H∞ State Feedback Controller Based on Dynamic Observer Design for Singular Fractional-Order Systems

Authors: minghui  wei
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-30 10:48:37

Abstract: This paper focuses on the problem of H∞ state feedback controller design based on dynamic observer for singular fractional-order systems (FOS), where the fractional derivative order α lies between 0 and 1. First, a new form of dynamic observer with a non-singular structure is proposed, which is easier to implement physically. Secondly, the bounded real lemma corresponding to H∞ norm of FOS is proposed via a set of linear matrix inequalities (LMIs). Compared to existing methods,the lemma employs real variable, which is easier to solve. Building upon the new lemma, the conditions for designing H∞ state feedback controller based on dynamic observer of FOS are derived. Finally, numerical example is presented to validate the effectiveness of the proposed method.

Main paper: 
 i. introduction: 
 In the past decade, fractional-order calculus has garnered considerable attention from physicists and engineers [1] . It has been observed that many systems across various interdisciplinary fields can be effectively described using fractional derivatives because these derivatives capture the historical evolution of functions and demonstrate stronger global correlations compared to integer derivatives. Numerous systems exhibit fractional-order dynamics, including viscoelastic systems [2] , dielectric polarization [3] , electrode-electrolyte polarization [4] , electromagnetic waves [5] , quantitative finance [6] , and the quantum evolution of complex systems [7] .Singular systems, also known as generalized systems, encompass both differential and algebraic equations [8] . This model accounts for physical constraints, static relationships, and broader impulsive behaviors due to improper transfer matrices. In contrast to non-singular systems, singular FOS provide a more precise representation of the physical properties of systems, offering direct and comprehensive descriptions [9] . Since their introduction in many fields of system design and control, singular FOS have received considerable attention. They have diverse applications in electrical systems, This study was funded by National Natural Science Foundation of China (grant number 62003223). large-scale interconnected networks, power grids, constrained mechanical systems, and chemical processes [10] - [12] .In control system design, state feedback controllers are typically designed to meet specific performance criteria, especially when access to all states of the considered system is unavailable, or when system output measurements cannot provide complete information about the internal system states. This poses challenges for the design of state feedback controllers [13] . Therefore, the theory of observer design has attracted widespread attention. Based on state estimates obtained from the observer, the observer based controller is used to generate control laws to stabilize unstable systems or ensure desired performance [14] , [15] . Recently, research activities focusing on observer-based control for FOS have been developed. In [22] , a novel observer-free synchronization method is introduced for a specific category of incommensurate fractionalorder systems. [23] studied the robust H ∞ observer control of linear time-invariant perturbed uncertain FOS. By analyzing the H ∞ norm of the FOS and considering the fractional derivative α, a new sufficient condition is proposed to ensure the stability of the estimation error system.H ∞ control plays a crucial role in control systems. The H ∞ optimization used under the presence of disturbances with bounded energy allows guaranteeing levels of disturbance attenuation. However, it is typically limited to integer-order systems [17] . In recent years, there have been developments extending the computation of H 2 and H ∞ norms to FOS. The H 2 norm of fractional transfer functions of implicit type is studied in [16] . [18] employs two methods based on a binary algorithm and LMI condition to compute the H ∞ norm of FOS and determine the Hamiltonian matrix. Using the generalized Kalman-Yakubovich-Popov (KYP) lemma, the bounded real lemmas for the H -norm and H ∞ norm of FOS are derived via a series of LMIs in [19] . Based on these analysis results, numerous studies have focused on designing H ∞ controllers and observers. [21] studied the finite time H ∞ control problem of fractional order neural networks using finite time stability theory and Lyapunov sample function method. The H ∞ control problem for singular FOS with order ranging from 0 to 1 is explored in [20] .In this work, the problem of designing H ∞ state feedback controller based on dynamic observers for singular FOS is studied. The main contributions can be summarized as follows:• A dynamic observer is proposed. Compared with [26] ,the observer in this paper has a non-singular structure, making it easier to implement. • Novel necessary and sufficient conditions for the bounded real lemma corresponding to H ∞ norm for singular FOS ranging 0 < α < 1 are proposed. Unlike previous approaches, such as [24] and [25] , the matrix variable is real, which is easier to solve. • Based on the bounded real lemma, the conditions for designing the dynamic observer are given via a set of LMIs. Notations: In the subsequent sections of the paper, A is a hermitian matrix if and only if A * = A and A > 0. Re(Q) and Im(Q) represent the real and imaginary parts of the complex Q, respectively.Sym(A) = A + A T . Proposition 1. A complex Hermitian matrix Q satisfies Q < 0 if and only if R e (Q) I m (Q) -I m (Q) R e (Q) < 0Consider the following singular FOS:EQUATIONin which α is the fractional-order, ranging from 0 < α < 1, x ∈ R n is the pseudo state vector, y ∈ R q is the output vector, z ∈ R r is the control output, u ∈ R m is the control input, w ∈ R p is disturbance input. A, B, B w , C, C z are the constant matries for the appropriate dimensions, E ∈ R n×n is the singular matrix, which is rank(E) < n. D α denotes the Caputo fractional derivativeEQUATIONIn this paper, it is assumed that E and C such thatEQUATIONThen, consider the following observer based controller EQUATIONEQUATIONwhere,Ē =   E 0 0 0 I 0 0 0 I   , Ā =   A + BK -BK 0 0 T A -D d C -C d 0 B d C A d   , B =   B w T B w 0   , C = C z + D z K -D z K 0 .The transfer function of system ( 6) isEQUATIONThe design problem of H ∞ state feedback controller based on dynamic observer is to design a controller such that the closed-loop system ( 6) is admissibility, and its transfer function satisfies ||G(s)|| ∞ < γ. Lemma 1. [27] Let γ be a scalar such that γ > 0. The singular FOS is admissible and satisfies the conditionEQUATIONwhere r = e jθ , θ = π 2 (1 -α). Lemma 2. [28] Let γ be a scalar such that γ > 0, the following statements hold the same significance:(i) there exists ĒP = P * ĒT ∈ C n×n > 0 such thatEQUATIONwhere r = e jθ , θ = π 2 (1 -α). (ii) there exists matrix M ∈ R n×n such thatEQUATIONEQUATIONwhere θ = π 2 (1 -α), a = 4cosθ, b = 4sinθ. Proof. Define Q 1 = ĒP 1 = R e (Q), Q 2 = ĒP 2 = I m (Q).According to Proposition 1, the conditionEQUATIONSince r = e jθ = cosθ+isinθ and r = e -jθ = cosθ-isinθ, then it yieldsEQUATIONNote that ĒP 1 is real symmetric matrix, while ĒP 2 is skewsymmetric matrix. ThereforeEQUATIONLet Q = r Q +r Q, M = rP + r P , then Q = ĒM , Then ( 9) is equivalent to (10) . From ( 13) and ( 14), we obtainEQUATIONCombining ( 15) and ( 12), it follows that (11) holds, which is equivalent to condition Q = ĒP 1 + ĒP 2 i > 0. This ends the proof.In this section, the problem of designing observer based controller is transformed into an optimization problem of LMIs. The conditions for designing H ∞ state feedback controller based on dynamic observer for singular FOS are provided in the form of LMIs.Theorem 1. Let the system (6) be admissible and γ > 0. For G(s) ∞ < γ if and only if there exists matrices X, Y , H, S, G, L and W such thatEQUATIONΩ 11 = Sym(A X +BS), Ω 22 = Sym(T A X -H), Ω 32 = A T T T -C T L T +W , Ω 33 = Sym(Y T A + GC), Ω 41 = C z X +D z S, Q =   EX 0 E 0 X I 0 Z Y   .EQUATIONwhere V and U are invertible, which satisfiesEQUATIONProof. According to Lemma 2, the singular FOS is admissible and satisfies G(s) ∞ < γ if there exists a matrixEQUATIONEQUATIONNext, the target is to linearize the condition (20) as described in [29] . First, M and M -1 asM =   X 0 V -T 0 X (I -XY T )V -T 0 U -U Y T V -T   , M -1 =   X -1 -X -1 U -1 0 Y T (I -Y T X)U -1 0 V T -V T X U -1   ,where V and U are invertible, which satisfies Z = Y X +V U , and defineF =   I 0 0 0 I 0 0 Y V   .Multiplying both sides of (20) with the diagonal matrix {F, I, I} and its transpose, we obtainEQUATIONwhereEQUATIONEQUATIONH = D d C X +C d U, W = Y T A X -Y D d C X +V B d CX -Y C d U + V A d U, G = V B d -Y D d .Combining (24) and (22) , it follows that (16) holds. Then multiplying both sides of (21) with the F and F T , and define Z = Y X +V U , this concludes the proof.Consider the fractional-order circuit depicted in Fig. 1 [30] , which can be characterized ase 1 = L 1 d α i 1 dt α + L 3 d α i 3 dt α + R 1 i 1 + R 3 i 3 , e 2 = L 2 d α i 1 dt α + L 3 d α i 3 dt α + R 2 i 2 + R 3 i 3 , i 3 = i 1 + i 2 . Let x(t) =   i 1 i 2 i 3   , z(t) = Cx(t). Then   L 1 0 L 3 0 L 2 L 3 0 0 0   D α x(t) =   -R 1 0 -R 3 0 -R 2 -R 3 1 1 -1   x(t) +   1 0 0 1 0 0   u(t) +   1 1 1   w, z(t) = 1 1 1 x(t).Consider the appropriate parameters as followsEQUATIONThen the solution to ( 5) isT =   1 -1 2 0 1 3 0 1 3   , N =   1 0 0 0 -1 1   .According to Theorem 1, we obtain that Fig. 2 shows state vector and state estimation vector. The maximum singular values of G(s) are plotted in Fig. 3 , peaking at approximately 0.4031. The state diagram of system (6) illustrated in Fig. 4 . It implies that the system is stable and admissible. A d =   -0.The problem of designing H ∞ state feedback controller based on dynamic observer for singular FOS is investigated in this paper. Firstly, a non-singular H ∞ state feedback controller based on dynamic observer is proposed. Additionally, a new bounded real lemma for the H ∞ norm of FOS is presented, which forms the foundation for designing the dynamic observer. Then, conditions for designing H ∞ state feedback controller based on dynamic observer for FOS are derived. Ultimately, the effectiveness of the proposed methodology is confirmed through a simulation example. 
 
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Reviews: review: title: title: This paper can be accepted.
main_review: main_review: 1. While the paper presents a solid approach to the design of H∞ state feedback controllers for singular fractional-order systems (FOS), the level of innovation could be strengthened. It would be helpful to clarify how the proposed method significantly advances or differs from existing approaches in the literature. What specific advantages does the new dynamic observer structure offer over traditional methods beyond ease of physical implementation?
2. It was noted that the formatting of the references is not consistent throughout the paper. Please ensure that all references adhere to a uniform citation style, as this is important for the overall professionalism and readability of the paper.
title: title: This paper can be accepted.
main_review: main_review: This paper is well written and the contributions are solid. However, I have some suggestions 
1. The definitions of some notations should be given, for example, $R^{r}$, $\mathbf C$.
2. The references should be updated. For example, some dynamic controller design results,  IEEE TFS 26 (6), 3301-3313. 
3. The motivation of the study of fractional-order systems should be strengthened.