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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 407409, 11 pages
http://dx.doi.org/10.1155/2012/407409
Research Article

Asymptotic Stability of the Golden-Section Control Law for Multi-Input and Multi-Output Linear Systems

1Institute of Mathematics and Systems Science, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
2Beijing Institute of Control Engineering, China Academy of Space Technology, Beijing 100190, China
3School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China

Received 16 September 2011; Accepted 19 December 2011

Academic Editor: Mark A. Petersen

Copyright © 2012 Duo-Qing Sun and Zhu-Mei Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the problem of the asymptotic stability of the characteristic model-based golden-section control law for multi-input and multi-output linear systems. First, by choosing a set of polynomial matrices of the objective function of the generalized least-square control, we prove that the control law of the generalized least square can become the characteristic model-based golden-section control law. Then, based on both the stability result of the generalized least-square control system and the stability theory of matrix polynomial, the asymptotic stability of the closed loop system for the characteristic model under the control of the golden-section control law is proved for minimum phase system.

1. Introduction

Since the late 1950s, the first adaptive control system was presented by the Massachusetts Institute of Technology, the adaptive control has been attracted extensively. Fruitful results have been achieved in theory and application. However, the adaptive control in practice has not been widely used. The reason is that there are some problems of the existing adaptive control theory in practical engineering applications, such as the following: the transient response is very poor, the number of parameters need to be estimated is too many, the convergence of parameter estimation is difficult to be guaranteed, and parameters that need to be adjusted artificially are too many [1].

To solve the above problems, Wu et al. [1, 2] presented an integrated and practical all-coefficient adaptive control theory and method based on characteristic models, which has been gradually improved in the application course of more than 20 years. This theory and method provide a new approach for the modeling and control of the complex systems. It should be mentioned that the theory and method have already been applied successfully to more than 400 systems belonging to nine kinds of engineering plants in the field of astronautics and industry. In particular, the engineering key points of the method are creatively applied to the reentry adaptive control of a manned spaceship, which the accuracy of the parachute-opening point of the spaceship reaches the advanced level of the world.

The method given by Wu is simple in design, easy to adjust and with strong robustness, and in some ways solves the above-mentioned problems. This method includes the following three aspects: (1) all-coefficient adaptive control method; (2) golden-section adaptive control law; (3) characteristic model [1].

It is worth mentioning that the golden-section control law is a new control law, which can solve the problem that adaptive control cannot guarantee the stability of a closed-loop system during the transient process when the parameters have not converged to their “true value.” The so-called golden-section control law means that the golden section ratio (0.382/0.618) is used to controller designs, see [1, 2] or Section 3 of this paper in detail.

For a second-order single-input and single-output (SISO) invariant linear system, Xie et al. [3, 4] proved that the golden-section control law has strong robust stability. The sufficient conditions for the stability of the closed-loop system based on golden-section control law for SISO and 2-input-2-output invariant linear systems have been reported; see, Qi et al. [5], Sun and Wu [6, 7], and Meng et al. [8]. Recently, Sun [9] gave sufficient conditions for the stability of the golden-section control system for 3-input-3-output invariant linear systems, but these conditions are difficult to verify. Among these references, we notice that the closed-loop control properties based on the golden-section control law are given by using the stability results of the generalized least-square control system and Jury stability criteria, aiming at the characteristic model of a second-order continuous SISO invariant linear system.

In summary, for the MIMO system, the stability of golden-section control systems is still an open question.

In this paper, for the MIMO linear system, we investigate the stability of the characteristic model-based golden-section control law by using the stability results of the multivariable generalized least-square control system.

2. Preliminaries

2.1. Introduction to Generalized Least-Square Controller

Consider the following system described by the linear vector difference equation:𝐴𝑧1𝑌(𝑘)=𝑧𝑑𝐵𝑧1𝑧𝑈(𝑘)+𝐶1𝜉(𝑘),(2.1) where 𝑈(𝑘) and 𝑌(𝑘) are the 𝑛×1 input and output vectors, respectively, and 𝜉(𝑘) is the 𝑛×1 zero-mean white-noise vector with covariance matrix 𝐸(𝜉(𝑘)𝜉𝑇(𝑘))=𝑟𝜉, 𝑑 denotes the system time delay, and 𝐴,𝐵, and 𝐶 are polynomial matrices in backward shift operator 𝑧1 given by𝐴𝑧1=𝐼+𝐴1𝑧1++𝐴𝑛𝑎𝑧𝑛𝑎,𝐵𝑧1=𝐵0+𝐵1𝑧1++𝐵𝑛𝑏𝑧𝑛𝑏,𝐵0𝐶𝑧nonsingular,1=𝐼+𝐶1𝑧1++𝐶𝑛𝑐𝑧𝑛𝑐,(2.2)

here 𝐴𝑖, 𝐵𝑖, and 𝐶𝑖 are matrix coefficients. The cost function to be considered is described by𝑃𝑧𝐽=𝐸1𝑧𝑌(𝑘+𝑑)𝑅1𝑌𝑟(𝑘)2+𝑄𝑧1𝑈(𝑘)2,(2.3) where 𝑌𝑟(𝑘) is an 𝑛-vector defining the known reference signal. 𝑃,𝑅, and 𝑄 are 𝑛×𝑛 weighting polynomial matrices. The notation 𝑋2=𝑋T𝑋 has been used.

The optimal control law is [10]𝐻𝑧1𝑧𝑈(𝑘)=𝐸1𝑌𝑟𝐺𝑧(𝑘)1𝑌(𝑘),(2.4) where𝐻𝑧1=𝐹𝑧1𝐵𝑧1+𝐶𝑧1𝑄𝑧1,𝐸𝑧(2.5)1𝐶𝑧=1𝑅𝑧1𝑄𝑧,(2.6)1=𝑃0𝐵0T1𝑄(0)T𝑄𝑧1,𝐵0=𝐵(0),𝑃0𝐶𝑧=𝑃(0),(2.7)1𝑃𝑧1=𝐹𝑧1𝐴𝑧1+𝑧𝑑𝐺𝑧1𝐶𝑧,(2.8)1𝐹𝑧1=𝐹𝑧1𝐶𝑧1,(2.9) here the order of the polynomial matrix 𝐹 is equal to 𝑑1.

2.2. Two Important Lemmas

Let an (𝑚×𝑚) nonsingular real matrix polynomial of 𝑛th-order 𝑃(𝑧) be given by𝑃(𝑧)=𝑎𝑛𝑧𝑛+𝑎𝑛1𝑧𝑛1+𝑎𝑛2𝑧𝑛2++𝑎1𝑧+𝑎0,(2.10) where 𝑎𝑛,𝑎𝑛1,𝑎𝑛2,,𝑎1, and 𝑎0 are (𝑚×𝑚) real matrices, 𝑚1.

We can construct an (𝑚𝑛×𝑚𝑛) symmetric matrix 𝐶=[𝑐𝑖𝑗] by the Christoffel-Darboux formula as follows. 𝑐𝑖𝑗’s are defined as𝑐𝑖𝑗=𝑖𝑘=1𝑎T𝑛+𝑘𝑖𝑎𝑛+𝑘𝑗𝑎T𝑗𝑘𝑎𝑖𝑘𝑐,𝑖𝑗,𝑗𝑖=𝑐T𝑖𝑗,𝑖>𝑗(2.11) for 𝑖,𝑗=1,2,,𝑛.

For example, when 𝑛=2 [11],𝑎C=T2𝑎2𝑎T0𝑎0𝑎T2𝑎1𝑎T1𝑎0𝑎T1𝑎2𝑎T0𝑎1𝑎T2𝑎2𝑎T0𝑎0.(2.12)

Lemma 2.1 (see [11, Theorem  1]). If 𝐶=[𝑐𝑖𝑗] defined in (2.11) is positive definite, then all the roots of the determinant of the matrix polynomial (2.10) lie inside the unit circle.

It is well known that the following lemma holds.

Lemma 2.2. Let 𝐴 be an (𝑛×𝑛) real symmetric matrix and 𝐼 be an (𝑛×𝑛) identity matrix. Then there exists a sufficiently small positive number 𝜀 such that 𝐼+𝜀𝐴 becomes a positive definite matrix.

3. Problem Formulation

Suppose that a second-order multi-input and multi-output dynamic process can be expressed as𝑌(2)(𝑡)+𝐴1𝑌(1)(𝑡)+𝐴0𝑌(𝑡)=𝐵1𝑈(1)(𝑡)+𝐵0𝑈(𝑡),(3.1) where 𝑈(𝑘) and 𝑌(𝑘) are the 𝑛×1 input and output vectors, respectively, and 𝐴0, 𝐴1, 𝐵0, and 𝐵1 are polynomial matrices. By using forward difference method, the corresponding difference equation can be given as𝑌(𝑘+1)=𝐴1𝑌(𝑘)+𝐴2𝑌(𝑘1)+𝐵0𝑈(𝑘)+𝐵1𝑈(𝑘1)+𝑒(𝑘+1),(3.2) where𝐴1=2𝐼𝐴1Δ𝑡,𝐴2=𝐼+𝐴1Δ𝑡𝐴0(Δ𝑡)2,𝐵0=𝐵1Δ𝑡,𝐵1=𝐵1Δ𝑡+𝐵0Δ𝑡2,(3.3) and 𝑒(𝑘) is the modeling error vector.

For a linear multi-input and multi-output constant high-order plant, if sampling period Δ𝑡 is sufficiently small, when the control requirement is position keeping or tracking, we can prove that its characteristic model can be also expressed with (3.2).

It is easily to be seen that (3.2) can be written as (2.1), that is,𝐴𝑧1𝑌(𝑘)=𝑧1𝐵𝑧1𝑧𝑈(𝑘)+𝐶1𝑒(𝑘),(3.4) where𝐴𝑧1=𝐼𝐴1𝑧1𝐴2𝑧2,𝐵𝑧1=𝐵0+𝐵1𝑧1,𝐶𝑧1=𝐼.(3.5)

The characteristic model-based golden-section control law is designed as follows:𝑈(𝑘)=(1+𝜆)1𝐵01𝑌𝑟(𝑘)𝐵1𝑈(𝑘1)𝑙1𝐴1𝑌(𝑘)𝑙2𝐴2𝑌(𝑘1),(3.6) where 𝜆0 is a constant; 𝑙1=0.382 and 𝑙2=0.618 are golden-section coefficients.

4. Main Result and Proof

In this section, by selecting a set of polynomial matrices in the objective function (2.3), for the characteristic model (3.2), we find the generalized least-square control law under the objective function. By comparison, we can see that this control law is just the same law as the golden-section control law based on characteristic model. Finally, based on the stability result of generalized least-square control law, we will give the stability result of the closed-loop of the golden-section control law.

For the objective function (2.3), we choose that𝑃𝑧1=𝐼𝑙2𝐴1𝑧1𝑙1𝐴2𝑧2𝑧,𝑅1=𝐼,𝑄𝑧1=𝜆𝐵0,𝐹𝑧1=𝐼.(4.1)

Note that 𝑑=1, and hence 𝐹(𝑧1)=𝐼. By 𝐶(𝑧1)=𝐼, 𝐹(𝑧1)=𝐼, and (2.9), we get 𝐶(𝑧1)=𝐼. Using 𝐶(𝑧1)=𝐼, 𝑅(𝑧1)=𝐼, and (2.6), 𝐸(𝑧1)=𝐼.

From 𝑃0=𝐼, 𝐵(0)=𝐵0, 𝑄(𝑧1)=𝜆𝐵0, and (2.7), it follows that𝑄𝑧1=𝑃0𝐵0T1𝑄(0)T𝑄𝑧1=𝐵T01𝜆𝐵0T𝜆𝐵0=𝐵T01𝐵T0𝜆𝐵0=𝜆𝐵0.(4.2)

By using Diophantine equation (2.8), we obtain that𝐼𝑙2𝐴1𝑧1𝑙1𝐴2𝑧2=𝐼𝐴1𝑧1𝐴2𝑧2+𝑧1𝐺𝑧1.(4.3)

Therefore,𝐺𝑧1=𝑙1𝐴1+𝑙2𝐴2𝑧1.(4.4)

Using (2.5), we have𝐻𝑧1=𝐹𝑧1𝐵𝑧1+𝐶𝑧1𝑄𝑧1=𝐵0+𝐵1𝑧1+𝐼𝜆𝐵0=𝐵0+𝐵1𝑧1+𝜆𝐵0=(1+𝜆)𝐵0+𝐵1𝑧1.(4.5)

According to (2.4), it follows that(1+𝜆)𝐵0+𝐵1𝑧1𝑈(𝑘)=𝐼𝑌𝑟𝑙(𝑘)1𝐴1+𝑙2𝐴2𝑧1𝑌(𝑘).(4.6)

That is,𝑈(𝑘)=(1+𝜆)1𝐵01𝑌𝑟(𝑘)𝐵1𝑈(𝑘1)𝑙1𝐴1𝑌(𝑘)𝑙2𝐴2𝑌(𝑘1).(4.7)

The above control law obtained by generalized least-square control law is exactly the characteristic model-based golden-section control law designed in (3.6). Hence, the stability of the closed-loop system based on the characteristic model-based golden-section control law is determined by the distribution of zero points on 𝑧-plane of the following equation:𝑧det𝐻1𝐴𝑧det1+𝑧𝑑𝐵𝑧1𝐻1𝑧1𝐺𝑧1=0.(4.8)

We note that𝐻𝑧1=𝐹𝑧1𝐵𝑧1+𝐶𝑧1𝑄𝑧1𝑧=𝐼𝐵1+𝐼𝜆𝐵0𝑧=𝐵1+𝜆𝐵0.(4.9)

Now, taking 𝜆=0, we have 𝐻(𝑧1)=𝐵(𝑧1), and, furthermore,𝐴𝑧1+𝑧𝑑𝐵𝑧1𝐻1𝑧1𝐺𝑧1𝑧=𝐴1𝑧+𝐵1𝐵𝑧11𝐶𝑧1𝑃𝑧1𝐹𝑧1𝐴𝑧1𝑧=𝐴1+𝑧𝐼𝑃1𝑧𝐼𝐴1𝑧=𝐴1+𝑃𝑧1𝑧𝐴1𝑧=𝑃1.(4.10)

Then, when 𝐵(𝑧1) is stable, the stability of the closed-loop system formed by the characteristic model-based golden-section control law is determined by the stability of 𝑃(𝑧1).

Theorem 4.1. Assume that (3.2) is a minimum phase system and 𝜆=0. Then, the closed-loop system involving the characteristic model-based golden-section control law (3.6) is asymptotic stable.

Remark 4.2. Since the corresponding relationship between the zero positions of continuous controlled objects and that of the discrete-time systems is complex, here the minimum phase system means that the zero points of the characteristic model (3.2) lie inside the unit circle.

Proof. First, we notice that 𝑃(𝑧1)=𝐼𝑙2𝐴1𝑧1𝑙1𝐴2𝑧2, and take 𝑎2=𝐼,𝑎1=𝑙2𝐴1,𝑎0=𝑙1𝐴2.(4.11) By (2.12), we have 𝐶=𝐼𝑙21𝐴T2𝐴2𝑙2𝐴1𝑙1𝑙2𝐴T1𝐴2𝑙2𝐴T1𝑙1𝑙2𝐴T2𝐴1𝐼𝑙21𝐴T2𝐴2.(4.12)
Now, we show that 𝐶is positive definite when Δ𝑡 is sufficiently small.
It is easy to see that 𝑁𝐶=𝐵𝑁T+𝛼𝐼𝛼𝐼𝑙21𝐴T2𝐴2𝑂𝑂𝛼𝐼𝑙21𝐴T2𝐴2,(4.13) where 𝐵=𝛼𝐼, 𝑁=𝑙2𝐴1𝑙1𝑙2𝐴T1𝐴2, 0.764=2𝑙1<𝛼<1𝑙21=0.8541, and 𝛼=1𝛼.
To show 𝐶 be positive definite, we will prove that 𝑁𝐵𝑁T𝛼𝐼,(4.14) and 𝛼𝐼𝑙21𝐴T2𝐴2 are all positive definite when Δ𝑡 is sufficiently small as follows.
First, we notice that 𝐼𝑂𝑁T𝐵1𝐼𝐵𝑁𝑁T𝐵𝛼𝐼𝐼1T𝑁=𝐵𝑂𝐼𝐵𝐵1T𝑁+𝑁𝑂𝛼𝐼𝑁T𝐵1𝑁=𝐵𝑂𝑂𝛼𝐼𝑁T𝐵1𝑁=𝛼𝐼𝑂𝑂𝛼𝐼𝑁T𝐵1𝑁.(4.15) We claim that 𝛼𝐼𝑁T𝐵1𝑁 is positive definite when Δ𝑡 is sufficiently small. In fact, since 𝑁T𝐵1𝑁=𝑙2𝐴T1𝑙1𝑙2𝐴T2𝐴1𝐵1𝑙2𝐴1𝑙1𝑙2𝐴T1𝐴2=𝑙2𝐴T1+𝑙1𝑙2𝐴T2𝐴1𝐵1𝑙2𝐴1+𝑙1𝑙2𝐴T1𝐴2=1𝛼𝑙2𝐴T1+𝑙1𝑙2𝐴T2𝐴1𝑙2𝐴1+𝑙1𝑙2𝐴T1𝐴2,𝑙2𝐴T1+𝑙1𝑙2𝐴T2𝐴1=𝑙22𝐼𝐴T1Δ𝑡+𝑙1𝑙2𝐼+𝐴T1Δ𝑡𝐴T0(Δ𝑡)22𝐼𝐴1Δ𝑡=2𝑙2𝐼𝑙2𝐴T1Δ𝑡2𝑙1𝑙2𝐼+𝑙1𝑙2𝐴1Δ𝑡+𝑙1𝑙2𝐴T1Δ𝑡𝐴T0(Δ𝑡)22𝐼𝐴1Δ𝑡=2𝑙21𝑙1𝐼𝑙2𝐴T1Δ𝑡+𝑙1𝑙2𝐴1Δ𝑡+𝑙1𝑙2𝐴T1Δ𝑡𝐴T0(Δ𝑡)22𝐼𝐴1Δ𝑡=2𝑙22𝐼𝑙2𝐴T1Δ𝑡+𝑙1𝑙2𝐴1Δ𝑡+𝑙1𝑙2𝐴T1Δ𝑡𝐴T0(Δ𝑡)22𝐼𝐴1Δ𝑡=2𝑙22𝐼𝑙2𝐴T1Δ𝑡+𝑙1𝑙2𝐴1𝑙Δ𝑡+1𝑙2𝐴T1Δ𝑡𝑙1𝑙2𝐴T0(Δ𝑡)22𝐼𝐴1Δ𝑡=2𝑙22𝐼𝑙2𝐴T1Δ𝑡+𝑙1𝑙2𝐴1Δ𝑡+2𝑙1𝑙2𝐴T1Δ𝑡2𝑙1𝑙2𝐴T0(Δ𝑡)2𝑙1𝑙2𝐴T1𝐴1(Δ𝑡)2+𝑙1𝑙2𝐴T0𝐴1(Δ𝑡)3=2𝑙22𝐼+𝑙2+2𝑙1𝑙2𝐴T1Δ𝑡+𝑙1𝑙2𝐴1Δ𝑡2𝑙1𝑙2𝐴T0+𝑙1𝑙2𝐴T1𝐴1(Δ𝑡)2+𝑙1𝑙2𝐴T0𝐴1(Δ𝑡)3,(4.16) we get 𝑁T𝐵11𝑁=𝛼4𝑙42𝐼+2𝑙22𝑙2+3𝑙1𝑙2𝐴1+𝐴T1𝑙Δ𝑡++1𝑙22𝐴T0𝐴1𝐴T1𝐴0(Δ𝑡)6,𝛼𝐼𝑁T𝐵11𝑁=𝛼𝐼𝛼4𝑙42𝐼+2𝑙22𝑙2+3𝑙1𝑙2𝐴1+𝐴T1𝑙Δ𝑡++1𝑙22𝐴T0𝐴1𝐴T1𝐴0(Δ𝑡)6=𝛼4𝑙42𝛼1𝐼𝛼2𝑙22𝑙2+3𝑙1𝑙2𝐴1+𝐴T1𝑙Δ𝑡++1𝑙22𝐴T0𝐴1𝐴T1𝐴0(Δ𝑡)6.(4.17) Using 𝛼>2𝑙1=2𝑙22 yields 𝛼4𝑙42/𝛼=(𝛼24𝑙42)/𝛼>0. Thus, we have 𝛼𝐼𝑁T𝐵1𝑁=𝛼24𝑙42𝛼𝐼+Δ𝑡𝛼24𝑙422𝑙22𝑙2+3𝑙1𝑙2𝐴1+𝐴T1𝑙1𝑙22𝐴T0𝐴1𝐴T1𝐴0(Δ𝑡)5.(4.18) By Lemma 2.2, it is follows that 𝛼𝐼𝑁T𝐵1𝑁 is positive definite when Δ𝑡 is sufficiently small. Therefore, there exists nonsingular matrix 𝐷 such that 𝐷T(𝛼𝐼𝑁T𝐵1𝑁)𝐷=𝐼. Furthermore, we have 1𝛼𝐼𝑂𝑂𝐷T𝛼𝐼𝑂𝑂𝛼𝐼𝑁T𝐵1𝑁1𝛼=𝛼𝐼𝑂𝑂𝐷𝛼𝐼𝑂𝑂𝐷𝑇𝛼𝐼𝑁T𝐵1𝑁1𝛼=.𝐼𝑂𝑂𝐷𝐼𝑂𝑂𝐼(4.19) That is, by the congruent transformation twice, 𝑁𝐵𝑁T𝛼𝐼(4.20) can be transformed into the identity matrix. From this, it is positive definite when Δ𝑡 is sufficiently small.
We also claim that 𝛼𝐼𝑙12𝐴T2𝐴2 is positive definite when Δ𝑡 is sufficiently small. In fact, it can be seen that 𝛼𝐼𝑙21𝐴T2𝐴2=𝛼𝐼𝑙21𝐼+𝐴T1Δ𝑡𝐴T0(Δ𝑡)2𝐼+𝐴1Δ𝑡𝐴0(Δ𝑡)2=𝛼𝑙21𝐼𝑙21𝐴1Δ𝑡+𝐴0(Δ𝑡)2𝐴T1Δ𝑡+𝐴T1𝐴1(Δ𝑡)2𝐴T1𝐴0(Δ𝑡)3+𝐴T0(Δ𝑡)2𝐴T0𝐴1(Δ𝑡)3+𝐴T0𝐴0(Δ𝑡)4.(4.21) Using 𝛼<1𝑙21 and 𝛼=1𝛼, we have 𝛼𝑙21>0. Thus, 𝛼𝐼𝑙21𝐴T2𝐴2=𝛼𝑙21𝑙𝐼21Δ𝑡𝛼𝑙21𝐴1𝐴T1+𝐴0Δ𝑡+𝐴T1𝐴1Δ𝑡𝐴T1𝐴0(Δ𝑡)2+𝐴T0Δ𝑡𝐴T0𝐴1(Δ𝑡)2+𝐴T0𝐴0(Δ𝑡)3.(4.22) By Lemma 2.2, 𝛼𝐼𝑙21𝐴T2𝐴2 is also positive definite when Δ𝑡 is sufficiently small.
Thus, it follows from (4.13) that 𝐶 is positive definite when Δ𝑡 is sufficiently small. By Lemma 2.1, 𝑃(𝑧1) is a stable matrix polynomial. Hence, the asymptotic stability of the closed-loop system follows immediately. This completes the proof.

Remark 4.3. It is well known that an eigenvalue of a matrix is a continuous function with respect to elements of the matrix. From this, (4.18), and (4.22), we can also see that 𝛼𝐼𝑁T𝐵1𝑁 and 𝛼𝐼𝑙21𝐴T2𝐴2 are positive definite when Δ𝑡 is sufficiently small.

Remark 4.4. By using a constructive proof of Lemma 2.2, we can determine minimum sampling period Δ𝑡 so that 𝛼𝐼𝑁T𝐵1𝑁 and 𝛼𝐼𝑙21𝐴T2𝐴2 are all positive definite.

Conjecture 4.5. If (3.2) is a nonminimum phase system, we may investigate the stability of the closed-loop system involving the characteristic model-based golden-section control law (3.6) by using the approach of the root locus in linear multivariable systems.

Acknowledgment

This work was supported by the National Nature Science Foundation of China under Grant 60874055, the Key Subject Construction Project, and the Funds for Creative Research Groups of Hebei Normal University of Science and Technology (no. CXTD2010-05).

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