Dynamic Output Feedback Stabilization of Singular Fractional-Order Systems

This paper is concerned with dynamic output feedback controller (DOFC) design problem for singular fractional-order systems with the fractional-order α satisfying 0 < α < 2. Based on the stability theory of fractional-order system, sufficient and necessary conditions are derived for the admissibility of the systems, which are more convenient to analytical design of stabilizing controllers than the existing results. A full-order DOFC is then synthesized based on the obtained conditions and the characteristics ofMoorePenrose inverse. Finally, a numerical example is presented to show the effectiveness of the proposed methods.


Introduction
During the past years, the study on fractional-order systems has become a hot research topic since they can concisely and precisely characterize many real-world physical systems with the introduction of fractional-order calculus [1,2].Up to date, considerable attention has been devoted to the stability analysis and controller design for fractional-order systems and many results have been published in the literature (see, e.g., [3][4][5][6][7][8][9] and the references therein).
On the other hand, singular systems have been extensively studied in the past years due to their applications in economics, circuits, and many other fields (see, e.g., [10][11][12][13][14] and the references therein).The research on singular fractional-order systems is much more complicated than that for state-space systems, because it requires considering not only stability, but also regularity and impulse elimination, while the latter do not appear in regular ones.Very recently, the study on singular fractional-order systems has received much attention.For example, sufficient and necessary condition of admissibility for fractional-order singular system was proposed in [15].However, it is noted that the conditions obtained in [15] are not convenient for matrix transformation and may cause difficulty in controller design.In addition, only systems with fractional-order  belonging to 0 <  < 1 are considered.Based on the obtained stability results in [15] and matrix's singular value decomposition (SVD), state and static output feedback controller design methods were proposed in [16] for singular fractional-order systems with fractional-order 0 <  < 1.In [17], the problem of robust stabilization for a class of uncertain singular fractional-order systems with fractional commensurate order 0 <  < 2 was studied.However, the systems in [17] are required to be normalizable, that is, rank [ ] = , which is very restrictive.In [18], the problem of robust stability and stabilization of interval uncertain descriptor fractional-order systems with fractional-order 1 ≤  < 2 was studied.However, similarly as in [17], only state-feedback controller was designed, which means that the obtained method cannot be applied to system with state being unavailable.Up to now, the problem of stability analysis and controller design for singular fractionalorder systems has not been fully investigated, which is very challenging and of great importance.This motivates us to carry out this study.
In this paper, we will study the dynamic output control problem for singular fractional-order systems with fractional commensurate order 0 <  < 2. The main contributions of this paper can be summarized as follows.
(1) New necessary and sufficient conditions will be derived for the stability of the focused systems, which 2 Mathematical Problems in Engineering are more convenient to analytical design of stabilizing controllers than the existing results.
(2) The desired full DOFC is designed in a whole framework.It is practically important to stabilize systems by output feedback controller since it is usually hard to sense all the states and feed them back.The effectiveness and applicability of proposed results are verified by a numerical example.
Notations.R × is the set of all  ×  real matrices.The superscripts "" and "+" represent the transpose and the Moore-Penrose inverse, respectively, and " * " denotes the term that is induced by symmetry.Sym{} is used to denote   + , ⊗ stands for the Kronecker product, and   represents initialized th order differintegration.

Problem Formulation and Preliminaries
Consider the following singular fractional-order system: where 0 <  < 2 is the fractional commensurate order; () ∈ R  , () ∈ R  , and () ∈ R  are the state vector, the control input, and the measurable output, respectively.The matrix  ∈ R × may be singular and it is assumed that rank() =  ≤ .The system matrices , , , and  are real matrices with appropriate dimensions.In this paper, the Caputo definition is adopted for fractional derivatives of order  of function () since this Laplace transform allows using initial values of classical integralorder derivatives with clear physical interpretations, which is defined as follows: where the fractional-order  − 1 <  ≤ ,  ∈ , and the gamma function Γ() = ∫ ∞ 0  −  −1 .For system (1), we are interested in designing a DOFC of the following form: where   () ∈ R  is the state vector of the controller.The matrices   ,   , and   are the controller matrices to be determined.
Augmenting the model of system (1) to include the states of DOFC (3), the closed-loop system is governed by where () = [  ()    ()]  and The unforced singular fractional-order system of (1) can be written as Definition 1 (see [15]).For singular fractional-order ( 6), the pair (, ) is said to be regular if there exists a constant scalar  ∈ C such that the pseudopolynomial det(   − ) is not identically zero.
Lemma 2 (see [15]).(1) For singular fractional-order ( 6), the pair (, ) is regular if and only if there exist two nonlinear matrices  and  such that where  1 and  are in Jordan canonical forms and  is nilpotent.
Lemma 3 (see [19]).For fractional-order, linear system with order  :   /  =  is asymptotically stable if and only if where  ∈ R × is a deterministic real matrix and spec() is the spectrum of .
The regularity and absence of impulses of the pair (, ) ensure the existence and uniqueness of an impulse-free solution to system (6).Based on Lemmas 2 and 3, we have the following lemma.
(2) Suppose the pair (, ) is regular and impulse-free; system (6) Proof.Noting the regularity and absence of impulses of the pair (, ) and using the decomposition as in [15] and Lemma 3, the desired result follows immediately.
Throughout the paper, we will use the following notion of admissibility for singular fractional-order system.Definition 5 (see [15]).The singular fractional-order system ( 6) is said to be admissible, if it is regular, impulse-free, and asymptotically stable.
The necessary and sufficient conditions for the stability of state-space fractional-order systems with the fractional-order  belonging to 0 <  < 2 are provided by the two following lemmas.

Admissibility Analysis of Singular Fractional-Order Systems
In this section, we will investigate the admissibility of closedloop system (3) with fractional-order  belonging to 0 <  < 1 and 1 ≤  < 2.
Next, we will show the asymptotical stability of system (4).
Necessity.Since (  ,   ) is admissible, there exist nonsingular matrices  and  such that By Lemma 7, there exist two real symmetric positive definite matrices  11 ∈  2×2 ,  = 1, 2, and two skewsymmetric matrices  21 ∈  2×2 ,  = 1, 2, such that Define We can easily verify that the conditions in ( 12)-( 15) hold.This completes the proof.Remark 9. Sufficient and necessary conditions are proposed for the admissibility of singular fractional-order systems with fractional-order  belonging to 0 <  < 1.The conditions are formulated in terms of nonstrict LMI with equality constraint.Compared with the results obtained in [15], the conditions proposed in Theorem 8 are more convenient to controller design.By Lemma 6, we have the following theorem for the admissibility of closed-system (4) with fractional-order  belonging to 1 ≤  < 2.
Theorem 10.Closed-loop system (4) with fractional-order  belonging to 1 ≤  < 2 is admissible, if and only if there exists nonsingular matrix   ∈  2×2 such that where Proof.The proof is similar to that of Theorem 8; we omit it.
Remark 11.To the author's best knowledge, it is for the first time that sufficient and necessary conditions are proposed for the admissibility of singular fractional-order systems with fractional-order 1 ≤  < 2. The results obtained in Theorems 8 and 10 can be easily extended to singular fractional-order systems with norm-bounded parameter uncertainties (i.e., in [17]).

Dynamic Output Controller Design
In this section, we study the design of DOFC for singular fractional-order system (1) with fractional-order  belonging to 0 <  < 1 and 1 ≤  < 2. The following lemma will be used in this section.
Lemma 12 (see [23]).The matrix inequality Based on the conditions obtained in Theorem 8, the following theorem is presented to construct the desired controller.
Theorem 13.Closed-system (3) with fractional-order  belonging to 0 <  < 1 is admissible, if there exist matrices , , Θ, Ω, and Λ with appropriate dimensions such that where Moreover, if the above conditions in (30) and (31) are feasible, the system matrices of an admissible DOFC in the form of (3) are given by Proof.By setting  11 =  21 =   and  12 =  22 = 0 and using Theorem 8, sufficient conditions for the admissibility of closed-loop system (4) can be obtained as Define Without loss of generality, it is assumed that matrices  and ( −1 − ) are nonsingular; if not,  and ( −1 − ) may be perturbed by Δ and Δ, respectively, with sufficient small norm such that  + Δ and ( −1 − ( + Δ)) are nonsingular and satisfy (31).From (30) and by using the properties of Moore-Penrose inverse, we can get Moreover, from (37), we have Considering the controller matrices given in (33), we obtain (34) from (39).This completes the proof.Remark 14.By using the characteristics of Moore-Penrose inverse, Theorem 13 provides sufficient conditions for the solvability of DOFC design problem for singular fractionalorder systems with fractional-order  belonging to 0 <  < 1.The matrices of desired DOFC can be constructed through the solutions of LMIs (30)-(31).
The following theorem is proposed for the design of DOFC for singular fractional-order systems with fractionalorder  belonging to 1 ≤  < 2.
Theorem 15.Closed-system (3) with fractional-order  belonging to 1 ≤  < 2 is admissible, if there exist matrices , , Θ, Ω, and Λ with appropriate dimensions such that Sym {Γ ⊗ Ψ} < 0, where Moreover, if the above conditions in (41)-( 42) are feasible, the system matrices of an admissible DOFC in the form of (3) are given by (44) Proof.The desired result can be carried out by employing the same technique used as in Theorem 13.

Illustrative Example
Consider the following singular fractional-order system described in (1) with parameters as follows: Case I (0 <  < 1).In this case, we choose  = 0.

Conclusion
In this paper, the problem of DOFC design for singular fractional-order systems with fractional-order  satisfying 0 <  < 2 has been investigated.Sufficient and necessary conditions for the admissibility of the systems have been derived, which are formulated in terms of LMIs.Based on the obtained conditions, the desired DOFC have been designed.An illustrative example has been given to show the effectiveness of the proposed method.

Figure 1 :
Figure 1: Time response of the closed-loop system with fractionalorder  = 0.5 under DOFC.

Figure 2 :
Figure 2: Time response of the closed-loop system with fractionalorder  = 1.5 under DOFC.