Unknown inputs' adaptive observer for a class of ... - Semantic Scholar
Mathematical and Computer Modelling 48 (2008) 1826–1839 www.elsevier.com/locate/mcm
Unknown inputs’ adaptive observer for a class of chaotic systems with uncertainties Samuel Bowong a,∗ , Jean Jules Tewa b a Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala,
P.O. Box 24157 Douala, Cameroon b Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812 Yaounde, Cameroon
Received 25 October 2006; received in revised form 16 August 2007; accepted 26 December 2007
Abstract This paper treats the adaptive synchronization problem of a class of uncertain chaotic systems with uncertainties and unknown inputs in the drive–response framework. A robust adaptive sliding mode observer-based response system is designed to synchronize a given chaotic system without the knowledge of upper bounds of uncertainties and unknown inputs. Further, the unknown inputs can be approximately recovered directly by the concept of equivalent control signal. To highlight our method, we improve the robustness of ciphering in a secure communication system. Computer simulation is also given for the purpose of illustration and verification. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Chaotic systems; Unknown inputs adaptive observers; Sliding mode; Equivalent control; Chaotic transmission
1. Introduction In the past decades, chaos has been an interesting topic in the field of nonlinear science. Among many studies on chaos, one important topic is the synchronization of chaotic systems. Synchronization is a fundamental phenomenon that enables coherent behavior in coupled systems. In 1990, Pecora and Carroll introduced a method [1] to synchronize two identical chaotic systems with different initial conditions. Since then, due to many potential applications in secure communication, biological science, chemical reaction, social science, and many other fields, the synchronization of coupled chaotic dynamical systems has been one of the most interesting topics in nonlinear science, and many theoretical and experimental results have been obtained. A significant result is the discovery of a variety of different synchronization phenomena, such as complete synchronization (CS) [1–3], phase synchronization (PS) [4,5], lag synchronization (LS) or anticipated synchronization (LS) [6,7], generalized synchronization (GS) [8–11], almostsynchronization (AS) [12,13] and the partial-state synchronization [12,13], reduced-order synchronization [14,15], etc. Many control methods have been derived for the synchronization of such systems including adaptive control [16–19], sliding mode control (or variable structure control) [17,20–22], impulsive control [23,24] and several others [25,26]. ∗ Corresponding author. Tel.: +237 99 96 41 64; fax: +237 22 31 95 84.
E-mail addresses: [email protected], [email protected] (S. Bowong), [email protected] (J.J. Tewa). c 2008 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.12.028
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More recently, chaos synchronization can be considered as an observer design problem, in the sense that the response system is the observer of the drive system [27]. Of particular interest is the connection between the observers for nonlinear systems and chaos synchronization which is also known as master–slave configuration. Thus, chaos synchronization problem can be posed as an observer design where the coupling signal is viewed as the output and the slave system as the observer. In other words, basically, chaos synchronization problem can be formulated as follows. Given a chaotic system, which is considered as the master (or driving) system, and another identical system, which is considered as the slave (or response) system, the aim is to force the response of the slave system to synchronize the master system [27–32]. The idea is to use the output of the master system to control the slave system so that the states of the slave system follow the states of the master system asymptotically. Unfortunately, uncertainties occur commonly in chaotic synchronization and many other real-world control systems. It has been noted that, when chaotic synchronization is applied in engineering applications such as communications, these chaos-based synchronization schemes are rather sensitive to noise and distortions in channels. This is an undesirable feature, as it is one of the main factors leading to failure of the intended synchronization. This effect should be taken into account when we want to evaluate the performance of a practical chaos synchronization scheme. One important reason for chaos synchronization is the successful application of chaos to secure communication. So far, many ideas and methods have been proposed to tackle the problem of chaotic secure communication including chaotic masking [33,35], chaotic shift keying [35,36], chaotic modulation [18,31,34] among many others (see [41,42,45] and the references therein). Moreover, in secure communication we need to estimate not only the state for chaos synchronization but also the input confidential message, which can be considered as to design an unknown input observer. At present, little attention has been given to adaptive synchronization between coupled chaotic systems with unknown inputs without the limitation of knowing the bounds of uncertainties and unknown inputs. As a consequence, adaptive robust observer-based synchronization of chaotic systems with unknown inputs in the presence of system’s uncertainties including parametric perturbations and external disturbances is an important issue. This paper considers the adaptive synchronization of a class of chaotic systems subject to uncertainties and unknown inputs in the drive–response framework. In the drive system, not only are the Lipschitz constants on nonlinear functions, but also the bounds on uncertainties and inputs are unknown. A robust adaptive sliding mode observer-based response system is therefore designed to synchronize the given uncertain chaotic system, and the unknown inputs can be approximately recovered in the sense of least mean square. If certain conditions are satisfied, the adaptation laws are chosen to estimate the bounds of unknown Lipschitz constants and uncertain parameters’ vector and to repress external disturbances. The Lyapunov stability theory ensures the global synchronization between the drive and response chaotic systems which results in the recovery of unknown inputs by the equivalent control, that is, the average behavior of discrete control law representing the effort necessary to maintain the motion on the sliding surface, even not only are the Lipschitz constants on nonlinear functions, but also the bounds on uncertainties and inputs are unknown. In order to highlight our method, we give an example of secure data communication. Comparing to existing results [16–32], the proposed adaptive synchronization scheme in this paper offers the following advantages. (i) The main objective of synchronization of master–slave chaotic systems with uncertainties is achieved with requiring any knowledge of upper bound of perturbations. The above properties make the proposed synchronization controller easy to implement. (ii) The unknown inputs can be approximately recovered by the equivalent control signal. Throughout this paper, it is noted that λ(A) denotes an eigenvalue of A, λmax (A) and λmin (A) represent, respectively, the max[λi (A)] and the min[λi (A)], i = 1, . . . , n. |x| represents the absolute value of x and kX k represents the Euclidian norm when X is a vector or the induced norm when X is a matrix. However, sign(W ) is the W sign function of the vector W ∈ Rn , and sign(W ) = kW k if W 6= 0 and sign(W ) = 0 if W = 0. 2. Problem formulation Consider the master chaotic system subject to unknown inputs, uncertainties and parametric perturbations: x˙ = Ax + B[ f (x, µ) + φ(x, d(t)) + u(t)], y = C x,
(1)
where x ∈ Rn is the state vector, y ∈ Rm is the output vector, µ ∈ R p is the parameter vector, f (x, µ) : Rn × R p → Rq is a nonlinear smooth function vector which depends nonlinearly on x and µ, d(t) ∈ Rs is the external disturbance
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Fig. 1. The coupled chaotic systems with uncertainties and unknown inputs.
vector and φ(x, d(t)) : Rn × Rs → Rq represents a nonlinear vector field that may include parametric perturbations and external disturbances. A ∈ Rn×n , B ∈ Rn×q and C ∈ Rm×n are known constant matrices. The parameter vector µ is assumed to be constant or varying slowly with time t. Generally speaking, the number of external disturbances φ(x, d(t)) is not greater than that of the output y, namely q ≤ m. If q < m, we can augment B, f (x, µ) and φ(x, d(t)) into B 0 = [B, 0] with 0 ∈ Rn×(m−q) , f 0 (x, µ) = [ f T (x, µ), 0T ] with 0 ∈ R(m−q) and φ 0 (x, d(t)) = [φ T (x, d(t)), 0T ] with 0 ∈ Rm−q , respectively. Clearly, we have B 0 f 0 (x, µ) = B f (x, µ) and B 0 φ 0 (x, d(t)) = Bφ(x, d(t)), respectively, and such replacement does not change the chaotic nature of system (1). Therefore, in system (1) we can suppose that q = m. Remark 1. A wide variety of chaotic systems can be represented by the form of (1) without the terms of φ(x, d(t)) and u(t). For example, R¨ossler system, MCK circuit, several types of Chua’s circuit, Lorenz system, L¨u system, Chen chaotic dynamical system, and almost all forced chaotic oscillators all belong to the class of systems defined by Eq. (1). Let the slave system based on the chaotic system (1) be given by: x˙ˆ = A xˆ + B f (x, ˆ µ) ˆ + Bv, yˆ = C x, ˆ
(2)
where xˆ indicates the dynamical estimate of the state x, µˆ ∈ R p is the parameter vector which is assumed to be different from µ, and v is the feedback coupling, which will be designed so as to achieve synchronization. In Fig. 1, the error y − yˆ is used for control, where y is sent from the master system through the channel. We define the error vector by e = x − x, ˆ
(3)
which is the state error between the master and the slave systems. Then, from (1)–(3), by adding and subtracting corresponding terms, the error dynamic equation is obtained as e˙ = (A − LC)e + LCe + B[ f (x, µ) − f (x, ˆ µ)] + B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ + Bφ(x, d(t)) + Bu(t) − Bv.
(4)
Our objective here is, without the requirement of the bounds of uncertainties and unknown inputs, to design an adaptive feedback coupling v using only the available signal y and the output of the observer-based slave system yˆ so that limt→∞ ke(t)k = 0, i.e., x(t) → x(t) ˆ as t → ∞.
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To ensure the achievement of synchronization’s objective, we make the following hypothesis. Assumption 1. There is a bounded region U ⊂ Rn containing the whole attractor of the master system (1) such that no orbit of system (1) ever leaves it. Further, the uncertain parameter µ, the uncertainties φ(x, d(t)) and the unknown inputs u(t) are norm bounded by three unknown positive constants µm , φm and u m , respectively, namely kµk ≤ µm , kφ(x, d(t))k ≤ φm and ku(t)k ≤ u m . Remark 2. Assumption 1 is reasonable for the boundedness of the chaotic attractor in the state space and the interaction of all trajectories inside the attractor. We are unable to give a mathematical proof of the boundedness of the trajectories starting in a subset of the total phase space, therefore we assume the boundedness of the trajectories in order to proceed with the discussion. Fortunately, most chaotic oscillators satisfy this property. Assumption 2. The pair (C, A) is observable. We can choose a feedback gain matrix L and two positive definite matrices P = P T and Q = Q T such that the following algebraic equations hold: (A − LC)T P + P(A − LC) = −Q,
(5)
B T P = C.
(6)
and
Equality (6) implies that the span of rows of B T P belongs to the span of rows of C. Remark 3. Finding a feedback gain matrix L satisfying Assumption 2 is not a trivial task. However, it was shown in [37] using the Kalman–Yakubovich–Popov lemma that if a gain matrix L can be chosen such that the transfer function matrix H (s) = C(s In − (A − LC))−1 B is strictly positive real, then there exist matrices P = P T and Q = Q T such that (5) and (6) are satisfied. Assumption 3. Let M ⊂ Rq be a region containing the relevant parameter values for which system (1) exhibits chaotic behavior. The nonlinear function f (x, µ) satisfy the following conditions: k f (x, µ) − f (z, µ)k ≤
w X
k fk kC x − C zkk ,
∀x, z ∈ U, ∀µ ∈ M,
(7)
k=0
k f (x, µ) − f (x, ν)k ≤ kµ kµ − νk,
∀x ∈ U, ∀µ, ν ∈ M,
(8)
where k fk , k = 0, 1, . . . , w and kµ are unknown positive constants and w in condition (7) is the high degree of the nonlinearity state of system (1). Note that the Lipschitz constants k fk and kµ can be subjected to uncertainties such as parametric variations, modelling error and external perturbations. Condition (7) is reasonable and not restrictive. The synchronization of chaotic systems is often based on adaptive observers’ design via linear feedback couplings derived using the output signal y of the master system to drive the response system. This technique needs the Lipschitz condition on the nonlinear terms which results in the convergence achieved by high gains of the feedback coupling. Also, note that in some practical systems, the Lipschitz constants are often selected to be larger, thus causing the gains of the feedback coupling to be higher, so that the obtained results are therefore conservative. The above remark motivates the redesign of adaptive observers via nonlinear feedback coupling using small gains to achieve the synchronization of chaotic systems with uncertainties and unknown inputs. The following examples show that condition (7) can be satisfied for some class of chaotic systems. Like the modified Chua circuit and the φ 6 -Duffing oscillator; all classes of chaotic systems for which the degree of the nonlinearity is greater than two or that has quadratic terms satisfy the condition (7). Consider the modified Chua circuit [38]: x˙1 1.43 10 0 x1 1 x˙2 = 1 −1 −1 x2 + 0 f (x1 , µ), (9) x˙3 0 16 0 x3 0
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where f (x1 , µ) = µx13 . Thus, if we define y = x1 , this system is obviously in the form given by Eq. (1) with φ(x, d(t)) = 0, u = 0 and C = (1, 0, 0). Then, one can prove that k f (x1 , µ) − f (z 1 , µ)k ≤ k f1 |x1 − z 1 | + k f3 |x1 − z 1 |3 ,
(10)
with k f1 = µ and k f3 = 3µr1r2 where r1 > 0 and r2 > 0 are such that |x1 (t)| ≤ r1 and |z 1 (t)| ≤ r2 . Also, the extended Duffing oscillator [39] is described by x˙1 0 1 x1 0 = + [ f (x1 , µ) + F0 cos ωt], x˙2 −1 −0.05 x2 1
(11)
where f (x1 , µ) = γ x15 . Also, if y = x1 , this system is obviously in the form given by Eq. (1) with φ(x1 , d(t)) = F0 cos ωt, u = 0 and C = (1, 0). Assuming that |x1 | ≤ r1 and |z 1 | ≤ r2 , one has that k f (x1 , µ) − f (z 1 , µ)k ≤ k f1 |x1 − z 1 | + k f3 |x1 − z 1 |3 + k f5 |x1 − z 1 |5 ,
(12)
where k f1 = 25γ r12r22 , k f3 = 5γ r1r2 and k f5 = γ . 3. Main results 3.1. The sliding mode observer-based scheme An adaptive feedback coupling is proposed as v=
w X
αˆ k (y − yˆ )ky − yˆ kk−1 + βˆ sign(y − yˆ ) + u(t), ˜
(13)
k=1
where u(t) ˜ is selected as: u(t) ˜ = (ρ + θˆ ) sign(y − yˆ ),
(14)
with ρ a tuning parameter. The adaptive gains are updated according to the following adaptation laws: α˙ˆ k = γk ky − yˆ kk+1 ,
k = 1, 2, . . . , w,
β˙ˆ = δky − yˆ k
and
θ˙ˆ = ηky − yˆ k,
(15)
where γk , k = 1, 2, . . . , w, δ and η are positive constants specified by the designer. The proposed adaptive feedback coupling (13)–(15) will guarantee the asymptotic synchronization for the master–slave chaotic systems (1) and (2) as shown in Fig. 1, and is proven in Theorem 1. Substituting the feedback couplings (13) and (14) into the error equation (4), one obtains e(t) ˙ = (A − LC)e + LCe + B[ f (x, µ) − f (x, ˆ µ)] + B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + B[φ(x, d(t)) + u(t)] − B αˆ k (B T Pe)kB T Pekk−1 − (ρ + βˆ + θˆ )B sign(B T Pe).
(16)
k=1
Now, we can establish the following result. Theorem 1. Consider the master–slave chaotic systems (1) and (2) with all the aforementioned assumptions. If the feedback coupling is given by (13) and (14) with the adaptive laws (15), then the the over-all system will be globally asymptotically synchronized on U, i.e., limt→∞ ke(t)k = 0. Proof. Define the following Lyapunov function candidate: V (t) = eT Pe +
w X 1 1 1 ˆ 2 + (θ − θˆ )2 , (αk − αˆ k )2 + (β − β) γ δ η k=1 k
(17)
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where α, β and θ are positive constants to be determined. Differentiating V (t) with respect to time yields V˙ (t) = eT [(A − LC)T P + P(A − LC)]e + 2eT P LCe + 2eT P B[ f (x, µ) − f (x, ˆ µ)] + 2eT P B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + 2eT B Pφ(x, d(t)) + 2eT B Pu(t) − 2 αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek k=1 w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ , − (αk − αˆ k )α˙ˆ k − (β − β) γ δ η k=1 k
≤ −λmin (Q)kek + 2kB T Pek kL T Pek + 2kB T Pek k f (x, µ) − f (x, ˆ µ)k + 2kB T Pek k f (x, ˆ µ) − f (x, ˆ µ)k ˆ + 2kB T Pek kφ(x, d(t))k + 2kB T Pek ku(t)k w w X X 2 −2 (αk − αˆ k )α˙ˆ k αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek − γ k k=1 k=1 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . − (β − β) δ η
(18)
Since systems (1) and (2) are contained in an attractor, we can suppose that the whole error system (16) can be in the domain Ω = {e ∈ Rn , kek ≤ 2r, r > 0} ⊂ U. Let kxk ˆ ≤ xˆm be satisfied for some xˆm > 0. Moreover, let us define ∆µm = kµ − µk. ˆ Then, Eq. (18) can be rewritten as follows: V˙ (t) ≤ −λmin (Q)kek2 + 2(2r kL T Pk + k f0 + kµ ∆µm + φm + u m )kB T Pek w w X X +2 k fk kB T Pekk+1 − 2 αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek k=1
k=1
w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . − (αk − αˆ k )α˙ˆ k − (β − β) γ δ η k=1 k
(19)
Thus, by setting αk = k f k ,
k = 1, . . . , w,
β = 2r kL T Pk + k f0 + kµ ∆µm + φm
and θ = u m ,
(20)
one has that ! w ˙ˆ k X α T k+1 V˙ (t) ≤ −λmin (Q)kek − 2 (αk − αˆ k ) kB Pek − γk k=1 ! ! ˙ˆ ˙ˆ β θ ˆ kB T Pek − ˆ kB T Pek − + 2(β − β) + 2(θ − θ) − 2ρkB T Pek. δ η 2
(21)
Thus, by applying the adaptation laws (15), one obtains V˙ (t) ≤ −λmin (Q)kek2 − 2ρkB T Pek, ≤ −λmin (Q)kek2 = −ω(t), where ω(t) = λmin
(22)
(Q)ke(t)k2 .
V (0) ≥ V (t) +
Z
Integrating Eq. (22) from zero to t yields Z t t ω(s)ds ≥ ω(s)ds.
0
(23)
0
As t goes R t to infinity, the above integral is always less than or equal to V (0). Since, V (0) is positive and finite, limt→∞ 0 ω(λ)dλ exists and is finite. Thus, according to the Barbalat Lemma [40], we obtain lim ω(t) = λmin (Q) lim ke(t)k2 = 0,
t→∞
(24)
t→∞
which implies that limt→∞ e(t) = 0. This achieves the proof.
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3.2. Reconstruction of the unknown inputs Recently, the concept of equivalent control has been utilized in the field of fault diagnosis, whose aim is to analyze possible faults in the actuators and sensors. From the works [41,42], we know that the concept of equivalent control can recover the actuator and sensor faults directly without using the residual signal, the basis for evaluating the faults. Here, if we regard the unknown inputs in chaotic systems as the faults, it will be shown that, provided a sliding motion can be attained, estimates of u(t) can be computed from approximating the so-called equivalent control required to maintain the sliding motion. In order to do so, suppose that there exists one positive definite matrix P = P T such that P = diag(P1 , P2 ) with P1 ∈ Rm×m and P2 ∈ R(n−m)×(n−m) . Therefore, Eqs. (5) and (6) can be replaced by the following conditions: (A − LC)T P + P(A − LC) = −Q,
(25)
P1 = Im .
(26)
and
In order to ensure the recovery of the unknown inputs u(t), the state error e and the matrix A can be decomposed into e = [e1T , eT2 ]T and A = [AT1 , AT2 ]T , respectively, where e1 ∈ Rm , e2 ∈ Rn−m , A1 ∈ Rm×n and A2 ∈ R(n−m)×n . Hence, the error dynamics (16) can be partitioned into e˙1 = A1 e + [ f (x, µ) − f (x, ˆ µ)] + [ f (x, ˆ µ) − f (x, ˆ µ)] ˆ + φ(x, d(t)) + u(t) w X − αˆ k (y − yˆ )ky − yˆ kk−1 − (ρ + βˆ + θˆ ) sign(y − yˆ ),
(27)
k=1
and (28)
e˙2 = A2 e. Now, we can prove the following result.
Theorem 2. If conditions (25) and (26) are satisfied, an ideal sliding motion takes place on the following hyperplane S = e1 (t) ∈ Rm , e1 (t) = 0 . (29) Proof. For the e1 -subsystem (27), let a Lyapunov function candidate be V1 (t) = e1T e1 +
w X 1 1 1 ˆ 2 + (θ − θˆ )2 , (αk − αˆ k )2 + (β − β) γ δ η k=1 k
(30)
where αk , k = 1, 2, . . . , w, β and θ are defined as in Eq. (20). The derivative of V1 (t) with respect to time is V˙1 (t) = 2eT1 A1 e + 2eT1 [ f (x, µ) − f (x, ˆ µ)] + 2eT1 [ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + 2eT1 φ(x, d(t)) + 2eT1 u(t) − 2 αˆ k ke1 kk+1 − 2(ρ + βˆ + θˆ )ke1 k k=1 w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ , (αk − αˆ k )α˙ˆ k − (β − β) − γ δ η k=1 k
≤ 2ke1 k kA1 k kek + 2ke1 k k f (x, µ) − f (x, ˆ µ)k + 2ke1 k k f (x, ˆ µ) − f (x, ˆ µ)k ˆ + (φm + u m )ke1 k − 2
w X
αˆ k ke1 kk+1
k=1
− 2(ρ + βˆ + θˆ )ke1 k −
w X k=1
2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . (αk − αˆ k )α˙ˆ k − (β − β) γk δ η
(31)
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According to the proof of Theorem 1, we know that the state error can be contained within a neighborhood of the origin which can be made as small as possible. Then, using conditions (7) and (8) and the adaptive laws (15), one has V˙1 (t) ≤ 4r kA1 kke1 k + (k f0 + kµ ∆µm )ke1 k + 2
w X
k fk ke1 kk+1
k=1
+ 2(φm + u m )ke1 k − 2
w X
αˆ k ke1 kk+1 − 2(ρ + βˆ + θˆ )ke1 k
k=1 w X ˆ 1 k − 2(θ − θˆ )ke1 k. −2 (αk − αˆ k )ke1 k2k − 2(β − β)ke
(32)
k=1
Thus, if αk , β and θ are defined as in Eq. (20), one has that V˙1 (t) ≤ −2[ρ − 2r λmax (A1 )]ke1 k = −ω1 (t),
(33)
where ω1 (t) = 2[ρ − 2r λmax (A1 )]ke1 k. Note that the tuning parameter ρ can be selected to be large enough to let ρ − 2r λmax (A1 ) > 0. Integrating Eq. (33) from zero to t yields Z t Z t V1 (0) ≥ V1 (t) + ω1 (s)ds ≥ ω1 (s)ds. (34) 0
0
As t goes R t to infinity, the above integral is always less than or equal to V1 (0). Since, V1 (0) is positive and finite, limt→∞ 0 ω1 (λ)dλ exists and is finite. Also, according to the Barbalat Lemma [40], we obtain lim ω1 (t) = 2[ρ − 2r λmax (A1 )] lim ke1 k = 0,
t→∞
t→∞
(35)
which implies that limt→∞ e1 (t) = 0 because ρ − 2r λmax (A1 ) > 0. Summing up the above analysis, an ideal sliding motion takes place on S [41–43] and this completes the proof. Theorem 2 implies that the hyperplane S can be attained. Therefore, we can conclude that e1 (t) = 0 and e˙1 (t) = 0. Now, assume that −βˆ sign(e1 ) can be approximated φ(x, d(t)). It then follows from subsystem (27) that 0 = [ f (x, µ) − f (x, ˆ µ)] ˆ + u eq (t),
(36)
ˆ sign(y − yˆ ) stands for the equivalent control action during the sliding motion. Since the state where u eq (t) = (ρ + θ) error can be contained within a neighborhood of the origin and the parameter vector µ is assumed to vary slowly with time t during the sliding motion, i.e., x(t) ≈ x(t) ˆ and µˆ ≈ µ, one can conclude that lim (u(t) − u eq (t)) ≈ 0,
t→∞
(37)
which means that the unknown input u(t) can be approximated by the equivalent control u eq (t). This equivalent control signal represents the average behavior of the discontinuous component u eq (t) and the effort necessary to maintain the motion on the sliding surface. From the works [41,42], it can be approximated by the following continuous function: u eq (t) =
(ρ + θˆ )(y − yˆ ) , ky − yˆ k + ε
(38)
where ε is a sufficiently small positive constant. For a sufficiently small value of ε, the continuous action (38) can approach the discontinuous action (ρ + θˆ ) sign(y − yˆ ) very well. Therefore, the unknown input u(t) can be recovered by the approximated equivalent control (38). 4. Illustrative example: Secure data transmission In this section, the three-dimensional modified Chua circuit is used to illustrate the effectiveness of the proposed robust adaptive observer scheme. Also, we propose a new encoding algorithm based on chaotic synchronization, in which the message is considered as an unknown input. When parameter µ is chosen to be −10, the modified Chua
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Fig. 2. Chaotic attractor of the modified Chua circuit.
Fig. 3. Gaussian random noise G(t) with mean zero and variance one.
circuit behaves chaotically as shown in Fig. 2. The initial conditions were selected to be (x1 (0), x2 (0), x3 (0)) = (0.1, 0, 0). From this figure, one can see that |x1 | ≤ 0.51, |x2 | ≤ 0.5 and |x3 | ≤ 1 so that r ≤ 1.1225. Furthermore, in real physical systems, it is impossible to neglect the effect of external noise. Because the chaotic system depends sensitively on a slight perturbation of signals, we have to ensure that the synchronization is robust to external noises. Here, we assume that the uncertainty arises from the unknown parameter µ or from the fluctuations of the parameter µ. So, we assume that µ is disturbed by d(t) = 0.001G(t) so that φ(x1 , d(t)) = 0.001x13 + 0.001G(t). The function G(t) is a Gaussian random noise with mean zero and variance one, as shown in Fig. 3. Further, we regard the message to be transmitted to the receiver as the unknown input u(t) = 0.01 sin(2t) which can be a low-amplitude perturbation added to the transmitter state so as to not modify its chaotic dynamics. We will use the solution x1 of (9) as the signal to be transmitted to the response system, i.e., y = x1 . Thus, the modified Chua circuit (9) with uncertainties can be rewritten as x˙1 1.43 10 0 x1 1 x˙2 = 1 −1 −1 x2 + 0 [µx13 + u(t) + φ(x1 , d(t))], x˙3 0 16 0 x3 0 ≡ Ax + B[ f (x, µ) + φ(x, d(t)) + u(t)], (39)
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y = x1 = [1, 0, 0]x, ≡ C x. From the above system, the driving signal does not embody the hidden message. In this case, the methods proposed in [33–36,43,44] cannot be used to recover this message. This is because these methods need the driving signal containing the message to synchronize the driving system and to recover the hidden message. Also in system (39), the linear part is completely observable, thereby permitting the choice of the gain matrix L = (2.43, 1.6207, −0.5519)T to make the transfer function H (s) = C[s I3 − (A(µ) − LC)]−1 B =
s 2 + s + 16 s 3 + 2s 2 + 23.2070s + 21.519
strictly positive real. Also, the eigenvalues of matrix A − LC are −0.969 and −0.5155 ± 4.6843i. Moreover, the following symmetric and positive definite matrices 2 0 0 1 0 0 and Q = 0 2 0 P = 0 17 1 0 0 2 0 1 1.125 are selected to satisfy Eqs. (25) and (26). As derived earlier, a robust adaptive sliding mode observer-based response system is designed as follows: ˙ xˆ 1 xˆ1 1 1.43 10 0 x˙ˆ 2 = 1 −1 −1 xˆ2 + 0 µˆ xˆ13 xˆ3 0 0 16 0 x˙ˆ 3 1 + 0 [αˆ 1 (x1 − xˆ1 ) + αˆ 3 (x1 − xˆ1 )3 + (ρ + βˆ + θˆ ) sign(x1 − xˆ1 )], 0
(40)
ˆ and θˆ (t) are given by where ρ is a tuning parameter. The adaptation laws αˆ 1 (t), αˆ 3 (t), β(t) α˙ˆ 1 = γ1 (x1 − xˆ1 )2 ,
α˙ˆ 3 = γ3 (x1 − xˆ1 )4 ,
β˙ˆ = δ|x1 − xˆ1 |
and
θ˙ˆ = η|x1 − xˆ1 |,
(41)
where γ1 , γ3 , δ and η are positive constants. Without loss of generality, one can consider that the initial conditions and parameters of the drive system are those of Fig. 2. The robust adaptive sliding mode observer-based response system (40) and (41) was simulated with the following parameters and initial conditions: γ1 = 10, γ3 = 1, δ = 0.01, η = 0.1, ρ = 1, (xˆ1 (0), xˆ2 (0), xˆ3 (0)) = ˆ (0, 0, 0), αˆ 1 (0) = 0, αˆ 3 (0) = 0, β(0) = 0 and θˆ (0) = 0. We also simulated (40) with µˆ = 10.5 so that the parameter of the slave system are chosen with a difference of 5% from the parameter of the master system. In this case, ∆µm = 0.5. Fig. 4 presents the state trajectories of the drive modified Chua circuit (solid line) and the slave modified Chua circuit (dashed line). It clearly appears that after a transient oscillatory period, the states x1 (t) and xˆ1 (t), x2 (t) and xˆ2 (t) and x3 (t) and xˆ3 (t) evolve together in an almost synchronous way. This implies that the modified Chua circuit response system (40) globally synchronizes the driving modified Chua circuit (39) though there exist unknown Lipschitz constants on nonlinear functions and unknown bounds on uncertainties and inputs. In other words, the proposed adaptive sliding mode observer-based response system (2) and (13)–(15) can effectively synchronize the chaotic system (1) in spite of uncertainties and unknown inputs. Now, it is important to demonstrate that the information signal can be “hidden” within the chaotic signal as it propagates in the hostile environment. The equivalent control u eq (t) is constructed as u eq (t) =
(ρ + θˆ )(x1 − xˆ1 ) , |x1 − xˆ1 | + ε
(42)
where ε is a sufficiently small positive constant. Fig. 5 shows the real and recovered messages. The recovered message (42) was simulated with ε = 0.001. In this figure, the real line stands for the real message u(t) while the dashed line represents the recovered message
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Fig. 4. State trajectories of x1 (t), x2 (t) and x3 (t) (solid line) and xˆ1 (t), xˆ2 (t) and xˆ3 (t) (dashed line), respectively.
Fig. 5. Time evolution of the real and recovered messages: u(t) (solid line) and u eq (t) (dashed line).
u eq (t) via the concept of equivalent control (42). One can see that the hidden message has been recovered with good accuracy. Also, one learns from this figure that the proposed scheme is effective to recover the hidden message in spite of uncertainties. Note that as the signal is transmitted through the hostile environment, it is secure since it requires
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Fig. 6. Time evolution of the adaptive parameters. (a) αˆ 1 , (b) αˆ 3 , (c) βˆ and (d) θˆ .
Fig. 7. Time evolution of the feedback coupling v.
that an interloper possesses an identical chaotic signal generator and feedback couplings in order to intercept the information. ˆ and θ(t) ˆ are depicted, respectively, in Fig. 6(a), The responses in time of the estimated parameters αˆ 1 (t), αˆ 3 (t), β(t) ˆ (b), (c) and (d). One can observe that the estimated gains αˆ 1 (t), αˆ 3 (t), β(t) and θˆ (t) increase with time and then
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converge at values which are optimal parameters suitable for the implementation process. Also, as predicted by our analysis, we only need small gains to synchronize the master and slave modified Chua circuits (39) and (40). The time evolution of the feedback coupling is displayed in Fig. 7. It clearly appears that after a transient period of oscillation, the feedback coupling signal converges to zero. 5. Conclusion In this paper, the problem of a class of uncertain chaotic systems is considered in the drive–response framework. For a class of chaotic systems with unknown Lipschitz constants on nonlinear functions, unknown bounds on uncertainties and unknown inputs, an adaptive sliding mode observer-based response system is constructed to globally synchronize the uncertain drive system. Further, the unknown inputs can be recovered by the equivalent control signal in the sense of least mean square. A secure transmission example is illustrated in order to highlight the proposed method. Computer simulation is also given for the purpose of illustration and verification. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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Unknown inputs’ adaptive observer for a class of chaotic systems with uncertainties Samuel Bowong a,∗ , Jean Jules Tewa b a Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala,
P.O. Box 24157 Douala, Cameroon b Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812 Yaounde, Cameroon
Received 25 October 2006; received in revised form 16 August 2007; accepted 26 December 2007
Abstract This paper treats the adaptive synchronization problem of a class of uncertain chaotic systems with uncertainties and unknown inputs in the drive–response framework. A robust adaptive sliding mode observer-based response system is designed to synchronize a given chaotic system without the knowledge of upper bounds of uncertainties and unknown inputs. Further, the unknown inputs can be approximately recovered directly by the concept of equivalent control signal. To highlight our method, we improve the robustness of ciphering in a secure communication system. Computer simulation is also given for the purpose of illustration and verification. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Chaotic systems; Unknown inputs adaptive observers; Sliding mode; Equivalent control; Chaotic transmission
1. Introduction In the past decades, chaos has been an interesting topic in the field of nonlinear science. Among many studies on chaos, one important topic is the synchronization of chaotic systems. Synchronization is a fundamental phenomenon that enables coherent behavior in coupled systems. In 1990, Pecora and Carroll introduced a method [1] to synchronize two identical chaotic systems with different initial conditions. Since then, due to many potential applications in secure communication, biological science, chemical reaction, social science, and many other fields, the synchronization of coupled chaotic dynamical systems has been one of the most interesting topics in nonlinear science, and many theoretical and experimental results have been obtained. A significant result is the discovery of a variety of different synchronization phenomena, such as complete synchronization (CS) [1–3], phase synchronization (PS) [4,5], lag synchronization (LS) or anticipated synchronization (LS) [6,7], generalized synchronization (GS) [8–11], almostsynchronization (AS) [12,13] and the partial-state synchronization [12,13], reduced-order synchronization [14,15], etc. Many control methods have been derived for the synchronization of such systems including adaptive control [16–19], sliding mode control (or variable structure control) [17,20–22], impulsive control [23,24] and several others [25,26]. ∗ Corresponding author. Tel.: +237 99 96 41 64; fax: +237 22 31 95 84.
E-mail addresses: [email protected], [email protected] (S. Bowong), [email protected] (J.J. Tewa). c 2008 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.12.028
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More recently, chaos synchronization can be considered as an observer design problem, in the sense that the response system is the observer of the drive system [27]. Of particular interest is the connection between the observers for nonlinear systems and chaos synchronization which is also known as master–slave configuration. Thus, chaos synchronization problem can be posed as an observer design where the coupling signal is viewed as the output and the slave system as the observer. In other words, basically, chaos synchronization problem can be formulated as follows. Given a chaotic system, which is considered as the master (or driving) system, and another identical system, which is considered as the slave (or response) system, the aim is to force the response of the slave system to synchronize the master system [27–32]. The idea is to use the output of the master system to control the slave system so that the states of the slave system follow the states of the master system asymptotically. Unfortunately, uncertainties occur commonly in chaotic synchronization and many other real-world control systems. It has been noted that, when chaotic synchronization is applied in engineering applications such as communications, these chaos-based synchronization schemes are rather sensitive to noise and distortions in channels. This is an undesirable feature, as it is one of the main factors leading to failure of the intended synchronization. This effect should be taken into account when we want to evaluate the performance of a practical chaos synchronization scheme. One important reason for chaos synchronization is the successful application of chaos to secure communication. So far, many ideas and methods have been proposed to tackle the problem of chaotic secure communication including chaotic masking [33,35], chaotic shift keying [35,36], chaotic modulation [18,31,34] among many others (see [41,42,45] and the references therein). Moreover, in secure communication we need to estimate not only the state for chaos synchronization but also the input confidential message, which can be considered as to design an unknown input observer. At present, little attention has been given to adaptive synchronization between coupled chaotic systems with unknown inputs without the limitation of knowing the bounds of uncertainties and unknown inputs. As a consequence, adaptive robust observer-based synchronization of chaotic systems with unknown inputs in the presence of system’s uncertainties including parametric perturbations and external disturbances is an important issue. This paper considers the adaptive synchronization of a class of chaotic systems subject to uncertainties and unknown inputs in the drive–response framework. In the drive system, not only are the Lipschitz constants on nonlinear functions, but also the bounds on uncertainties and inputs are unknown. A robust adaptive sliding mode observer-based response system is therefore designed to synchronize the given uncertain chaotic system, and the unknown inputs can be approximately recovered in the sense of least mean square. If certain conditions are satisfied, the adaptation laws are chosen to estimate the bounds of unknown Lipschitz constants and uncertain parameters’ vector and to repress external disturbances. The Lyapunov stability theory ensures the global synchronization between the drive and response chaotic systems which results in the recovery of unknown inputs by the equivalent control, that is, the average behavior of discrete control law representing the effort necessary to maintain the motion on the sliding surface, even not only are the Lipschitz constants on nonlinear functions, but also the bounds on uncertainties and inputs are unknown. In order to highlight our method, we give an example of secure data communication. Comparing to existing results [16–32], the proposed adaptive synchronization scheme in this paper offers the following advantages. (i) The main objective of synchronization of master–slave chaotic systems with uncertainties is achieved with requiring any knowledge of upper bound of perturbations. The above properties make the proposed synchronization controller easy to implement. (ii) The unknown inputs can be approximately recovered by the equivalent control signal. Throughout this paper, it is noted that λ(A) denotes an eigenvalue of A, λmax (A) and λmin (A) represent, respectively, the max[λi (A)] and the min[λi (A)], i = 1, . . . , n. |x| represents the absolute value of x and kX k represents the Euclidian norm when X is a vector or the induced norm when X is a matrix. However, sign(W ) is the W sign function of the vector W ∈ Rn , and sign(W ) = kW k if W 6= 0 and sign(W ) = 0 if W = 0. 2. Problem formulation Consider the master chaotic system subject to unknown inputs, uncertainties and parametric perturbations: x˙ = Ax + B[ f (x, µ) + φ(x, d(t)) + u(t)], y = C x,
(1)
where x ∈ Rn is the state vector, y ∈ Rm is the output vector, µ ∈ R p is the parameter vector, f (x, µ) : Rn × R p → Rq is a nonlinear smooth function vector which depends nonlinearly on x and µ, d(t) ∈ Rs is the external disturbance
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Fig. 1. The coupled chaotic systems with uncertainties and unknown inputs.
vector and φ(x, d(t)) : Rn × Rs → Rq represents a nonlinear vector field that may include parametric perturbations and external disturbances. A ∈ Rn×n , B ∈ Rn×q and C ∈ Rm×n are known constant matrices. The parameter vector µ is assumed to be constant or varying slowly with time t. Generally speaking, the number of external disturbances φ(x, d(t)) is not greater than that of the output y, namely q ≤ m. If q < m, we can augment B, f (x, µ) and φ(x, d(t)) into B 0 = [B, 0] with 0 ∈ Rn×(m−q) , f 0 (x, µ) = [ f T (x, µ), 0T ] with 0 ∈ R(m−q) and φ 0 (x, d(t)) = [φ T (x, d(t)), 0T ] with 0 ∈ Rm−q , respectively. Clearly, we have B 0 f 0 (x, µ) = B f (x, µ) and B 0 φ 0 (x, d(t)) = Bφ(x, d(t)), respectively, and such replacement does not change the chaotic nature of system (1). Therefore, in system (1) we can suppose that q = m. Remark 1. A wide variety of chaotic systems can be represented by the form of (1) without the terms of φ(x, d(t)) and u(t). For example, R¨ossler system, MCK circuit, several types of Chua’s circuit, Lorenz system, L¨u system, Chen chaotic dynamical system, and almost all forced chaotic oscillators all belong to the class of systems defined by Eq. (1). Let the slave system based on the chaotic system (1) be given by: x˙ˆ = A xˆ + B f (x, ˆ µ) ˆ + Bv, yˆ = C x, ˆ
(2)
where xˆ indicates the dynamical estimate of the state x, µˆ ∈ R p is the parameter vector which is assumed to be different from µ, and v is the feedback coupling, which will be designed so as to achieve synchronization. In Fig. 1, the error y − yˆ is used for control, where y is sent from the master system through the channel. We define the error vector by e = x − x, ˆ
(3)
which is the state error between the master and the slave systems. Then, from (1)–(3), by adding and subtracting corresponding terms, the error dynamic equation is obtained as e˙ = (A − LC)e + LCe + B[ f (x, µ) − f (x, ˆ µ)] + B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ + Bφ(x, d(t)) + Bu(t) − Bv.
(4)
Our objective here is, without the requirement of the bounds of uncertainties and unknown inputs, to design an adaptive feedback coupling v using only the available signal y and the output of the observer-based slave system yˆ so that limt→∞ ke(t)k = 0, i.e., x(t) → x(t) ˆ as t → ∞.
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To ensure the achievement of synchronization’s objective, we make the following hypothesis. Assumption 1. There is a bounded region U ⊂ Rn containing the whole attractor of the master system (1) such that no orbit of system (1) ever leaves it. Further, the uncertain parameter µ, the uncertainties φ(x, d(t)) and the unknown inputs u(t) are norm bounded by three unknown positive constants µm , φm and u m , respectively, namely kµk ≤ µm , kφ(x, d(t))k ≤ φm and ku(t)k ≤ u m . Remark 2. Assumption 1 is reasonable for the boundedness of the chaotic attractor in the state space and the interaction of all trajectories inside the attractor. We are unable to give a mathematical proof of the boundedness of the trajectories starting in a subset of the total phase space, therefore we assume the boundedness of the trajectories in order to proceed with the discussion. Fortunately, most chaotic oscillators satisfy this property. Assumption 2. The pair (C, A) is observable. We can choose a feedback gain matrix L and two positive definite matrices P = P T and Q = Q T such that the following algebraic equations hold: (A − LC)T P + P(A − LC) = −Q,
(5)
B T P = C.
(6)
and
Equality (6) implies that the span of rows of B T P belongs to the span of rows of C. Remark 3. Finding a feedback gain matrix L satisfying Assumption 2 is not a trivial task. However, it was shown in [37] using the Kalman–Yakubovich–Popov lemma that if a gain matrix L can be chosen such that the transfer function matrix H (s) = C(s In − (A − LC))−1 B is strictly positive real, then there exist matrices P = P T and Q = Q T such that (5) and (6) are satisfied. Assumption 3. Let M ⊂ Rq be a region containing the relevant parameter values for which system (1) exhibits chaotic behavior. The nonlinear function f (x, µ) satisfy the following conditions: k f (x, µ) − f (z, µ)k ≤
w X
k fk kC x − C zkk ,
∀x, z ∈ U, ∀µ ∈ M,
(7)
k=0
k f (x, µ) − f (x, ν)k ≤ kµ kµ − νk,
∀x ∈ U, ∀µ, ν ∈ M,
(8)
where k fk , k = 0, 1, . . . , w and kµ are unknown positive constants and w in condition (7) is the high degree of the nonlinearity state of system (1). Note that the Lipschitz constants k fk and kµ can be subjected to uncertainties such as parametric variations, modelling error and external perturbations. Condition (7) is reasonable and not restrictive. The synchronization of chaotic systems is often based on adaptive observers’ design via linear feedback couplings derived using the output signal y of the master system to drive the response system. This technique needs the Lipschitz condition on the nonlinear terms which results in the convergence achieved by high gains of the feedback coupling. Also, note that in some practical systems, the Lipschitz constants are often selected to be larger, thus causing the gains of the feedback coupling to be higher, so that the obtained results are therefore conservative. The above remark motivates the redesign of adaptive observers via nonlinear feedback coupling using small gains to achieve the synchronization of chaotic systems with uncertainties and unknown inputs. The following examples show that condition (7) can be satisfied for some class of chaotic systems. Like the modified Chua circuit and the φ 6 -Duffing oscillator; all classes of chaotic systems for which the degree of the nonlinearity is greater than two or that has quadratic terms satisfy the condition (7). Consider the modified Chua circuit [38]: x˙1 1.43 10 0 x1 1 x˙2 = 1 −1 −1 x2 + 0 f (x1 , µ), (9) x˙3 0 16 0 x3 0
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where f (x1 , µ) = µx13 . Thus, if we define y = x1 , this system is obviously in the form given by Eq. (1) with φ(x, d(t)) = 0, u = 0 and C = (1, 0, 0). Then, one can prove that k f (x1 , µ) − f (z 1 , µ)k ≤ k f1 |x1 − z 1 | + k f3 |x1 − z 1 |3 ,
(10)
with k f1 = µ and k f3 = 3µr1r2 where r1 > 0 and r2 > 0 are such that |x1 (t)| ≤ r1 and |z 1 (t)| ≤ r2 . Also, the extended Duffing oscillator [39] is described by x˙1 0 1 x1 0 = + [ f (x1 , µ) + F0 cos ωt], x˙2 −1 −0.05 x2 1
(11)
where f (x1 , µ) = γ x15 . Also, if y = x1 , this system is obviously in the form given by Eq. (1) with φ(x1 , d(t)) = F0 cos ωt, u = 0 and C = (1, 0). Assuming that |x1 | ≤ r1 and |z 1 | ≤ r2 , one has that k f (x1 , µ) − f (z 1 , µ)k ≤ k f1 |x1 − z 1 | + k f3 |x1 − z 1 |3 + k f5 |x1 − z 1 |5 ,
(12)
where k f1 = 25γ r12r22 , k f3 = 5γ r1r2 and k f5 = γ . 3. Main results 3.1. The sliding mode observer-based scheme An adaptive feedback coupling is proposed as v=
w X
αˆ k (y − yˆ )ky − yˆ kk−1 + βˆ sign(y − yˆ ) + u(t), ˜
(13)
k=1
where u(t) ˜ is selected as: u(t) ˜ = (ρ + θˆ ) sign(y − yˆ ),
(14)
with ρ a tuning parameter. The adaptive gains are updated according to the following adaptation laws: α˙ˆ k = γk ky − yˆ kk+1 ,
k = 1, 2, . . . , w,
β˙ˆ = δky − yˆ k
and
θ˙ˆ = ηky − yˆ k,
(15)
where γk , k = 1, 2, . . . , w, δ and η are positive constants specified by the designer. The proposed adaptive feedback coupling (13)–(15) will guarantee the asymptotic synchronization for the master–slave chaotic systems (1) and (2) as shown in Fig. 1, and is proven in Theorem 1. Substituting the feedback couplings (13) and (14) into the error equation (4), one obtains e(t) ˙ = (A − LC)e + LCe + B[ f (x, µ) − f (x, ˆ µ)] + B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + B[φ(x, d(t)) + u(t)] − B αˆ k (B T Pe)kB T Pekk−1 − (ρ + βˆ + θˆ )B sign(B T Pe).
(16)
k=1
Now, we can establish the following result. Theorem 1. Consider the master–slave chaotic systems (1) and (2) with all the aforementioned assumptions. If the feedback coupling is given by (13) and (14) with the adaptive laws (15), then the the over-all system will be globally asymptotically synchronized on U, i.e., limt→∞ ke(t)k = 0. Proof. Define the following Lyapunov function candidate: V (t) = eT Pe +
w X 1 1 1 ˆ 2 + (θ − θˆ )2 , (αk − αˆ k )2 + (β − β) γ δ η k=1 k
(17)
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where α, β and θ are positive constants to be determined. Differentiating V (t) with respect to time yields V˙ (t) = eT [(A − LC)T P + P(A − LC)]e + 2eT P LCe + 2eT P B[ f (x, µ) − f (x, ˆ µ)] + 2eT P B[ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + 2eT B Pφ(x, d(t)) + 2eT B Pu(t) − 2 αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek k=1 w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ , − (αk − αˆ k )α˙ˆ k − (β − β) γ δ η k=1 k
≤ −λmin (Q)kek + 2kB T Pek kL T Pek + 2kB T Pek k f (x, µ) − f (x, ˆ µ)k + 2kB T Pek k f (x, ˆ µ) − f (x, ˆ µ)k ˆ + 2kB T Pek kφ(x, d(t))k + 2kB T Pek ku(t)k w w X X 2 −2 (αk − αˆ k )α˙ˆ k αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek − γ k k=1 k=1 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . − (β − β) δ η
(18)
Since systems (1) and (2) are contained in an attractor, we can suppose that the whole error system (16) can be in the domain Ω = {e ∈ Rn , kek ≤ 2r, r > 0} ⊂ U. Let kxk ˆ ≤ xˆm be satisfied for some xˆm > 0. Moreover, let us define ∆µm = kµ − µk. ˆ Then, Eq. (18) can be rewritten as follows: V˙ (t) ≤ −λmin (Q)kek2 + 2(2r kL T Pk + k f0 + kµ ∆µm + φm + u m )kB T Pek w w X X +2 k fk kB T Pekk+1 − 2 αˆ k kB T Pekk+1 − 2(ρ + βˆ + θˆ )kB T Pek k=1
k=1
w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . − (αk − αˆ k )α˙ˆ k − (β − β) γ δ η k=1 k
(19)
Thus, by setting αk = k f k ,
k = 1, . . . , w,
β = 2r kL T Pk + k f0 + kµ ∆µm + φm
and θ = u m ,
(20)
one has that ! w ˙ˆ k X α T k+1 V˙ (t) ≤ −λmin (Q)kek − 2 (αk − αˆ k ) kB Pek − γk k=1 ! ! ˙ˆ ˙ˆ β θ ˆ kB T Pek − ˆ kB T Pek − + 2(β − β) + 2(θ − θ) − 2ρkB T Pek. δ η 2
(21)
Thus, by applying the adaptation laws (15), one obtains V˙ (t) ≤ −λmin (Q)kek2 − 2ρkB T Pek, ≤ −λmin (Q)kek2 = −ω(t), where ω(t) = λmin
(22)
(Q)ke(t)k2 .
V (0) ≥ V (t) +
Z
Integrating Eq. (22) from zero to t yields Z t t ω(s)ds ≥ ω(s)ds.
0
(23)
0
As t goes R t to infinity, the above integral is always less than or equal to V (0). Since, V (0) is positive and finite, limt→∞ 0 ω(λ)dλ exists and is finite. Thus, according to the Barbalat Lemma [40], we obtain lim ω(t) = λmin (Q) lim ke(t)k2 = 0,
t→∞
(24)
t→∞
which implies that limt→∞ e(t) = 0. This achieves the proof.
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3.2. Reconstruction of the unknown inputs Recently, the concept of equivalent control has been utilized in the field of fault diagnosis, whose aim is to analyze possible faults in the actuators and sensors. From the works [41,42], we know that the concept of equivalent control can recover the actuator and sensor faults directly without using the residual signal, the basis for evaluating the faults. Here, if we regard the unknown inputs in chaotic systems as the faults, it will be shown that, provided a sliding motion can be attained, estimates of u(t) can be computed from approximating the so-called equivalent control required to maintain the sliding motion. In order to do so, suppose that there exists one positive definite matrix P = P T such that P = diag(P1 , P2 ) with P1 ∈ Rm×m and P2 ∈ R(n−m)×(n−m) . Therefore, Eqs. (5) and (6) can be replaced by the following conditions: (A − LC)T P + P(A − LC) = −Q,
(25)
P1 = Im .
(26)
and
In order to ensure the recovery of the unknown inputs u(t), the state error e and the matrix A can be decomposed into e = [e1T , eT2 ]T and A = [AT1 , AT2 ]T , respectively, where e1 ∈ Rm , e2 ∈ Rn−m , A1 ∈ Rm×n and A2 ∈ R(n−m)×n . Hence, the error dynamics (16) can be partitioned into e˙1 = A1 e + [ f (x, µ) − f (x, ˆ µ)] + [ f (x, ˆ µ) − f (x, ˆ µ)] ˆ + φ(x, d(t)) + u(t) w X − αˆ k (y − yˆ )ky − yˆ kk−1 − (ρ + βˆ + θˆ ) sign(y − yˆ ),
(27)
k=1
and (28)
e˙2 = A2 e. Now, we can prove the following result.
Theorem 2. If conditions (25) and (26) are satisfied, an ideal sliding motion takes place on the following hyperplane S = e1 (t) ∈ Rm , e1 (t) = 0 . (29) Proof. For the e1 -subsystem (27), let a Lyapunov function candidate be V1 (t) = e1T e1 +
w X 1 1 1 ˆ 2 + (θ − θˆ )2 , (αk − αˆ k )2 + (β − β) γ δ η k=1 k
(30)
where αk , k = 1, 2, . . . , w, β and θ are defined as in Eq. (20). The derivative of V1 (t) with respect to time is V˙1 (t) = 2eT1 A1 e + 2eT1 [ f (x, µ) − f (x, ˆ µ)] + 2eT1 [ f (x, ˆ µ) − f (x, ˆ µ)] ˆ w X + 2eT1 φ(x, d(t)) + 2eT1 u(t) − 2 αˆ k ke1 kk+1 − 2(ρ + βˆ + θˆ )ke1 k k=1 w X 2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ , (αk − αˆ k )α˙ˆ k − (β − β) − γ δ η k=1 k
≤ 2ke1 k kA1 k kek + 2ke1 k k f (x, µ) − f (x, ˆ µ)k + 2ke1 k k f (x, ˆ µ) − f (x, ˆ µ)k ˆ + (φm + u m )ke1 k − 2
w X
αˆ k ke1 kk+1
k=1
− 2(ρ + βˆ + θˆ )ke1 k −
w X k=1
2 2 2 ˆ β˙ˆ − (θ − θˆ )θ˙ˆ . (αk − αˆ k )α˙ˆ k − (β − β) γk δ η
(31)
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According to the proof of Theorem 1, we know that the state error can be contained within a neighborhood of the origin which can be made as small as possible. Then, using conditions (7) and (8) and the adaptive laws (15), one has V˙1 (t) ≤ 4r kA1 kke1 k + (k f0 + kµ ∆µm )ke1 k + 2
w X
k fk ke1 kk+1
k=1
+ 2(φm + u m )ke1 k − 2
w X
αˆ k ke1 kk+1 − 2(ρ + βˆ + θˆ )ke1 k
k=1 w X ˆ 1 k − 2(θ − θˆ )ke1 k. −2 (αk − αˆ k )ke1 k2k − 2(β − β)ke
(32)
k=1
Thus, if αk , β and θ are defined as in Eq. (20), one has that V˙1 (t) ≤ −2[ρ − 2r λmax (A1 )]ke1 k = −ω1 (t),
(33)
where ω1 (t) = 2[ρ − 2r λmax (A1 )]ke1 k. Note that the tuning parameter ρ can be selected to be large enough to let ρ − 2r λmax (A1 ) > 0. Integrating Eq. (33) from zero to t yields Z t Z t V1 (0) ≥ V1 (t) + ω1 (s)ds ≥ ω1 (s)ds. (34) 0
0
As t goes R t to infinity, the above integral is always less than or equal to V1 (0). Since, V1 (0) is positive and finite, limt→∞ 0 ω1 (λ)dλ exists and is finite. Also, according to the Barbalat Lemma [40], we obtain lim ω1 (t) = 2[ρ − 2r λmax (A1 )] lim ke1 k = 0,
t→∞
t→∞
(35)
which implies that limt→∞ e1 (t) = 0 because ρ − 2r λmax (A1 ) > 0. Summing up the above analysis, an ideal sliding motion takes place on S [41–43] and this completes the proof. Theorem 2 implies that the hyperplane S can be attained. Therefore, we can conclude that e1 (t) = 0 and e˙1 (t) = 0. Now, assume that −βˆ sign(e1 ) can be approximated φ(x, d(t)). It then follows from subsystem (27) that 0 = [ f (x, µ) − f (x, ˆ µ)] ˆ + u eq (t),
(36)
ˆ sign(y − yˆ ) stands for the equivalent control action during the sliding motion. Since the state where u eq (t) = (ρ + θ) error can be contained within a neighborhood of the origin and the parameter vector µ is assumed to vary slowly with time t during the sliding motion, i.e., x(t) ≈ x(t) ˆ and µˆ ≈ µ, one can conclude that lim (u(t) − u eq (t)) ≈ 0,
t→∞
(37)
which means that the unknown input u(t) can be approximated by the equivalent control u eq (t). This equivalent control signal represents the average behavior of the discontinuous component u eq (t) and the effort necessary to maintain the motion on the sliding surface. From the works [41,42], it can be approximated by the following continuous function: u eq (t) =
(ρ + θˆ )(y − yˆ ) , ky − yˆ k + ε
(38)
where ε is a sufficiently small positive constant. For a sufficiently small value of ε, the continuous action (38) can approach the discontinuous action (ρ + θˆ ) sign(y − yˆ ) very well. Therefore, the unknown input u(t) can be recovered by the approximated equivalent control (38). 4. Illustrative example: Secure data transmission In this section, the three-dimensional modified Chua circuit is used to illustrate the effectiveness of the proposed robust adaptive observer scheme. Also, we propose a new encoding algorithm based on chaotic synchronization, in which the message is considered as an unknown input. When parameter µ is chosen to be −10, the modified Chua
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Fig. 2. Chaotic attractor of the modified Chua circuit.
Fig. 3. Gaussian random noise G(t) with mean zero and variance one.
circuit behaves chaotically as shown in Fig. 2. The initial conditions were selected to be (x1 (0), x2 (0), x3 (0)) = (0.1, 0, 0). From this figure, one can see that |x1 | ≤ 0.51, |x2 | ≤ 0.5 and |x3 | ≤ 1 so that r ≤ 1.1225. Furthermore, in real physical systems, it is impossible to neglect the effect of external noise. Because the chaotic system depends sensitively on a slight perturbation of signals, we have to ensure that the synchronization is robust to external noises. Here, we assume that the uncertainty arises from the unknown parameter µ or from the fluctuations of the parameter µ. So, we assume that µ is disturbed by d(t) = 0.001G(t) so that φ(x1 , d(t)) = 0.001x13 + 0.001G(t). The function G(t) is a Gaussian random noise with mean zero and variance one, as shown in Fig. 3. Further, we regard the message to be transmitted to the receiver as the unknown input u(t) = 0.01 sin(2t) which can be a low-amplitude perturbation added to the transmitter state so as to not modify its chaotic dynamics. We will use the solution x1 of (9) as the signal to be transmitted to the response system, i.e., y = x1 . Thus, the modified Chua circuit (9) with uncertainties can be rewritten as x˙1 1.43 10 0 x1 1 x˙2 = 1 −1 −1 x2 + 0 [µx13 + u(t) + φ(x1 , d(t))], x˙3 0 16 0 x3 0 ≡ Ax + B[ f (x, µ) + φ(x, d(t)) + u(t)], (39)
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y = x1 = [1, 0, 0]x, ≡ C x. From the above system, the driving signal does not embody the hidden message. In this case, the methods proposed in [33–36,43,44] cannot be used to recover this message. This is because these methods need the driving signal containing the message to synchronize the driving system and to recover the hidden message. Also in system (39), the linear part is completely observable, thereby permitting the choice of the gain matrix L = (2.43, 1.6207, −0.5519)T to make the transfer function H (s) = C[s I3 − (A(µ) − LC)]−1 B =
s 2 + s + 16 s 3 + 2s 2 + 23.2070s + 21.519
strictly positive real. Also, the eigenvalues of matrix A − LC are −0.969 and −0.5155 ± 4.6843i. Moreover, the following symmetric and positive definite matrices 2 0 0 1 0 0 and Q = 0 2 0 P = 0 17 1 0 0 2 0 1 1.125 are selected to satisfy Eqs. (25) and (26). As derived earlier, a robust adaptive sliding mode observer-based response system is designed as follows: ˙ xˆ 1 xˆ1 1 1.43 10 0 x˙ˆ 2 = 1 −1 −1 xˆ2 + 0 µˆ xˆ13 xˆ3 0 0 16 0 x˙ˆ 3 1 + 0 [αˆ 1 (x1 − xˆ1 ) + αˆ 3 (x1 − xˆ1 )3 + (ρ + βˆ + θˆ ) sign(x1 − xˆ1 )], 0
(40)
ˆ and θˆ (t) are given by where ρ is a tuning parameter. The adaptation laws αˆ 1 (t), αˆ 3 (t), β(t) α˙ˆ 1 = γ1 (x1 − xˆ1 )2 ,
α˙ˆ 3 = γ3 (x1 − xˆ1 )4 ,
β˙ˆ = δ|x1 − xˆ1 |
and
θ˙ˆ = η|x1 − xˆ1 |,
(41)
where γ1 , γ3 , δ and η are positive constants. Without loss of generality, one can consider that the initial conditions and parameters of the drive system are those of Fig. 2. The robust adaptive sliding mode observer-based response system (40) and (41) was simulated with the following parameters and initial conditions: γ1 = 10, γ3 = 1, δ = 0.01, η = 0.1, ρ = 1, (xˆ1 (0), xˆ2 (0), xˆ3 (0)) = ˆ (0, 0, 0), αˆ 1 (0) = 0, αˆ 3 (0) = 0, β(0) = 0 and θˆ (0) = 0. We also simulated (40) with µˆ = 10.5 so that the parameter of the slave system are chosen with a difference of 5% from the parameter of the master system. In this case, ∆µm = 0.5. Fig. 4 presents the state trajectories of the drive modified Chua circuit (solid line) and the slave modified Chua circuit (dashed line). It clearly appears that after a transient oscillatory period, the states x1 (t) and xˆ1 (t), x2 (t) and xˆ2 (t) and x3 (t) and xˆ3 (t) evolve together in an almost synchronous way. This implies that the modified Chua circuit response system (40) globally synchronizes the driving modified Chua circuit (39) though there exist unknown Lipschitz constants on nonlinear functions and unknown bounds on uncertainties and inputs. In other words, the proposed adaptive sliding mode observer-based response system (2) and (13)–(15) can effectively synchronize the chaotic system (1) in spite of uncertainties and unknown inputs. Now, it is important to demonstrate that the information signal can be “hidden” within the chaotic signal as it propagates in the hostile environment. The equivalent control u eq (t) is constructed as u eq (t) =
(ρ + θˆ )(x1 − xˆ1 ) , |x1 − xˆ1 | + ε
(42)
where ε is a sufficiently small positive constant. Fig. 5 shows the real and recovered messages. The recovered message (42) was simulated with ε = 0.001. In this figure, the real line stands for the real message u(t) while the dashed line represents the recovered message
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Fig. 4. State trajectories of x1 (t), x2 (t) and x3 (t) (solid line) and xˆ1 (t), xˆ2 (t) and xˆ3 (t) (dashed line), respectively.
Fig. 5. Time evolution of the real and recovered messages: u(t) (solid line) and u eq (t) (dashed line).
u eq (t) via the concept of equivalent control (42). One can see that the hidden message has been recovered with good accuracy. Also, one learns from this figure that the proposed scheme is effective to recover the hidden message in spite of uncertainties. Note that as the signal is transmitted through the hostile environment, it is secure since it requires
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Fig. 6. Time evolution of the adaptive parameters. (a) αˆ 1 , (b) αˆ 3 , (c) βˆ and (d) θˆ .
Fig. 7. Time evolution of the feedback coupling v.
that an interloper possesses an identical chaotic signal generator and feedback couplings in order to intercept the information. ˆ and θ(t) ˆ are depicted, respectively, in Fig. 6(a), The responses in time of the estimated parameters αˆ 1 (t), αˆ 3 (t), β(t) ˆ (b), (c) and (d). One can observe that the estimated gains αˆ 1 (t), αˆ 3 (t), β(t) and θˆ (t) increase with time and then
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converge at values which are optimal parameters suitable for the implementation process. Also, as predicted by our analysis, we only need small gains to synchronize the master and slave modified Chua circuits (39) and (40). The time evolution of the feedback coupling is displayed in Fig. 7. It clearly appears that after a transient period of oscillation, the feedback coupling signal converges to zero. 5. Conclusion In this paper, the problem of a class of uncertain chaotic systems is considered in the drive–response framework. For a class of chaotic systems with unknown Lipschitz constants on nonlinear functions, unknown bounds on uncertainties and unknown inputs, an adaptive sliding mode observer-based response system is constructed to globally synchronize the uncertain drive system. Further, the unknown inputs can be recovered by the equivalent control signal in the sense of least mean square. A secure transmission example is illustrated in order to highlight the proposed method. Computer simulation is also given for the purpose of illustration and verification. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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