Physics Letters A 374 (2010) 2835–2840

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Physics Letters A

www.elsevier.com/locate/pla

Feedback control design for Rössler and Chen chaotic systemsanti-synchronization

S. Hammami a,c,∗, M. Benrejeb a, M. Feki b, P. Borne c

a UR LARA Automatique, Ecole Nationale d’Ingénieurs de Tunis, BP 37, Tunis Le Belvédère, 1002, Tunisiab UR MECA, Ecole Nationale d’Ingénieurs de Sfax, BP 3038, Sfax, Tunisiac LAGIS, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d’Ascq Cedex, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 February 2010Received in revised form 7 April 2010Accepted 4 May 2010Available online 7 May 2010Communicated by A.R. Bishop

Keywords:Chaotic systemsAnti-synchronizationAggregation techniquesArrow form matrixFeedback stabilization

The proposed anti-synchronization conditions of coupled chaotic systems are based, in this Letter, on theuse of aggregation techniques for the stability study of the error dynamics. The schemes are, successfully,applied to coupled Rössler and Chen chaotic systems making the instantaneous characteristic matrixunder the arrow form. Numerical simulations are performed to illustrate the efficiency of the proposedapproach.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

After the pioneering work on chaotic systems of Pecora andCarroll [1], the concept of synchronization has been extendedto the generalized synchronization [2–4], the phase synchroniza-tion [2], the lag synchronization [5] and the anti-phase synchro-nization [6,7]. Anti-synchronization is a phenomenon that the statevectors of the synchronized systems have the same amplitude butopposite signs as those of the driving system. Therefore, the sumof two signals is expected to converge to zero when either anti-synchronization or anti-phase synchronization appears. Recently,several stability methods have been applied to anti-synchronizechaotic systems [8–12].

To lead stability study, one can start with a system description(respectively stability methods), then choose an adapted analysismethod (respectively system description). Then, the problem ofLyapunov functions construction and the problem of determiningthe largest stability domain, for example, can be transformed by asuitable system description choice [13–16].

Several approaches are considered in the literature using de-composition of a large scale system into subsystems, or of the

* Corresponding author at: UR LARA Automatique, Ecole Nationale d’Ingénieursde Tunis, BP 37, Tunis Le Belvédère, 1002, Tunisia.

E-mail addresses: sonia.hammami@enit.rnu.tn (S. Hammami),mohamed.benrejeb@enit.rnu.tn (M. Benrejeb), moez.feki@enig.rnu.tn (M. Feki),pierre.borne@ec-lille.fr (P. Borne).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.05.008

vector Lyapunov function or the vector norm taking into accountspecific theoretical or physical properties of the process.

Based on aggregation techniques and on the use of the Borne–Gentina practical stability criterion applied to continuous systems(Appendix A) [17,18], this Letter sets out to establish a new outputfeedback stabilizing approach for nonlinear continuous hierarchicalsystems. The proposed approach, which constitutes an extensionof the used state feedback stabilizing approach formulated in ourprevious work [4], is carried out through the determination of con-trol laws, guaranteeing the asymptotic stabilisability property bymaking the matrix description of the controlled system under thearrow form [16–19].

This arrow form representation was used in previous works onasymptotic and global stability of nonlinear systems (Appendix B)[16] and, recently, on chaotic systems synchronization [4]. It ap-pears very suitable for two-level hierarchical systems with manynonlinearities and can be extended without any difficulty to mul-tilevel hierarchical system description.

Indeed, the main purpose in this work is to design an adaptiveoutput feedback controller guaranteeing the asymptotic stabilitythen the anti-synchronization of nonlinear continuous error of twoidentical and different chaotic systems.

The Letter is organized as follows. After a brief description ofthe studied systems and the definition of the anti-synchronizationconcept, Section 2 investigates the design of the proposed anti-synchronous output feedback controller. The case of two identicalRössler systems is considered in Section 3. Section 4 deals, finally,

2836 S. Hammami et al. / Physics Letters A 374 (2010) 2835–2840

with non-identical anti-synchronization between Rössler and Chensystems case.

2. Main result

Consider a system with a general form, which is usually calledas master system:

(Sm):

{xm(t) = Am

(xm(t)

)xm(t)

ym(t) = Cmxm(t)(1)

and a controlled system named as slave system:

(Ss):

{xs(t) = As

(xs(t)

)xs(t) + Bsu

(xm(t), xs(t)

)ys(t) = Csxs(t)

(2)

with

xm(t) = [xm1(t) . . . xmn(t)

]T ∈ Rn

xs(t) = [xs1(t) . . . xsn(t)

]T ∈ Rn

the state vectors of master and slave systems, respectively,

ym(t) = [ym1(t) . . . yml(t)

]T ∈ Rl

ys(t) = [ys1(t) . . . ysl(t)

]T ∈ Rl

the outputs of the master and slave systems, respectively, u(xm(t),xs(t)) ∈ R

m the control input of the slave system, and Am ∈ Rn×n ,

As ∈ Rn×n, Bs ∈ R

n×m , Cm ∈ Rl×n, Cs ∈ R

l×n .Let the state error e(t) be:

e(t) = xs(t) + xm(t) (3)

and the output error signal ye(t), such that:

ye(t) = ys(t) + ym(t) (4)

It comes:

e(t) = xs(t) + xm(t) (5)

Definition. (See [20].) The two systems, (Sm) and (Ss), satisfy theproperty of anti-synchronization, between xm(t) and xs(t), if thereexists an anti-synchronous manifold:

N = {(xm(t), xs(t)

): xm(t) = −xs(t)

}(6)

such that all trajectories (xm(t), xs(t)) approach N as time goes toinfinity:

limt→∞

∥∥e(t)∥∥ = lim

t→∞∥∥xs(t) + xm(t)

∥∥ = 0 (7)

The aim of this work is to design an output feedback controller:

u(xm(t), xs(t)

) = −K(xm(t), xs(t)

)ye(t) (8)

guaranteeing the asymptotic stability of the error state, then theanti-synchronization to the coupled systems (Sm) and (Ss); K (.) isthe m × l instantaneous gain matrix, K (.) = {kij(.)}.

The closed loop slave chaotic system is then characterized bythe following state space description:

xs(t) = Ac(.)xs(t) (9)

Ac(.) = As(.) − Bs(.)K (.)Cs(.) (10)

The comparison system associated to the vector norm:

p(z) = [ |z1| . . . |zn|]T

(11)

with z = [z1 . . . zn]T , is defined by the following system:

z(t) = M(

Ac(.))z(t) (12)

such that the elements mij(.) of M(Ac(.)) are deduced from theones of the matrix Ac(.) by substituting the off-diagonal elementsby their absolute values:{

mii(.) = acii (.) ∀i = 1, . . . ,n

mij(.) = ∣∣acij (.)∣∣ ∀i, j = 1, . . . ,n, i �= j

(13)

When the nonlinearities are isolated in either one row or onecolumn of M(Ac(.)), the system (2) is then stabilized by (8) if thematrix M(Ac(.)) is the opposite of an M-matrix, or equivalently,by application of the Borne–Gentina stability criterion [16–18], thefollowing inequalities corresponding to (14) are satisfied:

(−1)h M(

Ac(.))(

1 2 . . . h1 2 . . . h

)> 0 ∀h = 1,2, . . . ,n. (14)

Taking into account that the arrow form choice for instanta-neous characteristic matrices makes sufficient stability conditions(14) easy to test, let design the control law u, so that the instan-taneous characteristic matrix of the closed loop system (9) beingunder this form, (Ac)a , is such as:

(Ac(.)

)a =

⎡⎢⎢⎢⎢⎢⎢⎣

ac11(.) 0 . . . 0 ac1n (.)

0 ac22(.). . .

... ac2n (.)...

. . .. . . 0

...

0 . . . 0 acn−1n−1(.) acn−1n(.)

acn1(.) . . . . . . acnn−1(.) acnn(.)

⎤⎥⎥⎥⎥⎥⎥⎦

(15)

This approach can be considered only if the following condi-tions are fulfilled:

aij(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csj(.) = 0

∀i, j = 1, . . . ,n − 1 for i �= j (16)

Indeed, it is obvious that a necessary condition to solve the pre-vious Eqs. (16) is expressed as follows:

ml � (n − 1)(n − 2) (17)

Then, stabilisability conditions (14) become:{acii (.) < 0 ∀i = 1,2, . . . ,n − 1

(−1)n det(M

(Ac(.)

))> 0

(18)

such that:

(−1)n det(M

(Ac(.)

))= (−1)

(acnn(.) −

n−1∑i=1

∣∣acni (.)acin (.)∣∣a−1

cii(.)

)

× (−1)n−1n−1∏j=1

ac jj (.) (19)

Thus, it is interesting to point out that the arrow form is welladapted to the use of practical Borne–Gentina stability criterion,needing principal minor computations.

Theorem. The system, described by (2) is stabilized by the control gainK (8), if the matrix Ac(.), defined by (10), is such that:

S. Hammami et al. / Physics Letters A 374 (2010) 2835–2840 2837

i. the conditions (16) are satisfied:

aij(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csj(.) = 0

∀i, j = 1, . . . ,n − 1 for i �= j

ii. the nonlinear elements are isolated in either one row or one column,iii. the first (n − 1) diagonal elements are such that:

aii(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csi(.) < 0

∀i = 1, . . . ,n − 1 (20)

iv. there exists ε > 0, such that:

ann(.) −l∑

s=1

(m∑

r=1

bnr(.)krs(.)

)csn(.)

−n−1∑i=1

∣∣∣∣∣(

ani(.) −l∑

s=1

(m∑

r=1

bnr(.)krs(.)

)csi(.)

)

×(

ain(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csn(.)

)∣∣∣∣∣×

(aii(.) −

l∑s=1

(m∑

r=1

bir(.)krs(.)

)csi(.)

)−1

� −ε (21)

Corollary. The system, described by (2) is stabilized by the control gainK (8), if the instantaneous characteristic matrix Ac(.), defined by (10),is such that:

i. the conditions (16) are satisfied:

aij(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csj(.) = 0

∀i, j = 1, . . . ,n − 1 for i �= j

ii. all the nonlinearities are located in either one row or one column,iii. the first (n − 1) diagonal elements are strictly negative,iv. the products of the off-diagonal elements acni (.)acin (.), ∀i = 1,

. . . ,n − 1, are such that:

(ani(.) −

l∑s=1

(m∑

r=1

bnr(.)krs(.)

)csi(.)

)

×(

ain(.) −l∑

s=1

(m∑

r=1

bir(.)krs(.)

)csn(.)

)> 0

∀i = 1, . . . ,n − 1 (22)

v. there exists ε > 0, for which the instantaneous characteristic poly-nomial P Ac (., λ) is such that:

P Ac (., λ)|λ=0 = det(λI − Ac(.)

)∣∣λ=0 � ε (23)

Proof. The proof of the above-mentioned corollary is inferred oftheorem one by taking into account the new added iv. conditions,which guarantee, through a simple transformation, the identity ofthe matrix Ac(.) with its overvaluing M(Ac(.)). �

3. Anti-synchronization of two identical Rössler systems case

Anti-synchronization of two identical coupled chaotic Rösslersystems is considered in this section.

Let us consider the master Rössler system (Sm) given by [21]:⎡⎣ x1m(t)

x2m(t)x3m(t)

⎤⎦ =

⎡⎣ 0 −1 −1

1 a 0x3m(t) 0 −c

⎤⎦

⎡⎣ x1m(t)

x2m(t)x3m(t)

⎤⎦ +

⎡⎣ 0

0b

⎤⎦ (24)

which drives a similar slave Rössler system (Ss) described by:⎡⎣ x1s(t)

x2s(t)x3s(t)

⎤⎦ =

⎡⎣ 0 −1 −1

1 a 0x3s(t) 0 −c

⎤⎦

⎡⎣ x1s(t)

x2s(t)x3s(t)

⎤⎦ +

⎡⎣ 0

0b

⎤⎦

+⎡⎣ 1 0 0

0 1 00 0 1

⎤⎦

⎡⎣ u1(t)

u2(t)u3(t)

⎤⎦ (25)

where u, u(t) = [u1(t) u2(t) u3(t)]T , is the active control functionand a, b and c are three parameters of the Rössler chaotic system,a = 0.398, b = 2 and c = 4.

For the following state error vector components:⎧⎨⎩

x1e(t) = x1s(t) + x1m(t)

x2e(t) = x2s(t) + x2m(t)

x3e(t) = x3s(t) + x3m(t)

(26)

the error system can be defined by:⎡⎢⎣

x1e(t)

x2e(t)

x3e(t)

⎤⎥⎦ =

⎡⎣ 0 −1 −1

1 a 0x3e(t) 0 −c

⎤⎦

⎡⎢⎣

x1e(t)

x2e(t)

x3e(t)

⎤⎥⎦

+⎡⎣ 0

02b − (x3s(t)x1m(t) + x3m(t)x1s(t))

⎤⎦

+⎡⎣ 1 0 0

0 1 00 0 1

⎤⎦

⎡⎣ u1(t)

u2(t)u3(t)

⎤⎦ (27)

The problem of chaos anti-synchronization between two identi-cal Rössler chaotic dynamical systems is solved here by the designof a state feedback structure K , and the choice of nonlinear func-tions f i , i = 1,2,3, such that [4]:

ui(t) = −3∑

j=1

kij(.)x j(t) − f i(.) ∀i = 1,2,3 (28)

It comes the following closed loop system:⎡⎣ x1e(t)

x2e(t)x3e(t)

⎤⎦ =

⎡⎢⎣

−k11(.) −(1 + k12(.)) −(1 + k13(.))

(1 − k21(.)) (a − k22(.)) −k23(.)

(x3e(t) − k31(.)) −k32(.) −(c + k33(.))

⎤⎥⎦

×⎡⎢⎣

x1e(t)

x2e(t)

x3e(t)

⎤⎥⎦

−⎡⎣ f1(.)

f2(.)

f3(.) + (x3s(t)x1m(t) + x3m(t)x1s(t)) − 2b

⎤⎦(29)

The nonlinear elements f i and kij have to be chosen, in thesame scheme as done in the previous section, to make the in-stantaneous characteristic matrix of the closed loop system in the

2838 S. Hammami et al. / Physics Letters A 374 (2010) 2835–2840

Fig. 1. Error dynamics (x1e , x2e , x3e and e) of the coupled Rössler system when theactive controller is deactivated.

Fig. 2. Partial time series of anti-synchronization for Rössler chaotic system.

arrow form, nonlinear parts cancelled and the closed loop errorsystem asymptotically stable.

From the possible solutions, let consider:{−1 − k12(.) = 01 − k21(.) = 0

(30)

and:⎧⎨⎩

f1(.) = 0f2(.) = 0

f3(.) = 2b − (x3s(t)x1m(t) + x3m(t)x1s(t)

) (31)

For the vector norm p(xe):

p(xe) = [ |x1e| |x2e| |x3e|]T

(32)

the overvaluing matrix is in arrow form and has nonnegative off-diagonal elements and nonlinearities isolated in either one row orone column.

By the use of the proposed theorem, stability and anti-synchronization properties are satisfied for the both followingsufficient conditions (33) and (34):{−k11(.) < 0(

a − k (.))< 0

(33)

22

−(c + k33(.)

) − (∣∣(x3e(t) − k31(.))(−(

1 + k13(.)))∣∣(−k11(.)

)−1

+ ∣∣(−k32(.))(−k23(.)

)∣∣(a − k22(.))−1)

< 0 (34)

Various choices of the instantaneous gain matrix K (.) are pos-sible, as:

K (.) = {kij(.)

} =⎡⎣ 2 −1 −1

1 5.398 0.5x3e(t) 0 2

⎤⎦ (35)

By considering the initial condition x(0) = [1 1.5 2]T , for theRössler error system (27) when the active controller is deactivated,it is obvious that the error states grow with time chaotically, asshown in Fig. 1, and after activating the controller, Fig. 2 showsthree parametrically harmonically excited 3D systems evolve inthe opposite direction. The trajectories of error system (29) implythat the asymptotical anti-synchronization has been, successfully,achieved.

4. Anti-synchronization between Rössler and Chen chaoticsystems

In order to observe the anti-synchronization behaviour ofRössler and Chen chaotic systems using a nonlinear controlscheme, let assume that Rössler system (24) drives Chen system.

The chaotic Chen dynamical system, considered as a slave sys-tem at this stage, is described by the following system model ofdifferential equations [22]:⎡⎣ x1s(t)

x2s(t)x3s(t)

⎤⎦ =

⎡⎣ −α α 0

γ − α γ 00 0 −β

⎤⎦

⎡⎣ x1s(t)

x2s(t)x3s(t)

⎤⎦

+⎡⎣ u1(t)

−x1s(t)x3s(t) + u2(t)x1s(t)x2s(t) + u3(t)

⎤⎦ (36)

where α, β and γ are three positive parameters, α = 35, β = 3and γ = 28, or by the state space representation:

xs(t) = As(xs(t)

)xs(t) + Bsu(t) (37)

with

As(xs(t)

) =⎡⎣ −α α 0

γ − α γ −x1s(t)0 x1s(t) −β

⎤⎦ (38)

For xs(0) = [1 1 0.5]T , Fig. 3 presents the time responses ofboth Rössler and Chen systems, before activating the controller.

As in the previous sections, our aim is to design the most ad-equate structure of controllers ui(t), ∀i = 1,2,3, making the slavesystem achieves anti-synchronism with the master system, thenthe dynamics of the chaotic Chen system being opposite of the dy-namics reached by the Rössler system.

Adding Eq. (24) to Eq. (36), it comes the following called errordynamical system:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

x1e(t) = −(x2m(t) + x3m(t)

)+ α

(x2s(t) − x1s(t)

) + u1(t)

x2e(t) = x1m(t) + ax2m(t)

+ (γ − α − x3s(t)

)x1s(t) + γ x2s(t) + u2(t)

x3e(t) = b + x3m(t)(x1m(t) − c

)+ x1s(t)x2s(t) − βx3s(t) + u3(t)

(39)

Proceeding as before, an adequate control law has to be de-signed to anti-synchronize asymptotically the trajectories of theslave attractor facing to those of the master ones.

S. Hammami et al. / Physics Letters A 374 (2010) 2835–2840 2839

Fig. 3. Error dynamics between the Rössler system and the Chen system when con-troller is switched off.

Let the signal output:

ye(t) = [y1e(t) y2e(t)

]T(40)

such that:[y1e(t)y2e(t)

]=

[x1e(t) + 2x2e(t)

−x1e(t) + x2e(t)

](41)

To satisfy the anti-synchronization property, between the twonon-identical systems (24) and (36), among various choices ofthe gain matrix K (.) as well as the nonlinear functions f i(.),∀i = 1,2,3, one possible solution is given easily by:

K =⎡⎣ −2 0

−1 −10 1

⎤⎦ (42)

and:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

f1(.) = −4x2e(t) − (x2m(t) + x3m(t)

)+ α

(x2s(t) − x1s(t)

)f2(.) = x1m(t) + ax2m(t)

+ (γ − α − x3s(t)

)x1s(t) + γ x2s(t)

f3(.) = −2x3e(t) + b + x3m(t)(x1m(t) − c

)+ x1s(t)x2s(t) − βx3s(t)

(43)

For the controller parameters (42) and (43), the state vectorsof Rössler system (24) and those of Chen system (36) have thesame absolute values but with opposite signs, as they achieve anti-synchronous states shown in Fig. 4; both chaotic systems oscillatein an anti-synchronized manner. Thus, the required asymptoticalanti-synchronization has been provided, thanks to the designedcontroller.

5. Conclusion

The proposed approach of nonlinear systems stabilization,based on the synthesis of an output feedback controller, is of greatsignificance in the design and applications of anti-synchronization,relatively to chaotic systems. Numerical simulations are also em-ployed to illustrate the effective performance of the proposedcontrol scheme that is outlined in this Letter, in order to anti-synchronize both chaotic identical and non-identical systems.

Fig. 4. Anti-synchronization dynamics between the Rössler system and the Chensystem when controller is activated.

Appendix A. Borne–Gentina practical stability criterion [16–18]

Let consider the nonlinear continuous system described in thestate space by:

x(t) = A(.)x(t) (A.1)

A(.) is an n × n matrix, A(.) = {aij(.)}, and x the state vector, x =[x1 x2 . . . xn]T , x ∈ R

n .If the overvaluing matrix M(A(.)) has its nonconstant elements

isolated in only one row, the verification of the Kotelyanski con-dition enables to conclude to the stability of the initial systemcharacterized by A(.).

As an example, if the nonconstant elements are isolated inthe last row of A(.), Kotelyanski lemma applied to the overvalu-ing matrix obtained by the use of the n regular vector normp(x), p(x) = [|x1| |x2| . . . |xn|]T , leads to the following stabilityconditions of initial system:

a11 < 0,

∣∣∣∣ a11 |a12||a21| a22

∣∣∣∣ > 0, . . . ,

(−1)n

∣∣∣∣∣∣∣∣∣

a11 |a12| . . . |a1n||a21| a22 . . . |a2n|

......

. . ....

|an1(.)| |an2(.)| . . . ann(.)

∣∣∣∣∣∣∣∣∣> 0 (A.2)

The Borne–Gentina practical stability criterion applied to con-tinuous systems generalizes the Kotelyanski lemma for nonlinearsystems and defines large classes of systems for which the linearconjecture can be applied, either for the initial system or for itscomparison system [23].

Appendix B. How to obtain arrow form matrix from Compagnonone [16]?

In this part, is arised the importance of arrow form choice forinstantaneous characteristic matrices to obtain useful sufficient sta-bility conditions for systems described by scalar differential equa-tions:

y(n) +n−1∑i=0

ai(

y, y′, . . . , y(n−1))

y(i) = u (B.1)

y is the output, y ∈ R, x the state vector x = [y y′ . . . y(n−1)]T ,x ∈ R

n , u the input, u ∈ R, and ai(x), ∀i = 0,1, . . . ,n − 1, coeffi-

2840 S. Hammami et al. / Physics Letters A 374 (2010) 2835–2840

cients of the instantaneous characteristic polynomial P A(x, λ) ofthe matrix A(x), such that:

P A(., λ) = λn +n−1∑i=0

ai(.)λi (B.2)

It is relevant to denote that the system (B.1) can be rewrittenin a state space controllable form as:

˙x(t) = A(x)x(t) + Bu(t) (B.3)

A(x) =

⎡⎢⎢⎢⎢⎢⎣

0 1 0 . . . 0...

. . .. . .

. . ....

0 00 . . . 0 1

−a0(x) . . . −an−1(x)

⎤⎥⎥⎥⎥⎥⎦ , B =

⎡⎢⎢⎢⎢⎢⎢⎣

0......

01

⎤⎥⎥⎥⎥⎥⎥⎦

(B.4)

The change of base defined by:

x = T x (B.5)

T =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 1 . . . 1 0

α1 α2 . . . αn−1...

......

......

αn−21 αn−2

2 . . . αn−2n−1 0

αn−11 αn−1

2 . . . αn−1n−1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(B.6)

with

αi, ∀i = 1,2, . . . ,n − 1, αi �= α j, ∀i �= j (B.7)

distinct arbitrary constant parameters, leads to the new state equa-tion:

x(t) = A(x)x(t) + Bu(t) (B.8)

Then the new characteristic matrix, denoted by A(x), is in theBenrejeb arrow form [19]:

A(x) = P−1 A(x)P (B.9)

A(x) =

⎡⎢⎢⎢⎣

α1 β1. . .

...

αn−1 βn−1γ1(x) . . . γn−1(x) γn(x)

⎤⎥⎥⎥⎦ (B.10)

with

βi =n−1∏

i, j=1i �= j

(αi − α j)−1 (B.11)

γi(x) = −P A(x,αi) ∀i = 1,2, . . . ,n − 1 (B.12)

γn(x) = −an−1(x) −n−1∑i=1

αi (B.13)

This arrow form matrix leads to interesting sufficient asymp-totic stability conditions for the system described by (B.1) with thenotations (B.7), (B.11)–(B.15).

The equilibrium state of the nonlinear system (B.1) is asymptot-ically stable if the conditions:

i. αi < 0, i = 1, . . . ,n − 1, αi �= α j, ∀i �= j, (B.14)

ii. there exists a small strictly positive parameter ε, such that:

γn(x) −n−1∑i=0

∣∣βiγi(x)∣∣α−1

i � −ε (B.15)

are satisfied.In the particular case where the (n − 1) products βiγi(x), ∀i =

1, . . . ,n − 1, are nonnegative, the condition (B.15) can be reducedand stated, by means of the instantaneous characteristic polyno-mial of the matrix A(x), in the following manner:

P A(x,0) � ε (B.16)

which constitutes a verification case of the validity of the linearAizerman conjecture [23].

These conditions, associated to aggregation techniques based onthe use of vector norms, have led to stability domains for a classof Lur’e Postnikov systems whereas, for example, Popov stabilitycriterion use failed.

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