Supplemental Stability Criteria with New Formulation for Linear Time-Invariant Fractional-Order Systems

: In this paper, new stability criteria for linear time-invariant fractional-order systems (LTIFOSs) based on linear matrix inequalities (LMIs) are derived. The solved variable of the existing LMI formulations is generalized to a complex one. In addition, based on the congruent transformation, a new LMI formulation is obtained, which is different from those in the existing literature. To deal with the above LMIs more conveniently with simulation software, the complex LMIs are converted to equivalent real LMIs. Finally, numerical examples are presented to validate the effectiveness of our theoretical results.


Introduction
Fractional-order systems (FOSs) have been receiving more and more attentions in recent years.Along with the deep research on FOSs, researchers have found that it is better to describe some phenomena with FOSs than integer-order systems [1,2].At present, FOSs are widely used in many areas such as engineering systems [3][4][5] and economic systems [6,7].A lot of significant achievements in the field of integer-order systems have been extended to FOSs, such as stability analysis [8,9] and feedback control [10,11].
The stability of FOSs is a basic and important issue, and it is also a current research hotspots.Since the stability of FOSs is different from that of integer-order systems [12,13], new stability analysis methods are required to accommodate the characteristics of FOSs [14].When the order of an FOS belongs to different intervals, the features of the stability domains for the FOS are different.For this, researchers divided the order of FOSs into the intervals (0, 1) and (1,2), and derived different results for the stability of FOSs separately within the above two intervals.
(1) For the case of the order belonging to the interval (0, 1), Martignon et al. [15] proposed that the stability of LTIFOSs can be analyzed based on the eigenvalues of the system matrix, but sometimes, it is difficult to calculate all the system eigenvalues, especially when designing stabilizable controllers.Later, Moze et al. [16] provided an LMIbased analysis approach for the stability of LTIFOSs.To simplify the results in [16], the authors in [17,18] gave some extra LMI formulations for the stability of LTIFOSs.
In [19], Zhang et al. generalized the results to the case of singular FOSs and provided some LMI formulations with two real variables for the admissibility of singular FOSs; (2) For the case of the order belonging to the interval (1, 2), Sabatier et al. [20] pointed out that the result in [15] is also valid.After this, Farges et al. [17] gave an LMI formulation for the stability of LTIFOSs.The LMI formulation in [17] is a special case of the ones in [21], which provided an LMI-based method to check if the system eigenvalues are located in the given LMI region.In [22], Zhang et al. generalized the results in [21] and supplemented some new LMI formulations for the stability of LTIFOSs.
In [23], the authors discussed some unified LMI formulations for the stability of LTIFOSs, which are suitable for the case of the order belonging to the interval (0, 2).Based on the above literature, we note that when the order belongs to the interval (0, 1), the LMI formulations of the stability for LTIFOSs in the existing results are complete and mature.However, when the order belongs to the interval (1, 2), these different LMI formulations restrict the solved variable into being real.Furthermore, it is worth considering whether a new LMI formulation for the stability of LTIFOSs exists.
Based on the above investigations, the stability criteria for LTIFOSs are enriched in this paper, and a new LMI formulation that is different from those in the existing literature is derived.The main contributions of this paper are as follows: (1) The solved variable P of the existing LMI formulations for the stability of LTIFOSs is generalized to be complex; (2) A new LMI formulation for the stability of LTIFOSs is derived, which is different from those in the existing literature.
This paper is organized as follows.In Section 2, some useful lemmas and definitions are introduced, and the problem statement is given.In Section 3, the solved variable of the existing LMI formulations for the stability of LTIFOSs is generalized to be complex, and a new LMI formulation for the stability of LTIFOSs is obtained, which is different from those in the existing literature.In Section 4, the effectiveness of our theoretical results is illustrated through numerical examples.In Section 5, this paper is summarized.Notation 1. S > 0 (< 0) indicates that S is a positive (negative) definite matrix.A T and A * represent the transposition of a matrix A and the conjugate transposition of A, respectively.Her(A) denotes the sum of A and its conjugate transpose.Γ(•) denotes the Gamma function with Γ(s) = ∞ 0 t s−1 e −t dt.A ⊗ B stands for the Kronecker product of A and B. spec(A) represents a set consisting of the eigenvalues of A. a = sin(α π 2 ) and b = cos(α π 2 ).For brevity, Θ is defined by

Problem Statement and Preliminaries
In this section, important lemmas are given in preparation for the main results of this paper.
Consider the following FOS described by: where α ∈ (1, 2) is the commensurate fractional order, x(t) ∈ R n is the system state, and A ∈ R n×n is the system matrix.In addition, D α is the Caputo fractional-order differential operator that is defined by: We use the Caputo derivative in this paper, since its Laplace transform allows using initial values like classical integer-order derivatives, which explain the physical phenomena more clearly.
The LMI region is often used to analyze the properties of LTI systems.The definition of the LMI region is given as follows: Definition 1 ([22]).Let L ∈ R p×p and M ∈ C p×p .Then, the region D is said to be an LMI region, where Lemma 1 ([22]).For a given region D ⊂ C defined by (2), spec(A) ⊂ D if there exists a Hermitian matrix P, such that P > 0 and In the following two lemmas, we introduce some characteristics of the Kronecker product.
Lemma 2 ([22]).For a given matrix A, there holds Lemma 3 ([24]).For a given matrix A, define Ω and Ω 0 by Remark 1. Lemma 3 establishes a relationship between the region of spec(A) and the region of spec(Θ ⊗ A).An example is provided to illustrate this relationship.Consider (1) with α = 1.5 and   The following lemma gives a method to change a complex LMI into an equivalent real one.Lemma 4. For a complex matrix P, P is a positive (or negative) definite matrix if ℜ(P) ℑ(P) −ℑ(P) ℜ(P) , ( 7) is a positive (or negative) definite matrix, where ℜ(P) and ℑ(P) denote the real and imaginary parts of P, respectively.

Main Results
In this section, new LMI formulations for the stability criteria are derived.
Theorem 1. System (1) is stable if the following LMI admits a positive definite matrix P ∈ C 2n×2n .
Remark 2. In [21], Chilali et al. derived the LMI formulation of (8) with the solved variable P being real.In Theorem 1, we generalized the solved variable P to be complex.It is noted that (8) can be changed into Summing (8) and (10) yields Let P + P = P 0 , and there exists P 0 ∈ R 2n×2n such that P 0 > 0 and (11) hold.This indicates that the result in [21] can also be obtained from Theorem 1.
Remark 3. If we define P of (8) by P = X + jY, then (8) can be changed into Theorem 2. System (1) is stable if the LMI admits a positive definite matrix P ∈ C 2n×2n .

Remark 4.
As an extension to Theorem 1, Theorem 2 gives an additional LMI formulation for the stability of LTIFOSs.The solved variable P in ( 14) is also complex.Note that (14) can be changed into By summing ( 16) and ( 14), and letting P 0 = P + P, one obtains a real matrix P 0 > 0 such that This indicates that the previous results can also be derived from Theorem 2.
Remark 5. From Lemma 4, system (1) is stable if there exist X ∈ R 2n×2n and Y ∈ R 2n×2n such that (12) and admits a positive definite matrix P ∈ C n×n .
Proof.Construct the stability region for FOSs by (2) with L = 0 2×2 and From Lemma 1, system (2) is stable if there exists P ∈ C n×n , such that P > 0 and With some computations, ( 20) is equivalently changed into (19).
Remark 6. Farges et al. [17] and Marir et al. [25] derived the stability criterion with the LMI formulation of (19).However, it is worth noting that P in the above two papers is restricted to be real.Different from [17] and [25], the solved variable P of (19) is generalized to be complex.Furthermore, it is seen that (19) can be changed into Summing ( 21) and (19) yields Her Θ ⊗ (A(P + P)) < 0. ( 22) Let P 0 = P + P.Then, (22) becomes where P 0 ∈ R n×n .Hence, a connection between Theorem 3 is established (see the results in [17,25]).
For more details, ( 23) is further changed into Her e jθ (AP 0 ) < 0, . This indicates that we can derive the result in [17] from Theorem 3.
Remark 7. Define P of (19) by X + jY.Then, (19) can be further changed into In the following theorem, an additional LMI formulation for the stability of LTIFOSs is presented, which is different from those in the existing literature.
admits a positive definite matrix P ∈ C n×n .
Proof.Let AP = [l ij ] n×n .Then, Let e i be a row vector with the i-th element being one and the other elements being zero.Define S 0 = e T With some trivial computations, there holds S Her (AP) ⊗ Θ S * = Her Θ ⊗ (AP) . Therefore, This ends the proof.
Remark 8.It is noted that Theorems 1-4 are all equivalent to the stability of LTIFOSs and, therefore, they are equivalent to each other.Moreover, in each theorem, we provide a new different LMI formulation for the stability of LTIFOSs.

Numerical Experiments
In the following discussions, numerical examples are shown to validate the effectiveness of our theoretical results.In Example 1, a relationship between the convergence speed of an LTIFOS and the eigenvalues of its system matrix is presented.In Examples 2-4, the solved variable P of ( 8) is verified as being complex.In Examples 5-7, the solved variable P of ( 25) is validated as being complex.
Example 1.Consider the following two systems: Through some computations, the matrix eigenvalues of (26) are 1 ± 0.3j, and the matrix eigenvalues of (27) are −1 ± 0.9j.It is intuitively seen from Figure 2 that the locations of the matrix eigenvalues of (27) are much closer to the boundary of the stability region.From Figures 3 and 4, it is seen that system (26) reaches stability at about 14 s, but system (27) does not reach stability until 20 s.The stability time of system ( 27) is longer than that of system (26).In Example 2, the eigenvalues of A are −1 ± 0.5j, indicating that system (1) is stable.Figure 5 is presented to show the stability of system (1).
By solving LMIs ( 12) and ( 13), one obtains a positive definite solution P, where  Example 3. Investigate system (1) using the following parameters: α = 1.5 and The eigenvalues of A are −2.5 ± 0.5j, indicating that system (1) in this example is stable.Figure 6 shows the stability of system (1) in this example.The eigenvalues of A are −3.5 ± 1.4142j, indicating that system (1) in Example 4 is stable.In Figure 7, it is intuitively seen that system (1) in this example is stable.The eigenvalues of A are −1.55 ± 0.3969j, indicating that system (1) is stable.The stability of (1) in this example is shown in Figure 8.By solving LMIs ( 12) and ( 24), one obtains a positive definite solution P, where P = 0.8566 −0.1816 − 0.1579j −0.1816 + 0.1579j 0.1579 .

Conclusions
In this paper, we derive some new stability criteria for LTIFOSs, which enrich the theoretical results on the stability of LTIFOSs.We generalize the solved variable of the existing LMI formulations to a complex one, and derive a new LMI formulation for the stability of LTIFOSs that is different from those in the existing literature.These results add new analytical tools to the stability theory of LTIFOSs, and fill a gap in this field.In future, we will focus on the stability region of time-delayed LTIFOSs and the LMI formulations for the stability of time-delayed LITFOSs.

Figure 1
Figure1shows the relationship between the locations of the eigenvalues of A and the locations of the eigenvalues of Θ ⊗ A.