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Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties
Commun Nonlinear Sci Numer Simulat 14 (2009) 2784–2792
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties Cheng-Fang Huang, Kuo-Hua Cheng, Jun-Juh Yan * Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 August 2008 Received in revised form 29 September 2008 Accepted 30 September 2008 Available online 7 October 2008 PACS: 05.45.Gg 05.45.Xt
a b s t r a c t This paper investigates the robust chaos synchronization problem for the four-dimensional energy resource systems. Based on the sliding mode control (SMC) technique, this approach only uses a single controller to achieve chaos synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero or into predictable bounds for matched and unmatched perturbations, respectively. Numerical simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Energy resource system Sliding mode control Chaos synchronization Perturbation
1. Introduction Over the last two decades, since the pioneering work of Pecora and Carroll in 1990 [1], much attention has been paid to the chaos synchronization in non-linear systems. Generally, designing a controller to force a system to mimic the behavior of another chaotic system is called synchronization. There is much prospect of chaos synchronization being applied in the vast areas of physics and engineering systems such as in power converters, chemical reactions, biological systems, information processing, especially in secure communication where a lot of progress has been achieved [2–6]. Nowadays, different techniques and methods have been proposed to achieve chaos synchronization such as adaptive control [7–11], sliding mode control (SMC) [12–14], impulsive control [15,16], linear control [17], optimal control [18], digital redesign control [19], and backstepping control [20,21]. On the other hand, the energy resource system is a kind of complex non-linear system and widely used in the industry. Study of chaos and its synchronization control in power systems is with considerable importance from the point of view of avoiding undesired behaviors such as power blackout [22–25]. Recently, in paper [26]. Sun et al. established a three-dimensional energy resource demand-supply system based on the real energy resources demand-supply in the East and the West of China. Furthermore, by adding a new variable to consider the renewable resources, a four-dimensional energy resource system was proposed in [27]. The dynamics behaviors of the four-dimensional energy resource system have been analyzed by means of the Lyapunov exponents and bifurcation diagrams. Also the same as the above-mentioned power systems, this four-dimensional energy resource system is with rich chaos behaviors. Furthermore, in chaos synchronization, most publications often assume that the * Corresponding author. Tel.: +886 7 6158000/4806; fax: +886 7 6158000/4899. E-mail address: [email protected] (J.-J. Yan). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.09.017
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synchronization system is without external disturbances. If external disturbance is considered, then, in general, only matched disturbance is considered. However, in practical applications, especially in energy resource systems, it is hard to avoid external disturbances due to uncontrollable environmental conditions. In other words, chaos synchronization systems may be undergoing either matched or unmatched disturbances. Thus, the derivation of a robust synchronization controller for the fourdimensional energy resource systems to resist the presence of disturbance is an important problem. In this paper, the SMC technique is applied to achieve the chaos synchronization for the four-dimensional energy resource system. In our design, a single controller is used enough to realize synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero with the matched perturbations or into predictable bounds with unmatched perturbations. Simulation results show the effectiveness of the proposed control scheme. Throughout this paper, it is noted that, jwj represents the absolute value of w and sign(r) is the sign function of r, if r > 0, sign(r) = 1; if r = 0, sign(r) = 0; if r < 0, sign(r) = 1. 2. System description In this paper, a four-dimensional energy resource demand-supply system is considered. The dynamics of this system is described by the following differential equations [27]:
x_ ¼ a1 xð1 x=MÞ a2 ðy þ zÞ d3 w; y_ ¼ b1 y b2 z þ b3 x½N ðx zÞ; z_ ¼ c1 c2 zx c1 c3 z; _ ¼ d1 x d2 w; w
ð1Þ
where x(t) is the energy resource shortage in A region, y(t) is the energy resource supply increment in B region. z(t) and w(t) are energy resource import in A region and renewable energy resource in A region, respectively. ai, bi, ci, di and M, N are positive real constant. The dynamics of this system has been extensively studied in [27]. The bifurcation diagram of system for the parameter value, M = 1.8, N = 1, a2 = 0.15, b1 = 0.06, b2 = 0.082, b3 = 0.07, c1 = 0.2, c2 = 0.5, c3 = 0.4, d1 = 0.1, d2 = 0.06, d3 = 0.07, a1 2 [0.05 0.15], is shown in Fig. 1. From Fig. 1, it shows that this energy resource system displays chaotic behavior when a1 = 0.1 and the chaotic attractors and state responses are shown in Fig. 2 with initial condition ½xð0Þ yð0Þ zð0Þ wð0Þ ¼ ½0:82 0:29 0:48 0:1. 3. Program formulation The aim of this section is to design a controller using the sliding mode control, such that the slave system is able to mimic the behavior of the master chaotic system. The master (or drive) system and slave (or response) system with consideration of external perturbations are defined below, respectively
x_ m ¼ a1 xm ð1 xm =MÞ a2 ðym þ zm Þ d3 wm ; y_ m ¼ b1 ym b2 zm þ b3 xm ½N ðxm zm Þ; z_ m ¼ c1 c2 zm xm c1 c3 zm ; _ m ¼ d1 xm d2 wm w
Fig. 1. Bifurcation diagram of x (a1 2 [0.06 0.12]).
ð2Þ
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Fig. 2. Attractor of energy resource system: (A) three-dimensional view (y–x–z) (B) three-dimensional view (y–x–w)(C) three-dimensional view (w–z–x).
and
x_ s ¼ a1 xm ð1 xs =MÞ a2 ðys þ zs Þ d3 ws þ p1 ðtÞ þ uðtÞ; y_ s ¼ b1 ys b2 zs þ b3 xs ½N ðxm zm Þ þ p2 ðtÞ; z_ s ¼ c1 c2 zm xs c1 c3 zs þ p3 ðtÞ;
ð3Þ
_ s ¼ d1 xs d2 ws þ p4 ðtÞ; w where X m ¼ ½xm ym zm wm T and X s ¼ ½xs ys zs ws T are state vectors of the master and slave systems, respectively, u(t) 2 R is the control input, pi(t), i = 1, 2, 3, 4, are the unavoidable external perturbations in practical systems and assumed bounded, i.e.
j pi ðtÞ j6 ai ;
i ¼ 1; 2; 3; 4;
ð4Þ
where ai > 0 are given. Generally, p1 is called the matched perturbation and pi, i = 2, 3, 4 are the unmatched perturbations.
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The control problem considered in this paper is that for different initial conditions of systems (2) and (3), the two coupled systems can achieve synchronization by designing an appropriate control input u(t) attached to the slave system (3). Now define the synchronization error vector between systems (2) and (3) as
e1 ðtÞ ¼ xm ðtÞ xs ðtÞ;
e2 ðtÞ ¼ ym ðtÞ ys ðtÞ; e3 ðtÞ ¼ zm ðtÞ zs ðtÞ; e4 ðtÞ ¼ zm ðtÞ zs ðtÞ
ð5Þ
so the error dynamics is determined by the following equation:
e_ 1 ¼ ða1 xm e1 Þ=M a2 ðe2 þ e3 Þ d3 e4 p1 u; e_ 2 ¼ b1 e2 b2 e3 b3 e1 ½N ðxm zm Þ p2 ; e_ 3 ¼ c1 c2 zm e1 c1 c3 e3 p3 ;
ð6Þ
e_ 4 ¼ d1 e1 d2 e4 p4 : Obviously, the problem in realizing synchronization now transfers to another problem on how to propose a control law u(t) to force error states ei(t), i = 1, 2, 3, 4 to zero with matched perturbations or into a predictable bound with unmatched perturbations. Consequently, to achieve the control goal via SMC, there are two basic steps for the design procedure: the first step is to construct an appropriate switching surface such that the sliding motion on the manifold can result in limt?1jei(t)j 6 qi, i = 1, 2, 3, 4 where qi are some predictable bounds, even with any initial states and unavoidable external perturbations.; the second step is to ensure the existence of the sliding mode r(t) = 0 by designing a SMC law. To complete the two design steps above, a switching surface r(t) in the state space is first defined as follows:
r ¼ e1 þ
Z
t
ke1 ðsÞds;
ð7Þ
0
where r 2 R and k > 0 is an designed parameter to guarantee the stability of system in the sliding mode. It is well known that when the system operates in the sliding mode, it satisfies the following fact [28,29]
r ¼ e1 þ
Z
t
ke1 ðsÞds ¼ 0:
ð8Þ
0
Furthermore, since the sliding manifold r(t) = 0 occurs in a finite time tr(it will be proved in Lemma 1 later), i.e., r (t) = 0 for t > tr, we have
r_ ðtÞ ¼ e_ 1 þ ke1 ¼ 0
ð9Þ
) e_ 1 ¼ ke1 :
ð10Þ
Therefore, from (10) the following equivalent sliding mode dynamics can be obtained as
e_ 1 ¼ ke1 ; e_ 2 ¼ b1 e2 b2 e3 b3 e1 ½N ðxm zm Þ p2 ; e_ 3 ¼ c1 c2 zm e1 c1 c3 e3 p3 ;
ð11Þ
e_ 4 ¼ d1 e1 d2 e4 p4 : Obviously, since the design parameter k > 0 is specified, the exponential stability of e1(t) is surely guaranteed. Therefore, there exists time t1 large enough such that e1(t1) approach to zero and the dynamics of system (11), when time t P t1, can be simplified as
e1 ¼ 0; e_ 2 ¼ b1 e2 b2 e3 p2 ; e_ 3 ¼ c1 c3 e3 p3 ; e_ 4 ¼ d2 e4 p4 :
ð12aÞ ð12bÞ ð12cÞ ð12dÞ
Solving the differential Eqs. (12c) and (12d) for e3 and e4 when t P t1, respectively, results in
Z
e3 ðtÞ ¼ ec1 c3 ðtt1 Þ e3 ðt 1 Þ
tt 1
ec1 c3 ðtt1 sÞ p3 ðsÞds;
t P t1
ð13Þ
0
and
e4 ðtÞ ¼ ed2 ðtt1 Þ e4 ðt 1 Þ
Z 0
tt 1
ed2 ðtt1 sÞ p4 ðsÞds;
t P t1 :
ð14Þ
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Since c1c3 and d2 are positive, the bounds for e3 and e4, respectively, are obtained as
j e3 ðtÞ j¼j ec1 c3 ðtt1 Þ e3 ðt 1 Þ
Z
tt 1
ec1 c3 ðtt1 sÞ p3 ðsÞds j
0
6 ec1 c3 ðtt1 Þ j e3 ðt1 Þ j þ max p3 ðtÞ j j ec1 c3 ðtt1 Þ 1 ec1 c3 ðtt1 Þ 6 ec1 c3 ðtt1 Þ j e3 ðt1 Þ j þa3 c 1 c3
Z
tt 1
ec1 c3 s ds j
ð15Þ
0
and
ed2 ðtt1 sÞ p4 ðsÞds 0 Z tt1 j e4 ðt 1 Þ j þ max j p4 ðtÞ j ed2 ðtt1 Þ ed2 s ds 0 1 ed2 ðtt1 Þ : j e4 ðt 1 Þ j þa4 d2
Z j e4 ðtÞ j¼ ed2 ðtt1 Þ e4 ðt 1 Þ 6 ed2 ðtt1 Þ 6 ed2 ðtt1 Þ
tt1
ð16Þ
Eqs. (15) and (16) with c1c3 > 0 and d2 > 0 show that
lim j e3 ðtÞ j6 q3 ¼
t!1
a3
ð17Þ
c 1 c3
and
lim j e4 ðtÞ j6 q4 ¼
t!1
a4 d2
ð18Þ
:
^ 3 for t P t2. Let Since e3(t) converges to q3 as shown in (17), there exists a finite time t2 such that j e3 ðtÞ j6 q
gðtÞ ¼ b2 e3 p2
ð19Þ
and solving the differential Eq. (12b) results in
e2 ðtÞ ¼ eb1 ðtt2 Þ e2 ðt 2 Þ
Z
tt 2
eb1 ðtt2 sÞ gðsÞds;
t P t2 :
ð20Þ
0
For t P t2, we have
^ 3 þ a2 j g 3 ðtÞ j¼j b2 e3 p2 j6 b2 q
ð21Þ
Therefore, for t P t2, the bound of je2(t)j satisfies
eb1 ðtt2 sÞ gðsÞds 0 1 eb1 ðtt2 Þ ^ 3 þ a2 Þ j e2 ðt 2 Þ j þðb2 q b1
Z j e2 ðtÞ j¼ eb1 ðtt2 Þ e2 ðt 2 Þ 6 eb1 ðtt2 Þ
tt2
ð22Þ
^ 3 converges to q3 and we obtain By taking t2 sufficiently large, q
lim e2 ðtÞ j6 q2 ¼
t!1
b2 q3 þ a2 1 b2 a3 ¼ þ a2 b1 c1 c3 b1
ð23Þ
Remark 1. According the discussion above, we can conclude that the tracking errors in the sliding manifold can be forced to some predictable bounds as follows:
lim j e1 ðtÞ j¼ 0; lim j e2 ðtÞ j6 q2 ¼
t!1
t!1
1 b2 a3 a3 a4 þ a2 ; lim j e3 ðtÞ j6 q3 ¼ ; lim j e4 ðtÞ j6 q4 ¼ : t!1 b1 c1 c3 c1 c3 t!1 d2
ð24Þ
Particularly, for the nominal systems (i.e., p1(t) = p2(t) = p3 (t) = p4(t) = 0) or systems with matched perturbations (i.e., p1 (t) – 0 and p2(t) = p3(t) = p4(t) = 0), the resulting tracking errors limt?1jei(t)j = 0, i = 1, 2, 3, 4 can be ensured. After proposing the appropriate switching surface, it follows the design of a robust SMC law u(t) in the slave system (3) such that the states of the error system (6) can be driven to the switching surface r(t) = 0 in a finite time tr. Before stating the synchronization scheme to guarantee the occurrence of sliding manifold in a finite time, the reaching condition of the sliding manifold is given below.
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Lemma 1. [30] If the reaching condition of rðtÞr_ ðtÞ 6 l j rðtÞ j for l > 0, r(t) – 0, is satisfied, then in spite of any initial condition r(0), the sliding manifold r(t) = 0 will occur in a finite time bounded by t r 6 jrlð0Þj. Proof. The proof is an immediate consequence in the work of Tang [30] and hence is omitted.
Fig. 3. The time response of switching surface r(t) for the controlled chaotic energy system.
Fig. 4. State responses of controlled master and slave energy resource systems: (A) state response of xm,xs. (B) state response of ym,ys. (C) state response of zm,zs. (D) state response of wm,ws.
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After stating Lemma 1, to achieve the reaching condition indicated in Lemma 1, the synchronization scheme is proposed as
uðtÞ ¼ ½ða1 xm =M kÞe1 þ a2 ðe2 þ e3 Þ þ d3 e4 þ g signðrðtÞÞ;
g > a1 :
ð25Þ
Theorem 1. For the error dynamics (6) with the control law (25), suppose the external perturbations are bounded by Eq. (4), the reaching condition in Lemma 1 is surely satisfied and the system trajectory (6) enters the sliding manifold in a finite time. Proof. Substituting (6) and (25) into the derivative of
rðtÞr_ ðtÞ yields
rr_ ¼ r½e_ 1 þ ke1 ; ¼ r½a1 xm e1 =M a2 ðe2 þ e3 Þ d3 e4 p1 þ ke1 u; ¼ r½p1 g signðrðtÞÞ; 6j p1 j j r j g r signðrðtÞÞ; 6 ðg a1 Þ j r j :
ð26Þ
Since g > a1 is specified in (25), we get
rr_ 6 ðg a1 Þ j rðtÞ j6 l j r j; where l ¼ g a1 > 0
ð27Þ
Furthermore, according to Lemma 1, the system trajectory enters the sliding manifold r(t) = 0 and r_ ðtÞ ¼ 0 for t P t r ¼ jrlð0Þj. The proof is achieved completely.
4. Numerical examples This section presents some numerical examples to verify and demonstrate the effectiveness of the proposed synchronization scheme. In all numerical simulations, the fourth-order Runge–Kutta method with fixed step size of 0.0001 in MATLAB software is used.
Fig. 5. Error states of controlled energy resource systems: (A) Error state of e1. (B) Error state of e2. (C) Error state of e3. (D) Error state of e4.
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Here, the system accords with (2) and (3). The parameters M = 1.8, N = 1, a1 = 0.1, a2 = 0.15, b1 = 0.06, b2 = 0.082, b3 = 0.07, c1 = 0.2,c2 = 0.5, c3 = 0.4, d1 = 0.1, d2 = 0.06, d3 = 0.07 are specified for simulation. Case I: for the nominal system, i.e., p1(t) = p2(t) = p3(t) = p4 (t) = 0. As mentioned in Section 3, the proposed design procedure can be summarized as follows: Step1: According to (7), the switching surface with k = 3 is given by
r ¼ e1 þ
Z
t
3e1 ðsÞds:
ð28Þ
0
Step2: According to (25), the sliding mode control law is obtained as follows:
uðtÞ ¼ ½ð0:09xm =1:8 3Þe1 þ 0:15ðe2 þ e3 Þ þ 0:07e4 þ g signðrðtÞÞ;
g ¼ 0:1 > 0
ð29Þ
In numerical simulations, the initial value of the master and slave systems are given, respectively, as ½ xm ð0Þ ym ð0Þ zm ð0Þ wm ð0Þ T ¼ ½ 0:1 0:8 0:2 0:1 T and ½ xs ð0Þ ys ð0Þ zs ð0Þ ws ð0Þ T ¼ ½ 0:4 0:1 0:6 0:3 T . The simulation results are shown in Figs. 3–5. Fig. 3 shows the corresponding r(t) for the controlled energy resource system under the proposed sliding mod control (29). The state and error state responses and are shown in Figs. 4 and 5, respectively. From the simulation results, it shows that the proposed controller (29) can drive the resulting tracking errors limt?1jei(t)j = 0,i = 1, 2, 3, 4, which fully coincide with theoretical results in this paper. Case II: for the non-nominal system with the assumption of p1 (t) = 0.5sin(3t), p2(t) = 0.01sin(2t), p3(t) = 0.02sin(4t), p4 (t) = 0.01cos(t). Under the same simulation conditions as in Case I, and the switching surface with k = 2 is given by
r ¼ e1 þ
Z
t
2e1 ðsÞds
ð30Þ
0
and the sliding mode control law is given as follows:
uðtÞ ¼ ½ð0:09xm =1:8 2Þe1 þ 0:15ðe2 þ e3 Þ þ 0:07e4 þ g signðrðtÞÞ;
g ¼ 1 > a1 ¼ 0:5:
ð31Þ
The time responses of the error states, under the proposed sliding mod control (31), are shown in Fig. 6 and harmonic forms within the boundary predicted by (24) can be seen due to the effect of unmatched perturbations, which also coincide with theoretical results.
Fig. 6. Error states of controlled energy resource systems: (A) Error state of e1. (B) Error state of e2. (C) Error state of e3. (D) Error state of e4.
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5. Conclusion In this paper, based on sliding mode control technique, a robust control design for synchronization of four-dimensional energy resource demand-supply systems has been proposed. By the new PI switching surface, it has been found the stability of the error dynamics in the sliding mode can be easily ensured. The error states can be driven to zero or into predictable bounds. Illustrative examples, including nominal (matched) and non-nominal (unmatched) cases, have been presented to demonstrate the validity of the proposed synchronization scheme. References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Diallo O, Koné Y. Melnikov analysis of chaos in a general epidemiological model. Nonlinear Anal, Real World Appl 2007;81:20–6. [3] Salarieh H, Alasty A. Adaptive synchronization of two chaotic systems with stochastic unknown parameters. Commun Nonlinear Sci Numer Simul 2009;14:508–19. [4] Ngueuteu GSM, Yamapi R, Woafo P. Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices. Commun Nonlinear Sci Numer Simul 2008;13:1213–40. [5] Yau HT. Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs. Nonlinear Anal: Real World Appl 2007. doi:10.1016/j.nonrwa.2007.08.002. [6] Chen M, Zhou D, Shang Y. A new observer-based synchronization scheme for private communication. Chaos Soliton Fract 2005;24:1025–30. [7] Bowong S. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simul 2007;12:976–85. [8] Efimov DV. Dynamical adaptive synchronization. Int J Adaptive Control Signal Process 2006;20:491–507. [9] Chiang TY, Lin JS, Liao TL, Yan JJ. Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity. Nonlinear Anal 2008;68:2629–37. [10] Lu J, Cao J. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos 2005;15:043901. [11] Cao J, Lu J. Adaptive synchronization of neural networks with or without time-varying delay. Chaos 2006;16:013133. [12] Yan JJ, Hung ML, Chiang TY, Yang YS. Robust synchronization of chaotic systems via adaptive sliding mode control. Phys Lett A 2006;356:220–5. [13] Si-Ammour A, Djennoune S, Bettayed M. Asliding mode control for linear fractional systems with input and state delays. Commun Nonlinear Sci Numer Simul 2009;14:2310–8. [14] Konishi K, Hirai M, Kokame H. Sliding mode control for a class of chaotic systems. Phys Lett A 1998;245:511–7. [15] Sun JT, Zhang YP, Wu QD. Impulsive control for the stabilization and synchronization of Lorenz systems. Phys Lett A 2002;298:153–60. [16] Chen SH, Yang Q, Wang CP. Impulsive control and synchronization of unified chaotic system. Chaos Soliton Fract 2004;20:751–8. [17] Rafikov M, Balthazar José Manoel. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simul 2008;13:1246–55. [18] Tian YP, Yu X. Stabilizing unstable periodic orbits of chaotic systems via an optimal principle. J Franklin Inst 2000;337:771–9. [19] Guo SM, Shieh LS, Chen G, Lin CF. Effective chaotic orbit tracker: a prediction-based digital redesign approach. IEEE Trans Circuits Syst I 2000;47:1557–60. [20] Wu T, Chen MS. Chaos control of the modified Chua’s circuit system. Physica D 2002;164:53–8. [21] Zhang J, Li C, Zhang H, Yu J. Chaos synchronization using single variable feedback based on backstepping method. Chaos Soliton Fract 2004;21:1183–93. [22] Kopell N, Washburn RB. Chaotic motions in the two-degree-of –freedom swing equations. IEEE Trans Circ Syst CAS 1982;29:738–46. [23] Abed EH, Varaiya PP. Nonlinear oscillations in power systems. Int J Electr Power Energy Syst 1984;6:37–43. [24] Chen HK, Lin TN, Chen JH. Dynamic analysis, controlling chaos and chaotification of a SMIB power system. Chaos Soliton Fract 2005;24:1307–15. [25] Shahverdiev EM, Hashimova LH, Hashimova NT. Chaos synchronization in some power systems. Chaos Soliton Fract 2006. doi:10.1016/ j.chaos.2006.09.071. [26] Sun M, Tian L, Ying F. An energy resources demand-supply system and its dynamical analysis. Chaos Soliton Fract 2007;32:168–80. [27] Sun M, Jia Q, Tian L. A new four-dimensional energy resources system and its linear feedback control. Chaos Soliton Fract 2007. doi:10.1016/ j.chaos.2007.01.125. [28] Itkis U. Control system of variable structure. New York: Wiley; 1976. [29] Utkin VI. Sliding mode and their applications in variable structure systems. Moscow: Mir Editors; 1978. [30] Tang Y. Terminal sliding mode control for rigid robots. Automatica 1998;34:51–6.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties Cheng-Fang Huang, Kuo-Hua Cheng, Jun-Juh Yan * Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 August 2008 Received in revised form 29 September 2008 Accepted 30 September 2008 Available online 7 October 2008 PACS: 05.45.Gg 05.45.Xt
a b s t r a c t This paper investigates the robust chaos synchronization problem for the four-dimensional energy resource systems. Based on the sliding mode control (SMC) technique, this approach only uses a single controller to achieve chaos synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero or into predictable bounds for matched and unmatched perturbations, respectively. Numerical simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Energy resource system Sliding mode control Chaos synchronization Perturbation
1. Introduction Over the last two decades, since the pioneering work of Pecora and Carroll in 1990 [1], much attention has been paid to the chaos synchronization in non-linear systems. Generally, designing a controller to force a system to mimic the behavior of another chaotic system is called synchronization. There is much prospect of chaos synchronization being applied in the vast areas of physics and engineering systems such as in power converters, chemical reactions, biological systems, information processing, especially in secure communication where a lot of progress has been achieved [2–6]. Nowadays, different techniques and methods have been proposed to achieve chaos synchronization such as adaptive control [7–11], sliding mode control (SMC) [12–14], impulsive control [15,16], linear control [17], optimal control [18], digital redesign control [19], and backstepping control [20,21]. On the other hand, the energy resource system is a kind of complex non-linear system and widely used in the industry. Study of chaos and its synchronization control in power systems is with considerable importance from the point of view of avoiding undesired behaviors such as power blackout [22–25]. Recently, in paper [26]. Sun et al. established a three-dimensional energy resource demand-supply system based on the real energy resources demand-supply in the East and the West of China. Furthermore, by adding a new variable to consider the renewable resources, a four-dimensional energy resource system was proposed in [27]. The dynamics behaviors of the four-dimensional energy resource system have been analyzed by means of the Lyapunov exponents and bifurcation diagrams. Also the same as the above-mentioned power systems, this four-dimensional energy resource system is with rich chaos behaviors. Furthermore, in chaos synchronization, most publications often assume that the * Corresponding author. Tel.: +886 7 6158000/4806; fax: +886 7 6158000/4899. E-mail address: [email protected] (J.-J. Yan). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.09.017
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synchronization system is without external disturbances. If external disturbance is considered, then, in general, only matched disturbance is considered. However, in practical applications, especially in energy resource systems, it is hard to avoid external disturbances due to uncontrollable environmental conditions. In other words, chaos synchronization systems may be undergoing either matched or unmatched disturbances. Thus, the derivation of a robust synchronization controller for the fourdimensional energy resource systems to resist the presence of disturbance is an important problem. In this paper, the SMC technique is applied to achieve the chaos synchronization for the four-dimensional energy resource system. In our design, a single controller is used enough to realize synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero with the matched perturbations or into predictable bounds with unmatched perturbations. Simulation results show the effectiveness of the proposed control scheme. Throughout this paper, it is noted that, jwj represents the absolute value of w and sign(r) is the sign function of r, if r > 0, sign(r) = 1; if r = 0, sign(r) = 0; if r < 0, sign(r) = 1. 2. System description In this paper, a four-dimensional energy resource demand-supply system is considered. The dynamics of this system is described by the following differential equations [27]:
x_ ¼ a1 xð1 x=MÞ a2 ðy þ zÞ d3 w; y_ ¼ b1 y b2 z þ b3 x½N ðx zÞ; z_ ¼ c1 c2 zx c1 c3 z; _ ¼ d1 x d2 w; w
ð1Þ
where x(t) is the energy resource shortage in A region, y(t) is the energy resource supply increment in B region. z(t) and w(t) are energy resource import in A region and renewable energy resource in A region, respectively. ai, bi, ci, di and M, N are positive real constant. The dynamics of this system has been extensively studied in [27]. The bifurcation diagram of system for the parameter value, M = 1.8, N = 1, a2 = 0.15, b1 = 0.06, b2 = 0.082, b3 = 0.07, c1 = 0.2, c2 = 0.5, c3 = 0.4, d1 = 0.1, d2 = 0.06, d3 = 0.07, a1 2 [0.05 0.15], is shown in Fig. 1. From Fig. 1, it shows that this energy resource system displays chaotic behavior when a1 = 0.1 and the chaotic attractors and state responses are shown in Fig. 2 with initial condition ½xð0Þ yð0Þ zð0Þ wð0Þ ¼ ½0:82 0:29 0:48 0:1. 3. Program formulation The aim of this section is to design a controller using the sliding mode control, such that the slave system is able to mimic the behavior of the master chaotic system. The master (or drive) system and slave (or response) system with consideration of external perturbations are defined below, respectively
x_ m ¼ a1 xm ð1 xm =MÞ a2 ðym þ zm Þ d3 wm ; y_ m ¼ b1 ym b2 zm þ b3 xm ½N ðxm zm Þ; z_ m ¼ c1 c2 zm xm c1 c3 zm ; _ m ¼ d1 xm d2 wm w
Fig. 1. Bifurcation diagram of x (a1 2 [0.06 0.12]).
ð2Þ
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Fig. 2. Attractor of energy resource system: (A) three-dimensional view (y–x–z) (B) three-dimensional view (y–x–w)(C) three-dimensional view (w–z–x).
and
x_ s ¼ a1 xm ð1 xs =MÞ a2 ðys þ zs Þ d3 ws þ p1 ðtÞ þ uðtÞ; y_ s ¼ b1 ys b2 zs þ b3 xs ½N ðxm zm Þ þ p2 ðtÞ; z_ s ¼ c1 c2 zm xs c1 c3 zs þ p3 ðtÞ;
ð3Þ
_ s ¼ d1 xs d2 ws þ p4 ðtÞ; w where X m ¼ ½xm ym zm wm T and X s ¼ ½xs ys zs ws T are state vectors of the master and slave systems, respectively, u(t) 2 R is the control input, pi(t), i = 1, 2, 3, 4, are the unavoidable external perturbations in practical systems and assumed bounded, i.e.
j pi ðtÞ j6 ai ;
i ¼ 1; 2; 3; 4;
ð4Þ
where ai > 0 are given. Generally, p1 is called the matched perturbation and pi, i = 2, 3, 4 are the unmatched perturbations.
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The control problem considered in this paper is that for different initial conditions of systems (2) and (3), the two coupled systems can achieve synchronization by designing an appropriate control input u(t) attached to the slave system (3). Now define the synchronization error vector between systems (2) and (3) as
e1 ðtÞ ¼ xm ðtÞ xs ðtÞ;
e2 ðtÞ ¼ ym ðtÞ ys ðtÞ; e3 ðtÞ ¼ zm ðtÞ zs ðtÞ; e4 ðtÞ ¼ zm ðtÞ zs ðtÞ
ð5Þ
so the error dynamics is determined by the following equation:
e_ 1 ¼ ða1 xm e1 Þ=M a2 ðe2 þ e3 Þ d3 e4 p1 u; e_ 2 ¼ b1 e2 b2 e3 b3 e1 ½N ðxm zm Þ p2 ; e_ 3 ¼ c1 c2 zm e1 c1 c3 e3 p3 ;
ð6Þ
e_ 4 ¼ d1 e1 d2 e4 p4 : Obviously, the problem in realizing synchronization now transfers to another problem on how to propose a control law u(t) to force error states ei(t), i = 1, 2, 3, 4 to zero with matched perturbations or into a predictable bound with unmatched perturbations. Consequently, to achieve the control goal via SMC, there are two basic steps for the design procedure: the first step is to construct an appropriate switching surface such that the sliding motion on the manifold can result in limt?1jei(t)j 6 qi, i = 1, 2, 3, 4 where qi are some predictable bounds, even with any initial states and unavoidable external perturbations.; the second step is to ensure the existence of the sliding mode r(t) = 0 by designing a SMC law. To complete the two design steps above, a switching surface r(t) in the state space is first defined as follows:
r ¼ e1 þ
Z
t
ke1 ðsÞds;
ð7Þ
0
where r 2 R and k > 0 is an designed parameter to guarantee the stability of system in the sliding mode. It is well known that when the system operates in the sliding mode, it satisfies the following fact [28,29]
r ¼ e1 þ
Z
t
ke1 ðsÞds ¼ 0:
ð8Þ
0
Furthermore, since the sliding manifold r(t) = 0 occurs in a finite time tr(it will be proved in Lemma 1 later), i.e., r (t) = 0 for t > tr, we have
r_ ðtÞ ¼ e_ 1 þ ke1 ¼ 0
ð9Þ
) e_ 1 ¼ ke1 :
ð10Þ
Therefore, from (10) the following equivalent sliding mode dynamics can be obtained as
e_ 1 ¼ ke1 ; e_ 2 ¼ b1 e2 b2 e3 b3 e1 ½N ðxm zm Þ p2 ; e_ 3 ¼ c1 c2 zm e1 c1 c3 e3 p3 ;
ð11Þ
e_ 4 ¼ d1 e1 d2 e4 p4 : Obviously, since the design parameter k > 0 is specified, the exponential stability of e1(t) is surely guaranteed. Therefore, there exists time t1 large enough such that e1(t1) approach to zero and the dynamics of system (11), when time t P t1, can be simplified as
e1 ¼ 0; e_ 2 ¼ b1 e2 b2 e3 p2 ; e_ 3 ¼ c1 c3 e3 p3 ; e_ 4 ¼ d2 e4 p4 :
ð12aÞ ð12bÞ ð12cÞ ð12dÞ
Solving the differential Eqs. (12c) and (12d) for e3 and e4 when t P t1, respectively, results in
Z
e3 ðtÞ ¼ ec1 c3 ðtt1 Þ e3 ðt 1 Þ
tt 1
ec1 c3 ðtt1 sÞ p3 ðsÞds;
t P t1
ð13Þ
0
and
e4 ðtÞ ¼ ed2 ðtt1 Þ e4 ðt 1 Þ
Z 0
tt 1
ed2 ðtt1 sÞ p4 ðsÞds;
t P t1 :
ð14Þ
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Since c1c3 and d2 are positive, the bounds for e3 and e4, respectively, are obtained as
j e3 ðtÞ j¼j ec1 c3 ðtt1 Þ e3 ðt 1 Þ
Z
tt 1
ec1 c3 ðtt1 sÞ p3 ðsÞds j
0
6 ec1 c3 ðtt1 Þ j e3 ðt1 Þ j þ max p3 ðtÞ j j ec1 c3 ðtt1 Þ 1 ec1 c3 ðtt1 Þ 6 ec1 c3 ðtt1 Þ j e3 ðt1 Þ j þa3 c 1 c3
Z
tt 1
ec1 c3 s ds j
ð15Þ
0
and
ed2 ðtt1 sÞ p4 ðsÞds 0 Z tt1 j e4 ðt 1 Þ j þ max j p4 ðtÞ j ed2 ðtt1 Þ ed2 s ds 0 1 ed2 ðtt1 Þ : j e4 ðt 1 Þ j þa4 d2
Z j e4 ðtÞ j¼ ed2 ðtt1 Þ e4 ðt 1 Þ 6 ed2 ðtt1 Þ 6 ed2 ðtt1 Þ
tt1
ð16Þ
Eqs. (15) and (16) with c1c3 > 0 and d2 > 0 show that
lim j e3 ðtÞ j6 q3 ¼
t!1
a3
ð17Þ
c 1 c3
and
lim j e4 ðtÞ j6 q4 ¼
t!1
a4 d2
ð18Þ
:
^ 3 for t P t2. Let Since e3(t) converges to q3 as shown in (17), there exists a finite time t2 such that j e3 ðtÞ j6 q
gðtÞ ¼ b2 e3 p2
ð19Þ
and solving the differential Eq. (12b) results in
e2 ðtÞ ¼ eb1 ðtt2 Þ e2 ðt 2 Þ
Z
tt 2
eb1 ðtt2 sÞ gðsÞds;
t P t2 :
ð20Þ
0
For t P t2, we have
^ 3 þ a2 j g 3 ðtÞ j¼j b2 e3 p2 j6 b2 q
ð21Þ
Therefore, for t P t2, the bound of je2(t)j satisfies
eb1 ðtt2 sÞ gðsÞds 0 1 eb1 ðtt2 Þ ^ 3 þ a2 Þ j e2 ðt 2 Þ j þðb2 q b1
Z j e2 ðtÞ j¼ eb1 ðtt2 Þ e2 ðt 2 Þ 6 eb1 ðtt2 Þ
tt2
ð22Þ
^ 3 converges to q3 and we obtain By taking t2 sufficiently large, q
lim e2 ðtÞ j6 q2 ¼
t!1
b2 q3 þ a2 1 b2 a3 ¼ þ a2 b1 c1 c3 b1
ð23Þ
Remark 1. According the discussion above, we can conclude that the tracking errors in the sliding manifold can be forced to some predictable bounds as follows:
lim j e1 ðtÞ j¼ 0; lim j e2 ðtÞ j6 q2 ¼
t!1
t!1
1 b2 a3 a3 a4 þ a2 ; lim j e3 ðtÞ j6 q3 ¼ ; lim j e4 ðtÞ j6 q4 ¼ : t!1 b1 c1 c3 c1 c3 t!1 d2
ð24Þ
Particularly, for the nominal systems (i.e., p1(t) = p2(t) = p3 (t) = p4(t) = 0) or systems with matched perturbations (i.e., p1 (t) – 0 and p2(t) = p3(t) = p4(t) = 0), the resulting tracking errors limt?1jei(t)j = 0, i = 1, 2, 3, 4 can be ensured. After proposing the appropriate switching surface, it follows the design of a robust SMC law u(t) in the slave system (3) such that the states of the error system (6) can be driven to the switching surface r(t) = 0 in a finite time tr. Before stating the synchronization scheme to guarantee the occurrence of sliding manifold in a finite time, the reaching condition of the sliding manifold is given below.
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Lemma 1. [30] If the reaching condition of rðtÞr_ ðtÞ 6 l j rðtÞ j for l > 0, r(t) – 0, is satisfied, then in spite of any initial condition r(0), the sliding manifold r(t) = 0 will occur in a finite time bounded by t r 6 jrlð0Þj. Proof. The proof is an immediate consequence in the work of Tang [30] and hence is omitted.
Fig. 3. The time response of switching surface r(t) for the controlled chaotic energy system.
Fig. 4. State responses of controlled master and slave energy resource systems: (A) state response of xm,xs. (B) state response of ym,ys. (C) state response of zm,zs. (D) state response of wm,ws.
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After stating Lemma 1, to achieve the reaching condition indicated in Lemma 1, the synchronization scheme is proposed as
uðtÞ ¼ ½ða1 xm =M kÞe1 þ a2 ðe2 þ e3 Þ þ d3 e4 þ g signðrðtÞÞ;
g > a1 :
ð25Þ
Theorem 1. For the error dynamics (6) with the control law (25), suppose the external perturbations are bounded by Eq. (4), the reaching condition in Lemma 1 is surely satisfied and the system trajectory (6) enters the sliding manifold in a finite time. Proof. Substituting (6) and (25) into the derivative of
rðtÞr_ ðtÞ yields
rr_ ¼ r½e_ 1 þ ke1 ; ¼ r½a1 xm e1 =M a2 ðe2 þ e3 Þ d3 e4 p1 þ ke1 u; ¼ r½p1 g signðrðtÞÞ; 6j p1 j j r j g r signðrðtÞÞ; 6 ðg a1 Þ j r j :
ð26Þ
Since g > a1 is specified in (25), we get
rr_ 6 ðg a1 Þ j rðtÞ j6 l j r j; where l ¼ g a1 > 0
ð27Þ
Furthermore, according to Lemma 1, the system trajectory enters the sliding manifold r(t) = 0 and r_ ðtÞ ¼ 0 for t P t r ¼ jrlð0Þj. The proof is achieved completely.
4. Numerical examples This section presents some numerical examples to verify and demonstrate the effectiveness of the proposed synchronization scheme. In all numerical simulations, the fourth-order Runge–Kutta method with fixed step size of 0.0001 in MATLAB software is used.
Fig. 5. Error states of controlled energy resource systems: (A) Error state of e1. (B) Error state of e2. (C) Error state of e3. (D) Error state of e4.
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Here, the system accords with (2) and (3). The parameters M = 1.8, N = 1, a1 = 0.1, a2 = 0.15, b1 = 0.06, b2 = 0.082, b3 = 0.07, c1 = 0.2,c2 = 0.5, c3 = 0.4, d1 = 0.1, d2 = 0.06, d3 = 0.07 are specified for simulation. Case I: for the nominal system, i.e., p1(t) = p2(t) = p3(t) = p4 (t) = 0. As mentioned in Section 3, the proposed design procedure can be summarized as follows: Step1: According to (7), the switching surface with k = 3 is given by
r ¼ e1 þ
Z
t
3e1 ðsÞds:
ð28Þ
0
Step2: According to (25), the sliding mode control law is obtained as follows:
uðtÞ ¼ ½ð0:09xm =1:8 3Þe1 þ 0:15ðe2 þ e3 Þ þ 0:07e4 þ g signðrðtÞÞ;
g ¼ 0:1 > 0
ð29Þ
In numerical simulations, the initial value of the master and slave systems are given, respectively, as ½ xm ð0Þ ym ð0Þ zm ð0Þ wm ð0Þ T ¼ ½ 0:1 0:8 0:2 0:1 T and ½ xs ð0Þ ys ð0Þ zs ð0Þ ws ð0Þ T ¼ ½ 0:4 0:1 0:6 0:3 T . The simulation results are shown in Figs. 3–5. Fig. 3 shows the corresponding r(t) for the controlled energy resource system under the proposed sliding mod control (29). The state and error state responses and are shown in Figs. 4 and 5, respectively. From the simulation results, it shows that the proposed controller (29) can drive the resulting tracking errors limt?1jei(t)j = 0,i = 1, 2, 3, 4, which fully coincide with theoretical results in this paper. Case II: for the non-nominal system with the assumption of p1 (t) = 0.5sin(3t), p2(t) = 0.01sin(2t), p3(t) = 0.02sin(4t), p4 (t) = 0.01cos(t). Under the same simulation conditions as in Case I, and the switching surface with k = 2 is given by
r ¼ e1 þ
Z
t
2e1 ðsÞds
ð30Þ
0
and the sliding mode control law is given as follows:
uðtÞ ¼ ½ð0:09xm =1:8 2Þe1 þ 0:15ðe2 þ e3 Þ þ 0:07e4 þ g signðrðtÞÞ;
g ¼ 1 > a1 ¼ 0:5:
ð31Þ
The time responses of the error states, under the proposed sliding mod control (31), are shown in Fig. 6 and harmonic forms within the boundary predicted by (24) can be seen due to the effect of unmatched perturbations, which also coincide with theoretical results.
Fig. 6. Error states of controlled energy resource systems: (A) Error state of e1. (B) Error state of e2. (C) Error state of e3. (D) Error state of e4.
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