Improving Delay-Range-Dependent Stability Condition for Systems with Interval Time-Varying Delay

This paper discusses the delay-range-dependent stability for systems with interval time-varying delay. Through defining the new Lyapunov-Krasovskii functional and estimating the derivative of the LKF by introducing new vectors, using free matrices and reciprocally convex approach, the new delay-range-dependent stability conditions are obtained. Two well-known examples are given to illustrate the less conservatism of the proposed theoretical results.


Introduction
It is well known that time-varying delays are frequently encountered in many practical control systems, and they are usually regarded as a source of instability and poor performance.So the stability issue of time delay systems has received considerable attention [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].In the last years, in order to further reduce conservatism of the stability results, some methods were developed, such as the delay-fraction approach in [1,2], free weighting matrices method in [3,4], the convex analysis method in [5,6], the reciprocally convex approach in [7], LKF constructing method with matrices that depend on the time delays [8], LKF constructing method with triple-integral terms in [9], LKF constructing method with quadruple-integral terms in [10], and simple LKF having quadratic terms multiplied by a higher degree scalar function [11,12].These methods reduced the conservatism of the stability results.But when delay is interval time-varying, the information of the delay derivative is not full used, which causes the conservatism of the stability results.
Motivated by recent methods, in this paper, we further discuss the stability of linear systems with interval timevarying delay.Firstly, a novel LKF is introduced.Then, by introducing new vectors, using free matrices and reciprocally convex approach, the derivative of LKF is estimated less conservatively, and as a result, the stability criterion is obtained in terms of LMI.Finally, two examples are given to illustrate the effectiveness of the proposed method.
Throughout the note, the used notations are standard.R  denotes the -dimensional Euclidean space, R × is a set of  ×  real matrix,   is the transpose of ,  > 0 ( < 0) means symmetric positive (negative) definite matrix, and * in the matrix denotes the symmetric element;  is the identity matrix of appropriate dimensions,   = ( + ),  ∈ [−ℎ, 0].Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulations
Consider the following time-delay system: where () ∈ R  is the state vector, the initial condition () is a continuously differentiable vector-valued function, ,  1 ∈ R × are known real constant matrices, and ℎ() is the timevarying delay satisfying where ℎ 1 , ℎ 2 , and  are constants.

Mathematical Problems in Engineering
To obtain the main results, the following lemmas are needed.

Main Results
In this section, the stability of system (1) is investigated.Through constructing a novel LKF and estimating the derivative of it, new stability condition is provided.
Then taking the time derivatives of   (  ) along the trajectory of system (1) yields where  2 −  3 > 0.

Mathematical Problems in Engineering 3
To reduce the conservatism of the main result, by using the Jesen inequality and Lemma 2 with introducing appropriate matrices  1 ,  2 ∈ R × , we can calculate V 3 (  ) as where by using Lemma 2 with matrix  1 , one can get and by using Lemma 2 with matrix  2 , one can get To the time-derivative of  4 (  ), it can be calculated as where by using Lemma 1, we can get Similar to the method in ( 12)-( 13), by introducing matrix  3 ∈ R × and using Lemma 2, we can estimate the following: In the last, in order to obtain the stability result base on LMI, we can deal with Γ 1 , Γ 2 as follows: where Also, for appropriate matrices  1 ,  2 , one can have Therefore, combining ( 10)-( 19), we can obtain where As we know, if Ξ − Υ()ΘΥ  () < 0 holds, then V(  ) < 0, which means that system (1) is asymptotically stable.So, by Lemma 3, there exists a matrix of appropriate dimension Π such that the following LMI holds: So, we give the main theorem of this paper as follows.
Remark 5. From ( 17) and ( 22), it can be seen that, in order to obtain the less conservative stability condition in terms of LMI, (), () are used, but free matrices Π = [Π  ] (1 ≤  ≤  ≤ 8) are also introduced, which increases the computational burden; this is a disadvantage of the proposed method.
, and using the convex combination again with < 0, the corresponding stability criterion can also be obtained.

Numerical Examples
In this section, the effectiveness of the obtained results in this paper is shown by the two well-known numerical examples.
The purpose is to compare the admissible upper bounds ℎ 2 given lower bound ℎ 1 and .
For various ℎ 1 , , the maximum upper bounds on delay ℎ 2 by different methods are also listed in Table 1.It can be seen that the result obtained in this paper is less conservative.
Example 2. Consider the linear system (1) with The purpose is to compare the admissible upper bounds ℎ 2 which guarantee the asymptotic stability of the above system for given lower bound ℎ 1 and .This example is used in many recent papers, such as [6,9], For  = 0.3, Table 2 lists the comparison of our results with some recent ones, and it is easy to see that the results obtained in this paper are less conservative.

Conclusions
In this note, the stability of interval time-varying delay systems has been discussed.Through constructing a novel LKF and using some new analysis methods, the delay-rangedependent stability criteria were derived.Compared with some previous stability conditions, the obtained main results in this paper have less conservatism.In the end, numerical examples were given to show the superiority of the obtained criteria and their improvements over the existing results.
1 ,  2 , . . .,   : R  → R have positive values in an open subset D of R  .Then, the reciprocally convex combination of   over D satisfies