Control and synchronizing nonlinear systems with delay based on a special matrix structure

Commun Nonlinear Sci Numer Simulat 19 (2014) 1072–1078

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Control and synchronizing nonlinear systems with delay based on a special matrix structure Jianbing Hu a,⇑, Lingdong Zhao a,b, Zhenguang Xie a a b

School of Electronics & Information, Nantong University, Nantong 226019, PR China College of Information Science and Technology, Donghua University, Shanghai 201620, PR China

a r t i c l e

i n f o

Article history: Received 7 September 2012 Received in revised form 16 January 2013 Accepted 19 August 2013 Available online 4 September 2013 Keywords: Nonlinear system Control and synchronizing Delay Special matrix

a b s t r a c t This work presents a direct approach to design stabilizing controller for nonlinear systems with delay based on a special matrix structure and proves the validity of the approach according to Lyapunov–Krasovskii stable theorem and Linear Matrix Inequality—LMI. Control Lorenz system and synchronizing Rössler system with delay are taken as examples to explain the approach. Numerical simulations confirm the effectiveness of the approach proposed.  2013 Elsevier B.V. All rights reserved.

1. Introduction Chaotic systems are very complex nonlinear systems that exhibit special characteristics such as sensitivity to initial conditions and system parameter variations, broad Fourier transform spectra, fractal properties of the motion in phase space, and strange attractors [1,2]. Due to these features and many useful applications in real-life processes, such as chemical reactions, power converters, secure communications, information processing, biological systems, and mechanics systems, control and synchronization of chaotic systems have been widely explored and studied [3–5]. A wide variety of approaches have been proposed to design the controller with linear or nonlinear feedback control [6], adaptive control [7], active control [8], or backstepping design [9]. Time-delay characteristics are frequently encountered in the most engineering systems such as physical and biological systems. It is well known that the existence of time delay in a system may cause system instability and oscillations. It is very difficult to achieve satisfactory performance for these systems. Stability of time-delay systems has been studied for decades and many results on this subject have been reported. In recent years, control and synchronizing chaotic systems with delay has attracted some researchers interests [10,11]. To date, a lot of approaches of synchronizing chaotic system without delay are extended to synchronize chaotic system with delay. Compared with chaotic system without delay, synchronizing chaotic system with delay is more complicated and difficult. How to design a suitable controller is a key problem of these approaches to realize chaotic control and synchronization. However, the authors usually present a controller and prove the controlled system stable to zero under the controller based on Lyapunov–Krasovskii stable theorem and Linear Matrix Inequality— LMI. But the design procedures of the controllers aren’t usually presented and the controllers are usually very complex and hard to understand. How to design a simple and easy controller become very important for synchronizing chaotic system with delay [12]. On this problem, Hu J.B. etc. proposes a direct method based on a special matrix structure [13] for ⇑ Corresponding author. Tel.: +86 15189401356. E-mail addresses: [email protected] (J. Hu), [email protected] (L. Zhao). 1007-5704/$ - see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.08.029

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chaotic system without delay [14]. Base on the similar idea, we extend the approach to synchronizing chaotic system with delay. We can directly design a controller to satisfy a special matrix to control and synchronize chaotic system with delay. What is important is that the designed controllers become very simple and the designed procedures become very easy. This paper is organized as follows: In Section 2, the main result is given. In Section 3, the method is utilized to control Lorenz system with delay. In Section 4, a controller is designed according to the approach to realize synchronizing Rössler system with delay. In Section 5, the conclusion is presented. 2. The main result In general, a controlled chaotic system with delay considered in this paper can be expressed:

_ XðtÞ ¼ f ðXðtÞÞ þ gðXðtÞ; Xðt  sÞÞ  uðtÞ

ð1Þ T

where f(X(t)), g(X(t), X(t  s)) is nonlinear function, s is the delay time, X = (x1, x2,    xn) are state variables and u(t) is control input. We can also express system (1) as:

_ XðtÞ ¼ AðXðtÞÞ  XðtÞ þ BðXðtÞ; Xðt  sÞÞ  Xðt  sÞ  uðtÞ

ð2Þ

where A(X) and B(X(t), X(t  s)) are called coefficient matrixes including state variable. The key problem is how to design the controller u(t) to make the controlled system (1) with delay stable to zero according to function (2). We define:

2

a11 6a 6 21 AðXðtÞÞ ¼ 6 6 .. 4 .

a12 .. .

   a1n

3

 .. .

7 7 7 7 5 ann

an1 2

c11 6c 6 21 CðXðtÞÞ ¼¼ 6 6 .. 4 .

ð3Þ

c12 .. .

   c1n

3

 .. .

7 6 7 6 7¼6 7 6 5 4

2

3

k1 ..

cnn

cn1

7 7 7AðXðtÞÞ 7 5

k2 . kn

and

2

b11

b12

6b 6 21 BðXðtÞ; Xðt  sÞÞ ¼ 6 6 .. 4 .

.. .

   b1n

3

 .. .

7 6d 7 6 21 7 DðXðtÞÞ ¼ 6 . 7 6 . 5 4 . bnn

bn1

2

d11

dn1

d12 .. .

   d1n

3

 .. .

7 6 7 6 7¼6 7 6 5 4

2

3

k1

7 7 7BðXðtÞ; Xðt  sÞÞ 7 5

k2 ..

dnn

. kn

ð4Þ

Theorem 1. If the control input u(t) can make matrixes C(X(t)) and D(X(t)) satisfy the following conditions: (1) (2) (3) (4)

"cij, dij  R ci,j =  cj,i (i – j) P P cii þ nj¼1 ðjdi;j j þ jdj;i jÞ=2 6 0 ðAll 2cii þ nj¼1 ðjdi;j j þ jdj;i jÞ aren0 t equal to zeroÞ "ki > 0

the controlled system (1) is gradually stable to zero.

Proof. Define fi ¼

Pn

j¼1 jdi;j j.

2

6 6 V ¼ ðx1 ; x2    xn Þ  6 6 4

Take Lyapunov–Krasovskii function candidate as

3

k1 k2 ..

2

7 6 Z t 7 6 7ðx1 ; x2    xn ÞT þ 6 ðx ; x    x Þ  1 2 n 7 6 ts 5 4

. kn

then the differential coefficient with time of V is

3

f1

7 7 7ðx1 ; x2    xn ÞT dt 7 5

f2 ..

. fn

ð5Þ

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2 V_ ¼ 2 

j¼n i¼n X X cij xi xj

!

i¼1 j¼1

j¼n i¼n X X cij xi  xj ¼2

3

k1

6 6 þ 2XðtÞ6 6 4

7 n X 7 7BðXðtÞÞ  ðXðt  sÞÞT þ fi ðx2i ðtÞ  x2i ðt  sÞÞ 7 5 i¼1

k2 ..

. kn

!

þ 2XðtÞDðXðtÞÞðXðt  sÞÞT þ f1 ðx21 ðtÞ  x21 ðt  sÞÞ; f2 ðx22 ðtÞ  x22 ðt  sÞÞ    fn ðx2n ðtÞ  x2n ðt  sÞÞ

i¼1 j¼1

ð6Þ According to the condition (2), it can be gained: j¼n i¼n X X 2 cij xi xj

!

i¼n X ¼2 cii x2i

i¼1 j¼1

! ð7Þ

i¼1

According to LMI, we can obtain:

2ðx1 ; x2    xn ÞDðXðtÞÞðx1 ; x2    xn ÞT ¼ 2

n X n n X n X X dij xi ðtÞxj ðt  sÞ 6 jdij jðx2i ðtÞ þ x2j ðt  sÞÞ i¼1 j¼1

i¼1 j¼1

n X n n X n X X jdij jðx2i ðtÞÞ þ jdji jx2i ðt  sÞÞ ¼ i¼1 j¼1

ð8Þ

i¼1 j¼1

Then:

V_ 6 2 

i¼n X cii x2i

! þ

i¼1

¼

i¼n X

i¼1 j¼1

2cii þ fi þ

i¼1

Since fi ¼

V_ 6

j¼1 jdji j,

i¼1

! ! n X jdij j x2i þ jdji j  fi x2i ðt  sÞ

i¼1

ð9Þ

j¼1

! ! ! ! n n i¼n n n X X X X X 2 2 2cii þ fi þ jdij j xi þ jdji j  fi xi ðt  sÞ ¼ 2cii þ jdji j þ jdij j x2i j¼1

According to cii þ

V_ 6

i¼1 j¼1

!

then:

i¼1

i¼n X

n X j¼1

Pn

i¼n X

n X n n X n n X X X jdij jðx2i ðtÞÞ þ jdji jx2i ðt  sÞÞ þ fi ðx2i ðtÞ  x2i ðt  sÞÞ

2cii þ

Pn

j¼1 ðjdi;j j

n X j¼1

j¼1

i¼1

j¼1

ð10Þ

j¼1

þ jdj;i jÞ=2 6 0, it obviously that:

! n X jdji j þ jdij j x2i 6 0

ð11Þ

j¼1

According to Lyapunov–Krasovskii stable theorem, system (1) is gradually stable to zero. The proof is completed. h

3. Control Lorenz system with delay In this section, controlling Lorenz chaotic system with delay is presented to explain how to design the controller according to the method proposed. Consider the following time-delayed Lorenz system [4]:

8 > < x_ 1 ðtÞ ¼ aðx2 ðt  sÞ  x1 ðtÞÞ x_ 2 ðtÞ ¼ cx1 ðtÞ  x3 ðtÞx1 ðtÞ  x2 ðtÞ > :_ x3 ðtÞ ¼ x2 ðtÞx1 ðtÞ  bx3 ðt  sÞ

ð12Þ

When system parameters a = 10, b = 8/3, c = 28, s = 1/6, system (12) has chaotic attracter. The chaotic attractor of the system (12) is shown in Fig. 1. in Fig. 1.To control system (12), transform the system (12) and design the controller as:

2

3 2 32 3 2 32 3 x1 ðtÞ 0 a 0 x1 ðt  sÞ a 0 0 x_ 1 ðtÞ 6_ 7 6 76 7 6 76 7 4 x2 ðtÞ 5 ¼ 4 c 1 x1 54 x2 ðtÞ 5 þ 4 0 0 0 54 x2 ðt  sÞ 5 þ uðtÞ 0 0 b 0 x1 0 x3 ðtÞ x3 ðt  sÞ x_ 3 ðtÞ To simplify the processing, we take k1 = 1, k2 = 1, k3 = 1 and get:

ð13Þ

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30

45

20

40 35 30

x3

x2

10 0

25 20

−10

15 10

−20

5

−30 −20

−15

−10

−5

0

5

10

15

0 −20

20

−15

−10

−5

0

5

10

15

20

x1

x1 Fig. 1. Chaotic attractor of time-delayed Lorenz system.

2_ 3 2 32 3 2 a 0 0 x1 ðtÞ 0 a x1 ðtÞ 6_ 7 6 76 7 6 4 x2 ðtÞ 5 ¼ 4 c 1 x1 54 x2 ðtÞ 5 þ 4 0 0

32

3 x1 ðt  sÞ 76 7 0 54 x2 ðt  sÞ 5 þ uðtÞ 0

0 x3 ðtÞ x3 ðt  sÞ 0 0 b 32 32 3 2 32 1 0 0 a 0 0 x1 ðtÞ 1 0 0 0 a 6 76 76 7 6 76 ¼ 4 0 1 0 54 c 1 x1 54 x2 ðtÞ 5 þ 4 0 1 0 54 0 0

x_ 3 ðtÞ

0

x1

2

0

0

0 1

x1

0

x3 ðtÞ

0

0 1

32

3 x1 ðt  sÞ 76 7 0 54 x2 ðt  sÞ 5 þ uðtÞ 0

ð14Þ

x3 ðt  sÞ

0 0 b

We design the controller according to the condition (2) of theorem as:

2

3 11 x1 ðtÞ  cx2 ðtÞ 6 7 uðtÞ ¼ 4 12 x2 ðtÞ 5 13 x3 ðtÞ

ð15Þ

Then:

2_ 3 2 a þ 11 x1 ðtÞ 6_ 7 6 c 4 x2 ðtÞ 5 ¼ 4 x_ 3 ðtÞ

c 1 þ 12

0 2

1 0 0 a þ 11 6 76 ¼ 4 0 1 0 54 c 0 1

32

0

x1 ðtÞ

3

2

0

a

76 7 6 x1 54 x2 ðtÞ 5 þ 4 0 0

13

x1 32

0

0

0

32

x1 ðt  sÞ

3

76 7 0 54 x2 ðt  sÞ 5

x3 ðt  sÞ 0 0 b 32 3 2 32 0 x1 ðtÞ 1 0 0 0 a 76 7 6 76 x1 54 x2 ðtÞ 5 þ 4 0 1 0 54 0 0

x3 ðtÞ c

1 þ 12 x1

13

x3 ðtÞ

0

0 1

3 x1 ðt  sÞ 76 7 0 54 x2 ðt  sÞ 5 0

0 0 b

32

ð16Þ

x3 ðt  sÞ

According to the condition (3) of Theorem 1, we can get: if f1 < 0, f2 < 1  |a|, f3 < 2|b|, it obviously

cii þ

n X ðjdi;j j þ jdj;i jÞ=2 6 0: j¼1

According to Theorem 1, the controlled system (12) is stable to zero. The initial value is taken as x1 = 3, x2 = 4, x3 = 14, and choose f1 = 2, f2 = 11, f3 = 6 With the controller designed, the numerical simulation is shown as Fig. 2. It is clear that x1 ? 0, x2 ? 0, x3 ? 0 as t ? 1 from Fig. 2 4. Synchronizing Rössler system with delay A double delayed Rössler system can be depicted as:

2

3 2 32 3 2 0 1 1 x1 ðtÞ a1 x_ 1 ðtÞ 6_ 7 6 76 7 6 0 4 x2 ðtÞ 5 ¼ 4 1 b1 54 x2 ðtÞ 5 þ 4 0 0 0 0 x1 ðtÞ  c x3 ðtÞ x_ 3 ðtÞ

a2 0 0

0

32

x1 ðt  s1 Þ

3

2

0

3

76 7 6 7 0 54 x2 ðt  s2 Þ 5 þ 4 0 5 0 b2 x3 ðt  s2 Þ

ð17Þ

where delay parameters are s1 and s2, a1, a2, b1, b2 and c are the usual parameters of Rössler system. When a1 = 0.2, a2 = 0.5, b1 = 0.2, b2 = 0.2, c = 5.7, s1 = 0.2 and s2 = 0.1, system (17) has chaotic attracter. The chaotic attractor of the system (17) is shown in Fig. 3.

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Fig. 2. State variables x1, x2, x3 of system (12) with time t.

15

70

10

60

5

50

x3

x2

0 −5

30

−10

20

−15

10

−20 −25 −15

40

−10

−5

0

5

10

0 −25

15

−20

−15

−10

−5

x1

0

5

10

15

x2

Fig. 3. Chaotic attractor of time-delayed Rössler system.

Take system (17) as the drive system and the response system is defined as:

2

3 2 32 3 2 0 1 1 y1 ðtÞ a1 y_ 1 ðtÞ 6_ 7 6 76 7 6 b1 0 4 y2 ðtÞ 5 ¼ 4 1 54 y2 ðtÞ 5 þ 4 0 0 x3 ðtÞ 0 y1 ðtÞ  c y3 ðtÞ y_ 3 ðtÞ

a2 0 0

0

32

y1 ðt  sÞ

3

2

0

3

2

u1 ðtÞ

3

76 7 6 7 6 7 0 54 y2 ðt  sÞ 5 þ 4 0 5  4 u2 ðtÞ 5 0 b2 y3 ðt  sÞ u3 ðtÞ

ð18Þ

where u(t) is the controller. Define the synchronizing errors between the drive system (17) and the response system (18) as:

e1 ¼ y1  x1 e2 ¼ y2  x2 e3 ¼ y3  x3 and we can get:

2

3 2 32 3 2 0 1 1 e1 ðtÞ a1 e_ 1 ðtÞ 6_ 7 6 76 7 6 0 b1 4 e2 ðtÞ 5 ¼ 4 1 54 e2 ðtÞ 5 þ 4 0 x3 ðtÞ 0 y1 ðtÞ  c e3 ðtÞ 0 e_ 3 ðtÞ

a2 0

32 3 2 3 0 e1 ðt  sÞ u1 ðtÞ 76 7 6 7 0 54 e2 ðt  sÞ 5  4 u2 ðtÞ 5

0

0

e3 ðt  sÞ

u3 ðtÞ

ð19Þ

J. Hu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1072–1078

1077

Fig. 4. The synchronization errors e1, e2, e3 with time t.

Take k1 = 1, k2 = 1, k3 = 1 and design the controller according to condition (2) of Theorem 1 as:

2

11

6 uðtÞ ¼ 4 0

0

1 þ x3 ðtÞ

12

0

0

0

13

32

e1 ðtÞ

3

76 7 54 e2 ðtÞ 5 e3 ðtÞ

ð20Þ

and then the error system can be obtained:

2

3 2 1 11 e_ 1 ðtÞ 6_ 7 6 b1  12 4 e2 ðtÞ 5 ¼ 4 1 x3 ðtÞ 0 e_ 3 ðtÞ

x3 ðtÞ

32

e1 ðtÞ

3

2

a1

76 7 6 54 e2 ðtÞ 5 þ 4 0 0 e3 ðtÞ y1 ðtÞ  c  13 0

a2

0

32

e1 ðt  sÞ

3

0

76 7 0 54 e2 ðt  sÞ 5

0

0

ð21Þ

e3 ðt  sÞ

According to condition (3) of Theorem 1, we can see that if f1 + 2|a1| + |a2| < 0, b1  f2 + |a2| < 0, y1(t)  c  f3 < 0, the synchronizing errors are stable to zero. That is to say that the response system realizes synchronization with the drive system. The initial conditions of the drive system (17) and the response system (18) are chosen as x1 = 2, x2 = 4, x3 = 10, y1 = 1, y2 = 2, y3 = 1 .The parameters of the controller (20) are selected as f1 = 1, f2 = 1, f3 = y1(t). Fig. 4 depicts the synchronization errors of state variables between the drive system and the response system almost surely converge to zero. The numerical simulations clearly verify the effectiveness of the controller designed. 5. Conclusion In this paper, a simple approach for designing controller to control and synchronize nonlinear system without delay is extended to nonlinear system with delay. The process of designing the controller becomes very simple and easy. It can also be extended to adaptive control and synchronize nonlinear system with delay and unknown parameters. Acknowledgements This work is supported by Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents, National Natural Science Foundation of China (61304062, 61004027, 61174065, 61174066, 61273103, 61171077) and The Basic Research Programs of Chinese Department of Transportation (2011-319-813-510). References [1] Balasubramaniam P, Nagamani G. A delay decomposition approach to delay-dependent robust passive control for Takagi–Sugeno fuzzy nonlinear systems. Circ Syst Signal Pr 2012;31:1319–41. [2] Baleanu D, Ranjbar A, Sadati SJ, Delavari RH, Abdeljawad T, Gejji V. Lyapunov–Krasovskii stability theorem for fractional systems with delay. Rom J Phys 2011;56:636–43. [3] Bhalekar S, Daftardar-Gejji V, Baleanu D, Magin R. Generalized fractional order bloch equation with extended delay. Int J Bifurcat Chaos 2012;22. [4] Chen LP, Wei SB, Chai Y, Wu RC. Adaptive projective synchronization between two different fractional-order chaotic systems with fully unknown parameters. Math Probl Eng 2012. [5] Liu H, Shen Y, Zhao XD. Delay-dependent observer-based H-infinity finite-time control for switched systems with time-varying delay. Nonlinear Anal Hybrid 2012;6:885–98. [6] Merrikh-Bayat F. Stability of fractional-delay systems: a practical approach. In: New trends in nanotechnology and fractional calculus applications, 2010. p. 163–170.

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