New Delay-Dependent Stability Conditions for Time-Varying Delay Systems

This paper addresses the delay-dependent stability for systems with time-varying delay. First, by taking multi-integral terms into consideration, newLyapunov-Krasovskii functional is defined. Second, in order to reduce the computational complexity of themain results, reciprocally convex approach and some special transformations are introduced, and new delay-dependent stability criteria are proposed, which are less conservative and have less decision variables than some previous results. Finally, two well-known examples are given to illustrate the correctness and advantage of our theoretical results.


Introduction
Time-varying delays are often source of the degradation of performance and the instability of practical control systems.In the last decades, the stability of time-delay systems was one of the hot issues in control theory, and many approaches are proposed to study this topic, such as the properly chosen LKFs [1][2][3], the delay-fraction approach [4][5][6], descriptor model transformation method [7], integral inequality lemma [8][9][10], free weighting matrices [11], and Jensen inequality [12,13].
Recently, in order to further reduce conservatism of the stability results, some new methods were developed.In [14], by constructing novel LKF with matrices that depended on the time delays, stability conditions that depend on both the upper and lower bounds on delay derivatives were obtained.In [15], the convex analysis method was proposed, and in [16], reciprocally convex approach was used to consider the stability of time-varying delay systems.In [17], new LKF including quadratic terms multiplied by a higher degree scalar function was proposed and this method was further improved in [18], where the quadratic convex combination technique was used.In [19], some triple-integral terms were introduced to LKF, and in [20], some quadruple-integral terms were used in LKF to discuss the stability of time-varying delay systems.These methods reduced the conservatism of the stability results, but more decision variables were introduced; that is, the advantage in conservatism is obtained at the cost of high computational complexity.The above analysis naturally leads us to find possible solutions to this problem.
In this paper, we further discuss the stability of linear systems with time-varying delay.The main contribution of this paper lies in two aspects.First, a novel LKF which contains some multi-integral terms is introduced; second, the derivative of LKF is estimated by using reciprocally convex approach and some special transformations, which reduces the conservatism as well as the computational burden of the main results.
Throughout the paper, the used notations are standard.R  denotes the -dimensional Euclidean space, R × is a set of  ×  real matrices,   is the transpose of ,  > 0 ( < 0) means symmetric positive (negative) definite matrix, * in the matrix denotes the symmetric element,  is the identity matrix of appropriate dimensions, and   = ( + ),  ∈ [−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 0 < ℎ and  ≥ 0 are constants.
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.Firstly, construct the following LKF candidate: where and matrices Next, let us define a vector as and   = [  1 0 0 0], block entry matrices   ( = 1, 2, . . ., 5) (e.g.,   2 = [0  0 0 0]).Taking the time derivatives of   (  ) along the trajectory of system (1) yields To the time derivative of  4 (  ), it can be calculated as To reduce the conservatism of the main results, the last two terms of V 4 (  ) were estimated by different methods.In [11], the free matrices were introduced.Reference [13] used convex combination by introducing an approximation on the difference between delay bounds.Reference [16] adopted reciprocally convex approach, which achieved performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables.Therefore, in the following, by introducing appropriate matrix  ∈ R 2×2 and using Lemma 1, we can estimate V 4 (  ) as where [   *  ] ≥ 0. To the time derivative of  5 (  ), on the first step, the following transformation is used: and then, in order to reduce the conservatism of the main result, the appropriate matrices  1 ,  2 ∈ R ×5 are introduced to establish the relations between different vectors of (), so the last two terms in ( 14) can be estimated as Therefore, combining ( 11)-( 15), we can obtain where It can be found that the function and the coefficient of second order is   ()[   1  −1  1 +   2  −1  2 ]() ≥ 0 since  > 0, which means the V(  ) is a convex quadratic function for ℎ().Therefore, apply Lemma 2, if the following holds: then we have which implies V(  ) < 0, which means that system (1) is asymptotically stable.So, we give the main theorem of this paper as follows.

Numerical Examples
In this section, the effectiveness of the obtained results in this paper is shown by two well-known numerical examples.
Example 7. Consider the linear system (1) with The purpose of this example is to compare the admissible upper bounds ℎ which guarantee the asymptotic stability of the above system by different methods.Using the method in [12], when  = 0.5,  = 0.9, and  > 1, the upper bounds for the time delay are ℎ = 2.08, ℎ = 1.66, and ℎ = 1.66; ours are ℎ = 2.33, ℎ = 1.87, and ℎ = 1.86.Table 1 also lists the comparison of our results with some recent ones in [2,7,11,15,16,18].From Table 1, it can be seen that our results are less conservative.
Moreover, the maximum upper bounds ℎ obtained by Theorem 3 are almost equal to those in [18], but the number of variables is 49 2 + 5 in [18], and ours is just 21 2 + 4.That is, the variables in [18] are around 2.2 times more than those in Theorem 3. Therefore, from both mathematical and practical points of view, our condition is more desirable than that in [18] even for some cases when the two methods give almost the same upper bound on the delay.
This example is used in many recent papers, such as [7,15,18].For various , the maximum upper bounds on delay ℎ by different methods are listed in Table 2.It can be seen that the result obtained in this paper is less conservative.

Conclusions
In this paper, the stability of time-varying delay systems has been discussed.Through constructing a novel LKF which contains multi-integral terms and using some new analysis methods, some delay-dependent stability criteria were derived.Compared with some previous stability conditions, the obtained main results in this paper have less conservatism and less decision variables.In the end, numerical examples were given to show the superiority of the obtained criteria and their improvements over the existing results.

Example 8 .
Consider the linear system (1) with 1 ,  2 , . . .,   : R  → R have positive values in an open subset D of R  .Then, the reciprocally convex combination of   over D satisfies

Table 1 :
Admissible upper bound ℎ for various  in Example 7.

Table 2 :
Admissible upper bound ℎ for various  in Example 8.