Abstract: This paper investigates the chaos control of uncertain unified chaotic systems. Adaptive control approach with three control signals is presented to stabilize states of the uncertain unified chaotic system at the zero equilibrium point. Since an adaptive controller based on dynamic compensation is employed, the faithful model of unified chaotic system is not necessarily required. By choosing proper controller parameters, chaotic phenomenon can be suppressed. In addition, the response speed of the closed-loop system is tunable. Sufficient conditions for the asymptotic stability of the adaptive approach are derived. Numerical simulation results confirm the adaptive control approach with three control inputs is valid in chaos control of uncertain unified chaotic systems.
INTRODUCTION
Chaos, generated from deterministic dynamics, exhibits unpredictable dynamic behaviors on the basis of its initial conditions and plays a significant part in nonlinear science. Since, Lorenz (1963) found the first chaotic attractor, chaos has been becoming a focus of nonlinear science in last few decades. Lots of chaotic systems, such as proposed by Chua et al. (1986), Chen and Ueta (1999) and Ueta and Chen (2000) and Lu system by Lu and Chen (2002) and Liu et al. (2004a) have been found in recent years. Lu et al. (2002) introduced a unified chaotic system, which contains the Lorenz system and Chen system as two extremes and the Lu system as a special case. Nowadays, chaos has a number of useful applications in secure communication, information processing, biological engineering, chemical processing, lasers and other areas (Chen and Dong, 1998; Ogorzalek, 1997). However, chaotic behaviors may result in destructive effects as well (Ogorzalek, 1997), therefore, the undesired chaotic phenomenon need to be regulated.
In 1990, based on linearization of the Poincare map, Ott et al. (1990) gave out the ‘Ott-Grebogi-Yorke (OGY) method’ to control chaos. It has turned out to be that chaos is controllable. From then on, various approaches have been applied to control chaos, such as ‘Pyragas method’ (Pyragas, 1992) (based on a time-delayed feedback), linear feedback (Lu and Lu, 2003; Rafikov and Balthazar, 2008), nonlinear feedback (Chen et al., 2004a), backstepping design technology (Ge et al., 2000; Wang and Ge, 2001). For unified chaotic system, there are also fruitful results reported by Chen et al. (2004b), Chen and Lu (2003). Impulsive control method (Ge et al., 2000) is utilized for the control and synchronization of unified chaotic systems. Adaptive control (Liu et al., 2004b; Hua et al., 2004), fuzzy control (Gao and Liu, 2007; Chen et al., 2007), passive control (Chen and Liu, 2010) and sliding mode control (Chiang et al., 2007; Ablay, 2009) etc., have been successfully applied in the control of unified chaotic systems.
Chaotic systems, like other dynamic systems, include unknown nonlinearities and time varying parameters, therefore, a control algorithm, which is robust to those uncertainties, is of importance. However, most of the control laws are obtained under the condition that parameters of the system are fully or partly known. In fact, it is difficult to get a faithful model for a chaotic system in engineering applications. The existing nonlinearities and uncertainties may result in a failed control. In addition, in most cases, the response speed of the closed-loop system is not directly related to the controller parameters. Chaos can be suppressed but the transient time is not desirable, i.e., the response speed is not tunable. Therefore, a model-free chaos control approach, whose response speed is tunable, is of great theoretical and practical value.
In this study, an adaptive control law based on the dynamic compensation is adopted for the control of unified chaotic systems. Approach utilized in this study is capable of rejecting the uncertainties of dynamic systems. Numeric and theoretical results are presented to confirm the adaptive control approach.
PROBLEM STATEMENT
The uncontrolled unified chaotic system [7] is described below:
| (1) |
where, x1, x2, x3 are state variables and α∈[0, 1] is the system parameter. When α∈[0, 0.8] system 1 is called the generalized Lorenz chaotic system. When α = 0.8, system 1 is Lu chaotic system and when αε(0.8, 1], system 1 is called the generalized Chen chaotic system. As α varies continuously from 0 to 1,system 1 continuously to be chaotic. Taking parameters α = 0, 0.8, 1, respectively, chaotic attractors of unified chaotic system for different parameters α are shown in Fig. 1.
The control objective is to stabilize the unified chaotic systems at the equilibrium point x = 0.
ADAPTIVE CONTROL APPROACH DESIGN
The controlled unified chaotic system can be written as follows:
| (2) |
where, u1, u2, u3 are control inputs. In this study, an adaptive controller based on the dynamic compensation (Tornambe and Valigi, 1994) is adopted. When relative degree is one, the adaptive control law can be written as:
| (3) |
where,
| Fig. 1(a-c): | (a) Lorenz chaotic attractor, (b) Lu chaotic attractor and (c) Chen chaotic attractor |
Ui0, i = 1-3, in system 2, are given in Eq. 4:
| (4) |
where, yir = 0, yi = xi, after identical transformation, Eq. 4 can be rewritten as:
| (5) |
where ci1 =-(k0i+h0i),ci2=k0ih0i, i=1-3. Substituting Eq. 5 into system 2, we derive closed-loop system 6.
| (6) |
Closed-loop system 6 can be written as compact form:
| (7) |
where, ς = (x1, x2, x3, ξ1, ξ2, ξ3)T, O(ς, t) = (0, -x1x3, x1x2, 0, 0, 0)T and:
where, I3x3 and 03x3 are 3th order identity matrix and zero matrix, respectively.
Lemma 1: Liu and Fei (2005) consider the nonlinear system:
| (8) |
where, x = (x1, x2,..., xn)T∈
| • | |
| • | ∀t≥0, A(t) is bounded |
| • | The zero solution of linear system |
According to Lemma 1, we have;
Theorem 1: For closed-loop system 7, controller parameters are chosen
such that linear system
Proof : For closed-loop system 7, O(ς,t)|ς = 0 = 0 is satisfied for any time. Since:
i.e.,
we have:
For:
according to Squeeze theorem we have:
| (9) |
which means condition 1) is satisfied. Matrix A in system 7 is a constant matrix, so condition 2) is also satisfied. The linear part of system 7 is:
| (10) |
| Fig. 2(a-f): | Controlled states of the unified chaotic systems (α = 0) (a, X1, X2, (c) X3, (d) U1, (e) U2 and (f) U3 |
| Table 1: | Characteristic parameters for different α values |
The characteristic polynomial of system matrix A in Eq. 10 is:
where:
| ε5 | = | -c31-3.7α-c21-c11+13.7 |
| ε4 | = | c32+c22+c12+4c31α+c31c21-11c31+c31c11+148.7α2-25.3c21 α-622α+28.7αc11+c21c11-12.7c21-240.7 |
| ε3 | = | -720-1730α-c12c31-2.7c11-26.7c21+270c31+12.7c223.7c12 +11c32+195α2+77αc11-70αc21+2.7c21c11+615αc31 +10c31c21+c31c11-28.7αc12-150α2c31+9.7α2c11-8.3α2c21 +50α3-29αc31c11+25αc31c21-c31c21c11+0.3αc21c11 +25.3αc22-c22c31-c22c11-4αc32-c32c21-c32c11 |
| ε2 | = | -0.3αc22c11-25αc22c31-25αc32c21+29αc12c31+c22c12-c12c31 +26.7c22+2.7c12-270c32-77αc12-2.712c21-0.3αc21c21 +29αc32c11+c12c31c21-9.7α2c12+70αc22+c22c31c11-10c22c31 -2.7c22c11+8.3α2c22-615αc32-10c32c21-c32c11+c32c21c11 +150α2c32+c32c22+c32c12 |
| ε1 | = | -c22c12c31+2.7c22c12+0.3αc22c12-29αc32c12+10c32c22 +c32c12+25αc32c22-c32c22-c32c12c21-c32c22c11 |
| ε0 | = | c32c22c11 |
If εi>0 (I = 0-5), according to Hurwitz criterion, f(λ) is of Hurwitz type if and only if following conditions hold (where εs = 0, when s<0 or s>6), i.e.,:
|
|
(11) |
Hence, by choosing proper controller parameters condition 3) is satisfied. For closed-loop system 7, conditions 1, 2 and 3) hold, according to Lemma 1, zero solution of system 7 is uniformly asymptotically stable. By this we conclude the proof of Theorem 1.
NUMERICAL SIMULATIONS
The numerical simulation results are given to confirm the adaptive control approach. In the simulations, 4th order Runge-Kutta method is used to solve the systems in time steps of 0.001. The initial conditions are chosen to be (x1(0), x2(0), x3(0))T = (1, 3, 2)T. Simulation time is 10s, control inputs are added at the beginning of the simulations. Controller parameters are chosen to be k01 = 100, h01 = 0.01; k02 = 56, h02 = 0.01; k03 = 16 and h03 = 0.01. For this selection, we have Table 1.
From Table 1, we can see clearly that εi>0 (I = 0-5) and μj>0 (j = 1-6), i.e., conditions given in (11) are satisfied. According to Theorem 1, closed-loop system 7 is uniformly asymptotically stable. The dynamic responses of the controlled unified chaotic systems and the corresponding control inputs are given in Fig. 2-4. As we can see from the figures, adaptive controllers (4) can regulate the unified chaotic systems to zero equilibrium point effectively.
| Fig. 3(a-f): | Controlled states of the unified chaotic systems (α = 0.8) (a) X1, (b) X2, (c) X3, (d) U1, (e) U2 and (f) U3 |
| Fig. 4(a-f): | Controlled states of the unified chaotic systems (α = 1) (a) X1, (b) X2, (c) X3, (d) U1, (e) U2 and (f) U3 |
| Fig. 5: | Varying α in interval [0, 1] |
| Fig. 6(a-f): | Controlled states of the uncertain unified chaotic systems by adaptive control approach (α = |1sin(20t)|) (a) X1, (b) X2, (c) X3, (d) U1, (e) U2 and (f) U3 |
In order to test the performance robustness of the closed-loop system 7, we suppose parameter α = |1sin(20t)| and thus αε[0, 1], system 1 is still chaotic. The varying parameter α is shown in Fig. 5.
From Fig. 6, we can see that the states still converge to zero even if α is varying in interval [0, 1].
CONCLUSION
In this study, adaptive control approach with three control inputs is presented to realize the control of unified chaotic systems. Sufficient conditions for stability of closed-loop systems and numerical results confirm the approach. The advantages of the approach can be summarized as follows:
| • | The exact model of the unified chaotic systems is not required. Adaptive controller adopted is capable of estimating and compensating the uncertainties effectively. By choosing proper values of h0 and k0, we can suppress the chaotic behavior as desired |
| • | The response speed of the closed-loop is tunable. Parameter h0 determines the response speed. To change the value of h0, the response speed is tunable |
| • | Theoretical and numeric results confirm that adaptive approach utilized in this paper can stabilize the uncertain unified chaotic systems successfully |
ACKNOWLEDGMENTS
This study was supported by National Natural Science Foundation of China (61170113), National Basic Research Program of China (973 Program) (2012CB821206) and the Research Foundation for Youth Scholars of Beijing Technology and Business University (QNJJ2011-40).