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

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: 𝑥̇_𝑖=^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗,𝑓_𝑗(0)=0,𝑗=1,…,𝑛, (1) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 is the system state vector and 𝑓_𝑖∶ℝ→ℝ,𝑖=1,…,𝑛, are time-invariant state-dependent nonlinearities with sector restrictions such as the following: 𝛾_𝑖𝜏²≤𝑓_𝑖(𝜏)𝜏≤𝛿_𝑖𝜏²,𝑖=1,…,𝑛. (2) For convenience, system (1) will often be written by using the following compact notation: 𝑥̇=𝐴𝑓(𝑥),𝑓(𝟎)=𝟎, (3) where 𝐴∶=(𝑎_𝑖𝑗) and 𝑓(𝑥)∶=(𝑓₁,…,𝑓_𝑛)^𝑇. And the sector conditions (2) will be denoted as follows: 𝐅(Γ,Δ), (4) where Γ∶=diag(𝛾₁,…,𝛾_𝑛) and Δ∶=diag(𝛿₁,…,𝛿_𝑛).

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 𝑃 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 𝑓∈𝐅(Γ,Δ), if there exist diagonal and positive-definite matrices 𝑃 and 𝑄, and a symmetric matrix 𝑊>−𝑃Γ such that the following LMI 𝐴^𝑇𝑃+𝑃𝐴−𝑄𝐴^𝑇1𝑊+2()Q1Γ+Δ𝑊𝐴+2𝑄(Γ+Δ)−ΓΔ𝑄<0 (5) is feasible, where Γ∶=diag(𝛾₁,…,𝛾_𝑛) and Δ∶=diag(𝛿₁,…,𝛿_𝑛).

Lemma 2 (KYP lemma, Rantzer [16]). Given 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, Π=Π^𝑇∈ℝ^{(𝑛+𝑚)×(𝑛+𝑚)}, with det(𝑗𝜔𝐼−𝐴)≠0 for all 𝜔∈ℝ, the following statements are equivalent.(i)^{(𝑗𝜔𝐼−𝐴)−1𝐵}_𝐼^∗Π^{(𝑗𝜔𝐼−𝐴)−1𝐵}_𝐼<0 for all 𝜔∈ℝ.(ii)There exists a matrix 𝑃=𝑃^𝑇∈ℝ^𝑛×𝑛 such that 𝐴^𝑇𝑃+𝑃𝐴^𝑇𝐵𝑃𝐵_𝑇𝑃0+Π<0. (6)

Lemma 3 (Schur complement, Boyd et al. [17]). The LMI 𝑆₁₁𝑆₁₂𝑆^𝑇₁₂−𝑆₂₂<0 (7) with 𝑆₁₁=𝑆^𝑇₁₁ and 𝑆₂₂=𝑆^𝑇₂₂ is equivalent to one of the following statements:(i)𝑆₂₂>0 and 𝑆₁₁+𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0;(ii)𝑆₁₁<0 and 𝑆₂₂+𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂>0.

2. Absolute Stability Criteria

To analyze the absolute stability of system (3), a Lur’e function 𝑉(𝑥)=𝑥^𝑇𝑊𝑥+2^𝑛_𝑖=1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)𝑑𝜏, (8) where 𝑃=diag(𝑝₁,…,𝑝_𝑛)>0 and 𝑊>−𝑃Γ, 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 𝑆-procedure (see [4] for details).

Note that the diagonal matrix 𝑃 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 𝑃 is unnecessary by finally showing that the Lur’e function 𝑉(𝑥) in (8) can still be taken as a Lyapunov candidate when some or even all entries 𝑝_𝑖 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∈𝐅(Γ,Δ), if 𝐴Γ is stable and there exist diagonal matrices 𝑃 and 𝑄 with 𝑄>0 and a symmetric matrix 𝑊 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)𝐴Γ   is stable; (ii)𝑊+𝑃Γ>0.

Proof. Define 𝐴Φ=^𝑇𝑃+𝑃𝐴−𝑄𝐴^𝑇1𝑊+2()𝑄1Γ+Δ𝑊𝐴+2𝑄(Γ+Δ)−ΓΔ𝑄. (9) Then the congruence transformation of Φ by the nonsingular matrix 𝑇₁=Γ𝐼𝐼0 (10) provides an equivalent inequality to (5)𝑇₁Φ𝑇^𝑇₁=Γ𝐴^𝑇𝑌₁+𝑌₁𝐴Γ𝑌₁𝐴+Γ𝐴^𝑇1𝑃−2()𝑄𝐴Γ−Δ_𝑇𝑌₁1+𝑃𝐴Γ−2𝑄(Γ−Δ)𝐴^𝑇𝑃+𝑃𝐴−𝑄<0, (11)where 𝑌₁=𝑊+𝑃Γ. By the Schur complement (Lemma 3), (11) implies the following: Γ𝐴^𝑇(𝑊+𝑃Γ)+(𝑊+𝑃Γ)𝐴Γ<0. (12) 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 𝑃.

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

Theorem 7. Suppose that 𝐴Γ is stable. Then, there exist diagonal matrices 𝑃 and 𝑄 with 𝑄>0 and a symmetric matrix 𝑊 such that the LMI (5) is feasible if, and only if, Re(Δ−Γ)𝑄𝐾_Γ𝑗𝜔+2𝑗𝜔𝑃𝐾_Γ𝑗𝜔−𝑄<0,∀𝜔∈ℝ, (13) where 𝐾_Γ(𝑠)=(𝑠𝐼−𝐴Γ)⁻¹𝐴.

Proof. Let Π=0Γ𝐴^𝑇1𝑃−2()𝑄1Γ−Δ𝑃𝐴Γ−2𝑄(Γ−Δ)𝐴^𝑇𝑃+𝑃𝐴−𝑄. (14) Then, inequality (11), which is equivalent to the LMI (5) as proved in Proposition 5, can be rewritten as follows: Π+Γ𝐴^𝑇𝑌₁+𝑌₁𝐴Γ𝑌₁𝐴𝐴^𝑇𝑌₁0<0. (15) From the KYP lemma (Lemma 2), (15) holds if, and only if, 𝑗𝜔𝐼−𝐴Γ⁻¹𝐴𝐼^∗Π𝑗𝜔𝐼−𝐴Γ⁻¹𝐴𝐼<0,∀𝜔∈ℝ, (16) which is equivalent to inequality (13) through direct calculation.
Furthermore, noticing that 𝑠𝑃𝐾_Γ(𝑠)=𝑠𝑃(𝑠𝐼−𝐴Γ)⁻¹𝐴=𝑃𝐼−𝑠⁻¹𝐴Γ⁻¹𝐴=𝑃𝐴𝐼−𝑠⁻¹Γ𝐴⁻¹=𝑃𝐴𝐼+𝑠⁻¹Γ𝐼−𝑠⁻¹𝐴Γ⁻¹𝐴=𝑃𝐴𝐼+Γ(𝑠𝐼−𝐴Γ)⁻¹𝐴, (17) inequality (13) is equivalent to Re{𝑀(𝑗𝜔)}>0,forall𝜔∈ℝ, where 𝑀(𝑠)=12𝑄(Γ−Δ)−𝑃𝐴Γ(𝑠𝐼−𝐴Γ)⁻¹1𝐴+2𝑄−𝑃𝐴. (18)

Remark 8. With the stability of 𝐴Γ, the frequency-domain inequality (FDI) (13) is equivalent to that the transfer function 𝑀(𝑠) 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 𝑃.

Proposition 9. Suppose that 𝐴Γ is stable, then Theorem 4 (or equivalently Theorem 7) ensures that 𝐴(Γ+Δ) is stable for any 0≤Δ≤Δ−Γ.

Proof. Suppose to the contrary that there exists a Δ, 0≤Δ≤Δ−Γ, such that 𝐴(Γ+Δ) is not stable. Then, there exists a real number 𝛼, 0≤𝛼≤1, such that 𝐴(Γ+𝛼Δ) has at least an eigenvalue on the imaginary axis, which we denote by 𝑗𝜔₀,𝜔₀∈ℝ. Then det𝑗𝜔₀𝐼−𝐴Γ−𝛼𝐴Δ=0. (19) Since 𝐴Γ is stable, det(𝑗𝜔₀𝐼−𝐴Γ)≠0. It follows from (19) that det𝐼−𝛼𝑗𝜔₀𝐼−𝐴Γ⁻¹𝐴Δ=0; (20) that is, det𝐼−𝛼𝐾_Γ𝑗𝜔₀Δ=0. (21) So there exists a vector 𝜂∈ℂ^𝑛,  𝜂≠0, such that 𝜂^∗𝐼−𝛼Δ𝐾^∗_Γ𝑗𝜔₀=0. (22) Thus 𝜂^∗Δ≠0, and 𝜂^∗=𝛼𝜂^∗Δ𝐾^∗_Γ𝑗𝜔₀,𝜂=𝛼𝐾_Γ𝑗𝜔₀Δ𝜂. (23) Equality (23) and the conditions 0≤Δ≤Δ−Γ and 0<𝛼≤1 indicate the following: 𝜂^∗Δ−2𝑄+(Δ−Γ)𝑄𝐾_Γ𝑗𝜔₀+𝐾^∗_Γ𝑗𝜔₀𝑄(Δ−Γ)Δ𝜂=𝜂^∗2−2Δ𝑄Δ+𝛼(Δ−Γ)𝑄Δ𝜂≥0. (24) On the other hand, it follows from the FDI (13) that 𝜂^∗Δ−𝑄+(Δ−Γ)𝑄𝐾_Γ𝑗𝜔₀+2𝑗𝜔₀𝑃𝐾_Γ𝑗𝜔₀−𝑄+𝐾^∗_Γ𝑗𝜔₀𝑄(Δ−Γ)−2𝑗𝜔₀𝐾^∗_Γ𝑗𝜔₀𝑃Δ𝜂<0. (25) From (23), 𝜂^∗Δ2𝑗𝜔₀𝑃𝐾_Γ𝑗𝜔₀2Δ𝜂=𝛼𝑗𝜔₀𝜂^∗𝜂Δ𝑃𝜂,_∗Δ2𝑗𝜔₀𝐾^∗_Γ𝑗𝜔₀𝑃2Δ𝜂=𝛼𝑗𝜔₀𝜂^∗𝑃Δ𝜂, (26) and since Δ𝑃=𝑃Δ, (25) indicates that 𝜂^∗Δ−2𝑄+(Δ−Γ)𝑄𝐾_Γ𝑗𝜔₀+𝐾^∗_Γ𝑗𝜔₀𝑄(Δ−Γ)Δ𝜂<0. (27) 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 𝑉(𝑥)=𝑥^𝑇𝑊𝑥+2^𝑛_𝑖=1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)𝑑𝜏, (28) where 𝑊 is a symmetric matrix and 𝑝_𝑖,  𝑖=1,…,𝑛, need not to be positive.
The time derivative of  𝑉(𝑥) along any trajectory of system (3) is given by the following: 𝑉̇(𝑥)=2𝑓(𝑥)^𝑇𝐴^𝑇𝑊𝑥+𝑓(𝑥)^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑓(𝑥), (29) where 𝑃=diag(𝑝₁,…,𝑝_𝑛). For all 𝑥_𝑖≠0, the sector condition (2) can be expressed as follows: 𝑠_𝑖𝑥_𝑖∶=𝑓_𝑖𝑥_𝑖−𝛾_𝑖𝑥_𝑖𝑓_𝑖𝑥_𝑖−𝛿_𝑖𝑥_𝑖≤0,∀𝑥_𝑖≠0,𝑖=1,…,𝑛. (30) Let 𝑄=diag(𝑞₁,…,𝑞_𝑛), where 𝑞_𝑖>0,𝑖=1,…,𝑛; then ^𝑛_𝑖=1𝑞_𝑖𝑠_𝑖𝑥_𝑖=𝑓(𝑥)^𝑇𝑄𝑓(𝑥)−𝑓(𝑥)^𝑇(Γ+Δ)𝑄𝑥+𝑥^𝑇(𝑥)ΓΔ𝑄𝑥≤0,∀𝑥,𝑓≠0. (31) Thus, if it is possible to show that 𝑉̇(𝑥)<^𝑛_𝑖=1𝑞_𝑖𝑠_𝑖𝑥_𝑖,∀𝑥,𝑓(𝑥)≠0, (32) it follows that 𝑉̇(𝑥)<0,forall𝑥≠0. Inequality (32) can be written as follows: 𝑓(𝑥)𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴−𝑄𝐴^𝑇1𝑊+2()𝑄1Γ+Δ𝑊𝐴+2𝑄()×𝑓(𝑥)𝑥𝑓(𝑥)𝑥Γ+Δ−ΓΔ𝑄<0,∀≠0, (33) which is equivalent to the existence of a feasible solution to the LMI (5).
Then (5) guarantees the negative definiteness of  𝑉̇(𝑥). In fact, (5) also guarantees that the Lyapunov candidate 𝑉(𝑥) defined in (28) is positive definite, even without the requirement of 𝑃>0. Rewrite 𝑉(𝑥) in (28) as follows: 𝑉(𝑥)=𝑥^𝑇𝑊𝑥+2^𝑛_𝑖=1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)𝑑𝜏=𝑥^𝑇𝑊𝑥+2^𝑛_𝑖=1𝑝_𝑖^𝑥𝑖₀𝛾_𝑖𝜏+𝑓_𝑖(𝜏)−𝛾_𝑖𝜏𝑑𝜏=𝑥^𝑇(𝑊+𝑃Γ)𝑥+2^𝑛_𝑖=1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)−𝛾_𝑖𝜏𝑑𝜏. (34) By Proposition 5, 𝑊+𝑃Γ>0. And ∫^𝑥𝑖₀(𝑓_𝑖(𝜏)−𝛾_𝑖𝜏)𝑑𝜏≥0 because of the sector condition (2). So, 𝑉(𝑥)>0,forall𝑥≠0, holds if 𝑝_𝑖≥0,𝑖=1,…,𝑛. In the case that there exists 𝑝_𝑖<0, we suppose that 𝑝_𝑖<0,  𝑖=1,…,𝑘, and 𝑝_𝑖≥0,  𝑖=𝑘+1,…,𝑛, without loss of generality. Then, we get the following inequality: 𝑉(𝑥)≥𝑥^𝑇(𝑊+𝑃Γ)𝑥+2^𝑘_𝑖=1𝑝_𝑖^𝑥𝑖₀𝛿_𝑖𝜏−𝛾_𝑖𝜏𝑑𝜏+2^𝑛_𝑖=𝑘+1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)−𝛾_𝑖𝜏𝑑𝜏=𝑥^𝑇𝑊+𝑃Γ+𝑃_𝑘Δ_𝑘−Γ_𝑘𝑥+2^𝑛_𝑖=𝑘+1𝑝_𝑖^𝑥𝑖₀𝑓_𝑖(𝜏)−𝛾_𝑖𝜏𝑑𝜏, (35) where 𝑃_𝑘=diag(𝑝₁,…,𝑝_𝑘,0,…,0), Δ_𝑘=diag(𝛿₁,…,𝛿_𝑘,0,…,0), and Γ_𝑘=diag(𝛾₁,…,𝛾_𝑘,0,…,0). The congruence transformation of Φ in (9) defined by the nonsingular matrix 𝑇₂=Γ+Δ_𝑘−Γ_𝑘𝐼𝐼0 (36) provides an equivalent inequality to the LMI (5): Ψ=𝑇₂Φ𝑇^𝑇₂=𝐴^𝑇𝑌₂+𝑌₂𝐴Ψ₁₂Ψ^𝑇₁₂𝐴^𝑇𝑃+𝑃𝐴−𝑄<0, (37) where 𝐴=𝐴[Γ+(Δ_𝑘−Γ_𝑘)],  𝑌₂=𝑊+𝑃Γ+𝑃_𝑘(Δ_𝑘−Γ_𝑘),   and Ψ₁₂=𝑌₂𝐴+𝐴^𝑇𝑃+(1/2)(Δ−Γ)𝑄−(Δ_𝑘−Γ_𝑘)𝑄.   From Proposition 9, 𝐴 is stable; then it follows from (37) that 𝑌₂=𝑊+𝑃Γ+𝑃_𝑘Δ_𝑘−Γ_𝑘>0. (38) Thus, by (35), 𝑉(𝑥)>0,forall𝑥≠0. 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 𝐴Δ instead of 𝐴Γ, the following corollary can be deduced.

Corollary 10. The zero solution of the nonlinear system (3) is GAS for all 𝑓∈𝐅(Γ,Δ), if 𝐴Δ is stable (or equivalently, 𝑊+𝑃Δ>0) and there exist diagonal matrices 𝑃 and 𝑄 with 𝑄>0 and a symmetric matrix 𝑊 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: 𝑥̇(𝑡)=𝐴𝑓(𝑥(𝑡)),𝑓(𝟎)=𝟎, (39) where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)(𝑥(𝑡=𝑓,𝑓))¹𝑥₁(𝑡)𝑓₂𝑥₂(𝑡), (40) and 𝐴 is given in Table 1. The nonlinear function 𝑓 is supposed to belong to the sector [Γ,Δ], where Γ=−20000−200,Δ=Γ+𝛿00𝛿. (41) The purpose is to find a maximum upper bound 𝛿_max such that system (39) is absolute stable for all 𝛿<𝛿_max. Using Theorem 2 in [4] and Theorem 4 in this paper, the corresponding 𝛿_max for system (39) with different 𝐴 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 𝑓_𝑖𝑥_𝑖(𝑡)=𝐺_𝑏𝑖𝑥_𝑖(𝑡)+12𝐺_𝑎𝑖−𝐺_𝑏𝑖𝑥_𝑖(𝑡)+1−𝑥_𝑖(𝑡)−1,𝑖=1,2, (42) where 𝐺_𝑎1=−200,𝐺_𝑏1=−60.01,𝐺_𝑎2=−200,and𝐺_𝑏2=−60.01. Then, the nonlinearity 𝑓 belongs to the sector [Γ,Δ], where Γ=−20000−200,Δ=−60.0100−60.01; (43) that is, 𝛿=139.99 in (41). The states of system (39) with 𝐴=¹⁰⁻⁸₁₅₋₃ and 𝑓 as given in (42) are presented in Figure 1, from which it is observed that the origin of the system is asymptotically stable.

Table 1 

The matrix 𝐴 and the maximum allowed 𝛿_max.

Figure 1 

The state of system (39) with 𝐴=10−815−3 and 𝛿=139.99.

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: (𝑀)𝑥̇(𝑡)(𝑥(𝑡,(𝑆)𝑧̇(𝑡)(𝑧(𝑡(𝑥(𝑡)(𝑡,=𝐴𝑓))=𝐴𝑓))+𝐾−𝑧)) (44) with master 𝑀, slave 𝑆, and feedback matrix 𝐾∈𝑅^𝑛×𝑛. The aim of synchronization is then to obtain ‖𝑥(𝑡)−𝑧(𝑡)‖→0 for time 𝑡→∞. Defining the error signal 𝑒=𝑥−𝑧, one obtains the following error system: 𝑒̇(𝑡)=𝐴𝜂(𝑒(𝑡))−𝐾(𝑒(𝑡)), (45) where 𝜂(𝑒(𝑡))∶=𝑓(𝑧(𝑡)+𝑒(𝑡))−𝑓(𝑧(𝑡)). Assume a sector condition (Γ,Δ) on 𝜂(⋅), with Γ=diag(𝛾₁,…,𝛾_𝑛) and Δ=diag(𝛿₁,…,𝛿_𝑛), which gives the following inequalities for 𝜂: 𝛾_𝑖𝜏²≤𝜂_𝑖(𝜏)𝜏≤𝛿_𝑖𝜏²,𝑖=1,…,𝑛. (46)

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 𝑃 and 𝑄 with 𝑄>0, a symmetric matrix 𝑊, and a feedback matrix 𝐾 such that 𝐴Γ−𝐾 is stable, and𝐴Φ=^𝑇𝑃+𝑃𝐴−𝑄𝐴^𝑇1𝑊+2()1Γ+Δ𝑄−𝑃𝐾𝑊𝐴+2𝑄(Γ+Δ)−𝐾^𝑇𝑃−𝑊𝐾+𝐾^𝑇𝑊−ΓΔ𝑄<0. (47)

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 𝐾, condition (47) is a linear matrix inequality problem (LMI) in 𝑃, 𝑄, and 𝑊. The overall design problem can be formulated as the optimization problem [18]: min_{𝐾,𝑃,𝑄,𝑊}𝜆_maxΦ, (48) where 𝜆_max[⋅] 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 𝑃.

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.