ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(2011) No.1,pp.74-79
Generalized Theorem of The Alternatives-Based Approach to Lyapunov Stability Analysis for LTI Systems Jianhong Wang ∗ School of Science, Nantong University, Nantong, Jiangsu, 226007, China (Received 10 October 2009, accepted 13 September 2010)
Abstract: Theorem of the alternatives for linear matrix inequality based on the separation theorems are proposed to provide systematic and unified proofs of necessary and sufficient conditions for solvability of Lyapunov inequality on 𝑃 . As an application of the Theorem of the alternatives, we investigate the conditions for the existence of feasible solutions to Lyapunov inequality on 𝑃 and obtain a restatement of the celebrated Lyapunov stability theorem for linear time-invariant systems. Furthermore, the effectiveness of the proposed method is illustrated by a numerical example. Keywords: Theorem of the alternatives; linear matrix inequality; Lyapunov stability; Linear time-invariant systems; LMI optimization
1
Introduction
A wide variety of problems in systems and control theory can be cast or recast as the problem of minimizing a linear function of a variable 𝑥 ∈ R𝑚 subject to a matrix inequality: 𝑚𝑖𝑛 𝑐𝑇 𝑥 𝑠.𝑡 𝐹 (𝑥) ⪯ 0
(1)
where 𝐹 (𝑥) ≜ 𝐹0 + 𝑥1 𝐹1 + ⋅ ⋅ ⋅ + 𝑥𝑚 𝐹𝑚 The problem data are the vector 𝑐 ∈ R𝑚 , 𝐹0 , ⋅ ⋅ ⋅ , 𝐹𝑚 ∈ 𝐻 𝑛 , the space of 𝑛 × 𝑛 Hermitian matrices. We assume that the matrices 𝐹𝑖 , 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑚 are linearly independent. We call the inequality 𝐹 (𝑥) ⪯ 0 a linear matrix inequality (LMI) and the problem (1) a LMI optimization.We will also consider strict LMI 𝐹 (𝑥) = 𝐹0 + 𝑥1 𝐹1 + ⋅ ⋅ ⋅ + 𝑥𝑚 𝐹𝑚 ≺ 0 The inequality ≺ indicates that, for Hermitian matrices 𝐴 and 𝐵, 𝐴 ≺ 𝐵 (respectively, 𝐴 ⪯ 𝐵), 𝐴 − 𝐵 is Hermitian negative definite (respectively, negative semidefinite). The sets {𝑥∣𝐹 (𝑥) ⪯ 0} and {𝑥∣𝐹 (𝑥) ≺ 0}are convex sets on the variable 𝑥. For convenience, we introduce some notation to make the LMI easier to state. The natural inner product between two elements 𝐴, 𝐵 ∈ H𝑛 is 𝐴 ∙ 𝐵 = 𝐵 ∙ 𝐴 = 𝑇 𝑟𝑎𝑐𝑒𝐴∗ 𝐵 = Σ𝑛𝑖=1 Σ𝑛𝑗=1 𝐴𝑖𝑗 𝐵𝑖𝑗 = ⟨𝐴, 𝐵⟩. We define the ∑𝑚 𝑎𝑑𝑗 linear operator ℱ : R𝑚 → H𝑛 by ℱ(𝑥) = (𝑌 ) = (𝐹𝑖 ∙ 𝑌 )𝑚 𝑖=1 = 𝑖=1 𝑥𝑖 𝐹𝑖 , so the adjoint of ℱ is given by ℱ (𝐹1 ∙ 𝑌, ⋅ ⋅ ⋅ , 𝐹𝑚 ∙ 𝑌 )𝑇 , a mapping from H𝑛 to R𝑚 . Using this notation, we can rewrite LMIs as 𝐹 (𝑥) = 𝐹0 + ℱ𝑥 ⪯ 0 and 𝐹 (𝑥) = 𝐹0 + ℱ𝑥 ≺ 0 ∗ Corresponding
author.
E-mail address:
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For symmetry reason, we will call (1) the primal optimization problem, and call 𝑥 the primal vector. Introducing the dual variable 𝑌 ∈ H𝑛 , 𝑌 ર 0, we can form the Lagrangian function 𝐿(𝑥, 𝑌 ) = 𝑐𝑇 𝑥 + ⟨𝐹0 + ℱ𝑥, 𝑌 ⟩ = 𝐹0 ∙ 𝑌 + ⟨𝑥, 𝑐 + ℱ 𝑎𝑑𝑗 𝑌 ⟩ The so-called dual function 𝑔(𝑌 ) associated with (1) is defined as { 𝐹0 ∙ 𝑌, if 𝑐 + ℱ 𝑎𝑑𝑗 𝑌 = 0 𝑔(𝑌 ) = 𝑖𝑛𝑓𝑥 𝐿(𝑥, 𝑌 ) = −∞, otherwise Introducing a slack variable 𝑋, we can express (1) as 𝑚𝑖𝑛{𝑐𝑇 𝑥 : 𝐹0 + ℱ𝑥 + 𝑋 = 0, 𝑋 ર 0}.
(3)
The dual problem associated with (6) is 𝑚𝑎𝑥{𝐹0 ∙ 𝑌 : ℱ 𝑎𝑑𝑗 𝑌 + 𝑐 = 0, 𝑌 ર 0}. Though the form of LMIs appear very specialized, it turns out that it is widely encountered in systems and control theory. Examples include: multicriterion LQG, synthesis of linear state feedback for multiple or nonlinear plants (multi-model control), optimal state-space realizations of transfer matrices, norm scaling, robustness analysis and robust controller design, gain-scheduled controller design, and many others. For an overview of these results we refer to the books [1-2]. Since solving LMI is a convex optimization problem, such formulations offer a numerically tractable means of attacking problems that lack an analytical solution. In addition, efficient interior-point algorithms are now available to solve the generic LMI problems with a polynomial-time worst-case complexity. Consequently, reducing a control design problem to an LMI can be considered as a practical solution to this problem. We consider the linear time-invariant (LTI) system 𝑥˙ = 𝐴𝑥
(4)
where 𝐴 ∈ C𝑛×𝑛 . The Lyapunov inequality on a Hermitan 𝑃 ≻ 0, i.e., LMI of the form 𝐴∗ 𝑃 + 𝑃 𝐴 ≺ 0 plays a fundamental role in establishing the stability of systems (8); see any text on linear systems, for instance,[3]. The LMI convex optimization approach is a powerful tool in solving many control problems. For example, see[4-10]. Moreover, there exists a rich literature based on convex optimization to re-interpret the existing results and derive new results in system theory. Rantzer [11] uses ideas from convexity theory to give a new proof of the Kalman-YakubovichPopov Lemma. Henrion and Meinsma [12] apply semidefinite programming [13] to provide a new proof of a generalized form of Lyapunov’s matrix inequality on the location of the eigenvalues of a matrix in some region of the complex plane. Park [14] investigates the dynamic output feedback control for delay differential systems of neutral type based on convex optimization approach. Balakrishnan and vandenberghe [15] apply semidefinite programming duality to derive a new criterion for the controllability of an LTI system realization. In this paper, we give a restatement of the celebrated Lyapunov stability theorem for linear time-invariant systems based on generalized Theorem of the Alternatives.
2
Theorem of the alternatives for LMIs
In this section, we utilize the separation theorems [16] to obtain the so-called Theorem of the alternatives for Lyapunov inequality on 𝑃 , which is very important for the later discussion of Lyapunov stability analysis. Lemma 1 Exactly one of the following statements is true. 1.There exists 𝑥 ∈ R𝑚 such that 𝐹0 + ℱ𝑥 ≺ 0, 2.There exists 𝑌 ∈ H𝑛 such that
0 ∕= 𝑌 ર 0, 𝐹0 ∙ 𝑌 ≥ 0, ℱ 𝑎𝑑𝑗 𝑌 = 0.
Proof. (1)Note that the two statements contradict each other: 0 > ⟨𝐹0 + ℱ𝑥, 𝑌 ⟩ = ⟨𝐹0 , 𝑌 ⟩ + ⟨𝑥, ℱ 𝑎𝑑𝑗 𝑌 ⟩ ≥ 0
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International Journal of Nonlinear Science, Vol.11(2011), No.1, pp. 74-79
Therefore at most one of the two statements is true. (2)Suppose statement 1 is false, i.e., 𝐹0 + ℱ𝑥 ≺ 0 has no solution, then 𝐹0
∈ /
𝒞 = {𝑈 : ℱ𝑥 + 𝑈 ≺ 0, 𝑥 ∈ R𝑚 }
Since 𝒞 is open, nonempty, and convex, there must be a hyperplane strictly separating 𝐹0 from 𝒞, i.e., there exists a nonzero 𝑌 ∈ H𝑛 that satisfies 𝐹0 ∙ 𝑌 > 𝑈 ∙ 𝑌 for all 𝑈 ∈ 𝒞. In other words, 𝑌 ∕= 0 and ⟨𝐹0 , 𝑌 ⟩ > ⟨−ℱ𝑥 − 𝑋, 𝑌 ⟩ = −⟨𝑥, ℱ 𝑎𝑑𝑗 𝑌 ⟩ − ⟨𝑋, 𝑌 ⟩ for all 𝑥 ∈ R𝑚 , 𝑋 ≻ 0. Therefore 𝑌 ∈ H𝑛 satisfies 0 ∕= 𝑌 ર 0, 𝐹0 ∙ 𝑌 ≥ 0, ℱ 𝑎𝑑𝑗 𝑌 = 0 Which means that statement 2 is true. This completes the proof. Let ℋ denote the set of block diagonal Hermitian ∑ matrices with given dimensions, ℋ = H𝑛1 × ⋅ ⋅ ⋅ × H𝑛𝑟 , with inner 𝑟 product ⟨𝑑𝑖𝑎𝑔(𝐴1 , ⋅ ⋅ ⋅ , 𝐴𝑟 ), 𝑑𝑖𝑎𝑔(𝐵1 , ⋅ ⋅ ⋅ , 𝐵𝑟 )⟩ = 𝑘=1 𝑇 𝑟𝑎𝑐𝑒𝐴𝑘 𝐵𝑘 . Suppose that 𝒱 is a finite dimensional vector space and ℱ of LMI (2) is defined as ℱ : 𝒱 → ℋ. Applying these notation to the previous result, we similarly obtain Lemma 2 Exactly one of the following statements is true. 1.There exists 𝑥 ∈ 𝒱 such that 𝐹0 + ℱ𝑥 ≺ 0, 2.There exists 𝑌 ∈ ℋ such that
0 ∕= 𝑌 ર 0, 𝐹0 ∙ 𝑌 ≥ 0, ℱ 𝑎𝑑𝑗 𝑌 = 0.
An application of the Lemma 2 is the following theorem. Theorem 3 Exactly one of the following statements is true. 1.There exists 𝑃 ∈ H𝑛 such that 𝑃 ≻ 0, 𝐴∗ 𝑃 + 𝑃 𝐴 ≺ 0, 2.There exists 𝑌1 , 𝑌2 ∈ H𝑛 such that 0 ∕= 𝑑𝑖𝑎𝑔(𝑌1 , 𝑌2 ) ર 0, 𝑌1 𝐴∗ + 𝐴𝑌1 − 𝑌2 = 0. Proof. Using the above Lemma 2. Let ℱ : H𝑛 → H𝑛 × H𝑛 is defined by ℱ(𝑃 ) = 𝑑𝑖𝑎𝑔(𝐴∗ 𝑃 + 𝑃 𝐴, −𝑃 ). Then, it is easily verified that ℱ 𝑎𝑑𝑗 : H𝑛 × H𝑛 → H𝑛 is given by ℱ 𝑎𝑑𝑗 (𝑌 ) = 𝑌1 𝐴∗ + 𝐴𝑌1 − 𝑌2 , where 𝑌 = 𝑑𝑖𝑎𝑔(𝑌1 , 𝑌2 ). This completes the proof.
3
Lyapunov stability
In this section, we give a restatement of the celebrated Lyapunov stability theorem for linear time-invariant systems. Proposition 4 The following system: ⎧ ⎨ 𝑃 ≻0 𝐴∗ 𝑃 + 𝑃 𝐴 ≺ 0 ⎩ 𝑃 ∈ H𝑛
(5)
has solution if and only if all eigenvalues of 𝐴 are with negative real part. Proof. By the Theorem 3, it is clear that a necessary and sufficient condition for system (5) has no solution: there exist 𝑌1 , 𝑌2 ∈ H𝑛 such that 0 ∕= 𝑑𝑖𝑎𝑔(𝑌1 , 𝑌2 ) ર 0, 𝑌1 𝐴∗ + 𝐴𝑌1 − 𝑌2 = 0 We now show that this condition is equivalent to 𝐴 having eigenvalues with non-negative real part.
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Suppose that 𝐴 has an eigenvalue with non-negative real part, i.e., there exist 0 ∕= 𝑣 ∈ C𝑛 , 𝜎 ≥ 0, 𝑤 ∈ R with 𝐴𝑣 = (𝜎 + 𝑗𝑤)𝑣. It is easily shown that 𝑌1 = 𝑣𝑣 ∗ , 𝑌2 = 2𝜎𝑣𝑣 ∗ satisfy (6). ∗ Conversely, suppose that (6) holds. For 𝑌1 ∈ H𝑛+ , we have an unitary matrix(𝑈 such that ( ) ) 𝑌1 = 𝑈 ∧0 𝑈 , where ∧1 ∧1 ∧0 = , ∧1 = 𝑑𝑖𝑎𝑔(𝜆1 , ⋅ ⋅ ⋅ , 𝜆𝑟 ) and 𝜆𝑖 > 0.So that 𝐴𝑌1 = 𝑈 𝑈 ∗ 𝐴𝑈 𝑈 ∗ . Let 𝐵 = 𝑈 ∗ 𝐴𝑈 = 0 0 ( ) 𝐵1 𝐵2 , where 𝐵1 is a 𝑟 × 𝑟 matrix. Thus 𝐴𝑌1 = 𝑈 𝐵 ∧0 𝑈 ∗ , i.e., 𝐵∧0 is unitarily similar to 𝐴𝑌1 . From (6), we 𝐵3 𝐵4 note that the symmetric part of 𝐴𝑌1 is Hermitian positive semidefinite, so that the symmetric part of 𝐵∧0 is also Hermitian positive semidefinite. Since 𝐴 is unitarily similar to 𝐵 and thus we only need to prove 𝐵 having eigenvalues with non) ( 𝑌11 𝑌12 ∗ ∗ ∗ ∗ negative real part. By 𝑌1 𝐴 + 𝐴𝑌1 − 𝑌2 = 0, we have 𝐵 ∧0 + ∧0 𝐵 = 𝑈 𝑌2 𝑈 .Let𝑌3 = 𝑈 𝑌2 𝑈 = ∗ 𝑌12 𝑌22 ,then ) ( ( ) 𝑌11 𝑌12 𝐵1 ∧1 + ∧1 𝐵1∗ ∧1 𝐵3∗ = . ∗ 𝐵3 ∧1 0 𝑌12 𝑌22 This implies 𝑌22 = 0. Since 𝑌2 is Hermitian positive semidefinite ( and thus)𝑌3 is also Hermitian positive semidefinite. 𝑉1 𝑉 2 𝑛×𝑛 ∗ Therefore, there exist 𝑉 ∈ C such that 𝑌3 = 𝑉 𝑉 .Let 𝑉 = , then 𝑉3 𝑉 4 (
𝑌11 ∗ 𝑌12
𝑌12 0
)
( =
𝑉1 𝑉1∗ + 𝑉2 𝑉2∗ 𝑉3 𝑉1∗ + 𝑉4 𝑉2∗
𝑉1 𝑉3∗ + 𝑉2 𝑉4∗ 𝑉3 𝑉3∗ + 𝑉4 𝑉4∗
)
∗ = 0. So that we must have 𝐵3 ∧1 = 0, i.e., 𝐵3 = 0. Therefore, which(implies that)𝑉3 = 𝑉4 ( = 0 and 𝑌12 )= 𝑌12 𝐵1 𝐵2 𝐵1 ∧1 0 𝐵= , 𝐵∧0 = . We have thus show that the symmetric part of 𝐵1 ∧1 is Hermitian positive 0 𝐵4 0 0 √ √ semidefinite and the eigenvalues of 𝐵1 are the eigenvalues of 𝐵. Let ∧ = 𝑑𝑖𝑎𝑔( 𝜆1 , ⋅ ⋅ ⋅ , 𝜆𝑟 ), then we must have ∧1 = ∧2 = ∧∧∗ and 𝐵1 ∧1 = 𝐵1 ∧2 = ∧ ∧−1 𝐵1 ∧ ∧. We can write 𝑆 = ∧−1 𝐵1 ∧, i.e., 𝐵1 ∧1 = ∧𝑆∧. Then, 𝑆 = ∧−1 𝐵1 ∧1 ∧−1 = ∧−1 𝐵1 ∧1 (∧−1 )∗ . We have thus show that the symmetric part of 𝑆 is Hermitian positive semidefinite. For 𝐵1 ∧1 = 𝐵1 ∧ ∧ = ∧𝑆∧, we have 𝐵1 ∧ = ∧𝑆. The eigenvalues of S are all in the closed righthalf plane because S is the sum of a skew-Hermitian and a Hermitian positive semidefinite matrix. Therefore ∧ spans a (nonempty) invariant subspace of 𝐵1 associated with a set eigenvalues of 𝐵1 with non-negative real part. Therefore 𝐵1 have eigenvalue with non-negative real part and thus 𝐵 and 𝐴 have eigenvalue with non-negative real part. Thus, system (5) has solution if and only if all eigenvalues of 𝐴 are with negative real part. This completes the proof.
4
LMI feasibility problem
In this section, we explain how to solve a LMI feasibility problem, i.e., in the case of 𝑐 = 0. First of all, the LMI feasibility problem is defined by 𝑓∑ 𝑖𝑛𝑑 𝑥 ∈ 𝑅𝑚 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑚 𝐹0 + 𝑖=1 𝐹𝑖 𝑥𝑖 + 𝑋 = 0 𝑋 ર 0 where 𝑥 = (𝑥1 , 𝑥2 , ⋅ ⋅ ⋅ , 𝑥𝑚 ) and 𝑥𝑖 ∈ 𝑅 for all 𝑖. This problem can be converted into the standard form LMI optimization (3) by introducing a slack variable 𝑡 such as 𝑚𝑖𝑛 𝑡 ∑𝑚 𝑠.𝑡 𝐹0 + 𝑖=1 𝐹𝑖 𝑥𝑖 + 𝑋 − 𝑡𝐼 = 0 𝑋ર0 Therefore, 𝑥 and 𝑐 become
(7)
𝑥 := (𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑚 , 𝑡)𝑇 , 𝑐 := (0, ⋅ ⋅ ⋅ , 0, 1)𝑇 .
By solving the problem (7), if we obtain 𝑡 < 0, then we conclude that the problem is feasible. To avoid the numerical problem, in practice instead of solving the problem (7) derectly, we solve the problem 𝑚𝑖𝑛 𝑡 ∑𝑚 𝑠.𝑡 𝐹0 + 𝑖=1 𝐹𝑖 𝑥𝑖 + 𝑋 − 𝑡𝐼 = 0 𝑋 ર 0, 𝑡 + 1 > 0
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International Journal of Nonlinear Science, Vol.11(2011), No.1, pp. 74-79
In articular, we consider the following feasibility problem of the LMI: 𝑓 𝑖𝑛𝑑
𝑋 ∈ 𝐻 2 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴∗ 𝑋 + 𝑋𝐴 ≺ 0, 𝑋 ≻ 0
(8)
where 𝐴 ∈ 𝐶 2×2 is given. To avoid the numerical problem, we replace 𝑋 ≻ 0 with 𝑋 ≻ 𝐼 in (8). Then, the problem (8) is converted into 𝑚𝑖𝑛 𝑐𝑇 𝑥 (9) 𝑠.𝑡 𝐹0 + 𝐹1 𝑥1 + 𝐹2 𝑥2 + 𝐹3 𝑥3 + 𝐹4 𝑡 + 𝑋 = 0 (
where
𝑥1 𝑥2
𝑥2 𝑥3
)
( := 𝑋,
𝑎11 𝑎21
𝑎12 𝑎22
) := 𝐴,
𝑐 := (0, 0, 0, 1)𝑇 , 𝑥 := (𝑥1 , 𝑥2 , 𝑥3 , 𝑡)𝑇 . ⎛ ⎜ ⎜ 𝐹0 := ⎜ ⎜ ⎝
⎛
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 −1
⎛
⎞
⎜ ⎜ 𝐹1 := ⎜ ⎜ ⎝
⎟ ⎟ ⎟, ⎟ ⎠
2𝑎11 𝑎12 𝑎12 0 0 0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
⎞ ⎟ ⎟ ⎟, ⎟ ⎠
⎞ ⎞ ⎛ 0 0 0 0 0 0 𝑎11 + 𝑎22 0 𝑎21 ⎜ ⎜ 𝑎21 2𝑎22 0 0 0 ⎟ 0 0 0 ⎟ 2𝑎12 ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0 1 0 ⎟, 0 0 0 0 0 ⎟ 𝐹2 := ⎜ 𝐹3 := ⎜ ⎟, ⎝ ⎝ 0 1 0 0 ⎠ 0 0 0 1 0 ⎠ 0 0 0 0 0 0 0 0 0 ⎛ ⎞ 1 0 0 0 0 ⎜ 0 1 0 0 0 ⎟ ⎜ ⎟ 0 0 1 0 0 ⎟ 𝐹4 := ⎜ ⎜ ⎟. ⎝ 0 0 0 1 0 ⎠ 0 0 0 0 1 Note that the 1st diagonal block of 𝐹 matrices corresponds with 𝐴∗ 𝑋 +𝑋𝐴 ≺ 𝑡𝐼, the 2nd diagonal block corresponds with 𝑋 − 𝐼 ≻ −𝑡𝐼, and the 3rd diagonal block corresponds with 𝑡 + 1 > 0, respectively.
5
2𝑎21 𝑎11 + 𝑎22 0 0 0
An illustrative example
Example 5 Consider the following LTI system 𝑥˙ = 𝐴𝑥 ) √ √ −1 1 If we choose 𝐴 = , then two eigenvalues of 𝐴 are −2 + 3 and −2 − 3 with negative real part. In this 2 −3 case, it is easy to show that ( ) 7/4 5/8 𝑃 = ≻0 5/8 3/8 (
satisfy ∗
𝐴 𝑃 + 𝑃𝐴 = (
(
−1 0
0 −1
) ≺ 0.
) 0 1 , which has positive eigenvalues, 𝜆(𝐴) = 1, 3. By using SDP-M to −3 4 solve the LMI optimization (9), we obtain that the primal objective function value and the dual objective function value are 𝑜𝑏𝑗𝑉 𝑎𝑙𝑃 𝑟𝑖𝑚𝑎𝑙 = 8.7748519224301846𝑒 − 01 On the other hand, we choose 𝐴 =
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79
and 𝑜𝑏𝑗𝑉 𝑎𝑙𝐷𝑢𝑎𝑙 = 8.7748517260429215𝑒 − 01, respectively. Since 𝑡 > 0, we conclude that the problem is not feasible.
6
Conclusions
In this paper, by using the separation theorems, theorem of the alternatives for Lyapunov inequality on 𝑃 is obtained. This result provide systematic and unified proofs of necessary and sufficient conditions for solvability of Lyapunov inequality on 𝑃 . As an application of the Theorem of the alternatives, we investigate the conditions for the existence of feasible solutions to Lyapunov inequality on 𝑃 and obtain a restatement of the celebrated Lyapunov stability theorem for linear time-invariant systems. A numerical example is given to show the effectiveness of the proposed method.
7
Acknowledgment
Research was supported by the Natural Science Foundation of China ( No.61004027)and the Nature Science Foundation of Universities (No.09KJD510002) from Jiangsu Province.
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