergence along the iteration axis, as far as the probabilities of the data dropout and communication delay are known a priori. Owing to the essence of feedforward-based control ILC can perform trajectory-tracking tasks while both the data-dropout and the one-step delay phenomena are taken into consideration. Theoretical analysis and simulations validate the effectiveness of the ILC algorithm for network-based control tasks.

1. Introduction

Iterative leaning control (ILC) is a control method that achieves perfect trajectory tracking when the system operates repeatedly. ILC has made significant progresses over the past two decades [1–3] and covered a wide scope of research issues such as continuous-time nonlinear system control [4], discrete-time nonlinear system [5], the initial reset problem [6, 7], stochastic process control [8], state delays [9], and data dropout [10].

On the other hand, the research on networked control systems has attracted much attention [11, 12] over the past few years. In network control, two frequently encountered issues are data dropout and communication delays, which are causes of poor performance of remote control systems. A central research area in remote control systems is to evaluate and compensate data dropout and time-delay factors [13–16]. Since data dropout and delay are random and time varying by nature, the existing control methods for deterministic data dropout and communication delays cannot be directly applied. Significant research efforts have been made on the control problems for networked systems with random data dropout and communication delays that are modeled in various ways in terms of the probability and characteristics of sources and destinations, for instance [10, 17].

It is in general still an open research area in ILC when remote control systems problems are concerned, except for certain pioneer works that address linear systems associated with either random data dropout [10, 18] or random communication delays [17, 19–21]. This paper investigates the implementation of ILC in a remote control systems setting, specifically focusing on compensation when both random data dropout and delays occur at the communication channels between the plant output and the controller.

Since ILC is in principle a feedforward technique, it is possible to send the controller signal before the task is executed. This would not be possible for feedback-based control systems. Hence, the data dropout can be circumvented to certain extent by using network protocols that assure the delivery of data packets. Likewise, the large delay due to large data package can also be avoided when the package is used for repeated task executions, namely, in future executions. ILC task is carried out in a finite-time interval, hence the time-domain stability is not a concern. Thus, unlike most network control works that focus on the stability issue, ILC can be applied to address trajectory-tracking tasks and the learning convergence is achieved in the iteration domain.

On the other hand, the use of data in the feedforward fashion would require the temporal analysis and management of data packages as well as resending the missing data package, which may not be available in certain remote control systems tasks. In this work, we adopt an ILC scheme that uses pastcontrol signals, as well as the error signals that are perturbed by the data dropout and communication delay. The ILC law adopts classical D-type algorithm and a revised learning gain that takes into consideration the probabilities of both data-dropout and communication-delay factors. As a result, the output tracking errors can be made to converge along the iteration axis. The ILC scheme can be applied to linear discrete-time plants with trajectory-tracking tasks.

The paper is organized as below. Section 2 formulates the remote control systems problem. Sections 3 and 4 prove the convergence property of ILC for linear discrete-time plants. Section 5 presents a numerical examples.

Throughout the paper, the following notations are used. Let ℰ[⋅] be the expected value of a random variable, 𝒫[⋅] the probability of an event, ‖⋅‖₂ the Euclidean norm of a vector, and ‖⋅‖ the maximal singular value of a matrix. Let 𝐳(𝑡) is a discrete time signal with 𝑡∈{0,1,…,𝑇}. For any 𝑎>1 and any 𝜆>1, define ‖𝐳‖_(𝜆,𝑎)≜sup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡‖𝐳(𝑡)‖₂, (1.1) where [0,𝑇]={0,1,…,𝑇}.

2. Problem Formulation

Consider a deterministic discrete-time linear time-invariant dynamics system: 𝐱_𝑖(𝑡+1)=𝐴𝐱_𝑖(𝑡)+𝐵𝐮_𝑖(𝑡),𝐲_𝑖(𝑡)=𝐶𝐱_𝑖(𝑡), (2.1) where “𝑖” and “𝑡” denote the iteration index and discrete time, respectively. 𝐱_𝑖(𝑡)∈ℝ^𝑛, 𝐮_𝑖(𝑡)∈ℝ^𝑝, and 𝐲_𝑖(𝑡)∈ℝ^𝑟 for all 𝑡∈[0,𝑇] are system states, inputs, and outputs, respectively, at the 𝑖th iteration. 𝐴, 𝐵, and 𝐶 are constant matrices with appropriate dimensions.

The schematic diagram of the remote control systems under consideration is shown in Figure 1.

Figure 1 

The schematic diagram of the remote control system.

It should be noted that the open-loop system from the ILC input to the plant output is deterministic. The randomness occurs during the data transmission from the plant output to the ILC input. There are two approaches in analyzing the closed-loop system. The first approach is to treat the entire closed-loop system as a random or stochastic process. In such circumstances, the topology of the overall system keeps changing and the control process is either a Markovian jump process or a switching process. Another approach, which is adopted in this work, is to retain the essentially deterministic structure of the original open-loop system, meanwhile model the random data dropout and communication delay into two random factors with known probability distributions. As a consequence, the signals used in ILC, 𝐲_𝑖(𝑡) are the modulated plant output with the two random factors.

When the control process is deterministic, an effective ILC law for the linear system (2.1) is 𝐮_𝑖+1(𝑡)=𝐮_𝑖(𝑡)+𝐿𝐞_𝑖(𝑡+1), (2.2) where 𝐮_𝑖+1(𝑡) and 𝐮_𝑖(𝑡) are control inputs at the (𝑖+1)th and 𝑖th iterations, namely, the present trial and the previous trial, respectively. 𝐞_𝑖(𝑡+1)=𝐲_𝑑(𝑡+1)−𝐲_𝑖(𝑡+1) is the output tracking error at the time (𝑡+1)th time instance of the 𝑖th iteration. 𝐿 is a learning gain matrix.

Remark 2.1. Note that in the ILC law (2.2), the control signal of the present iteration, 𝐮_𝑖+1(𝑡), consists of both the pastcontrol input, 𝐮_𝑖(𝑡) and the past error with one-step temporal advance, 𝐞_𝑖(𝑡+1). The current-cycle feedback errors, such as 𝐞_𝑖+1(𝑡), are not used. Since ILC does not require the current-cycle feedback nor the temporal stability, it is an effective control method for remote control systems problems with random data dropout and communication delay.

To facilitate the ILC design and convergence analysis, data dropout and one-step communication delay are formulated. First formulate the data-dropout problem. Denote 𝛾(𝑡) a stochastic variable with Bernoulli distribution taking binary values 0 and 1, where 𝛾(𝑡)=0 denotes an occurrence of data dropout and 𝛾(𝑡)=1 denotes a normal data communication. The probabilities of 𝛾(𝑡) are 𝒫𝛾(𝑡)=1=ℰ𝛾(𝑡)=𝒫𝛾(𝑡)𝛾(𝑡𝛾,=0=1−ℰ)=1−𝛾, (2.3) where 𝛾>0 is a known constant. Here, we assume that 𝛾(𝑡) is a stationery stochastic process, thus the data dropout rate is independent of the time 𝑡. In subsequent derivations, we treat 𝛾 as time invariant.

When the data dropout occurs in multiple communication channels, we can similarly define ℰ[𝛾_𝑗]=𝛾_𝑗>0 for the 𝑗th communication channel. Thus, denote Γ=diag𝛾_𝑗=𝛾₁0⋯00𝛾₂⋯0⋮⋮⋱⋮00⋯𝛾_𝑟, (2.4) the corresponding mathematical expectation is ℰ[Γ]=Γ, (2.5) where Γ>0 is known a priori.

Due to the data dropout, the plant output received by the controller at the (𝑖+1)th iteration is Γ_𝑖+1𝐲_𝑖+1(𝑡). (2.6) Generally speaking, the occurrences of data dropouts at two iterations are uncorrelated, thus independent. On the other hand, ILC law at the current iteration, the (𝑖+1)th iteration, uses only signals of the previous iteration, namely, 𝑖th iteration, as shown in (2.2). Thus 𝐲_𝑖+1(𝑡) with the control input 𝐮_𝑖+1(𝑡) contains data dropouts upto the 𝑖th iteration. Therefore, Γ_𝑖+1 and 𝐲_𝑖+1(𝑡) are independent, that is, ℰΓ_𝑖+1𝐲_𝑖+1(𝑡)=ℰΓ_𝑖+1ℰ𝐲_𝑖+1(𝑡=)Γ_𝑖+1ℰ𝐲_𝑖+1(𝑡). (2.7) Without the loss of generality, we assume ℰ[Γ_𝑖]=Γ, namely, the data dropout rate is invariant at different iterations.

Next formulate the one-step communication delay problem. Denote 𝑤(𝑡) is a random delay factor with Bernoulli distribution, which takes binary values 0 and 1 that indicate, respectively, the presence and absence of an one-step communication delay. Here we assume that 𝑤(𝑡) is a stationery stochastic process, thus the occurrence of the communication delay is independent of the time 𝑡. In subsequent derivations we treat 𝑤 as time invariant. With multiple communication channels, we define matrix 𝑊=diag(𝑤_𝑗), where 𝑤_𝑗 denotes the occurrence of communication delay at the 𝑗th communication channel. Denote ℰ[𝑤]=𝑤 and ℰ[𝑊]=𝑊. The plant output received by ILC with possible communication delay is formulated by 𝐲^𝑜_𝑖(𝑡)=𝑊_𝑖𝐲_𝑖(𝑡)+1−𝑊_𝑖𝐲_𝑖(𝑡−1), (2.8) where 𝑊_𝑖 is the communication delay at the 𝑖th iteration. Without the loss of generality, we assume ℰ[𝑊_𝑖]=𝑊, namely, the probability of the communication delay is invariant at different iterations. Analogous to data dropout, assume that communication delay at any two iterations are independent, then 𝑊_𝑖+1 and 𝑊_𝑖 are independent, so are 𝑊_𝑖+1 and 𝐲_𝑖+1(𝑡) because 𝐲_𝑖+1(𝑡) contains communication delays upto the 𝑖th iteration through the ILC law (2.2).

It is worthwhile noting that stochastic variables 𝛾 and 𝑤 are not completely independent. A delayed or nondelayed communication occurs only when 𝛾=1, that is, no data dropout. Hence, we should have the condition probability for data transmission without delay Prob𝛾=1,𝑤=1=𝒫𝛾=1𝒫𝑤=1∣𝛾=1=𝛾𝑤, (2.9) and the condition probability for data transmission with one-step delay𝒫𝒫=𝛾=1,𝑤=0=𝒫𝛾=1𝑤=0∣𝛾=1=𝒫𝛾=11−𝒫𝑤=1∣𝛾=1𝛾(1−𝑤). (2.10) As a consequence, we haveℰ𝛾𝑤=𝛾𝑤. (2.11) The relationship between data drop out and communication delay, (2.11), can be extended to multiple channels at the 𝑖th iteration ℰΓ_𝑖𝑊_𝑖=Γ𝑊. (2.12)

At the 𝑖th iteration, the output signals perturbed by data dropout and one-step communication delay can be expressed as 𝐲_𝑖(𝑡)=Γ_𝑖𝐲^𝑜_𝑖(𝑡)=Γ_𝑖𝑊_𝑖𝐲_𝑖(𝑡)+𝐼−𝑊_𝑖𝐲_𝑖(𝑡−1), (2.13) where 𝐼 is a unity matrix of appropriate dimensions. The mathematical expectation of 𝐲_𝑖(𝑡) can be derived using the independence property between Γ_𝑖, 𝑊_𝑖, and 𝐲_𝑖, as well as the relationship  (2.12) ℰ𝐲_𝑖(𝑡)=ℰΓ_𝑖𝑊𝐲_𝑖(𝑡)+(𝐼−𝑊)𝐲_𝑖(=𝑡−1)Γ𝑊ℰ𝐲_𝑖(𝑡)+𝐼−𝑊ℰ𝐲_𝑖(𝑡−1). (2.14)

The objective of control design is to seek an appropriate ILC law that can take into consideration data dropout and communication delay concurrently. The following ILC law is adopted 𝐮_𝑖+1(𝑡)=𝐮_𝑖(𝑡)+𝐿𝑒_𝑖̃(𝑡+1), (2.15) where 𝐞̃_𝑖(𝑡+1)=Γ_𝑖𝐲_𝑑(𝑡+1)−𝐲_𝑖(𝑡+1)=Γ_𝑖𝐲_𝑑(𝑡+1)−𝑊_𝑖𝐲_𝑖(𝑡+1)−𝐼−𝑊_𝑖𝐲_𝑖(𝑡)=Γ_𝑖𝑊_𝑖𝐞_𝑖(𝑡+1)+𝐼−𝑊_𝑖𝐞_𝑖(𝑡)+𝐼−𝑊_𝑖𝜹(𝑡), (2.16) where 𝜹(𝑡)≜𝐲_𝑑(𝑡+1)−𝐲_𝑑(𝑡).

3. Convergence Analysis for Left-Invertible Systems 𝑟≥𝑝

In this section, we derive the convergence property of the ILC (2.15) in the presence of data dropout and communication delays.

In ILC, the learning convergence can be derived in terms of either the output tracking error, 𝐞_𝑖(𝑡), or the input tracking error, Δ𝐮_𝑖(𝑡). In this section, we prove the learning convergence property of Δ𝐮_𝑖(𝑡).

Assumption 3.1. For a given output reference trajectory 𝐲_𝑑(𝑡), which is realizable, there exists a unique desired control input 𝐮_𝑑(𝑡)∈ℝ^𝑝 such that𝐱_𝑑(𝑡+1)=𝐴𝐱_𝑑(𝑡)+𝐵𝐮_𝑑(𝑡),𝐲_𝑑(𝑡)=𝐶𝐱_𝑑(𝑡), (3.1) where 𝐮_𝑑(𝑡) is uniformly bounded for all 𝑡∈[0,𝑇]. It is assumed that for all𝑖∈ℤ₊,𝐱_𝑖(0) is a random variable with ℰ[𝐱_𝑖(0)]=𝐱₀=𝐱_𝑑(0).

Define the input and state errors Δ𝐮_𝑖+1≜𝐮_𝑑(𝑡)−𝐮_𝑖+1(𝑡),Δ𝐱_𝑖+1≜𝐱_𝑑(𝑡)−𝐱_𝑖+1(𝑡), (3.2) then from (2.1) and (3.1), we have Δ𝐱_𝑖(𝑡+1)=𝐴Δ𝐱_𝑖(𝑡)+𝐵Δ𝐮_𝑖(𝑡),𝐞_𝑖(𝑡)=𝐶Δ𝐱_𝑖(𝑡). (3.3)

From (2.15), using the relationship (2.12), we haveℰ𝐞̃_𝑖+1(𝑡+1)=ℰΓ_𝑖𝐲_𝑑(𝑡+1)−𝑊_𝑖𝐲_𝑖(𝑡+1)−𝐼−𝑊_𝑖𝐲_𝑖(𝑡Γ)=ℰ_𝑖𝑊_𝑖𝐞_𝑖(𝑡+1)+𝐼−𝑊_𝑖𝐞_𝑖(𝑡)+𝐼−𝑊_𝑖𝐲_𝑑(𝑡+1)−𝐲_𝑑(𝑡)=Γ𝑊ℰ𝐞_𝑖(𝑡+1)+𝐼−𝑊ℰ𝐞_𝑖(𝑡)+𝐼−𝑊ℰ[𝛿(𝑡)]. (3.4)

Theorem 3.2. Suppose that the update law (2.15) is applied to the networked control system and satisfied the Assumption 3.1. If there exist 𝜌 satisfying 𝐼_𝑝−𝐿Γ𝑊𝐶𝐵≤𝜌<1, (3.5) then the input error along the iteration axis, ℰ[Δ𝐮_𝑖(𝑡)], converges to a bound that is proportional to the factor 𝜹(𝑡).

Proof. First, subtracting 𝐮_𝑑(𝑡) from both sides of the ILC law (2.15) yields Δ𝐮_𝑖+1(𝑡)=Δ𝐮_𝑖(𝑡)−𝐿𝐞̃_𝑖(𝑡+1). (3.6) Applying the ensemble operator ℰ[⋅] to both sides of (3.6) and substituting the relationship (3.4) with 𝐞_𝑖(𝑡)=𝐶Δ𝐱_𝑖(𝑡), we obtain ℰΔ𝐮_𝑖+1(𝑡)=ℰΔ𝐮_𝑖(𝑡)−𝐿Γ𝑊𝐶ℰΔ𝐱_𝑖(𝑡+1)−𝐿Γ𝐼−𝑊𝐶ℰΔ𝐱_𝑖(𝑡)−𝐿Γ𝐼−𝑊𝜹(𝑡). (3.7) Substituting the state error dynamics (3.3) into (3.7) leads to the following relationship: ℰΔ𝐮_𝑖+1(𝑡)=𝐼_𝑝−𝐿Γ𝑊𝐶𝐵ℰΔ𝐮_𝑖(𝑡)−𝐿Γ𝑊𝐶𝐴+Γ𝐼−𝑊𝐶ℰΔ𝐱_𝑖(𝑡)−𝐿Γ𝐼−𝑊𝜹(𝑡). (3.8) Define 𝜌≜‖𝐼_𝑝−𝐿Γ𝑊𝐶𝐵‖.
Now let us handle the second term on the right hand side of (3.8), which is related to Δ𝐱_𝑖(𝑡). Applying the ensemble operation to the following relationship: Δ𝐱_𝑖(𝑡)=𝐴^𝑡𝐱_𝑑(0)−𝐱_𝑖(0)+^𝑡−1_𝑘=0𝐴^{𝑡−1−𝑘}𝐵Δ𝐮_𝑖(𝑘), (3.9)
Substituting the relation (3.9) into (3.8), taking the norm ‖⋅‖₂ on both sides, the following relationship is derived:ℰΔ𝐮_𝑖+1(𝑡)₂≤𝜌ℰΔ𝐮_𝑖(𝑡)₂+𝐿Γ𝑊𝐶𝐴+𝐿Γ𝐼−𝑊𝐶𝑎^𝑡𝐱_𝑑(0)−ℰ𝐱_𝑖(0)₂+𝐿Γ𝑊𝐶𝐴+𝐿Γ𝐼−𝑊𝐶‖𝐵‖^𝑡−1_𝑘=0𝑎^{𝑡−1−𝑘}ℰΔ𝐮_𝑖(𝑘)₂+𝐿Γ𝐼−𝑊‖𝜹(𝑡)‖₂, (3.10) where 𝑎≥‖𝐴‖ and in this work we choose 𝑎>1 if ‖𝐴‖≤1.
In order to handle the exponential term with 𝑎^𝑡 in (3.11), we introduce the 𝜆 norm. From Assumption 3.1, multiplying both sides of (3.10) by 𝑎^−𝜆𝑡 and taking the supermum over [0,𝑇] yieldsup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡ℰΔ𝐮_𝑖+1(𝑡)₂≤𝜌sup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡ℰΔ𝐮_𝑖(𝑡)₂+𝛽₁sup_{[]𝑡∈0,𝑇}𝑎^{−𝜆𝑡𝑡−1}_𝑘=0𝑎^{𝑡−1−𝑘}ℰΔ𝐮_𝑖(𝑘)₂+𝛽₂sup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡‖𝜹(𝑡)‖₂, (3.11) where 𝛽₁=‖𝐿Γ𝑊𝐶𝐴+𝐿Γ(𝐼−𝑊)𝐶‖‖𝐵‖, 𝛽₂=‖𝐿Γ(𝐼−𝑊)‖, and 𝜂_𝑖=‖𝐱_𝑑(0)−ℰ[𝐱_𝑖(0)]‖ that is independent of 𝑡. Since ‖𝜹(𝑡)‖ is bounded, so is 𝜇=sup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡‖𝜹(𝑡)‖₂. (3.12)
Substituting the properties of Lemma A.1 into (3.11) yields ℰΔ𝐮_{𝑖+1(𝜆,𝑎)}≤𝜌+𝛽₁1−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆ℰ−𝑎Δ𝐮_{𝑖(𝜆,𝑎)}+𝛽₂𝜇. (3.13)
Since 0≤𝜌<1, it is possible to choose 𝜆 sufficiently large such that 𝜌₁=𝜌+𝛽₁1−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆−𝑎<1. (3.14) Therefore we can rewrite (3.13) asℰΔ𝐮_{𝑖+1(𝜆,𝑎)}≤𝜌₁ℰΔ𝐮_{𝑖(𝜆,𝑎)}+𝛽₂𝜇, (3.15) which implies lim_𝑖→∞ℰΔ𝐮_{𝑖(𝜆,𝑎)}≤𝛽₂𝜇1−𝜌₁. (3.16)

Note that 𝜇 is proportional to 𝜹(𝑡), namely, the maximum difference between 𝐲_𝑑(𝑡+1)−𝐲_𝑑(𝑡) in 𝑡∈[0,𝑇], which is bounded and small when the reference trajectory is smooth or the sampling interval is sufficiently small. When the probability associated with the data communication delay, 𝑊, is known a priori, we can further revise the reference trajectory to an augmented one, such that the resulting 𝜹(𝑡)=0.

Corollary 3.3. Revising the original reference 𝐲_𝑑(𝑡) into an augmented one 𝐲^𝑎_𝑑(𝑡)=𝑊𝐲_𝑑(𝑡)+(𝐼−𝑊)𝐲_𝑑(𝑡−1), then 𝜹(𝑡)=0 and the ILC (2.15) ensures a zero-tracking error.

Proof. Note that ℰ[𝐲^𝑜_𝑖(𝑡)]=𝑊ℰ[𝐲_𝑖(𝑡)]+(𝐼−𝑊)ℰ[𝐲_𝑖(𝑡−1)]. Suppose that 𝐲_𝑖(𝑡)=𝐲_𝑑(𝑡), then the delay-perturbed output should be 𝑊ℰ[𝐲_𝑑(𝑡)]+(𝐼−𝑊)ℰ[𝐲_𝑑(𝑡−1)]. In other words, the augmented reference trajectory for 𝐲^𝑜_𝑖(𝑡+1) should be 𝐲^𝑎_𝑑(𝑡)=𝑊𝐲_𝑑(𝑡)+(𝐼−𝑊)𝐲_𝑑(𝑡−1). As a result, 𝐲^𝑜_𝑖(𝑡)=𝐲^𝑎_𝑑(𝑡) implies 𝐲_𝑖(𝑡)=𝐲_𝑑(𝑡). Now replacing 𝐲_𝑑(𝑡+1) in (2.16) with 𝐲^𝑎_𝑑(𝑡+1), we can derive 𝐞̃_𝑗(𝑡+1)=Γ_𝑗𝐲^𝑎_𝑑(𝑡+1)−𝐲^𝑜_𝑗(𝑡+1)=Γ_𝑗𝑊_𝑗𝐲_𝑑(𝑡+1)−𝑊_𝑗𝐲_𝑗(𝑡+1)+𝐼−𝑊_𝑗𝐲_𝑑(𝑡−1)−𝐼−𝑊_𝑗𝐲_𝑗(𝑡)=Γ_𝑗𝑊_𝑗𝐞_𝑗(𝑡+1)+Γ_𝑗𝐼−𝑊_𝑗𝐞_𝑗(𝑡). (3.17) Comparing the above expression with (2.16), we conclude that 𝜹(𝑡)=0, subsequently 𝜇=0, which implies a zero-tracking error according to (3.16).

4. Convergence Analysis for Right-Invertible Systems 𝑟≤𝑝

In this section, we prove the learning convergence property of 𝐞_𝑖(𝑡).

Assumption 4.1. (𝐼_𝑟−𝐶𝐵𝐿Γ𝑊)⁻¹ always exists.

Theorem 4.2. Suppose that the update law (2.15) is applied to the networked control system and satisfied the Assumption 4.1. If𝜌^′≜𝐼_𝑟−𝐶𝐵𝐿Γ𝑊<1, (4.1) then the tracking error along the iteration axis, ℰ[𝐞_𝑖(𝑡)], converges to a bound that is proportional to the factor 𝜹(𝑡).

Proof. First note the relationship: 𝐞_𝑖+1(𝑡+1)=𝐲_𝑑(𝑡+1)−𝐲_𝑖+1(𝑡+1)=𝐞_𝑖(𝑡+1)+𝐲_𝑖(𝑡+1)−𝐲_𝑖+1(𝑡+1)=𝐞_𝑖(𝑡+1)+𝐶𝐴𝐱_𝑖(𝑡)−𝐱_𝑖+1(𝑡)+𝐶𝐵𝐮_𝑖(𝑡)−𝐮_𝑖+1(𝑡,𝐱_𝑖)(𝑡)=𝐴^𝑡𝐱_𝑖(0)+^𝑡−1_𝑘=0𝐴^{𝑡−1−𝑘}𝐵𝐮_𝑖(𝑘). (4.2)(4.3) Substituting ILC law (2.15), (2.16), and (4.3) into (4.2) yields 𝐞_𝑖+1(𝑡+1)=𝐞_𝑖(𝑡+1)+𝐶𝐴^𝑡+1𝐱_𝑖(0)−𝐱_𝑖+1(0)−𝐶^𝑡−1_𝑘=0𝐴^𝑡−𝑘𝐵𝐿𝐞̃_𝑖(𝑘+1)−𝐶𝐵𝐿𝐞̃_𝑖(𝑡+1)=𝐞_𝑖(𝑡+1)+𝐶𝐴^𝑡+1𝐱_𝑖(0)−𝐱_𝑖+1(0)−𝐶^𝑡−1_𝑘=0𝐴^𝑡−𝑘𝐵𝐿Γ_𝑖𝑊_𝑖𝐞_𝑖(𝑘+1)+𝐼−𝑊_𝑖𝐞_𝑖(𝑘)+𝐼−𝑊_𝑖𝜹(𝑘)−𝐶𝐵𝐿Γ_𝑖𝑊_𝑖𝐞_𝑖(𝑡+1)+𝐼−𝑊_𝑖𝐞_𝑖(𝑡)+𝐼−𝑊_𝑖𝜹(𝑡=𝐼_𝑟)−𝐶𝐵𝐿Γ_𝑖𝑊_𝑖𝐞_𝑖(𝑡+1)+𝐶𝐴^𝑡+1𝐱_𝑖(0)−𝐱_𝑖+1(0)−𝐶^𝑡_𝑘=0𝐴^𝑡−𝑘𝐵𝐿Γ_𝑖𝑊_𝑖𝐞_𝑖(𝑘+1)−𝐶^𝑡−1_𝑘=0𝐴^{𝑡−𝑘−1}𝐵𝐿Γ_𝑖𝐼−𝑊_𝑖𝐞_𝑖(𝑘)−𝐶^𝑡_𝑘=0𝐴^𝑡−𝑘𝐵𝐿Γ_𝑖𝐼−𝑊_𝑖𝜹(𝑘). (4.4)Assumption 4.3. Assume ℰ[𝐱_𝑖+1(0)]−ℰ[𝐱_𝑖(0)]=0.Applying the ensemble operator ℰ[⋅] to both sides of (4.4) and substituting the relationship (4.2), we obtain ℰ𝐞_𝑖+1(𝑡+1)=𝐼_𝑟−𝐶𝐵𝐿Γ𝑊ℰ𝐞_𝑖(𝑡+1)−𝐶^𝑡−1_𝑘=0𝐴^{𝑡−𝑘−1}𝐵𝐿Γ𝑊ℰ𝐞_𝑖(𝑘+1)−𝐶^𝑡_𝑘=0𝐴^𝑡−𝑘𝐵𝐿Γ𝐼−𝑊ℰ𝐞_𝑖(𝑘)−𝐶^𝑡_𝑘=0𝐴^𝑡−𝑘𝐵𝐿Γ𝐼−𝑊ℰ[𝜹(𝑡)]. (4.5) Taking the norm ‖⋅‖₂ on both sides of (4.5), the following relationship is derived ℰ𝐞_𝑖+1(𝑡+1)₂≤𝐼_𝑟−𝐶𝐵𝐿Γ𝑊ℰ𝐞_𝑖(𝑡+1)₂+‖𝐶‖^𝑡−1_𝑘=0𝑎^{𝑡−𝑘−1}‖𝐵𝐿‖Γ𝑊ℰ[𝐞_𝑖(𝑘+1)]₂+‖𝐶‖^𝑡_𝑘=0𝑎^𝑡−𝑘‖𝐵𝐿‖Γ𝐼−𝑊ℰ𝐞_𝑖(𝑘)₂+‖𝐶‖^𝑡_𝑘=0𝑎^𝑡−𝑘‖𝐵𝐿‖Γ𝐼−𝑊‖ℰ[𝜹(𝑡)]‖₂, (4.6) where 𝑎≥‖𝐴‖ and in this work we choose 𝑎>1 if ‖𝐴‖≤1.
In order to handle the exponential term with 𝑎^𝑡 in (4.5), we introduce the 𝜆 norm. Multiplying both sides of (4.5) by 𝑎^−𝜆𝑡 and taking the supermum over [0,𝑇] yieldsup_{𝑡∈[0,𝑇]}𝑎^−𝜆𝑡ℰ𝐞_𝑖+1(𝑡)₂≤sup_{𝑡1∈[]0,𝑇}𝐼−𝐶𝐵𝐿Γ𝑊sup_{[]𝑡∈0,𝑇}𝑎^−𝜆𝑡ℰ𝐞_𝑖(𝑡)₂+‖𝐶‖sup_{[]𝑡∈0,𝑇}𝑎^{𝑡−𝜆𝑡}_𝑘=1𝑎^𝑡−𝑘‖𝐵𝐿‖Γ𝑊ℰ𝐞_𝑖(𝑘)₂+‖𝐶‖sup_{[]𝑡∈0,𝑇}𝑎^{−𝜆𝑡𝑡−1}_𝑘=0𝑎^{𝑡−𝑘−1}‖𝐵𝐿‖Γ𝐼−𝑊ℰ𝐞_𝑖(𝑘)₂+‖𝐶‖sup_{[]𝑡∈0,𝑇}𝑎^{𝑡−𝜆𝑡}_𝑘=0𝑎^𝑡−𝑘‖𝐵𝐿‖Γ𝐼−𝑊‖ℰ[𝜹(𝑡)]‖₂. (4.7) Substituting the properties of Lemma A.2 into (4.7) yields ℰ𝐞_{𝑖+1(𝜆,𝑎)}≤𝜌^′+𝛽₄1−𝑎^{−(𝜆−1)𝑇}𝑎^(𝜆−1)ℰ𝐞−1^{𝑖(𝜆,𝑎)}+𝛽₅𝜇, (4.8) where 𝛽₃≜‖𝐶‖‖𝐵𝐿‖Γ and 𝛽₄≜‖𝐶‖‖𝐵𝐿‖Γ(𝐼−𝑊)((1−𝑎^{−(𝜆−1)𝑇})/(𝑎^(𝜆−1)−1)).
Since 0≤𝜌^′<1, it is possible to choose 𝜆 sufficiently large such that𝜌₂=𝜌^′+𝛽₃1−𝑎^{−(𝜆−1)𝑇}𝑎^(𝜆−1)−1<1. (4.9) Therefore, we can rewrite (4.8) as ℰ𝐞_{𝑖+1(𝜆,𝑎)}≤𝜌₂ℰ𝐞_{𝑖(𝜆,𝑎)}+𝛽₄𝜇, (4.10) which implies lim_𝑖→∞ℰ𝐞_{𝑖(𝜆,𝑎)}≤𝛽₄𝜇1−𝜌₂. (4.11)

5. Numerical Examples

Consider the following linear discrete-time system: 𝐱_𝑖(𝑡+1)=𝐱0.50−0.251.000.150.30−0.50−0.750.25−0.25^𝑖(𝑡)+001𝑢_𝑖(𝑡),𝑦_𝑖(𝑡)=001.0𝐱_𝑖(𝑡), (5.1) with the initial condition 𝐱_𝑖(0)=0. The desired trajectory is 𝑦_𝑑(𝑡)=sin(2𝜋𝑡/50). The tracking period is {0,1,…,49}. The control profile of the first iteration is 𝑢₀(𝑡)=0 for 𝑡=0,1,…,49. Two sets of probabilities for the data dropout rate and communication delay are considered, which are 𝛾=0.9, 𝑤=0.9, 𝛾=0.6, and 𝑤=0.6, respectively. The learning gain is 𝐿=0.5, which yields ‖𝐼−𝐿𝛾𝑤𝐶𝐵‖=0.595<1, and ‖𝐼−𝐿𝛾𝑤𝐶𝐵‖=0.82<1 with respect to the two sets of probabilities. The tracking performance of two ILC algorithms is given in Figure 2, where Max Error denotes the maximum absolute error of each iteration.

(a)

(b)

Figure 2 

The tracking error profiles for the discrete-time linear system with data dropout and one-step communication delay. (a) is learning results with the data dropout rate 𝛾=0.9 and communication delay rate 𝑤=0.9. (b) is learning results with the data dropout rate 𝛾=0.6 and communication delay rate 𝑤=0.6.

6. Conclusion

In this work, we address a class of networked control system problems with random data dropout and communication delay. D-type ILC is applied to handle this remote control systems problem with repeated tracking tasks. Through analysis, we illustrate the desired convergence property of the ILC. Although we focus on one-step communication delay in this work, the results could be extended to multiple delays, which is one of our ongoing research topics. In our future work, we will also explore the extension to more generic nonlinear dynamic processes.

Appendix

Lemma A.1. For all  𝑎>1, for all 𝜆>1, for all 𝑖∈ℤ₊, the inequality: sup_{[]𝑡∈0,𝑇}𝑎^{−𝜆𝑡𝑡−1}_𝜏=0𝑎^{𝑡−1−𝜏}ℰΔ𝐮_𝑖(𝜏)₂≤1−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆ℰ−𝑎Δ𝐮_{𝑖(𝜆,𝑎)} (A.1) holds.

Proof. Consequently sup_{[]𝑡∈0,𝑇}𝑎^{−𝜆𝑡𝑡−1}_𝜏=0𝑎^{𝑡−1−𝜏}ℰΔ𝐮_𝑖(𝜏)₂=𝑎⁻¹sup_{𝑡∈[0,𝑇]}𝑎^{−𝑡(𝜆−1)𝑡−1}_𝜏=0𝑎^−𝜆𝜏ℰΔ𝐮_𝑖(𝜏)𝑎^{(𝜆−1)𝜏}≤𝑎⁻¹ℰΔ𝐮_{𝑖(𝜆,𝑎)}sup_{[]𝑡∈0,𝑇}𝑎^{−𝑡(𝜆−1)}𝑎^{(𝜆−1)𝑡}−1𝑎^𝜆−1≤−11−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆ℰ−𝑎Δ𝐮_{𝑖(𝜆,𝑎)}. (A.2)

Lemma A.2. For all  𝑎>1, for all 𝜆>1, for all 𝑖∈ℤ₊, the inequalitiessup_{[]𝑡∈0,𝑇}𝑎^{−𝜆𝑡𝑡−1}_𝑘=0𝑎^{𝑡−1−𝑘}ℰ𝐞_𝑖(𝑘)₂≤1−𝑎^{−(𝜆−1)𝑇}𝑎^(𝜆−1)ℰ𝐞−1^{𝑖(𝜆,𝑎)},sup_{𝑡∈[0,𝑇]}𝑎^{𝑡−𝜆𝑡}_𝑘=1𝑎^{𝑡−1−𝑘}ℰ𝐞_𝑖(𝑘)₂≤1−𝑎^{−(𝜆−1)𝑇}𝑎^(𝜆−1)ℰ𝐞−1^{𝑖(𝜆,𝑎)} (A.3) hold.

Proof. Consequently sup_{𝑡∈[0,𝑇]}𝑎^{−𝜆𝑡𝑡−1}_𝑘=0𝑎^{𝑡−1−𝑘}ℰ𝐞_𝑖(𝑘)₂≤sup_{[]𝑡∈0,𝑇}𝑎^{𝑡−𝜆𝑡}_𝑘=0𝑎^𝑡−𝑘ℰ𝐞_𝑖(𝑘)₂=sup_{[]𝑡∈0,𝑇}𝑎^{𝑡−𝑡(𝜆−1)}_𝑘=0𝑎^−𝜆𝑘ℰ𝐞_𝑖(𝑘)𝑎^{(𝜆−1)𝑘}≤ℰ𝐞_{𝑖(𝜆,𝑎)}sup_{[]𝑡∈0,𝑇}𝑎^{−𝑡(𝜆−1)}𝑎^{(𝜆−1)𝑡}−1𝑎^𝜆−1≤−11−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆−1ℰ𝐞−1^{𝑖(𝜆,𝑎)},sup_{𝑡∈[0,𝑇]}𝑎^{𝑡−𝜆𝑡}_𝑘=1𝑎^{𝑡−1−𝑘}ℰ𝐞_𝑖(𝑘)₂≤sup_{𝑡∈[0,𝑇]}𝑎^{𝑡−𝜆𝑡}_𝑘=0𝑎^𝑡−𝑘ℰ𝐞_𝑖(𝑘)₂=sup_{𝑡∈[0,𝑇]}𝑎^{𝑡−𝑡(𝜆−1)}_𝑘=0𝑎^−𝜆𝑘ℰ𝐞_𝑖(𝑘)𝑎^{(𝜆−1)𝑘}≤ℰ𝐞_{𝑖(𝜆,𝑎)}sup_{𝑡∈[0,𝑇]}𝑎^{−𝑡(𝜆−1)}𝑎^{(𝜆−1)𝑡}−1𝑎^𝜆−1≤−11−𝑎^{−(𝜆−1)𝑇}𝑎^𝜆−1ℰ𝐞−1^{𝑖(𝜆,𝑎)}. (A.4)

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants no. 60736021 and no. 60721062), The 973 Program of China (Grant no. 2009CB320603).