General Solution and Observability of Singular Differential Systems with Delay

Abstract

We study singular differential systems with delay. A general description for the solutions of singular differential systems with delay is given and a necessary and sufficient condition for exact observability of singular differential systems with delay is derived.

1. Introduction

In many practical systems, such as economic systems, power systems, biological systems, ecosystems, and so on, due to the transmission of information need time span, usually there are the phenomena of delay. Up to now, many authors have paid much attention to the study for delay, and many excellent results have been given [1–5]. We notice that in some systems we must consider their character of dynamic state and static state synchronously, these result in many good consequences for singular systems that have been obtained recently [6–9].

We must emphasize that there are a lot of systems in which there exist the phenomena of delay and singular simultaneously, we call such systems as the singular differential systems with delay. These systems have many special characters. If we want to describe them more exactly, more accurately design them, and more effectively control them, we must pay tremendous endeavor to investigate them, but that is obviously very difficult. In [3, 10–19], authors have discussed such systems, and got some consequences. But in the study for such systems, there are still many problems to be considered.

Generally, the singular differential control systems with delay can be written as 𝐸𝑥̇(𝑡)(𝑥(𝑡)()(𝑡))𝑥(𝑡)(𝑡)=𝐹,𝑥𝑡−𝜏,𝑢,𝑡,𝑡≥0,=𝜑,−𝜏≤𝑡≤0, (1) where𝑥(𝑡)∈𝑅^𝑛 is a state vector, 𝑢(𝑡)∈𝑅^𝑙 is a control vector, 𝐸∈𝑅^𝑛×𝑛 is a singular matrix, 𝜑∈𝐶=𝐶([−𝜏,0],𝑅^𝑛) is an admissible initial state functional, 𝐶=𝐶([−𝜏,0],𝑅^𝑛) is the Banach space of continuous functions mapping the interval [−𝜏,0] into 𝑅^𝑛 with the topology of uniform convergence, the norm of an element 𝜑 in 𝐶 can be designated as |𝜑|=sup_{𝜃∈[−𝜏,0]}|𝜑(𝜃)|, and 𝐹∈𝐶¹(𝑅^𝑛×𝑅^𝑛×𝑅^𝑙×𝑅,𝑅^𝑛).

The autonomous linear form of (1) can be written as 𝐸𝑥̇(𝑡)(𝑡)()(𝑡)(𝑡)𝑥(𝑡)(𝑡)=𝐴𝑥+𝐵𝑥𝑡−𝜏+𝐷𝑢+𝑓,𝑡≥0,=𝜑,−𝜏≤𝑡≤0, (2) where 𝑥(𝑡)∈𝑅^𝑛 is a state vector [6, 7], 𝑢(𝑡)∈𝑅^𝑙 is a control vector [6, 7], 𝐸∈𝑅^𝑛×𝑛 is a singular matrix, 𝐴,𝐵∈𝑅^𝑛×𝑛, and𝐷∈𝑅^𝑛×𝑙 are constant matrix, 𝑓(𝑡)∈𝑅^𝑛 is a sufficiently smooth function, and 𝜑(𝑡) is the initial state function.

Usually, the output functions of (2) can be written as: 𝑦(𝑡)=𝐶𝑥(𝑡)+𝐻𝑢(𝑡), (3) where 𝑦(𝑡)∈𝑅^𝑚 is an output vector [6, 7], 𝑥(𝑡)∈𝑅^𝑛 is a state vector, 𝑢(𝑡)∈𝑅^𝑙 is a control vector, and 𝐶∈𝑅^𝑚×𝑛 and 𝐻∈𝑅^𝑚×𝑙 are constant matrices.

Definition 1. Singular differential control systems with delay (2) and (3) are observable on [𝑡₁,𝑡₂] if the initial function 𝜑(𝑡) in [−𝜏,0] can be uniquely determined from the knowledge of the control 𝑢(⋅) and observation 𝑦(⋅) over [𝑡₁,𝑡₂].

Remark 2. The observability concept in Definition 1 comes from [4], which is sometimes called exact observability.

Recently, many authors have discussed the observability of singular differential control systems without delay [6–8], but for such systems the discussion is much easier than that for singular differential control systems with delay. This is because singular differential control systems without delay can be changed as two autocephalous systems [7, 8] as follows: 𝑧̇₁(𝑡)=𝐴₁𝑧₁(𝑡)+𝐺₁𝑢(𝑡)+𝑓₁(𝑡),𝑁𝑧̇₂(𝑡)=𝑧₂(𝑡)+𝐵𝑢(𝑡)+𝑓₂(𝑡), (4) and they can be discussed, respectively. But for singular differential systems with delay (2), the canonical forms are 𝑥̇₁(𝑡)=𝐴₁𝑥₁(𝑡)+𝐵₁₁𝑥₁(𝑡−𝜏)+𝐵₁₂𝑥₂(𝑡−𝜏)+𝐶₁𝑢(𝑡)+𝑓₁(𝑡)𝑁𝑥̇,𝑡≥0,₂(𝑡)=𝑥₂(𝑡)+𝐵₂₁𝑥₁(𝑡−𝜏)+𝐵₂₂𝑥₂(𝑡−𝜏)+𝐶₂𝑢(𝑡)+𝑓₂(𝑡)𝑥,𝑡≥0,₁(𝑡)=𝜑₁(𝑡)𝑥,−𝜏≤𝑡≤0,₂(𝑡)=𝜑₂(𝑡),−𝜏≤𝑡≤0, (5) which generally cannot be discussed, respectively.

In papers [10, 16], we have had some results about the controllability of singular control systems with control-delay and the controllability of singular control systems with state-delay, respectively. As the corresponding concept of controllability, observability is obviously and important concept. As we know, up to now, there exists hardly any discussion on the observability of singular differential systems with delay.

In this paper, we study singular differential systems with delay. Firstly, for the singular differential systems with delay, the general solutions will be given. Then, for the observability of singular differential systems with delay, we will give criterion.

2. General Solutions of Singular Differential Systems with Delay

In the study of the observability of singular differential systems with delay, we will use some concepts and conclusions. Now, we give some of them, and specially give the general solutions of singular differential systems with delay (2).

Definition 3. Let 𝐸 be a square matrix, if there exists a matrix 𝐸^𝑑 satisfy 1⁰𝐸𝐸^𝑑=𝐸^𝑑𝐸,2⁰𝐸^𝑑𝐸𝐸^𝑑=𝐸^𝑑,3⁰(𝐼−𝐸^𝑑𝐸)𝐸^𝑙=0, we call 𝐸^𝑑 the Drazin inverse matrix of matrix 𝐸, simply 𝐷-inverse matrix. Where 𝑙 is the index of matrix 𝐸; it is the smallest nonnegative integer which makes rank(𝐸^𝑙+1)=rank(𝐸^𝑙) be true.

From [3] or [7], we have the following.

Lemma 4. For any square matrix 𝐸, its Drazin inverse matrix 𝐸^𝑑 is existent and unique. If the Jordan normalized form of 𝐸 is 𝐸=𝑇^𝐽10_0𝐽0𝑇⁻¹, then 𝐸^𝑑=𝑇^𝐽1−10₀₀𝑇⁻¹. Here, 𝐽₀ is a nilpotent matrix, 𝐽₁ and 𝑇 are invertible matrices.
Consider the following systems 𝐸𝑥̇(𝑡)(𝑡)()𝑥(𝑡)(𝑡)=𝐴𝑥+𝐵𝑥𝑡−𝜏,𝑡≥0,=𝜑,−𝜏≤𝑡≤0, (6) and systems: 𝐸𝑥̇(𝑡)(𝑡)()(𝑡)(𝑡)𝑥(𝑡)=𝐴𝑥+𝐵𝑥𝑡−𝜏+𝐷𝑢+𝑓,𝑡≥0,≡0,−𝜏≤𝑡≤0. (7) It is not difficult to prove the following.

Lemma 5. If𝑥(𝑡) and 𝑥(𝑡) are, respectively, the solution to (6) and (7), then 𝑥(𝑡)=𝑥(𝑡)+𝑥(𝑡) is the solution to (2).

Definition 6. Let 𝑋(𝑡)∈𝑅^𝑛×𝑛, 𝑋(𝑡) is called as the first class foundation solution of singular differential systems with delay, if it satisfies matrix equations the following: 𝐸𝑋̇(𝑡)(𝑡)()𝑋(𝑡)==𝐴𝑋+𝐵𝑋𝑡−𝜏,𝑡≥0,𝐸𝐸^𝑑,𝑡=0,0,−𝜏≤𝑡<0. (8)

Definition 7. Let 𝑌(𝑡)∈𝑅^𝑛×𝑛, 𝑌(𝑡) is called as the second class foundation solution of singular differential systems with delay, if it satisfies matrix equations the following: 𝐸𝑌̇(𝑡)=𝐴𝑌(𝑡)+𝐵𝑌(𝑡−𝜏)+𝐼−𝐸𝐸^𝑑𝛿(𝑡)𝑌(𝑡)=,𝑡≥0,𝐼−𝐸𝐸^𝑑,𝑡=0,0,−𝜏≤𝑡<0, (9) where 𝛿(𝑡) is a delta function or impulse function (see [3] or [7]).

Lemma 8. For delta function 𝛿(𝑡), one has ^𝑡₀𝛿(𝑡−𝑠)𝑓(𝑠)𝑑𝑠=𝑓(𝑡). (10)
This Lemma is well known. The proof will be omitted.

Lemma 9. For any square matrix 𝐸, one has: (𝐸+𝐼)𝐼−𝐸𝐸^𝑑𝐼+𝐸𝐼−𝐸𝐸^𝑑−1=𝐼−𝐸𝐸^𝑑. (11)

Proof . Let 𝐼 be an identity matrix with appropriate dimension, and 𝐽𝐸=𝑇¹00𝐽₀𝑇⁻¹, (12) from Lemma 4, we have 𝐸^𝑑𝐽=𝑇¹⁻¹0𝑇00⁻¹. (13)
For 𝐽₀ is a nilpotent matrix, 𝐽₀+𝐼 is invertible. Consider the following 𝐼−𝐸𝐸^𝑑=𝑇𝑇𝐼00𝐼⁻¹𝐽−𝑇¹00𝐽₀𝑇⁻¹𝑇𝐽₁⁻¹0𝑇00⁻¹=𝑇𝑇𝐼00𝐼⁻¹−𝑇𝑇𝐼000⁻¹=𝑇𝑇000𝐼⁻¹,(𝐸+𝐼)𝐼−𝐸𝐸^𝑑𝐽=𝑇¹+𝐼00𝐽₀𝑇+𝐼⁻¹𝑇𝑇000𝐼⁻¹=𝑇000𝐽₀𝑇+𝐼⁻¹,𝐼+𝐸𝐼−𝐸𝐸^𝑑=𝑇𝑇𝐼00𝐼⁻¹𝐽+𝑇¹00𝐽₀𝑇⁻¹𝑇𝑇000𝐼⁻¹=𝑇𝐼00𝐽₀𝑇+𝐼⁻¹,𝐼−𝐸𝐸^𝑑=𝑇𝑇000𝐼⁻¹=𝑇000𝐽₀𝑇+𝐼⁻¹𝑇𝐼00𝐽₀+𝐼⁻¹𝑇⁻¹=(𝐸+𝐼)𝐼−𝐸𝐸^𝑑𝐼+𝐸𝐼−𝐸𝐸^𝑑−1. (14) That means (11) is true.

Definition 10. If det(𝜆𝐸−𝐴)≢0, we call matrix couple (𝐸,𝐴) regular. If (𝐴,𝐸) is regular, we call system (2) regular.

Remark 11. Using the method of [3], we can prove that if (𝐸,𝐴) is regular, systems (2) (for any consistent initial state function 𝜑(𝑡)), (8), and (9) are solvable.

Theorem 12. Suppose that matrix couple (𝐴,𝐸) is regular, 𝑥(𝑡) is the solution of (6); then when 𝑡≥𝜏, 𝑥(𝑡)=𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+⁰_−𝜏𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)𝑑𝜃, (15) when 0≤𝑡≤𝜏, 𝑥(𝑡)=𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+^𝑡−𝜏_−𝜏𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)𝑑𝜃, (16) where 𝑋(𝑡) is the first class of foundation solution of singular differential systems with delay, 𝑌(𝑡) is the second class of foundation solution of singular differential systems with delay.
To prove Theorem 12, we can partition homogeneous singular differential systems with delay (6) into two classes of systems as follows: 𝐸𝑥̇(𝑡)(𝑡)()𝐸𝑥̇(𝑡)(𝑡)=𝐴𝑥+𝐵𝑥𝑡−𝜏,𝑡≥𝜏,=𝐴𝑥+𝐸𝐸^𝑑()𝑥(𝑡)𝐵𝜑𝑡−𝜏,0≤𝑡≤𝜏,=𝐸𝐸^𝑑𝜑(𝑡)𝐸𝑥̇(𝑡)(𝑡)()𝐸𝑥̇(𝑡)(𝑡)+,−𝜏≤𝑡≤0,=𝐴𝑥+𝐵𝑥𝑡−𝜏,𝑡≥𝜏,=𝐴𝑥𝐼−𝐸𝐸^𝑑()𝑥(𝑡)=𝐵𝜑𝑡−𝜏,0≤𝑡≤𝜏,𝐼−𝐸𝐸^𝑑𝜑(𝑡),−𝜏≤𝑡≤0. (17)(18)
We can easily prove the following.

Lemma 13. If 𝑥₁(𝑡), and 𝑥₂(𝑡)are respectively the solution of (17) and (18), then 𝑥(𝑡)=𝑥₁(𝑡)+𝑥₂(𝑡) is the solution of (6).

Lemma 14. Suppose that matrix couple (𝐴,𝐸) is regular, then the solution of (17) can be written as 𝑥₁(𝑡)=𝑋(𝑡)𝜑(0)+∫⁰_−𝜏𝑋(𝑡−𝜃−𝜏)𝐸^𝑑(𝜃)𝑋(𝑡)𝜑(0)𝐵𝜑𝑑𝜃,𝑡≥𝜏,+∫^𝑡₀𝑋(𝑡−𝜃)𝐸^𝑑𝐵𝜑(𝜃−𝜏)𝑑𝜃,0≤𝑡≤𝜏, (19) where 𝑋(𝑡) is the first class of foundation solution of singular differential systems with delay.

Proof . Since 𝑋(𝑡) is the first class of foundation solution of singular differential systems with delay, 𝑋(𝑡) satisfies matrix equations (8). Take Laplace transform for (8), we have ^∞₀𝑒^−𝜆𝑡𝐸𝑋̇(𝑡)𝑑𝑡=𝐸^∞₀𝑒^−𝜆𝑡𝑋̇(𝑡)𝑑𝑡=𝐸𝜆^∞₀𝑒^−𝜆𝑡𝑋(𝑡)(0)(𝑋(𝑡𝑑𝑡−𝐸𝑋=𝜆𝐸𝐿))−𝐸𝐸𝐸^𝑑,_∞0𝑒^−𝜆𝑡(𝐴𝑋(𝑡)+𝐵𝑋(𝑡−𝜏))𝑑𝑡=𝐴^∞₀𝑒^−𝜆𝑡𝑋(𝑡)𝑑𝑡+𝐵^∞₀𝑒^−𝜆𝑡𝑋()(𝑋(𝑡𝑡−𝜏𝑑𝑡=𝐴𝐿))+𝐵^∞_−𝜏𝑒^{−𝜆𝑡−𝜆𝜏}𝑋(𝑡)(𝑋(𝑡𝑑𝑡=𝐴𝐿))+𝐵𝑒^{−𝜆𝜏∞}₀𝑒^−𝜆𝑡𝑋(𝑡)=𝑑𝑡𝐴+𝐵𝑒^−𝜆𝜏𝐿(𝑋(𝑡)). (20)
That is, 𝜆𝐸𝐿(𝑋(𝑡))−𝐸𝐸𝐸^𝑑=𝐴+𝐵𝑒^−𝜆𝜏𝐿(𝑋(𝑡)). (21)
Because (𝐸,𝐴)is regular, for real(𝜆)large enough, 𝜆𝐸−𝐴−𝐵𝑒^−𝜆𝜏 is invertible, so we have 𝐿(𝑋(𝑡))=𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}𝐸𝐸𝐸^𝑑. (22)
If 𝑥₁(𝑡) is the solution of (17), we have ^∞₀𝑒^−𝜆𝑡𝐸𝑥̇₁(𝑡)𝑑𝑡=𝐸^∞₀𝑒^−𝜆𝑡𝑥̇₁(𝑡)𝑑𝑡=𝐸𝜆^∞₀𝑒^−𝜆𝑡𝑥₁(𝑡)𝑑𝑡−𝐸𝑥₁(0)=𝜆𝐸𝐿𝑥₁(𝑡)−𝐸𝐸𝐸^𝑑𝜑(0),_∞0𝑒^−𝜆𝑡𝐸𝑥̇₁(𝑡)𝑑𝑡=^𝜏₀𝑒^−𝜆𝑡𝐸𝑥̇₁(𝑡)+𝑑𝑡_∞𝜏𝑒^−𝜆𝑡𝐸𝑥̇₁(𝑡)𝑑𝑡=𝐴^𝜏₀𝑒^−𝜆𝑡𝑥₁(𝑡)+𝑑𝑡_𝜏0𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡+𝐴^∞_𝜏𝑒^−𝜆𝑡𝑥₁(𝑡)𝑑𝑡+𝐵^∞_𝜏𝑒^−𝜆𝑡𝑥₁(𝑡−𝜏)𝑑𝑡=𝐴^∞₀𝑒^−𝜆𝑡𝑥₁(𝑡)𝑑𝑡+𝐵𝑒^{−𝜆𝜏∞}₀𝑒^−𝜆𝑡𝑥₁(𝑡)+𝑑𝑡_𝜏0𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑()𝑥𝑡−𝜏𝑑𝑡=𝐴𝐿₁(𝑡)+𝐵𝑒^−𝜆𝜏𝐿𝑥₁(𝑡)+^𝜏₀𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (23) That is, 𝜆𝐸−𝐴−𝐵𝑒^−𝜆𝜏𝐿𝑥₁(𝑡)=𝐸𝐸𝐸^𝑑𝜑(0)+^𝜏₀𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (24) Since 𝐸^𝑑=𝐸𝐸^𝑑𝐸^𝑑, from (22), we can see that 𝐿𝑥₁(𝑡)=𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}𝐸𝐸𝐸^𝑑𝜑(0)+^𝜏₀𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}×𝐸𝐸𝐸^𝑑𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑()(𝑋(𝑡𝜑(0)+𝑡−𝜏𝑑𝑡=𝐿))_𝜏0𝐿(𝑋(𝑡))𝐸^𝑑𝐵𝑒^−𝜆𝑠𝜑(𝑠−𝜏)𝑑𝑠. (25)
Define an auxiliary function 𝜔∶[−𝜏,∞)→[0,1] as follows: 𝜔(𝑡)=0,𝑡≥0,1,𝑡<0, (26) we have ^𝜏₀𝐿(𝑋(𝑡))𝐸^𝑑𝐵𝑒^−𝜆𝑠𝜑()(𝑋(𝑡𝑠−𝜏𝑑𝑠=𝐿))^𝜏₀𝑒^−𝜆𝑡𝐸^𝑑()(𝑋(𝑡𝐵𝜑𝑡−𝜏𝑑𝑡=𝐿))^∞₀𝑒^−𝜆𝑡𝐸^𝑑()𝜔()(𝑋(𝑡𝐿𝐸𝐵𝜑𝑡−𝜏𝑡−𝜏𝑑𝑡=𝐿))^𝑑𝐵𝜑(𝑡−𝜏)𝜔(𝑡−𝜏)=𝐿^𝑡₀𝑋(𝑡−𝜃)𝐸^𝑑𝐵𝜑(𝜃−𝜏)𝜔(𝜃−𝜏)𝑑𝜃. (27) Then, we have 𝑥₁(𝑡)(𝑡)𝜑(0)+=𝑋_𝑡0𝑋(𝑡−𝜃)𝐸^𝑑𝐵𝜑(𝜃−𝜏)𝜔(𝜃−𝜏)𝑑𝜃. (28) When 𝑡≥𝜏, 𝑥₁(𝑡)(𝑡)𝜑(0)+=𝑋_0−𝜏𝑋(𝑡−𝜃−𝜏)𝐸^𝑑𝐵𝜑(𝜃)𝑑𝜃, (29) when 0≤𝑡≤𝜏, 𝑥₁(𝑡)(𝑡)𝜑(0)+=𝑋_𝑡0𝑋(𝑡−𝜃)𝐸^𝑑𝐵𝜑(𝜃−𝜏)𝑑𝜃. (30) The proof of Theorem 12 is completed.

Lemma 15. Suppose that matrix couple (𝐴,𝐸) is regular, then the solution of (18) can be written as 𝑥₂(𝑡)=𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝜑(0)+∫⁰_−𝜏𝑌(𝑡−𝜃−𝜏)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(𝜃)𝑌(𝑡)×𝐵𝜑𝑑𝜃,𝑡≥𝜏,𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝜑(0)+∫^𝑡₀𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝜑(𝜃−𝜏)𝑑𝜃,0≤𝑡≤𝜏, (31) where 𝑌(𝑡) is the second class of foundation solution of singular differential systems with delay.

Proof. Since 𝑌(𝑡) is the second class of foundation solution of singular differential systems with delay, 𝑌(𝑡) satisfies matrix equations (9). Take Laplace transform for (9), we have ^∞₀𝑒^−𝜆𝑡𝐸𝑌̇(𝑡)𝑑𝑡=𝐸^∞₀𝑒^−𝜆𝑡𝑌̇(𝑡)𝑑𝑡=𝐸𝜆^∞₀𝑒^−𝜆𝑡𝑌(𝑡)(0)(𝑋(𝑡𝑑𝑡−𝐸𝑌=𝜆𝐸𝐿))−𝐸𝐼−𝐸𝐸^𝑑,_∞0𝑒^−𝜆𝑡𝐴𝑌(𝑡)+𝐵𝑌(𝑡−𝜏)+𝐼−𝐸𝐸^𝑑𝛿(𝑡)𝑑𝑡=𝐴^∞₀𝑒^−𝜆𝑡𝑌(𝑡)𝑑𝑡+𝐵^∞₀𝑒^−𝜆𝑡𝑌(𝑡−𝜏)𝑑𝑡+𝐼−𝐸𝐸^𝑑=𝐴𝐿(𝑌(𝑡))+𝐵^∞_−𝜏𝑒^{−𝜆𝑡−𝜆𝜏}𝑌(𝑡)𝑑𝑡+𝐼−𝐸𝐸^𝑑=𝐴𝐿(𝑌(𝑡))+𝐵𝑒^{−𝜆𝜏∞}₀𝑒^−𝜆𝑡𝑌(𝑡)𝑑𝑡+𝐼−𝐸𝐸^𝑑=𝐴+𝐵𝑒^−𝜆𝜏𝐿(𝑌(𝑡))+𝐼−𝐸𝐸^𝑑. (32) That is 𝜆𝐸𝐿(𝑌(𝑡))−𝐸𝐼−𝐸𝐸^𝑑=𝐴+𝐵𝑒^−𝜆𝜏𝐿(𝑌(𝑡))+𝐼−𝐸𝐸^𝑑. (33) Because (𝐸,𝐴) is regular, for real(𝜆)large enough, 𝜆𝐸−𝐴−𝐵𝑒^−𝜆𝜏 is invertible, so, we have 𝐿(𝑌(𝑡))=𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}(𝐼+𝐸)𝐼−𝐸𝐸^𝑑. (34) If 𝑥₂(𝑡) is the solution of (18), we have ^∞₀𝑒^−𝜆𝑡𝐸𝑥̇₂(𝑡)𝑑𝑡=𝐸^∞₀𝑒^−𝜆𝑡𝑥̇₂(𝑡)𝑑𝑡=𝐸𝜆^∞₀𝑒^−𝜆𝑡𝑥₂(𝑡)𝑑𝑡−𝐸𝑥₂(0)=𝜆𝐸𝐿𝑥₂(𝑡)−𝐸𝐼−𝐸𝐸^𝑑𝜑(0),_∞0𝑒^−𝜆𝑡𝐸𝑥̇₂(𝑡)𝑑𝑡=^𝜏₀𝑒^−𝜆𝑡𝐸𝑥̇₂(𝑡)+𝑑𝑡_∞𝜏𝑒^−𝜆𝑡𝐸𝑥̇₂(𝑡)𝑑𝑡=𝐴^𝜏₀𝑒^−𝜆𝑡𝑥₂(𝑡)𝑑𝑡+^𝜏₀𝐼−𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡+𝐴^∞_𝜏𝑒^−𝜆𝑡𝑥₂(𝑡)𝑑𝑡+𝐵^∞_𝜏𝑒^−𝜆𝑡𝑥₂(𝑡−𝜏)𝑑𝑡=𝐴^∞₀𝑒^−𝜆𝑡𝑥₂(𝑡)𝑑𝑡+𝐵𝑒^{−𝜆𝜏∞}₀𝑒^−𝜆𝑡𝑥₂(𝑡)+𝑑𝑡_𝜏0𝐼−𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑()𝑥𝑡−𝜏𝑑𝑡=𝐴𝐿₂(𝑡)+𝐵𝑒^−𝜆𝜏𝐿𝑥₂(𝑡+_𝜏0)𝐼−𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (35) That is, 𝜆𝐸−𝐴−𝐵𝑒^−𝜆𝜏𝐿𝑥₂(𝑡)=𝐼−𝐸𝐸^𝑑(0)+𝐸𝜑_𝜏0𝐼−𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (36) Then, 𝐿𝑥₂(𝑡)=𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}𝐼−𝐸𝐸^𝑑(0)+𝐸𝜑_𝜏0𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}×𝐼−𝐸𝐸^𝑑𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (37) From Lemma 9, we have 𝐿𝑥₂(𝑡)=𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}(𝐸+𝐼)𝐼−𝐸𝐸^𝑑×𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(0)+𝐸𝜑_𝜏0𝜆𝐸−𝐴−𝐵𝑒^{−𝜆𝜏−1}()×𝐸+𝐼𝐼−𝐸𝐸^𝑑𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝑒^−𝜆𝑡𝜑(𝑡−𝜏)𝑑𝑡. (38) from (34) and (38), we know 𝐿𝑥₂(𝑡)=𝐿(𝑌(𝑡))𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(0)+𝐸𝜑_𝜏0𝐿(𝑌(𝑡))𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝑒^−𝜆𝜃𝜑(𝜃−𝜏)𝑑𝜃. (39) Use auxiliary function 𝜔∶[−𝜏,∞)→[0,1] as above, we have ^𝜏₀𝐿(𝑌(𝑡))𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝑒^−𝜆𝜃𝜑()(𝑌(𝑡𝜃−𝜏𝑑𝜃=𝐿))×^𝜏₀𝑒^−𝜆𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1()(𝑌(𝑡×𝐵𝜑𝜃−𝜏𝑑𝜃=𝐿))_∞0𝑒^−𝜆𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1()𝜔()(𝑌(𝑡𝐿×𝐵𝜑𝜃−𝜏𝜃−𝜏𝑑𝜃=𝐿))𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝜑(𝑡−𝜏)𝜔(𝑡−𝜏)=𝐿^𝑡₀𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝜑(𝜃−𝜏)𝜔(𝜃−𝜏)𝑑𝜃. (40) Then, we have 𝑥₂(𝑡)=𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(0)+𝐸𝜑_𝑡0𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝜑(𝜃−𝜏)𝜔(𝜃−𝜏)𝑑𝜃. (41) When 𝑡≥𝜏, 𝑥₂(𝑡)=𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(0)+𝐸𝜑_0−𝜏𝑌(𝑡−𝜃−𝜏)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐵𝜑(𝜃)𝑑𝜃, (42) when 0≤𝑡≤𝜏, 𝑥₂(𝑡)=𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1(0)+𝐸𝜑_𝑡0𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃−𝜏)𝑑𝜃. (43) This will complete the proof of Lemma 15.

From Lemma 13, 14 and 15 we can easily see that Theorem 12 is true.

Just as the proof of Theorem 12, we can prove that.

Theorem 16. Suppose that matrix couple (𝐴,𝐸) is regular, then the solution of (7) can be written as 𝑥(𝑡)=^𝑡₀𝑋(𝑡−𝜃)𝐸^𝑑(𝜃)(𝜃+𝐷𝑢+𝑓)𝑑𝜃_𝑡0𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃, (44) where 𝑋(𝑡) is the first class of foundation solution of singular differential systems with delay and 𝑌(𝑡) is the second class of foundation solution of singular differential systems with delay.

From Lemma 5 and Theorems 12 and 16, we have the following.

Theorem 17. Suppose that matrix couple (𝐴,𝐸) is regular, then 𝑥(𝑡) is the solution of (2), when 𝑡≥𝜏, 𝑥(𝑡)=𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+⁰_−𝜏𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)+𝑑𝜃_𝑡0𝑋(𝑡−𝜃)𝐸^𝑑()×+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃, (45) when 0≤𝑡≤𝜏, 𝑥(𝑡)=𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+^𝑡₀𝑋(𝑡−𝜃)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑()+𝜃−𝜏𝑑𝜃_𝑡0𝑋(𝑡−𝜃)𝐸^𝑑()×+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃, (46) where 𝑋(𝑡) is the first class of foundation solution of singular differential systems with delay and 𝑌(𝑡) is the second class of foundation solution of singular differential systems with delay.

Remark 18. As the illustration in Section 2, if (𝐸,𝐴) is regular, by the method of step by step, we can prove that both (8) and (9) are uniquely solvable. We can see that for any consistent initial function 𝜑(𝑡), Theorem 17 will be true. Such expression of general solution of (2) will be very useful in the study of the characteristic of (2), such as the observability, the controllability, the optimal control, and so on.

3. Observability of Singular Differential Systems with Delay

Now, let us discuss observability of singular differential systems with delay (2) and (3). We work over two cases: 𝑡≥𝜏 and 0≤𝑡<𝜏.

Case 1 (𝑡≥𝜏). From (3) and Theorem 17, we have 𝑦(𝑡)=𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+⁰_−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)+𝑑𝜃_𝑡0𝐶𝑋(𝑡−𝜃)𝐸^𝑑()×+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃+𝐻𝑢(𝑡). (47) That is, 𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+⁰_−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)(𝑡)−𝑑𝜃=𝑦^𝑡₀𝐶𝑋(𝑡−𝜃)𝐸^𝑑()×+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃−𝐻𝑢(𝑡). (48) From that, we can see that the problem of system observability can be addressed when 𝑢(⋅) and 𝑓(⋅) are identically zero. Thus, we can assume, with no loss of generality, that 𝑢(⋅)≡0,𝑓(⋅)≡0 and study the observability of systems (6) and𝑦(𝑡)=𝐶𝑥(𝑡). (49) Then, we have 𝑦(𝑡)=𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+⁰_−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)𝑑𝜃. (50) From Lemma 8, we have ^𝑙₀𝐶𝑋()()×𝑡−𝜃+𝜏+𝑌𝑡−𝜃+𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑()𝛿()𝑋()()×𝜃−𝜏𝑙−𝜃𝑑𝜃=𝐶𝑡−𝑙+𝜏+𝑌𝑡−𝑙+𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(𝑙−𝜏). (51) Let 𝑙=𝜏; then ^𝜏₀𝐶𝑋()()×𝑡−𝜃+𝜏+𝑌𝑡−𝜃+𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑()𝛿()𝑋(𝑡)(𝑡𝜃−𝜏𝜏−𝜃𝑑𝜃=𝐶+𝑌)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0), (52) that is, 𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)=⁰_−𝜏𝐶𝑋()()×𝑡−𝜃+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝛿(−𝜃)𝜑(𝜃)𝑑𝜃. (53) Taking it into (50), we have 𝑦(𝑡)=⁰_−𝜏𝐶𝑋()()×𝑡−𝜃+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝛿()𝜑(𝜃)+−𝜃𝑑𝜃_0−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)=𝑑𝜃_0−𝜏𝐶𝑋()()×𝑡−𝜃+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝛿()()𝐸−𝜃+𝑋𝑡−𝜃−𝜏^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)𝑑𝜃. (54) Let 𝑊()=𝑋()()×𝑡,𝜃𝑡−𝜃+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝛿()()𝐸−𝜃+𝑋𝑡−𝜃−𝜏^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵. (55) Then, 𝑦(𝑡)=⁰_−𝜏𝐶𝑊(𝑡,𝜃)𝜑(𝜃)𝑑𝜃. (56) So, we have ⁰_−𝜏𝑊^𝑇(𝑡,𝑠)𝐶^𝑇𝑦(𝑡)=𝑑𝑠_{0−𝜏0−𝜏}𝑊^𝑇(𝑡,𝑠)𝐶^𝑇()𝜑(𝜃)=𝐶𝑊𝑡,𝜃𝑑𝜃𝑑𝑠_{0−𝜏0−𝜏}𝑊^𝑇(𝑡,𝑠)𝐶^𝑇()𝜑(𝜃)=𝐶𝑊𝑡,𝜃𝑑𝑠𝑑𝜃_{0−𝜏0−𝜏}𝑊^𝑇(𝑡,𝜃)𝐶^𝑇()𝜑(𝑠)=𝐶𝑊𝑡,𝑠𝑑𝜃𝑑𝑠_{0−𝜏0−𝜏}𝑊^𝑇(𝑡,𝜃)𝐶^𝑇𝐶𝑊(𝑡,𝑠)𝑑𝜃𝜑(𝑠)𝑑𝑠. (57) That is, ⁰_−𝜏^𝑡1_𝜏𝑊^𝑇(𝑡,𝑠)𝐶^𝑇𝑦(𝑡)=𝑑𝑡𝑑𝑠_{0−𝜏𝑡1𝜏0−𝜏}𝑊^𝑇(𝑡,𝜃)𝐶^𝑇𝐶𝑊(𝑡,𝑠)𝑑𝜃𝑑𝑡𝜑(𝑠)𝑑𝑠. (58) Let 𝑄𝑡₁,𝑠=^𝑡1_𝜏⁰_−𝜏𝑊^𝑇(𝑡,𝜃)𝐶^𝑇𝐶𝑊(𝑡,𝑠)𝑑𝜃𝑑𝑡. (59) Then, ⁰_−𝜏^𝑡1_𝜏𝑊^𝑇(𝑡,𝑠)𝐶^𝑇𝑦(𝑡)=∫𝑑𝑡𝑑𝑠_0−𝜏𝑄𝑡₁,𝑠𝜑(𝑠)𝑑𝑠. (60) From that, we have the main result of this paper as follows.

Theorem 19. Singular differential system with delay (6) and (49) is observable in [𝜏,𝑡₁] if the observability matrix 𝑄𝑡₁,𝑠=^𝑡1_𝜏⁰_−𝜏𝑊^𝑇(𝑡,𝜃)𝐶^𝑇𝐶𝑊(𝑡,𝑠)𝑑𝜃𝑑𝑡 (61) has rank𝑛 for all 𝑠∈[−𝜏,0]. Where 𝑊()=𝑋()()×𝑡,𝜃𝑡−𝜃+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝛿()()𝐸−𝜃+𝑋𝑡−𝜃−𝜏^𝑑𝐵+𝑌(𝑡−𝜃−𝜏)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵, (62) where 𝑋(𝑡) is the first class of foundation solution of singular differential system with delay and 𝑌(𝑡) is the second class of foundation solution of singular differential system with delay.

Proof . If the observability matrix 𝑄𝑡₁,𝑠=^𝑡1_𝜏⁰_−𝜏𝑊^𝑇(𝑡,𝜃)𝐶^𝑇𝐶𝑊(𝑡,𝑠)𝑑𝜃𝑑𝑡 (63) has rank 𝑛 for all 𝑠∈[−𝜏,0], then 𝑄⁻¹(𝑡₁,𝑠) exists, and from (60), we can take 𝜑(𝑠)=𝑄⁻¹𝑡₁,𝑠^𝑡1_𝜏𝑊^𝑇(𝑡,𝑠)𝐶^𝑇𝑦(𝑡)𝑑𝑡, (64) the singular differential system with delay (6) and (49) is observable in [𝜏,𝑡₁].

Theorem 20. If there exist 𝑠₁,𝑠₂∈[−𝜏,0], and 𝑠₁<𝑠₂ such that when 𝑠∈(𝑠₁,𝑠₂), the observability matrix 𝑄(𝑡₁,𝑠), rank𝑄(𝑡₁,𝑠)<𝑛, then singular differential system with delay (6) and (49) is unobservable.

Proof. For rank𝑄(𝑡₁,𝑠)<𝑛 (𝑠∈(𝑠₁,𝑠₂)), there exists 𝛼(𝑠)∈𝑅^𝑛, 𝛼(𝑠)≠0, such that 𝑄(𝑡₁,𝑠)𝛼(𝑠)=0.
Let 𝜑₁(𝑠) satisfy (60) for any 𝑦(𝑡), and 𝜑₂(𝑠)=𝜑₁(𝑠),𝑠∉𝑠₁,𝑠₂,𝜑₁(𝑠)+𝛼(𝑠),𝑠∈𝑠₁,𝑠₂. (65)
Obviously, for 𝑠∈[−𝜏,0], 𝜑₁(𝑠)≠𝜑₂(𝑠), and ⁰_−𝜏𝑄𝑡₁,𝑠𝜑₂(𝑠)=𝑑𝑠_0−𝜏𝑄𝑡₁,𝑠𝜑₁(𝑠)𝑑𝑠+^𝑠2_𝑠1𝑄𝑡₁𝛼(𝑠)=,𝑠𝑑𝑠_0−𝜏𝑄𝑡₁,𝑠𝜑₁(𝑠)=𝑑𝑠_{0−𝜏𝑡1𝜏}𝑊^𝑇(𝑡,𝑠)𝐶^𝑇𝑦(𝑡)𝑑𝑡𝑑𝑠. (66) That is for any 𝑦(𝑡), we can have more than one 𝜑(𝑠) in (60), so singular differential system with delay (6) and (49) is unobservable.

Case 2 (0≤𝑡<𝜏). From (3) and Theorem 17, we have 𝑦(𝑡)=𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+^𝑡−𝜏_−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑𝐵+𝑌(𝑡−𝜃−𝜏)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵(𝜃)+×𝜑𝑑𝜃_𝑡0𝐶𝑋(𝑡−𝜃)𝐸^𝑑+𝑌(𝑡−𝜃)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃+𝐻𝑢(𝑡). (67) That is, 𝐶𝑋(𝑡)+𝑌(𝑡)𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐸𝜑(0)+^𝑡−𝜏_−𝜏𝐶𝑋(𝑡−𝜃−𝜏)𝐸^𝑑()×𝐵+𝑌𝑡−𝜃−𝜏𝐼+𝐸𝐼−𝐸𝐸^𝑑−1𝐵𝜑(𝜃)(𝑡)−𝑑𝜃=𝑦^𝑡₀𝐶𝑋(𝑡−𝜃)𝐸^𝑑()×+𝑌𝑡−𝜃𝐼+𝐸𝐼−𝐸𝐸^𝑑−1×𝐷𝑢(𝜃)+𝑓(𝜃)𝑑𝜃−𝐻𝑢(𝑡). (68) From (68), we can easily find that when 𝜃∈(𝑡−𝜏,0), we cannot get any information about 𝜑(𝜃). That is, the state initial function 𝜑(𝑡) in [−𝜏,0] cannot be uniquely determined from the knowledge of the control 𝑢(⋅) and observation 𝑦(⋅) over [0,𝜏), singular differential control systems with delay (2) and (3) are not observable on [0,𝜏).

4. Conclusion

In this paper, we study linear singular differential systems with delay in the state equations. We give a general description of the solutions and as the use of such result, we derive a necessary and sufficient condition for exact observability in finite time.

In [7], the problem, finding general solutions of singular systems with delay, was presented for the first time. But the result and the approach of our paper are brand-new, which may have significant advantages in comparison with the results of [7]. For example, it can be easily used to study the observability of singular control systems with delay.

For the observability of singular control systems with delay, we study it in two cases. When 𝑡≥𝜏, we give a necessary and sufficient condition. When 0≤𝑡<𝜏, we point out that singular differential control systems with delay (2) and (3) are not observable. These will be very useful in the study of singular control systems with delay, and that could also be regarded as the example of the use of the general description of the solutions of singular differential systems with delay.

Acknowledgments

This research has been supported by the National Natural Science Foundation of China (no. 11371027 and no. 11071001), Doctoral Fund of the Ministry of Education of China (no. 20093401110001), and the Major Program of Educational Commission of Anhui Province of China (no. KJ2010ZD02).