A universal framework of GKYP lemma for singular fractional order systems

1 A uni v ersal frame work of GKYP lemma for singular fractional order systems Y uman Li, Y iheng W ei, Y uquan Chen, and Y ong W ang Abstract —The well-known GKYP is widely used in system analysis, b ut f or singular systems, especially singular fractional order systems, there is no corr esponding theory , for which many control problems for this type of system can not be optimized in the limited frequency ranges. In this paper , a universal framework of finite frequency band GKYP lemma for singular fractional order systems is established. Then the bounded real lemma in the sense of L ∞ is derived f or different frequency ranges. Furthermore, the corresponding controller is designed to impro ve the L ∞ performance index of singular fractional order systems. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results. Index T erms —Singular fractional order systems, GKYP lemma, L ∞ norm, Bounded real lemma. I . I N T RO D U C T I O N The singular fractional order system (SFOS) is a research hotspot in recent years. It plays an important role in many applications, such as optimization problems, economics, con- strained mechanics, biology , aircraft, robot dynamics, electric networks and large systems. Man y scholars have paid attention to the analyses and control of SFOSs in many aspects, such as stability [1 – 3], admissibility [4], iterative learning control [5] and feedback control [6, 7]. The L ∞ norm of the transfer function matrix is an important performance index for SFOSs, which plays a significant role in energy calculation, controller design and filter analysis [8–10]. Generally speaking, there are many methods to solve L ∞ norm, among which the most effecti ve method is to establish the L ∞ bounded real lemma by using KYP lemma and GKYP lemma [11]. The KYP lemma was first proposed in [12], which effec- tiv ely establishes the connection between time domain method and frequency domain method, and transforms the complex frequency domain inequality into a simple time domain linear matrix inequality . It enables comprehensive control of the system over the full frequency range. The KYP lemma was generalized to singular systems by Xu who gave a series of theories of positiv e real control for singular systems in [13]. Subsequently , there were many studies based on KYP lemma for singular systems [14 – 16] and people paid more and more attention to the research of singular systems [17 – 21]. More recently , Bhaw al et al. established a ne w KYP lemma for strong passiv e singular systems [22]. Y . Li, Y . W ei, Y . Chen and Y . W ang are with the Department of Automation, Univ ersity of Science and T echnology of China, Hefei 230027, China. E-mail: lym2014@mail.ustc.edu.cn; neudawei@ustc.edu.cn; cyq@mail.ustc.edu.cn; yongwang@ustc.edu.cn Although the KYP lemma has been well generalized to singular systems, GKYP lemma has not. The GKYP lemma was first proposed by Iwasak in 2005 to solve control prob- lems in dif ferent frequency ranges [23]. Compared with KYP lemma, GKYP lemma is more practical, because in actual control problems, systems are usually required to meet dif fer- ent performance specifications in different frequency ranges. The proposal of GKYP lemma provided ne w ideas for many control problems, thus there are plenty of researches on GKYP lemma [24 – 27]. Recently , the GKYP lemma has also been extended to fractional order systems [11, 28]. But for singular systems, the research in this area is remarkably lagging. In [29], Mei first designed the controller of singular perturbed systems based the GKYP lemma of non-singular systems by using the fast and slo w subsystem decomposition method. Based on this idea, many researches on the finite frequency control of singularly perturbed systems have been de veloped [30 – 32]. Howe ver , this kind of normalization method could only apply to a part of SFOSs which can be normalized to non-singular systems, and could not handle systems that do not satisfy the normalization condition. Up to now , there is no universal GKYP lemma for singular systems, regardless of fractional or integer-order systems. This leads to the fact that researches of singular integer and fractional order systems are limited to the whole frequency band, or only those systems which can be normalized into non-singular systems are able to be controlled in a limited frequency band. Motiv ated by these findings, the GKYP lemma of SFOSs in the finite frequency ranges without normalization restriction is established in this paper, which can also be applied to singular integer order systems. Furthermore, the L ∞ bounded real theorem and L ∞ controller are obtained. The establish- ment of these theories will fundamentally solv e the finite frequency control problem of singular systems, and provide a new perspectiv e for other researches of SFOSs, like H ∞ control and optimal tracking problem in finite frequenc y bands, the optimization of uncertain SFOSs in dif ferent frequency ranges and the decoupling of large-scale systems’ singular representation. The rest of this paper is arranged as follo ws. In Section II, some basic f acts on this work are pro vided. Section III deriv es the uni versal framework of finite frequency band GKYP lemma for the SFOS. In section IV, L ∞ bounded real lemma of SFOS is derived in different frequency ranges. Then an L ∞ controller is designed in Section V. The numerical simulation is performed in Section VI. Finally , conclusions in Section VII close the paper . Notations. F or matrix X , the symbols X T and X ∗ represent 2 the transpose and complex conjugate transpose, respecti vely . Expression X > 0 ( X < 0) indicates that X is positiv e (negati ve) definite. The symbol sym( X ) is an abbre viation for X + X ∗ , and δ max ( X ) represents the maximum singular value of X . The symbols C m × n and R m × n stand for sets of m × n complex and real matrices, respecti vely . H n denotes the set of n × n complex Hermitian matrices. The trace and rank of a matrix X are represented by tr( X ) and rank( X ) , respectiv ely . Re( X ) and Im( X ) denote the real and imaginary parts of X . The operator ⊗ is the Kronecker’ s product. The con vex hull and the interior of a set S are denoted by co( S ) and int( S ) , respectiv ely . The set N represents a set of matrices that satisfy N = N ∗ ≤ 0 . I I . P R E L I M I NA R I E S A. Singular fractional order system model Consider a singular fractional order system  E D α x ( t ) = Ax ( t ) + B u ( t ) , y ( t ) = C x ( t ) + Du ( t ) , (1) where x ( t ) ∈ R n is the state variable of the system, u ( t ) ∈ R m is the control input, y ( t ) ∈ R p is the measured output, B , C and D are constant matrices with appropriate dimensions, D α represents the Caputo fractional deri vati ve, α is the commensurate order of the SFOS and 0 <α< 2 , E , A ∈ R n × n , and rank( E ) = r < n . If the SFOS is relaxed at t = 0 , the transfer function matrix between u ( t ) and y ( t ) is G ( s ) = C ( s α E − A ) − 1 B + D . (2) B. S-pr ocedure In deriving the process of GKYP lemma, S-procedure is a very important tool. Giv en Θ , M ∈ H q , if the regularity , M > 0 , is assumed, there exists the equiv alence ξ ∗ Θ ξ < 0 , ∀ ξ ∈ C q such that ξ 6 = 0 , ξ ∗ M ξ ≥ 0 . ⇔ ∃ τ ∈ R such that τ ≥ 0 , Θ + τ M < 0 . T o generalize the above S-procedure, paper [23] re writes it with a dif ferent notation. Define set S 1 specified by M as follows S ( M ) = { S ∈ H q : S > 0 , tr ( M S ) ≥ 0 } , (3) S 1 ( M ) = { S ∈ S ( M ) : rank ( S ) = 1 } , (4) where M = { τ M : τ ∈ R , τ ≥ 0 , M ∈ H q } . Then, the S-procedure can be stated as tr(Θ S 1 ) < 0 ⇔ (Θ + M ) ∩ int( N ) 6 = Ø . (5) Clearly , the S-procedure is completely specified by the set M . If in the equation (5), “ ⇒ ” and “ ⇐ ” are simultaneously established, the S-procedure is considered to be lossless. The lossless condition of S-procedure has been demonstrated in [23] as following. Definition 1. M ⊂ H q is said to be i) admissible if it is a nonempty closed conve x cone and in t ( N ) ∩ M = Ø ; ii) rank-one separable if S = co( S 1 ) . Lemma 1. [23] Let M ⊂ H q be defined by (3) and (4). Then for any matrix Θ ∈ H q , if and only if the set M is admissible and rank-one separable, the strict S-pr ocedur e is lossless. Remark 1. Lemma 1 shows that when M is chosen to be admissible and rank-one separable, no matter which Θ is selected, the S-pr ocedure will be lossless. This is important in the following subsequent pr oof pr ocess. Lemma 2. [23] Let M ⊂ H q be a rank-one separable set. Then the set F ∗ M F + P is rank-one separable for an arbitrary F ∈ C q × p and subset P ⊂ H p of positive semi-definite matrices containing the origin. C. F r equency range Definition 2. A curve on the complex plane is a collection of infinitely many points λ ( t ) ∈ C q × p continuously parame- terized by t for t 0 ≤ t ≤ t f wher e t 0 , t f ∈ R ∪ {±∞} and t 0 < t f . A set of comple x numbers Λ ⊆ C is said to r epr esent a curve (or curves) if it is a union of a finite number of curve(s). W ith Φ , Ψ ∈ H 2 being given matrices, Λ is defined as Λ (Φ , Ψ) = { λ ∈ C : δ ( λ, Φ) = 0 , δ ( λ, Ψ) ≥ 0 } , (6) wher e δ ( λ, Φ) =  λ I  ∗ Φ  λ I  According to [28], when Φ and Ψ take different forms, Λ can represent a specific frequency range. Lemma 3. F or the continuous-time setting fr actional or der system, one has Φ =  0 e j θ e − j θ 0  , (7) Λ = { (j ω ) α : ω ∈ Ω } , (8) wher e θ = π 2 (1 − α ) and Ω is a subset of r eal numbers which is determined by the choice of Ψ . F or differ ent frequency , we can get a table as follows Low F r equency Middle F r equency High F r equency Ω 0 ≤ ω ≤ ω L 0 ≤ ω 1 ≤ ω ≤ ω 2 0 ≤ ω H ≤ ω Ψ  − 1 0 0 ω 2 α L   − 1 ω c ω c − ω α 1 ω α 2   1 0 0 − ω 2 α H  wher e ω c = j α ( ω α 1 + ω α 2 ) 2 . In or der to ensure that ω belongs to the main Riemann sheet, her e ω must be nonne gative. This is determined by the particularity of the fractional or der system. Pr oof: Substituting matrix Φ into the definition of δ ( λ, Φ) , ther e is δ ( λ, Φ) =  λ I  ∗  0 e j θ e − j θ 0   λ I  = cos θ Re ( λ ) + sin θ Im ( λ ) = 0 3 Ther efore, matrix Φ r epr esents that the object consider ed her e is a continuous-time system. When Ψ =  − 1 0 0 ω 2 α L  , ther e is ( λ + ω α L ) ( λ − ω α L ) ≤ 0 ⇒ λ ≤ ω α L . Hence, Λ repr esents the low frequency . When Ψ =  1 0 0 − ω 2 α H  , ther e is ( λ + ω α H ) ( λ − ω α H ) ≥ 0 ⇒ λ ≥ ω α H . Hence, Λ r epr esents the high fr equency . As for the middle fr equency situation, there is ( λ − ω α 1 ) ( λ − ω α 2 ) ≤ 0 ⇒ ω α 1 ≤ λ ≤ ω α 2 . It is worth noting that here , ω c is used to guarantee the matrix Ψ ∈ H 2 . Since λ = ω α , the conclusion is reac hed. Remark 2. Note that when Ψ ≥ 0 , Ω can r epresent the infinite fr equency range. If Ψ = 0 2 , there is no constraint on λ , so it can r epresent the full band. If Ψ > 0 , depending on the pr operty of positive definite matrix, for any non-zer o vector X , there is X ∗ Ψ X > 0 . Let X =  λ I  , and one can then find that there is no constraint on λ by simple calculation, so Ω can also r epr esent the full frequency range when Ψ > 0 . In this study , Ψ = 0 2 is simply taken. I I I . G K Y P L E M M A F O R S F O S In this section, the appropriate S 1 and M will be chosen to deriv e the GKYP lemma for SFOFs by the S-procedure tool, and the conclusion will be strictly proved. According to the standard KYP lemma in [12] and the transfer function of SFOS in (2), the set S 1 which represents the positiv e definiteness of the SFOS should be given as S 1 =  ξ ξ ∗ : ξ =  [(j ω ) α E − A ] − 1 B I  η , η ∈ C m , η 6 = 0 ω ∈ R + ∪ { + ∞}  . (9) This set can be described as S 1 =  ξ ξ ∗ : ξ ∈ G λ , λ ∈ Λ  , G λ = { ξ ∈ C n + m : ξ 6 = 0 , Γ λ F ξ = 0 } , (10) where Λ = (j R + ) α ∪ {∞} and Γ λ =  [ I n − λI n ] ( λ ∈ C )  0 − I n  ( λ = ∞ ) , F =  A B E 0  . (11) Considering the general frequency range Λ in (6), the Λ in (10) is defined as Λ =  Λ , if Λ is bounded ; Λ ∪ {∞} , otherwise . (12) Now , the main technical steps in achieving the GKYP lemma for the SFOS are to represent the set S 1 in (10) as (4) through choosing a suitable M . At the same time, in order to ensure that the S-procedure is lossless, it is also necessary to indicate that the selected set M possesses the attributes in Definition 1. Lemma 4. [28] Let Φ 0 , Ψ 0 ∈ H 2 and a nonsingular matrix T ∈ C 2 × 2 be given and define Φ , Ψ ∈ H 2 as follows Φ = T ∗ Φ 0 T , Ψ = T ∗ Ψ 0 T , (13) Φ 0 =  0 e j θ e − j θ 0  , Ψ 0 =  α β e j θ β e − j θ γ  , (14) wher e α, β , γ ∈ R , α ≤ γ , and γ ≥ 0 . Consider Γ λ in (11), Λ in (6) and Λ in (12). Suppose Λ repr esents curves. F or a given vector ζ ∈ C 2 n , the following two conditions are equivalent. i) Γ λ ζ = 0 holds for some λ ∈ Λ (Φ , Ψ) . ii) Γ s ( T ⊗ I ) ζ = 0 holds for some s ∈ Λ (Φ 0 , Ψ 0 ) . Lemma 5. [28] Let Φ 0 , Ψ 0 be defined in (14), Γ λ in (11), Λ in (6) and Λ in (12). Suppose Λ r epr esents curves. The following conditions ar e equivalent. i) Γ s η = 0 for some s ∈ Λ (Φ 0 , Ψ 0 ) ; ii) η ∗ (Φ 0 ⊗ P + Ψ 0 ⊗ Q ) η ≥ 0 for all P , Q ∈ H n , Q ≥ 0 . Theorem 1. Let F ∈ C 2 n × ( n + m ) and define Γ λ and Λ as (11) and (12). The matrices Φ , Ψ ∈ H 2 ar e given such that Λ in (6) r epresents curves. Then the set S 1 defined in (10) can be r epr esented by (5) and (6) with M = { F ∗ (Φ ⊗ P + Ψ ⊗ Q ) F : P , Q ∈ H n , Q ≥ 0 } . (15) Pr oof: Let S 1 be defined by (10) and S 2 be defined to be S 1 in (5) with M in (15). F 0 = ( T ⊗ I ) F . Then, for a nonzero vector ξ , one has ξ ξ ∗ ∈ S 1 ⇔ Γ λ F ξ = 0 for some λ ∈ Λ (Φ , Ψ) ⇔ Γ s F 0 ξ = 0 for some s ∈ Λ (Φ 0 , Ψ 0 ) ⇔ ξ ∗ F ∗ 0 (Φ 0 ⊗ P + Ψ 0 ⊗ Q ) ξ F 0 ≥ 0 for all P , Q ∈ H n , Q ≥ 0 ⇔ ξ ξ ∗ ∈ S 2 , (16) where the first and fourth equiv alences can be deri ved from the definition, the second equiv alence holds based on Lemma 4, and the third equiv alence holds due to Lemma 5. Note that for a singular fractional order system, the matrix F that affects the set M is different from the normal system , so we need to specifically prov e that M satisfies Definition 1. Theorem 2. Let F =  A B E 0  and Φ , Ψ ∈ H 2 be given such that Λ in (6) r epresents curves. Define Γ λ by (11) , Λ by (12), and the set M by (15). Then the set M is admissible and rank-one separable. Pr oof: The proof process is divided into the following two steps. Step 1. The set M is a closed conv ex cone by definition. When M ∈ M > 0 , the set S 1 is not empty . According to Lemma 11 in [23], M is admissible. Step 2. According to Lemma 4, we get Φ ⊗ P + Ψ ⊗ Q =( T ⊗ I ) ∗  αQ P e j θ + β e j θ Q P e − j θ + β e − j θ Q γ Q  ( T ⊗ I ) , (17) 4 where α < 0 < γ or γ ≥ α ≥ 0 . When γ ≥ α ≥ 0 , let V =  e − j θ/ 2 0 0 e j θ/ 2  ( T ⊗ I )  A B E 0  , (18) X = P + β Q, (19) Y = [( T ⊗ I ) F ] ∗  αQ 0 0 γ Q  ( T ⊗ I )  A B E 0  . (20) Then, the set M can be expressed as M = V ∗ M X V + Y with M X defined as M X =  0 X X 0  : X ∈ H n  . (21) When α < 0 < γ , let W =  √ − αI e − j ϕ/ 2 0 0 √ γ I e j ϕ/ 2  ( T ⊗ I )  A B E 0  , (22) X = ( P + β Q ) √ − αγ , Y = Q. (23) Then, the set M can be expressed as M = W ∗ M X Y W with M X Y defined as M X Y =    − Y X X Y   : X, Y ∈ H n , Y ≥ 0  . (24) Since M X and M X Y are proved rank-one separable in [23], it then follows from Lemma 2 that M is rank-one separable. This ends the proof. Theorem 3. (GKYP lemma for SFOS) Let A ∈ R n × n , B ∈ R n × m , Θ ∈ H n + m and Φ , Ψ ∈ H 2 be given. Define Λ and Λ by (6) and (12). Γ λ is defined in (11) and S λ is defined as the null space of Γ λ F . Suppose Λ repr esents curves on the right half complex plane, the following statements are equivalent i) S ∗ λ Θ S λ < 0 , ∀ λ ∈ Λ (Φ , Ψ) . ii) Ther e exist P , Q ∈ H n such that Q > 0 and  A B E 0  ∗ (Φ ⊗ P + Ψ ⊗ Q )  A B E 0  + Θ < 0 . (25) Pr oof: i) holds if and only if tr(Θ S 1 ) < 0 where S 1 is defined in (10) with M in (15). By Theorem 2, the set M is admissible and rank-one separable. Hence, according to (5), condition i) is equi v alent to Θ + M < 0 . Substituting the set M in (15) into the abov e inequality , it is concluded that (25) holds when there exist P, Q ∈ H n such that Q ≥ 0 . Since the inequality in (25) is strict, the positivity of Q can be enhanced to Q > 0 without loss of generality . I V . L ∞ B O U N D E D R E A L L E M M A S F O R T H E S F O S In this section, the L ∞ bound real lemmas for the SFOS in different frequency ranges will be derived. As is known, for a matrix singular G ( s ) , the L ∞ norm of G ( s ) is defined as k G (s) k L ∞ = sup ω ∈ R σ max ( G (j ω )) , where σ max is the maximum singular value. It has the following property . Lemma 6. [11] F or a matrix function G(s), there holds k G (s) k L ∞ = sup ω ≥ 0 σ max ( G (j ω )) , (26) Theorem 4. (L-BR Lemma for SFOSs at Low F r equency) Consider an SFOS whose transfer function G(s) is (2). If the L ∞ performance bound is given as δ > 0 , then for all ω belong to the principal Riemann sheet and ω ∈ Ω L = { ω ∈ R + : 0 ≤ ω ≤ ω L } , k G (s) k L ∞ < δ holds if and only if ther e e xist P , Q ∈ H n , Q > 0 , such that   sym( X E ) − A T QA + W Y ∗ C T Y − δ I − B T QB D T C D − δ I   < 0 , (27) wher e X = e j θ A T P , Y = − B T QA + e j θ B T P E , W = E T ω 2 α L QE , θ = π 2 (1 − α ) . Pr oof: For low frequency , according to Lemma 3, let Φ =  0 e j θ e − j θ 0  , Ψ =  − 1 0 0 ω 2 α L  . Then, according to Lemma 4, Λ (Φ , Ψ) can represent a curve on complex plane with the frequenc y range Ω L . Let λ ( ω ) = e j π 2 α ω α , and then G (j ω ) = C [ λ ( ω ) E − A ] − 1 B + D . According to the definition of σ max , one has sup ω ∈ R σ max ( G (j ω )) < δ ⇔ G ∗ (j ω ) G (j ω ) − δ 2 I < 0 , ∀ ω ∈ Ω L ⇔  H ( λ ) I  ∗ θ  H ( λ ) I  < 0 , ∀ λ ∈ Λ (Φ , Ψ) . (28) where H ( λ ) = ( λE − A ) − 1 B , θ =  C T C C T D D T C D T D − δ 2 I  . According to Theorem 3, the last part of (28) is equiv alent to the following LMI with P , Q ∈ H n , Q > 0 .  A B E 0  ∗ (Φ ⊗ P + Ψ ⊗ Q )  A B E 0  + θ < 0 . (29) Substituting the definitions of Φ and Ψ into the above equa- tion, it follows  A B E 0  ∗  − Q e j θ P e − j θ P ω 2 α Q   A B E 0  + θ < 0 . (30) Let X = e j θ A T P , Y = − B T QA + e j θ B T P E , W = E T ω 2 α L QE and B δ = − δ 2 I − B T QB . The LMI (30) can be simplified as  sym ( X E ) − A T QA + W Y ∗ Y B δ  +  C T D T   C D  < 0 . (31) Then according to the Schur complement theorem in [33], LMI (27) is finally achiev ed. This completes the proof. Theorem 5. (L-BR Lemma for SFOSs at Middle F r equency) Consider an SFOS whose transfer function G(s) is (2). If L ∞ performance bound is given as δ > 0 , then for all ω belong to the principal Riemann sheet and ω ∈ Ω M = 5 { ω ∈ R + : 0 < ω 1 < ω < ω 2 } , k G (s) k L ∞ < δ holds if and only if ther e exist P , Q ∈ H n , Q > 0 , such that   sym( X E ) − A T QA − W Y ∗ C T Y − δ I − B T QB D T C D − δ I   < 0 , (32) wher e X = e j θ A T P , Y = − B T QA + e j θ B T P E , W = E T ω α 1 ω α 2 QE , θ = π 2 (1 − α ) . Pr oof: The theorem of middle frequency can be proved similar to the proof of lo w frequency . According to Lemma 3, the curve Λ (Φ , Ψ) here is chosen as Φ =  0 e j θ e − j θ 0  , Ψ = " − 1 j α ω α 1 + ω α 2 2 ( − j) α ω α 1 + ω α 2 2 − ω α 1 ω α 2 # . (33) Then following the proof of Theorem 4, Theorem 5 can be prov ed. T o avoid duplication, the remaining proof process is omitted here. Theorem 6. (L-BR Lemma for SFOSs at High F r equency) Consider an SFOS whose transfer function G(s) is (2). If L ∞ performance bound is given as δ > 0 , then for all ω belong to the principal Riemann sheet and ω ∈ Ω H = { ω ∈ R + : 0 ≤ ω H ≤ ω } , k G (s) k L ∞ < δ holds if and only if ther e e xist P , Q ∈ H n , Q > 0 , such that   sym( X E ) + A T QA − W Y ∗ C T Y − δ I + B T QB D T C D − δ I   < 0 , (34) wher e X = e j θ A T P , Y = − B T QA + e j θ B T P E , W = E T ω 2 α H QE , θ = π 2 (1 − α ) . Pr oof: The curve Λ (Φ , Ψ) in high frequency is chosen as Φ =  0 e j θ e − j θ 0  , Ψ =  1 0 0 − ω 2 α H  . (35) Similar to the proof process of the previous two theorems, Theorem 6 can be proved to be true. Theorem 7. (L-BR Lemma for SFOSs at Full F r equency) Consider an SFOS whose transfer function G(s) is (2). Given a pr escribed L ∞ performance bound δ > 0 , then k G ( s ) k L ∞ = sup ω ∈ R σ max ( G (j ω )) < δ , wher e ω belongs to the principal Riemann sheet and Ω ∈ Ω 0 = R + ∪ { + ∞} , holds if and only if ther e exists P ∈ H n , such that   sym( X A ) X B C T ( X B ) ∗ − δ I D T C D − δ I   < 0 , (36) wher e X = e − j θ E T P ∗ , θ = π 2 (1 − α ) . Pr oof: The curve Λ (Φ , Ψ) in infinite frequency is chosen as Φ =  0 e j θ e − j θ 0  , Ψ =  0 0 0 0  . (37) Using these two matrices for calculation, Theorem 7 can be prov ed to be correct. Remark 3. This is the first time to establish the universal frame work of the GKYP lemma for the SFOS. When the or der α = 1 , it de generates into the GKYP lemma of the inte ger or der singular system. Ther efor e, the conclusion her e is more general and universal. Besides, for singular systems, the GKYP lemma established her e is completely new and the conclusion is pr oven systematically for the first time, r egar dless of fractional or inte ger order systems. V . L ∞ C O N T RO L L E R S Y N T H E S I S F O R T H E S F O S As the L ∞ performance index of the system matrix is very important to a system, we can design the controller to modify the system which does not meet our requirements. Considering the SFOS in (1), state feedback will be used to design the controller . Let u ( t ) = v ( t ) + K x ( t ) , where v ( t ) ∈ R n is the exogenous input. Then the closed loop system has the transfer function from v to y as G v y ( s ) = ( C + D K ) [ s α E − ( A + B K )] − 1 B + D . (38) Theorem 8. Consider the SFOS system (1) with transfer function G v y ( s ) in (38). If and only if there exist P ∈ R n × n , Q ∈ R m × n , such that for X = e j θ P , Y = e j θ Q and θ = π 2 (1 − α ) , the following LMI is established   sym( AX E + B Y E ) E T ( C X + D Y ) ∗ B ( C X + D Y ) E − δ I D B T D T − δ I   < 0 , (39) ther e holds k G v y k L ∞ < δ . The state feedback controller can be derived as K = Y X − 1 . Pr oof: In order to facilitate the design of the controller, we should make some mathematical deformation for Theorem 7. Considering the duality of the system, let A 1 = A T , B 1 = C T , C 1 = B T , D 1 = D T , and X 1 = X ∗ = e j θ P E , and then (31) can be replaced by  A ∗ C ∗ E 0  ∗ (Φ ⊗ P + Ψ ⊗ Q )  A ∗ C ∗ E 0  + θ < 0 , (40) where θ =  C T C C T D D T C D T D − δ 2 I  . By the deriv ation step by step, the LMI (36) in Theorem 7 is transformed into   sym( A 1 X 1 ) ( C 1 X 1 ) ∗ B 1 C 1 X 1 − δ I D 1 B T 1 D T 1 − δ I   < 0 . (41) By comparing G v y in (38) with the original transfer function in (2), there is A ⇒ A + B K , C ⇒ C + D K . In order to make sure that K = Y X − 1 is real and available, X needs to be nonsingular . Let X 1 = e j θ P E = X 2 E , and then the matrix X 2 is in vertible, and the LMI (39) is reached. Since the controller gain must be real, the conclusion needs to be further prov en. Suppose there exists e P ∈ H n satisfying the LMI (39). According to the property of Hermitian matrix, e P = e P ∗ , one has Re( e P ) = 1 2 ( e P + e P ∗ ) . 6 Let P = Re( e P ) , and then for each e P ∈ H n satisfying the LMI (39), we can find a corresponding P ∈ R n × n also satisfies the abov e condition. V I . S I M U L A T I O N S T U DY Example 1. Consider a singular fractional or der system with the following state space repr esentation         1 0 0 0  D 0 . 5 x =  1 2 2 − 1  x +  1 0  u, y =  1 1  x. Under low fr equency conditions, let 0 ≤ ω ≤ 100 H z , the maximum singular values ar e shown in F ig. 1. frequency(rad/s) 0 10 20 30 40 50 60 70 80 90 100 maximum singular values 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Fig. 1. Maximum singular values of Example 1. It shows that the k G ( s ) k L ∞ index of Example 1 is about 0.85 in the fr equency rang e. Due to Theor em 4, letting δ =0 . 9 , one has U =  − 3 . 1443 − 2 . 8870 − 2 . 8870 0  , V =  0 . 0035 1 . 4166 1 . 4166 5908 . 0319  This implies that k G ( s ) k L ∞ < 0 . 9 is con vinced. Howe ver , when setting δ =0 . 8 , LMI (27) cannot be solved because 0.8 is less than the max value shown in F ig. 1. It means that Theor em 4 is correct. Further , by using the mincx solver in MA TLAB, it can be got that k G ( s ) k L ∞ = 0 . 8486 , which is corr esponds to the maximum value shown in Fig . 1. Example 2. One newer area of r esear ch which must be mentioned is the application of SFOSs to network theory . W esterlund et al. first pr oposed a new linear capacitor model in [34], and after that, many material scientists have studied and pro ved the existence of fractional capacitance and other fractional components fr om differ ent perspectives [35 – 37]. The fr actional capacitance is based on Curies empirical law which states that the current thr ough a capacitor is i ( t ) = u 0 h 1 t α wher e h 1 and α are constants, u 0 is the DC voltage applied at t = 0 , and 0 < α < 1 , ( α ∈ R ) . F or an input voltage u ( t ) , the curr ent is i ( t ) = C 0 D α u ( t ) , wher e C 0 is the capacitance of the capacitor , which is r elated to a kind of dielectric. Another constant α (or der) is r elated to the loss of the capacitor . Let C 0 = 1 , α = 0 . 2 . Applying this special capacitor to the following cir cuit in Fig . 2(a) and using the equivalent cir cuit in F ig. 2(b), a singular fractional or der system is obtained as         1 0 0 0  D 0 . 2 x =  0 1 1 0  x +  0 1  u, y =  0 β R  x, wher e x =  u c i c  T . The singularity of the coefficient matrix reflects the fact that unless u c (0) = − u (0) and i c (0) = 0 , there will be an impulse when the circuit is turned on at t = 0 . C + u ( t ) _ + y ( t ) _ R β ( a ) + u ( t ) _ + y ( t ) _ R β i c i c _ u c + ( b ) Fig. 2. The circuit structure diagram of Example 2. T o simplify the calculation, let R = 1 /β . The maximum singular values of this SFOS are shown in F ig. 3. At the same time, accor ding to the L-BR lemma at infinite frequency band in Theorem 7, there is k G ( s ) k L ∞ = + ∞ . Therefor e, it can be indir ectly judged that this system is unstable and the theor em is corr ect. 0 10 20 30 40 50 60 70 80 90 100 frequency(rad/s) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 maximum singular values Fig. 3. Maximum singular values of the SFOS in Example 2. 7 Example 3. Consider a singular fractional or der system with the following state space repr esentation         1 0 0 0  D 0 . 5 x =  1 2 2 − 1  x +  1 1  u, y =  2 1  x + 0 . 2 u. The maximum singular values are shown in Fig . 4. By using the mincx solver in MA TLAB, there is k G ( s ) k L ∞ = 2 . 8971 . If the r equired performance index is given as k G ( s ) k L ∞ < 1 , then a contr oller method should be designed to achie ve the tar get. Accor ding to Theorem 8, one has K = [ 4 . 850 − 3 . 084 ] . After feedback, k G c ( s ) k L ∞ = 0 . 7964 , which meets the r equirements of the performance indicators. The maximum singular values of the closed-loop system ar e shown in F ig. 5. frequency(rad/s) 0 10 20 30 40 50 60 70 80 90 100 maximum singular values 1.5 2 2.5 3 Fig. 4. Maximum singular values of the open-loop system in Example 3. frequency(rad/s) 0 10 20 30 40 50 60 70 80 90 100 maximum singular values 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Fig. 5. Maximum singular values of the closed-loop system in Example 3. V I I . C O N C L U S I O N S In this paper , a universal frame work of finite frequency band GKYP lemma has been well dev eloped for the SFOS. This is the first time to fundamentally study the singular fractional- order system in finite frequency ranges without normalization constraints, thus the GKYP lemma on it is brand ne w . Based on the proposed GKYP lemma, the L ∞ bounded real lemma of SFOSs in different frequency ranges is obtained, and the controller is designed to ef fectively satisfy the performance index. The future research directions include the analysis and synthesis of SFOSs with time-delay , the control and optimization of uncertain SFOSs in different frequency ranges, the passivity of SFOS, the decoupling of large-scale systems’ singular representation and the applications of finite frequency SFOSs in power , mechanical and aerospace systems. A C K N O W L E D G E M E N T The work described in this paper was fully supported by the National Natural Science Foundation of China (No. 61601431, No. 61573332), the Anhui Provincial Natural Science Founda- tion (No. 1708085QF141), the Fundamental Research Funds for the Central Univ ersities (No. WK2100100028) and the General Financial Grant from the China Postdoctoral Science Foundation (No. 2016M602032). R E F E R E N C E S [1] I. N’Doye, M. Zasadzinski, M. Darouach, and N.-E. 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