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

We consider the robust asymptotical stabilization of uncertain a class of descriptor fractional-order systems. In the state matrix, we require that the parameter uncertainties are time-invariant and norm-bounded.We derive a sufficient condition for the systemwith the fractional-order α satisfying 1 ≤ α < 2 in terms of linear matrix inequalities (LMIs). The condition of the proposed stability criterion for fractional-order system is easy to be verified. An illustrative example is given to show that our result is effective.


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][3][4], and discrete descriptor systems [5][6][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][12][13][14], robotics [15][16][17][18], finance system [19][20][21], and many others [22][23][24][25][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 , 1 <  < 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  satisfying 0 <  < 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 fractionalorder 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 , 1 ≤  < 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  satisfying 1 ≤  < 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 × stands for the set of  by  matrices with real entries,   stands for the transpose of , {} denotes the expression   + ,   denotes the identity matrix of order , diag( 1 ,  2 , . . .,   ) denotes the diagonal matrix, and • will be used in some matrix expressions to indicate a symmetric structure; i.e., if given matrices

Preliminary Results
Consider the following class of linear fractional-order systems: where 0 <  < 2 is the fractional-order, () ∈ R  is the state vector,  ∈ R × is a constant matrix, and  0    represent the fractional-order derivative, which can be expressed as where Γ(⋅) is the Euler Gamma function.For convenience, we use   to replace  0    in the rest of this paper.It is well known that system (2) is stable if [30][31][32]     arg (spec ()) >   2 where 0 <  < 2 and spec() is the spectrum of all eigenvalues of .
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  belongs to 1 ≤  < 2.
It is well known that the following system Further we have that the uncertain descriptor fractionalorder 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  and a matrix  such that the following LMI is satisfied.In this case, the gain matrix  is given by Assume that ( 6) is normalizable; by applying LMI (11), we obtain  ∈ R × such that rank( + ) = .Consider the feedback control for (6) in the following form: where  ∈ R × is one gain matrix such that the obtained normalized system is asymptotically stable.Then we have the closed-loop system: that is, where 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  satisfying 1 ≤  < 2 in terms of LMIs.
Lemma 6 (see [41]).Let , , and  be real matrices of appropriate sizes.Then, for any

Main Result
In this section, we present a new sufficient condition to design the gain matrix .In the following theorem, Δ  and Δ  are given nonsingular matrices, such that From now on, we denote Δ = Δ −1  ΔΔ −1  , M =  1   Δ  , and N = Δ    .It is obvious that ΔΔ  ≤   .Thus, for any  1 > 0 and  2 > 0, by using Lemmas 5 and 6 and ΔΔ  ≤   , we have and that is, Remark 7. Note that, when  = 2, we have  1 +  2 ≤ 2 and That is, for any real scalar  > 0, and two matrices  ∈ R × 1 and  ∈ R  2 × , we cannot obtain real scalars  1 > 0 and  2 > 0 such that where Theorem 8. Assume that ( 6) is normalizable; then there exists a gain matrix  such that the uncertain descriptor fractionalorder system (6) with fractional-order 1 ≤  < 2 controlled by the controller ( 13) is asymptotically stable, if there exist matrices  ∈ R × ,  =   > 0 ∈ R × and two real scalars  1 > 0 and  2 > 0, such that where with  =  − (/2) and matrices  and  are given by LMI (11).Moreover, the gain matrix  is given by Proof.Suppose that there exist matrices  ∈ R × ,  =   > 0 ∈ R × and two real scalars  1 > 0 and  2 > 0 such that (29) holds.It is easy to derive that By using the Schur complement of ( 29), one obtains Write  =  −1 .It follows from applying (25) that By using the above inequality (34) and Lemma 1, we obtain Therefore, system ( 6) is asymptotically stable.This ends the proof.
Note that if we choose Δ  =   and Δ  =   in LMI (29), It is easy to see the following: (1) For given , when  1 −  > 0, it is always true that  1 +  2 −  > 0; that is, there do not exist  1 and  2 such that  < 0. Therefore, Theorem 8 is not a special case of Lemma 4 [28, Theorem 2], when Δ  =   and Δ  =   . (

A Numerical Example
In this section, we assume that the matrix of uncertain parameters Δ 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 fractionalorder system described in (6) with parameters as follows: where  = 1.23.
It is easy to check that rank() = 2 < 3; that is,  is singular.By applying the LMI (11) It follows from ( 16) that Based on those results, it is debatable whether or not system (6) is stable.
In the second way, we compute  0 , ,  1 ,  2 , and  by using Theorem 8; we choose It is easy to check that (51) Therefore, system (6) is stable.

Conclusion
In this paper, the robust asymptotical stability of uncertain descriptor fractional-order systems (6) with the fractionalorder  belonging to 1 ≤  < 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.
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].