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The sliding mode control and optimization are investigated for a class of nonlinear
neutral systems with the unmatched nonlinear term. In the framework of Lyapunov stability theory,
the existence conditions for the designed sliding surface and the stability bound

Time delays often arise in the processing state, input, or related variables of dynamic systems. Inparticular, when the state derivative also contains time-delay, the considered systems are called as neutral systems [

On the other hand, as an important robust control approach, sliding mode control strategy has many advantages such as fast response, insensitiveness to parametric uncertainties and external disturbances, and unnecessariness for online identification. Hence, the sliding mode control for dynamic systems has received attention extensively [

To the best of the authors’ knowledge, the sliding control and optimization for uncertain neutral systems have not yet been investigated, which motivates the present study. One contribution of this paper is the optimization for the upper bound of the unmatched nonlinear term. The other contribution lies in the introduction of the scalars

The sliding mode control problem formulation is described in Section

Consider the following class of nonlinear neutral systems:

Without loss of the generality, assume that

The following state transformation is similar to [

Furthermore, the system (

On account of the second equation of (

Based on (

In view of (

Our purpose is to design a sliding surface

sliding motion (

the system (

For any constant matrix

Let

The sliding motion (

First, transform the sliding motion (

Choose

Obviously,

Pre- and postmultiplying

By Schur complement lemma and Lemma

According to Schur complement lemma, LMI (

Thus, one can obtain that

This implies that the sliding motion (

By conditions (

Furthermore, for some desired value

It should be pointed out that the norm of the gain matrix is implicitly bounded by (

It is obvious that (

If the convex optimization problem is solvable, the bound

Suppose that the optimization problem (

From the sliding surface (

In view of

In this section, three examples are presented to illustrate the design approach of sliding surface and reaching motion control and show their advantages.

Consider parts of parameters in [

For the system (

Under the following initial condition

State response of the open-loop system (

State response of the open-loop system (

State response of the closed-loop system (

State response of the closed-loop system (

Control signal in (

Trajectories of sliding variable in sliding surface (

In order to compare with [

In addition, to remove the nonlinear term

As for the constant delays, their derivatives,

Utilizing the parameters in [

Under the initial

State response of the closed-loop system (

State response of the closed-loop system (

Control signal in (

Trajectories of sliding variable in sliding surface (

Under the initial

State response of the closed-loop system (

State response of the closed-loop system (

Control signal in (

Trajectories of sliding variable in sliding surface (

Seen from Figures

The sliding mode control and optimization problem of nonlinear neutral systems with time-varying delays are complex and challenging. In the framework of the Lyapunov stability theory, based on the methods of singular value decomposition and descriptor system model transformation, the sliding surface and the reaching control law are designed, which can be obtained by solving the optimization problem. Also, the upper bound

In future research, it is expected to investigate the optimization of reaching motion control for a class of nonlinear neutral systems with time-varying delays.

This research is supported by Natural Science Foundation of China (no. 61104106), Science Foundation of Department of Education of Liaoning Province (no. L2012422), and Start-up Fund for Doctors of Shenyang University (no. 20212340).