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Mathematics and Computers in Simulation 80 (2010) 2245–2257
Adaptive sliding mode control of chaotic dynamical systems with application to synchronization Sara Dadras, Hamid Reza Momeni ∗ Automation and Instruments Lab, Electrical Engineering Department, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran Received 5 February 2009; received in revised form 7 February 2010; accepted 2 April 2010 Available online 21 April 2010
Abstract We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results. © 2010 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Uncertain chaotic system; Synchronization; Sliding mode control; Adaptive control; Chattering phenomenon
1. Introduction Dynamic systems described by nonlinear differential equations can be extremely sensitive to initial conditions. This phenomenon is known as deterministic chaos, which means that, although the system mathematical description is deterministic, its behavior is still unpredictable [14]. Chaos phenomenon is an interesting subject in the nonlinear systems. The concept of controlling and synchronization of chaotic systems have attracted many interests, since the evolutionary work on chaos control was first presented by Ott et al. in 1990 [24], followed by the Pyragas time-delayed auto-synchronization control scheme [28]; and the pioneering work on the synchronization of identical chaotic systems evolving from different initial conditions was first introduced by Pecora and Carroll [26], the same year. The possible applications of chaos control and synchronization arouse many research activities in recent years. This fact has motivated researchers to seek for various effective methods to achieve these goals [3,7,22,33,36]. So, the chaos control and synchronization as well as their application have become the hotspot in nonlinear fields. In literature, from the viewpoint that control theory is the origin of the techniques of chaos control, synchronization of chaos has evolved somewhat in its own right [18]. From the other point of view, there are some works which deal with the problem of chaos synchronization in the framework of nonlinear control theory [5,16,23]. So, the study of chaos control and chaos synchronization unify due to this fact. ∗
Corresponding author. Tel.: +98 21 82883375. E-mail addresses: s [email protected] (S. Dadras), momeni [email protected] (H.R. Momeni).
0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.04.005
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In recent years, several strategies to control and synchronize chaos have been developed, such as linear and nonlinear feedback control [6,17,29], adaptive control [11,15,34,35,37], sliding mode control [9,20]. A widely considered controlling method consists in adding an input control signal to attempt to stabilize an unstable equilibrium point or an unstable periodic orbit. This input control signal can be constructed using linear state feedback or nonlinear state feedback [12]. In the past several decades, the sliding mode control (SMC) has been effectively applied to control the systems with uncertainties because of the intrinsic nature of robustness of sliding mode [27]. However, the SMC suffers from the problem of chattering, which is caused by the high-speed switching of the controller output in order to establish a sliding mode. The undesirable chattering may excite the high-frequency system response and result in unpredictable instabilities [32]. The adaptive techniques have been widely applied to control and synchronize chaotic systems [15,31]. Recently, researchers have utilized the adaptive techniques together with the sliding mode control for many engineering systems to smooth the output from a sliding mode controller and alleviate the chattering in the pure SMC [8,27]. In this paper, we address chaos control and synchronization using sliding mode theory. In the second section, dynamics of the system is described and an appropriate sliding surface is selected. In the third section, the adaptive sliding mode control (ASMC) scheme is briefly introduced. The proposed scheme is fairly simple in comparison with other works [21,38] and decreases the cost and complexity of the closed-loop system. Besides, this controller reduces the chattering phenomenon and guarantees some properties, such as the robust performance and stability properties in presence of parameter uncertainties and external disturbance. Then, stability of the proposed scheme is analyzed. In Section 5, Genesio system [13] is considered to verify the validity of proposed control scheme by a computer simulation, respectively. Many approaches have been presented for the synchronization of chaotic systems, but in most of them have been assumed that the master and slave system are the same [7,39]. Hence, the synchronization of two different chaotic systems plays a significant role in practical applications [10,25,30,31]. This problem becomes more difficult in presence of environmental disturbance, measurement noise, or if the two chaotic systems have some uncertainties. In Section 4, synchronization of two different chaotic systems via the ASMC is investigated. Theoretical results are verified via simulating the synchronization of Genesio system [13] and Arneodo system [1]. At the end, conclusion is presented. 2. System description for uncertain chaotic system with ASMC Generally, the nonlinear differential equations are only an approximate description of the actual plant due to the presence of various uncertainties. Let the chaotic dynamical systems be represented in the Brunovsky form [2] by the following differential equations: x˙ i = xi+1 , 1 ≤ i ≤ n − 1, (1) x˙ n = f0 (X, t) + f (X, t) + d(t) + u(t) X = [x1 , x2 , ..., xn ]T ∈ Rn , T
where X(t) = [x1 (t), x2 (t), . . . , xn (t)]T = [x(t), x˙ (t), . . . , x(n−1) (t)] ∈ Rn is the state vector, f0 (X,t) is given as nonlinear function of X and t, f(X,t) is time-varying, not precisely known, and uncertain part of chaotic system, u(t) ∈ R is the control input, and d(t) is the external disturbance of system (1). In general the uncertain term f(X,t) and disturbance term d(t) are assumed bounded, i.e.: |f (X, t)| ≤ α
and
|d(t)| ≤ β,
(2)
where α and β are positive. The control problem is to induce the system to track an n-dimensional desired vector Xd (i.e. the original nth order (n−1) T
tracking problem of state xd (t)), Xd = [xd1 , xd2 , . . . , xdn ]T = [xd , x˙ d , . . . , xd continuous function on [t0 , ∞). Let’s the tracking error be as:
(n−1)
E(t) = X(t) − Xd (t) = [x(t) − xd (t), x˙ (t) − x˙ d (t), . . . , x(n−1) (t) − xd = [e1 (t), e2 (t), . . . , en (t)]T .
] ∈ Rn , which belong to a class of T
T
(t)] = [e(t), e˙ (t), . . . , e(n−1) (t)]
(3)
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The control goal considered in this paper is such that for any given target orbit Xd (t), a SMC is designed, such that the resulting state response of tracking error vector satisfies: lim E(t) = lim X(t) − Xd (t) → 0,
t→∞
(4)
t→∞
where ||·|| is the Euclidean norm of a vector [19]. In the traditional SMC, a switching surface representing a desired system dynamics is considered as follows: s = en +
n−1
(5)
ci ei .
i=1
The switching surface parameters ci are selected to be positive such that the polynomial λn−1 + cn−1 λn−2 + . . . + c2 λ + c1 is Hurwitz (i.e. all the roots of the characteristic polynomial have negative real parts with desirable pole placement). The control u is designed to guarantee that the states are hitting on the sliding surface s = 0 (i.e. to satisfy the reaching condition s˙s < 0). When the closed-loop system is in the sliding mode, it satisfies s˙ = 0 and then the control law is obtained by u = ueq + ur = ueq + K sgn(s) = −f0 (X, t) − f (X, t) − d(t) −
n−1
(n)
ci ei+1 + xd (t) + K sgn(s).
(6)
i=1
where K is the switching gain. In practical system, the system uncertainty f(X,t) and external disturbance d(t) are unknown and the implemented overall control input is modified as: u = −f0 (X, t) −
n−1
(n)
ci ei+1 + xd (t) + K sgn(s).
i=1
Fig. 1. The chaotic trajectories of uncertain Genesio system.
(7)
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A useful method that is usually used to reduce the chattering which is caused by the traditional SMC is the adaptive technique. Here, an adaptive method is proposed to solve the chattering problem. The adaptive sliding mode control law for system (1) is ˆ s), ˆ λ, u = −f0 (X, t) − γξ(
(8)
ˆ s) is a hyperbolic function: where ξ(λ, ˆ s) = tanh(λs). ˆ ξ(λ, and the adaptive law is designed as: ∂x γˆ˙ = k1 eξ , ∂u ˆλ˙ = k2 e(1 − ξ 2 )s ∂x , ∂u
(9)
(10)
where k1 and k2 are positive constants. 3. Stability analysis
Theorem 1. Consider the system (1) is controlled by u(t) in (7). Then the condition s˙s < 0 is guaranteed and the error trajectories converge to the sliding surface (5).
Fig. 2. The time response of system states, sliding surface and the control input for regulating to the desired orbit xd = sin(2t).
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Proof Using (1), (5), and (7), direct differentiation of s yields: s˙ = e˙ n +
n−1
ci e˙ i
(11)
i=1
Multiplying both sides of Eq. (11) with s yields: n−1 n−1 (n) s˙s = s e˙ n + ci e˙ i = s f0 (X, t) + f (X, t) + d(t) + ueq + ur − xd (t) + ci ei+1 i=1
i=1
= s[f (X, t) + d(t) + Ksgn(s)] ≤ α |s| + β |s| + K |s| = −[−K − (α + β)] |s| .
(12)
If we select K < −(α + β) in Eq. (12), one can conclude that the reaching condition (s˙s < 0) is always satisfied and the error trajectories converge to the sliding surface. When the control scheme drive the trajectories to the sliding mode s = 0, it can be declared that the error declines asymptotically on the sliding surface (5), because each root of the characteristic polynomial has negative real part. Thus the proof is achieved completely. Theorem 2. Consider the uncertain chaotic dynamical system (1). The closed-loop system with the control law (8) and (9), and the adaptation law (10) will be asymptotically stable and the tracking error is eventually diminished. Proof Using the Lyapunov stability method, we can choose a Lyapunov function candidate, such as: V =
1 2 e . 2
(13)
Fig. 3. Adaptive parameters γˆ and λˆ versus time for regulating to the desired orbit xd = sin(2t).
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By direct differentiation of Eq. (13), we can obtain: dV ∂V ∂e ∂x ∂u ∂γˆ ∂V ∂e ∂x ∂u ∂λˆ = + . dt ∂e ∂x ∂u ∂γˆ ∂t ∂e ∂x ∂u ∂λˆ ∂t
(14)
Thus, we have ∂V ∂e ∂x ∂ ∂γˆ ∂λˆ ∂ ˆ ˆ ˆ ˆ V˙ = λs))) λs))) (−f0 (X, t) − γ(tanh( + (−f0 (X, t) − γ(tanh( ∂e ∂x ∂u ∂γˆ ∂t ∂t ∂λˆ = −e
∂x ˆ λ], ˆ˙ ˆ γˆ˙ + γs(1 ˆ [(tanh(λs)) − tanh2 (λs)) ∂u
(15)
Substituting Eq. (10) into (15) yields: 2 2 2 2 ∂x 2 ∂x ˆ ˆ 2 γs ˙ ˆ 2 ≤ −ω(t) ≤ 0, (tanh(λs)) − k2 e (1 − tanh2 (λs)) V = −k1 e ∂u ∂u
(16)
ˆ 2 + k2 (1 − tanh2 (λs)) ˆ 2 γs ˆ 2 ] is positive. Now by integrating (16), from zero to t, where ω(t) = e2 (∂x/∂u)2 [k1 (tanh(λs)) one has:
t V (0) ≥ V (t) + ω(λ) dλ (17) 0
Fig. 4. The time response of system states, sliding surface and the control input for regulating to the desired orbit xd = 0.
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Since V˙ ≤ 0 while V (0) − V (t) ≥ 0 and finite, as defined, it can be concluded that lim t→∞ Therefore, according to Barbalat lemma [32], we find:
2 ∂x 2 2 ˆ ˆ ˆ 2] = 0 lim ω(λ) = lim e2 γs [k1 (tanh(λs)) + k2 (1 − tanh2 (λs)) t→∞ t→∞ ∂u
t 0
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ω(λ) dλ exists and is finite.
(18)
So, from Eq. (18), it can be inferred that s reaches zero in finite time, and e = 0. Therefore, it can be declared that the tracking error will be finally eliminated. 4. Synchronization via adaptive sliding mode controller Next, we consider the chaos synchronization, from a dynamical control perspective. In this case, the chaos synchronization can be regarded as a model-tracking problem, in which the response system, can track the drive system asymptotically. We consider the slave system (1) to follow the master chaotic system with the following dynamics: y˙ i = yi+1 , 1 ≤ i ≤ n − 1, (19) y˙ n = f0 (Y, t) + fm (Y, t) + dm (t) Y = [y1 , y2 , . . . , yn ]T ∈ Rn , then states of the master and slave system will be synchronized. It is worth to notice that Y (t) = T [y1 (t), y2 (t), . . . , yn (t)]T = [y(t), y˙ (t), . . . , y(n−1) (t)] ∈ Rn is the state vector of the master system, f0 (Y,t) is the known nonlinear dynamic part of master system, f(Y,t) is time-varying, not precisely known, and uncertain part of master system, and dm (t) is the external disturbance of system (19). This hypothesis is not restrictive as it can be satisfied by many chaotic systems in literature, such as Genesio system [13], Arneodo system [1] and Duffing–Holmes system [4].
Fig. 5. Adaptive parameters γˆ and λˆ versus time for regulating to the desired orbit xd = 0.
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As it was said in the previous section, the uncertain term fm (Y,t) and disturbance term dm (t) are assumed bounded, i.e.: |fm (Y, t)| ≤ αm
and
|dm (t)| ≤ βm ,
(20)
where αm and βm are positive. Thus, to obtain synchronization, we can modify the error Eq. (3) as follows: T
E = Xs − Ym = [x1s − y1m , x2s − y2m , . . . , xns − ynm ]T = [e, e˙ , . . . , en−1 ] = [e1 , e2 , . . . , en ]T
(21)
If the error states off the coupled system are defined as Eq. (21), then the dynamic equations of these errors can be determined directly by subtracting Eq. (19) from Eq. (1), to yield: e˙ i = ei+1 , 1 ≤ i ≤ n − 1, (22) e˙ n = F + F + D(t) + u where F = f0s (X, t) − f0m (Y, t) F = fs (X, t) − fm (Y, t) D = ds (t) − dm (t)
(23)
in which, F is known nonlinear function, F is time-varying, not precisely known, and uncertain part of error dynamical system (22), and D is the external disturbance of the error system (22). The same as what we do before in Eqs. (5)–(7), we have ˆ s) = −f0s (Xs , t) + f0m (Ym , t) − γξ( ˆ s). ˆ λ, ˆ λ, u = −F − γξ(
Fig. 6. The chaotic trajectories of uncertain Arneodo system.
(24)
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ˆ s) is defined as: where ξ(λ, ˆ s) = tanh(λs). ˆ ξ(λ,
(25)
and the adaptive laws are designed as: ˙γˆ = k3 eξ ∂xs , ∂u ∂xs λˆ˙ = k4 e(1 − ξ 2 )s , ∂u
(26)
where k3 and k4 are positive constants. Proof of the stability of above formula is the same as Theorem 1 and Theorem 2. 5. Simulation results This section of the paper presents two illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. The simulation results are carried out using the MATLAB software. The fourth order Runge–Kutta integration algorithm was performed to solve the differential equations. A time step size 0.001 was employed. Consider the Genesio chaotic system as follows: ⎧ ⎪ ⎨ x˙ 1 = x2 x˙ 2 = x3 (27) ⎪ ⎩ 2 x˙ 3 = −cx1 − bx2 − ax3 + x1 ,
Fig. 7. The time response of master system and slave system states.
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where x1 , x2 , x3 are state variables, and a, b and c are the positive real constants satisfying ab < c. For instance, the Genesio system is chaotic for the parameters a = 1.2, b = 2.92, c = 6. Consider that system (27) is perturbed by an uncertainty term f(X,t) and excited by a disturbance term d(t), added to its third equation. The uncertain Genesio system can be written as: ⎧ ⎪ ⎨ x˙ 1 = x2 x˙ 2 = x3 (28) ⎪ ⎩ 2 x˙ 3 = −cx1 − bx2 − ax3 + x1 + Δf (X) + d(t) + u(t). From (8), the continuous control input is determined as: ˆ s). ˆ λ, u = cx1 + bx2 + ax3 − x12 − γξ(
(29)
The system is perturbed by an uncertainty term f (X, t) = 0.1 sin(πx1 ) sin(2πx2 ) sin(3πx3 ) and d(t) = 0.1 cos(3t), where |Δf(X)| ≤ α = 0.1 and |d(t)| ≤ β = 0.1. The uncontrolled uncertain system (28) (i.e. u(t) = 0) is also chaotic and retains one positive Lyapunov exponent as shown in Fig. 1. The simulation is done with the initial value [x1 x2 x3 ]T = [3 −4 2]T , c1 = 10, c2 = 6, k1 = 10, k2 = 15, λˆ 0 = 15, and γˆ 0 = 30. The simulation results are shown in Figs. 2–5 under the proposed ASMC (29) with the adaptation algorithm (9) and (10). Fig. 2 represents respectively the state time response, sliding surface dynamics and control input for tracking a periodic orbit xd = sin(2t). It can be seen that the dynamic of system states are stabilized to a periodic motion. Fig. 4 shows the state time responses, corresponding s(t) and control input for regulating to xd = 0. It can be seen that chattering does not appear, due to continuous control. The time responses of adaptation parameters are shown in Figs. 3 and 5 respectively.
Fig. 8. Adaptive parameters γˆ and λˆ and the control input versus time.
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Next, consider the Arneodo chaotic system: ⎧ ⎪ ⎨ y˙ 1 = y2 y˙ 2 = y3 ⎪ ⎩ y˙ 3 = −b1 y1 − b2 y2 − b3 y3 + b4 y13 + fm (Y ) + dm (t)
(30)
with b1 = −5.5, b2 = 3.5, b3 = 1, b4 = −1, is the master system, and Genesio chaotic system (28) is the slave. The state trajectories of the uncertain Arneodo system (30) are depicted in Fig. 6. From (24), the continuous control input is determined as: 2 3 ˆ s). ˆ λ, − b1 y1m − b2 y2m − b3 y3m + b4 y1m − γξ( u = cx1s + bx2s + ax3s − x1s
(31)
The control parameters for the controller (31) were chosen as k3 = 5, k4 = 15, λˆ 0 = 15, and γˆ 0 = 10. The simulation is done with the initial value [x1s x2s x3s ]T = [3 −4 2]T , [y1m y2m y3m ]T = [1 −1 0]T and the sliding surface parameters are c1 = 10, c2 = 6. The slave system is perturbed by an uncertainty term fs (Xs , t) = 0.1 sin(πx1s ) sin(2πx2s ) sin(3πx3s ) and ds (t) = 0.1 sin(3t), where |Δf(Xs )| ≤ α = 0.1 and |ds (t)| ≤ β = 0.1 and these parameters for the master system are set as f (Ym , t) = 0.2 sin(y1m ) sin(y2m y3m ) and dm (t) = 0.2 sin(2t) where |Δf(Ym )| ≤ αm = 0.2 and |dm (t)| ≤ βm = 0.2. The trajectories of the states of master and slave system are shown in Fig. 7. It can be seen that the slave system (Genesio–Tesi chaotic system) response immediately follows the reference trajectories given by Arneodo chaotic system. The evolution of the two adaptation parameters and the control input are depicted in Fig. 8. Finally, Fig. 9 shows the sliding surface and time responses of the synchronization error states. From the simulation results, it can be concluded that the obtained theoretic results are feasible and efficient for controlling uncertain chaotic dynamical system and synchronizing two different uncertain chaotic systems.
Fig. 9. The time response of synchronization error states, sliding surface.
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6. Conclusion This work presents the control and synchronization of chaos by designing the adaptive sliding mode controller. In the proposed approach, by applying appropriate control signal based on adaptive update law, a continuous control signal is achieved and stability of the system is guaranteed. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of Lyapunov stability theory. As the simulations show, the new controller could track and stabilize the desired trajectory within a short time not only to a fixed point, but also to an arbitrary orbit. The most distinguished feature of our controller is that it can be applied to synchronize two different chaotic systems. Meanwhile, numerical simulations illustrate the robustness of the controller against model uncertainty and noise disturbance; and it is a good solution to the chattering problem in the traditional sliding mode control. References [1] A. Arneodo, P.H. Coullet, E.A. 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Mathematics and Computers in Simulation 80 (2010) 2245–2257
Adaptive sliding mode control of chaotic dynamical systems with application to synchronization Sara Dadras, Hamid Reza Momeni ∗ Automation and Instruments Lab, Electrical Engineering Department, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran Received 5 February 2009; received in revised form 7 February 2010; accepted 2 April 2010 Available online 21 April 2010
Abstract We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results. © 2010 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Uncertain chaotic system; Synchronization; Sliding mode control; Adaptive control; Chattering phenomenon
1. Introduction Dynamic systems described by nonlinear differential equations can be extremely sensitive to initial conditions. This phenomenon is known as deterministic chaos, which means that, although the system mathematical description is deterministic, its behavior is still unpredictable [14]. Chaos phenomenon is an interesting subject in the nonlinear systems. The concept of controlling and synchronization of chaotic systems have attracted many interests, since the evolutionary work on chaos control was first presented by Ott et al. in 1990 [24], followed by the Pyragas time-delayed auto-synchronization control scheme [28]; and the pioneering work on the synchronization of identical chaotic systems evolving from different initial conditions was first introduced by Pecora and Carroll [26], the same year. The possible applications of chaos control and synchronization arouse many research activities in recent years. This fact has motivated researchers to seek for various effective methods to achieve these goals [3,7,22,33,36]. So, the chaos control and synchronization as well as their application have become the hotspot in nonlinear fields. In literature, from the viewpoint that control theory is the origin of the techniques of chaos control, synchronization of chaos has evolved somewhat in its own right [18]. From the other point of view, there are some works which deal with the problem of chaos synchronization in the framework of nonlinear control theory [5,16,23]. So, the study of chaos control and chaos synchronization unify due to this fact. ∗
Corresponding author. Tel.: +98 21 82883375. E-mail addresses: s [email protected] (S. Dadras), momeni [email protected] (H.R. Momeni).
0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2010.04.005
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In recent years, several strategies to control and synchronize chaos have been developed, such as linear and nonlinear feedback control [6,17,29], adaptive control [11,15,34,35,37], sliding mode control [9,20]. A widely considered controlling method consists in adding an input control signal to attempt to stabilize an unstable equilibrium point or an unstable periodic orbit. This input control signal can be constructed using linear state feedback or nonlinear state feedback [12]. In the past several decades, the sliding mode control (SMC) has been effectively applied to control the systems with uncertainties because of the intrinsic nature of robustness of sliding mode [27]. However, the SMC suffers from the problem of chattering, which is caused by the high-speed switching of the controller output in order to establish a sliding mode. The undesirable chattering may excite the high-frequency system response and result in unpredictable instabilities [32]. The adaptive techniques have been widely applied to control and synchronize chaotic systems [15,31]. Recently, researchers have utilized the adaptive techniques together with the sliding mode control for many engineering systems to smooth the output from a sliding mode controller and alleviate the chattering in the pure SMC [8,27]. In this paper, we address chaos control and synchronization using sliding mode theory. In the second section, dynamics of the system is described and an appropriate sliding surface is selected. In the third section, the adaptive sliding mode control (ASMC) scheme is briefly introduced. The proposed scheme is fairly simple in comparison with other works [21,38] and decreases the cost and complexity of the closed-loop system. Besides, this controller reduces the chattering phenomenon and guarantees some properties, such as the robust performance and stability properties in presence of parameter uncertainties and external disturbance. Then, stability of the proposed scheme is analyzed. In Section 5, Genesio system [13] is considered to verify the validity of proposed control scheme by a computer simulation, respectively. Many approaches have been presented for the synchronization of chaotic systems, but in most of them have been assumed that the master and slave system are the same [7,39]. Hence, the synchronization of two different chaotic systems plays a significant role in practical applications [10,25,30,31]. This problem becomes more difficult in presence of environmental disturbance, measurement noise, or if the two chaotic systems have some uncertainties. In Section 4, synchronization of two different chaotic systems via the ASMC is investigated. Theoretical results are verified via simulating the synchronization of Genesio system [13] and Arneodo system [1]. At the end, conclusion is presented. 2. System description for uncertain chaotic system with ASMC Generally, the nonlinear differential equations are only an approximate description of the actual plant due to the presence of various uncertainties. Let the chaotic dynamical systems be represented in the Brunovsky form [2] by the following differential equations: x˙ i = xi+1 , 1 ≤ i ≤ n − 1, (1) x˙ n = f0 (X, t) + f (X, t) + d(t) + u(t) X = [x1 , x2 , ..., xn ]T ∈ Rn , T
where X(t) = [x1 (t), x2 (t), . . . , xn (t)]T = [x(t), x˙ (t), . . . , x(n−1) (t)] ∈ Rn is the state vector, f0 (X,t) is given as nonlinear function of X and t, f(X,t) is time-varying, not precisely known, and uncertain part of chaotic system, u(t) ∈ R is the control input, and d(t) is the external disturbance of system (1). In general the uncertain term f(X,t) and disturbance term d(t) are assumed bounded, i.e.: |f (X, t)| ≤ α
and
|d(t)| ≤ β,
(2)
where α and β are positive. The control problem is to induce the system to track an n-dimensional desired vector Xd (i.e. the original nth order (n−1) T
tracking problem of state xd (t)), Xd = [xd1 , xd2 , . . . , xdn ]T = [xd , x˙ d , . . . , xd continuous function on [t0 , ∞). Let’s the tracking error be as:
(n−1)
E(t) = X(t) − Xd (t) = [x(t) − xd (t), x˙ (t) − x˙ d (t), . . . , x(n−1) (t) − xd = [e1 (t), e2 (t), . . . , en (t)]T .
] ∈ Rn , which belong to a class of T
T
(t)] = [e(t), e˙ (t), . . . , e(n−1) (t)]
(3)
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The control goal considered in this paper is such that for any given target orbit Xd (t), a SMC is designed, such that the resulting state response of tracking error vector satisfies: lim E(t) = lim X(t) − Xd (t) → 0,
t→∞
(4)
t→∞
where ||·|| is the Euclidean norm of a vector [19]. In the traditional SMC, a switching surface representing a desired system dynamics is considered as follows: s = en +
n−1
(5)
ci ei .
i=1
The switching surface parameters ci are selected to be positive such that the polynomial λn−1 + cn−1 λn−2 + . . . + c2 λ + c1 is Hurwitz (i.e. all the roots of the characteristic polynomial have negative real parts with desirable pole placement). The control u is designed to guarantee that the states are hitting on the sliding surface s = 0 (i.e. to satisfy the reaching condition s˙s < 0). When the closed-loop system is in the sliding mode, it satisfies s˙ = 0 and then the control law is obtained by u = ueq + ur = ueq + K sgn(s) = −f0 (X, t) − f (X, t) − d(t) −
n−1
(n)
ci ei+1 + xd (t) + K sgn(s).
(6)
i=1
where K is the switching gain. In practical system, the system uncertainty f(X,t) and external disturbance d(t) are unknown and the implemented overall control input is modified as: u = −f0 (X, t) −
n−1
(n)
ci ei+1 + xd (t) + K sgn(s).
i=1
Fig. 1. The chaotic trajectories of uncertain Genesio system.
(7)
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A useful method that is usually used to reduce the chattering which is caused by the traditional SMC is the adaptive technique. Here, an adaptive method is proposed to solve the chattering problem. The adaptive sliding mode control law for system (1) is ˆ s), ˆ λ, u = −f0 (X, t) − γξ(
(8)
ˆ s) is a hyperbolic function: where ξ(λ, ˆ s) = tanh(λs). ˆ ξ(λ, and the adaptive law is designed as: ∂x γˆ˙ = k1 eξ , ∂u ˆλ˙ = k2 e(1 − ξ 2 )s ∂x , ∂u
(9)
(10)
where k1 and k2 are positive constants. 3. Stability analysis
Theorem 1. Consider the system (1) is controlled by u(t) in (7). Then the condition s˙s < 0 is guaranteed and the error trajectories converge to the sliding surface (5).
Fig. 2. The time response of system states, sliding surface and the control input for regulating to the desired orbit xd = sin(2t).
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Proof Using (1), (5), and (7), direct differentiation of s yields: s˙ = e˙ n +
n−1
ci e˙ i
(11)
i=1
Multiplying both sides of Eq. (11) with s yields: n−1 n−1 (n) s˙s = s e˙ n + ci e˙ i = s f0 (X, t) + f (X, t) + d(t) + ueq + ur − xd (t) + ci ei+1 i=1
i=1
= s[f (X, t) + d(t) + Ksgn(s)] ≤ α |s| + β |s| + K |s| = −[−K − (α + β)] |s| .
(12)
If we select K < −(α + β) in Eq. (12), one can conclude that the reaching condition (s˙s < 0) is always satisfied and the error trajectories converge to the sliding surface. When the control scheme drive the trajectories to the sliding mode s = 0, it can be declared that the error declines asymptotically on the sliding surface (5), because each root of the characteristic polynomial has negative real part. Thus the proof is achieved completely. Theorem 2. Consider the uncertain chaotic dynamical system (1). The closed-loop system with the control law (8) and (9), and the adaptation law (10) will be asymptotically stable and the tracking error is eventually diminished. Proof Using the Lyapunov stability method, we can choose a Lyapunov function candidate, such as: V =
1 2 e . 2
(13)
Fig. 3. Adaptive parameters γˆ and λˆ versus time for regulating to the desired orbit xd = sin(2t).
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By direct differentiation of Eq. (13), we can obtain: dV ∂V ∂e ∂x ∂u ∂γˆ ∂V ∂e ∂x ∂u ∂λˆ = + . dt ∂e ∂x ∂u ∂γˆ ∂t ∂e ∂x ∂u ∂λˆ ∂t
(14)
Thus, we have ∂V ∂e ∂x ∂ ∂γˆ ∂λˆ ∂ ˆ ˆ ˆ ˆ V˙ = λs))) λs))) (−f0 (X, t) − γ(tanh( + (−f0 (X, t) − γ(tanh( ∂e ∂x ∂u ∂γˆ ∂t ∂t ∂λˆ = −e
∂x ˆ λ], ˆ˙ ˆ γˆ˙ + γs(1 ˆ [(tanh(λs)) − tanh2 (λs)) ∂u
(15)
Substituting Eq. (10) into (15) yields: 2 2 2 2 ∂x 2 ∂x ˆ ˆ 2 γs ˙ ˆ 2 ≤ −ω(t) ≤ 0, (tanh(λs)) − k2 e (1 − tanh2 (λs)) V = −k1 e ∂u ∂u
(16)
ˆ 2 + k2 (1 − tanh2 (λs)) ˆ 2 γs ˆ 2 ] is positive. Now by integrating (16), from zero to t, where ω(t) = e2 (∂x/∂u)2 [k1 (tanh(λs)) one has:
t V (0) ≥ V (t) + ω(λ) dλ (17) 0
Fig. 4. The time response of system states, sliding surface and the control input for regulating to the desired orbit xd = 0.
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Since V˙ ≤ 0 while V (0) − V (t) ≥ 0 and finite, as defined, it can be concluded that lim t→∞ Therefore, according to Barbalat lemma [32], we find:
2 ∂x 2 2 ˆ ˆ ˆ 2] = 0 lim ω(λ) = lim e2 γs [k1 (tanh(λs)) + k2 (1 − tanh2 (λs)) t→∞ t→∞ ∂u
t 0
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ω(λ) dλ exists and is finite.
(18)
So, from Eq. (18), it can be inferred that s reaches zero in finite time, and e = 0. Therefore, it can be declared that the tracking error will be finally eliminated. 4. Synchronization via adaptive sliding mode controller Next, we consider the chaos synchronization, from a dynamical control perspective. In this case, the chaos synchronization can be regarded as a model-tracking problem, in which the response system, can track the drive system asymptotically. We consider the slave system (1) to follow the master chaotic system with the following dynamics: y˙ i = yi+1 , 1 ≤ i ≤ n − 1, (19) y˙ n = f0 (Y, t) + fm (Y, t) + dm (t) Y = [y1 , y2 , . . . , yn ]T ∈ Rn , then states of the master and slave system will be synchronized. It is worth to notice that Y (t) = T [y1 (t), y2 (t), . . . , yn (t)]T = [y(t), y˙ (t), . . . , y(n−1) (t)] ∈ Rn is the state vector of the master system, f0 (Y,t) is the known nonlinear dynamic part of master system, f(Y,t) is time-varying, not precisely known, and uncertain part of master system, and dm (t) is the external disturbance of system (19). This hypothesis is not restrictive as it can be satisfied by many chaotic systems in literature, such as Genesio system [13], Arneodo system [1] and Duffing–Holmes system [4].
Fig. 5. Adaptive parameters γˆ and λˆ versus time for regulating to the desired orbit xd = 0.
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As it was said in the previous section, the uncertain term fm (Y,t) and disturbance term dm (t) are assumed bounded, i.e.: |fm (Y, t)| ≤ αm
and
|dm (t)| ≤ βm ,
(20)
where αm and βm are positive. Thus, to obtain synchronization, we can modify the error Eq. (3) as follows: T
E = Xs − Ym = [x1s − y1m , x2s − y2m , . . . , xns − ynm ]T = [e, e˙ , . . . , en−1 ] = [e1 , e2 , . . . , en ]T
(21)
If the error states off the coupled system are defined as Eq. (21), then the dynamic equations of these errors can be determined directly by subtracting Eq. (19) from Eq. (1), to yield: e˙ i = ei+1 , 1 ≤ i ≤ n − 1, (22) e˙ n = F + F + D(t) + u where F = f0s (X, t) − f0m (Y, t) F = fs (X, t) − fm (Y, t) D = ds (t) − dm (t)
(23)
in which, F is known nonlinear function, F is time-varying, not precisely known, and uncertain part of error dynamical system (22), and D is the external disturbance of the error system (22). The same as what we do before in Eqs. (5)–(7), we have ˆ s) = −f0s (Xs , t) + f0m (Ym , t) − γξ( ˆ s). ˆ λ, ˆ λ, u = −F − γξ(
Fig. 6. The chaotic trajectories of uncertain Arneodo system.
(24)
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ˆ s) is defined as: where ξ(λ, ˆ s) = tanh(λs). ˆ ξ(λ,
(25)
and the adaptive laws are designed as: ˙γˆ = k3 eξ ∂xs , ∂u ∂xs λˆ˙ = k4 e(1 − ξ 2 )s , ∂u
(26)
where k3 and k4 are positive constants. Proof of the stability of above formula is the same as Theorem 1 and Theorem 2. 5. Simulation results This section of the paper presents two illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. The simulation results are carried out using the MATLAB software. The fourth order Runge–Kutta integration algorithm was performed to solve the differential equations. A time step size 0.001 was employed. Consider the Genesio chaotic system as follows: ⎧ ⎪ ⎨ x˙ 1 = x2 x˙ 2 = x3 (27) ⎪ ⎩ 2 x˙ 3 = −cx1 − bx2 − ax3 + x1 ,
Fig. 7. The time response of master system and slave system states.
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where x1 , x2 , x3 are state variables, and a, b and c are the positive real constants satisfying ab < c. For instance, the Genesio system is chaotic for the parameters a = 1.2, b = 2.92, c = 6. Consider that system (27) is perturbed by an uncertainty term f(X,t) and excited by a disturbance term d(t), added to its third equation. The uncertain Genesio system can be written as: ⎧ ⎪ ⎨ x˙ 1 = x2 x˙ 2 = x3 (28) ⎪ ⎩ 2 x˙ 3 = −cx1 − bx2 − ax3 + x1 + Δf (X) + d(t) + u(t). From (8), the continuous control input is determined as: ˆ s). ˆ λ, u = cx1 + bx2 + ax3 − x12 − γξ(
(29)
The system is perturbed by an uncertainty term f (X, t) = 0.1 sin(πx1 ) sin(2πx2 ) sin(3πx3 ) and d(t) = 0.1 cos(3t), where |Δf(X)| ≤ α = 0.1 and |d(t)| ≤ β = 0.1. The uncontrolled uncertain system (28) (i.e. u(t) = 0) is also chaotic and retains one positive Lyapunov exponent as shown in Fig. 1. The simulation is done with the initial value [x1 x2 x3 ]T = [3 −4 2]T , c1 = 10, c2 = 6, k1 = 10, k2 = 15, λˆ 0 = 15, and γˆ 0 = 30. The simulation results are shown in Figs. 2–5 under the proposed ASMC (29) with the adaptation algorithm (9) and (10). Fig. 2 represents respectively the state time response, sliding surface dynamics and control input for tracking a periodic orbit xd = sin(2t). It can be seen that the dynamic of system states are stabilized to a periodic motion. Fig. 4 shows the state time responses, corresponding s(t) and control input for regulating to xd = 0. It can be seen that chattering does not appear, due to continuous control. The time responses of adaptation parameters are shown in Figs. 3 and 5 respectively.
Fig. 8. Adaptive parameters γˆ and λˆ and the control input versus time.
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Next, consider the Arneodo chaotic system: ⎧ ⎪ ⎨ y˙ 1 = y2 y˙ 2 = y3 ⎪ ⎩ y˙ 3 = −b1 y1 − b2 y2 − b3 y3 + b4 y13 + fm (Y ) + dm (t)
(30)
with b1 = −5.5, b2 = 3.5, b3 = 1, b4 = −1, is the master system, and Genesio chaotic system (28) is the slave. The state trajectories of the uncertain Arneodo system (30) are depicted in Fig. 6. From (24), the continuous control input is determined as: 2 3 ˆ s). ˆ λ, − b1 y1m − b2 y2m − b3 y3m + b4 y1m − γξ( u = cx1s + bx2s + ax3s − x1s
(31)
The control parameters for the controller (31) were chosen as k3 = 5, k4 = 15, λˆ 0 = 15, and γˆ 0 = 10. The simulation is done with the initial value [x1s x2s x3s ]T = [3 −4 2]T , [y1m y2m y3m ]T = [1 −1 0]T and the sliding surface parameters are c1 = 10, c2 = 6. The slave system is perturbed by an uncertainty term fs (Xs , t) = 0.1 sin(πx1s ) sin(2πx2s ) sin(3πx3s ) and ds (t) = 0.1 sin(3t), where |Δf(Xs )| ≤ α = 0.1 and |ds (t)| ≤ β = 0.1 and these parameters for the master system are set as f (Ym , t) = 0.2 sin(y1m ) sin(y2m y3m ) and dm (t) = 0.2 sin(2t) where |Δf(Ym )| ≤ αm = 0.2 and |dm (t)| ≤ βm = 0.2. The trajectories of the states of master and slave system are shown in Fig. 7. It can be seen that the slave system (Genesio–Tesi chaotic system) response immediately follows the reference trajectories given by Arneodo chaotic system. The evolution of the two adaptation parameters and the control input are depicted in Fig. 8. Finally, Fig. 9 shows the sliding surface and time responses of the synchronization error states. From the simulation results, it can be concluded that the obtained theoretic results are feasible and efficient for controlling uncertain chaotic dynamical system and synchronizing two different uncertain chaotic systems.
Fig. 9. The time response of synchronization error states, sliding surface.
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6. Conclusion This work presents the control and synchronization of chaos by designing the adaptive sliding mode controller. In the proposed approach, by applying appropriate control signal based on adaptive update law, a continuous control signal is achieved and stability of the system is guaranteed. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of Lyapunov stability theory. As the simulations show, the new controller could track and stabilize the desired trajectory within a short time not only to a fixed point, but also to an arbitrary orbit. The most distinguished feature of our controller is that it can be applied to synchronize two different chaotic systems. Meanwhile, numerical simulations illustrate the robustness of the controller against model uncertainty and noise disturbance; and it is a good solution to the chattering problem in the traditional sliding mode control. References [1] A. Arneodo, P.H. Coullet, E.A. 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