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This paper is concerned with the sliding mode control for a class of linear systems with time-varying delays. By utilizing a novel Lyapunov-Krasovskii functional and combining it with the delay fractioning approach as well as the free-weighting matrix technology, a sufficient condition is established such that the resulting sliding mode dynamics is asymptotically stable. Then, a sliding mode controller for reaching motion is synthesized to guarantee that the trajectories of the resulting closed-loop system can be driven onto a prescribed sliding surface and maintained there for all subsequent time. A numerical example is provided to illustrate the effectiveness of the proposed design approach.

Since time delays often appear in various engineering, biological, and economical systems, which are frequently a source of instability or poor performance, the analysis and synthesis of time-delay systems have been one of the most active research areas in system science over the past decades; see, for example, [

In the process of stability analysis and synthesis, a Lyapunov function (or functional) is usually involved, and more or less tight techniques to bound some cross terms are used. Since the general form of this functional leads to a complicated system of partial differential equations [

On the other hand, the sliding mode control (SMC) scheme is well known for its robustness against incompletely modeled dynamics, disturbances, time delays, and nonlinearities [

In this context, we are interested in investigating the delay-dependent sliding mode control problem of a class of time-varying delay systems. By dividing the time-delay interval into multiple segments, using the Lyapunov functional technology combined with matrix inequality technology, a new delay-dependent sufficient condition for the existence of linear sliding surfaces is proposed. To obtain a less conservative delay-dependent condition, some slack matrix variables based on the Newton-Leibniz formula are introduced. And an explicit parameterization of the desired sliding surface is also given. Then, a sliding mode controller for reaching motion is synthesized to guarantee that the trajectories of the resulting closed-loop system can be driven onto a prescribed sliding surface and maintained there for all subsequent time.

The rest of the paper is organized as follows. Section

Consider a class of linear time-delay systems given by

The following preliminary assumptions are made for system (

The pair

The perturbation term

Matrix

The aim of this paper is to design a SMC controller

We first transform the original system (

Let

It is obvious that the first equation of system (

When the system trajectories reach onto the sliding surface

In this paper, we are interested in designing a reaching motion control law

The corresponding sliding motion (

The system (

To facilitate further developments, we introduce the following lemma that will be frequently used in deriving the main results.

For symmetrical matrix

Due to the influence of the sliding manifold on system stability and transient performance, the design and analysis of the sliding manifold has become one of the main issues in the sliding mode control [

For given positive scalars

We define the following Lyapunov-Krasovskii functional

Due to this relation, one can introduce the following zero equation:

Moreover, it follows from (

According to (

By Schur complement (Lemma

Thus, the proof is completed.

It should be pointed out that Theorem

Note that the criteria for sliding mode dynamics analysis problem in Theorem

For given positive scalars

Moreover, if the conditions mentioned above are feasible, the matrix

By performing a congruence transformation

Similarly, performing a congruence transformation to (

Performing a congruence transformation to (

According to the definition of

Hence, the proof is completed.

After switching surface design, the next important aspect of sliding mode control is to guarantee the existence of a sliding mode.

Now, we are in the position to design a SMC law, by which the trajectories of the time-delay system (

Suppose that the conditions of (

We will complete the proof by showing that the control law (

According to SMC theory, when the system trajectories reach onto the sliding surface, it follows that

From the sliding surface

Therefore, by

Choose the following Lyapunov functional

By taking the derivative of

Combining (

In this section, a simulation example is used to demonstrate the effectiveness of our proposed theoretical results. Consider the following time-delay system (

States of the open-loop system.

According to (

The existence of a feasible solution shows that there exists a delay-dependent LPK functional for checking the stability of the sliding mode dynamics in (

According to Theorem

Set

The simulation results of the closed-loop system with (

States of the closed-loop system.

Sliding surface function.

Control input.

The problem of sliding mode control for time-delay systems has been studied in this paper. The main contribution of this paper lies in that a novel Lyapunov-Krasovskii functional that reflects the delay-fractioning nature is constructed in order to reduce the possible conservatism introduced by the time delays. It should be pointed out that the delay partitioning technique of this paper is different from others in the literature, and one of the advantages of the method is that it is not necessary to represent the time delay model with two parts, the constant part and the time varying part. Firstly, by constructing a novel Lyapunov-Krasovskii functional based on the delay partitioning approach, a sufficient condition is given to guarantee the asymptotic stability of the sliding mode dynamics. Furthermore, a sliding mode control law is proposed to ensure the reachability of the system’s trajectories to the predefined sliding surface. Finally, a numerical example has been provided to demonstrate the effectiveness of the proposed methods.

The work is supported by National Natural Science Foundation of China (NSFC 61304108, NSFC 51307035, NSFC 61273094) and the Project for Distinguished Young Scholars of the Basic Research Plan in Shenzhen City under Contract no. JCJ201110001. The authors are very thankful to the reviewers for their valuable suggestions and comments.