Abstract

A novel combined convex method is developed for the stability of linear systems with a time-varying delay. A new delay-dependent stability condition expressed in terms of linear matrix inequalities (LMIs) is derived by employing a dedicated constructed Lyapunov-Krasovskii functional (LKF), utilizing the Wirtinger inequality and the reciprocally convex approach to handle the integral term of quadratic quantities. Different from the previous convex techniques which only tackle the time-varying delay, our method adopts the idea of combined convex technique which can tackle not only the delay but also the delay variation. Four well-known examples are illustrated to show the effectiveness of the proposed results.

1. Introduction

In recent years, the stability of the time-delayed linear system is one of the hot issues in control theory, for time delay occurs in different physical, industrial, and engineering systems, such as aircraft, biological systems, population dynamics, and neural networks. It is well-known that time delay is often a source of the degradation of performance and/or the instability of the time-delayed linear system. Hence, the problem of the stability analysis of time-delayed systems has attracted considerable attention in recent years. For more details, see the literature [1–29].

Currently, many researchers have devoted time and effort to the stability analysis of linear-systems with time delay, and a great number of results on delay-dependent stability conditions for time-delayed systems have been reported in the briefs [6, 8, 11, 17, 20, 22, 28, 29] because it is well known that delay-dependent stability criteria which include the information on the size of time delay are generally less conservative than delay-independent ones when the size of time delay is small. The objective of the stability analysis is to find a less conservative condition to enlarge the feasibility region of stability criteria such that it guarantees asymptotic stability of time-delayed systems as large as possible. In order to reduce the conservatism of the stability criteria for linear time-delayed system, integral inequality lemma was used by Park and Ko [9]. He et al. presented some less conservative stability conditions using free-weighting matrix in [11, 23]. Descriptor model transform method was presented by Fridman and Shaked in [12]; Jensen’s inequality and delay decomposition method were used in [17, 20, 21] and [30], respectively. Jensen’s inequality introduces an undesirable conservatism in the stability conditions, so, some Wirtinger inequalities which allow consideration of more accurate integral inequalities are introduced by Seuret and Gouaisbaut to deal with the derivative of LKF recently in [18]. Notice that the reciprocal convex approach presented in [24] has been a popular method. Although this method can be more effective than earlier convex techniques in studying the time-varying delay systems, it still needs more improvements since it cannot tackle the delay variation or more complicated cases [15].

In the light of the discussion above, in this paper, the combined convex method which was presented in [31, 32] is further developed for the stability of the linear systems with time-varying delay. With the new method, both the time-varying delay and the variation of the delay can be tackled. We notice that some important terms are ignored during the construction of the LKF because of limitation of the previous method. First, we construct a new LKF and use reciprocal convex approach and Wirtinger inequality to handle the integral term of quadratic quantities, and then we derive the stability condition in terms of the sum of two first-order convex functions with respect to the time-varying delay and its variation. Second, a novel delay-dependent stability criterion is presented in terms of LMIs which can be solved efficiently by convex optimization algorithm. Finally, four well-known examples are given to illustrate the effectiveness of the proposed method.

Throughout this paper, the following notations will be used: represents the transposition of matrix , denotes -dimensional Euclidean space, and is the set of all real matrices. means that is positively definite. Symbol represents the elements below the main diagonal of a symmetric block matrix. is defined as .

2. Problem Statement

Consider the following linear systems with time-varying delay: where is the state vector, are constant matrices with appropriate dimensions, is the time-varying delay, and it is assumed to satisfy the following:C1:, , for all ;C2:, for all .

The initial condition is a continuously differentiable function on .

For C1, let us define in the following set: where denotes the convex hull, , and . Then, there exists a parameter such that can be expressed as convex combination of the vertices as follows: If a matrix is affinely dependent on , then can be expressed as convex combinations of the vertices From (4), if a stability condition is affinely dependent on , then it needs only to check the vertex values of instead of checking all values of [33].

Before deriving the main results, the following lemmas are stated, which will be used in the proof of the main results.

Lemma 1 (see [18]). For a given matrix , the following inequality holds for all continuously differentiable functions in : where .

Lemma 2 (see [22]). Suppose that , , are the constant matrices of appropriate dimensions, , and , then holds, if the following four inequalities hold simultaneously:

The reciprocally convex combination inequality provided in Park et al. [24] is used in this paper. This inequality has been reformulated by Seuret and Gouaisbaut [18] and is stated in Lemma 3.

Lemma 3. For given positive integers , , a scalar in the interval , a given matrix , and two matrices and in . Define, for all vectors in , the function given by the following: Then, if there exists a matrix in such that , then the following inequality holds:

3. Main Result

The main objective of this section is to achieve a less conservative condition such that it can guarantee the stability of system (1) under the constraint C1. First, we estimate the derivative of Lyapunov functional less conservatively by constructing a new augmented LKF; then, with the Wirtinger inequality and the newly developed combined convex technique, the improved stability results are derived, which are less conservative than some existing ones.

For simplicity of matrix representation, we set block entry matrices (e.g., ) and we define the following: and construct the following LKF: where and here , , , , and .

Theorem 4. For given scalar , and with C1, the system (1) is asymptotically stable, if there exist symmetric positive definite matrices , , , , and and any matrices , such that the following LMIs are feasible: where

Proof. Taking the derivative of with respect to along the solutions of system (1) yieldsFinally, is easily obtained as Using Lemmas 1 and 3 yields
Now, combining with (16)–(18), we get where It is clear that implies , which means that system (1) is asymptotically stable. Notice that is affine and a convex combination on satisfying , then can be expressed as convex combinations of the vertices , . Utilizing Lemma 2, then holds, if and only if (12) and (13) hold. This completes our proof.

Remark 5. Recently, the reciprocally convex optimization technique and Wirtinger inequality to reduce the conservatism of stability criteria for linear systems with time-varying delay were proposed in [24, 25, 28] and [18, 21], respectively, and these methods were utilized in (18). In Lemma 1, it can be noticed that the term is equal to Jensen’s inequality and that the newly appeared term can reduce the LKF enlargement of the estimation. The usage of reciprocally convex optimization method avoids the enlargement of and while only introducing matrix . Then, the convex optimization method is used to handle . During the proof procedure above, the dedicated constructed LKF (11) has full information on the systems.