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

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.

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][6][7][8][9][10][11][12][13][14][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 Mathematical Problems in Engineering 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.

Absolute Stability Criteria
To analyze the absolute stability of system (3), a Lur' e function where  = diag( 1 , . . .,   ) > 0 and  > −Γ, 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  ∈ F(Γ, Δ), if Γ is stable and there exist diagonal matrices  and  with  > 0 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: Then the congruence transformation of Φ by the nonsingular matrix provides an equivalent inequality to ( 5) where  1 =  + Γ.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 Γ is stable.Then, there exist diagonal matrices  and  with  > 0 and a symmetric matrix  such that the LMI (5) is feasible if, and only if, where  Γ () = ( − Γ) −1 .
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 Re{()} > 0, for all  ∈ R, where Remark 8.With the stability of Γ, 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 .
Analogous to the proof of Theorem 4 while considering the stability of Δ instead of Γ, the following corollary can be deduced.Corollary 10.The zero solution of the nonlinear system (3) is GAS for all  ∈ F(Γ, Δ), if Δ is stable (or equivalently,  + Δ > 0) and there exist diagonal matrices  and  with  > 0 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 [Γ, Δ], where The purpose is to find a maximum upper bound  max such that system (39) is absolute stable for all  <  max .Using Theorem 2 in [4] and Theorem 4 in this paper, the corresponding  max for system (39) with different  is listed in that is,  = 139.99 in (41).The states of system (39) with  = [ 10 −8 15 −3 ] and  as given in (42) are presented in Figure 1, from which it is observed that the origin of the system is asymptotically stable.

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 masterslave synchronization scheme with linear full static state feedback, one has the following: where (()) := (() + ()) − (()).Assume a sector condition (Γ, Δ) on (⋅), with Γ = diag( 1 , . . .,   ) and Δ = diag( 1 , . . .,   ), 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  > 0, a symmetric matrix , and a feedback matrix  such that Γ −  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]: min ,,, where  max [⋅] denotes the maximal eigenvalue of a symmetric matrix.Comparing with the synchronization criteria given in the literature [10][11][12][13][14][15], Theorem 12 is less conservative by relaxing the restriction of positiveness on matrix .

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.

Table 1 :
The matrix  and the maximum allowed  max .