IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 2016 477

Robust Output Feedback Control for FractionalOrder Nonlinear Systems with Time-varying Delays

Changchun Hua, Tong Zhang, Yafeng Li, and Xinping Guan

Abstract—Robust controller design problem is investigated fora class of fractional order nonlinear systems with time varyingdelays. Firstly, a reduced-order observer is designed. Then,an output feedback controller is designed. Both the designedobserver and controller are independent of time delays. Bychoosing appropriate Lyapunov functions, we prove the designedcontroller can render the fractional order system asymptoticallystable. A simulation example is given to verify the effectivenessof the proposed approach.

Index Terms—Fractional order systems, time-varying delays,Laypunov function, backstepping.

I. INTRODUCTION

Fractional calculus is an ancient concept, which can bedated back to the end of 17th century, the time when the classi-cal integer order calculus was established. It is a generalizationof the ordinary differentiation and integration to arbitrary or-der[1]. Although it has a long history, it has not attracted muchattention until recently in the control field. It is found thatmany systems with memory feature or complex material canbe more concisely and actually described by fractional orderderivatives, such as the diffusion process, the heat transferprocess and the effect of the frequency in induction machines.It also has been proved that fractional order controllers, likefractional order PID controllers and fractional order modelreference adaptive controllers, can capture much better effectand robustness[2]. For some basic theory of fractional ordercalculus and fractional order systems, one can refer to [1−6]and the references therein.

Stability analysis is one of the most fundamental andessential issues for the control system. In [7], Matignonfirstly studied the stability of fractional-order linear differ-ential systems with the Caputo definition. Since then, manyfurther achievements have been obtained[8−11]. In [8−9], theauthors presented the sufficient and necessary conditions forthe asymptotical stability of fractional order interval systemswith fractional order α satisfying 0 < α < 1 and 1 < α < 2,

Manuscript received September 8, 2015, accepted March 3, 2016. This workwas partially supported by National Natural Science Foundation of China(61290322, 61273222, 61322303, 61473248, 61403335), Hebei ProvinceApplied Basis Research Project (15967629D), and Top Talents Project ofHebei Province and Yanshan University Project (13LGA020).

Citation: Changchun Hua, Tong Zhang, Yafeng Li, Xinping Guan. Ro-bust output feedback control for fractional order nonlinear systems withtime-varying delays. IEEE/CAA Journal of Automatica Sinica, 2016, 3(4):477−482

Changchun Hua, Tong Zhang, and Yafeng Li are with the Institute ofElectrical Engineering, Yanshan University, Qinhuangdao 066004, China (e-mail: cch@ysu.edu.cn; 13230929204@163.com; y.f.li@foxmail.com).

Xinping Guan is with the Department of Automation, Shanghai Jiao TongUniversity, Shanghai 200030, China (e-mail: xpguan@ysu.edu.cn).

respectively. Reference [10] designed both state and outputfeedback controllers for fractional order linear systems intriangular form by introducing appropriate transformations ofcoordinates. Based on Gronwall-Bellman lemma and sectorbounded condition, the stability and stabilization of fractionalorder linear systems subject to input saturation were studied in[11]. For the nonlinear fractional order systems, the stabilityanalysis is much more difficult than that of the linear systems.We can find some sufficient conditions in [12−14]. In [15], theauthors introduced Mittag-Leffler stability by using Lyapunovdirect method.

Time delay is an inherent phenomenon in the interconnectedsystems or processes, which makes the stability analysis andcontroller design challenging. Robust output control problemfor a class of nonlinear time-delay systems was studied in[16]. The necessary and sufficient stability conditions for linearfractional-order differential equations and linear time-delayedfractional differential equations have already been obtainedin [17−19]. Reference [20] investigated the stability of α-dimensional linear fractional-order differential systems withorder 1 < α < 2. References [21−22] contain the stabilityanalysis of fractional order nonlinear time delay system basedon Lyapunov direct method and by using properties of Mittag-Leffler function and Laplace transform.

In recent years, the backstepping technique has attractedmuch attention as a powerful method for controlling the strictfeedback nonlinear systems. There are a few works usingbackstepping technique to handle fractional order systems.Using Lyapunov indirect method, the authors of [23] presenteda new method to design an adaptive backstepping controllerfor triangular fractional order nonlinear systems. In [24], a newadaptive fractional-order backstepping method is proposed fora class of commensurate fractional order nonlinear systemswith uncertain constant parameters. However, for fractionalorder nonlinear systems with time-varying delays, there isnone related work. Motivated by the mentioned situation, wedevote to solve the stabilization problem of fractional ordernonlinear systems with time-varying delays.

The contributions of this paper are as follows: 1) A reduced-order observer is designed to estimate the state of the system;2) Based on the backstepping method, we design a robustoutput feedback controller for a class of fractional ordernonlinear time-varying delay systems; 3) With a novel classof fractional Lyapunov functions, we prove the stability offractional order nonlinear systems.

The remainder of this paper is organized as follows: SectionII presents some basic concepts about fractional order calculusand the stability of fractional order nonlinear systems. In Sec-

478 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 2016

tion III, as the main part of this note, an adaptive controlleris designed by using the backstepping method for fractionalorder nonlinear time-varying delayed systems. An example ispresented to show the effectiveness of the proposed controllerin Section IV. Finally, Section V gives the conclusion of thispaper.

Notations . Throughout this paper, R denotes the set ofreal numbers, Rn for n-dimensional Euclidean vector spaceand Rn×n for the space of n×n real matrices. XT and X−1

represent the transpose and the inverse of matrix X , respec-tively. I denotes the unit matrix with proper dimensions. Forany matrix A ∈ Rn×n, λi(A) stands for the i-th eigenvalueof A. For simplicity, C

0 Dαt is mentioned as Dα.

II. PRELIMINARIES

In this section, we provide some basic knowledge of frac-tional calculus and fractional order systems (details can befound in [1−2]). There are several definitions of fractionalorder derivatives, among which the Riemann-Liouville andCaputo definitions are well known and most commonly used.In this paper, we choose the Caputo definition for the fractionalorder derivatives. The Caputo derivative and fractional integralare defined as follows.

Definition 1 (Caputo fractional derivative). The Caputofractional derivative of order α ≥ 0 for a function f : [0,∞] →R is defined as

0Dαt f(t) =

1Γ(n− α)

∫ t

0

f (n)(s)(t− s)α+1−n

ds, t > 0, (1)

where n is the first integer that is larger than α and Γ(·) isthe well known Gamma function which is defined as follows:

Γ(t) =∫ ∞

0

xt−1e−xdx.

Definition 2 (Fractional integral). The fractional integralof order α ≥ 0 for a function f : [0,∞] → R is defined as

0Iαt f(t) =

1Γ(α)

∫ t

0

f(s)(t− s)1−α

ds, t > 0. (2)

Definition 3 (Class-K function). A continuous function γ :[0, t) → [0,∞) is said to belong to class-K function if it isstrictly increasing and γ(0) = 0.

Here are some lemmas we could use in this paper.Lemma 1 (General Leibnitz’s rule). If f and g are

differentiable and continuous functions, the product f · g isalso differentiable and its α-th (α ≥ 0) derivative is given by

Dα(f · g) = (Dαf) · g +∞∑

k=1

Γ(α + 1)Dα−kf ·Dkg

Γ(k + 1)Γ(α− k + 1). (3)

According to Lemma 1, the α-th order time derivative ofV (x) = 2xTx can be extended as DαV (x) = (Dαx)Tx+ xTDαx + 2γ, where x is a column vector and γ =∑∞

k=1Γ(α+1)(Dα−kx)TDkx

Γ(k+1)Γ(α−k+1) .Lemma 2 (Fractional comparison principle). Let Dαx(t)

≥ Dαy(t) and x(0) = y(0), where α ∈ (0, 1). Then x(t) ≥y(t). In particular, if x(t) = c, where c is a constant, Dαc =0 and y(0) = c, we will have y(t) ≤ c.

Theorem 1[15]. Let x = 0 be an equilibrium point forDαx(t) = f(t, x) and D ⊂ Rn be a domain containing x = 0.Let V (t, x) : [0,∞)×D → R be a continuously differentiablefunction such that for ∀t ≥ 0, ∀x ∈ D, 0 < α < 1,

W1(x) ≤ V (t, x) ≤ W2(x),DαV (t, x) ≤ −W3(x), (4)

where W1(x), W2(x), and W3(x) are class-K functions on D.Then x = 0 is asymptotically stable.

Lemma 4[12]. Let x(t) ∈ Rn be a differentiable andcontinuous function. Then, for ∀t ≥ t0 and ∀α ∈ (0, 1)

12Dα(xT(t)x(t)) ≤ xT(t)Dαx(t). (5)

Lemma 5 (Schur complement lemma). The linear matrixinequality (LMI)

M =[

A BBT C

]< 0,

where A = AT, C = CT and C is invertible, is equivalent to

C < 0, A−BC−1BT < 0.

III. MAIN RESULTS

Consider the following fractional order nonlinear systemwith 0 < α < 1 and time-varying delays:

Dαx1 = x2 + F1(x1) + H1(y(t), y(t− d1(t))),Dαxi = xi+1 + Fi(xi) + Hi(y(t), y(t− di(t))),Dαxn = u + Fn(xn) + Hn(y(t), y(t− dn(t))),y = x1,

(6)

where x(t) = [x1 (t) , x2 (t) , . . . , xn (t)]T ∈ Rn is the stateand xi(θ) = φi(θ), θ ∈ [−di(0), 0), i = 1, . . . , n, x(0) = 0,u(t) ∈ R and y(t) ∈ R are the control input and the outputof the system, respectively; xi(t) = [x1(t), x2(t), . . . , xi(t)]T;Fi (·) and Hi (·) are smooth nonlinear functions and Fi(0, . . . ,0) = Hi (0, 0) = 0; di(t) is the time-varying delay and thereexists positive scalar ηi such that di(t) ≤ ηi < 1.

We impose the following assumptions on system (6).Assumption 1. Nonlinear functions Hi(ξ1, ξ2) (i = 1, 2,

. . . , n) satisfy the following inequality:

|Hi(ξ1, ξ2)| ≤ Hi1(ξ1)ξ1 + Hi2(ξ2)ξ2, (7)

where Hi1(·) and Hi2(·) are known functions.Let Hi1(ξ1) = 2H2

i1(ξ1)ξ1, Hi2(ξ2) = 2H2i2(ξ2)ξ2, then we

can have the following inequality:

|Hi(ξ1, ξ2)|2 ≤ Hi1(ξ1)ξ1 + Hi2(ξ2)ξ2. (8)

Assumption 1 is very common in nonlinear time delaysystems, by which the term y(t−di(t)) can be separated fromthe delay-function, so that we can handle the delay problems.

Assumption 2. For nonlinear functions Fi(·), there existsome positive scalars li such that the following inequalitieshold for i = 1, 2, 3, . . . , n:

|Fi(ζi)− Fi(ζi)| ≤ li

∥∥∥ζi − ζi

∥∥∥ , (9)

where ζi = [ζ1, ζ2, . . . , ζi]T, ζi = [ζ1, ζ2, . . . , ζi]T and li is aknown positive parameter.

We can use the following expression for Fi(xi)

HUA et al.: ROBUST OUTPUT FEEDBACK CONTROL FOR FRACTIONAL ORDER NONLINEAR SYSTEMS WITH TIME-VARYING DELAYS 479

Fi(xi) = ni1x1 + Fi(xi), i = 1, . . . , n, (10)

where ni1 is a constant which could be zero.In this paper, we focus on solving the following problem.

For system (6) satisfying Assumptions 1 and 2, design areduced-order observer based memoryless output feedbackcontroller to render the closed-loop system stable.

Considering system (6) with unmeasured state variables, wepropose the following reduced-order observer:

Dαλi(t) = λi+1(t) + ki+1x1(t) + Fi(xi(t))−ki(λ2(t) + k2y(t) + F1(y(t))),

Dαλn(t) = u(t) + Fn(xn(t))−kn(λ2(t) + k2y(t) + F1(y(t))),

xi(t) = λi + kiy(t), i = 2, . . . , n,

(11)

where xi(t) = [x1(t), x2(t), . . . , xi(t)]T and parameters ki (i= 2, . . . , n) are to be specified later.

Similar to (10), we can change Fi(xi(t)) into

Fi(xi(t)) = ni1x1 + Fi(xi(t)).

Remark 1. In this paper, we introduce the reduced-orderobserver instead of the full-order one. In this way, some of thestates can be derived from the real output, making the resultsmore precise and simplifying the structure and computationcomplexity. To our best knowledge, it is the first time thatthe reduced-order observer is introduced to fractional ordernonlinear systems.

The estimation errors are defined as

ei(t) = xi(t)− xi(t). (12)

From (11) and (12), we can get

Dαe(t) = Ae(t) + F (t) + H(t), (13)

where

e(t) = [e2(t), . . . , en(t)]T,

A =

−k2 1 0 · · · 0−k3 0 1 · · · 0

......

.... . .

...−kn 0 0 · · · 0

,

F (t) = [F2(x2(t))− F2(x2(t)), . . . ,

Fn(xn(t))− Fn(xn(t))]T, (14)

H(t) = [H2(y(t), y(t− d2)))− k2H1(y(t), y(t− d1))),

. . . ,Hn(y(t), y(t− dn)))− knH1(y(t), y(t− d1)))]T.(15)

Next, we extend the backstepping technique to the fractionalorder nonlinear system with time-varying delays described by(6). The virtual controllers αi (i = 1, . . . , n−1) are developedat each step. Finally, at step n, the actual controller u isdesigned. First, we introduce the following transformation ofstates.

z1(t) = y(t),zi(t) = λi(t)− αi−1, i = 2, . . . , n. (16)

Then we choose the Lyapunov function as

V = Ve + Vz + Vd, (17)

whereVe = I1−αeTPe, (18)

and P is a real symmetric positive matrix.

Vz = I1−αn∑

i=1

Mi, (19)

Mi =12z2i ,

Vd =

(2

n∑

i=2

k2i + 1

)∫ t

t−d1(t)

11− η1

H12(y(t))y(t)dt

+ 2n∑

i=2

∫ t

t−di(t)

11− ηi

Hi2(y(t))y(t)dt. (20)

Next, we will give the derivative of Ve, Vz , Vd, and V inturn. From Lemma 4, the derivative of Ve is:

Ve = Dαe(t)TPe(t)

≤ (Dαe(t))TPe(t) + eT(t)PDαe(t) (21)

= (Ae(t) + F (t) + H(t))TPe(t)

+ eT(t)P (Ae(t) + F (t) + H(t))

= eT(t)(ATP + PA)e(t) + FT(t)Pe(t)

+ eT(t)PF (t) + HT(t)Pe(t) + eT(t)PH(t) (22)

According to Assumption 2, we can get

FT(t)F (t) =n∑

i=2

(Fi(xi(t))− Fi(xi(t)))2

≤n∑

i=2

l2i∥∥xi − xi

∥∥2

≤ ρ∥∥xn − xn

∥∥2,

where ρ =∑n

i=2 l2i .

Ve ≤ eT(t)(ATP + PA)e(t) + ρeT(t)e(t)

+ HT(t)H(t) + 2eT(t)PPe(t)

≤ eT(t)(ATP + PA + ρI + 2PP )e(t) + HT(t)H(t)

≤ eT(t)(ATP + PA + ρI + 2PP )e(t)

+ 2n∑

i=2

H2i + 2

n∑

i=2

k2i H2

1 (23)

and P and ki (i = 2, . . . , n) satisfy

ATP + PA + ρI + 2PP < − n

2ε1I. (24)

To solve inequality (24), we decompose A = A + kB with

A =[

0 I(n−2)×(n−2)

0 0

], k =

k2

...kn

(n−1)×1

,

B = [−1, 0, . . . , 0]1×(n−1).

480 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 2016

According to Lemma 5, inequality (24) is equivalent to thefollowing LMI[

PA + WB + BTWT + ATP + (ρ + n2ε1

)I P

P −0.5I

]

< 0, (25)

where W = Pk. Further, we can use LMI toolbox in Matlabto obtain P and k.

The derivative of Vz is:

Vz =n∑

i=1

DαMi. (26)

Next, by the backstepping method, the virtual controllers αi

(i = 1, 2, . . . , n−1) and controller u are designed respectively.Step 1.

DαM1 =12Dαz2

1 ≤ z1Dαz1

= z1(z2 + α1 + k2x1 + e2 + F1(y(t))+ H1(y(t), y(t− d1(t))))

≤(

14

+ε12

)z21 +

12ε1

e22 + H2

1 (y(t), y(t− d1(t)))

+ z1(α1 + k2x1 + F1(x1(t))) + z1z2, (27)

where ε1 is a positive constant.Choose

α1 =−(

c1 +14

+ε12

+ k2 + n11

)z1 − F1(y(t))

−(

2n∑

i=2

k2i + 1

)(H11(y(t)) +

11− η1

H12(y(t)))

− 2n∑

i=2

(Hi1(y(t)) +

11− ηi

Hi2(y(t)))

=−K1z1 − α1, (28)

where ci (i = 1, 2, . . . , n) are positive constants,

K1 = c1 +14

+ε12

+ k2 + n11

α1 = F1(y(t)) +

(2

n∑

i=2

k2i + 1

)(H11(y(t))

+1

1− η1H12(y(t))) + 2

n∑

i=2

(Hi1(y(t))

+1

1− ηiHi2(y(t))).

Step 2.

DαM2 =12Dαz2

2 ≤ z2Dαz2

= z2(z3 + α2 + k3x1 + F2(x2(t))− k2(λ2 + k2x1 + F1(x1))−Dαα1)

≤ z2z3 +1

2ε1e22 + z2(α2 + k3x1 + F2(x2(t))

− k2(λ2 + k2x1 + F1(x1)) + Dαα1

+ K1(z2 + α1 + k2x1 + F1(y(t))) +K1ε1

2z22 .

(29)

Choose

α2 =− z1 −(

c2 +K1ε1

2

)z2 − k3x1 − n21z1

− F2(x2(t))−K1(z2 + α1 + k2x1 + F1(y(t)))+ k2(λ2 + k2x1(t) + F1(x1(t)))−Dαα1

=−K2z1 − α2, (30)

where

K2 = 1 + (K1 − k2)(n11 + k2) + n21 + k3

α2 =(

c2 +K1ε1

2

)z2 + F2(x2(t))

+ (K1 − k2)(z2 + α1 + F1(y(t))) + Dαα1.

Step iii.

DαMi =12Dαz2

i ≤ ziDαzi

= zi(zi+1 + αi + ki+1x1 + Fi(xi(t))− ki(λ2 + k2x1(t) + F1(x1(t)))−Dααi−1)

= zizi+1 +1

2ε1e22 + zi(αi + ki+1x1 + Fi(xi(t)))

− ki(λ2 + k2x1(t) + F1(x1(t))) + Dααi−1

+ Ki−1(z2 + α1 + k2x1 + F1(y(t)))) +Ki−1ε1

2z2i .

(31)

Choose

αi =− zi−1 −(

ci +Ki−1ε1

2

)zi − ki+1x1 − ni1z1

− Fi(xi(t)) + ki(λ2 + k2x1(t) + F1(x1(t)))−Dααi−1 −Ki−1(z2 + α1 + k2x1 + F1(y(t)))

=−Kiz1 − αi, (32)

where

Ki = (Ki−1 − ki)(n11 + k2) + ni1 + ki+1

αi = zi−1 +(

ci +Ki−1ε1

2

)zi + Fi(xi(t))

+ (Ki−1 − ki)(z2 + α1 + F1(y(t))) + Dααi−1.

Step nnn.

DαMn =12Dαz2

n ≤ znDαzn

= zn(u + Fn(xn(t))− kn(λ2 + k2x1(t)+ F1(x1(t)))−Dααn−1)

=1

2ε1e22 +

Kn−1ε12

z2n + zn(u + Fn(xn(t))

+ Dααn−1 − kn(λ2 + k2x1(t) + F1(x1(t)))+ Kn−1(z2 + α1 + k2x1 + F1(y(t)))). (33)

Then

u = − zn−1 −(

cn +Kn−1ε1

2

)zn − Fn(xn(t))

+ kn(λ2 + k2x1(t) + F1(x1(t)))−Kn−1(z2 + α1 + k2x1 + F1(x1(t)))−Dααn−1.

(34)

HUA et al.: ROBUST OUTPUT FEEDBACK CONTROL FOR FRACTIONAL ORDER NONLINEAR SYSTEMS WITH TIME-VARYING DELAYS 481

The derivative of Vd is:

Vd =

(2

n∑

i=2

k2i + 1

)

×(

ddt

(∫ t

t−d1(t)

11− η1

H12(y(t))y(t)dt

))

+ 2n∑

i=2

(ddt

(∫ t

t−di(t)

11− ηi

Hi2(y(t))y(t)dt

))

≤(

2n∑

i=2

k2i + 1

)(1

1− η1H12(y(t)

)y(t)

−H12(y(t− d1(t)))y(t− d1(t)))

+ 2n∑

i=2

(1

1− ηiHi2(y(t))y(t)

−Hi2(y(t− di(t)))y(t− di(t))). (35)

Then, we have

V = Ve + Vz + Vd

≤ eT(t)(

ATP + PA +(

ρ +n

2ε1

)I + 2PP

)e(t)

−n∑

i=1

ciz2i . (36)

SoV ≤ −W (e(t), zn), (37)

where zi = [z1, z2, . . . , zn] and W (·) and W1(·) are class-Kfunctions.

According to the definition of the Caputo fractional deriva-tive, Lemma 2 and (37)

DαV = I1−αV ≤ −I1−αW (e(t), zn) ≤ −W1(e(t), zn)

Finally, we present the main result of this paper as follows:Theorem 2. For a system described by (6) satisfying

Assumptions 1 and 2, controller (34) can render the closed-loop system asymptotically stable.

IV. SIMULATION

In this section, an example is given to show the effectivenessof the proposed controller.

Consider the following system:{

Dαx1(t) = x2(t)− 0.8x1(t) + 0.5x21(t− d1(t)) sin t,

Dαx2(t) = u− 0.8x2(t) + 0.5x31(t− d2(t)) sin t,

where d1(t) = 0.5(1 + sin t), d2(t) = 0.5(1 + cos t). Wecan see that the aforementioned system satisfies the aboveassumptions with P = I , η1 = η2 = 0.5, l = 0.8, ρ = 0.64,n11 = −0.8, F1(x1) = 0, n21 = 0, F2(x2) = −0.8x2,H11(t) = H21(t) = 0, H12(t − d1(t)) = 0.25x3

1(t − d1(t))and H22(t−d1(t)) = 0.25 x5

1(t−d1(t)). Choosing ε1 = 1, k2

= 2, c1 = 0.05 and c2 = 0.8, we can obtain the reduced-orderobserver and the function β1(x1(t)):

Dαλ2(t) = u− 2.8λ2(t)− 4x1(t),

α1(x1(t)) = −2x1(t)− 4.5x31(t)− x5

1(t).

Then, K1 = 2, α1 = 4.5x31 + x5

1, the controller can bedesigned as

u(t) = 0.6x1(t)− λ2(t) + 1.8α1(x1(t))−Dαα1.

The simulation results are shown in Figs. 1 and 2, fromwhich we can see that the constructed controller renders theclosed-loop system stable.

Fig. 1. The output response of the closed-loop system.

Fig. 2. The trajectories of x2 and x2.

V. CONCLUSION

In this paper, we study the controller design problem forfractional order nonlinear time-varying delay systems, us-ing the well known backstepping method. Also, we extendthe Lyapunov method to fractional order systems. Both thedesigned observer and controller are independent of timedelays. Through the simulation presented in Section IV, theeffectiveness of the proposed controller has been verified. Asput in [25], fractional order systems have a memory feature,which could make difficulties in the process of controllerdesign. In the future, we will further consider the memoryfeature and its influence. Based on the result of this paper, wewill study the stability and stabilization problems for fractionalnonlinear delayed systems with fractional order 1 < α < 2and the stability of fractional order nonlinear systems withfractional order α > 2.

482 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 2016

REFERENCES

[1] Podlubny I. Fractional Differential Equations: An Introduction to Frac-tional Derivatives, Fractional Differential Equations, Some Methods ofTheir Solution and Some of Their Applications. San Diego: AcademicPress, 1999.

[2] Pan I, Das S. Intelligent Fractional Order Systems and Control. BerlinHeidelberg: Springer, 2013.

[3] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applicationsof Fractional Differential Equations, Volume 204 (North-Holland Mathe-matics Studies). New York: Elsevier Science Inc., 2006.

[4] Lakshmikantham V, Leela S, Vasundhara Devi J. Theory of FractionalDynamic Systems. Cambridge, UK: Cambridge Scientific Publishers,2009.

[5] Diethelm K. The Analysis of Fractional Differential Equations. Berlin:Springer, 2010.

[6] Lakshmikantham V, Vatsala A S. Basic theory of fractional differentialequations. Nonlinear Analysis: Theory, Methods and Applications, 2008,69(8): 2677−2682

[7] Matignon D. Stability results for fractional differential equations withapplications to control processing. Computational Engineering in Sys-tems Applications, 1996, 2: 963−968

[8] Lu J G, Chen Y Q. Robust stability and stabilization of fractional-orderinterval systems with the fractional order α: the 0 ¿ α ¿ 1 case. IEEETransactions on Automatic Control, 2010, 55(1): 152−158

[9] Lu J G, Chen G R. Robust stability and stabilization of fractional-orderinterval systems: an LMI approach. IEEE Transactions on AutomaticControl, 2009, 54(6): 1294−1299

[10] Zhang X F, Liu L, Feng G, Wang Y Z. Asymptotical stabilization offractional-order linear systems in triangular form. Automatica, 2013,49(11): 3315−3321

[11] Lim Y H, Oh K K, Ahn H S. Stability and stabilization of fractional-order linear systems subject to input saturation. IEEE Transactions onAutomatic Control, 2013, 58(4): 1062−1067

[12] Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A. Lyapunovfunctions for fractional order systems. Communications in NonlinearScience and Numerical Simulation, 2014, 19(9): 2951−2957

[13] Wen X J, Wu Z M, Lu J G. Stability analysis of a class of nonlinearfractional-order systems. IEEE Transactions on Circuits and Systems II:Express Briefs, 2008, 55(11): 1178−1182

[14] Delavari H, Baleanu D, Sadati J. Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynamics, 2012, 67(4):2433−2439

[15] Li Y, Chen Y Q, Podlubny I. Stability of fractional-order nonlineardynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications, 2010,59(5): 1810−1821

[16] Hua C C, Guan X P, Shi P. Robust backstepping control for a class oftime delayed systems. IEEE Transactions on Automatic Control, 2005,50(6): 894−899

[17] Bonnet C, Partington J R. Coprime factorizations and stability offractional differential systems. Systems and Control Letters, 2000, 41(3):167−174

[18] Deng W H, Li C P, Lv J H. Stability analysis of linear fractionaldifferential system with multiple time delays. Nonlinear Dynamics,2007, 48(4): 409−416

[19] Kheirizad I, Tavazoei M S, Jalali A A. Stability criteria for a class offractional order systems. Nonlinear Dynamics, 2010, 61(1−2): 153−161

[20] Zhang F R, Li C P. Stability analysis of fractional differential systemswith order lying in (1, 2). Advances in Difference Equations, 2011, 2011:Article ID 213485

[21] Wen Y H, Zhou X F, Zhang Z X, Liu S. Lyapunov method for nonlinearfractional differential systems with delay. Nonlinear Dynamics, 2015,82(1−2): 1015−1025

[22] Yu W, Li T Z. Stability analysis of fractional-order nonlinear systemswith delay. Mathematical Problems in Engineering, 2014, 2014(4): Ar-ticle ID 301235

[23] Wei Y H, Chen Y Q, Liang S, Wang Y. A novel algorithm on adaptivebackstepping control of fractional order systems. Neurocomputing, 2015,165: 395−402

[24] Ding D S, Qi D L, Meng Y, Xu L. Adaptive Mittag-Leffler stabilizationof commensurate fractional-order nonlinear systems. In: Proceedings ofthe 53rd IEEE Conference on Decision and Control (CDC). Los Angeles,CA: IEEE, 2014. 6920−6926

[25] Syta A, Litak G, Lenci S, Scheffler M. Chaotic vibrations of the duffingsystem with fractional damping. Chaos: An Interdisciplinary Journal ofNonlinear Science, 2014, 24(1): 013107

Changchun Hua received the Ph. D. degree inelectrical engineering from Yanshan University, Qin-huangdao, China, in 2005. He was a research fellowin National University of Singapore from 2006 to2007. From 2007 to 2009, he worked in CarletonUniversity, Canada, funded by Province of OntarioMinistry of Research and Innovation Program. From2009 to 2011, he worked in University of Duisburg-Essen, Germany, funded by Alexander von Hum-boldt Foundation. Now he is a full professor at Yan-shan University, China. He is the author or coauthor

of more than 110 papers in mathematical, technical journals, and conferences.He has been involved in more than 10 projects supported by the NationalNatural Science Foundation of China, the National Education CommitteeFoundation of China, and other important foundations. His research interestsinclude nonlinear control systems, control systems design over network,teleoperation systems, and intelligent control. Corresponding author of thispaper.

Tong Zhang received the B. S. degree in electricalengineering from Yanshan University, Qinhuangdao,China, in 2014. She is currently working toward theM. S. degree in electrical engineering. Her researchinterests include fractional order system control.

Yafeng Li received the B. S. degree in electricalengineering from Hebei University of Science andTechnology, Shijiazhuang, China, in 2013. He is cur-rently working toward the Ph. D. degree in YanshanUniversity. His research interests include nonlinearsystem control and multi-agent system control.

Xinping Guan received the B. S. degree in math-ematics from Harbin Normal University, Harbin,China, and the M. S. degree in applied mathematicsand the Ph. D. degree in electrical engineering, bothfrom Harbin Institute of Technology, in 1986, 1991,and 1999, respectively. He is with the Departmentof Automation, Shanghai Jiao Tong University. Heis the (co)author of more than 200 papers in math-ematical, technical journals, and conferences. As(a)an (co)-investigator, he has finished more than20 projects supported by National Natural Science

Foundation of China (NSFC), the National Education Committee Foundationof China, and other important foundations. He is Cheung Kong ScholarsProgramme Special appointment professor. His current research interestsinclude networked control systems, robust control and intelligent control forcomplex systems and their applications. Dr. Guan is serving as a reviewerof Mathematic Review of America, a member of the Council of ChineseArtificial Intelligence Committee, and chairman of Automation Society ofHebei Province, China.