Stability Criteria of Interval Time-Varying Delay Systems and Their Application

The stability for a class of uncertain linear systems with interval time-varying delays is studied. Based on the delay-dividing approach, the delay interval is partitioned into two subintervals. By constructing an appropriate Lyapunov-Krasovskii functional and using the convex combinationmethod and the improved integral inequality, the delay-dependent stability criteria with less conservation are derived. Finally, some numerical examples are given to show the effectiveness and superiority of the proposedmethod.

In recent years, some scholars have put forward many effective methods in order to reduce the conservation of the existing results, solve the time-delay problem of system, and make the system more stable.For example, in the process of analyzing the time-delay system, the free weighting matrix method is used in [5], which reduces the conservation of the fixed weighting matrix.In [6], the delay-dividing approach is adopted.However, too many partition intervals increase the computational complexity and simulation time, which lead to the decrease of the system operating efficiency.In the construction of functional, [8] is introduced to the tripleintegral terms.The conclusion shows that there is no obvious decrease in the conservation of the results after adding the item.When dealing with integral terms which are generated in the process of functional derivation, one common point of the above reference is the use of Jensen's inequality.Jensen's inequality is simple and convenient, but it has a certain conservation.Integral inequality of the Wirtinger type is introduced by [9].Under the premise that conservation of the results is not affected, the number of decision variables to be used is small.But the method is mainly used in such case that time delay is not decomposed.Therefore, it is meaningful to obtain less conservative stability criterion by combining the delay-dividing approach and integral inequality.
Motivated by the above research, this paper considers the problem of delay-dependent stability for uncertain systems with interval time-varying delay.By constructing an appropriate Lyapunov-Krasovskii functional and using the convex combination method and the improved integral inequality, a new less conservative delay-dependent stability criterion is proposed.The proposed method is verified by the classical numerical examples and applied to the WSCC 3-machine 9bus system.The results suggest that the proposed method is less conservative than some known results.

Description of Linear Uncertain Time-Delay Systems
Consider the following uncertain time-delay system: where () ∈   is the state vector of system and () is defined as the continuous initial real function on the interval [−ℎ 2 , 0].Time-delay function ℎ() is differentiable and satisfies the following conditions: where ℎ 1 , ℎ 2 ,  are constants; ,  are real constant matrices with corresponding dimensions; Δ(), Δ() are uncertain parameter matrices with appropriate dimensions and denote uncertainty of the time-varying, satisfying the following conditions: where ,

Delay-Dependent Stability Theorem and Main Results
Firstly, some related lemmas are given in this section.
In view of Lemma 4 and Schur complement, ( 19) can be expressed as where , and the definitions of symbols Ω 0 , Ω 1 , Φ, Ψ 1 are the same as (11).
When the time-delay function ℎ() is not differentiable or the time-delay-variation rate  is unknown, in Lyapunov-Krasovskii functional (13), removing integral terms ∫  −ℎ()  T () 6 (), the following conclusion can be obtained by using the time-delay segmentation technique.
Remark .The delay-dividing technique requires 0 <  < 1, and the maximum allowable delay bound is related to the accuracy of .For given , according to Theorem 6 and Corollary 7, we can obtain the corresponding maximum allowable delay bound, and maximum value of these is taken as the maximum allowable delay bound of system.By improving the accuracy of , the maximum allowable delay bound of system can be increased, and the time-delay information is more fully utilized.Furthermore, the conservation of system will reduce and the computational complexity will increase.

The Analysis of Simulation Examples
In this section, the validity of Theorem 6 and Corollary 7 is verified by the classical numerical examples; then they are applied to the WSCC 3-machine 9-bus system for example analysis.
Example .Consider the uncertain linear system (1) described by the matrices as When the accuracy of  is 0.1, ℎ 1 = 0 and time-delayvariation rates  are equal to 0.3, 0.5, and 0.9, respectively; the maximum allowable delay bound ℎ 2 for system (1) is obtained by using Theorem 6.The results are shown in Table 1; it is clear that the proposed stability criterion is less conservative than those in [16][17][18].
Example .Consider the time-delay system (1) with the following parameters: International Journal of Engineering Mathematics (30) When the accuracy of  is 0.01, ℎ 1 = 1 and time-delayvariation rates  are equal to 0.5 and 0.9, respectively; delay bound ℎ 2 for ensuring stability of system (1) is obtained by using Theorem 6.When  is unknown, delay bound ℎ 2 is got by using Corollary 7. The results are shown in Table 2. Compared with [16,19,20], the method of this paper is less conservative.
Example .Consider WSCC 3-machine 9-bus system, the wiring diagram of the system is shown in Figure 1.Generator G1 is infinite bus, and there is a time delay in the loop controlled by G3.Branches and parameter nodes are detailed in [21].
The generator equation of system can be expressed as The coefficient matrices in the system model are as follows:     When the accuracy of  is 0.1, ℎ 1 = 0 and time-delay variation rates  are equal to 0.3, 0.5, and 0.9, respectively; the maximum allowable delay bound ℎ 2 is obtained by using [16,18] and Theorem 6.The results are shown in Table 3. Obviously, our criterion leads to much less conservative results.

Conclusion
The paper investigates the stability of uncertain linear systems with interval time-varying delay.According to the delaydividing approach, the delay interval is partitioned into two subintervals and a new Lyapunov-Krasovskii functional is constructed, which makes use of the information on the some delayed sufficiently.The delay-dependent stability criteria are presented by using convex combination technique and improved integral inequality.In the example of WSCC 3machine 9-bus system, the calculation results show that the upper bound of time delay is larger than that of the previous references.Therefore, the delay-dependent stability criterion presented in this paper is less conservative.Based on the work of this paper, then the analytical method of the uncertain time-delay systems can be extended to the nonlinear uncertain time-delay systems, and the stability of nonlinear uncertain time-delay systems will be analyzed.