Mathematical Problems in Engineering

Volume 2013, Article ID 326560, 6 pages

http://dx.doi.org/10.1155/2013/326560

## Absolute Stability and Master-Slave Synchronization of Systems with State-Dependent Nonlinearities

^{1}Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, China^{2}College of Sciences, North China University of Technology, Beijing 100144, China^{3}Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China

Received 3 April 2013; Accepted 7 May 2013

Academic Editor: Wenwu Yu

Copyright © 2013 Rong Li et al. 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

This paper is concerned with the problems of absolute stability and master-slave synchronization of systems with state-dependent nonlinearities. The Kalman-Yakubovich-Popov (KYP) lemma and the Schur complement formula are applied to get novel and less conservative stability conditions. A numerical example is presented to illustrate the efficiency of the stability criteria. Furthermore, a synchronization criterion is developed based on the proposed stability results.

#### 1. Introduction and Preliminaries

This paper considers the problems of absolute stability and master-slave synchronization of dynamical systems described by the following differential equation: where is the system state vector and , are time-invariant state-dependent nonlinearities with sector restrictions such as the following: For convenience, system (1) will often be written by using the following compact notation: where and . And the sector conditions (2) will be denoted as follows: where and .

The problems of stability and synchronization of systems of form (3) play an important role in nonlinear systems theory. It has been found that system (3) has connections with problems in system theory and computation in fields as diverse as Hopfield neural networks [1], Lotka-Voltera ecosystems [2], and systems with saturation nonlinearities [3], among others. A rather recent contribution to the stability analysis of system (3) is [4].

As illustrated in [4], some well-known stability results, such as diagonal stability and passivity-based methods (the circle and the Popov criteria), can be used as stability criteria for system (3) with some particular sector conditions. However, while bringing simplicity, these stability criteria may also introduce conservativeness to the problem. By using a Lur’e function as a Lyapunov function candidate, [4] introduced a new absolute stability test for system (3) which was proved to be much less conservative than both diagonal stability and passivity-based methods. For the sake of convenience, we put this stability test in Lemma 1.

On the other hand, the problems of absolute stability and synchronization of Lur’e systems have been widely studied [5–15]. Thanks to the results of [9], we found that the stability criteria proposed in [4] can be further improved by relaxing the restriction of positiveness on matrix in Lemma 1, which, as illustrated by a numerical example, can further reduce the conservativeness of the stability test of [4]. Last but not least, a new synchronization criterion for systems of form (3) is developed based on the proposed stability results.

The Kalman-Yakubovic-Popov (KYP) lemma will be used in this paper to establish the equivalence relationship between the frequency-domain conditions and time-domain inequalities. The Schur complement formula will also be applied in the process of proof. They are both presented in lemmas below for the convenience of reading.

Lemma 1 (Theorem 2 of [4]). *The zero solution of system (3) is globally asymptotically stable (GAS) for all , if there exist diagonal and positive-definite matrices and , and a symmetric matrix such that the following LMI
**
is feasible, where and . *

Lemma 2 (KYP lemma, Rantzer [16]). *Given , , , with for all , the following statements are equivalent.*(i)* for all .*(ii)*There exists a matrix such that
*

Lemma 3 (Schur complement, Boyd et al. [17]). *The LMI
**
with and is equivalent to one of the following statements:*(i)* and ;*(ii)* and . *

#### 2. Absolute Stability Criteria

To analyze the absolute stability of system (3), a Lur’e function where and , was taken in [4] as a Lyapunov function candidate. The stability condition of Lemma 1 was then deduced by the analysis of the time derivative of (8) incorporating the -procedure (see [4] for details).

Note that the diagonal matrix in Lemma 1 is allowed to be only positive, so as to ensure the nonnegativeness of the Lyapunov function (6) in [4]. The main goal of this section is to prove that the restriction of positiveness on matrix is unnecessary by finally showing that the Lur’e function in (8) can still be taken as a Lyapunov candidate when some or even all entries are nonpositive. We will start with a revised time-domain criterion for the absolute stability of system (3).

Theorem 4. *The zero solution of the nonlinear system (3) is GAS for all , if is stable and there exist diagonal matrices and with and a symmetric matrix such that the LMI (5) is feasible. *

Before proving this theorem, some needed results and some discussions on the frequency-domain interpretation to LMI (5) are first introduced.

Proposition 5. * Under the condition of inequality (5), the following two statements are equivalent:*(i) *is stable; *(ii).

*Proof. *Define
Then the congruence transformation of by the nonsingular matrix
provides an equivalent inequality to (5)where . By the Schur complement (Lemma 3), (11) implies the following:
This inequality directly leads to the result of the proposition.

*Remark 6. *Proposition 5 shows that the conditions of Theorem 4 guaranteeing the GAS of system (3) are consistent with Lemma 1, except for the restriction of positiveness on matrix .

The next theorem reveals a frequency domain interpretation to the LMI (5).

Theorem 7. *Suppose that is stable. Then, there exist diagonal matrices and with and a symmetric matrix such that the LMI (5) is feasible if, and only if,
**
where . *

*Proof. *Let
Then, inequality (11), which is equivalent to the LMI (5) as proved in Proposition 5, can be rewritten as follows:
From the KYP lemma (Lemma 2), (15) holds if, and only if,
which is equivalent to inequality (13) through direct calculation.

Furthermore, noticing that
inequality (13) is equivalent to , where

*Remark 8. *With the stability of , the frequency-domain inequality (FDI) (13) is equivalent to that the transfer function as given in (18) is strictly positive real (SPR), which is consistent with the frequency-domain criterion of [4], except for the restriction of positiveness on matrix .

Proposition 9. *Suppose that is stable, then Theorem 4 (or equivalently Theorem 7) ensures that is stable for any . *

*Proof. *Suppose to the contrary that there exists a , , such that is not stable. Then, there exists a real number , , such that has at least an eigenvalue on the imaginary axis, which we denote by . Then
Since is stable, . It follows from (19) that
that is,
So there exists a vector , , such that
Thus , and
Equality (23) and the conditions and indicate the following:
On the other hand, it follows from the FDI (13) that
From (23),
and since , (25) indicates that
It contradicts to (24). The proof is completed.

Now, we are ready to prove Theorem 4.

*Proof of Theorem 4. *Take for Lyapunov function candidate
where is a symmetric matrix and , , need not to be positive.

The time derivative of along any trajectory of system (3) is given by the following:
where . For all , the sector condition (2) can be expressed as follows:
Let , where ; then
Thus, if it is possible to show that
it follows that . Inequality (32) can be written as follows:
which is equivalent to the existence of a feasible solution to the LMI (5).

Then (5) guarantees the negative definiteness of . In fact, (5) also guarantees that the Lyapunov candidate defined in (28) is positive definite, even without the requirement of . Rewrite in (28) as follows:
By Proposition 5, . And because of the sector condition (2). So, , holds if . In the case that there exists , we suppose that , , and , , without loss of generality. Then, we get the following inequality:
where , , and . The congruence transformation of in (9) defined by the nonsingular matrix
provides an equivalent inequality to the LMI (5):
where , , and . From Proposition 9, is stable; then it follows from (37) that
Thus, by (35), . By the canonical Lyapunov theory, system (3) is absolutely stable. Now, we complete the proof of Theorem 4.

Analogous to the proof of Theorem 4 while considering the stability of instead of , the following corollary can be deduced.

Corollary 10. *The zero solution of the nonlinear system (3) is GAS for all , if is stable (or equivalently, ) and there exist diagonal matrices and with and a symmetric matrix such that the LMI (5) is feasible. *

In the following, a numerical example is presented to illustrate the effectiveness of the proposed stability criteria.

*Example 11. *Consider the following system:
where
and is given in Table 1. The nonlinear function is supposed to belong to the sector , where
The purpose is to find a maximum upper bound such that system (39) is absolute stable for all . Using Theorem 2 in [4] and Theorem 4 in this paper, the corresponding for system (39) with different is listed in Table 1, from which it is shown that Theorem 4 in this paper is less conservative than Theorem 2 in [4].

Take
where . Then, the nonlinearity belongs to the sector , where
that is, in (41). The states of system (39) with and as given in (42) are presented in Figure 1, from which it is observed that the origin of the system is asymptotically stable.

#### 3. Master-Slave Synchronization

The absolute stability criteria proposed in the last section can be applied to the master-slave synchronization of coupled systems of form (3). Using two identical systems in a master-slave synchronization scheme with linear full static state feedback, one has the following: with master , slave , and feedback matrix . The aim of synchronization is then to obtain for time . Defining the error signal , one obtains the following error system: where . Assume a sector condition on , with and , which gives the following inequalities for :

Following a similar approach to the stability analysis, a synchronization criterion for the systems in (44) is obtained.

Theorem 12. *The zero solution of the error system (45) is GAS, which implies that system (44) synchronizes, if there exist diagonal matrices and with , a symmetric matrix , and a feedback matrix such that is stable, and*

*Proof. *This theorem can be completed by the method analogous to that employed in the last section, so its proof is omitted here.

*Remark 13. *For a given feedback matrix , condition (47) is a linear matrix inequality problem (LMI) in , , and . The overall design problem can be formulated as the optimization problem [18]:
where denotes the maximal eigenvalue of a symmetric matrix. Comparing with the synchronization criteria given in the literature [10–15], Theorem 12 is less conservative by relaxing the restriction of positiveness on matrix .

#### 4. Conclusion

In this paper, the absolute stability criteria for systems with state-dependent sector nonlinearities provided in [4] are further studied. By relaxing some restrictions, revised stability criteria are proposed, which further reduce the conservativeness of the stability conditions as shown in a numerical example. In addition, the feasibility of the derived LMIs actually implies some FDI conditions bearing the same forms as those in the circle criterion and the Popov criterion. Finally, based on the proposed stability results, a synchronization criterion is developed in a master-slave synchronization scheme.

#### Acknowledgment

This work is supported by the National Science Foundation of China under Grant no. 11101256.

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