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In this paper, we deal with the problem of stability and stabilization for linear parameter-varying (LPV) systems with time-varying time delays. The uncertain parameters are assumed to reside in a polytope with bounded variation rates. Being main difference from the existing achievements, the representation of the time derivative of the time-varying parameter is under a polytopic structure. Based on the new representation, delay-dependent sufficient conditions of stability and stabilization are, respectively, formulated in terms of linear matrix inequalities (LMI). Simulation examples are then provided to confirm the effectiveness of the given approach.

Linear time-varying (LPV) systems, which depend on unknown but measurable time-varying parameters, have received much attention. This is mainly due to the context of gain scheduled control of nonlinear systems. In the framework of gain scheduled control, the design can be regarded as applicable for the underlying LPV system, which is a weighted combination of linearized systems. This type of system was first presented in [

On the other hand, time delay is frequently encountered in control systems [

In the light of the above, this paper investigates the stability and stabilization of polytopic LPV systems with parameter-varying time delays. Firstly, an innovative representation for the rate of variation of the parameter is stated. Secondly, based on this representation and parameter-dependent Lyapunov functionals, delay-dependent sufficient conditions for the stability and stabilization are derived in terms of LMIs. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Consider a polytopic system with time parameter-varying time delays described by state-space equations.

The time-varying parameter

The time derivative

The delay is bounded and the function

In this paper, we will establish stability and stabilization conditions for system (

If parameter

To complete the proof of Lemma

The proof is completed.

The main difference for studying the stability problems of LPV time-dependent systems is how to represent the derivative of the time-varying parameter.

We note, for instance, the results presented in [

We mention in a second case the results given in [

The main advantage of the new representation

Now, we use Lyapunov-Krasovskii functional and Lemma

Let

Consider the unforced system (

Consider the following Lyapunov-Krasovskii type functional:

Let

The proof is completed.

In this paper, we consider the following control law:

Before giving a solution to the robust stabilization problem of system (

Let

There exists a matrix

Consider the system matrices

there exist matrices

for the above matrices

For necessity, we perform multiplication on the left by

From conditions (

Applying Schur complement to the above inequality, we have

For Sufficiency, defining

The proof is completed.

If there exist a sufficient large scalar

Let

Multiplying condition (

The proof is completed.

In this section, we give two examples to demonstrate the effectiveness of the proposed methods.

Let us consider a polytopic system in the form of (

System response:

Let us consider a polytopic system in the form of (

System response:

This paper develops the stability and stabilization of polytopic LPV systems with parameter-varying time delays. For this purpose, an innovative representation for the rate of variation of the parameter is addressed. By using this representation and parameter-dependent Lyapunov functionals, delay-dependent sufficient conditions for the stability and stabilization are then derived in terms of LMIs. Two examples are given to illustrate the effectiveness of the methods presented in this paper.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Fu Chen carried out the molecular genetic studies, participated in the sequence alignment, and drafted the manuscript. Shugui Kang conceived of the study and participated in its design and coordination. Fangyuan Li gave some important insights and revised the first draft.

This work is supported by the National Natural Science Foundation of China (Grants no. 61803241 and no. 11871314).