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A New Sufficient Condition for Checking the Robust Stabilization of Uncertain Descriptor Fractional-Order Systems.

1. Introduction

Descriptor systems arise naturally in many applications such as aerospace engineering, social economic systems, and network analysis. Sometimes we also call descriptor systems singular systems. Descriptor system theory is an important part in control systems theory. Since 1970s, descriptor systems have been wildly studied, for example, descriptor linear systems [1], descriptor nonlinear systems [2-4], and discrete descriptor systems [5-7]. In particular, Dai has systematically introduced the theoretical basis of descriptor systems in [8], which is the first monograph on this subject. A detailed discussion of descriptor systems and their applications can be found in [9,10].

It is well known that fractional-order systems have been studied extensively in the last 20 years, since the fractional calculus has been found many applications in viscoelastic systems [11-14], robotics [15-18], finance system [19-21], and many others [22-26]. Studying on fractional-order calculus has become an active research field. To the best of our knowledge, although stability analysis is a basic problem in control theory, very few works existed for the stability analysis for descriptor fractional-order systems.

Many problems related to stability of descriptor fractional-order control systems are still challenging and unsolved. For the nominal stabilization case, N'Doye et al. [27] study the stabilization of one descriptor fractional-order system with the fractional-order [alpha], 1 < [alpha] < 2, in terms of LMIs. N'Doye et al. [28] derive some sufficient conditions for the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order a satisfying 0 < [alpha] < 2. Furthermore, Ma et al. [29] study the robust stability and stabilization of fractional-order linear systems with positive real uncertainty. Note that, in Example 1, by applying Theorem 2 [27], it is harder to determine whether the uncertain descriptor fractional-order system (6) is asymptotically stable. Therefore, it is valuable to seek sufficient conditions, for checking the robust asymptotical stabilization of uncertain descriptor fractional-order systems.

In this paper, we study the stabilization of a class of descriptor fractional-order systems with the fractional-order [alpha], 1 [less than or equal to] [alpha] < 2, in terms of LMIs. We derive a new sufficient condition for checking the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order a satisfying 1 [less than or equal to] [alpha] < 2, in terms of LMIs. It should be mentioned that, compared with some prior works, our main contributions consist in the following: (1) we assume that the matrix of uncertain parameters in the uncertain descriptor fractional-order system is diagonal. Thus, compared with the results in [28], our conclusion, Theorem 8, is more feasible and effective and has wider applications; (2) compared with some stability criteria of fractional-order nonlinear systems, for example, in [9, 22], our method is easier to be used.

Notations: throughout this paper, [R.sup.mxx] stands for the set of m by n matrices with real entries, [M.sup.T] stands for the transpose of M, Sym{X} denotes the expression [X.sup.T] + X, [I.sub.n] denotes the identity matrix of order n, diag([a.sub.1], [a.sub.2], ..., [a.sub.n]) denotes the diagonal matrix, and * will be used in some matrix expressions to indicate a symmetric structure; i.e., if given matrices [H.sub.1] = [H.sup.T.sub.1] [member of] [R.sup.mxm] and [H.sub.2] [member of] [H.sup.T.sub.2] [member of] [R.sup.nxn], then

[mathematical expression not reproducible]. (1)

2. Preliminary Results

Consider the following class of linear fractional-order systems:

[mathematical expression not reproducible], (2)

where 0 < [alpha] < 2 is the fractional-order, x(t) [member of] [R.sup.n] is the state vector, A [member of] [R.sup.nxn] is a constant matrix, and [mathematical expression not reproducible] represent the fractional-order derivative, which can be expressed as

[mathematical expression not reproducible], (3)

where [GAMMA](x) is the Euler Gamma function. For convenience, we use [D.sup.[alpha]] to replace [mathematical expression not reproducible] in the rest of this paper. It is well known that system (2) is stable if [30-32]

[absolute value of arg (spec(A))] > [alpha] [pi]/2 (4)

where 0 < [alpha] < 2 and spec(A) is the spectrum of all eigenvalues of A.

The next lemma, given by Chilali et al. [33], contains the necessary and sufficient conditions of (4) in terms of LMI, when the fractional-order a belongs to 1 [less than or equal to] [alpha] < 2.

Lemma 1 (see [33]). Let A [member of] [R.sup.nxn] be a real matrix and 1 [less than or equal to] [alpha] < 2. Then [absolute value of arg(spec(A))] > ([pi]/2)[alpha] if and only if there exists P >0 such that

[mathematical expression not reproducible]. (5)

Consider the following uncertain descriptor fractional-order systems:

[ED.sup.[alpha]] x (t) = (A + [[DELTA].sub.A]) x (t) + Bu (t)

x (0) = [x.sub.0] (6)

where 1 [less than or equal to] [alpha] < 2, x(t) [member of] [R.sup.n] is the semistate vector, u(t) [member of] [R.sup.m] is the control input, E [member of] [R.sup.nxn] is singular, A [member of] [R.sup.nxn] and B [member of] [R.sup.nxm] are constant matrices, and the time-invariant matrix [[DELTA].sub.A] corresponds to a norm-bounded parameter uncertainty, which is the following form:

[[DELTA].sub.A] = [M.sub.A] [DELTA] [N.sub.A] (7)

where [M.sub.A] and [N.sub.A] are real constant matrices of appropriate sizes, and the uncertain matrix [DELTA] = [([[gamma].sub.ij]).sub.pxq] satisfies

[DELTA][[DELTA].sup.t] [less than or equal to] [I.sub.p]. (8)

Remark 2. Condition [DELTA][[DELTA].sup.t] [less than or equal to] [I.sub.p] is rational because a lot of system uncertainties satisfy this inequality. Besides, this condition can also be used in many literatures, for example, in [9, 34-39].

It is well known that the following system

[ED.sup.[alpha]]x (t) ? Ax (t) + Bu (t)

x (0) = [x.sub.0] (9)

is normalizable if and only if

rank [E B] = n. (10)

Further we have that the uncertain descriptor fractional-order systems (6) is normalizable if and only if the nominal descriptor fractional-order system (9) is normalizable.

Lemma 3 (see [28], Theorem 1). System (6) is normalizable if and only if there exist a nonsingular matrix P and a matrix Y such that the following LMI

EP + BY + [P.sup.T][E.sup.T] + [Y.sup.T][B.sup.T] < 0 (11)

is satisfied. In this case, the gain matrix L is given by

L = [YP.sup.-1]. (12)

Assume that (6) is normalizable; by applying LMI (11), we obtain L [member of] [R.sup.mxn] such that rank(E + BL) = n. Consider the feedback control for (6) in the following form:

u(t) = -[LD.sup.[alpha]]x (t) + Kx (t), (13)

where K [member of] [R.sup.mxn] is one gain matrix such that the obtained normalized system is asymptotically stable. Then we have the closed-loop system:

(E + BL) [D.sup.[alpha]] x (t) = (A + [[DELTA].sub.A] + BK) x (t), (14)

that is,

[D.sup.[alpha]]x (t) = ([A.sub.1] + [B.sub.1]K + [E.sub.1] [DELTA].sub.A]) x (t), (15)

where

[E.sub.1] = [(E + BL).sup.-1],

[A.sub.1] = [E.sub.1]A,

[B.sub.1] = [E.sub.1]B. (16)

To facilitate the description of our main results, we need the following results.

In [28], N'Doye et al. derive a sufficient condition for the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order a satisfying 1 [less than or equal to] [alpha] < 2 in terms of LMIs.

Lemma 4 (see [28], Theorem 2). Assume that (6) is normalizable; then there exists gain matrix K such that the uncertain descriptor fractional-order system (6) with fractional-order 1 [less than or equal to] [alpha] < 2 controlled by the control (13) is asymptotically stable, if there exist matrices X [member of] [R.sup.mxn], [P.sub.0] = [P.sup.T.sub.0] > 0 [member of] [R.sup.nxn] and a real scalar [delta] > 0, such that

[mathematical expression not reproducible] (17)

where

[mathematical expression not reproducible], (18)

with [theta] = [pi] - [alpha]([pi]/2) and matrices P and Y are given by LMI (11).

Moreover, the gain matrix K is given by

K = [XP.sup.-1.sub.0]. (19)

Lemma 5 (see [40]). For any matrices X and Y with appropriate sizes, we have

[X.sup.T]Y + Y[T.sup.T] [less than or equal to] [epsilon][X.sup.T]X + [[epsilon].sup.-1][Y.sup.T]Y, (20)

for any [epsilon] > 0.

Lemma 6 (see [41]). Let X, Y, and Z be real matrices of appropriate sizes. Then, for any x [member of] [R.sup.n],

max{[([x.sup.T]XPYx).sup.2] : [F.sup.T]F [less than or equal to] I}

= ([x.sup.T][XX.sup.T]x) ([x.sup.T][Y.sup.T]Yx). (21)

3. Main Result

In this section, we present a new sufficient condition to design the gain matrix K. In the following theorem, [[DELTA].sub.M] and [[DELTA].sub.N] are given nonsingular matrices, such that

[[DELTA].sup.-1.sub.M] [DELTA][[DELTA].sup.-1.sub.M] ([[DELTA].sup.-1.sub.M] [DELTA][[DELTA].sup.-1.sub.M]).sup.T] [less than or equal to] [I.sub.p]. (22)

From now on, we denote [mathematical expression not reproducible]. Thus, for any [[epsilon].sub.1] > 0 and [[epsilon].sub.2] > 0, by using Lemmas 5 and 6 and [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (23)

and

[mathematical expression not reproducible], (24)

that is,

[mathematical expression not reproducible]. (25)

Remark 7. Note that, when [delta] = 2, we have [[epsilon].sub.1] + [[epsilon].sub.1] [less than or equal to] 2 and

1/[[epsilon].sub.1] + 1/[[epsilon].sub.2] [greater than or equal to] 2 > 1/[delta] = 1/2. (26)

That is, for any real scalar [delta] > 0, and two matrices [mathematical expression not reproducible], we cannot obtain real scalars e1 >0 and [[epsilon].sub.2] > 0 such that

([mathematical expression not reproducible], (27)

where

([mathematical expression not reproducible], (28)

Theorem 8. Assume that (6) is normalizable; then there exists a gain matrix K such that the uncertain descriptor fractional-order system (6) with fractional-order 1 [less than or equal to] [alpha] < 2 controlled by the controller (13) is asymptotically stable, if there exist matrices X [member of] [R.sup.mxn], P = [P.sup.T] > 0 [member of] [R.sup.mxn] and two real scalars [[epsilon].sub.1] > 0 and [[epsilon].sub.2] > 0, such that

[mathematical expression not reproducible], (29)

where

[mathematical expression not reproducible], (30)

ith [theta] = [pi] - [alpha]([pi]/2) and matrices P and Y are given by LMI (11).

Moreover, the gain matrix K is given by

K = [XP.sup.-1]. (31)

Proof. Suppose that there exist matrices X [member of] [R.sup.mxn], P = [P.sup.T] > 0 [member of] [R.sup.mxn] and two real scalars [[epsilon].sub.1] > 0 and [[epsilon].sub.2] > 0 such that (29) holds. It is easy to derive that

[mathematical expression not reproducible]. (32)

By using the Schur complement of (29), one obtains

[mathematical expression not reproducible]. (33)

Write K = [XP.sup.-1]. It follows from applying (25) that

[mathematical expression not reproducible]. (34)

By using the above inequality (34) and Lemma 1, we obtain

[absolute value of arg (spec ([A.sub.1] + [B.sub.1]K + [E.sub.1][M.sub.A][DELTA][N.sub.A]] > [pi]/2 [alpha]. (35)

Therefore, system (6) is asymptotically stable. This ends the proof.

Remark 9. Write

[mathematical expression not reproducible] (36)

Note that if we choose [[DELTA].sub.M] = [I.sub.p] and [[DELTA].sub.N] = [I.sub.q] in LMI (29),

[mathematical expression not reproducible]. (37)

It is easy to see the following:

(1) For given [delta], when [[epsilon].sub.1] - [delta] > 0, it is always true that [[epsilon].sub.1] + [[epsilon].sub.2] - [delta] > 0; that is, there do not exist [[epsilon].sub.1] and [[epsilon].sub.2] such that T < 0. Therefore, Theorem 8 is not a special case of Lemma 4 [28, Theorem 2], when [[DELTA].sub.M] = [I.sub.p] and [[DELTA].sub.N] = [I.sub.q].

(2) For given [[epsilon].sub.1] and [[epsilon].sub.2], when [[epsilon].sub.1] - [delta] < 0, there exists [[epsilon].sub.2] such that T is positive definite; that is, there exists 8 such that T > 0. Since conditions in Lemma 4 and Theorem 8 are both sufficient, we cannot derive Lemma 4 by applying Theorem 8; that is, Theorem 8 is not a generalization of Lemma 4 [28, Theorem 2].

4. A Numerical Example

In this section, we assume that the matrix of uncertain parameters A in the uncertain descriptor fractional-order system (6) is diagonal. We provide a numerical example to illustrate that Theorem 8 is feasible and effective with wider applications.

Example 1. Consider the uncertain descriptor fractional-order system described in (6) with parameters as follows:

[mathematical expression not reproducible], (38)

where [alpha] = 1.23.

It is easy to check that rank(E) = 2 < 3; that is, E is singular. By applying the LMI (11), we obtain

[mathematical expression not reproducible], (39)

and the gain matrix L

[mathematical expression not reproducible]. (40)

It follows from (16) that

[mathematical expression not reproducible]. (41)

Firstly, we compute [P.sub.0], X, [delta], and K by using Lemma 4 [28, Theorem 2]. A feasible solution of LMI (11) is as follows:

[mathematical expression not reproducible]. (42)

We choose

[mathematical expression not reproducible], (43)

It follows from (15) that

[mathematical expression not reproducible], (44)

and the arguments of all eigenvalues of S are

3.1416,

3.1416,

0. (45)

Based on those results, it is debatable whether or not system (6) is stable.

In the second way, we compute [P.sub.0], X, [[epsilon].sub.1], [[epsilon].sub.2], and K by using Theorem 8; we choose

[mathematical expression not reproducible]. (47)

It is easy to check that

[mathematical expression not reproducible], (48)

It follows that a feasible solution of LMI (11) is

[mathematical expression not reproducible]. (49)

asymptotically stabilizing state-feedback gain is

[mathematical expression not reproducible]. (50)

and the arguments of all eigenvalues of S are

3.1416,

3.1416,

3.1416. (51)

Therefore, system (6) is stable.

5. Conclusion

In this paper, the robust asymptotical stability of uncertain descriptor fractional-order systems (6) with the fractional-order a belonging to 1 [less than or equal to] [alpha] < 2 has been studied. We derive a new sufficient condition for checking the robust asymptotical stabilization of (6) in terms of LMIs. Out results can be seen as a generalization of [28, Theorem 2]. By adding appropriate parameters into LMIs, our result has wider applications. One special numerical example has shown that our results are feasible and easy to be used.

https://doi.org/10.1155/2018/4980434

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author was supported partially by China Postdoctoral Science Foundation [Grant no. 2015M581690], the National Natural Science Foundation of China [Grant no. 11361009], High level innovation teams and distinguished scholars in Guangxi Universities, the Special Fund for Scientific and Technological Bases and Talents of Guangxi [Grant no. 2016AD05050], and the Special Fund for Bagui Scholars of Guangxi. The second author was supported partially by the National Natural Science Foundation of China [Grant no. 11701320] and the Shandong Provincial Natural Science Foundation [Grant no. ZR2016AM04].

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Hongxing Wang (iD) (1,2) and Aijing Liu (3)

(1) School of Mathematics, Southeast University, Nanjing 210096, China

(2) School of Science, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning 530006, China

(3) School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Correspondence should be addressed to Hongxing Wang; winghongxing0902@163.com

Received 14 April 2018; Accepted 11 June 2018; Published 18 July 2018

Academic Editor: Xinguang Zhang
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Publication:Journal of Function Spaces
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