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

Authors: Rong Li; Bo Liu; Chao Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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:(1)x˙i=∑j=1n‍aijfj(xj),fj(0)=0,j=1,…,n,
where x=(x1,…,xn)T∈ℝn is the system state vector and fi:ℝ→ℝ,i=1,…,n, are time-invariant state-dependent nonlinearities with sector restrictions such as the following:
(2)γiτ2≤fi(τ)τ≤δiτ2,i=1,…,n.
For convenience, system  (1) will often be written by using the following compact notation:
(3)x˙=Af(x),f(0)=0,
where A:=(aij) and f(x):=(f1,…,fn)T. And the sector conditions  (2) will be denoted as follows:
(4)F(Γ,Δ),
where Γ:=diag(γ1,…,γn) and Δ:=diag(δ1,…,δn).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
(5)[ATP+PA-QATW+12(Γ+Δ)QWA+12Q(Γ+Δ)-ΓΔQ]<0
is feasible, where Γ:=diag(γ1,…,γn) and Δ:=diag(δ1,…,δn).Lemma 2 (KYP lemma, Rantzer [16]).
GivenA∈ℝn×n, B∈ℝn×m, Π=ΠT∈ℝ(n+m)×(n+m), with det(jωI-A)≠0 for all ω∈ℝ, the following statements are equivalent.(i)
[(jωI-A)-1BI]*Π[(jωI-A)-1BI]<0 for all ω∈ℝ.(ii)
There exists a matrixP=PT∈ℝn×n such that
(6)[ATP+PATPBBTP0]+Π<0.Lemma 3 (Schur complement, Boyd et al. [17]).
The LMI(7)[S11S12S12T-S22]<0
with S11=S11T and S22=S22T is equivalent to one of the following statements:(i)
S22>0 and S11+S12S22-1S12T<0;(ii)
S11<0 and S22+S12TS11-1S12>0.


## 2. Absolute Stability Criteria

To analyze the absolute stability of system  (3), a Lur’e function
(8)V(x)=xTWx+2∑i=1n‍pi∫0xi‍fi(τ)dτ,
where P=diag(p1,…,pn)>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 matrixP 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 pi 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(9)Φ=[ATP+PA-QATW+12(Γ+Δ)QWA+12Q(Γ+Δ)-ΓΔQ].
Then the congruence transformation of Φ by the nonsingular matrix
(10)T1=[ΓII0]
provides an equivalent inequality to  (5)(11)T1ΦT1T=[ΓATY1+Y1AΓY1A+ΓATP-12(Γ-Δ)QATY1+PAΓ-12Q(Γ-Δ)ATP+PA-Q]<0,where Y1=W+PΓ. By the Schur complement (Lemma 3),  (11) implies the following:
(12)ΓAT(W+PΓ)+(W+PΓ)AΓ<0.
This inequality directly leads to the result of the proposition.Remark 6.
Proposition5 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 thatAΓ 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,
(13)Re{(Δ-Γ)QKΓ(jω)+2jωPKΓ(jω)-Q}<0,∀ω∈ℝ,
where KΓ(s)=(sI-AΓ)-1A.Proof.
Let(14)Π=[0ΓATP-12(Γ-Δ)QPAΓ-12Q(Γ-Δ)ATP+PA-Q].
Then, inequality  (11), which is equivalent to the LMI  (5) as proved in Proposition 5, can be rewritten as follows:
(15)Π+[ΓATY1+Y1AΓY1AATY10]<0.
From the KYP lemma (Lemma 2),  (15) holds if, and only if,
(16)[(jωI-AΓ)-1AI]*Π[(jωI-AΓ)-1AI]<0,∀ω∈ℝ,
which is equivalent to inequality  (13) through direct calculation.
Furthermore, noticing that(17)sPKΓ(s)=sP(sI-AΓ)-1A=P(I-s-1AΓ)-1A=PA(I-s-1ΓA)-1=PA[I+s-1Γ(I-s-1AΓ)-1A]=PA[I+Γ(sI-AΓ)-1A],
inequality  (13) is equivalent to Re{M(jω)}>0,forallω∈ℝ, where
(18)M(s)=[12Q(Γ-Δ)-PAΓ](sI-AΓ)-1A+(12Q-PA).Remark 8.
With the stability ofAΓ, 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 thatAΓ 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 aΔ~, 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
(19)det(jω0I-AΓ-αAΔ~)=0.
Since AΓ is stable, det(jω0I-AΓ)≠0. It follows from  (19) that
(20)det(I-α(jω0I-AΓ)-1AΔ~)=0;
that is,
(21)det(I-αKΓ(jω0)Δ~)=0.
So there exists a vector η∈ℂn,  η≠0, such that
(22)η*(I-αΔ~KΓ*(jω0))=0.
Thus η*Δ~≠0, and
(23)η*=αη*Δ~KΓ*(jω0),η=αKΓ(jω0)Δ~η.
Equality  (23) and the conditions 0≤Δ~≤Δ-Γ and 0<α≤1 indicate the following:
(24)η*Δ~[-2Q+(Δ-Γ)QKΓ(jω0)+KΓ*(jω0)Q(Δ-Γ)]Δ~η=η*[-2Δ~QΔ~+2α(Δ-Γ)QΔ~]η≥0.
On the other hand, it follows from the FDI  (13) that
(25)η*Δ~[-Q+(Δ-Γ)QKΓ(jω0)+2jω0PKΓ(jω0)-Q+KΓ*(jω0)Q(Δ-Γ)-2jω0KΓ*(jω0)P]Δ~η<0.
From  (23),
(26)η*Δ~2jω0PKΓ(jω0)Δ~η=2αjω0η*Δ~Pη,η*Δ~2jω0KΓ*(jω0)PΔ~η=2αjω0η*PΔ~η,
and since Δ~P=PΔ~,  (25) indicates that
(27)η*Δ~[-2Q+(Δ-Γ)QKΓ(jω0)+KΓ*(jω0)Q(Δ-Γ)]Δ~η<0.
It contradicts to  (24). The proof is completed.Now, we are ready to prove Theorem4.Proof of Theorem4.
Take for Lyapunov function candidate(28)V(x)=xTWx+2∑i=1n‍pi∫0xi‍fi(τ)dτ,
where W is a symmetric matrix and pi,  i=1,…,n, need not to be positive.
The time derivative ofV(x) along any trajectory of system  (3) is given by the following:
(29)V˙(x)=2f(x)TATWx+f(x)T(ATP+PA)f(x),
where P=diag(p1,…,pn). For all xi≠0, the sector condition  (2) can be expressed as follows:
(30)si(xi):=(fi(xi)-γixi)(fi(xi)-δixi)≤0,∀xi≠0,i=1,…,n.
Let Q=diag(q1,…,qn), where qi>0,i=1,…,n; then
(31)∑i=1n‍qisi(xi)=f(x)TQf(x)-f(x)T(Γ+Δ)Qx+xTΓΔQx≤0,∀(x,f(x))≠0.
Thus, if it is possible to show that
(32)V˙(x)<∑i=1n‍qisi(xi),∀(x,f(x))≠0,
it follows that V˙(x)<0,forallx≠0. Inequality  (32) can be written as follows:
(33)(f(x)x)T[ATP+PA-QATW+12(Γ+Δ)QWA+12Q(Γ+Δ)-ΓΔQ]×(f(x)x)<0,∀(f(x)x)≠0,
which is equivalent to the existence of a feasible solution to the LMI  (5).
Then  (5) guarantees the negative definiteness of  V˙(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:
(34)V(x)=xTWx+2∑i=1n‍pi∫0xi‍fi(τ)dτ=xTWx+2∑i=1n‍pi∫0xi‍(γiτ+fi(τ)-γiτ)dτ=xT(W+PΓ)x+2∑i=1n‍pi∫0xi‍(fi(τ)-γiτ)dτ.
By Proposition 5, W+PΓ>0. And ∫0xi‍(fi(τ)-γiτ)dτ≥0 because of the sector condition  (2). So, V(x)>0,forallx≠0, holds if pi≥0,i=1,…,n. In the case that there exists pi<0, we suppose that pi<0,  i=1,…,k, and pi≥0,  i=k+1,…,n, without loss of generality. Then, we get the following inequality:
(35)V(x)≥xT(W+PΓ)x+2∑i=1k‍pi∫0xi‍(δiτ-γiτ)dτ+2∑i=k+1n‍pi∫0xi‍(fi(τ)-γiτ)dτ=xT(W+PΓ+Pk(Δk-Γk))x+2∑i=k+1n‍pi∫0xi‍(fi(τ)-γiτ)dτ,
where Pk=diag(p1,…,pk,0,…,0), Δk=diag(δ1,…,δk,0,…,0), and Γk=diag(γ1,…,γk,0,…,0). The congruence transformation of Φ in  (9) defined by the nonsingular matrix
(36)T2=[Γ+(Δk-Γk)II0]
provides an equivalent inequality to the LMI  (5):
(37)Ψ=T2ΦT2T=[A~TY2+Y2A~Ψ12Ψ12TATP+PA-Q]<0,
where A~=A[Γ+(Δk-Γk)],  Y2=W+PΓ+Pk(Δk-Γk),   and Ψ12=Y2A+A~TP+(1/2)(Δ-Γ)Q-(Δk-Γk)Q.   From Proposition 9, A~ is stable; then it follows from  (37) that
(38)Y2=W+PΓ+Pk(Δk-Γk)>0.
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 Theorem4 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:(39)x˙(t)=Af(x(t)),f(0)=0,
where
(40)x(t)=[x1(t)x2(t)],f(x(t))=[f1(x1(t))f2(x2(t))],
and A is given in Table 1. The nonlinear function f is supposed to belong to the sector [Γ,Δ], where
(41)Γ=[-20000-200],Δ=Γ+[δ00δ].
The purpose is to find a maximum upper bound δmax such that system  (39) is absolute stable for all δ<δmax. Using Theorem 2 in [4] and Theorem 4 in this paper, the corresponding δmax 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(42)fi(xi(t))=Gbixi(t)+12(Gai-Gbi)(|xi(t)+1|-|xi(t)-1|),i=1,2,
where Ga1=-200,Gb1=-60.01,Ga2=-200,andGb2=-60.01. Then, the nonlinearity f belongs to the sector [Γ,Δ], where
(43)Γ=[-20000-200],Δ=[-60.0100-60.01];
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 matrixA and the maximum allowed δmax.
A
[
2
-
3
1
-
1
]
[
5
-
4
5
-
3
]
[
10
-
8
15
-
3
]
δ
max from [4]
86.85
71.65
93.33
δ
max from Theorem 4
99.99
79.99
139.99Figure 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:
(44)(M)x˙(t)=Af(x(t)),(S)z˙(t)=Af(z(t))+K(x(t)-z(t)),
with master M, slave S, and feedback matrix K∈Rn×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:
(45)e˙(t)=A(η(e(t)))-K(e(t)),
where η(e(t)):=f(z(t)+e(t))-f(z(t)). Assume a sector condition (Γ¯,Δ¯) on η(·), with Γ¯=diag(γ1¯,…,γn¯) and Δ¯=diag(δ1¯,…,δn¯), which gives the following inequalities for η:
(46)γi¯τ2≤ηi(τ)τ≤δi¯τ2,i=1,…,n.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, and(47)Φ¯=[ATP+PA-QATW+12(Γ+Δ)Q-PKWA+12Q(Γ+Δ)-KTP-(WK+KTW)-ΓΔQ]<0.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 matrixK, 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]:
(48)minK,P,Q,Wλmax[Φ¯],
where λmax[·] 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.



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