Automatica 59 (2015) 216–223 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper New sufficient conditions for observer-based control of fractional-order uncertain systems✩ Salim Ibrir a , Maamar Bettayeb b,c a King Fahd University of Petroleum and Minerals, Electrical Engineering Department, Dhahran, Saudi Arabia b University of Sharjah, Department of Electrical and Computer Engineering, University City, Sharjah, United Arab Emirates c King Abdulaziz University, Jeddah, Saudi Arabia article info Article history: Received 19 January 2014 Received in revised form 24 January 2015 Accepted 21 May 2015 Keywords: Fractional-order systems Observer-based control Convex-optimizations Linear Matrix Inequalities (LMIs) abstract New simple linear matrix inequalities are proposed to ensure the stability of a class of uncertain fractionalorder linear systems by means of a fractional-order deterministic observer. It is shown that the conditions of existence of an observer-based feedback can be split into a set of linear matrix inequalities that are numerically tractable. The presented results show that it is possible to decouple the conditions containing the bilinear variables into separate conditions without imposing equality constraints or considering an iterative search of the controller and the observer gains. Simulations results are given to approve the efficiency and the straightforwardness of the proposed design. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction Fractional-order calculus has a long history and its serves as a modern powerful tool in analyzing various physical phenomena. The interest in understanding systems governed by fractionalorder differential equations has grown up during the last decades and many associated results have been appeared, see e.g., Farges, Moze, and Sabatier (2010), Manabe (1960), Matignon (1996), Oustaloup (1983, 1995) and Trigeassou, Maamri, Sabatier, and Oustaloup (2011). It was found that diffusion processes and biological systems can be modeled in terms of fractional-order differential equations, see e.g., Oustaloup (2014) and Sabatier, Agrawal, and Machado (2007). Additionally, the use of fractional-order derivatives and integrals in feedback design has been successful to a large extent in improving the robustness of the closed-loop systems. Nevertheless, fractional differential equations have not yet received the same attention as ordinary differential equations in the investigation of their stability, simulation, and analysis. Owing to the lack of effective analytic methods for the time-domain analysis and simulation of linear feedback fractional-order systems, ✩ The material in this paper was partially presented at the 19th IFAC World Congress Conference, August 24–29, 2014, Cape Town, South Africa. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (S. Ibrir), [email protected] (M. Bettayeb). http://dx.doi.org/10.1016/j.automatica.2015.06.002 0005-1098/© 2015 Elsevier Ltd. All rights reserved. a numerical simulation scheme is developed in Hwang, Leu, and Tsay (2002). Exact calculation of fractional-order derivatives of some particular polynomial signals is discussed in Samadi, Ahmad, and Swamy (2004). Stability of dynamical systems, represented by fractional order derivatives, has been investigated using the Routh–Hurwitz criteria, the pole placement method, and Lyapunov strategies. For linear fractional-order systems, it was found that the stability is equivalent to the repartition of the system poles in a restricted area of the complex plane. Based on this key formulation of stability and the use of convex-optimization algorithms, stated as linear-matrix-inequality conditions, numerous sufficient conditions have been proposed to ensure robust stability of some classes of fractional-order type systems, see e.g., Ahn and Chen (2008), Farges et al. (2010) and Sabatier, Moze, and Farges (2010). A considerable interest has been also devoted to stability and stabilizability of special classes of fractional-order systems, see e.g., Li, Chen, and Podlubny (2010) and Wen, Wu, and Lu (2008). A new Lyapunov stability analysis of fractional differential equations is discussed in Trigeassou et al. (2011). The problem of pseudo-state feedback stabilization of fractional-order systems using LMI setting was addressed in Farges et al. (2010). In the recent paper (Lan, Huang, & Zhou, 2012), the authors have presented a numerical scheme for stabilization of uncertain commensurate (1 < α < 2) fractional-order systems by means of dynamic output feedback. In Lan and Zhou (2013), observerbased control of a class of uncertain fractional-order systems 0 < α < 1 is studied using convex-optimization tools. Other S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 recent works on stabilization of triangular fractional-order systems can be traced in Zhang, Liu, Feng, and Wang (2013) and references therein. In this paper, we devote our attention to the control of commensurate fractional-order-pseudo-state systems subject to bounded uncertainties and partially-state measurements where the non integer differentiation order is between zero and one. Additionally, we assume that the system uncertainties are randomly distributed in the state matrix and the output matrices as well. By decoupling the necessary conditions into a set of matrix inequalities, we show that the search of the observer and the controller gains can be transformed into a convex optimization problem. A set of sufficient linear-matrix-inequality conditions are developed to ensure the existence of a pseudo-state observer-based controller assuring the asymptotic stability of the pseudo-state system under consideration. It is shown that the developed results are less conservative by demonstration of a case study. In the particular case where the system uncertainty are null, sufficient conditions for stability by dynamic output feedback is given when the non-integer differentiation order is between 0 and 2. Detailed proofs are presented and the efficiency of the proposed design is testified by numerical simulations. 2. Preliminaries Throughout this paper we note by R, R>0 , and C the set of real number, the set of positive real numbers, and the set of complex numbers, respectively. The notation A > 0, with A being a Hermitian matrix (respectively, A < 0), means that the matrix A is positive definite (respectively, negative definite). A′ is the matrix transpose of A. X ⋆ stands for the complex conjugate transpose of the matrix X . The notation X̄ stands for the matrix conjugate of the complex matrix X . The star element in a given matrix stands for any element that is induced by conjugate transposition. The spec(A) denotes the set of the eigenvalues of the matrix A. We note by I and 0 the identity matrix of appropriate dimension and the null matrix of appropriate dimension, respectively. ℜ(Z ) stands for the real part of the complex matrix/number while ℑ(Z ) denotes the imaginary part of the complex matrix/number. The notation [a]G stands for the integer part of the real a. In this paper, Riemann–Liouville fractional differentiation definition is used. Referring to Samko, Kilbas, and Marichev (1987), the fractional integral of a continuously differentiable function f (t ) is defined by: t 1 I α f (t ) = Γ (α) (t − τ )α−1 f (τ )dτ , t >0 (1) 0 where α ∈ R>0 denotes the fractional-integration order, and Γ (α) = +∞ e−x xα−1 dx. (2) 0 The order α fractional derivative of a function f (t ), with α ∈ R>0 , is consequently defined by: dm m−α I f (t ) Dα f (t ) = dt m = 1 d Γ (m − α) dt t (t − τ )m−α−1 f (τ ) dτ ; t > 0 (3) 0 where m is the smallest integer verifying ‘‘m − 1 < α < m’’. In Annaby and Mansour (2012), it has been reported that the Riemann–Liouville fractional derivative of order α coincides with the definition of the Grünwald–Letnikov definition, that is DαGL f (t ) = lim T →0 t T 1 G Tα k=0 Fig. 1. Stability domain of fractional-order linear systems 0 < α < 1. where α k = α(α − 1) · · · (α − k + 1) , k! 1, (−1)k α k f (t − kT ), t > 0, (4) for; k ̸= 0, (5) for; k = 0. The Grünwald–Letnikov definition (4) is a generalization of the ordinary discretization formulas for integer-order derivatives. Depending on the value of the fractional-differentiation order ‘‘α ’’, several stability theorems have been stated, see the results in Matignon (1996) for 0 < α < 1 and Sabatier et al. (2010) for 1 < α < 2. Theorem 2.1 (Moze, Sabatier, & Oustaloup, 2005). Let A ∈ Rn×n be a real matrix. Then, the fractional-order system: Dα x(t ) = A x(t ), 1 < α < 2, (6) π is asymptotically stable, that is, | arg(spec (A))| > α 2 if and only if there exists a symmetric and positive definite matrix P verifying (AP + PA′ ) sin(θ ) ⋆ (AP − PA′ ) cos(θ ) <0 ′ (AP + PA ) sin(θ ) (7) where θ = (1 − α2 )π . The following result concerns the stabilizability of fractional-order linear systems by means of pseudo-state feedback where its proof is given in Farges et al. (2010). As it has been reported in the literature, the stability of fractional-order linear systems is one particular case of domain stability where the eigenvalues of the system should be located in a specific region of the complex plane as shown in Fig. 1. Theorem 2.2 (Farges et al., 2010). The fractional-order system: Dα x(t ) = A x(t ) + B u(t ), where 0 < α < 1, A ∈ Rn×n , and B ∈ Rn×m , is stabilizable by pseudo-state feedback u = Y (rX + r̄ X̄ )−1 x iff ∃X = X ⋆ ∈ Cn×n > 0 and Y ∈ Rm×n such that (rX + r̄ X̄ )′ A′ + A(rX + r̄ X̄ ) + BY + Y ′ B′ < 0, where r = e m 217 i(1−α) π2 (8) , i = −1. 2 The Schur Complement lemma along with the result of the following lemma are extensively used in the proof of the main statement. Lemma 2.3 (Boyd, Ghaoui, Feron, & Balakrishnan, 1994). Given real matrices H, E and Y < 0 of appropriate dimensions, the inequality: Y + HF (t )E + E ′ F ′ (t )H ′ < 0 holds for all F (t ) satisfying F ′ (t )F (t ) ≤ I if and only if there exists an ε > 0 such that Y + ε HH ′ + ε −1 E ′ E < 0. Theorem 2.2 will serve as a starting result for further development. The details are given in the following sections. 218 S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 ′ −1 −1 ′ (rX + r̄ X̄ ) − (rX + r̄ X̄ ) . Since rZ + r̄ Z̄ 3. Observer-based stabilization − The objective of this section is to develop new LMI conditions for stabilization of pseudo-state fractional-order linear systems subject to state and output uncertainties. The ultimate goal of the proposed development is to decouple the necessary conditions guaranteeing the convergence of the observer-based feedback into two sets of convex sufficient conditions that are numerically tractable. 2 cos(θ )ℜ(Z ) − 2 sin(θ )ℑ(Z ) then, (12) holds true. The positivedefiniteness property of the matrix Z is verified by pre-multiplying inequality (11) by (rX + r̄ X̄ )−1 and post-multiplying the same inequality by (rX ′ + r̄ X̄ ′ )−1 . 3.1. Main result Consider the fractional-order-pseudo-state system of commensurate type described by the following dynamics: Dα x = (A + MA FA NA ) x + B u, (9) y = (C + MC FC NC ) x, where α is the non-integer differentiation order (0 < α < 1), x = x(t ) ∈ Rn is the pseudo state vector, u = u(t ) ∈ Rm is the control input, and y = y(t ) ∈ Rp is the system measured output. The real-valued matrices A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rp×n are assumed to be known and constant. The uncertain parameters of the systems are regrouped in the constant matrices FA and FC while we assume that the remaining real-valued matrices MA , MC , NA , and NC are known. Consequently, the uncertain matrices 1A = MA FA NA ∈ Rn×n and 1C = MC FC NC ∈ Rp×n are norm-bounded. Additionally, based upon the knowledge of the upper and the lower bounds of the uncertain parameters of the system, we assume that FA′ FA ≤ I and FC′ FC ≤ I. In this paper, we focus on observerbased control of pseudo-state-fractional-order systems without involving the true state of the fractional order system in the design. Therefore, it is assumed that the pair (A + 1A, B) is controllable and the pair (A + 1A, C + 1C ) is observable in the sense of Kalman for all possible bounded uncertainties 1A and 1C . In the sequel, the time argument is omitted for notation simplicity. In this subsection, we present sufficient conditions under which an observer-based controller of the following form: Dα x̂ = A x̂ + B u + L(C x̂ − y), u = K x̂ (10) could make the closed-loop system: Dα x = (A + 1A) x + B u, u = K x̂ globally stable, where L and K are some design matrices to be determined later. Before presenting the main result, let us introduce the following result. Proposition 3.1. Let r = eiθ where i2 = −1, θ = (1 − α) π2 , 0 < α < 1. If there exists a Hermitian matrix X = X ⋆ > 0 such that 1 (rX + r̄ X̄ ) + (rX + r̄ X̄ )′ cos(θ) i + (rX + r̄ X̄ ) − (rX + r̄ X̄ )′ > 0 (11) sin(θ ) 1 4 sin(θ ) Now, we are ready to present the main result of the paper. Theorem 3.2. Consider system (9) where the uncertain matrices 1A and 1C are not null. Define θ = (1 − α) π2 and r = eiθ such that i2 = −1. If there exist a set of matrices X1 = X1⋆ ∈ Cn×n , X2 = X2⋆ ∈ Cn×n , W ∈ Rn×n , Y1 ∈ Rm×n , Y2 ∈ Rn×p and a set of positive scalars ε1 , ε2 , ε3 , and ε4 such that the following linear matrix inequalities hold simultaneously: X1 > 0, 1 cos(θ ) Z = (12) 4 cos(θ ) − (rX + r̄ X̄ ) i 4 sin(θ ) −1 P1 + P1′ − W L11 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ K11 ⋆ ⋆ ⋆ + (rX + r̄ X̄ ) ′ −1 ⋆ ⋆ P1′ NC′ 0 0 −ε2 I P2 MA (−2 + ε1 )I ⋆ ⋆ (15) P1′ NA′ 0 0 <0 0 −ε4 I ⋆ ′ > 0, ε3 I Y2 MC 0 (−2 + ε2 )I (16) 0 <0 0 −W ⋆ (17) where P1 = (rX1 + r̄ X̄1 ), P2 = (rX2 + r̄ X̄2 ), L11 = P1 A + AP1 + BY1 + Y1′ B′ + ε4 MA MA′ , ′ ′ ′ ′ (18) ′ ′ K11 = A P2 + P2 A + Y2 C + C Y2 then, the observer-based feedback u = Y1 P1−1 x̂ stabilizes system (9) with x̂ = x̂(t ) being the state vector of the pseudo-state fractionalorder observer: Dα x̂ = Ax̂ + B u + (rX2′ + r̄ X̄2′ )−1 Y2 (C x̂ − y), (19) u = Y1 (rX1 + r̄ X̄1 )−1 x̂. Proof. Let e = x̂ − x be the observation error. Then, the closedloop dynamics of system (9) and observer (19) under the feedback u = Y1 P1−1 x̂ are given by the composite system: Dα x e = A + 1A + BY1 P1−1 BY1 P1−1 −1 −1 −1A − P ′ 2 Y2 1C x ∆ = Aclosed . A + P ′ 2 Y2 C x e (20) According to the result of Theorem 2.2, the composite system (20) is stable if there exists a Hermitian matrix Z = Z ⋆ > 0 of dimensions 2n × 2n such that ′ ∆ A∆ closed (rZ + r̄ Z̄ ) + (rZ + r̄ Z̄ ) Aclosed < 0. −1 ′ ′ −1 (rX + r̄ X̄ ) − (rX + r̄ X̄ ) . 1 4 cos(θ) (2ε3 − 1)I P1′ NA′ 0 −ε1 I BY1 −I (14) (P2 − P2′ ) > 0, I ⋆ sin(θ ) ′ ′ (13) (rX + r̄ X̄ )−1 + (rX ′ + r̄ X̄ ′ )−1 and ℑ(Z ) = (21) (P2 + P2 ) + (P2 − P2 ) > 0 Note that if the condition: holds true then, by the use of the result of Proposition 3.1, ∃ Z2 > 0 such that (rX2 + r̄ X̄2 )−1 = rZ2 + r̄ Z̄2 , and therefore, 1 cos(θ ) To verify (12), notice that rX + r̄ X̄ is a real matrix; therefore, ℜ(Z ) = (P2 + P2′ ) + W = W ′ > 0, i e Proof. Setting the matrix Z as 1 X2 > 0, then, there exists a Hermitian matrix Z = Z ⋆ > 0 such that (rX + r̄ X̄ )−1 = rZ + r̄ Z̄ . = diag rX1 + r̄ X̄1 , (rX2 + r̄ X̄2 )−1 ′ i sin(θ ) ′ = diag P1 , P2−1 = rdiag X1 , Z2 + r̄diag X̄1 , Z̄2 . (22) S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 If X1 = X1⋆ > 0 and X2 = X2⋆ > 0 then, by taking Z = diag X1 , Z2 > 0, inequality (21) holds true if the following holds: W = W11 W12 W22 ⋆ where Ω = R22 + ε2 Y2 MC M ′ C Y ′ 2 + P ′ 1 P1−1 . Consequently, G3 < 0 if the following holds true: −1 < 0, (23) R11 + ε2−1 P ′ 1 N ′ C NC P1 ⋆ ⋆ where the entries of the matrix W are: W12 = BY1 P1−1 P2−1 − P ′ 1 1A′ − P ′ 1 1C ′ Y2′ P2−1 , W22 = AP2 ′ −1 ′ +P A +P 2 ′ −1 2 (24) (Y2 C + C Y2 )P2 . BY1 P1−1 P2−1 − P ′ 1 1C ′ Y2′ P2−1 W22 W11 ⋆ 0 F N P MA A A 1 − 0 − NA P1 ′ 0 FA 0 MA ′ . (25) Then, based on the result of Lemma 2.3 and (25), inequality W < 0 holds if there exists ε1 > 0 such that the following holds true: G1 = BY1 P1−1 P2−1 − P ′ 1 1C ′ Y2′ P2−1 W22 W11 ⋆ 0 0 MA + ε1 MA + ε1 ′ −1 G2 = ⋆ = 0 I G1 ⋆ P ′2 I 0 P2 ⋆ 0 < 0. (26) <0 (27) where R11 = W11 +ε1−1 P ′ 1 NA′ NA P1 , R22 = P ′ 2 W22 +ε1 MA M ′ A P2 . Similarly, the matrix inequality G2 < 0 can be rewritten as follows: G2 = −1 R11 ⋆ − BY1 P1 R22 0 − F N P Y2 MC C C 1 P ′1N ′C F ′C 0 0 0 G3 = ⋆ (28) 0 0 0 M ′ C Y2′ + ε2 Y2 MC R + ε2−1 P ′ 1 N ′ C NC P1 BY1 P1−1 = 11 ⋆ R22 + ε2 Y2 MC M ′ C Y ′ 2 < 0. (29) The matrix inequality G3 < 0 is not linear. In order to decouple the LMI variables, notice that the matrix G3 can be decomposed as follows: G3 = I 0 0 ′ −1 P I × 0 0 1 R11 + ε −1 P ′ 1 N ′ C NC P1 2 ⋆ I ⋆ 0 0 P1−1 I 2 +P < 0, ′ −1 −1 P1 1 (32) <0 ε3 > 0, (33) −1 P ′ 1 P1−1 < ε32 W −1 =⇒ ε32 P1 P1′ > W . (34) Then, ε > W if P1 + P 1 − W − ε3 I > 0 which is equivalent by the Schur complement lemma to 2 ′ 3 P1 P1 −2 ′ P1 + P ′ 1 − W I I ε 2 3I > 0. (35) Since ε32 > 2ε3 − 1 then, the linear matrix inequality (15) is a sufficient condition to fulfill (35). Furthermore, using the Schur complement, inequality R22 + ε2 Y2 MC M ′ C Y ′ 2 + ε32 W −1 < 0 is verified if the following holds: R22 + ε2 Y2 MC M ′ C Y ′ 2 ⋆ ε3 I < 0. −W (36) The last inequality is equivalent by the Schur complement lemma to the following inequality: M ′ C Y2′ < 0. P ′1N ′C BY1 P1−1 NC P1 + ε2−1 0 R22 R11 ′ must hold simultaneously to ensure G3 < 0. Let us now introduce new scalar LMI variable: ε3 > 0 and a new LMI variable W ∈ Rn×n −1 such that P ′ 1 P1−1 < ε32 W −1 . This last inequality does not restrict the design of P1 because any positive definite matrix can be upperbounded by another positive definite matrix. Then, the second inequality of (32) is verified if the following condition holds true, −1 that is R22 + ε2 Y2 MC M ′ C Y ′ 2 + ε32 W −1 < 0, P ′ 1 P1−1 < ε32 W −1 . Since Applying again Lemma 2.3, a necessary and sufficient condition to fulfill (28) is given by: ⋆ (31) Ω and, P ′ 1 NA′ NA P1 0 BY1 P1−1 − P ′ 1 1C ′ Y2′ R22 R11 BY1 −I P1 + P ′ 1 ≤ ε32 P1 P ′ 1 + ε3−2 I , It is obvious that G1 < 0 if the following matrix inequality is verified: R11 + ε2−1 P ′ 1 N ′ C NC P1 R22 + ε2 Y2 MC M C Y ′ ⋆ ′ By expanding the matrix W as W = −1 ′ ′ 0 0 < 0. BY1 −I This implies immediately that the following conditions: W11 = (A + 1A)P1 + P1′ (A + 1A)′ + BY1 + Y1′ B′ , −1 219 BY1 −I ⋆ 0 0 Ω (30) P ′ 2 A + A′ P2 + Y2 C + C ′ Y2′ ⋆ ⋆ ⋆ P ′ 2 MA −ε1−1 I ⋆ ⋆ Y2 MC 0 −ε2−1 I ⋆ ε3 I 0 < 0. (37) 0 −W Since −ε1−1 < ε1 − 2 and −ε2−1 < ε2 − 2 then, the linear matrix inequality (17) is a sufficient condition to fulfill (37). The first inequality of (32) is equivalent by the Schur Complement lemma to inequality (38) as given in Box I. Using Lemma 2.3, we could find ε4 > 0 such that 1AP1 + P ′ 1 1A′ ≤ ε4 MA MA′ + ε4−1 P ′ 1 NA′ NA P1 . Therefore, the condition given by (39) in Box I is a sufficient condition to fulfill (38). Finally, one can easily verify that (39) is equivalent by the Schur Complement lemma to (16). This ends the proof. In case where the system uncertainties are null, the computation of the controller and the observer gains are summarized in the following statement. Corollary 3.3. Consider the pseudo-state fractional-order system (9) with 1A = 0 and 1C = 0. Let θ = (1 − α) π2 and r = eiθ where i2 = −1. If there exist two complex, Hermitian, and positive definite matrices X1 = X1⋆ ∈ Cn×n , X2 = X2⋆ ∈ Cn×n , a real, symmetric, and positive definite matrix W ∈ Rn×n , a positive scalar ε , and two 220 S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 (A + 1A)P1 + P ′ 1 (A + 1A)′ + BY1 + Y1′ B′ BY1 P1′ NA′ ⋆ −I 0 ⋆ ⋆ −ε1 I ⋆ ⋆ ⋆ AP1 + P ′ 1 A′ + BY1 + Y1′ B′ + ε4 MA MA′ + ε4−1 P ′ 1 NA′ NA P1 ⋆ ⋆ ⋆ P1′ NC′ 0 < 0. 0 −ε2 I BY1 −I ⋆ ⋆ P1′ NA′ 0 −ε1 I (38) P1′ NC′ 0 <0 0 −ε2 I ⋆ (39) Box I. matrices Y1 ∈ Rm×n , Y2 ∈ Rn×p such that the following linear matrix inequalities hold true: 1 cos(θ) (P2 + P2′ ) + P1 + P1′ − W ⋆ i sin(θ ) I (40) > 0, (2ε − 1)I (41) P1′ A′ + AP1 + BY1 + Y1′ B′ BY1 −I A′ P2 + P2′ A + Y2 C + C ′ Y2′ εI < 0, −W ⋆ (P2 − P2′ ) > 0, ⋆ < 0, (42) (43) where P1 = (rX1 + r̄ X̄1 ) and P2 = (rX2 + r̄ X̄2 ) then, the observerbased feedback u = Y1 P1−1 x̂ stabilizes system (9) when all the uncertainties are null with x̂ being the state trajectory of the pseudostate fractional-order observer: Dα x̂ = A x̂ + B u + (rX2′ + r̄ X̄2′ )−1 Y2 (C x̂ − y), (44) u = Y1 (rX1 + r̄ X̄1 )−1 x̂. Proof. The new conditions are obtained by putting MA , NA , MC , and NC equal to zero. The outline of the proof remains unchanged as shown in the proof of Theorem 3.2. Proof. First of all, it is important to stress that even Theorem 2.1 is a necessary and a sufficient condition for checking a system stability with 1 < α < 2, it can be also used for stabilizability design for 0 < α < 1. In other words, conditions (7) are only sufficient conditions for 0 < α ≤ 1. This comes from the fact that inequality (7) is satisfied if and only if the diagonal block A′ P + PA is negative definite with sin(α π2 ) being always positive for 0 < α < 1. Having A′ P + PA < 0 implies that all the eigenvalues of A are located in the left-hand side of the complex plane which leads to the conclusion that | arg(spec(A))| > α π2 , 0 < α < 1, see Matignon (1996) for more details. The system and the observer dynamics are given by the following composite system: Dα x A + BK = e 0 BK A + LC x x = Aclosed . e e (49) The design of the observer-controller gains L and K is conditioned by the stability of the fractional-order augmented system (49). Based on the stability result of fractional-order systems, see the statement of Theorem 2.2, one can conclude that system (49) is stable if there exists a matrix X = X ′ = P −1 > 0, of appropriate dimension, such that the following holds: 3.2. Sufficient conditions with equality constraint A1 ⋆ A2 A1 < 0 or, Π1 ⋆ Π2 < 0, Π1 (50) Here, it is showed that the observer-based conditions for stabilization of fractional-order systems can be reformulated as linear matrix inequalities subject to equality constraint. The obtained result can be applied to fractional order-systems where the degree of differentiation α can take non-integer values inside the interval ]0 2[. The whole design is given by the following statement. where A1 = (Aclosed X + XA′closed ) sin(θ ), A2 = (Aclosed X − XA′closed ) cos(θ ), θ = (1 − α2 )π , Π1 = (PAclosed + A′closed P ) sin(θ ), and Π2 = (PAclosed − A′closed P ) cos(θ ). Let us now take the partition of P as P = diag(P1 , P2 ) where P1 ∈ Rn×n , P2 ∈ Rn×n . Let Z ∈ Rm×m be any full-rank matrix and let Y ∈ Rn×p be any realvalued matrix. Then, by introducing the following constraint: Theorem 3.4. Consider the pseudo-state fractional-order system P1 B = BZ , α D x = A x + B u, y = C x, (45) where 0 < α < 2, A ∈ R , B ∈ R , and C ∈ R pseudo-state observer-based controller as: n×n α D x̂ = A x̂ + B u + L(C x̂ − y), α n×m p×n . Define the u = K x̂. Let θ = (1 − 2 )π . If there exist two symmetric matrices P1 ∈ R (46) n×n > 0, P2 ∈ Rn×n > 0, a matrix Y ∈ Rn×p , a matrix K̄ ∈ Rm×n and a matrix Z ∈ Rm×m such that the following set of linear matrix inequality hold simultaneously: R1 ⋆ R2 R1 < 0, P1 B = BZ , (47) where R1 and R2 are defined in (48) as given in Box II. Then, the pseudo-state system (45) is stable under the action of the observer controller u = K x̂ where x̂ is the state vector of the fractional-order pseudo-state observer (46) with K = Z −1 K̄ and L = P2−1 Y . (51) and set the controller and the observers gains as K = Z −1 K̄ , L = P2−1 Y then, A′ P1 + K̄ ′ B′ Aclosed P = K̄ ′ B′ ′ 0 A′ P2 + C ′ Y ′ . (52) This implies that the stability of the fractional-order system (45) is dependent on the solvability of the linear matrix inequalities (47). This ends the proof. The statement of Theorem 3.4 summarizes the design of dynamic-output stabilizing feedbacks for larger class of fractional order linear systems. However, the constraints imposed on the matrix P1 introduce some conservatism. Extensive simulations have also shown that the LMIs of Theorem 3.4 are not feasible for some single-input fractional-order systems. S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 A′ P1 + P1 A + K̄ ′ B′ + BK̄ R1 = sin(θ ) ⋆ BK̄ sin(θ ) 221 A′ P2 + P2 A + YC + C ′ Y ′ sin(θ ) , (48) BK̄ cos(θ ) P1 A − A P1 − K̄ B + BK̄ cos(θ ) . R2 = P2 A − A′ P2 + YC − C ′ Y ′ cos(θ ) −K̄ ′ B′ cos(θ ) ′ ′ ′ Box II. 4. Numerical simulation and comparison given by: x((k + 1)T ) = Consider the pseudo-state fractional-order system D0.85 x1 = γ1 x1 + x2 , D0.85 x2 = γ2 x2 + γ3 u, D ξ= y= 1 0 0 0 1 0 0 0 0 1 + 1A ξ + 0 v, −2 1 (54) 0 + 1C ξ 0 + BT α u(kT ); where v is the new control input, MA = 0 0.4 0 0 0 0 0.35 0 0.3 0 0 0 0.3 0 0 0.4 0 , NA = , MC = [0.3 0 0.2], and NC = [0.5 0 0.99]. Under the aforementioned conditions, the LMIs of Theorem 3.2 were found feasible, where ε1 = 1.033, ε2 = 1.1211, ε3 = 1.2661, ε4 = 4.5991 and 1.0068 −0.31098 − 0.077746 i 0.68714 ⋆ 1.4665 −1.2708 − 0.31771 i 1.3144 ⋆ ⋆ ⋆ X1 = ⋆ ⋆ X2 = W = 1.9817 −0.70339 −0.54847 −0.70339 1.363 −0.47823 −0.30851 − 0.077 i −0.30734 − 0.076 i , 1.362 −0.61199 − 0.153 i 0.62248 + 0.15562 i , 1.1443 −0.54847 −0.47823 , 2.0494 Y1 = −0.85468 −2.2352 −0.90173 , Y2 = (55) −3.101 −1.5662 . 0.80779 The numerical simulation of the observer-based controller is done by discretization of the pseudo-state models using the Grünwald–Letnikov definition (4) where the sampling period is ′ set to T = 10−4 (s), FA = I, and FC = 1 0 0 . To get an approximate discrete model for the fractional-order continuoustime system, we set Dα x(kT ) ≃ k+1 1 T α i=0 (−1)i α k x((k + 1 − i)T ). (56) The numerical approximation scheme (56) generates an error of order T . Using (56), the approximated discrete-time system is k ≥ 1, (57) and for k = 0 x((k + 1)T ) = (A + 1A)T α + α I x(kT ) + BT α u(kT ). (58) The initialization of the pseudo-state system is realized by integrating the uncertain system from −10 ≤ t ≤ 0 where we assume that a constant input u(t ) = 1 is applied. After integration of the system for 10 (s) using the approximate schemes (57)–(58) with null initial conditions at t = −10 (s), we have peaked the values of the pseudo-state x at the end of the simulation and set them as the initial conditions for the simulationsof the observer-based con- troller. By setting x1 (0) x2 (0) u(0) = 71.699 8.2722 1 and x̂1 (0) x̂2 (0) u(0) = 0 0 1 ,the histories of the system states and the observation errors are given in Fig. 2. The spectrums of the vertices of the closed-loop matrix A∆ closed are shown in Fig. 3. To discuss and compare our results with similar published results, we restrict our comparison to commensurate fractional order systems with 0 < α < 1. In order to apply the results in the same conditions and ensure a fair comparison without state augmentation, all the system uncertainties are put to zero while comparing the results. Let us reconsider the pseudo-state fractional-order system (53) where γ1 = γ2 = 0, γ3 = γ4 = 1, and γ5 = 0. This yields to the following dynamics: 0.4 i i =2 where γi ; 1 ≤ i ≤ 5 are positive, constant, and uncertain parameters having the following upper and lower bounds: 0 ≤ {γ1 , γ2 } ≤ 0.12, 1 ≤ γ3 ≤ 1.12, 1 ≤ γ4 ≤ 1.15 and 0 ≤ γ5 ≤ 0.297. System (53) is not stable and y is assumed here to be dependent on both the state x1 and u. By adding the dynamics: ′ ′ D0.85 u = −2u + v to system (53) and put ξ = x u , the fractional-order system takes the following form: 0.85 (A + 1A)T α + α I x(kT ) k +1 i α − (−1) x((k + 1 − i)T ) (53) y = γ4 x1 + γ5 u, D0.85 x = 0 0 1 0 x+ u, 0 1 y= 1 0 x. (59) When the system uncertainty is equal to zero, the LMIs given by Theorem 2 in Lan and Zhou (2013) cannot be verified due to the constraint imposed on one of the LMI variable. As a second comparison, we found that the LMIs given by Theorem 3.2 stated in Ref. Lan and Zhou (2011) are not feasible for the same example (59). The presented results, given by the statement of Theorem 3.2 in this paper, show a useful decomposition of the observer and the controller gains. This decomposition, that decouples the design of the controller and the observer gains, is not penalized by a stringent conditions on the design of the matrices P1 and P2 . The first two conditions (14) and (15), in the statement of Theorem 3.2, translate some necessary constraints on the design of the matrices P1 and P2 . These constraints are not conservative because the constraint (14) and (15) are easily verified for 0 < α < 1, where cos(θ ) > 0, ε3 > 0, and W > 0. The last inequality (17) defines the condition that must satisfy the observer gain independently from the input matrices. Now, we show the usefulness of Theorem 3.4 in handling stabilizability issues with 222 S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 α= 19 , 20 the solutions of the LMIs of Theorem 3.4 are 589 0 4110 878 P1 = ⋆ 2051 ⋆ ⋆ 13649 −153 7052 419 1255 P2 = ⋆ 1613 ⋆ ⋆ 0 −683 , 3882 878 2051 −209 977 −559 , 1568 911 (61) 846 −6274 589 4110 Z = ⋆ Fig. 2. The pseudo states, the control input and the observation errors. 0 < α < 2. Consider the fractional order system 3 2 D x = −1 α 0 y= 1 0 1 0 1 2 2 0 0 1 u, 1 941 1549 −923 Y = , 2017 , 148 3732 987 1 1 x + 0 0 0 (60) 1 x. System (60) is unstable for some values of α < 1 and other values of α > 1. For α = 32 , the solutions of the LMIs of Theorem 3.4 are found feasible, but they are not shown for space limitation. For −935 184 −384 1318 K̄ = −121 1187 −328 1211 . −552 4085 1063 1913 As illustrated in Fig. 4, the closed-loop eigenvalues are located in the stability domain which affirms that the sufficient conditions of Theorem 3.4 can be used for stabilizability of commensurate fractional-order systems with 0 < α < 2. Fig. 3. The spectrums of the vertices of A∆ closed for α = 0.85 and α = 0.4. Fig. 4. The spectrums of A and Aclosed for α = 0.95 and α = 1.5. S. Ibrir, M. Bettayeb / Automatica 59 (2015) 216–223 5. Conclusion New LMI conditions are proposed to solve the problem of observer-based stabilization of a class of fractional-orderpseudo-state systems. The presented results are mainly devoted to uncertain commensurate fractional-order systems with noninteger differentiation order between 0 and 1. For fractional-orderpseudo-state systems without uncertainties, a unified result is given for differentiation order between 0 and 2. The transformation of the initial non-convex optimization problem into a set of convex optimization sub-problems makes the design simple and straightforward. Additionally, the proposed design reveals less conservative as demonstrated through case studies. Extension of the obtained results to other classes of fractional-order systems remains an open problem for further investigation. References Ahn, H. S., & Chen, Y. (2008). Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica, 44(11), 2985–2988. Annaby, M. H., & Mansour, Z. S. (2012). In J. M. Morel, & B. Teissier (Eds.), q-fractional calculus and equations. Berlin: Springer. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. Philadelphia: SIAM. Farges, C., Moze, M., & Sabatier, J. (2010). 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Zhang, X., Liu, L., Feng, G., & Wang, Y. (2013). Asymptotical stabilization of fractional-order linear systems in triangular form. Automatica, 49(11), 3315–3321. Salim Ibrir received his B.Eng. degree in Avionics from Blida Institute of Aeronautics, Algeria, in 1991. He obtained his M.Sc. from INSA de Lyon, France, in 1994 and his Ph.D. degree from Paris-11 University, in 2000, both in Electrical Engineering. From 1999 to 2000, he was a research associate ATER in the Department of Physics of Paris-11 University. He held several research positions in diverse North American universities before joining The University of Trinidad and Tobago as an Associate Professor. He had been with the University of The West Indies before joining the Department of Electrical Engineering at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. He has served as a member in many international conferences and IEEE committees. He has co-authored more than 100 publications that appeared in prestigious journals, conferences, and book chapters. His current research interests are in the areas of modeling and control of fractional-order systems, model reduction, state estimation, nonlinear observer-based control, adaptive control of non-smooth nonlinear systems, robust system theory and applications, time delay systems, hybrid systems, ill-posed problems in estimation, and Aero-Servo-Elasticity. Maamar Bettayeb received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from the University of Southern California, Los Angeles, in 1976, 1978, and 1981, respectively. He worked as a Research Scientist at the Bellaire Research Center at Shell Oil Development Company, Houston, Texas, USA, during 1981–1982. From 1982 to 1988, he directed the Instrumentation and Control Laboratory of the High Commission for Research in Algeria. In 1988, he joined the Electrical Engineering Department at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, where he stayed until 2000. He has been Professor at the University of Sharjah, UAE, since August 2000. He is also currently a Distinguished Adjunct Professor at the College of Engineering, King Abdulaziz University, Jeddah, KSA. He has published over 250 journal and conference papers in the fields of control and signal processing. He has also supervised over 50 M.Sc. and Ph.D. students. His research interests are in optimal control, model reduction, signal and image processing, networked control systems, fractional dynamic modeling and control, soft computing, wavelets, renewable energies, and engineering education.
