An improved stability criterion for linear time-varying delay systems

ABSTRACT This paper considers the stability problem of linear systems with time-varying delays. A modified Lyapunov–Krasovskii functional (LKF) is constructed, which consists of delay-dependent matrices and double integral items under two time-varying subintervals. Based on the modified LKF, a less conservative stability criterion than some previous ones is derived. Furthermore, to obtain a tighter bound of the integral terms, the quadratic generalized free-weighting matrix inequality (QGFMI) is fully applied to different delay subintervals, which further reduces the conservatism of the stability criterion. Finally, three numerical examples are presented to show the effectiveness of the proposed approach.


Introduction
The control theory of system is always a hot topic [1][2][3][4][5][6][7][8]. Time delays are of frequent occurrence in many practical systems, which often result in the major source of poor performance and instability. The stability problems of time-delayed systems have been a hot research topic. In this paper, the stability problems of linear systems with time-varying delays will been further analysed via the LKF method application. The linear systems with time-varying delays are described as: where x(t) ∈ R n is the state vector of the system. A and A 1 are real constant matrices with appropriate dimensions. The time-varying delay h(t) is continuous-time functional and satisfies the following conditions: where h and μ are positive constants.
To obtain stability criteria of system (1) by using the Lyapunov theorem, the main efforts are concentrated on the following several directions, one is finding an appropriate positive definite functional with a negative definite time derivative along the trajectory of system, e.g.LKF with delay partitioning approach [9,10], LKF with augmented terms [11][12][13][14][15][16], LKF with triple-integral and quadruple-integral terms [17][18][19], and so on. The other is reducing the upper bounds of the time derivative of LKF as much as possible by developing various inequality techniques, such as Jensen inequality [20], Wirtinger-based inequality [21], auxiliary function based inequality [22][23][24][25], Bessel-Legendre inequality [26], etc. Besides, further increasing the freedom of solving LMI, additional freeweighting-matrix technique is frequently introduced into the derivatives of LKF, for instance, the generalized zero equality [27,28], the one or second-order reciprocally convex combinations [29][30][31], the free-weightingmatrix approach [32], and so on. Based on those studies in [21,22,26,33], the tighter inequality seems to lead to less conservative results. Recently, Zhang [34] considered the effect of the LKFs and discussed the relationship between the tightness of inequalities and the conservatism of criteria. The results illustrate the integral inequality that makes the upper bound closer to the true value does not always lead to deduction of a less conservative stability condition if the LKF is not properly constructed. Thus, it is crucial to construct a proper LKF. Recently, to fully utilize the information of delay derivative, a novel LKF with delay-dependent matrix was constructed by Kwon et al. [35] as follows: where P > 0, R > 0 and Compared with the general delay-independent LKF, the Lyapunov matrices Q 1 (t), Q 2 (t) of the above LKF are delay-dependent, which can make further use of the information of time delay and derivative of delay. Kwon et al. have illustrated that the stability conditions based on the LKF (3) with delay-dependent matrices is less conservative than those based on the LKF without delay-dependent matrices. However, Thus, all of the above terms (7)-(9) are nonlinear function of d(t) and the inequalities (11)- (14) and (26) in Theorem 1 of [35] are not LMIs, although the Schur complement was used. Indeed, the first element of W 1 , W 2 , G 3 , G 4 and d(t)/3, (d − d(t))/3 lead to d 2 (t) and d 3 (t) contained in the terms (7)- (9). Although the terms (8)- (9) are transformed via the Schur complement application, the final forms r[i,j] still remain d 2 (t). Therefore, Theorem 1 of [35] is incorrect due to the above reasons.
Inspired by the above discussions, this paper constructs the following modified LKF (4) to analyse the stability of the linear time-delayed system (1). with where, P, Q 1 (t), Q 2 (t),Q a ,Q b , R 1 and R 2 are positive definite matrices. The main contributions of this paper are summed up as follows: • To avoid introducing the two additional pos- and h 3 (t), this paper introduces some necessary state variables into the LKF, which not only increases the freedom for solving LMIs, but also removes the non-linear terms with h 2 (t) and h 3 (t) in [35]. Finally, based on the LKF (10) proposed in this paper, a new less conservative stability criterion is obtained than some of the recent existing ones.
This paper is organized as follows. Section 2 gives the problem statement and provides some lemmas. Section 3 presents a new stability criterion for the linear time-varying delay system. Numerical examples and conclusions are drawn in Sections 4 and 5, respectively.
Notation: The notation P > 0 (< 0) mean that matrix P is positive (negative) definite. I represents an identity matrix with the corresponding dimension. * denotes the symmetric terms in a block matrix and diag{· · · } denotes a block-diagonal matrix.

Problem formulation
The main purpose of this paper is to derive the stability criterion for the system (1) satisfying the condition (2) by using a modifyed LKF. To further explain the topic of this paper, the following research flow chart is given in Figure 1. To achieve this purpose, the following lemma is important.

Lemma 2.1 (QGFMI [35]):
For given any matrices X, Y, a positive matrix R and a continuous differentiable where η 0 is an any vector, and

Main results
The main objective of this section is to describe the effect of the proposed LKF on reducing the conservatism of the stability criterion. A novel theorem is derived by employing the LKF proposed in previous section and a corollary for comparison is also given. For the sake of simplicity on matrix representation, the notations of several symbols and matrices are defined as: Theorem 3.1: The system (1) satisfying the conditions (2) is stable for given values of h ≥ 0, μ < 1, if there exist real positive definite matrices P ∈ R 5n×5n , (R 1 , a [1,4]μ a [1,2] h d b [3,4] b [3,2] [1,3] a [1,1] h d b [2,3] b [2,1] where the definitions of some symbols are shown in Appendix 1. Proof: Construct an LKF (10). The time derivative of V(t) with respect to time along the trajectory of the system (1) is as follows: For additional symmetric matrices Q a , Q b , R a and R b , the following zero equations are satisfied Taking the zero inequalities inV 2 andV 4 , we have the following integral terms fromV 2 toV 4 It follows from Lemma 1 with an augmented vector γ (s) that For any appropriately dimensioned matrices N ∈ Finally, from the above derivation, we havė Therefore, LMIs (11) which together with Schur complement equivalence imply thatV(t) < 0. This shows that system (1) is stable from Lyapunov stability theory. This completes the proof.

Remark 3.1:
It is important to reduce the conservativeness of stability conditions by introducing some delay-dependent matrices into the single-integral terms of LKF [35]. Unfortunately, the main result of [35] was incorrect due to the neglect of the terms with h 2 (t) even h 3 (t) discussed in the section 1. In this paper, all matrices inequations of Theorem 3.1 are LMI by introduc- and a zero equation (20), which can remove the terms with h 2 (t) even h 3 (t). Moreover, the double integral items V 4 (t) of the LKF (10)

divide the time delay interval [0, h] into two subintervals ([0, h(t)] and [h(t), h]) instead of using the item t t−h γ T (s)Rγ (s) ds directly. It further makes full use of the information of timevarying delays h(t), h − h(t)
and their derivative. Thus, the QGFMI technique can be fully used in each subinterval, which can further reduce the conservatism of the stability conditions.

Remark 3.2:
It is worth noting that in [35], to bound the integral item for −μ ≤ḣ(t) ≤ μ, (μ > 0) via the QGFMI technique, the following addition zero equation is introduced Then, the above integral item can be rewritten as the following form: The first two items on the right can be bounded via the QGFMI technique (for details see Equation (17)  γ T (s)Q b γ (s) ds due to their positive definiteness. Thus, to avoid introducing the two positive define integral terms, we give the modified LKF (10) with a double integral item V 3 (t).
To illustrate the effectiveness of delay-dependent matrices in reducing the conservativeness of stability conditions, we give the following corollary by choosing the following LKF:Ṽ where, P, Q 1 , Q 2 , R 1 and R 2 are positive definite matrices. [1,2] h d˜ b [3,2]

Corollary 3.1: The system (1) satisfying the conditions (2) is stable for given values of h
⎡ ⎣Π [h,ḣ(t)] h d˜ a [1,1] where the definitions of some symbols are shown in Appendix 2.
Proof: The steps of proof are similar to those of Theorem 3.1, so it is omitted here.

Numerical example
In this section, two numerical examples will be provided to illustrate the effectiveness and superiority of Theorem 3.1, in which the maximum allowable upper bounds (MAUBs) are carefully compared, including the numbers of decision variables (NoVs).

Remark 4.1:
In this paper, the strict LMIs (11) and (12) can be solved easy by using the Matlab LMI-toolbox. The search steps are: • Step 1. For given values of h ≥ 0, μ < 1, LMIs (11) and (12) can be solved to find the MAUB h h 1 ;  • Step 2. Solve LMIs (11) and (12) by progressively increasing h from h 1 obtained in Step 1 until LMIs in (11) and (12) are infeasible. The obtained MAUB value is denoted as h 2 ; • Step 3. Based on the search steps above, if there is no h 2 in Stept 2, the best result is h 1 ; otherwise, the best result is h 2 .
Example 4.1: Consider the following systems: The comparative results among some previous stability conditions, Theorem 3.1 and Corollary 3.1 are listed in Table 1. It is clear that the MAUBs obtained by Theorem 3.1 and Corollary 3.1 are all larger than those obtained by other conditions proposed in [33,[35][36][37][38][39]. As expected, the MAUBs obtained by Theorem 3.1 are the largest for different values μ, which matches the explanation in Remark 3.1. On the other hand, the number of decision values in our criteria is larger than those in [33,35,36,39] but less than those in [37,38].
To confirm the obtained result (h = 3.006), the simulation result is shown in Figure 2, which shows that the state responses of the system (28) with h(t) = 3.006 2 + 3.006 2 sin( 0.1t 3.006 ) converge to zero under the random initial state.

Example 4.2:
Consider the following systems:  For different lower bounds of delay derivative μ, the MAUBs are shown in Table 2 by utilizing Theorem 3.1, Corollary 3.1 and some methods in the literature. From Table 2, Theorem 3.1 gives larger MAUB than Corollary 3.1 and some existing ones. And the number of decision values in our criteria is larger than those in [11,33,35,39,41] and less than those in [12,37,38,40]. To confirm the obtained result (h = 8.3352), the simulation result is shown in Figure 3, which shows that the state responses of the system (29) The MAUBs are shown in Table 3 for different μ by utilizing Theorem 3.1, Corollary 3.1 and some methods in the literature. From Table 3, Theorem 3.1 gives larger MAUB than Corollary 3.1 and some existing ones.  and 2, which can show our stability criterion is less conservative than some existing ones.

Conclusions
In this paper, the stability of time-varying delay systems is studied. A modified Lyapunov functional with delaydependent matrices is proposed and used to derive the stability criterion. The novel stability condition can also be generalized to some other time-delayed control systems, for example, time-delayed Lur' e systems, time-delayed neural networks, time-delayed neutral-type systems, and so on. However, there is still some distances from practical application because of the complexity of the control theory, which can be something that we will continue to study in the future.

Disclosure statement
No potential conflict of interest was reported by the authors.