An iterative learning control scheme is applied to a class of linear discrete-time switched systems with arbitrary switching rules. The application is based on the assumption that the switched system repetitively operates over a finite time interval. By taking advantage of the super vector approach, convergence is discussed when noise is free and robustness is analyzed when the controlled system is disturbed by bounded noise. The analytical results manifest that the iterative learning control algorithm is feasible and effective for the linear switched system. To support the theoretical analysis, numerical simulations are made.

A switched system consists of a family of subsystems described by differential equations or difference equations, whose switching rules are usually considered to be arbitrary. The switched systems belong to the hybrid systems and have attracted flourishing investigations in the latest decade [

Fortunately, there exists a kind of efficient control scheme, named as iterative learning control (ILC), which has also drawn increasing attention for its simple control structure and perfect learning performance. One of the advantages of ILC is that it requires less knowledge of the controlled system in the procedure of learning. It utilizes the tracking error information of the previous operations to compensate for the current control input so as to generate an upgraded control input for the next operation. By this successive learning process, the tracking performance of the controlled system is improved. In view of the above properties of ILC, it can be used in the switched systems for tracking a given target. However, to authors’ knowledge, there are few efforts on the study of the ILC strategies for the switched systems. So far, few literatures [

In literature [

In the literatures [

Motivated by the drawbacks of the literatures [

The rest of the paper is organized as follows. Section

Consider a class of linear discrete time-invariant single-input, single-output switched systems described as follows:

For a given desired trajectory

Analysis of this paper is basis of the following primary assumptions [

All operations start at identical initial state; that is,

The desired trajectory

For the given desired trajectory

In this paper, a P-type ILC scheme is considered as

In this section, the super vector approach is employed to analyze the learning performance of the algorithm (

Given an arbitrary

For the sake of expression simplicity, the following “super vectors” are denoted as

It is noted that the presentation in the form of super vector reflects the dynamical properties of the system (

Consequently, the control objective of ILC in the form of super vector can be equivalently described as searching such an input super vector sequence

In the form of the super vector, the updating law (

Assume that the ILC algorithm (

By the definition of the tracking error, it is easy to derive

This completes the proof.

It is noted that the elements of the matrix

The assumption (

When considering measurement disturbance

There are two types of errors, namely, the contaminated tracking error

The signal

Consider (

Taking the Euclidean norm of both sides of (

It is seen from estimation (

The super vector approach employed in the paper is a tool to theoretically analyze the learning performance of the addressed ILC algorithm. Since it converts the original two-dimensional problem into a one-dimension linear input-output response problem, it can deal with the ILC problem only in the iterative domain regardless of time domain. This brings a great convenience for designing a controller and for theoretical analysis.

Compared with the existing ILC algorithms for the linear switched systems in the literatures [

It is necessary to point out that the study on the ILC algorithms for switched systems is more arduous than that for nonswitched systems, since the dynamical behaviors of switched systems are more complex. As seen that a switched system obeys diverse subdynamics, compared with an ILC algorithm for a nonswitched system, the ILC scheme for a switched system requires much more memory to store the switching rules as well as the subdynamics. It would be a dimensional disaster in the case when the system dimension is higher and the sampling number is much larger. However, the memory requirement will be solved sooner or later with the advancing of information technology. Next, hinted by the manner of the existing references [

To manifest the validity and the effectiveness of the algorithm, a simple example is considered, which has served as an example in the literature [

Arbitrary switching sequence

The system dynamics are given as

Assume that the system (

Tracking error in iteration domain.

The system outputs at the 5th trial, the 10th trial, and the 15th trial.

Suppose that the system (

A sequence of random disturbance of arbitrary trial in time domain.

Tracking error in the iteration domain.

The system outputs at the 10th trial and the 15th trial are exhibited in Figure

The system outputs of the 10th trial and the 15th trial.

In this paper, a P-type ILC algorithm is applied to a kind of discrete switched systems with an arbitrary switching rule. By the super vector approach, the convergence performance is firstly discussed and then the robustness is analyzed when the system is randomly interfered by bounded measurement noise for the case that the switching rule is fixed once it is randomly selected for the first operation. Results manifest that the convergence and robustness can be guaranteed under appropriate conditions. However, for the case when the switching rule is fired randomly both in time axis and iteration direction, the analysis of the convergence and the robustness remain a hanging issue. In addition, the learning performance of the ILC scheme for linear time-varying switched system or nonlinear switched system is a challenging topic.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors sincerely appreciate the supports of the National Natural Science Foundation of China under Grants nos. F010114-60974140 and 61273135.