Sliding mode control of uncertain unified chaotic systems

Nonlinear Analysis: Hybrid Systems 3 (2009) 531–535

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Sliding mode control of uncertain unified chaotic systems Günyaz Ablay Nuclear Engineering Program, The Ohio State University, Columbus, OH 43210, USA

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Article history: Received 31 March 2009 Accepted 6 April 2009 Keywords: Chaos control Uncertain chaotic systems Sliding mode Unified chaotic systems

abstract This paper investigates the chaos control of the uncertain unified chaotic systems by means of sliding mode control. A proportional plus integral sliding surface is introduced to obtain a sliding mode control law. To confirm the validity of the proposed method, numerical simulations are presented graphically. Published by Elsevier Ltd

1. Introduction Chaos is a periodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions [1]. The fundamental characteristics of chaotic behavior come from the internal structure of the systems, and chaotic behaviors are more complicated than limit cycle behaviors. Today, chaos has been seen to have many useful applications in many engineering systems such as in chemical reactors, genetic control systems, power converters, lasers, biological systems, and secure communication systems [1,2]. Chaos can be useful in propagation of mixing in processes, such as in convective heat transfer. However, chaotic behavior may lead to undesirable effects as well, such as uncontrolled oscillations in a power grid, and may need to be regulated [2]. After chaos control was introduced in [3], it has turned out to be an important area of nonlinear science, and various control approaches have been proposed. The Ott–Grebogi–Yorke (OGY) method [3], variable structure control [4], nonlinear feedback control [5], and some other methods [6,7] have been successfully applied to chaotic systems. The sliding mode control (SMC) scheme is one of these methods [8,9], and recently there has been a great deal of attention given to using SMC for controlling chaos. SMC is an effective methodology for controlling systems with variable structures and provides a systematic approach to the problem of maintaining stability and consistent performance in the face of modeling imprecision [10–12]. SMC is particularly preferred due to its capability to tolerate disturbances and dynamic model uncertainties. Chaotic systems include nonlinearities and often some parameters which cannot be exactly defined [13,14]. Therefore, a robust control method such as SMC would have the advantage of the capability to handle such uncertainties. In this study, a proportional plus integral (PI) sliding surface is introduced, and by satisfying the reachability condition, a suitable sliding mode control law is developed for chaos control of uncertain unified chaotic systems which was recently introduced by Lü et al. [15]. The unified chaotic system that has a single adjustable parameter exhibits all chaotic behaviors of Lorenz [16], Chen [17] and Lü [18] chaotic systems which all have three parameters. Lorenz type systems, or the unified chaotic systems, can be seen in atmospheric sciences, laser devices, and some other systems related to convection. This work researches chaos control of the uncertain chaotic systems by means of sliding mode control. In the Section 2, chaos control of an uncertain unified chaotic system is presented. In Section 3, numerical simulations are provided to confirm the validity of the method, and finally, conclusions are given.

E-mail address: [email protected]. 1751-570X/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.nahs.2009.04.002

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G. Ablay / Nonlinear Analysis: Hybrid Systems 3 (2009) 531–535

2. Sliding mode control of uncertain unified chaotic systems The unified chaotic system is introduced by Lü et al. [15] with a single parameter: x˙ 1 = (25α + 10)(x2 − x1 ) x˙ 2 = (28 − 35α)x1 + (29α − 1)x2 − x1 x3 x˙ 3 = x1 x2 − (8 + α)x3 /3

(

(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 called the Lü chaotic system, and when α ∈ (0.8, 1], system (1) is called the generalized Chen chaotic system. The controlled unified chaotic system can be rewritten as follows: x˙ 1 = (25α + 10)(x2 − x1 ) + u1 x˙ 2 = (28 − 35α)x1 + (29α − 1)x2 − x1 x3 + u2 x˙ 3 = x1 x2 − (8 + α)x3 /3 + u3

(

(2)

where u1 , u2 , u3 are control inputs. We have the following matrices:

" −10 A=

10 −1 0

28 0

0 0 ; −8/3

#

" B=

1 0 0

0 1 0

0 0 ; 1

#

25α(x2 − x1 ) −35α x1 + 29α x2 − x1 x3 x1 x2 − α x3 /3

" g =

# (3)

where A is the system matrix, B is the control matrix, and g represents the system nonlinearities plus parametric uncertainties in the system. The control problem is to get the state x = [x1 x2 x3 ]T to track a specific time varying state xd = [xd1 xd2 xd3 ]T in the presence of nonlinearities and parameter uncertainties on system parameter α . Therefore, the tracking error is defined as e = x − xd

(4) T

where e = [e1 e2 e3 ] is the tracking error vector. The error dynamics may be written as below: e˙ = x˙ − x˙ d = Ax + Bg + Bu − x˙ d .

(5)

Now, a time varying proportional plus integral (PI) sliding surface σ (e, t ) ∈ R3 is defined by the scalar equation σ = σ (e, t ) as

σ = Ke −

t

Z

K (A − BL)e(τ ) dτ

(6)

0

where K ∈ R3x3 , which must satisfy det(KB) 6= 0, is a gain matrix, and L ∈ R3x3 , which must have a stable A − BL, is a gain matrix, namely, the eigenvalues λi (i=1,2,3) of the matrix A − BL are negative (λi < 0). In the sliding mode, the sliding surface and its derivative must satisfy σ = σ˙ = 0 [11,12];

σ˙ = KBg + KBLe + KBu + KAxd − K x˙ d = 0.

(7)

Since KB is nonsingular, the equivalent control ueq which is the solution of σ˙ = 0 is thus ueq = − gˆ + Le − (KB)−1 [KAxd − K x˙ d ]





(8)

where g is not exactly

known,

but estimated as gˆ , and the estimation error on g is assumed to be bounded by some known function G such that g − gˆ ≤ G. To satisfy the sliding condition in the presence of imperfections, a discontinuous term is added to the equivalent control, which defines the following control law: u = ueq − (KB)−1 [ε + kKBGk] sign(σ )

(9)

where ε > 0, and the sign function is defined as sign(σ ) = 1 if σ > 0, sign(σ ) = 0 if σ = 0, sign(σ ) = −1 if σ < 0. Theorem 1. The control law (9) satisfies the following reachability condition:

σ T σ˙ < 0.

(10)

Proof. By substituting (7) and (9) into (10), we obtain

σ T σ˙ = σ T [KBg + KBLe + KAxd − K x˙ d + KBu] 
 = σ T KB g + Le + ueq − (KB)−1 [ε + kKBGk] sign(σ ) + KAxd − K x˙ d ≤ −ε kσ k .

(11)

The following equation is obtained for ε > 0, which indicates satisfying the reachability condition:

σ T σ˙ < 0. 

(12)

G. Ablay / Nonlinear Analysis: Hybrid Systems 3 (2009) 531–535

533

60

50

x3

40

30

20

10

0 -30

-20

-10

0 x1

10

20

30

Fig. 1. The phase portrait of the uncertain unified system.

50

a

x1 0

-50

b

0

5

10

15

20

25

30

35

40

45

50

50 x2 0

-50

c

0

5

10

15

20

25

30

35

40

45

50

60 x3 40 20 0

0

5

10

15

20

25

30

35

40

45

50

Fig. 2. The time responses of the state variables of the uncertain unified system.

3. Numerical simulations Here the numerical results are given to confirm the validity of the proposed method. In the numerical simulations, the gain matrix K is selected as K = diag(1, 1, 1) such that KB = diag(1, 1, 1) is nonsingular. The desired eigenvalues of the matrix A − BL are taken as P = −5 −5.001 −5.0001 , and by using the pole placement method, the gain matrix L is

−5 found as L =

28 0

10 4.001 0

0 0 2.3334



. As a result, the matrix K (A − BL) is computed as K (A − BL) = diag(−5, −5.001, −5.0001).

The PI switching surfaces are obtained as

Z t    σ1 = e1 + 5e1 (τ )dτ    Z0 t  σ2 = e2 + 5.001e2 (τ )dτ  0  Z  t   σ3 = e3 + 5.0001e3 (τ )dτ . 0

(13)

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G. Ablay / Nonlinear Analysis: Hybrid Systems 3 (2009) 531–535

a

1 x1

0.5 0 -0.5

b

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4 x2

2 0 -2

c

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2 x3

1 0 -1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 3. The time responses of the state variables of the uncertain unified system when the control signals are activated at the time t = 0.

a

1 σ1

0.5 0 -0.5

b

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4 σ2

2 0 -2

c

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2 σ3

1 0 -1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 4. The time responses of the sliding surfaces of the uncertain unified system when the control signals are activated at the time t = 0.

The initial conditions of the system are taken as [x1 (0), x2 (0), x3 (0)] = [1, 3, 2]. The constant controller coefficient ε is selected as ε = 5. In the simulation, we suppose that the system parameter varies as α = |1 sin(20t )|. The reference states xd1 , xd2 , xd3 are selected as xd1 = xd2 = xd3 = xd . Therefore, the control signals may be defined as

 [20(x2 − x1 ) + x˙ d + (−5e1 + 10e2 ) + (|20(x2 − x1 )| + ε) sign(σ1 )]  u = −   1 −x1 x3 − 15x1 + 15x2 + 27xd + x˙ d + (28e1 + 4.001e2 ) + 4.001e2 u2 = − + (|−x1 x3 − 30x1 + 25x2 | + ε) sign(σ2 )   u3 = − [x1 x2 − 0.8x3 /3 − 8/3xd + x˙ d + 2.3334e3 + (|x1 x2 − 0.9x3 /3| + ε) sign(σ3 )] .

(14)

In Fig. 1, the phase portrait of the uncertain unified system is presented. In Fig. 2, the time responses of the state variables of the uncertain Lorenz system are given. Fig. 2(a) displays x1 , Fig. 2(b) displays x2 , and Fig. 2(c) displays x3 . In Fig. 3, the reference states are taken as xd = 0, and the state vectors x1 , x2 , and x3 converge to zero quickly after control signals are activated at the time t = 0. Fig. 3(a) shows x1 , Fig. 3(b) shows x2 , and Fig. 3(c) shows x3 . Fig. 4 illustrates the time responses of the sliding surfaces σ1 , σ2 and σ3 . Fig. 4(a) displays σ1 , Fig. 4(b) displays σ2 , and Fig. 4(c) displays σ3 . In Fig. 5, the reference

G. Ablay / Nonlinear Analysis: Hybrid Systems 3 (2009) 531–535

a

535

2 x1 xd

0

-2

b

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4 x2 2

xd

0 -2

c

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2 x3 xd

0

-2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 5. The time responses of the state variables of the uncertain unified system for xd = 1 sin(2.4t ) when the control signals are activated at the time t = 0.

states are taken as xd = 1 sin(2.4t ), and the state vectors x1 , x2 , and x3 converge to xd quickly after control signals are activated at the time t = 0. Fig. 3(a) shows x1 and xd , Fig. 3(b) shows x2 and xd , and Fig. 3(c) shows x3 and xd . 4. Conclusions This study proposes a sliding mode control method for chaos control of the uncertain unified chaotic systems. A sliding mode control law is developed by using a PI switching surface, and the reachability condition is satisfied. I believe that this method will be generalized. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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