Abstract
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 combination method 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 proposed method.
1. Introduction
Time delay arises in many systems like manufacturing, telecommunications, chemical industry, power, transportation, and so on. It is generally regarded as a main source of instability and poor performance, which has a negative impact on the performance of the system [1–3]. Therefore, the stability of time-delay systems has always been the focus of attention [4–20].
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 triple-integral 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 9-bus system. The results suggest that the proposed method is less conservative than some known results.
2. 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 . Time-delay function is differentiable and satisfies the following conditions:where , , are constants; , are real constant matrices with corresponding dimensions; , are uncertain parameter matrices with appropriate dimensions and denote uncertainty of the time-varying, satisfying the following conditions:where , , are real constant matrices of appropriate dimensions; is a uncertain matrix with measurable element and satisfies .
Time-delay interval is divided into two sections, namely, , where
3. Delay-Dependent Stability Theorem and Main Results
Firstly, some related lemmas are given in this section.
Lemma 1 (see [11]). For any positive-definite matrix , scalar and the integral term of vector function has definition; the following inequality holds:
Lemma 2 (see [12]). Suppose , where . Then, for any symmetric matrix , the following integral inequality holds:where , , .
Lemma 3 ([13] Schur complement). For given real matrices , , of appropriate dimensions, satisfying , , then the following conditions are equivalent:(a);(b), ;(c), .Lemma 3 is mainly used to transform nonlinear matrix inequalities into linear matrix inequalities.
Lemma 4 (see [14]). Given matrices , , and of appropriate dimensions, then , for all satisfying , if and only if there exists a positive number such thatLemma 4 is mainly used to deal with uncertainty matrix.
Lemma 5 (see [15]). Suppose , where . Then, for any constant matrices , , and with proper dimensions, the following matrix inequalityholds, if and only if
Proof. Substituting , then and . Therefore, we haveAccording to convex combination method, the conclusion is proved.
Theorem 6. For any given constant , , if there are positive-definite matrices , , , , , , , , , , positive numbers , , and matrices , () of appropriate dimensions, satisfying the following matrix inequality:then system (1) is asymptotically stable, where
Proof. The Lyapunov-Krasovskii functional is constructed as follows:where , .
The derivative of along trajectories of systems (1) isBy Lemma 1, we obtainFunction is time-varying and for , must fall in or , so we need to discuss two cases.
The first case, when , thenUsing Lemma 2, it can be obtained thatwhere, , , .
According to (13)–(17), we havewhere, , , , , , and the definitions of symbols , , , , , , , , , , , , are the same as (11).
By Lemma 3, if , thenIn view of Lemma 4 and Schur complement, (19) can be expressed aswhere , and the definitions of symbols , , , are the same as (11).
Besides, based on the Lemma 5, (20) is equivalent toFor (21), using Schur complement, we can getNamely, when , (22) are equivalent to linear matrix inequalities (11); therefore . Then system (1) is asymptotically stable.
The second case, when , thenBy Lemma 2, we havewhere .
Similar to the first case, we can obtainwhere , and the definitions of symbols , , , are the same as (11).
Similarly,Namely, when , (26) are equivalent to linear matrix inequalities (11), hence . Then system (1) is asymptotically stable. Combined with the above two cases, as long as the linear matrix inequality (11) is satisfied, system (1) is asymptotically stable. The proof is now completed.
When the time-delay function is not differentiable or the time-delay-variation rate is unknown, in Lyapunov-Krasovskii functional (13), removing integral terms , the following conclusion can be obtained by using the time-delay segmentation technique.
Corollary 7. For any given constant , , if there are positive-definite matrices , , , , , , , , , positive numbers , , and matrices , () of appropriate dimensions, satisfying the following matrix inequalitythen system (1) is asymptotically stable, whereThe definitions of other symbols are the same as (11).
Remark 8. The delay-dividing technique requires , 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.
4. 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 1. Consider the uncertain linear system (1) described by the matrices asWhen the accuracy of is 0.1, and time-delay-variation rates are equal to 0.3, 0.5, and 0.9, respectively; the maximum allowable delay bound 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 [16–18].
Example 2. Consider the time-delay system (1) with the following parameters:When the accuracy of is 0.01, and time-delay-variation rates are equal to 0.5 and 0.9, respectively; delay bound for ensuring stability of system (1) is obtained by using Theorem 6. When is unknown, delay bound 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 3. 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 asBy calculating, the state variables at the equilibrium point are obtained:
The coefficient matrices in the system model are as follows:
When the accuracy of is 0.1, and time-delay variation rates are equal to 0.3, 0.5, and 0.9, respectively; the maximum allowable delay bound 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.
5. Conclusion
The paper investigates the stability of uncertain linear systems with interval time-varying delay. According to the delay-dividing 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 3-machine 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Program of National Natural Science Foundation, China (Grant no. 61473325).