^{1, 2}

^{3}

^{1}

^{1}

^{2}

^{3}

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

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 [

On the other hand, the research on networked control systems has attracted much attention [

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 [

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

Throughout the paper, the following notations are used. Let

Consider a deterministic discrete-time linear time-invariant dynamics system:

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

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,

When the control process is deterministic, an effective ILC law for the linear system (

Note that in the ILC law (

To facilitate the ILC design and convergence analysis, data dropout and one-step communication delay are formulated. First formulate the data-dropout problem. Denote

When the data dropout occurs in multiple communication channels, we can similarly define

Due to the data dropout, the plant output received by the controller at the

Next formulate the one-step communication delay problem. Denote

It is worthwhile noting that stochastic variables

At the

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

In this section, we derive the convergence property of the ILC (

In ILC, the learning convergence can be derived in terms of either the output tracking error,

Define the input and state errors

From (

Suppose that the update law (

First, subtracting

Now let us handle the second term on the right hand side of (

Substituting the relation (

In order to handle the exponential term with

Substituting the properties of Lemma

Since

Note that

Revising the original reference

Note that

In this section, we prove the learning convergence property of

Suppose that the update law (

First note the relationship:

In order to handle the exponential term with

Since

Consider the following linear discrete-time system:

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

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.

For all

Consequently

For all

Consequently

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).