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This paper investigates the integral sliding mode control problem for Lur’e singularly perturbed systems with sector-constrained nonlinearities. First, we design a proper sliding manifold such that the motion of closed-loop systems with a state feedback controller along the manifold is absolutely stable. To achieve this, we give a matrix inequality-based absolute stability criterion; thus the above problem can be converted into a matrix inequality feasibility problem. In addition, the gain matrix can also be derived by solving the matrix inequality. Then, a discontinuous control law is synthesized to force the system state to reach the sliding manifold and stay there for all subsequent time. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.

Singularly perturbed systems are commonly encountered due to the presence of small parasitic parameters such as small time constants and moments of inertia. These are often the source of the increased order and stiffness of systems [

For robust control strategy, sliding mode control (SMC) provides an efficient way to control high-order nonlinear dynamic systems operating under uncertain conditions, which has many advantages like ease of implementation, fast response, and reduction in the order of the state equation. The basic idea of SMC is to drive the state trajectory of the system onto some specified smooth manifold (sliding surface) passing through the zero state in the state space and maintain the trajectory on it for all subsequent time. Recently, considerable attention has been paid to it and significant advances have been made on this regard [

Motivated by the above works, we, in this paper, consider integral SMC for Lur’e singularly perturbed systems, although the absolute stability of such a system without control input has been studied in [

The rest of the paper is organized as follows. Section

Consider the following Lur’e singularly perturbed systems given by

The structure of the sector condition in the form of (

We now give some basic results before continuing our discussion, which will be useful for the stability analysis of the sliding mode dynamics.

Assume that

For any

If there exists a diagonal matrix

In this section, we will first construct a proper integral sliding manifold such that the motion of system (

In the integral SMC approach, a law of the form

The matrix product

The term

In this paper, we restrict ourselves to control functions in the form of static linear feedback

To determine the motion equation at the sliding manifold, we use the equivalent control method [

By substituting

The following result presents a sufficient condition via linear matrix inequalities technique which guarantees the absolute stability of the sliding mode dynamics (

If there exists a scalar

By Schur’s Complement Lemma, we can obtain that inequality (

The above Theorem gives the first step of the SMC for Lur’e singularly perturbed system (

Next, we will synthesize a proper SMC law to globally drive the system state trajectories onto the predefined switching surface

In this paper, the discontinuous control

Consider Lur’e singularly perturbed system (

To analyze the reachability of the specific switching surface, we choose

For the choice of

In this section, we present two numerical examples to illustrate the effectiveness of previous derived results.

Consider the following Lur’e singularly perturbed system (

States of the closed-loop system.

State of the sliding mode variable.

Consider the following inverted pendulum controlled by a DC-Motor plant via a gear train in [^{2},

In case of

States of the closed-loop system.

State of the sliding mode variable.

From these numerical studies, it is clear that the presented integral sliding mode control method eliminated the effects of sector-constrained conditions and guaranteed the asymptotic stability of the closed-loop systems. In addition, there is no equality constraint involved in matrix inequality; thus numerical problem when computing matrix inequality can be avoided. The effectiveness of the proposed method is shown clearly.

This paper has considered integral sliding mode control for Lur’e singularly perturbed systems with sector-constrained condition. We have constructed a proper integral sliding manifold and proposed a stability criterion expressed in terms of

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

This paper is supported by the National Science Foundation of China (11171113), the Soft Science Research Program of Henan Province (142400411358), and the Natural Science Foundation of Henan Province (142300410324 and 142300410464).