Sliding Mode Control for Uncertain Input Delay Fractional Order Systems

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Abstract: A sliding mode control is developed for robust stabilization of fractional-order input-delay linear systems in presence of uncertainties and external ...
Sliding mode control for uncertain input delay fractional order systems Amar Si Ammour ∗ , Said Djennoune ∗ , Malek Ghanes ∗∗ , Jean-Pierre Barbot ∗∗ and Maamar Bettayeb ∗∗∗ ∗

Laboratoire de Conception et Conduite des Syst` emes de Production University Mouloud Mammeri of Tizi-Ouzou Tizi-Ouzou, Algeria (e-mail: siammour [email protected]; s [email protected]). ∗∗ Laboratoire ECS, Ecole Nationale Sup´ erieure de l’Electronique et ses Applications, Cergy-Pontoise, France (email: [email protected]; [email protected]). ∗∗∗ Department of Electrical & Computer Engineering, University of Sharjah, United Arab Emirates and Distinguished Adjunct Professor, King Abdulaziz University, KSA (email: [email protected]). Abstract: A sliding mode control is developed for robust stabilization of fractional-order input-delay linear systems in presence of uncertainties and external disturbances. First, a fractional-order state predictor is used to compensate the delay in the input control. Second, a robust sliding mode control is proposed in order to stabilize the system and to thwart the effect of model uncertainties and external disturbances. The sliding mode controller is designed by considering a sliding surface defined by fractional order integral. Conditions for the existence and finite time reaching of the sliding mode in presence on bounded uncertainties and bounded external disturbances are established. Further, the stability of the closed loop system on the sliding mode is analyzed by using recent results on Lyapunov stability theory for fractional order systems. A numerical example is given to illustrate some theoretical results. Keywords: Fractional order systems, Sliding mode control, Input-delay systems. 1. INTRODUCTION Sliding mode control technique has been recognized as a powerful tool to design robust control for linear and non linear systems. This technique possesses many attractive feature such as finite-time stability and robustness with respect to disturbances and to some classes of model uncertainties (Utkin [1992], Perruquetti and Barbot [2002], Bartolini et al. [2008]). The sliding mode control principle consists on to force the system trajectories to reach a suitably desired sliding surface by injecting a discontinuous control. In recent years, fractional order systems have been a growing interest and have enjoyed wide success in modeling physical phenomena and in the synthesis of robust controllers. Several books, (Podlubny [1999], Kilbas et al. [2006], Sabatier et al. [2007], Monje & al. [2010]), provide a good source of references on fractional calculus and its applications. In control theory, many studies have been made to extend the most control design methods well known in the classical integer order modeling to fractional order systems. For linear systems, in the frequency domain, a fractional order Tilt Integral Derivative (TID) controller (Lurie [1994]) and the well known CRONE (Commande Robuste Ordre Non Entier) controllers (Sabatier et al. [2002]) have been firstly developed. Later, fractional order PID controller (Podlubny [1999]), fractional lead-lag compensator (Monje et al. [2005]) are proposed. Other well-known control strategies designed for integer order

systems have been extended to fractional order systems. State space fractional design methods based on pole placement are developed in (Farges et al. [2010]). The sliding mode control (SMC) has been also extended to fractional order systems (Calderon et al. [2006], El-Khazali et al. [2006], Efe and Kasnakoglu. [2008], Tavazoei and Haeri [2008], Si-Ammour et al. [2009], Dadras and Momeni [2010], Pisano et al. [2010], Pisano et al. [2011], Dadras and Momeni [2011]). In (Calderon et al. [2006]), a conventional PID based on sliding mode strategy is designed for linear fractional order systems. For nonlinear systems, a single input fractional order model described by a chain of integrators is considered in (Efe and Kasnakoglu. [2008]). Sliding mode method is applied to control and to synchronize fractional order nonlinear chaotic systems (El-Khazali et al. [2006], Tavazoei and Haeri [2008]). Sliding surface defined on a manifold expressed with fractional order integral is introduced in (Si-Ammour et al. [2009]) and in (Pisano et al. [2010]). Higher order sliding mode controllers and observers have been developed in (Pisano et al. [2010]) and (Pisano et al. [2011]). In (Dadras and Momeni [2010]), the authors have used LMI techniques to design a sliding mode controller for non linear systems. In this paper, the sliding mode control is designed for fractional order linear systems with input delay in presence of bounded nonlinear uncertainties and bounded external disturbances. The sliding mode control is designed within sliding surface defined by fractional order integral. Conditions for the existence and

finite time reaching of the sliding mode are established. The stability of the closed loop system on the sliding mode is analyzed by using recent results on Lyapunov stability theory for fractional order systems, (Li et al. [2009], Sadati et al. [2010]). The rest of the paper is organized as follows. Section 2 is devoted to basic definitions of fractional order integrals and derivatives. The proposed sliding mode controller is presented in Section 3. The design consists on two steps. First, a state predictor is applied to compensate for the delay of the system (Si-Ammour et al. [2009]). Second, the sliding mode controller is developed by considering a sliding surface expressed with fractional order integral. Conditions of the existence of the sliding mode are established in Section 4. The stability of the sliding mode dynamic is analyzed in Section 5. A numerical example is given in Section 6 to illustrate the effectiveness of the proposed method. Finally, some concluding remarks are given in Section 7.

• I 0 f (t) = f (t) and I α I β f (t) = I α+β f (t), f (t) ∈ L1 [a b], ∀α, β ∈ ℜ+ • Dα I β f (t) = Dα−β f (t), ∀α, β ∈ ℜ+ • Dα (c1 f1 (t) + c2 f2 (t)) = c1 Dα f1 (t) + c2 Dα f2 (t) • Dα (f (t)g(t)) = f (t)Dα g(t)+ ∞ ∑ Γ(1 + α) Dk f (t)Dα−k g(t) Γ(1 + k)Γ(1 − k + α) k=1

In the sequel, we use some special functions encountered in fractional calculus. • The Mittag-Leffler function ∞ ∑ Eα (z) = k=0

1 zk Γ(kα + 1)

• The generalized Mittag-Leffler function ∞ ∑ 1 Eα,β (z) = zk Γ(kα + β)

(5a)

(5b)

k=0

2. BASIC DEFINITIONS Let L1 [a b] denotes the space of Lebesgue integrable realvalued functions f (t) of the variable t, which represents the time, on the interval [a, b], a, b ∈ ℜ+ , such that 0 ≤ a < b < ∞ and ℜ+ denotes the non negative real numbers set. Let AC[a b] be the space of functions f (t) which are continuous on [a b] and we denote by AC k the space of real-valued function f (t) which have continuous derivative up to order k − 1 such that f (k−1) (t) ∈ AC[a b] where f (i) (t) is the i-th integer order derivative of f (t). Definition 1. (Kilbas et al. [2006]) Let f (t) ∈ L1 [a b] be a function of the variable t, t ∈ [a, b]. The fractional integral of order α ∈ ℜ+ is defined by the Riemann-Liouville integral ∫ t 1 α Ia f (t) = (t − τ )α−1 f (τ )dτ (1) Γ(α) a where the Euler’s Gamma function Γ(α) is defined as ∫ ∞ Γ(α) = ν α−1 e−ν dν (2) 0

Three definitions of fractional order derivatives are introduced in literature, namely the Riemann-Liouville’s derivative, the Caputo’s derivative and the Grunwald-Letkinov’s ˙ derivative, (Podlubny [1999], Kilbas et al. [2006]). Throughout this paper the Riemann-Liouville’s derivative is used. Let α ∈ ℜ+ be the fractional order of the derivative such that s − 1 < α < s, s denotes any integer and a denotes the initial time. Definition 2. (Kilbas et al. [2006]) The Riemann-Liouville fractional derivative of order α of f (t) ; t ∈ [a b] is defined as ∫ t ds 1 RL α (t − τ )s−α−1 f (τ )dτ (3) a Dt f (t) = Γ(s − α) dts a The initial time is taken zero. For simplicity we use the following notation: ∆ RL α 0 Dt f (t) =

Dα f (t) (4) Some useful properties of the Riemann-Liouville fractional integral and derivative are summarized below (Kilbas et al. [2006]). Proposition 1. (Kilbas et al. [2006]) Let f (t) be a real-valued function of the variable t ∈ [a b] ⊂ ℜ+ , then

• The generalized exponential function ezα = z α−1 Eα,α (z)

(5c)

3. SLIDING MODE CONTROL FOR FRACTIONAL ORDER UNCERTAIN SYSTEMS WITH INPUT DELAY Consider the fractional order uncertain system with delay in input described by the following state space model Dα x(t) = Ax(t) + Bu(t) + Bd u(t − τ ) + φ0 (x(t), t) + φ1 (t) (6) x(t) ∈ ℜn is the state vector, u(t) ∈ ℜm is the input vector, τ is the input delay assumed to be constant and known, φ0 (x(t), t) : ℜn × ℜ+ → ℜn represents the uncertainties and φ1 (t) : ℜ+ → ℜn the external disturbance. A, B and Bd are constant matrices with appropriate dimensions. The initialinput function is u0 (s) = ν(s), −τ ≤ s ≤ 0. The initial conditions are taken as (Hartley and Lorenzo [2002]): lim Dα−1 x(t) = x0 (7) t→0

Assumption 1. ∥φ0 (x(t), t)∥ ≤ k0 ∥x(t)∥ , k0 > 0 (8) ∥φ1 (t)∥ ≤ k1 , k1 > 0 (9) Assumption 2. φ0 (x, t) is sufficiently continuous derivable with respect to x and to t, and φ0 (0, t) = 0, ∀t ≥ 0. φ1 (t) is sufficiently continuous derivable with respect to t. Remark 1. In this paper, we consider the commensurate fractional order systems. The fractional order derivatives of the n state variables are equal. In the sequel, we consider 0 < α < 1. Stability of linear and non linear fractional order systems is investigated by several authors (Matignon [1996], Momani and Hadid [2004], Li et al. [2009], Sadati et al. [2010], Sabatier et al. [2010]). For linear systems, a condition based on the argument principle is established by Matignon (Matignon [1996]). Subsequently, Linear Matrix Inequality conditions have been derived in (Sabatier et al. [2010]). In the case of nonlinear fractional systems, Mittag-Leffler stability is introduced and Lyapunov stability theory is developed in (Li et al. [2009], Sadati et al. [2010], Momani and Hadid [2004]). The main theorems used in this paper are recalled below Theorem 1. (Matignon [1996]) The linear fractional order system Dα x(t) = Ax(t) 0 < α < 1

where A ∈ ℜn×n , is asymptotically stable if and only if the following condition holds π |Arg (λi (A))| > α , i = 1, 2, . . . , n 2 where λi (A) denotes the i − th eigenvalue of A Definition 3. (Khalil [2002]) A continuous function π : [0 t) → [0 ∞) is said to belong to class-κ functions if it is strictly increasing and π(0) = 0. Theorem 2. (Li et al. [2009]) Consider the fractional order non linear system Dα x(t) = f (x(t), t) (10) n Let x = 0 be an equilibrium of (10) and D ⊂ ℜ be a domain containing the origin. Let V (x(t), t) be a continuously differentiable function and locally Lipschitz with respect to x ∈ D and class-κ functions πi , (i = 1, 2, 3) such that π1 (∥x(t)∥) ≤ V (x(t), t) ≤ π2 (∥x(t)∥)

(11)

D V (x(t), t) ≤ −π3 (∥x(t)∥) (12) then, the equilibrium x = 0 is asymptotically stable, i.e., lim x(t) = 0. α

3.3 Sliding mode control As in the case of integer order systems, the design of sliding mode control law is done in two steps. The first step is to force the system trajectories to reach the surface and the second step constrains these trajectories to slide over the surface towards the equilibrium point with a desired dynamic. The first step objective is achieved by a discontinuous control ud (t) while the second step objective is guaranteed by an equivalent continuous control ueq (t). Finally, the global sliding mode control u(t) is constructed by the sum of ud and ueq . Here, the equivalent and discontinuous control laws are chosen as: ˆ −1 {CAˆ ueq (t) = −(C B) x(t) + ρσ(t)} (18) −1 ˆ ud (t) = −(C B) {∥C∥ δ(x, t) + k2 sign(σ(t))} (19) ρ > 0; k2 > 0; δ(x, t) = k0 ∥x(t)∥ + k1 (20) u(t) = ueq (t) + ud (t) (21) where { 1 if σ(t) > 0 sign(σ(t)) = = 0 if σ(t) = 0 −1 if σ(t) < 0

t→∞

4. EXISTENCE OF SLIDING MODE

3.1 State predictor In (Si-Ammour et al. [2009]), the input delayed fractional order system is transformed to free input delay fractional order system by introducing a state predictor. The state predictor is formulated by the following relation: ∫ t −s) x ˆ(t) = x(t) + eA(t−τ Bd u(s)ds (13) α t−τ

Substituting (13) in (6), we obtain free-delay system ˆ Dα x ˆ(t) = Aˆ x(t) + Bu(t) + φ0 (x(t), t) + φ1 (t)

(14)

where ˆ = B + eAτ B α Bd

(15)

3.2 Sliding surface Let the desired sliding mode surface be given by the fractional integral form (Si-Ammour et al. [2009], Pisano et al. [2010]) σ(ˆ x(t)) = CI

1−α

x ˆ(t), σ ∈ ℜ

m

(16)

The matrix C ∈ ℜm×n is chosen so that the controlled systems possesses a desired closed-loop dynamic. Applying the properties of Proposition 1, the derivative of σ with respect the to time t is ˆ σ(t) ˙ = CAˆ x(t) + C Bu(t) + Cφ0 (x(t), t) + Cφ1 (t) (17) The standard assumption of sliding mode control is considered. Assumption 3. (1) rank(B) = m. ˆ is invertible. (2) C B Remark 2. The last assumption can be made directly on the ˆ = CB + CeAτ systems matrices. In fact, we have C B α Bd . If CB is invertible, then ( ) ˆ = CB Im + (CB)−1 CeAτ CB α Bd Because of the presence of the identity matrix Im , the invertˆ is almost invertible. ibility of CB implies that C B

Under the sliding mode control (18)-(21), the trajectories of the fractional order delayed systems converge to the sliding surface σ = 0 as is established in the following theorem. Theorem 3. Let Assumptions 1 − 2 be satisfied. Then, under the control law (18) − (21), the sliding mode exists, i.e., the reaching condition of sliding mode control σ T σ˙ ≤ 0 is satisfied. Proof Substituting the the control law (18) − (21) in (17) yields to σ(t) ˙ = −ρσ(t) − {∥C∥ δ(x, t) + k2 sign(σ(t))} +Cφ0 (x(t), t)) + Cφ1 (t) (22) Let V (σ) = 12 σ T (t)σ(t) be a Lyapunov function, then V˙ (σ) = σ T {−ρσ − ∥C∥ δ(x, t) − k2 sign(σ)} +σ T {Cφ0 (x(t), t) + Cφ1 (t)} From (20) and Assumption 1 2 V˙ (σ) ≤ −ρ ∥σ∥ − k2 ∥σ∥ < 0 √ V˙ (σ) ≤ −ρV (σ) − k2 V (σ) It follows that reachability condition towards the sliding mode is satisfied, then the sliding mode exists. Furthermore, the sliding surface is reached in finite time. This completes the proof. 2 Then

5. STABILITY ANALYSIS OF THE SLIDING MODE DYNAMIC Once the trajectories of the controlled system reach the surface, we must ensure that the system dynamics in sliding mode is stable. The following Lemma will be useful for the sequel. Lemma 1. (Pisano et al. [2010]) Consider the vector η(t) ∈ ℜm . Let 1 − α ∈ (0, 1). If there exists t1 , 0 < t1 < ∞ such that I 1−α η(t) = 0, ∀t ≥ t1 (23) then lim η(t) = 0 (24) t→∞

Consider the coordinates transformation [ ] z (t) z(t) = T x ˆ(t) = 1 z2 (t)

(25)

with z1 ∈ ℜ(n−m) and z2 ∈ ℜm and the transformation matrix T is chosen such that ] [ ¯ ˆ = ¯0 ˆ=B (26) TB ˆm B ¯ˆ where B m is an m × m invertible matrix. Assumption 4. Matching Condition: [ ] 0 T φ1 (t) = (27) φ12 (t)

Lemma 2. Let the system be on the sliding mode. Then, there exists a positive constant µ ≥ 0 such that ∥x(t)∥ ≤ µ ∥z1 (t)∥ , ∀t ≥ 0 (42) Proof On the sliding mode u(t) = −K x ˆ(t), then (13) gives ∫ t −s) x(t) = x ˆ(t) + eA(t−τ Bd K x ˆ(s)ds α

(43)

t−τ

Using (25) and (39), it follows ∫ t −s) eA(t−τ Bd KL1 z1 (s)ds x(t) = L1 z1 (t) + α

(44)

t−τ

where L1 = W1 − W2 D2−1 D1 with T −1 = [W1 W2 ] From the Razumikhin theorem (Hale and Lune [1993]), the assumption In the new coordinates, the input-delay free system becomes V (z1 (s)) ≤ q 2 V (z1 (t)), q ≥ 1, t − τ ≤ s ≤ t α D z1 (t) = A¯11 z1 (t) + A¯12 z2 (t) + φ01 (x(t), t) (28) implies ¯ ˆm u(t)+φ02 (x(t), t)+φ12 (t) Dα z2 (t) = A¯21 z1 (t)+A¯22 z2 (t)+B ∥z1 (s)∥ ≤ qγ ∥z1 (t)∥ , t−τ ≤s≤t √ (29) λmax (P1 ) where γ = λmin (P1 ) . Then, (44) becomes where { ∫ τ [ ] [ ]

}

A(−s) A¯11 A¯12 φ01 (x, t) −1 ¯ ∥x(t)∥ ≤ ∥z (t)∥ ∥L ∥ 1 + qγ ∥B ∥ ∥K∥

e

ds 1 1 d α A = T AT = ¯ ¯ , T φ0 (x, t) = φ02 (x, t) A12 A22 0 (30) Since τ is a constant number then, there exists a constant µ1 > 0 such that Let ∫ τ

η(t) = C x ˆ(t) = CT −1 z(t) (31)

A(−s)

e

ds ≤ µ1 α Let us partition CT −1 as 0 CT −1 = [D1 D2 ] (32) Then ∥x(t)∥ ≤ µ ∥z1 (t)∥ ¯ˆ ¯ˆ ˆ = D2 B ˆ Since C B m , then the assumption that C B and Bm are with µ = | ∥L1 ∥ (1 + qγµ1 ∥Bd ∥ ∥K∥). This completes the non singular implies that D2 is non singular too. From (31), it proof. 2. follows We need the following assumption z2 (t) = −D2−1 D1 z1 (t) + D2−1 η(t) (33) Assumption 5. ∞ Substituting (33) in (28), one obtains the reduced sliding mode ∑ ( k )T α−k Γ(1 + α) ∆ dynamic equation Ψ= D z1 (t) D z1 (t) Γ(1 + k)Γ(1 − k + α) Dα z (t) = A¯ z (t) + A¯ D−1 η(t) + φ (x(t), t) (34) 1

r 1

12

k=1

01

2

2

where

A¯12 D2−1 D1

A¯r = A¯11 − (35) On the sliding mode, σ(t) = σ(t) ˙ = 0, then from (18) and (19) we have ueq (t) = −K x ˆ(t) (36) where ( )−1 ˆ K = CB CA (37) and ud (t) = 0 (38) Furthermore, once the sliding mode is reached, i.e., σ = I 1−α CT −1 z = 0 and from Lemma 1, η(t) is asymptotically vanishing. Then, (33) implies z2 (t) =

−D2−1 D1 z1 (t)

(39)

and Dα z1 (t) = A¯r z1 (t) + φ01 (x(t), t)

(40)

To investigate the stability of System (40), we introduce the Lyapunov function T

V (z1 (t)) = z1 (t) P1 z1 (t) (41) where P1 is an (n − m) × (n − m) positive definite symmetric matrix. The following Lemma will be used in the statement of the closed loop system stability.

≤ ψ ∥z1 (t)∥ ; ψ ≥ 0 (45) Remark 3. A condition which is almost equivalent to (45) has been considered in previous works (Efe and Kasnakoglu. [2008], Dadras and Momeni [2010]). The stability of the closed loop system on the sliding mode is established by the following result. Theorem 4. Let Assumptions (1) − (3) be satisfied, if the sliding matrix C is designed such that the matrix A¯r , given by (35), is Hurwitz, i.e., satisfies the Lyapunov equation A¯Tr P1 + P1 A¯r = −Q1 (46) where Q1 is a positive definite symmetric matrix and if λmin (Q1 ) 2

−ψ (47) µλmax (P1 ) where λmax (.) and λmin (.) stand for the maximum and the minimum eigenvalues of (.), respectively, then the closed loop system on the sliding mode is asymptotically stable, 0 ≤ k0
0. Then, from Theorem 2 lim z1 (t) = 0. Relation (42) implies that

0

-5 0

1

2

3

4

5

6

4

5

6

4

5

6

times (s)

Fig. 1. State trajectories

t→∞

x(t) = 0. This completes the proof.2

Proof If the condition (48) is satisfied, the reduced closed loop system (34) on the sliding mode becomes Dα z1 (t) = A¯r z1 (t) (50) From Theorem 1, the closed-loop system (50) is asymptotically stable if the condition (49) is satisfied. This completes the proof. 2 6. NUMERICAL EXAMPLE Consider the following numerical example Dα x(t) = Ax(t) + Bu(t) + Bd u(t − τ ) + φ0 (x(t), t) + φ1 (t) where [ ] [ ] [ ] 0 1 0 0 0 A = 0 0 1 ; B = 2 ; Bd = 2 ; 1 1 0 0.1 0.1 [ ] [ ] x1 sin(t) 0 0 φ0 (x, t) = 0.1 x1 cos(t) ; φ1 (t) = x3 0.5sin(t) ˆ = Some computations yield to the following result C B 5.6532 ̸= 0. The uncontrolled system is unstable. Let us choose the sliding matrix C as C = [3 4 1] with this choice, we can check that the matrix [ ] −0.54300 −0.2460 A¯r = −0.3394 −0.9780 is an Hurwitz matrix and possesses the following eigenvalues λ(A¯r ) = {−0.4, −1.2} The design parameters are chosen as ρ = 2; k0 = 0.1; k1 = 0.5 The simulation results for τ = 0.5 and α = 0.7 are presented in Figures 1, 2 and 3. The initial conditions on the state variables

0 Amplitude

Corollary 1. Suppose that all assumptions in Theorem 4 hold. In addition, if the uncertainties satisfy the matching condition, i.e., [ ] 0 T φ0 (x(t), t) = (48) φ02 (x(t), t) and if the following condition holds ( ) Arg λi (A¯r ) > α π , , i = 1, 2, . . . n − m (49) 2 then the closed loop system on the sliding mode is asymptotically stable.

5

-5 -10

-15 0

1

2

3 time (s)

Fig. 2. The equivalent control ueq (t) 1 0.5 Amplitude

lim

t→∞

0 -0.5 -1 -1.5 -2 0

1

2

3 time (s)

Fig. 3. The discontinuous control ud (t) are x0 = [5 10 − 5]T . The designed controller asymptotically stabilizes the system. The discontinuous controller ud given by the relation (19) permits to steer the system trajectories toward the sliding surface in finite time, in presence of uncertainty and external disturbances, with robustness properties. This can be depicted in Figure 4 which displays the time history of the sliding variable σ(t). Figure 1 shows the time evolution of the state variables. The state variables tend globally and asymptotically to zero. The objective of the control is well achieved with good closed loop performances. 7. CONCLUSION In this paper, a robust sliding mode controller for fractional order input delay systems in presence of uncertainties and external disturbances is developed. In order to take into account the input delay, the uncertainties and the external disturbances,

50

Amplitude

40 30 20 10 0 -10 0

1

2

3

4

5

6

time (s)

Fig. 4. The sliding surface σ(t) a suitable sliding mode controller is proposed. Conditions of the existence of the sliding mode are established. The stability of the reduced-order dynamics in the sliding mode has been investigated. The present results can be served for further improvements that will be pursued in the future. The case on unknown delay and the generalization to non commensurate fractional order systems are, for instance, some issues to be studied. REFERENCES Bartolini, G., Fridman, L., Pisano, A. and Usai, E. (2008). Modern sliding mode control theory: New perspectives and applications. Springer Lectures Notes in Control and Information Sciences, Vol. 375. Calderon, A. J., Vinagre, B. M., and Feliu, V. (2006). Fractional order control strategies for power electronic Buck converters. Signal Processing, 86, pp. 2803-2819. Dadras, S. and Momeni, H. R. (2011). Fractional sliding mode observer design for a class of uncertain fractional order nonlinear systems. IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011,Orlando, FL, USA, December 12-15. Dadras, S. and Momeni, H. R. (2010). LMI-based fractional order surface design for integral sliding-mode control of uncertain fractional order nonlinear systems. Proceeding of FDA’10, The 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, October 18-20. Efe, M. O. and Kasnakoglu, C. A. (2008). Fractional adaptation law for sliding Mode control. International Journal of Adaptive Control and Signal Processing, 22, pp. 968-986. El-Khazali, R., Ahmad, W. and Al-Assaf, Y. (2006). Sliding mode control of generalized fractional chaotic systems. International Journal of Bifurcations and Chaos, 16 (10), pp. 3113-3152. Farges, C., Moze, M. and Sabatier, J. (2010). Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica, 46(10), pp. 1730-1734. Hale, J. K. and Lune, S. M. V. (1993). Introduction to functional differential equation. New-York: Springer. Hartley, T. T. and Lorenzo, C. L. (2002). Dynamics and control of initialized fractional order systems. Nonlinear Dynamics, 29, pp. 201-233. Khalil, H. K. (2002). Nonlinear systems. (3rd ed.), Prentice Hall. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Application of Fractional Differential Equations North Holland Mathematics Studies, Editor Jan van Mill, Elsevier.

Li, Y., Chen, Y. Q. and Podlubny, I. (2009). Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica,45, pp. 1965-1969. Lurie, B. J. (1994). Three-parameters tunable Tilt-Integral Derivative (TID) controller. US Patent, US5371, 670. Matignon, D. (1996). Stability Results on Fractional Differential Equations with Applications to Control Processing. In IAMCS, IEEE SMC Proceedings Conference, Lille, France. Momani, S. and Hadid, S. (2004). Lyapunov stability solutions of fractional integrodifferential equations. International Journal of Mathematics and Mathematical Sciences, 47, pp. 2503-2507. Monje, C. A., Vinagre, B. M., Chen, Y. Q., Feliu, V., Lanusse, P. and Sabatier, J. (2005). Optimal tunings for fractional ˜ ⃝, c A., Tenreiro Machado, J.A., PIλ Dµ . In: Le MehautA Trigeassou, J. C., Sabatier, J. (eds), Fractional Differentiation and its Applications, U Books, Augsburg, Germany, pp. 675686. Monje, C. A., Chen, Y. Q., Vinagre, B. M., Xue, D. and Feliu, V. (2010). Fractional order systems and control: Fundamentals and Applications Springer. Perruquetti, W. and Barbot, J. P. (2002). Sliding mode control in engineering. Marcel Dekker, Inc., New-York. Pisano, A., Jelicic, Z. and Usai, E. (2010). Sliding mode control approaches to the robust regulation of linear multivaraible fractional order dynamics. International Journal of Robust and Nonlinear Control, 20(18), pp. 2021-2044. Pisano, A., Usai, E., Rapai´ c, M. and Jelicic, Z. (2011). Second order sliding mode control approaches to disturbance estimation and fault detection in fractional order systems. 18th IFA World Congress, Milano, Italy, August 28-September 2. Podlubny, I. (1999). Fractional differential equations. Academic Press, San Diego. Podlubny, I. (1999). fractional order system and PIDµ controllers. IEEE Transactions on Automatic Control, 44(1), pp. 208-214. Sabatier, J., Moze, M. and Farges, C. (2010). LMI stability conditions for fractional order systems. Computers and Mathematics with Applications, 59, pp. 1594-1609. Sabatier, J., Agrawal, O. and Machado, J. T. (2007). Advances in fractional calculus: Theoretical developments and applications in physic and engineering. Springer, Berlin. Sabatier, J., Oustaloup, A. Turricha, A. G. and Lanusse, P. (2002). CRONE control: Principles and extension to timevariant plants with asymptotically constant coefficients. Nonlinear Dynamics, Kluwer Academic Publishers, 29(1-4), pp 363-385. Sadati, S. J., Baleanu, D., Ranjbar, A., Ghaderi, R. and Abdeljawad (Maraaba), T. (2010). Mittag-Leffler stability theorem for fractional nonlinear systems with delay. Abstract and Applied Analysis, doi: 10.1155/2010/108651, Article ID 108651. Si-Ammour, A., Djennoune, S. and Bettayeb, M. (2009). A sliding mode control for linear fractional systems with input and state delays. Communications in Nonlinear Sciences and Numerical Simulations, 14(5), pp. 2310-2318. Tavazoei, M. S. and Haeri, M. (2008). Synchronization of chaotic fractional order systems via active sliding mode controller. Physica A., (378) 1, pp. 57-70. Utkin, V. (1992). Sliding modes in control and optimization. Springer-Verlag.