Finite time stabilization of chaotic systems via single input more

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Physics Letters A 375 (2010) 119–124 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Finite time stabilization of chaotic systems via single input Rongwei Guo a , U.E. Vincent b,∗,1 a b School of Mathematics and Physics, Shandong Institute of Light Industry, Jinan 250353, PR China Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom a r t i c l e i n f o a b s t r a c t In this Letter, we propose a single control input approach for stabilizing three-dimensional chaotic systems in a finite time. The method is more general and is derived from the finite-time stability theory and adaptive control technique; and can stabilize almost all well-known three-dimensional chaotic systems without a prior knowledge of the feedback gain. To show the wider applicability of our method, we give illustrations using different chaotic systems with different structure. Numerical simulations are also used to verify the effectiveness of the technique. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. Article history: Received 21 April 2010 Received in revised form 14 October 2010 Accepted 20 October 2010 Available online 30 October 2010 Communicated by A.R. Bishop Keywords: Chaos Chaos control Finite time stabilization 1. Introduction In the last two decades, extensive studies have been done on several aspects of nonlinear dynamical systems, including chaotic dynamics. Being a unique and very relevant nonlinear phenomenon, chaos has been intensively investigated in the context of several specific problems arising in Physics, mathematics, engineering science, and secure communication, etc. An important challenge in chaos theory is the control, including stabilization of chaotic systems to steady states or regular behaviour. In fact, this is particularly significant because most realistic systems need to be operated in the regular regimes. In 1990, Ott, Grebogi, and Yorke (OGY) [1] proposed the first approach of chaos control. The emergence of the OGY approach has generated enormous research activities, creating an entire new research domain in nonlinear dynamics. Indeed, chaos control theory has advanced far beyond theoretical perspectives to experimental realizations (see [2–7] and the references therein). For this reason, designing simple and available control inputs is extremely relevant for experimental chaos control. Furthermore, it will also be more reasonable based on practical perspective, to realize stabilization of chaotic systems in a given time with faster convergence. Achieving faster convergence in control systems requires finite time control techniques, which have demonstrated better robustness, as well as disturbance rejection properties [8]. Finite time stability which is basically concerned with finite time control, implies the optimality in settling time of the controlled system [9]. Notably, finite time convergent property is strictly a unique characteristics of only nonsmooth or non-Lipschitz continuous autonomous systems. For instance, the solution of the following scalar system [10] ˙ x = −x 3 , 1 x(0) = x0 , 2 in forward time, which can be described by ⎧ 2 ⎨ (x 3 − 2 t ) 3 , 0 2 0 3 x(t ) = ⎩ 0, t t 2 3 3 x , 2 0 3 3 x , 2 0 Corresponding author. E-mail addresses: rongwei_guo@163.com (R. Guo), u.vincent@lancaster.ac.uk (U.E. Vincent). 1 Permanent: Department of Physics, Faculty of Science, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye, Nigeria. 0375-9601/$ – see front matter Crown Copyright doi:10.1016/j.physleta.2010.10.037 * converges to the unique equilibrium x = 0 in a finite time; implying that nonsmooth or at least non-Lipschitz continuous feedback is a requirement for achieving finite time stabilization, even ˙ if the controlled plant x = f (x, u , t ) is smooth. Previous research on finite-time control have been largely concerned with the stabilization and stabilizability of nonlinear systems (see for example [8,9,11–16]). Although these studies have achieved some successes, they do not take into account the chaotic dynamics states which is known to alter the dynamics of nonlinear systems considerably. Furthermore, the control functions appears unrealistic in terms of possible experimental applications. Recently, research works were done on chaos synchronization based on finite time (see for instance [10,17] and the references therein). Basically, two chaotic systems are synchronized by appropriate controller, which is equivalent to the asymptotic stabilization of the error system (i.e. the difference between the master and slave system). This idea was employed by Gao et al. [18] and very © 2010 Published by Elsevier B.V. All rights reserved. Author's personal copy 120 R. Guo, U.E. Vincent / Physics Letters A 375 (2010) 119–124 recently by Wang et al. [19] to realize finite-time chaos control in the Duffing–Holmes oscillator and uncertain unified chaotic system, respectively. Gao et al. [18], used a sliding-mode (or variable structure) control with nonlinearity to achieve finite-time control, arguing that there often exist nonlinearities in some engineering systems inputs; so that nonlinear effects cannot be overlooked in controller design consideration. However, the above claim is not always applicable. For instance, nonlinearity in control input(s) often give rise to the problem of controller complexities; and could pose difficulties for practical applications. The problem of controller complexity is a crucial issue in control theory and has gained considerable research attention in the recent times [20–24]. It has been shown that simple control inputs in the form of simple limiter are easy-to-implement and are effective method of stabilizing irregular fluctuations [20,21]. In addition, the sliding-mode control employed in [18] demonstrates a discontinuous control law, and would allow the phenomenon of chattering to appear [26,25]. On the other hand, the recent paper by Wang [19] proposes a finite-time chaos control based on the combination of finite-time stability theory and stability theory of cascaded-connected system. However, the proposed approach requires three control inputs to stabilize the unified chaotic systems in finite-time. Clearly, the control input obtained in [19] is very complex compared to some existing results on the unified chaotic system [23,24]. Motivated by the above challenges, we propose in this Letter a simple single variable control approach to achieve finite-time chaos stabilization. Our method which is more general and derived from the finite-time stability theory and adaptive control mechanism can stabilize almost all well-known three-dimensional chaotic systems. Compared to previous approaches, we employ a time-varying feedback gain k1 which automatically converges to a suitable constant k∗ . The estimate of k1 is unknown in advance; and the finite-time controller only includes a single input for a class of three-dimensional chaotic systems compared to previous results [18,19]. Finally, the controllers obtained have similar form—implying that the method is universal for a class of chaotic systems. To show the wider applicability of our method, we give illustration using four different chaotic systems with numerical simulations to verify the effectiveness of the technique. The rest of the Letter is organized as follows. In Section 2 some preliminary definitions are given. Section 3 gives the main results. In Section 4, several three-dimensional chaotic systems stabilized using our results. The Letter is concluded in Section 5. 2. Preliminary definition and lemma Finite-time stabilization means that the state of the chaotic system can converge to the origin after a finite-time. The precise definition of finite time stabilization and a lemma are given below. For details see [10,17]. Definition 1. Consider the following chaotic system Lemma 1. Assume that a continuous, positive-definite function V (t ) satisfies the following differential inequality: ˙ V (t ) −cV η (t ), ∀t t 0 , V (t 0 ) 0 (2) where c > 0, 0 < η < 1 are all constants. Then, for any given t 0 , V (t ) satisfies the following inequality: V 1−η (t ) and V 1−η (t 0 ) − c (1 − η)(t − t 0 ), t0 t t1 (3) V (t ) = 0, ∀t t1 (4) with t 1 given by t1 = t0 + V 1−η (t 0 ) c (1 − η) . (5) 3. Main results In this section, we investigate the finite time stabilization of chaotic systems, and present the main results of this Letter. For system (1) to satisfy and fit into the new design and analysis, the following assumption is needed: Assumption 1. There is a nonsingular coordinate transformation y = T x, such that system (1) can be rewritten as ˙ z1 = g 1 ( z1 , z2 ), ˙ z2 = g 2 ( z1 , z2 ), (6) ˙ where z1 = y 1 , z2 = ( y 2 , y 3 ) T , the second equation satisfies z2 = g 2 (0, z2 ), with the vector function g 2 ( z1 , z2 ) being smooth in a ˙ neighborhood of z1 = 0, and the subsystem z2 = g 2 (0, z2 ) is uniformly exponentially stable about the origin z2 = 0 for all z. Remark 1. It should be pointed out that not all the threedimensional chaotic systems are given as (6) in their original forms. Therefore, we should make a nonsingular coordinate transformation y = T x, which can adjust the array order of the variables (x1 , x2 , x3 ) to make the original systems (with new variables ( y 1 , y 2 , y 3 )) take the form of (6) by letting z1 = y 1 , z2 = ( y 2 , y 3 )T . Thus, Assumption 1 is reasonable, and system (6) is very general, which contains most well-known three-dimensional chaotic systems [24,27]. Remark 2. The vector function g 2 ( z1 , z2 ) being smooth in a neighborhood of z1 = 0, i.e., there is a positive constant λ1 such that g 2 ( z1 , z2 ) − g 2 (0, z2 ) λ1 z 1 . ˙ The subsystem z2 = g 2 (0, z2 ) is uniformly exponentially stable about the origin z2 = 0 for all z, which implies that there are a Lyapunov function V 0 ( z2 ) and two positive numbers λ2 , λ3 , such that ˙ V 0 ( z2 ) = (1) ∂ V 0 ( z2 ) g 2 (0, z2 ) ∂ z2 λ3 z2 , −λ2 z2 2 , ˙ x = f (x), ∂ V 0 ( z2 ) ∂ z2 where x = (x1 , x2 , x3 ) T ∈ R3 , f (x) = ( f 1 (x), f 2 (x), f 3 (x)) T : R3 → R3 is a smooth nonlinear vector function. Without loss of the generality, let xe = 0 be an equilibrium point of the system (1). If there exists a constant T > 0, such that t →T lim x(t ) = 0, respectively. Since the system (6) is chaotic, and g 1 ( z1 , z2 ) is a smooth function, there exists a positive number λ4 , such that g 1 ( z1 , z2 ) λ4 z 1 . In order to stabilize the chaotic system (6) within a finite time, we add the following controller u 1 to the system (6), the controlled system (6) would be given as and x(t ) ≡ 0 if t T , then the stabilization of the system (1) is achieved in a finite-time. ˙ z1 = g 1 ( z1 , z2 ) + u 1 , where ˙ z2 = g 2 ( z1 , z2 ), (7) Author's personal copy R. Guo, U.E. Vincent / Physics Letters A 375 (2010) 119–124 121 u 1 = k1 z1 − c 1 u 1 = 0, V 1 ( z) η z1 if z1 = 0, , c 1 > 0, 0 < η < 1, if z1 = 0, (8) Remark 4. If xe = 0 is an equilibrium point of the chaotic system, this method can be also easily used by carrying out a coordinate transformation y = x − xe . 4. Applications and k1 is adapted according to the following update law: ˙ k1 = −γ where as 2 z1 , (9) γ is an arbitrary positive real number, and V 1 (z) is given 1 2 2 z1 + V 0 ( z2 ). V 1 ( z) = In this section, several three-dimensional chaotic systems, having different structures are used as examples to illustrate how to apply the results obtained and obtain finite-time stabilization of a class of chaotic systems. Example 1. The unified chaotic system [28] is written as: (10) Let the system (7) and (9) be the augment system, and introduce a Lyapunov function, ˙ x1 = (25α + 10)(x2 − x1 ), ˙ x2 = (28 − 35α )x1 + (29α − 1)x2 − x1 x3 , ˙ x3 = x1 x2 − 8+α 3 x3 , (14) V ( z, k1 ) = V 1 ( z) + where 11 2γ (k1 + L )2 , (11) L = (λ4 + α ), α λ 2 λ2 1 3 . 4 λ2 (12) It is easy to show that V ( z, k1 ) V 1 ( z), and there exists a constant η l = cc1 > 0 such that V η ( z, k1 ) lV 1 ( z). According to our previous results [23] and the finite-time control theory, it is easy to obtain the following result: Theorem 1. Starting from any initial values of the augment system, the orbits ( z(t ), k1 (t )) T converge to (0, k∗ ) T within a finite time, where k∗ is a negative constant depending on the initial value. That is, system (6) can be finite timely stabilized by the above controller u 1 . Proof. Differentiating the function V ( z, k1 ) along the trajectories of the augment system, we obtain where α ∈ [0, 1]. System (14) is chaotic for α ∈ [0, 1]. When α ∈ [0, 0.8], system (14) reduces to the general Lorenz system; when α = 0.8, it becomes the general Lü system; and when α ∈ [0.8, 1], system (14) is the general Chen system. It is easy to show that: if x2 = 0, the following two-dimensional subsystem of the system (14) is obtained: ⎧ ˙ ⎨ x1 = (25α + 10)x1 , 8+α ⎩ x3 = − ˙ x3 , 3 ˙ ˙ V = z1 z1 + ∂ V 0 ( z2 ) 1 ˙ g 2 ( z1 , z2 ) + (k1 + L )k1 ∂ z2 γ = z1 ( g 1 ( z1 , z2 ) + k1 z1 ) ∂ V 0 ( z2 ) η 2 g 2 ( z1 , z2 ) − (k1 + L ) z1 − c 1 V 1 ( z) + ∂ z2 ∂ V 0 ( z2 ) 2 g 2 ( z1 , z2 ) − g 2 (0, z2 ) = z1 g 1 ( z1 , z2 ) − Lz1 + ∂ z2 ∂ V 0 ( z2 ) η g 2 (0, z2 ) − c 1 V 1 ( z) + ∂ z2 2 2 λ4 z1 − (λ4 + α ) z1 + λ1 λ3 z1 2 = −α z1 + λ1 λ3 z1 z2 − λ2 z2 2 z 2 − λ2 z 2 2 − c 1 V 1 ( z) η − c 1 V 1 ( z) η −c 1 V 1 ( z) −cV η ( z, k1 ). According to Lemma 1, the augment system can be finite timely stabilized by the controller u 1 , thus, Theorem 1 is obtained. That is, the chaotic system (6) can be finite timely stabilized by the above controller u 1 . 2 Remark 3. From the proof of Theorem 1, it is easy to obtain the settling time T is η which is uniformly exponentially stable around the origin (x1 , x3 ) = (0, 0) for all x1 , and x3 regardless of the value of the parameter α . Therefore, there exists a nonsingular coordinate transformation y = T x, which makes the new system (with variable y) has the form of system (4) wherein y 1 = x2 , y 2 = x1 , y 3 = x3 . We point out that the coordinate transformation y = T x is chosen by inspection as follows. First, we let x1 = 0 and examine ˙ ˙ whether the rest subsystems, namely, (x2 = f 2 (0, x2 , x3 ), x3 = f 3 (0, x2 , x3 )) is uniformly exponentially stable about the origin (x2 , x3 ) = (0, 0) or not, and so on. Otherwise, we let x2 = 0 and examine again whether the rest subsystems of the chaotic system, ˙ ˙ (x1 = f 1 (x1 , 0, x3 ), x3 = f 3 (x2 , 0, x3 )) are uniformly exponentially stable about the origin, (x1 , x3 ) = (0, 0) or not, and so on. For example, if x2 = 0, and the rest subsystems are uniformly exponentially stable about the origin, (x1 , x3 ) = (0, 0) then, the coordinate transformation y = T x is chosen as y 1 = x2 , y 2 = x1 , y 3 = x3 . In principle, the coordinate transformation y = T x only adjusts the array order of the variables (x1 , x2 , x3 ) such that the original system (with new variables ( y 1 , y 2 , y 3 )) satisfy the condition of Assumption 1 by setting z1 = y 1 , z2 = ( y 2 , y 3 ) T ; thereby allowing for the application of our main results to stabilize the chaotic systems within a finite time. Accordingly, the controlled new unified chaotic system with a finite time control signal is given as, ˙ y1 = − y1 − y2 y3 + u1 , ˙ y 1 = (25α + 10)( y 1 − y 2 ), ˙ y3 = y1 y2 − 8+α 3 y3 , (15) (t 0 ) T = t0 + , c (1 − η) V1 1 −η where u 1 is given as (13) u 1 = k1 z1 − c 1 u 1 = 0, V 1 ( z) z1 η i.e., for any arbitrary initial value z0 , the system (6) can be finite timely stabilized by the above controller u 1 within the time T . , c 1 > 0, 0 < η < 1 , if z1 = 0, if z1 = 0, Author's personal copy 122 R. Guo, U.E. Vincent / Physics Letters A 375 (2010) 119–124 Fig. 1. Stability of the controller signal, showing convergence to origin within a finite time. Fig. 3. Asymptotic stabilization of the novel chaotic system to origin within a finite time. It is easy to show that, if x2 = 0, the following two-dimensional subsystem of the system (16) is obtained: ˙ x1 = −ax1 , ˙ x3 = −hx3 which is uniformly exponentially stable about the origin x1 = 0, x3 = 0 for all x1 and x3 , therefore, there exists a nonsingular coordinate transformation y = T x, i.e., y 1 = x2 , y 2 = x1 , y 3 = x3 , which makes the new system 16 (with variable y) have the form of system (4). Accordingly, the controlled new system with a finite time controller is given as, ˙ y1 = − y2 y3 + u1 , Fig. 2. Stabilization of the unified chaotic system with (α = 0) to origin within a finite time. ˙ y 2 = −ay 2 + y 1 + by 1 y 3 , ˙ y 3 = dy 1 y 2 − hy 3 . where u 1 is given as η (17) k1 is adapted according to the update law (9), and V 1 ( z) = 1 2 2 z1 + V 0 ( z2 ) = 1 2 3 2 T z1 + z2 z2 = i =1 y2 . i u 1 = k1 z1 − c 1 u 1 = 0, V 1 ( z) origin within T = According to Theorem 1, system (15) can be stabilized in finite time. To confirm that the control is stable and stabilization is achieved in finite time, we give numerical simulations with the following choices of initial conditions: [ y 1 (0), y 2 (0), y 3 (0)] = [−2, 4, 3] and k1 (0) = −1, c = 4, η = 1 . Throughout, the sixth2 order Runge–Kutta method has been used to solve ordinary differential equations with adaptive step-size algorithm in order to further ensure that numerical chattering is avoided. First, we illustrate the stability of the control signal u 1 in Fig. 1. Notice the speed and asymptotic convergence of the control signal. In Fig. 2, we show that the √ trajectory of the system (15) also converges to 29 2 z1 if z1 = 0, , c 1 > 0, 0 < η < 1, if z1 = 0, k1 is adapted to the update law (9), and V 1 ( z) = 1 2 2 z1 + V 0 ( z2 ) = 1 2 3 2 T z1 + z2 z2 = i =1 y2 . i ≈ 2.6926. According to Theorem 1, system (17) can be finite time stabilized. To illustrate the validity of the above conclusion, we carry out simulations with the following choices of initial conditions: [ y 1 (0), y 2 (0), y 3 (0)] = [−3, 2, 2] and k1 (0) = −1, c = 4, η = 1 . 2 Here and for the rest examples, we omit the controller signal for brevity. It can be observed from Fig. 3 that the trajectory of the sys√ tem (17) converges to the origin within T = 17 2 ≈ 2.0616. Example 2. The novel chaotic system [29]: Example 3. The Genesio chaotic system [30]: ˙ x1 = −ax1 + x2 + bx2 x3 , ˙ x2 = cx2 − x1 x3 + x3 , ˙ x3 = dx1 x2 − hx3 , (16) ˙ x1 = x2 , ˙ x2 = x3 , ˙ x3 = ax1 + bx2 + cx3 + x2 , 1 (18) where a, b, and c are the system’s parameters. System (18) is chaotic when the parameters take the values a = −6, b = −2.92, and c = −1.2. It is easy to show that the system (18) has the form of system (4). Then, according to the results in this Letter, thee we where a, b, c, d, and h are parameters. System (16) is chaotic when the parameters take the values a = 3, b = 2.7, c = 4.7, d = 3, h = 9. The system (16) has five equilibrium points which are all unstable saddle focus-nodes. Author's personal copy R. Guo, U.E. Vincent / Physics Letters A 375 (2010) 119–124 123 Fig. 4. Asymptotic stabilization of the Genesio chaotic system to origin within a finite time. Fig. 5. Asymptotic stabilization of the Tigan system to origin within a finite time. can write the controlled new system with a finite time controller as, ˙ y 1 = (c − a) y 2 − ay 2 y 3 + u 1 , ˙ y 2 = a( y 1 − y 2 ), ˙ y 3 = y 1 y 2 − by 3 , where u 1 is given as ˙ x1 = x2 + u 1 , ˙ x2 = x3 , ˙ x3 = ax1 + bx2 + cx3 + where u 1 is given as η (21) x2 , 1 (19) u 1 = k1 z1 − c 1 u 1 = 0, V 1 ( z) z1 η , c 1 > 0, 0 < η < 1 , if z1 = 0, u 1 = k1 z1 − c 1 u 1 = 0, V 1 ( z) z1 if z1 = 0, , c 1 > 0, 0 < η < 1, if z1 = 0, if z1 = 0, k1 is adapted to the update law (9), and k1 is adapted to the update law (9), and V 1 ( z) = 3 1 2 2 z1 + V 0 ( z2 ) = 1 2 3 2 T z1 + z2 z2 = i =1 y2 . i V 1 ( z) = 1 2 2 z1 + V 0 ( z2 ) = 1 2 2 z1 + T z2 z2 = i =1 x2 . i According to the Theorem 1, system (18) can be stabilized in finite time. Again, we show the validity of the above conclusion, with some simulation results shown in Fig. 4. The initial condition were selected as follows: [x1 (0), x2 (0), x3 (0)] = [5, −4, 3] and k1 (0) = −1, c = 4, η = 1 . 2 From Fig. 4, we observe that the trajectory of the system (19) √ converges to the origin within T = 17 2 According to the Theorem 1, the system (21) can be stabilized in finite time. For the controlled system (21), we give numerical example using the following initial condition: [ y 1 (0), y 2 (0), y 3 (0)] = [−2, 4, 3] and k1 (0) = −1, c = 4, η = 1 . From Fig. 5, it is clear 2 that √ trajectory of the system (21) converges to origin within the T= 22 2 ≈ 2.3452. 5. Conclusion In conclusion, we have investigated the stabilization of threedimensional chaotic systems based on finite time stability theory, and proposed a single input control law to realize finite-time chaos stabilization goal. Our proposed control is very simple and could be easily realized experimentally compared to previous techniques that uses complex control functions. In fact, our method which combines the adaptive feedback method [23] with the finite-time stability theory has at least three advantages, namely: (i) an estimate of the feedback gain k1 is not needed in advance, rather k1 is a time-varying feedback gain which would automatically converge to a suitable constant k∗ for arbitrary initial value of the controlled system; (ii) for a class of three-dimensional chaotic systems, the finite-time controller only includes single input and (iii) the method is universal, that is, for a class of chaotic systems, the controllers obtained are similar in form. We have demonstrated the performance of our technique using several examples of threedimensional chaotic systems with different structure. Numerical simulations have been provided in all cases to illustrate the effectiveness and validity of this method. We believe that our approach could be extended to other systems with more complex structure and possibly, higher-order (hyperchaotic) systems. ≈ 3.5355. Example 4. The last example is the Tigan system [31] given as: ˙ x1 = a(x2 − x1 ), ˙ x2 = (c − a)x1 − ax1 x3 , ˙ x3 = x1 x2 − bx3 , (20) where a = 2.1, b = 0.6, and c = 30. It is easy to show that if x2 = 0, the following two-dimensional subsystem of the system (20) is obtained: ˙ x1 = −ax1 , ˙ x3 = −bx3 which is uniformly exponentially stable about the origin x1 = 0, x3 = 0 for all x1 , x3 . Therefore, there exist a nonsingular coordinate transformation y = T x, i.e., y 1 = x2 , y 2 = x1 , y 3 = x3 , which makes new system (with variable y) take the form of system (6). Then, according to the results in this Letter, the controlled new system with a finite time controller is given as, Author's personal copy 124 R. Guo, U.E. 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