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Robust interval observers for global Lipschitz uncertain chaotic systems Marcelo Moisan a,∗ , Olivier Bernard b a

EMEL S.A., BP 8330097, Santiago, Chile

b

INRIA COMORE, BP 93, 06902 Sophia–Antipolis, France

article

info

Article history: Received 11 February 2009 Received in revised form 8 July 2010 Accepted 5 August 2010 Available online 28 September 2010 Keywords: Interval observers Observer-based synchronization Uncertain systems Chaotic systems

abstract In this paper we develop and apply a robust interval observer to estimate the unknown variables of uncertain chaotic systems. We consider that bounds of the uncertainties are known and propose an observer that computes guaranteed upper and lower bounds for the unknown variables. We show that the proposed observer converges asymptotically toward a bounded error, leading to an original scheme of observer-based synchronization. The method is applied to the estimation of the variables of Chua’s chaotic system. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Uncertain dynamics and perturbations are common drawbacks that have to be considered when estimating unknown variables of a system, especially when dealing with biological systems [1,2]. This is why the development of robust state observers is required. The task can be even more challenging when the system presents a chaotic behavior. Nowadays, the analytical study of chaos is used to explain some behavior in biology, chemical processes and physics [3]. Features of chaotic behavior, such as sensitivity to initial conditions and exponential divergence of nearby trajectories also make it potentially useful in the field of signal encryption and secure communication [4]. The idea consists of using the synchronization of chaotic systems. Since its introduction in [5], the problem of chaos synchronization has received great attention in the research community. Numerous achievements in this field have been made, introducing new methods in order to achieve the synchronization of two chaotic systems. Conceptually, given a master (drive) chaotic system and a second identical slave (response) system, the objective is to force the slave system to synchronize the behavior of the master system [5]. In order to solve this problem, the theory of automatic control has played a fundamental role. Methods based on continuous control have been widely discussed [6,3]. On the other hand, chaos synchronization has also been stated as an estimation problem:

∗

Corresponding author. Tel.: +56 2 7873495. E-mail address: [email protected] (M. Moisan).

0167-6911/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2010.08.005

the slave chaotic system is replaced by an observer, which will estimate (synchronize) the unknown variables of the master chaotic system. In [7], theoretical observability concepts are presented, giving notions and links to the system synchronization problem. Synchronization is achieved in [8] using observers up to a scalar signal, and high gains observers are used in [9]. Simultaneous state estimation is associated to message masking in [10], and the discrete time treatment of observer-based synchronization is studied in [11]. However, most of the previously mentioned techniques are usually formulated considering perfect knowledge of the chaotic system. This assumption can be quite difficult to fulfil considering realistic systems: mismatch between master and slave systems parameters can be found and moreover, transmitted signals or available information can be affected by noise (above all in the field of communications). Therefore, new methods more robust to uncertainties are needed. On this matter, some efforts have been already presented. For example, in [12] the control of uncertain chaotic systems is faced. In [13] a novel approach considering the simultaneous synchronization and identification of uncertain parameters is proposed for Rössler’s system. In this paper we consider the uncertainties from a purely deterministic point of view, that is, we assume bounds for the uncertain quantities. Then, we use these bounds to construct the so-called interval observer. An interval observer is a robust process that provides guaranteed bounds of the unknown part of an uncertain dynamical system. Interval observers are based on so called framers. These framers consist of two coupled differential systems that keep the partial order with respect to the system to be estimated. If the

688

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

framers provide a bounded estimate for bounded uncertainty, then this estimate is an interval observer. Interval observers [14] are commonly formulated considering monotonicity properties of differential systems. This theory was originally introduced in [14,15] and has been successfully applied to the analysis of various biological systems. Since the introduction of interval observers most of the efforts have focused in exploiting the characteristic of guaranteed upper and lower estimates [16–18]. This paper is structured as follows. In Section 2 we introduce some hypotheses about the system, definitions and an example of a chaotic system. Sections 3 and 4 are devoted to the theoretical aspects related to the formulation of an interval observer. This is performed in two steps: first we develop an interval observer under perfect knowledge of the dynamics of the system. Convergence conditions are given for this case. In Section 4 we deduce a framer which takes into account uncertainties on parameters and noise in the measurements. Asymptotic bounds on the estimation error are given in order to prove that the framer is indeed an interval observer. Each case is illustrated with the application to Chua’s chaotic system example. 2. Class of system and example We consider a dynamical system that can be written in the form:

(S ) :

x˙ (t ) = A(κ)x(t ) + ψ(x, κ); y(t ) = Cx(t )

x(0) = x0

(1)

where x ∈ Ω ⊂ Rn is the state vector of the system (x0 ∈ Ω0 ⊂ Ω is the initial condition), A ∈ Rn×n . The continuous and differentiable mapping ψ(x, κ) concentrates the non-linearity of the system. We will assume in the sequel that ψ(x, κ) is a global Lipschitz function on the considered domain Ω , with respect to x:

∀(x1 , x2 ) ∈ Ω 2 , |ψ(x1 , κ) − ψ(x2 , κ)| ≤ Γ |x1 − x2 |

(2)

where Γ is the Lipschitz constant. System (1) is also characterized by a set of parameters κ . This last point is treated in detail hereafter, when considering uncertainties in the observer formulation. Furthermore, we consider that a linear combination of the state vector is measured, with C ∈ Rn+ . Note that the same principle of observer design straightforwardly applies to a system output of dimension p. For sake of simplicity we focus here on the single output case. For the formulation and analysis of the observers, we make the following hypothesis on system (1). Hypothesis 1. The trajectories of (1) initialized at x0 in the bounded set Ω0 stay in a bounded domain Ω : ∀x0 ∈ Ω0 , |x(t , x0 )| ≤ xmax for all positive time. 2.1. Example: Chua’s chaotic system Chua’s system is one of the most cited and studied paradigms of chaotic behavior. The double scroll attractor that features this system is generated from a simple electrical circuit. The dimensionless equations that represent Chua’s system are the following [19]: x˙ 1 = α[x2 − x1 (1 + b) − h(x1 )] x˙ 2 = x1 − x2 + x3 x˙ 3 = −β x2 − γ x3 .

(3)

1 2

(a − b)[|s + 1| − |s − 1|]

−α(1 + b)

α 1 −1 A= 0 −β −α h(x1 ) 0 ψ(x) = .

(4)

0 1

and

−γ (5)

0 For the forthcoming analysis, we will consider that the variable y = x1 + x2 + x3 is measured, therefore C = [1 1 1]. Property 1. The solutions of system (3) remain bounded for all time. Proof. See for example [19,3], where a qualitative study of Chua’s equations can be found. Property 2. The function ψ1 (x) = −α h(x1 ) is monotone increasing and, moreover, positively bounded: ψ1 (x) ≤ −α(a − b). Proof. The proof is straightforward, under the assumptions given for the parameters α , a and b. In the following sections, we develop the interval observer in two steps: first, we consider the case of perfect knowledge of system (1). Then, we consider the case when system parameters are not accurately known and measurements are biased. We show how to obtain a guaranteed stable and error bounded estimation of the chaotic state. 3. Interval observers with perfect knowledge Before introducing the observers, let us recall the following definition. Definition 1. A square matrix B is said to be cooperative if its offdiagonal entries are nonnegative [20]: bij ≥ 0, ∀i ̸= j. Remark 1. The operator ≤ applied between vectors or matrices should be understood as a set of inequalities applied component by component. 3.1. A perfect knowledge interval observer Let us start our analysis giving some properties of any global Lipschitz mapping ψ(x) : Rn → Rq , that will be useful in the formulation of the observer. Property 3. The differentiable global Lipschitz function ψ(x) can be written as the difference of two differentiable global Lipschitz functions f (x) and g (x) which are two increasing functions of x:

ψ(x) = f (x) − g (x). Proof. Let us consider fj (x) = γ i xi and gj (x) = fj (x) − ψj (x), where γj ≥ 0 is the Lipschitz constant of function ψj and j = {1, . . . , q}. It is clear that gj (x) is increasing since:

∑n

∂ gj ∂ψj =γ − ≥ 0, ∂ xi ∂ xi then Property 3 holds.

The nonlinear feature is given by the function: h(s) =

where α , β , γ ∈ R+ and a, b ∈ R− , with a < b, are system parameters. Chua’s system (3) has the form of system (1) considering x = [x1 x2 x3 ]T ∈ R3 with:

∀i = {1, . . . , n}

Property 4. For any global Lipschitz function ψ(x), there exists a differentiable global Lipschitz function ψ(xa , xb ) : Rn × Rn → Rq ,

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

such that:

• ψ( x,x) = ψ(x) ≤0 • ∂∂ψ ≥ 0 and ∂∂ψ xa xb Proof. From Property 3, it holds that ψ(x) = f (x) − g (x) and therefore, mapping ψ can be written as

ψ(xa , xb ) = f (xa ) − g (xb )

(6)

with f and g monotone increasing.

The main consequence of this property is that we have, for the bounds of the argument x: x

−

−

+

+

−

(7)

Moreover, using the generalized Taylor formula, the upper difference can be written as follows:

∂ψ ψ(x , x ) − ψ(x, x) = (τ v + (1 − τ )u)dτ (v − u) ∂v 0 + x x where v = x− and u = x . From Eq. (6), we have +

∫

−

ψ(x+ , x− ) − ψ(x, x) =

Remark 2. It is rather straightforward to find appropriate Θi , and Fi : it suffices that the Fi off-diagonal terms are large enough so that Ai off-diagonal elements become positive. However, among the possible choices, some are more efficient in terms of interval size, especially the ones that ensure stability (see Property 2 in the sequel). Proof. We write the error dynamics by comparing Eqs. (12) and (1). This leads to the following system:

(Oe ) :

0

∂f (τ x+ + (1 − τ )x)dτ ∂x

∫

1

1

− 0

(8)

= [N1 (x , x) − N2 (x, x )](v − u). −

Aˆ =

[

e˙ k |t =t0 =

Similarly, for the lower bound we have: (10)

q×n

where the nonnegative matrices Ni ∈ R+ , i = {1, . . . , 4} are straightforwardly derived from (9) (and from the similar expression for the lower bounds). This allows us introduce the following property. Property 5. Consider the global Lipschitz function ψ(x) : R → R and its associate function ψ(xa , xb ) : Rn × Rn → Rq (see Property 4). n

∂ψ

(11)

where e+ = x+ − x and e− = x − x− . Proof. Given that x− ≤ x ≤ x+ , then Eq. (7) holds. Therefore, the proof is straightforwardly derived from Eqs. (9) and (10). Now let us consider the following system: + + + − + + − x˙ = Ax + ψ(x , x ) + Θ1 (y − y) + F1 (x − x ) − − − + − + − (O1 ) : x˙ = Ax + ψ(x , x ) − Θ2 (y − y ) − F2 (x − x ) + x (0) = x+ x− (0) = x− 0 , 0

e e−

, it can be rewritten in the

F1 A2

] and

[

(14)

] ψ(x+ , x− ) − ψ(x, x) . ψ(x, x) − ψ(x− , x+ )

2n −

aˆ ki ei + φk (x+ , x− , x) ≥ 0

(15)

i̸=k

where the components of matrix Aˆ are denoted aˆ ij . As a consequence ek will stay nonnegative and finally e will remain nonnegative for any time t. Moreover, considering the bounds given by Eq. (11) we have a ˜ where: differential inequality of the form e˙ ≤ Ae, A˜ =

[

A + Θ1 C + F1 + N1 F2 + N3

F1 + N2 A + Θ2 C + F2 + N4

]

.

(16)

q

If the Jacobian matrix ∂v ∈ Rq×2n is bounded then, for bounds of the q×n argument x− ≤ x ≤ x+ , there exist matrices Ni ∈ R+ , i = {1, . . . , 4} such that:

ψ(x+ , x− ) − ψ(x, x) ≤ N1 e+ + N2 e− ψ(x, x) − ψ(x− , x+ ) ≤ N3 e+ + N4 e−

A1 F2

φ(x+ , x− , x) =

(9)

ψ(x, x) − ψ(x− , x+ ) = [N3 (x, x− ) − N4 (x+ , x)](v − u)

+

By initial condition hypothesis, e(0) ≥ 0. Let us consider the first time instant t0 when one of the component of vector e is equal to zero. Let us e.g. assume that it is the kth component of ek . We have for this error component:

] ∂g (τ x + (1 − τ )x− )dτ (v − u) ∂x

+

(13)

ˆ + φ(x+ , x− , x), where: compact form e˙ = Ae

1

[∫

e˙ + = A1 e+ + F1 e− + ψ(x+ , x− ) − ψ(x) e˙ − = A2 e− + F2 e+ + ψ(x) − ψ(x− , x+ ).

Considering the error dynamics e =

≤ x ≤ x ⇒ ψ(x , x ) ≤ ψ(x, x) ≤ ψ(x , x ). +

689

Proposition 2. If matrix A˜ is stable and condition of Proposition 1 are fulfilled, then system (12) is an interval observer; it converges asymptotically towards the solution of (1). Proof. It follows that A˜ is cooperative because each Ni is nonnegˆ + φ(x+ , x− ) ≤ Ae, ˜ which implies that ative. On the other hand Ae ⋆ e (t ) ≥ e(t ) ≥ 0, where e⋆ is the solution of the system e˙ ⋆ = ˜ ⋆. Ae Remark 3. Conditions to have at the same time cooperativity and stability for linear systems are discussed in [21,18]. It is worth noting that, under some conditions, there exists a linear change of variables which can transform a stable linear system into a cooperative stable system. In the general linear case this change of variable is time varying.

(12)

where Θi = (θ1i , . . . , θni )T ∈ Rn are two gain vectors, x− and x+ ∈ Rn and y+ = Cx+ , y− = Cx− . F1 and F2 are nonnegative matrices. Proposition 1. For any Θ1 , Θ2 ∈ Rn and nonnegative matrices F1 and F2 such that the matrices A1 = A + Θ1 C + F1 and A2 = A + Θ2 C + F2 are cooperative, then system (12) is a framer of system (1): + if system (12) is initialized such that x− 0 ≤ x0 ≤ x0 , then for any nonnegative time t, x− (t ) ≤ x(t ) ≤ x+ (t ).

3.2. Application to Chua’s system For this case, the nonlinear part ψ1 (x1 ) = −α h(x1 ) of Chua’s system is perfectly known. Since it is monotone increasing (from Property 2), we consider ψ 1 (s) = α h(s) and thus:

ψ 1 (x− ) ≤ ψ1 (x) ≤ ψ 1 (x+ ).

(17)

For Chua’s equations, we have considered α = 11.85, β = 14.9, γ = 0.29, a = −1.14 and b = −0.71, and the initial condition

690

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

Fig. 1. Interval estimation of variables x1 , x2 and x3 , considering perfect knowledge of Chua’s system.

x0 = [−0.1 0.2 0.05]T [19]. We run the observer of Eq. (12) with − T T the initialization x+ 0 = [5 5 5] and x0 = [−5 − 5 − 5] . We take the gain value Θ = Θ1 = Θ2 = [−α − 1 0]T , and a matrix F = F1 = F2 computed such that:

fij =

|aij + θi | 0

if aij + θi < 0 otherwise.

for i, j = {1, . . . , 3},

i ̸= j

This allows to write matrix Aˆ of Eq. (14) as follows: −α(2 + b) 0 0 0 0 α 0 − 2 0 0 0 0 0 0 −γ 0 β 0 Aˆ = 0 0 α −α(2 + b) 0 0 0 0 0 0 −2 0 0 β 0 0 0 −γ

We will show that a framer for this class of system will depend on the sign of the state estimates. For this purpose let us introduce the following matrices:

σn+×n = diag[pos(x+ )]

and

σn−×n = diag[pos(x− )]

(22)

where (18)

(19)

which is cooperative and stable. The interval estimates performed by the observer can be seen in Figs. 1 and 2. 4. Interval observers with uncertainties

pos(xi ) =

1 0

if xi ≥ 0 otherwise.

(23)

In other words, σ + (x+ ) denotes a diagonal matrix where the element σk will be 1 if the kth component of the vector x+ is nonnegative and 0 if not. In a similar way, the diagonal matrix σ − (x− ) is constructed over the sign of the components of the vector x− . Now we write a candidate observer equation as follows:

+ x˙ = Bx+ + ψ(x+ , x− ) + Θ1 (y+ − y) + F1 (x+ − x− ) (O2 ) : x˙ − = Bx− + ψ(x− , x+ ) − Θ2 (y − y− ) − F2 (x+ − x− ) + x (0) = x+ x− (0) = x− 0 , 0 (24)

Now we take into account bounded uncertainties in the system parameters and measurements.

B = Aσ + + A(I − σ + ) and

4.1. Uncertain parameters Let us consider Eq. (1) in the case where κ is uncertain, but known to exist in the intervals κi ∈ [κ i , κ i ]. Under this consideration, matrix A is not known anymore but can be bounded by two known matrices A and A, such that A ≤ A ≤ A. We denote M1 and M2 ∈ Rn+×n , the two positive matrices such that: M1 = A − A

and

M2 = A − A.

(20)

For a simpler notation, we define

R = A − A = M1 + M2 which is a nonnegative matrix.

where:

(21)

B = Aσ − + A(I − σ − ).

(25)

Of course, the function ψ(.) is parameterized by [κ, κ]. This parameterization is not explicited here to lighten the notations. Proposition 3. If there exist Θ1 , Θ2 ∈ Rn and nonnegative matrices F1 and F2 such that the matrices A1 = A + Θ1 C + F1 and A2 = A + Θ2 C + F2 are cooperative, then system (24) is a framer for system (1). Proof. Let us construct the error dynamics. After some algebraic arrangements in Eq. (25) we have: B = [A − M2 + Rσ + ]

and

B = [A − M1 − Rσ + ].

(26)

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

691

Fig. 2. Convergence interval of variables x1 , x2 and x3 , considering perfect knowledge of Chua’s system.

The error system has thus the following form:

+ e˙ = (A + Θ1 C + F1 )e+ + F1 e− + φ + (.) + (Rσ + − M2 )x+ (Oe2 ) : − ˙ e = (A + Θ2 C + F2 )e− + F2 e+ + φ − (.) + (Rσ − − M1 )x− . + e The error dynamics e = e− can be rewritten in the form: ˆ + φ(x+ , x− , x) + H e˙ = Ae

x+ x−

[

(27)

4.1.1. Application to Chua’s system For the nonlinear part of Chua’s model, given by Eq. (4), we have to take into account the sign of the estimates. This leads to:

] (28)

where Aˆ and φ(x+ , x− , x) remain the same as in Eq. (14) and

[ H=

R σ + − M2 0

0

Rσ − − M1

]

ij

m1

−mij2

1

(33)

2

.

(29)

1

+ + ψ 1 ( x+ 1 ) = − (α a − α b)[|x1 + 1| − |x1 − 1|]

(34)

2

(30)

otherwise.

− [Rσ + − M2 ]ij x+ − M1 ]ij x− i ≥ 0 and [R σ i ≥ 0 + x and therefore H x− = H u ≥ 0.

(31)

2n − = {ˆaki ei + hki ui } + φk (x+ , x− , x) ≥ 0.

(35)

2

if x− 1 < 0: 1

− − ψ 1 ( x− 1 ) = − (α a − α b)[|x1 + 1| − |x1 − 1|].

2

(36)

1

(32)

i̸=k

1

− − ψ 1 ( x− 1 ) = − (α a − α b)[|x1 + 1| − |x1 − 1|]

Under this definition of functions ψ 1 and ψ , and considering

To finish the proof we consider the first time instant t0 when a component ek of the vector error e cancels. Writing the derivative e˙ k (t0 ), we get:

This proves the positivity of the error system.

if x+ 1 < 0:

if x− 1 ≥ 0:

if pos(x+ j ) = 1

Then it is clear that the kth column of Rσ + − M2 has the same sign as the kth component of x+ . The same statement applies to matrix Rσ − − M1 with respect to x− . This implies that:

e˙ k |t =t0

if x+ 1 ≥ 0: + + ψ 1 ( x+ 1 ) = − (α a − α b)[|x1 + 1| − |x1 − 1|]

The term H is related to the parametric uncertainty. From (21), we ij ij have Rij = m1 + m2 , and therefore:

[Rσ + − M2 ]ij =

Remark 4. In general, the choice of κ i or κ i to construct bounds of ψ(x, κ) depends exclusively on the nonlinearity structure. To simplify the writing, in the sequel we will assume that the term φ(x+ , x− , x) remains bounded by a positive constant vector φ = [φ 1 φ 2 ]T .

− Property 2, the boundedness of vector φ(x+ 1 , x1 , x) is clear.

4.2. Biased output In practice, measurements are corrupted with noise. We assume that this noise is bounded: y = Cx(1 + δ),

|δ| ≤ ∆ < 1.

(37)

692

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

Fig. 3. Interval estimation of variables x1 , x2 and x3 , for Chua’s system with parametric uncertainty and biased measurements.

In other words, the original measured set of variables Cx is corrupted by a bounded multiplicative perturbation δ . Depending on the sign of the measurements y, we can bound the actually unknown quantity Cx considering: y=

y 1 + ε∆

≤ Cx ≤

1

y 1 − ε∆

=y

(38)

Again, we use the compact notation of Eq. (28) to write the error dynamics. Then we have: Aˆ =

[

[ H =

Rσ + − M2

[

Proof. Let us focus on the correction terms. For the upper bound now we have:

[

+

y −y =

1+δ

+

Cx − Cx

= Ce+ − Cx

]

ε∆ + δ . 1 − ε∆

(39)

ε∆ − δ . 1 + ε∆

(40)

(41)

η2 =

−η2 Θ2 C

x . x

(44)

(45)

sign(η1 ) = sign(η2 ) = sign(Cx)

ε∆ − δ . 1 + ε∆

−η1 Θ1 C 0

0

−η2 Θ2 C

][ ]

x ≥ 0, x

∀Θi < 0.

(46)

With this last result the vector H is positive for bounded uncertainties in the system parameters and biased measurements. To complete the proof, we just need to verify that matrix Aˆ is cooperative. Indeed, if matrices F1 , F2 , A + Θ1 C + F1 and A + Θ2 C + F2 are cooperative, then it is straightforward that Aˆ is also cooperative, under the hypothesis of sign for Θi and Fi . 4.3. From framers to interval observers

where: and

]

][ ]

0

0

x+ x−

sign(y) = sign(η1 ) = sign(η2 ).

[

+ e˙ = (A + Θ1 C )e+ + (Rσ + − M2 )x+ −η Θ Cx + φ + (.) + F1 (e+ + e− ) (Oe3 ) : − 1 1 − − − e˙ = (A + Θ2 C )−e + (Rσ +− M−1 )x −η2 Θ2 Cx + φ (.) + F2 (e + e ) ε∆ + δ 1 − ε∆

(43)

We already proved that the vectors (Rσ + − M2 )x+ and (Rσ − − M1 )x− remain by construction positive for all time. Now, provided gains vectors Θ1 , Θ2 ∈ Rn− and considering Eqs. (38) and (42) we verify that:

⇒

Now we compute the error dynamics.

η1 =

]

Moreover, from Eq. (37) we have that the output y and Cx have the same sign, then:

1 − ε∆

Similarly, the lower bound correction can be written as: y − y− = Ce− − Cx

Rσ − − M1

−η1 Θ1 C

n

][

0

0

+ Proposition 4. If there exist Θ1 , Θ2 ∈ R− , and nonnegative matrices F1 and F2 such that the matrices A1 = A + Θ1 C + F1 and A2 = A + Θ2 C + F2 are cooperative, then system (24) is a framer for system (1), under parametric uncertainty and biased measurements.

F1 A + Θ2 C + F 2

and

if y ≥ 0

where ε = −1 otherwise. These bounds are known quantities that let us formulate a new framer according to the following proposition.

A + Θ1 C + F 1 F2

(42)

Until this point we have obtained guaranteed upper and lower bounds of system (1), because of the positivity of the error systems.

M. Moisan, O. Bernard / Systems & Control Letters 59 (2010) 687–694

693

Fig. 4. Convergence interval of variables x1 , x2 and x3 , for Chua’s system with parametric uncertainty and biased measurements.

Now, in order to derive an interval observer, we analyse the stability of the system (28), with Aˆ and H as defined in (43) and (44). For our purposes, the following lemma will be useful. Lemma 1. Consider the system e˙ = Ne + b(t ),

e(0) = e0 .

(47)

n×n

If matrix N ∈ R is cooperative and stable, and moreover, there exists a constant vector B ∈ Rn+ such that b(t ) ≤ B, then the solution of system (47) is upper bounded by the solution of the system z˙ = Nz +B, z (0) = e0 , which admits one stable equilibrium eeq = −N −1 B. Proof. See [14] for a detailed demonstration.

Lemma 2. If there exist Θ1 , Θ2 ∈ Rn− , and nonnegative matrices F1 and F2 such that the matrix Aˆ (see Eq. (43)) is cooperative and stable, then system (24) defines an interval observer, and the estimation error is ultimately bounded by: eeq = −N

−1

[

(R − η1 Θ1 C )xmax + φ 1 (R − η2 Θ2 C )xmax + φ 2

] (48)

where

[ N =

A + Θ1 C + Rσ + + F1 F2

F1 A + Θ2 C + R σ − + F 2

]

.

(49)

Proof. From Eq. (44) and considering that x+ = x + e+ and x− = x − e− the term H can be written as:

[ H =

Rσ + − M2

0

][

0 Rσ − − M1 [ ] (Rσ + − M2 − η1 Θ1 C )x + (Rσ − − M1 − η2 Θ2 C )x

e+ e−

]

(50)

then system (28) can be written in the form of (47) considering:

[ b=

(Rσ + − M2 − η1 Θ1 C )x + φ + (.) (Rσ − − M1 − η2 Θ2 C )x + φ − (.)

] (51)

and matrix N as defined in Eq. (49). From Proposition 4, matrix N is also cooperative. Indeed, matrices G, σ + and σ − are nonnegative. On the other hand, the term b is bounded by a positive vector. Considering Eqs. (21), (22) and (42), the following positive bounds can be deduced: (i) Rσ + − M2 ≤ R (ii) η1 ≤

2∆ 1−∆

= η1

and and

Rσ − − M1 ≤ R

η2 ≤

2∆ 1+∆

= η2 .

(52)

Using Hypothesis 1 and the previous bounds, it is possible to apply Lemma 1, and therefore we end up with the error bound of Eq. (48). 4.4. Application to Chua’s system Let us consider the observer given by Eq. (24). Now all the parameters of Chua’s system are supposed to be uncertain, with known upper and lower bounds: α ∈ [11.49, 12.21], β ∈ [14.45, 15.35], γ ∈ [0.29, 0.31] and a ∈ [−1.17, −1.1], b ∈ [−0.73 − 0.69] which corresponds to a ±5% range with respect to their real values. We have considered an uniformly distributed noise signal characterized by ∆ = 0.07. For this simulation we have considered the same values for the gain Θ and matrix F as in Section 3, which enables to obtain stable upper and lower bounds on the system state. Simulation results can be seen in Figs. 3 and 4. Even though the estimates performed for the perfect knowledge case and the uncertain case (see Figs. 1 and 3 respectively), are characterized by a strong peaking at the beginning of the simulation, the observer successfully converges. For the perfect knowledge case, the observer converges asymptotically to zero, as a direct consequence of the selected gain values. For the case where

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uncertainties are considered, the observer converges toward a bounded error, which depends on the range of the uncertainties. It is worth noting that for this case the variable x2 converges toward a very narrow interval, since its dynamics are not affected by parametric uncertainties (see Eq. (3)), performing interval estimates that are less sensitive to the noise in the available output. 5. Conclusions An interval observer has been presented in order to estimate the unmeasured variables of a chaotic system. This method is suitable for observer–based synchronization of chaotic systems, where uncertainties of parameters and noise in the measured variables have to be taken into account. Interval observers offer a way to deal with uncertainties with many advantages when compared with classical observers: for example, they work in a purely deterministic framework and they allow to online assess the error of the performed estimates. The proposed techniques can now be assessed in the framework of message encryption in highly biased communication channels. References [1] O. Bernard, G. Sallet, A. Sciandra, Nonlinear observers for a class of biological systems. Application to validation of a phytoplanktonic growth model, IEEE Trans. Automat. Control 43 (1998) 1056–1065. [2] L. Mailleret, O. Bernard, J.-P. Steyer, Robust nonlinear adaptive control for bioreactors with unknown kinetics, Automatica 40 (8) (2004) 365–383. [3] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering, Addison-Wesley, 1994. [4] T. Yang, L. Chua, Secure communication via chaotic parameter modulation, IEEE Trans. Circuits Syst.-I 43 (9) (1996) 817–819.

[5] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (10) (1990) 821–824. [6] T. Kapitaniak, L.O. Chua, G.Q. Zhong, Experimental synchronization of chaos using continuous control, Int. J. Bifurcations Chaos 2 (6) (1994) 483–488. [7] H. Nijmeijer, I. Mareels, An observer looks at synchronization, IEEE Trans. Circuits Syst.-I 44 (10) (1997) 882–890. [8] G. Grassi, S. Mascolo, Nonlinear observer design to synchronize hyperchaotic systems via scalar signal, IEEE Trans. Circuits Syst.-I 44 (10) (2003) 237–251. [9] W. Yu, High–Gain Observer for chaotic synchronization and secure communications, Int. J. Commun. Syst. 18 (2005) 487–500. [10] M. Boutayeb, M. Darouach, H. Rafaralahy, Generalized state–space observers for chaotic synchronization and secure communications, IEEE Trans. Circuits Syst.-I 49 (3) (2002) 345–349. [11] G. Millerioux, J. Daafouz, Input independent chaos synchronization of switched systems, IEEE Trans. Automat. Control 49 (7) (2004) 1182–1187. [12] Z. Li, G. Chen, S. Shi, C. Han, Robust adaptive tracking control for a class of uncertain chaotic systems, Phys. Lett. A 310 (2003) 40–43. [13] S. Chen, J. Hu, C. Wang, J. Lü, Adaptive synchronization of uncertain rössler hyperchaotic system based on parameter identification, Phys. Lett. A 321 (2004) 50–55. [14] J.-L. Gouzé, A. Rapaport, Z. Hadj-Sadok, Interval observers for uncertain biological systems, Ecological modell. 133 (2000) 45–56. [15] A. Rapaport, J.-L. Gouzé, Parallelotopic and practical observers for nonlinear uncertain systems, Int. J. Control 76 (3) (2003) 237–251. [16] O. Bernard, J.-L. Gouzé, Closed loop observers bundle for uncertain biotechnological models, J. Process Control 14 (2004) 765–774. [17] M. Moisan, O. Bernard, J.-L. Gouzé, Near optimal interval observers bundle for uncertain bioreactors, Automatica 45 (1) (2009) 291–295. [18] F. Mazenc, O. Bernard, Interval observers for linear systems with additive disturbances, Proceedings of the NOLCOS conference, Bologna, Italy, 2010. [19] G. Leonov, D. Ponomarenko, V. Smirnova, Global stability and instability of canonical chua’s circuits, in: R.N. Madan (Ed.), Chua’s Circuit: A Paradigm for Chaos, World Scientific, 1993, pp. 725–739. (series B). [20] H.L. Smith, Monotone Dynamical Systems: An introduction to the theory of competitive and cooperative systems, Amer. Math. Soc. 75 (1995). [21] F. Mazenc, O. Bernard, Asymptotically stable interval observers for planar systems with complex poles, IEEE Trans. Automat. Control 55 (2010) 523–527.

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