Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 473165, 12 pages
http://dx.doi.org/10.1155/2015/473165

Research Article

Control for Nonlinear Systems with Time-Varying Delay Using Matrix-Based Quadratic Convex Approach

C. Emharuethai and P. Niamsup

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 30 November 2014; Revised 18 February 2015; Accepted 24 March 2015

Academic Editor: Qingling Zhang

Copyright © 2015 C. Emharuethai and P. Niamsup. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

control problem for nonlinear system with time-varying delay is considered by using a set of improved Lyapunov-Krasovskii functionals including some integral terms, and a matrix-based on quadratic convex, combined with Wirtinger's inequalities and some useful integral inequality. controller is designed via memoryless state feedback control and new sufficient conditions for the existence of the state feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.

1. Introduction

The phenomena of time delays are often encountered in many practical systems such as process control systems, manufacturing systems, networked control systems, and economic systems. The existence of these delays may be the source of instability and serious deterioration in the performance of the closed-loop systems. In real world systems especially, the delay should be assumed to be time-varying satisfying and is not necessarily restricted to be 0, namely, interval time-varying delay. Stability analysis of time-delay system has been investigated extensively in the past decades [1–25].

As of time delays, it is well known that the nonlinear perturbations can also cause instability and poor performance of practical systems. Therefore, the stability problem of time-delay systems with nonlinear perturbations has received increasing attention; see [9, 13, 16] and the references cited therein.

control problem has been widely used to minimize the effects of the external disturbances. The purpose of the problem is to design an controller to robustly stabilize the systems while guaranteeing a prescribed level of disturbance attenuation in the sense for the systems with external disturbances. A delay-dependent controller ensures asymptotic stability and a prescribed performance level of the closed-loop systems. The performance indexes and the upper bound of the delay are usually two performance indexes to be used to evaluate the conservatism of the derived condition. The conservatism of the delay-dependent control is measured by the allowable delay size or performance level bound obtained.

Recently, an improved robust stability and performance analysis criterion has been reported [3–5, 7, 10–12, 15, 17, 18]. In [17], control problem for uncertain linear system with state delay and parameter uncertainties has been studied, but the time-varying delay is only bounded above by a constant. control problem for system with interval time-varying delay has been considered in [7], by employing free weighting matrices approach. However, some useful terms in estimating the derivative of Lyapunov-Krasovskii functional are ignored which might lead to some conservatism. In [15], control problem for nonlinear systems with interval time-varying delay has been studied by using Jensen’s inequality to estimate some integral terms of Lyapunov-Krasovskii functional and deriving delay-dependent sufficient condition for the existence of control by using reciprocally convex combination technique.

In the study of time-delay system, several approaches have been proposed in order to reduce conservatism. For example, it is well known that choosing appropriate Lyapunov-Krasovskii functional and using improved bounding techniques to estimate time-derivative of Lyapunov-Krasovskii functional lead to improvement of stability region. Furthermore, free weighting matrices approach; delay decomposition approach; and convex optimization and reciprocally convex optimization techniques have been widely used to reduce conservatism of stability criterion; see [3, 7, 8, 10, 11, 13, 14, 16]. Recently, the so-called matrix-based quadratic convex approach has been introduced to derive stability criterion for time-delay system which was shown to reduce conservatism; for example, it gives better maximum allowable upper bound for time-varying delay than some other existing approaches; see [20–22].

Motivated by the above discussions, in this paper, matrix-based quadratic convex approach will be used to study control problem for system with interval time-varying delay and nonlinear perturbations. To the best of our knowledge, this is one of the first reports of such investigation. By introducing new augmented Lyapunov-Krasovskii functional which has not been considered yet in stability analysis of control problem, a delay-dependent stability criterion and performance analysis are derived in terms of linear matrix inequalities (LMIs). This new Lyapunov-Krasovskii functional consists of integral terms of the form    which allows us to use the matrix-based quadratic convex approach introduced in [20–22]. With the use of this new Lyapunov-Krasovskii functional, matrix-based quadratic convex approach combined with some improved bounding techniques for integral terms such as Wirtinger-based integral inequality [14, 20], some new cross terms will be introduced which enhance the feasible stability criterion. Through two numerical examples, it is shown that the obtained stability criterion may give a larger maximum allowable upper bound of time-varying delay than some existing results.

2. Mathematical Model and Preliminaries

The following notations will be used in this paper: denotes the set of all nonnegative real numbers; denotes the -dimensional space with the Euclidean norm ; denotes the space of all matrices of -dimensions.

denotes the transpose of matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; .

, ; denotes the set of all -valued continuous functions on ; denotes the set of all the -valued square integrable functions on .

Matrix is called semipositive definite if , for all ; is positive definite if for all ; means . The symmetric term in a matrix is denoted by .

Consider the following system with time-varying delays and control input:

where is the state; is the control input; is a disturbance input; and is the observation output. The delay is time-varying continuous function which satisfies

Let . The nonlinear functions , satisfy the following growth condition:

Definition 1. Given , the control problem for system (1) is to seek if there exists a memoryless state feedback controller such that we have the following.

(i)The zero solution of the closed-loop system, where ,

 is asymptotically stable.(ii)The performance

 of the closed-loop system (5) is guaranteed for all nonzero and a prescribed under the condition . In this case, we say that the feedback control asymptotically stabilizes the system.

We introduce the following technical lemmas, which will be used in the proof of our results.

Lemma 2 (see [14]). For a given matrix , the following inequality holds for any continuously differentiable function :

where

Remark 3. It is clear to see that the inequality in Lemma 2 provides a tighter lower bound for than Jensen’s inequality since for . Thus, inequality (7) is an improvement over Jensen’s inequality.

Before we introduce some useful integral inequalities, we denote

Lemma 4 (see [20]). For a given scalar and any real matrices and and a vector such that the integration concerned below is well defined, the following inequality holds for any vector-valued function and matrices and satisfying ,

where is defined in (9).

Lemma 5 (see [21]). Let be a continuous function satisfying . For any real matrix and a vector such that the integration concerned below is well defined, the following inequality holds for any and real matrices , satisfying , ,

where

with () being defined in (9).

Lemma 6 (see [20]). Let be a continuous function satisfying . For any real matrix and a vector such that the integration concerned below is well defined, the following inequality holds for any real matrix satisfying ,

where and

with () being defined in (9).

Remark 7. In Lemma 6, when , the inequality reduces to similar one in [14]. It contains slack matrix variable with dimension comparing to slack matric variable with dimension introduced in [8].

Lemma 8 (see [20]). Let , , and be real symmetric matrices and a continuous function satisfy , where and are constants satisfying . If , then

or

3. Main Results

In this section, we give a design of memoryless feedback control for system (1). First, we present delay-dependent asymptotical stabilizability analysis conditions for the nonlinear system with time-varying delay (1). Now, we operate the matrix-based quadratic convex approach with the integral inequalities in [20] to formulate a new stability criterion for system (1). For our goal, we choose the following Lyapunov-Krasovskii functional:

where denotes the function defined on the interval . Set , , , and

where , , , , , , and are real matrices to be determined. Before introducing the main theorem, for simplicity, we set

Theorem 9. Given , then system (1) is asymptotically stabilizable and satisfies for all nonzero if there exist positive definite matrices , , , , , , , , , , , , , , and such that the following hold:

where and

with () denoting the th row-block vector of the identity matrix and

Moreover, the feedback control is given by

Proof. Taking the derivative of along the solution of system (1), we obtain

From (4) and Cauchy inequality, we get the following inequalities:

Similarly,

By using the following identity relation:

we obtain the following:

From (27) and (29), (32), is estimated as

where

With the consideration of the three terms of , we obtained the following inequality:

Therefore, the estimation of is as follows:

where is defined in (22). Similarly, is estimated as

where

By Lemmas 5 and 6, we obtain the following:

where and

Thus,

where is given in (23). From Lemmas 2 and 4, we obtain

from which it follows that

where

Hence, from (41) and (43), we obtain

where . From (33), (36), and (45), we obtain along the solution of system (1) as

where

Therefore, we have

where

and is defined in (24). Observe that may be rewritten as

where and , are -independent real matrices. By Lemma 8, if and the inequalities in (21) hold, then , , . Moreover, may be rewritten as a convex combination of as follows:

where and , are -independent real matrices. By utilizing the Schur complement lemma, it follows from (21), (50), and (51) that holds, from which it follows from inequality (48) that

Letting and from

there exists a scalar such that

Therefore, system (1) with is asymptotically stable. To complete the proof of theorem, next we consider the performance . By assuming that , , it follows from definition of that

From (52), we obtain

From estimations of and in (48) and (55), we obtain

Integrating both sides of the above equation from to , we get

It follows that

Therefore, under zero initial condition , , by letting in (59), we get

which gives . This completes the proof.

When , , and , the nonlinear function is , and (1) reduces to

where , . In the following, we present a stability criterion for the case when . We consider the following Lyapunov-Krasovskii functional candidate:

Corollary 10. For given scalars , , , , and , (61) is asymptotically stable if there exist symmetric positive definite matrices , , , , , , , , , and , such that the following LMIs hold:

where

with .

Proof. The proof is the same as in Theorem 9 by using Lyapunov-Krasovskii functional (62). The proof is omitted.

4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of theoretical results.

Example 1. Consider the nonlinear system with interval time-varying delays (1) which was considered in [15], where

, , , , , and . By using LMI Toolbox in MATLAB, the LMIs in Theorem 9 are feasible. From (26) the controller feedback gain can be computed as

For simulation, we choose , , , and is the Gaussian noise which is the set of random numbers with fluctuation range between −1 and 1. Figure 1 shows the trajectories of solutions and of system (1) without feedback control and Figure 2 shows the trajectories of solutions and of the system with feedback control . Moreover, in Tables 1 and 2, by using Theorem 9, we give the minimum allowable value with , and with , , for some given and , respectively.

Table 1: The minimum allowable value of disturbance attenuation with and .

Method

Theorem 9
0.1
0.3
0.2377

0.1
0.5
0.2474

Method

Theorem 9
0.1
0.3
0.8991

0.1
0.5
0.9643

Table 2: The minimum allowable value of disturbance attenuation with and .

Method

Theorem 9
0.1
0.5
0.2487

Figure 1: The trajectories of and of (1) in Example 1 without feedback control.

Figure 2: The trajectories of and of (1) in Example 1 with feedback control.

Example 2. Consider the following nonlinear system with interval time-varying delays which was considered in [18]:

where

, and , . By using LMI Toolbox in MATLAB, the LMIs in Corollary 10 are feasible. Table 3 shows the maximum allowable upper bound with different values of and .

Table 3: The maximum allowable upper bound in Example 2 with .

Method

By Corollary 10
−0.1
0.1
2.1443

−0.3
0.3
2.1126

−0.5
0.5
2.0833

−0.8
0.8
2.0473

−1
1
2.0332

Example 3. Consider the following nonlinear system with interval time-varying delays which was considered in [13]:

where

, and , . By using LMI Toolbox in MATLAB, the LMIs in Corollary 10 are feasible. For comparison with other existing results, we now calculate the admissible maximum allowable upper bounds of by setting for various values of . Table 4 shows that the maximum allowable upper bounds of with given obtained from Corollary 10 are greater than the upper bounds obtained in [13, 23, 25].

Table 4: Comparison of maximum allowable upper bound in Example 3 with .

Method

[25]
1.009
0.714

[23]
1.284
1.209

[13]
1.287
1.279

Corollary 10
1.7706
1.7355

5. Conclusions

In this paper, we have investigated the control problem for a class of nonlinear systems with interval time-varying delay. A new Lyapunov-Krasovskii functional is constructed to obtain new delay-dependent sufficient condition for the control and asymptotic stability condition in terms of LMIs. Numerical examples are given to illustrate the effectiveness of the theoretical results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author is supported by Thai Government Scholarships in the Area of Science and Technology (Ministry of Science and Technology, Bangkok, Thailand). The second author is supported by Chiang Mai University, Chiang Mai, Thailand.

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