This paper mainly studies the problem of the robust stability analysis for sampleddata system with long time delay. By constructing an improved LyapunovKrasovskii functional and employing some free weighting matrices, some new robust stability criteria can be established in terms of linear matrix inequalities. Furthermore, the proposed equivalent criterion eliminates the effect of free weighing matrices such that numbers of decision variables and computational burden are less than some existing results. A numerical example is also presented and compared with previously proposed algorithm to illustrate the feasibility and effectiveness of the developed results.
In the last few decades, there has been much interest in long time delay system. This is due to its key role in theory research and practical application, such as welding industry, communication networks, and electrical power system. Lots of relevant research results to long time delay systems have been reported in the literature. To mention a few, modeling of long time delay system was considered in [
In the recent years, increasing attention has been devoted to the problem of sampleddata system with long time delay, in which time delay of the plant is usually longer than a sampling period. When such problem is researched in the discretetime framework, concentrated augmentation approach and direct distribution approach can be chosen. However, in concentrated augmentation approach, time delay is not taken into consideration in process of deriving stability criterion and designing desired controller [
In this paper, we make an attempt to solve the robust stability problem of sampleddata systems with long time delay. Some free weighting matrices and an equivalent criterion are introduced, in order to reduce numbers of decide variable and computation burden. Numerical examples are also presented to illustrate the feasibility and effectiveness of the developed results.
Consider a continuous plant of sampleddata system with long time delay:
Sampler is timedriven with a constant sampling period
The time delay
For given matrices
Discretizing system (
Choosing appropriate variables
Modeling of sampleddata system with long time delay has been reported in [
The stability condition presented in this section is based on system (
System (
Letting
Defining
Since
There exist uncertainties in the stability condition of Theorem
System (
With Theorem
Substitution of the matrices
A congruence transformation is applied to (
Definiteness of a matrix is invariant under congruent transformation by a full rank matrix. For instance, if
System (
Consider a nonsingular transformation matrix
Using Schur complement, we can express the rightside LMI of (
Hence,
Theorem
System (
The process of proof is similar to Theorem
The equivalent stability criterion presented in Theorems
Comparison of decision variables.
Revelent theorem  Number of decision variables 

By [ 

By Theorem 

By Theorem 

Considering the system
The problem of relevant normbounded uncertain parameter is further solved in Theorem
State response curve of
In this correspondence paper, we have presented an equivalent stability criterion with less number of LMI variables for a robust stability criterion reported in Theorem
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported in part by the National Natural Science Funds for Young Scholar (no. 51307045), and the authors are indebted to the editor and reviewers for their valuable comments and suggestions.