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**Delay and Its Time-Derivative Dependent Stable Criterion for Differential-Algebraic Systems** ()

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*Applied Mathematics*,

**7**, 1124-1133. doi: 10.4236/am.2016.710100.

Received 21 April 2016; accepted 21 June 2016; published 24 June 2016

1. Introduction

Differential-algebraic systems, also referred to as singular systems, descriptor systems or generalized state-space systems, arise in a variety of practical systems such as chemical processes, nuclear reactors, biological systems, electrical networks and economy systems. Differential-algebraic systems include not only dynamic equations but also static equations [1] [2] ; the study of such systems is much more complicated than that for standard state- space systems. The existence and uniqueness of a solution to a given differential-algebraic system are not always guaranteed and the system can also have undesired impulsive behavior, which can lead to the instability and poor performance [3] .

Because of the extensive applications in many practical systems, a great number of fundamental notions and results in control and system theory based on standard state-space systems have been extended successfully to differential-algebraic systems. In recent years, much attention has been focused on stability, robust stability and control problems for differential-algebraic systems, and some results have been derived using the time domain method [4] - [11] . The existing results can be classified into two types: delay-dependent conditions, which include information on the size of delays [12] [13] , and delay-independent conditions, which are applicable to delays of arbitrary size [14] [15] . Since the stability of systems depends explicitly on the time-delay, the delay-independent conditions are more conservative, especially for small delays. While the delay-dependent conditions are usually less conservative, and the conservatism will dependent on the chosen of Lyapunov functional, the inequality bounding technique and so on. Two approaches were used to prove the stability of the system in the existing literature. The first approach consists of decomposing the system into fast and slow subsystems and the stability of the slow subsystem is proved using some Lyapunov functional. Then, the fast variables are expressed explicitly by an iterative equation in terms of the slow variables [16] . The second approach consists of constructing a Lyapunov-Krasovskii functional that corresponds directly to the descriptor form of the system [17] . Some other methods have also been provided to reduce the conservative, for example, convex analysis method [18] and delay partitioning approach [19] [20] . To the best of our knowledge, most of the existing delay-dependent asymptotically stable criteria only depend on the upper bound of the delay-derivative, and to the differential-algebraic systems, the delay and its time-derivative dependent stability criterion has not established, which motivates this paper.

This article deals with the problem of asymptotic stability for a class of linear differential-algebraic system with time-varying delay. The obtained criteria depend not only on the upper bound but also on the lower bound of the delay derivative. Based on the Lyapunov functional method and the delay partitioning approach, some delay and its time-derivative dependent stable criteria are obtained. One numerical example is provided to demonstrate the effectiveness of the proposed results. All the developed results are in the LMI framework which makes them more interesting since the solutions are easily obtained using existing powerful tools like the LMI toolbox of Matlab or any equivalent tool.

Notation: Throughout this paper, represents the transpose of A; The symbol “*” in matrix inequality denotes the symmetric term of the matrix; means A is a symmetrical positive (negative) definite matrix; stands for; I is a unit matrix.

2. Problem Statement

Consider the following differential-algebraic system:

(1)

where is the state vector, the matrix may be singular, and we assume that

, A and are constant matrices with appropriate dimensions. is time-varying delay, , , and it is assumed to satisfy, where d_{1}, d_{2} are constants. The initial con- dition is given by, , , where W is the space of absolutely continuous function: with the square integrable derivative and with the normal

.

The following definition, lemmas and notation are introduced, which will be used in the proof of the main results.

Definition 1 ( [6] ) System (1) is said to be regular if the characteristic polynomial, is not identically zero.

2) System (1) is said to be impulse-free if.

Lemma 1 ( [4] ) Let H, F and G be real matrices of appropriate dimensions then, for any scalar for all matrices F satisfying, we have:

3. Main Results

Theorem 1 System (1) is asymptotically stable for all differentiable delays with, if there exist symmetric positive-definite matrices Q, , , , , , satisfying

, and appropriately dimensioned matrices, , , , , ,

, and such that LMIs:, for and, and,

for and, where. and are denoted in (8) and (11), respectively.

Proof. The proof of this theorem is divided into two parts. First, we prove that the results, when, and, respectively, ensure that the derivative of Lyapunov functional is negative. Finally, we prove the results given in the first part ensure that system is regular and impulse-free.

First of all, we divide the delay interval into two segments: and, where we denote

, , and. Then, system (1) can be represented as

(2)

where is the characteristic function of

Consider the following Lyapunov functional:

(3)

where

In addition, we define the continuous function as the following form:

(4)

When, one can obtain

To obtain the main results, we consider the following three cases:, and.

Case 1:

Using the fact that, where, we easily obtain the following inequalities by Jensen’s inequality:

where, , ,. Then, the time-

derivative of along the solution of (2) is given by

(5)

Letting, and adding the following terms

(6)

to (5) gives

(7)

for some scalar, if

(8)

where

Inequality (8) contains two variables which make it difficult to solve by LMI tool. In order to overcome this difficulty, we seek the sufficient conditions for inequality (8). When and, the inequality (8) leads to the following LMIs:

(9)

and

(10)

where

Note that we have omitted the zero row and zero column in and. Letting

, the latter two LMIs imply (8), which is because

Thus, is convex in.

When, it follows from (9) that

(11)

The LMI (11) implies (9), since

Thus, is convex in. Similarly, we can obtain that is also convex in.

Case 2:

By the definition of the characteristic function, we apply the above arguments and representations with and. In addition, we replace (6) with the following equations:

and letting. It is easy to obtain that

for some scala, where

(12)

where

Similar to the case I, we obtain the results when and, which can be marked as and, respectively. Further, we can verify is convex in, and are also convex in.

From Case 1 and Case 2, we have

(13)

for some scalar.

Case 3:

(14)

Therefore, when,.

Now the asymptotic stability of system (1) can not be obtained yet, since the existence and uniqueness of a solution to system (1) are not always guaranteed and the system may have undesired impulsive behavior. In the following, we will prove that the above-mentioned results ensure the regular and impulse-free. It follows form (8), that

(15)

Pre- and post-multiplying (15) by and, respectively, we obtain

(16)

Since, there must exist two invertible matrices G and such that

(17)

Similar to (17), we define

Then, using, it can be shown that. Pre- and post-multiplying (16) by and H, we can formulate the following inequality easily

(18)

where will be irrelevant to the following discussion. Form (18) we get, and thus is nonsingular, which implies that is not identically zero and

. Hence, by Definition 1, the above-mentioned results guarantee system (1) is regular and impulse-free. This completes the proof.

When the matrix E is nonsingular, the result in this case can be obtained by setting E equal to I with the appropriate transformations. The corresponding result is given by the following corollary:

Corollary 1 System (1) with is asymptotically stable for all differentiable delays with, if there exist symmetric positive-definite matrices Q, , , , , , , and appropriately dimensioned matrices, , , , , , ,

and such that LMIs:, for and, and, for and

, where. and are denoted in (8) and (11), respectively.

When is unknown, let and, we have the following corollary.

Corollary 2 System (1) is asymptotically stable for all differentiable delays with, if there exist symmetric positive-definite matrices Q, , , , , , P satisfying, and appropriately dimensioned matrices, , , , , , ,

and such that LMIs:, for and, and, for and

, where. and are denoted in (8) and (11), respectively.

Remark 1. It should be pointed out that if is big enough, delay partitioning of may improve the

results. In addition, the better results may be obtained if we divide the delay interval into N

segments.

4. A Numerical Example

In this section, a numerical example will be presented to show the validity of the main results derived above.

Example Consider the following linear differential-algebraic system described by systems (1) with

For, choosing and and applying Theorem 1, the maximum values of is 5.27, which guarantees that the system is asymptotically stable.

5. Conclusion

In this paper, the asymptotic stability of differential-algebraic system with time-varying delay has been investi- gated. Some delay and its time-derivative dependent asymptotically stable criteria have been obtained by de- composing time-varying delay in a convex set. The obtained criteria depend not only on the upper but also on the lower bound of the delay derivative. One numerical example has been given to illustrate the effectiveness of the proposed main results.

Acknowledgements

We thank the Editor and the referee for their comments. This work was supported by A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department (16ZA0146) and the Doctoral Research Foundation of Southwest University of Science and Technology (13zx7141).

Conflicts of Interest

The authors declare no conflicts of interest.

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