A semi-global finite-time convergent observer for a class of nonlinear systems with bounded trajectories

Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

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A semi-global finite-time convergent observer for a class of nonlinear systems with bounded trajectories Weisong Tian a,∗ , Haibo Du a,b , Chunjiang Qian a a

Department of Electrical and Computer Engineering, University of Texas at San Antonio, TX, 78249, USA

b

School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

article

info

Article history: Received 20 September 2011 Accepted 12 December 2011 Keywords: Nonlinear systems Finite-time convergent observer Homogeneous domination approach

abstract This paper considers the problem of designing finite-time convergent observers for a class of lower-triangular nonlinear systems with bounded solution trajectories. Using the homogeneous domination approach, we construct an observer with homogeneous structure and saturation design, whose states will converge to the real states in a finite time by adjusting the observer gain. Several application examples of this finite-time convergent observer are discussed in this paper. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we consider a class of nonlinear systems in the form of: x˙ 1 = x2 + f1 (x1 )

.. . x˙ n−1 = xn + fn−1 (x1 , x2 , . . . , xn−1 ) x˙ n = fn (x1 , x2 , . . . , xn ) y = x1

(1)

where x = (x1 , . . . , xn ) is the system state and y ∈ R is the system output. The nonlinear functions fi (x1 , . . . , xi ) for i = 1, 2, . . . , n, are C 1 (continuously differentiable). The objective of this paper is to design a finite-time convergent observer for (1). The observer will also be used to design a slave system which synchronizes with a chaotic system in a finite time even when some of the states of the chaotic system are not available. The problem of estimating the unmeasurable states of a nonlinear system from its output has been receiving considerable attention and a number of results have been achieved. One existing method is based on the linearizations of the nonlinear systems [1–3]. The work [4] introduced a locally convergent nonlinear observer by using a back-stepping method, whose estimation error goes to zero when the initial value of estimation error is not too large. A high gain globally convergent observer for some nonlinear systems with bounded solutions was designed in [5]. Recently, new design methods based on homogeneous systems theory were developed to construct observers. The work [6] introduced a nonlinear observer with a homogeneous structure even for a linear system. The new construction of the observer enables us to relax the long-existing linear growth condition used for nonlinear observer design. Later, the works [7,8] extended the homogeneous domination approach introduced in [6] to more general nonlinear systems. In practice, the finite time synchronization is desirable since the finite-time convergence not only has a faster rate but also demonstrate better robustness and disturbance rejection T

∗

Corresponding author. Tel.: +1 210 458 7068; fax: +1 210 458 5947. E-mail addresses: [email protected] (W. Tian), [email protected] (H. Du), [email protected] (C. Qian).

1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.12.021

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properties [9]. The work [10] constructed a class of output feedback controllers that can achieve global finite-time stability for the double integrator system. The papers [8,11] introduced some observers with a fast convergence rate, i.e., the estimates converge to the original states in a finite time, rather than asymptotically. In this paper, we are interested in designing a finite-time convergent observer for a class of lower-triangular nonlinear systems with no control input. Instead of pursuing global finite-time convergence, we first investigate when the estimated states are finite-time convergent to the real states of system (1) for small initial errors. By pursuing this less ambitious goal, we can show that there is no need of growth conditions imposed on the nonlinearities [8,11]. Then, we will also show that for the systems with bounded solution trajectories, the requirement of small initial condition can be lifted by introducing and adjusting a scaling gain in the homogeneous observer. The rest of this paper is organized as follows. In Section 2, we introduce some useful lemmas and existing homogeneous design methods. In Section 3, we construct a low-order homogeneous observer for lower-triangular systems and we prove it is finite-time convergent when the initial error is small. Then, we integrate the homogeneous observer with a saturation method to relax the conditions on the initial error. The application of the finite-time convergent observer is discussed in Section 4. Our conclusion is included in Section 5. 2. Preliminaries In this section, we list some useful definitions and lemmas which will be constantly used in proving the main results. 2.1. Weighted homogeneity Listed below are the definitions of homogeneous systems with weighted dilation (refer to Refs. [12–15] for details). Weighted homogeneity: For fixed coordinates (x1 , . . . , xn ) ∈ Rn and real numbers ri > 0 (i = 1, 2, . . . , n),

• the dilation ∆ε (x) = (ε r1 x1 , . . . , ε rn xn ), ∀ε > 0, with ri being called as the weights of the coordinates; • a function V ∈ C (Rn , R) is said to be homogeneous of degree τ (τ ≥ − min{r1 , . . . , rn }), if there is a real number τ ∈ R such that

∀x ∈ Rn \ {0}, ε > 0, V (∆ε (x)) = ε τ V (x1 , x2 , . . . , xn ); • a vector field f ∈ C (Rn , Rn ) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that for i = 1, 2, . . . , n ∀x ∈ Rn \ {0}, ε > 0,

fi (∆ε (x)) = ε τ +ri fi (x).

Lemma 2.1. Given a dilation weight ∆ = (r1 , r2 , . . . , rn ), suppose V1 (x) and V2 (x) are respectively homogeneous functions of degree τ1 and τ2 . Then V1 (x)V2 (x) is also homogeneous with respect to the same dilation weight ∆. Moreover, the homogeneous degree of V1 (x)V2 (x) is τ1 + τ2 . Lemma 2.2. Assume that V : Rn → R is a homogeneous function of degree τ with respect to the dilation weight ∆. Then the following holds:

• ∂∂ xVi is homogeneous of degree τ − ri with ri being the homogeneous weight of xi . • For a positive definite function W (x) which is also homogeneous of degree τ1 with respect to the dilation weight ∆, there is a constant c such that V (x) ≤ cW τ /τ1 (x). 2.2. Finite-time stability Finite-time stability: Consider the following system x˙ = f (x),

t ≥ 0, x ∈ Rn ,

(2)

where f : R → R is continuous and f (0) = 0. The origin is said to be finite-time stable if all the solutions are (i) Lyapunov stable and (ii) finite-time convergent, i.e., for any given initial condition x0 , there exists a time instance T (x0 ) such that x(t , x0 ) ≡ 0, t ≥ T (x0 ). A more rigorous definition of finite-time stability can be found in [16]. To determine the finite-time stability of the nonlinear system (2), we introduce the following lemma in [16]. n

n

Lemma 2.3. For system (2), suppose that there exists a positive definite and proper function V (x) : Rn → R such that V˙ (x)|(2) + c (V (x))α ≤ 0 for constant c > 0 and α ∈ (0, 1). Then V (x) approaches zero in a finite time and system (2) is globally finite-time stable. 2.3. A finite-time convergent observer Next, we review an existing result on homogeneous observers. In a recent work [8], a solution was given for the problem of global output feedback stabilization for a class of nonlinear systems whose nonlinearities are bounded by both low-order

W. Tian et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

1829

and high-order terms. The paper [8] also introduced a homogeneous observer whose states converge to the unmeasurable states in a finite time. Specifically, for the following system: x˙ 1 = x2 , . . . ,

x˙ n−1 = xn ,

x˙ n = u,

y = x1 ,

(3)

given a ratio of even/odd integers τ ∈ (−1/n, 0), a homogeneous observer is constructed as follows x˙ˆ 1 = xˆ 2 + c1 [x1 − xˆ 1 ]r2

.. . x˙ˆ n−1 = xˆ n + cn−1 [x1 − xˆ 1 ]rn x˙ˆ n = u + cn [x1 − xˆ 1 ]rn+1

(4)

where r1 = 1, ri+1 = ri + τ , and ci are appropriately chosen constants, for i = 1, 2, . . . , n. Remark 2.1. Note that in this paper, we require τ = −q/p with a positive even integer q and a positive odd integer p. By doing this, one can obtain the following benefit: ri = 1 + (i − 1)τ (for i = 1, . . . , n + 1), will be odd in both denominator and numerator, which will simplify the notation of the observer since (−s)ri = −(s)ri . However, in the case when τ cannot be represented as an even/odd ratio, a similar result can be obtained for any real number τ ∈ (−1/n, 0) by defining [s]ri = sign(s)|s|ri instead of sri as shown in [17]. With the above in mind, defining ei = xi − xˆ i , i = 1, 2, . . . , n, the error dynamics become: r

e˙ 1 = e2 − c1 e12

.. . r e˙ n−1 = en − cn−1 e1n r

e˙ n = −cn e1n+1 .

(5)

Lemma 2.4 ([8]). There exist constants ci , i = 1, . . . , n such that the error dynamics (5) are globally finite-time stable. In addition, there exists a homogeneous Lyapunov function V (e) of degree k with respect to the dilation weight ∆ = {r1 , . . . , rn } such that

 V˙ (e)|(5) =

r



e2 − c1 e12

 .. ∂ V (e)    .   = −ω(e) ∂ e en − cn−1 er1n  rn+1 −cn e1

(6)

where the function ω(e) is positive definite and homogeneous of degree k + τ with respect to ∆. 2.4. Saturation function Unity saturation function: sat (s) =

1, s,

−1,

for s ≥ 1 for − 1 < s < 1 for s ≤ −1.

Next we list a lemma which characterizes a nice property of the saturation function. Lemma 2.5 ([5]). Given real numbers s1 , s2 and m > 0, if |s1 | ≤ m, the following holds:

  s  2   s1 − m · sat  ≤ |s1 − s2 |. m

(7)

3. Finite-time homogeneous observer design Based on the homogeneous observer (4) introduced in the previous section, we can construct a finite-time convergent observer for system (1): x˙ˆ 1 = xˆ 2 + f1 (x1 ) + c1 (x1 − xˆ 1 )r2

.. . x˙ˆ n−1 = xˆ n + fn−1 (x1 , xˆ 2 , . . . , xˆ n−1 ) + cn−1 (x1 − xˆ 1 )rn x˙ˆ n = fn (x1 , xˆ 2 , . . . , xˆ n ) + cn (x1 − xˆ 1 )rn+1 where constants ci ’s are selected using Lemma 2.4 and ri ’s are defined the same as those in (4).

(8)

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W. Tian et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

By defining the error as ei = xi − xˆ i , we obtain the following error dynamics r

e˙ 1 = e2 − c1 e12 + f1 (x1 ) − f1 (x1 )

.. . r e˙ n−1 = en − cn−1 e1n + fn−1 (x1 , x2 , . . . , xn−1 ) − fn−1 (x1 , xˆ 2 , . . . , xˆ n−1 ) r

e˙ n = −cn e1n+1 + fn (x1 , x2 , . . . , xn ) − fn (x1 , xˆ 2 , . . . , xˆ n ).

(9)

Before moving on, we give the following definition of semi-global finite-time stabilization by output feedback. Definition 3.1 (Semi-Global Finite-Time Stabilization by Output Feedback [18]). Given any compact subset Ωx ⊂ R2n , find, if possible, a linear output dynamic compensator x˙ˆ = G(ˆx, y, u),

u = u(ˆx),

(10)

such that all the solutions of the closed-loop system (1), (10) are locally finite-time stable and x(0) x(t ) ∈ Ωx ⇒ lim ˆ = 0. xˆ (0) t →+∞ x(t )









(11)

Theorem 3.1. Suppose that K is a compact, positively invariant set for system (1). For the initial conditions xˆ i (0) of (8) which are close to xi (0) in (1), the error dynamics (9) are finite-time stable. Proof. The error dynamics can be written as:





 .. 

 



e˙ 1

r

e2 − c1 e12

..





 



f1 (x1 ) − f1 (x1 )

..



. . . .  e˙ =  + ˙  =  en − cn−1 er1n  fn−1 (x1 , x2 , . . . , xn−1 ) − fn−1 (x1 , xˆ 2 , . . . , xˆ n−1 ) e n −1 r e˙ n fn (x1 , x2 , . . . , xn ) − fn (x1 , xˆ 2 , . . . , xˆ n ) −cn e1n+1 

(12)

To determine the stability of (12), we adopt the homogeneous Lyapunov function V (e) defined in (6). Taking the derivative of V (e) along (12), we have

V˙ |(12)

r

  f1 (x1 ) − f1 (x1 )   . .  ∂ V (e)  .. ..  ∂V   . =  +  r n ∂ e en − cn−1 e1  ∂ e fn−1 (x1 , x2 , . . . , xn−1 ) − fn−1 (x1 , xˆ 2 , . . . , xˆ n−1 ) r fn (x1 , x2 , . . . , xn ) − fn (x1 , xˆ 2 , . . . , xˆ n ) −cn e1n+1 

e2 − c1 e12



(13)

For i = 1, 2, . . . , n, by the smoothness of the functions fi ’s and the mean value theorem, there is a smooth function Hi (x1 , x2 , . . . , xi , xˆ 2 , . . . , xˆ i ) such that

|fi (x1 , x2 , . . . , xi ) − fi (x1 , xˆ 2 , . . . , xˆ i )| ≤ Hi (x1 , x2 , . . . , xi , xˆ 2 , . . . , xˆ i )(|e2 | + · · · + |ei |) ≤ H˜ i (x1 , . . . , xi )(|e2 | + · · · + |ei |) ˜ i is a smooth function for small errors ∥e∥. Noting that 0 < ri < ri−1 < r2 < r1 = 1, one obtains where H 1. Thus, here we can use this property to lead to the following equations:

(14) ri+1 r2

< 1, . . . ,

ri+1 ri

<

|fi (x1 , x2 , . . . , xn ) − fi (x1 , xˆ 2 , . . . , xˆ n )|   ri+1 r r ri+1 1− i+1 1− i+1 = H˜ i (x1 , . . . , xi ) |e2 | r2 · |e2 | r2 + · · · + |ei | ri · |ei | ri    ri+1 r r ri+1 1− i+1 1− i+1 ≤ H˜ i (x1 , . . . , xi ) |e2 | r2 + · · · + |ei | ri |e2 | r2 + · · · + |ei | ri .

(15)

Substituting (6) and (15) into (13), we have: V˙ |(12) ≤ −ω(e) +

    n  ri+1 ri+1 ri+1 ri+1   ∂V    |e2 | r2 + · · · + |ei | ri H˜ i (·) |e2 |1− r2 + · · · + |en |1− ri .  ∂e  i=2

i

(16)

W. Tian et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

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  ri+1 ri+1   On the other hand, by Lemma 2.1,  ∂∂ eV  (|e2 | r2 + · · · + |ei | ri ) is homogeneous of degree k + τ . Thus, by Lemma 2.2, we i can find a constant c such that

   ri+1 ri+1  ∂V    |e2 | r2 + · · · + |ei | ri ≤ c · ω(e)  ∂e  i

(17)

for the positive definite homogeneous function ω(e) defined in (6). Substituting (17) into (16), we have

   n r 1 r  1− i+ 1− ir+1 r ˙V |(12) ≤ −ω(e) 1 − c ˜ i 2 + · · · + |e | Hi (x1 , . . . , xi ) |e2 | . i

(18)

i=2

Note that the power 1 − c

n 

ri+1 ri



> 0 for i = 1, . . . , n, which implies, when the error ei is small enough, the function

˜ i (x1 , . . . , xi ) |e2 | H

r 1− ir+1 2

r 1 1− i+ ri

+ · · · + |e i |



<1

(19)

i=2

for the bounded x in K . With the help of (19), (18) becomes: V˙ |(12) ≤ −ˆc ω(e),

(20)

where cˆ is a positive constant. Note that ω(e) is homogeneous of degree k + τ and the positive definite function V (e) is homogeneous degree of k. According to Lemma 2.2, there is a constant k such that V˙ |(12) ≤ −ˆc kV (e)

k+τ k

.

Because τ < 0, by Lemma 2.3, we can conclude that the error dynamics (12) is finite-time stable as long as ∥e∥ is small enough.  The finite-time observer (8) requires the initial error to be small. When this condition cannot be guaranteed, we can integrate the finite-time observer with the saturation method introduced in [5]. Define a new observer as: r

x˙ˆ 1 = xˆ 2 + f1 (sat N (ˆx1 )) + Lc1 e12

.. . r x˙ˆ n−1 = xˆ n + fn−1 (sat N (ˆx1 ), . . . , sat N (ˆxn−1 )) + Ln−1 cn−1 e1n r

x˙ˆ n = fn (sat N (ˆx1 ), . . . , sat N (ˆxn )) + Ln cn e1n+1 ,

(21)

where sat N (x) = N · sat N , with the bound N for |xi | (i.e., |xi | ≤ N) and the gain L ≥ 1 will be determined later. Note that we still use the constants ci ’s and ri ’s defined in (4) or (8). Defining εi = ei /Li−1 , i = 1, 2, . . . , n, we have the following new error dynamic system:

x

f1 (x1 ) − f1 (sat N (ˆx1 ))





.. ε2 − ε ε˙ 1    .    ..  ..     fn−1 (x1 , . . . , xn−1 ) − fn−1 (sat N (ˆx1 ), . . . , sat N (ˆxn−1 ))  . . .  = L ε˙ =  +      εn − cn−1 ε1rn   ε˙ n−1   Ln−2 rn+1   ε˙ n −cn ε1 fn (x1 , . . . , xn ) − fn (sat N (ˆx1 ), . . . , sat N (ˆxn )) 





r c1 12



(22)

Ln−1 As shown in the following theorem, the error system (22) can be rendered finite-time stable without requiring small errors between the estimated states and real states. Theorem 3.2. Suppose that the solution trajectories of (1) are bounded with the bound N. The error dynamics (22) is finite-time stable for a large enough L. Proof. Similar as the proof of Theorem 3.1, we also adopt the homogeneous Lyapunov function (6) by replacing e by ε . Specifically, we will have the function V (ε) with a homogeneous degree k and its derivative along (22) is



V˙ (ε)|(22)

f1 (x1 ) − f1 (sat N (ˆx1 ))



.. ε2 − ε     .   .  ∂V  ..  ∂V  f ( x , . . . , x ) − f ( sat (ˆ x ), . . . , sat (ˆ x ))  . n − 1 1 n − 1 n − 1 N 1 N n − 1 =L  +  ∂ε εn − cn−1 ε1rn  ∂ε    Ln−2 rn+1   −cn ε1 fn (x1 , . . . , xn ) − fn (sat N (ˆx1 ), . . . , sat N (ˆxn )) 

r c1 12



Ln−1

(23)

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By the mean-value theorem, there exists a vector ξ ∈ Rn×1 with each element ξi ∈ [xi , sat N (ˆxi )], such that:

 i   ∂ fi   |fi (x1 , . . . , xi ) − fi (sat N (ˆx1 ), . . . , sat N (ˆxi ))| ≤  (ξ ) |xj − sat N (ˆxj )| ∂x j =1 ≤ D1

i 

|xj − sat N (ˆxj )|,

(24)

j =1

where the constant D1 is obtained due to the fact that [xi , sat N (ˆxi )] is bounded. In what follows, we show that for any rij ∈ (0, 1) there is a constant D2 > 0 such that the following inequality holds:

  xj − sat N (ˆxj ) ≤ D2 |xj − xˆ j |rij .

(25)

To prove inequality (25), we consider two different cases. Case 1: When |xj − xˆ j | ≥ 1. Due to the boundedness of the |xi |’s, we can always find two positive constants M1 and M2 such that: |xj | ≤ M1 , |sat N (ˆxj )| ≤ M2 . Thus,

|xj − sat N (ˆxj )| ≤ M1 + M2 ≤ (M1 + M2 )|xj − xˆ j |rij .   Case 2: When |xj − xˆ j | ≤ 1. By Lemma 2.5, we have xj − sat N (ˆxj ) ≤ |xj − xˆ j |. By the fact that 0 < rij < 1 and |xj − xˆ j | < 1, we know |xj − xˆ j | ≤ |xj − xˆ j |rij . Inequality (25) follows immediately by combining Cases 1 and 2. Substituting (25) into (24) yields

|fi (x1 , . . . , xi ) − fi (sat N (ˆx1 ), . . . , sat N (ˆxi ))| Li−1

D2 D1

≤



Li−1

|ε1 |

 ≤ D3 |ε1 |

ri +τ r1

ri +τ r1

+ |Lε2 |

+ |ε2 |

ri +τ r2

ri +τ r2

+ · · · + |Li−1 εi |

+ · · · + |εi |

ri +τ ri



ri +τ ri



,

(26)

where D3 = D1 D2 . With the help of Lemma 2.4 and (26), (23) becomes V˙ (ε)|(22) ≤ −Lω(ε) +

   n  ri +τ  ri +τ ri +τ  ∂ V (ε)    D3 |ε1 | r1 + |ε2 | r2 + · · · + |εi | ri ,  ∂ε 

(27)

i =1

where ω(ε) is the positive definite function with the homogeneous degree k + τ . On the other hand, by Lemma 2.1, it can be verified that the term

ri +τ ri +τ ri +τ n  ∂ V (ε)  r1 + |ε2 | r2 + · · · + |εi | ri ) is homogeneous with degree of k + τ . Thus, by i=1  ∂ε  D3 (|ε1 |

Lemma 2.2, we can find a positive constant M such that

 D3

|ε1 |

ri +τ r1

+ |ε2 |

ri +τ r2

+ · · · + |εi |

ri +τ ri



≤ M · ω(ε)

(28)

for the positive definite homogeneous function ω(e). Substituting (28) into (27) yields V˙ (ε)|(22) ≤ −ω(ε)(L − M ) ≤ −(L − M )kV (ε)

k+τ k

.

(29)

Clearly, if the gain L is large enough, we can guarantee (29) is negative definite. Since τ < 0, by Lemma 2.3, we can conclude that the error dynamic system (22) is finite-time stable.  To illustrate the advantage of this design, we make a comparison with the finite-time observer (8). Example 3.1. Consider the following system: x˙ 1 = x2 ,

x˙ 2 = x21 − x31 .

(30)

Now we use the two methods (8) and (21) to design observers for this system respectively as follows:

 Observer #1 :

5

x˙ˆ 1 = xˆ 2 + e17

 3

x˙ˆ 2 = xˆ 22 − xˆ 31 + e17 ,

Observer #2 :

5

x˙ˆ 1 = xˆ 2 + Le17

3

x˙ˆ 2 = sat N (ˆx2 )2 − sat N (ˆx1 )3 + L2 e17 .

When the difference between x(0) and xˆ (0) is relatively large, for example, x(0) = [3, 3]T , xˆ (0) = [15, 15]T , Observer #1 will not converge to (30) as shown in Fig. 1.

W. Tian et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

1833

Fig. 1. Errors between Observer #1 and (30).

Fig. 2. Errors between Observer #2 and (30).

However, by adjusting the gain function L, it is possible to make Observer #2 convergent. Fig. 2 shows when L = 5, the error dynamics convergent to zero in a finite time. 4. Applications In this section, we discuss the application of this finite-time observer. We design a slave system to synchronize with a chaotic system in the form of (1), whose partial states, i.e., x2 , . . . , xn , are not available. Since chaotic systems usually have bounded states, the finite-time convergent observer designed in the previous section can be applied to solve the synchronization problem as described in the following corollary. Corollary 4.1. For a chaotic system in the form of (1), a slave system in the form of (21) can be designed to synchronize with the chaotic system in a finite time. Remark 4.1. The slave system constructed by using the finite-time convergent observer has two advantages. First, the slave system does not need the information of full states of the chaotic system. As a matter of fact, only the output, i.e., y = x1 , is needed for the design of the salve system. Second, the slave system will synchronize with the chaotic system in a finite time. Specifically, the states of the slave system will be exactly the same as the states of the chaotic system after a finite time. In what follows, we consider two well-known chaotic systems as illustrative examples. Example 4.1 (Duffing Systems). Consider the Duffing system [19] given by x˙ 1 = x2 x˙ 2 = 1.8x1 − 0.1x2 − x31 + 1.1 cos(t ).

(31)

According Corollary 4.1, we can construct a slave system based on (21). Specifically, by choosing L = 4, c1 = c2 = 1, τ = − 72 , the slave system is designed as follows: 5

x˙ˆ 1 = xˆ 2 + 4e17

3

x˙ˆ 2 = 1.8sat N (ˆx1 ) − 0.1sat N (ˆx2 ) − sat N (ˆx1 )3 + 1.1 cos(t ) + 42 e17

(32)

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Fig. 3. Phase portrait of the Duffing system and the slave system (32).

Fig. 4. Errors between the slave system (32) and the Duffing system.

where N = 5. The computer simulation is shown in Figs. 3 and 4 which show that the slave states converge to the master states in a finite time. Example 4.2 (The Modified Chaotic Rössler System). This chaotic system [20] is described by: u˙ = −v − w;

v˙ = u + av + 0.01u ln(w);

w ˙ = c + w(u − b),

(33)

where a = 0.2, b = 5.7, c = 0.2. Based on the coordinate transformation introduced in [20], we define x1 = ln(w), x2 = u, x3 = −v , as well as the system’s output y = x1 , under which system (33) can be rewritten as: x˙ 1 = x2 + 0.2 exp(−x1 ) − 5.7 x˙ 2 = x3 − exp(x1 ) x˙ 3 = −x2 + 0.2x3 − 0.01x1 x2 . By choosing the gain L = 2 and τ = −

(34) 2 , 9

the slave system is designed as: 7

x˙ˆ 1 = xˆ 2 + 0.2 exp(−sat N (ˆx1 )) − 5.7 + 2e19 5

x˙ˆ 2 = xˆ 3 − exp(sat N (ˆx1 )) + 4e19

3

x˙ˆ 3 = −sat N (ˆx2 ) + 0.2sat N (ˆx3 ) − 0.01sat N (ˆx1 )sat N (ˆx2 ) + 8e19 .

(35)

The simulation results by using observer (35) are shown in Figs. 5 and 6, with the initial values x(0) = [2, 1, 1], xˆ (0) =

[1, 2, 2].

5. Conclusion For a class of nonlinear systems with bounded states, we first design a finite-time convergent observer whose states converge to the real states in a finite time for small initial errors. Later, to relax the restriction on the initial error, the observer is integrated with the saturation method and homogeneous domination approach. The observer introduced in this paper provides a solution to the output synchronization problem for a class of chaotic systems. Two well-known chaotic systems, Duffing and Rössler systems, are considered as illustrative examples.

W. Tian et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1827–1836

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Fig. 5. Synchronization of the modified chaotic Rössler system.

Fig. 6. Errors between the chaotic system (34) and slave system (35).

Acknowledgments This work is supported in part by U.S. National Science Foundation under Grant No. HRD-0932339 and Valero Research Excellence Funds. References [1] [2] [3] [4] [5]

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