Design of fuzzy state feedback controller for robust stabilization of uncertain fractional-order chaotic systems

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Journal of the Franklin Institute 351 (2014) 5480–5493 www.elsevier.com/locate/jfranklin

Design of fuzzy state feedback controller for robust stabilization of uncertain fractional-order chaotic systems$ Xia Huanga,n, Zhen Wangb, Yuxia Lia, Junwei Luc a

College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China b College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China c School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China Received 11 June 2013; received in revised form 30 August 2014; accepted 30 September 2014 Available online 16 October 2014

Abstract In this paper, the stabilization problem of uncertain fractional-order chaotic systems is investigated in the case where the fractional order α satisfies 0oαo1 and 1 r αo2. Firstly, the uncertain fractional-order chaotic system is described by the so-called fractional-order T–S fuzzy model, and then the fuzzy state feedback controller is correspondingly designed. Secondly, sufficient conditions are derived for the robust asymptotical stability of the closed-loop control systems in those two cases. These criteria are expressed in terms of linear matrix inequalities (LMIs), and the feedback gain matrices can be formulated into the solvability of the relevant LMIs. The proposed controller overcomes some defects in traditional control techniques and is easy to implement. Finally, two numerical examples are presented to demonstrate the effectiveness and the feasibility of the robust stabilizing controller and the robust asymptotical stability criteria. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

☆ This work was supported by the National Natural Science Foundation of China under Grant 61473178, 61473177, 61004078. n Corresponding author. E-mail address: [email protected] (X. Huang).

http://dx.doi.org/10.1016/j.jfranklin.2014.09.023 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Although fractional calculus has a history almost as long as classical calculus, it was once thought of as a pure mathematical problem and therefore was not applied to engineering practice. Fractional calculus is now widely accepted and is understood to be the generalization of ordinary calculus to noninteger-order case [1]. Fractional-order systems are receiving more attention due to the fact that many real-world physical systems can be well described with the help of fractional-order models and that an analog fractance circuit can be designed to implement any fractional-order integral and fractional-order derivative operator. Nowadays, fractional-order systems have found a wide variety of practical applications in the field of physics and engineering [2–4]. With the aid of numerical simulation, it has been shown that chaotic behaviors of integer-order nonlinear systems would tend to be preserved when the order becomes fractional. Some models of fractional-order variants of the integer-order chaotic systems have been established as a consequence, such as fractional-order Chua's system [5], fractional-order Lorenz system [6], fractional-order Rössler system [7], fractional-order Liu system [8], and fractional-order chaotic cellular neural networks [9]. In addition, the lowest orders, which ensure the existence of chaotic attractors, are determined respectively. It is well known that chaos would cause performance degradation or collapse of the system. In recent years, great efforts have been made to suppress chaos when it is undesirable. Chaos control consists of stabilizing the unstable equilibria or controlling chaos to unstable periodic orbits (UPOs). A number of methods have been developed for chaos control, including OGY method [10], Pyragas continuous control [11], linear and nonlinear feedback control [12], adaptive control [13,14], backstepping method [15], impulsive control [16], and sliding mode control [17]. It should be noted that not all of these control methods are applicable to fractionalorder systems due to the complexity of the fractional-order operators. Some of them have been attempted to control fractional-order chaotic systems. For instance, sliding mode control (SMC) is employed to control a specified fractional-order chaotic system in [18–20]. It is well known that the shortcoming of SMC method is that the chattering phenomenon is unavoidable. For this purpose, a fuzzy logic controller is integrated in order to reduce the chattering effect as did in [19]. However, it is not difficult to see that the design of the controller is complicated. Sometimes, linear feedback control is adopted in order to simplify the design of the controllers [21,22]. But it should be noted that the stability analysis of a fractional-order nonlinear system is substantially different from that of the integer-order one, that is, it is very difficult to construct a Lyapunov-like function to make stability analysis. (For this reason, some Lyapunov function based control methods, such as backstepping method [23], are difficult to apply to fractionalorder systems.) In these circumstances, linearization approach is usually employed to analyze the stability of the closed-loop system. Obviously, the obtained stability criteria are local. The uncertainty or vagueness is unavoidable in mathematical modeling, T–S fuzzy control is considered as an alternative for its ability of dealing with uncertainties and nonlinearities in this situation. In the past two decades, T–S fuzzy model has received considerable attention because it provides a systematic method to represent a complex nonlinear system exactly as a weighted sum of some linear subsystems via fuzzy membership functions. Meanwhile, T–S fuzzy model based control has been successfully applied to many famous integer-order chaotic systems [24–27]. However, T–S fuzzy model based control of fractional-order chaotic systems remains a challenging problem. In [28], the stabilization of fractional order chaotic systems has been studied by integrating the T–S fuzzy model and adaptive adjustment mechanism (AAM).

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However, the considered system is assumed to be with derivative order αo1 and be uncertaintyfree. The advantage of fuzzy fractional-order systems is that fuzzy fractional-order models provide a method of using local linear fractional-order systems combining with fuzzy IF–THEN rules to achieve nonlinearities. Thus, one can apply the stability analysis methods for fractionalorder linear systems to the fuzzy fractional-order systems to simplify the design of the controllers. In view of the fact that T–S fuzzy model is in favor of making stability analysis and controller synthesis, the aim of this paper is to introduce the extended T–S fuzzy-model-based control to stabilize uncertain fractional-order chaotic system. The extended fractional-order T–S fuzzy model is established firstly, and then the fuzzy state feedback controller is designed. Two LMIbased conditions are given to guarantee the stability of the closed-loop control system with the fractional order α satisfying 0oαo1 and 1 r αo2, and the feedback gain matrices could be easily solved by LMI formulation. It is worthwhile to mention that LMI approach has been proposed in [29–32] in order to solve the robust stability and stabilization problem of fractionalorder systems or uncertain T–S fuzzy systems. This paper is organized as follows. In Section 2, the problem is formulated. The extended T–S fuzzy model is presented and fuzzy state feedback controller is designed. In Section 3, stability analysis of the closed-loop control system is made and LMI-based stability criteria are derived. Simulation results are shown in Section 4. Finally, Section 5 concludes the paper. Notations: Throughout this paper, without special statement, matrices are supposed to have compatible dimensions. Sym(X) denotes the expression X þ X T , and  is the Kronecker product of two matrices and has the property of ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ. The identity matrix and zero matrix are denoted by In and 0, respectively. diagfa1 ; a2 ; …; an g denotes the blockdiagonal matrix. The notation A40 ðAo0Þ means that A is a positive (negative) definite matrix. XT and X  1 represent the transpose and the inverse of matrix X, respectively. ⋆ are entries inferred by symmetry.

2. Problem formulation and preliminaries To facilitate system analysis and controller synthesis, an extended T–S fuzzy model, namely, fractional-order T–S fuzzy model is employed to represent the dynamical behaviors of the fractional-order chaotic system, and meantime, a fuzzy state-feedback controller is designed to achieve the stabilization of the closed-loop system. The procedure of the fuzzy modeling and controller design will be presented in this section. First consider the controlled uncertain fractional-order chaotic system described by the following IF–THEN rules: Rule i: IF z1 ðtÞ is Fi1 AND z2 ðtÞ is Fi2 AND, …, AND zm(t) is Fim, THEN Dαt xðtÞ ¼ ðAi þ ΔAi ðtÞÞxðtÞ þ uðtÞ;

i ¼ 1; 2; …; r;

ð1Þ

in which α ð0oαo2Þ is the fractional derivative and Rule i ði ¼ 1; 2; …; rÞ denotes the ith IF– THEN rule and r is the total number of the fuzzy rules; zj ðtÞ AR and F ij ði ¼ 1; 2; …; r; j ¼ 1; 2; …; m) represent the premise variable and the fuzzy set of the ith rule, respectively; xðtÞ A Rn and uðtÞA Rn are the state and control vectors at time t, respectively; Ai A Rnn are the system matrices for the ith subsystem, ΔAi ðtÞA Rnn represent the time-varying uncertain matrices of appropriate dimensions, which may describe the identification errors of system parameters or may denote the approximation errors between the original system and the

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mathematical model. Throughout this paper, we make the following assumption on the uncertain matrices ΔAi . Assumption. The uncertain matrices ΔAi are assumed to satisfy ΔAi ðtÞ ¼ E i F i ðtÞH i ;

i ¼ 1; 2; …; r;

where E i ; H i are known constant matrices with appropriate dimensions and Fi(t) are unknown time-varying matrix functions satisfying F Ti ðtÞF i ðtÞr I;

i ¼ 1; 2; …; r:

Using the singleton fuzzifier, product fuzzy inference and weighted average defuzzifier, the uncertain fractional-order T–S fuzzy model can be represented by r

Dαt xðtÞ ¼ ∑ wi ðzðtÞÞfðAi þ ΔAi ÞxðtÞg þ uðtÞ;

ð2Þ

i¼1

in which wi ðzðtÞÞ ¼ μi ðzðtÞÞ=∑ri ¼ 1 μi ðzðtÞÞ, μi ðzðtÞÞ ¼ ∏m j ¼ 1 μij ðzj ðtÞÞ, and μij ðzj ðtÞÞ is the membership function for the fuzzy set Fij and zðtÞ ¼ ðz1 ðtÞ; z2 ðtÞ; …zm ðtÞÞ. Obviously, wi ðzðtÞÞ satisfies the conditions 0r wi ðzðtÞÞ r 1 and ∑ri ¼ 1 wi ðzðtÞÞ ¼ 1. It should be remarked that many fractional-order systems with parameter uncertainties can be expressed by Eq. (2), including chaotic fractional-order Lorenz system, chaotic fractional-order Lü system, chaotic fractional-order Liu system, hyperchaotic fractional-order Lorenz system, and hyperchaotic fractional-order cellular neural networks. Next, we consider the stabilization problem of uncertain fractional-order fuzzy system (2). In order to stabilize system (2), a fuzzy state feedback controller is designed as follows. Rule i: IF z1 ðtÞ is Fi1 AND z2 ðtÞ is Fi2 AND, …, AND zm(t) is Fim, THEN uðtÞ ¼ K i xðtÞ;

i ¼ 1; 2; …; r;

in which K i A Rnn are the feedback gain matrices to be designed later. Then the overall controller can be represented by r

uðtÞ ¼ ∑ wi ðzðtÞÞK i xðtÞ;

ð3Þ

i¼1

where wi ðzðtÞÞ have the same definition as those in model (2). Thus, the resulting closed-loop control system (2) with Eq. (3) is r

Dαt xðtÞ ¼ ∑ wi ðzðtÞÞðAi þ E i F i ðtÞH i þ K i ÞxðtÞ

ð4Þ

i¼1

Now, the design of the controller turns into the design of the feedback gain matrices Ki to ensure that the closed-loop system (4) is asymptotically stable. In what follows, we will introduce a definition and some lemmas which will be used in the next section. The Caputo definition for fractional derivative is used in this paper since only the Caputo definition has the same form as for integer-order differential equations in the initial conditions.

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Definition 1 (Podlubny [1]). The Caputo fractional derivative of order α of a continuous function f : Rþ -R is defined as 8 R t f ðmÞ ðτÞ 1 > > > dτ; m  1oαom; < Γðm  αÞ 0 ðt  τÞα  mþ1 α Dt f ðt Þ ¼ m > > > d m f ðt Þ; α ¼ m; : dt where Γ is Γ-function, and Z 1 ΓðzÞ ¼ e  t t z  1 dt;

Γðz þ 1Þ ¼ zΓðzÞ:

0

Lemma 1 (Matgnon [33]). Let A0 A Rnn be a real matrix. Then, fractional-order linear system Dαt xðtÞ ¼ A0 xðtÞ is asymptotically stable where 0 r αo2, if and only if   argðspecðA0 ÞÞ4 απ 2 where specðA0 Þ is the spectrum of A0. Lemma 2 (Chilali et al. [34]). Let A0 ARnn be a real matrix and θ ¼ π  απ=2. Then, jargðspecðA0 ÞÞj4απ=2 where 1 r αo2 if and only if there exists P40 such that " # ðA0 P þ PAT0 Þ sin θ ðA0 P  PAT0 Þ cos θ o0 ð5Þ ⋆ ðA0 P þ PAT0 Þ sin θ Lemma 3 (Lu and Chen [29]). Let A0 A Rnn be a real matrix and 0oαo1. Then, fractionalorder system Dαt xðtÞ ¼ A0 xðtÞ is asymptotically stable if and only if there exist two real symmetric positive definite matrices P11 A Rnn ; P21 ARnn , and two skew-symmetric matrices P12 A Rnn ; P22 A Rnn such that 2

2

∑ ∑ SymfΘij  ðA0 Pij Þgo0

i¼1j¼1

"

P11  P12

where

"

Θ11 ¼ " Θ21 ¼

# P12 40; P11 π  α 2π  cos 2 α sin

π  2α   cos π2 α sin

"

P21  P22

# P22 40; P21

 #  cos π2 α   ; sin π2 α π  # α π2  ; sin 2 α

cos

" Θ12 ¼ " Θ22 ¼

π  α 2 π   sin 2 α

π  # α 2π  ; cos 2 α

π  α π2   sin 2 α

π  # 2α  :  cos π2 α

cos

 cos

sin

sin

Lemma 4 (Xu et al. [35]). Let E; H and F(t) be real matrices of appropriate dimensions with F (t) satisfying F T ðtÞFðtÞr I. Then, for any scalar ϵ40, we have EFðtÞH þ H T F T ðtÞE T r ϵ  1 EE T þ ϵH T H: Lemma 5 (Schur complement [36]). The following linear matrix inequality (LMI)   Q S 40; ST R

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where Q ¼ QT ; R ¼ RT , is equivalent to Q40;

R  ST Q  1 S40;

ðiiÞ R40;

Q  SR  1 ST 40:

ðiÞ or

3. Stabilization criteria In this section, our objective is to design the feedback gain matrices K i ði ¼ 1; 2; …; rÞ to asymptotically stabilize system (4), which can be guaranteed by the following two theorems. Case I: 0oαo1. Theorem 1. The closed-loop system (4) with fractional derivative 0oαo1 is asymptotically stable if there exist matrices X i A Rnn ; i ¼ 1; 2; …; r, a symmetric positive definite matrix P A Rnn , and two real positive scalars α1 ; α2 , such that " # Υ 11 Υ 12 o0 ð6Þ ⋆ Υ 22 where 2

2

j¼1

j¼1

Υ 11 ¼ ∑ SymfΘj1  ðAi P þ X i Þg þ ∑ αj ðI 2  E i E Ti Þ; Υ 12 ¼ ½I 2  ðH i PÞT I 2  ðH i PÞT ;

i ¼ 1; 2; …; r;

i ¼ 1; 2; …; r;

Υ 22 ¼  diagfα1 ; α2 g  I 2n : Moreover, the state feedback gain matrices are given by K i ¼ X i P  1 ; i ¼ 1; 2; …; r. Proof. Set P12 ¼ 0, P22 ¼ 0, P11 ¼ P21 ¼ P40 in Lemma 3. Obviously, the last two LMIs in Lemma 3 hold. The first LMI in Lemma 3 is reduced to 2

∑ SymfΘi1  A0 Pgo0:

i¼1

Replacing A0 by ∑ri ¼ 1 wi ðzðtÞÞðAi þ Ei F i ðtÞH i þ K i Þ in Eq. (4) and in view of the properties of Kronecker product A  ðB þ CÞ ¼ A  B þ A  C and ðkAÞ  B ¼ kðA  BÞ ¼ A  ðkBÞ, we have   2 r ∑ Sym Θj1  ∑ wi ðzðtÞÞðAi þ E i F i ðtÞH i þ K i ÞP j¼1

i¼1



¼ ∑ ∑ wi Sym Θj1  ðAi P þ X i þ Ei F i H i PÞ ; 2

r

ð7Þ

j¼1i¼1

where Xi ¼ KiP. In consideration of the facts ðA  BÞT ¼ AT  BT and ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ and the Assumption, we obtain ðI 2  F i ÞðI 2  F i ÞT ¼ ðI 2  F i ÞðI 2  F Ti Þ ¼ I 2  ðF i F Ti Þr I:

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It follows from Lemma 3 that Θij ΘTij ¼ I 2 ði; j ¼ 1; 2Þ. In particular, we have Θj1 ΘTj1 ¼ I 2 ðj ¼ 1; 2Þ. Combined with Lemma 4, we have SymfΘj1  ðEi F i H i PÞg ¼ SymfðΘj1  E i ÞðI 2  F i ÞðI 2  H i PÞg r αj ðΘj1  E i ÞðI 2  F i ÞðI 2  F i ÞT ðΘj1  E i ÞT þ αj 1 ðI 2  H i PÞT ðI 2  H i PÞ r αj ðΘj1  E i ÞðΘj1  Ei ÞT þ αj 1 ðI 2  H i PÞT ðI 2  H i PÞ r αj ðI 2  Ei E Ti Þ þ αj 1 ðI 2  H i PÞT ðI 2  H i PÞ:

ð8Þ

Substituting Eq. (8) into Eq. (7), we obtain 2

r

∑ SymfΘj1  ∑ wi ðzðtÞÞðAi þ Ei F i ðtÞH i þ K i ÞPg

j¼1

i¼1

n o r ∑ ∑ wi SymfΘj1  ðAi P þ X i Þg þ αj ðI 2  E i E Ti Þ þ αj 1 ðI 2  H i PÞT ðI 2  H i PÞ 2

r

j¼1i¼1 r

2

i¼1

j¼1

¼ ∑ wi ∑

n

o SymfΘj1  ðAi P þ X i Þg þ αj ðI 2  Ei ETi Þ þ αj 1 ðI 2  H i PÞT ðI 2  H i PÞ

By the Schur complement, it was not hard to see that inequality 2

2

2

j¼1

j¼1

j¼1

∑ SymfΘj1  ðAi P þ X i Þg þ ∑ αj ðI 2  E i E Ti Þ þ ∑ αj 1 ðI 2  H i PÞT ðI 2  H i PÞo0

is equivalent to LMI (6). Taking 0 r wi ðzðtÞÞ r 1 and ∑ri ¼ 1 wi ðzðtÞÞ ¼ 1 into consideration, we have 2

r

j¼1

i¼1

∑ SymfΘj1  ∑ wi ðzðtÞÞðAi þ Ei F i ðtÞH i þ K i ÞPgo0

By Lemma 3, the closed-loop system (4) with fractional-order 0oαo1 is asymptotically stable. This completes the proof. □ Case II: 1 r αo2. Theorem 2. The closed-loop system (4) with fractional derivative 1r αo2 is asymptotically stable if there exist matrices X i A Rnn ; i ¼ 1; 2; …; r, a symmetric positive definite matrix P A Rnn , and real scalars εi , i ¼ 1; 2; …; r, such that 3 2 Γ 11 Γ 12 ðH i PÞT 0 7 6 6 ⋆ Γ 22 0 ðH i PÞT 7 7o0 6 ð9Þ 6H P 0  εi I 0 7 5 4 i 0 HiP 0  εi I where Γ 11 ¼ Γ 22 ¼ ðAi P þ PATi þ X i þ X Ti Þ sin θ þ εi E i E Ti ; Γ 12 ¼ ðAi P  PATi þ X i  X Ti Þ cos θ; i ¼ 1; 2; …; r

i ¼ 1; 2; …; r

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θ ¼ π  απ=2: Moreover, the state feedback gain matrices are given by K i ¼ X i P  1 ; i ¼ 1; 2; …; r. Proof. It follows from Lemmas 1 and 2 that system Dαt xðtÞ ¼ A0 xðtÞ is asymptotically stable if and only if Eq. (5) holds. Replacing A0 in Eq. (5) by ∑ri ¼ 1 wi ðzðtÞÞðAi þ E i F i ðtÞH i þ K i Þ, we have " # ðA0 P þ PAT0 Þ sin θ ðA0 P  PAT0 Þ cos θ ðPAT0  A0 PÞ cos θ ðA0 P þ PAT0 Þ sin θ 2 r wi ðAi P þ PATi þ K i P þ PK Ti Þ sin θ 6 i∑ 6 ¼1 ¼6 r 4 ∑ wi ðPATi  Ai P þ PK Ti  K i PÞ cos θ

3 r ∑ wi ðAi P  PATi þ K i P  PK Ti Þ cos θ 7 i¼1 7 7 r 5 T T ∑ wi ðAi P þ PAi þ K i P þ PK i Þ sin θ

i¼1

2

r

wi ðE i F i H i P þ PH Ti F Ti ETi Þ sin θ 6 i∑ 6 ¼1 þ6 r 4 ∑ wi ðPH Ti F Ti E Ti  Ei F i H i PÞ cos θ i¼1

r

¼ ∑ wi

"

ðAi P þ

PATi

þ KiP þ

i¼1

3 ∑ wi ðE i F i H i P  PH Ti F Ti E Ti Þ cos θ 7 i¼1 7 7 r 5 ∑ wi ðE i F i H i P þ PH Ti F Ti E Ti Þ sin θ r

i¼1

PK Ti Þ sin θ

ðAi P PATi þ K i P  PK Ti Þ cos θ

#

ðPATi  Ai P þ PK Ti  K i PÞ cos θ ðAi P þ PATi þ K i P þ PK Ti Þ sin θ (" #) r Ei F i H i P sin θ Ei F i H i P cos θ þ ∑ wi Sym  E i F i H i P cos θ E i F i H i P sin θ i¼1 i¼1

In view of Lemma 4 and Assumption, we have (" #) E i F i H i P cos θ Ei F i H i P sin θ Sym  E i F i H i P cos θ Ei F i H i P sin θ (" #" #) #" Ei sin θ E i cos θ 0 Fi 0 HiP ¼ Sym  E i cos θ Ei sin θ 0 Fi 0 HiP " #" #T " Ei sin θ E i cos θ E i cos θ HiP Ei sin θ r εi þ εi 1  E i cos θ Ei sin θ  E i cos θ Ei sin θ 0

0 HiP

#T "

ð10Þ

HiP

0

0

HiP

#

ð11Þ Substituting Eq. (11) into Eq. (10), we obtain "

ðA0 P þ PAT0 Þ sin θ

ðA0 P  PAT0 Þ cos θ

#

ðPAT0  A0 PÞ cos θ ðA0 P þ PAT0 Þ sin θ (" # r ðAi P þ PATi þ K i P þ PK Ti Þ sin θ ðAi P  PATi þ K i P  PK Ti Þ cos θ r ∑ wi ðPATi Ai P þ PK Ti K i PÞ cos θ ðAi P þ PATi þ K i P þ PK Ti Þ sin θ i¼1 " #" #T " #T " #) Ei sin θ E i cos θ E i cos θ H P 0 H P 0 Ei sin θ i i þεi þ εi 1  Ei cos θ E i sin θ  Ei cos θ E i sin θ 0 HiP 0 HiP

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("

r

¼ ∑ wi

ðAi P  PATi þ K i P  PK Ti Þ cos θ

ðPATi  Ai P þ PK Ti  K i PÞ cos θ

ðAi P þ PATi þ K i P þ PK Ti Þ sin θ þ εi Ei E Ti

i¼1

" þεi 1

HiP

0

0

HiP

#T "

HiP

0

0

HiP

#)

By Schur Complement, we have " ðAi P þ PATi þ K i P þ PK Ti Þ sin θ þ εi E i E Ti ðPATi  Ai P þ PK Ti  K i PÞ cos θ " þ

#

ðAi P þ PATi þ K i P þ PK Ti Þ sin θ þ εi Ei ETi

εi 1

HiP 0

0 HiP

#T "

HiP 0

#

ðAi P PATi þ K i P  PK Ti Þ cos θ ðAi P þ PATi þ K i P þ PK Ti Þ sin θ þ εi E i E Ti

# 0 o0 HiP

is equivalent to LMI (9). Taking 0 r wi ðzðtÞÞ r 1 and ∑ri ¼ 1 wi ðzðtÞÞ ¼ 1 into consideration, we obtain " # ðA0 P þ PAT0 Þ sin θ ðA0 P  PAT0 Þ cos θ o0 ðPAT0  A0 PÞ cos θ ðA0 P þ PAT0 Þ sin θ Thus, by Lemma 1 and Lemma 2, the closed-loop system (4) with fractional-order 1r αo2 is asymptotically stable. □ Remark 1. When α ¼ 1, that is, when the fractional-order chaotic system is reduced to an integer-order chaotic system, it is not difficult to see that Theorem 2 in this paper agrees with some existing results in the literature, including [25,37,38]. 4. Numerical simulations Consider the chaotic fractional-order Lorenz system Dαt x1 ¼ aðx2  x1 Þ þ u1 ðtÞ; Dαt x2 ¼ cx1  x2  x1 x3 þ u2 ðtÞ; Dαt x3 ¼ x1 x2  bx3 þ u3 ðtÞ:

ð12Þ

As shown in Fig. 1, system (12) displays chaotic behavior with α ¼ 0:995, a ¼ 10, b ¼ c ¼ 28, uðtÞ ¼ 0, and the initial conditions ðx0 ; y0 ; z0 ÞT ¼ ð  1; 2;  1ÞT . Firstly, we determine the T–S fuzzy model representation of system (12). Suppose jx1 j r d and the value of d is set to be d¼ 20 according to the numerical simulation in Fig.1. In fact, this assumption is reasonable because chaotic system is dissipative, and therefore the state variables are bounded.   Take l¼ 2, the  membership functions are selected as μF1 ðx1 Þ ¼ 12 1 þ x1 =d ,  μF 2 ðx1 Þ ¼ 12 1  x1 =d , i.e., two subsystems are employed to locally describe system (12). Thus, the controlled T–S fuzzy model of system (12) with uncertainty is defined as follows: 8 3,

2

Dαt xðtÞ ¼ ∑ μi ðzðtÞÞfðAi þ ΔAi ÞxðtÞg þ uðtÞ; i¼1

X. Huang et al. / Journal of the Franklin Institute 351 (2014) 5480–5493 40

60

20

40

0

x3

x2

5489

20

−20 −40 −20

−10

0

10

0 −20

20

−10

0

x1

10

20

x1

60 50

x3

x3

40

20

0 50 0

0 −40

−20

0

20

x2

40

−50

x2

−20

0

20

x1

Fig. 1. Chaotic attractors in system (12) with α ¼ 0:995, a ¼ 10; b ¼ 83 ; c ¼ 28.

in which

2

a 6 A1 ¼ 4 c 0

3 0 7  d 5;

a 1

b

d

2

 0:02 6 E1 ¼ E2 ¼ 4 0:03

2

a 6 A2 ¼ 4 c 0

0:02 0

0 0

0

 0:01

0

a 1 d

3 0 7 d 5; b

ΔAi ¼ E i F i ðtÞH i ; i ¼ 1; 2;

3 7 5;

H 1 ¼ H 2 ¼ I 3 , F 1 ðtÞ ¼ F 2 ðtÞ ¼ diagf sin 0:1t; sin 0:1t; sin 0:1tg, i.e., the parameter uncertainties in system (12) are Δa ¼ 0:02 sin 0:1t, Δb ¼ 0:01 sin 0:1t, Δc ¼ 0:03 sin 0:1t, respectively. Based on Theorem 1, a feasible solution for LMI (6) can be solved by virtue of MATLAB LMI toolbox as 2 3 0:0695 0:0000 0:0000 6 7 P ¼ 4 0:0000 0:0695 0:0000 5; α1 ¼ 1:1811; α2 ¼ 1:1811; 0:0000 2

0:3990

0:0000

6 X 1 ¼ 4  1:3197  0:0000

0:0695

 1:3197

 0:0000

3

 0:2263

7  0:0000 5;

 0:0000

 0:1101

2

0:3990

6 X 2 ¼ 4  1:3197 0:0000

 1:3197  0:2263 0:0000

0:0000

3

7 0:0000 5:  0:1101

5490

X. Huang et al. / Journal of the Franklin Institute 351 (2014) 5480–5493

x1

2 0 −2

0

2

4

0

2

4

0

2

4

t

6

8

10

6

8

10

6

8

10

x2

2 0 −2

t

x3

1 0 −1

t

Fig. 2. Time response of the states of the controlled system (12) with α ¼ 0:995. 40

60

40

0

x3

x2

20

20

−20 −40 −20

−10

0

10

0 −20

20

−10

0

10

20

x1

x1 60 50

x3

x3

40

20

0 −40

0 50 0 −20

0

20

40

x2

−50

−20

0

x1

x2 Fig. 3. Chaotic attractors in system (12) with α ¼ 1:05, a ¼ 10; b ¼ 83 ; c ¼ 28.

Thus, the feedback gain matrices K 1 ; K 2 are calculated by 2 3 5:7431  18:9949  0:0000 6 7 K 1 ¼ X 1 P  1 ¼ 4  18:9949  3:2578  0:0000 5;  0:0000

 0:0000

 1:5843

20

X. Huang et al. / Journal of the Franklin Institute 351 (2014) 5480–5493

5491

x1

1 0 −1

0

2

4

0

2

4

0

2

4

t

6

8

10

6

8

10

6

8

10

x2

2 0 −2

t

x3

1 0 −1

t

Fig. 4. Time response of the states of the controlled system (12) with α ¼ 1:05.

2

5:7431 6 1 K2 ¼ X 2 P ¼ 4  18:9949 0:0000

 18:9949  3:2578 0:0000

3 0:0000 7 0:0000 5:  1:5843

Adams–Bashforth–Moulton predictor–corrector algorithm is used in the simulation process. Set the initial conditions as ðx0 ; y0 ; z0 ÞT ¼ ð 1; 2;  1ÞT in the numerical simulation. Fig. 2 shows the time responses of the states of the controlled system (12) with α ¼ 0:995. From Fig. 2, we can see that the controlled system (12) converges to zero fast. In what follows, we consider the case where 1 r αo2: In fact, when α ¼ 1:05, system (12) also displays chaotic behaviors, as shown in Fig. 3. Based on Theorem 2, a feasible solution for LMI (9) can be solved in much the same way 2 3 0:1583  0:0000 0:0000 6 7 P ¼ 4  0:0000 0:1583 0:0000 5; ε1 ¼ 1:4251; ε2 ¼ 1:4251; 0:0000 0:0000 0:1583 2

0:8681

6 X 1 ¼ 4  4:4331 0:0000

 1:5830  0:5570  3:1668

Thus, the feedback gain matrices 2 5:4825 6 1  K 1 ¼ X 1 P ¼ 4 27:9973 0:0000 2 5:4825 6 1 K2 ¼ X 2 P ¼ 4  27:9973 0:0000

0:0000

3

3:1668 7 5;  0:2926

2

6 X 2 ¼ 4  4:4331  0:0000

K 1 ; K 2 are  9:9973  3:5180  20:0000  9:9973  3:5180  20:0000

0:8681

0:0000

3

7 20:0000 5;  1:8477 3 0:0000 7 20:0000 5:  1:8477

 1:5830  0:5570 3:1668

0:0000

3

 3:1668 7 5:  0:2926

5492

X. Huang et al. / Journal of the Franklin Institute 351 (2014) 5480–5493

Set the initial values as ðx0 ; y0 ; z0 ÞT ¼ ð1;  2; 1ÞT in the numerical simulation. Fig. 4 shows the time responses of the states of controlled system (12) with α ¼ 1:05. From Fig. 4, we can find that the controlled system (12) is asymptotically stable and its states converge to zero. 5. Conclusions In this paper, the stabilization problem of uncertain fractional-order chaotic systems is investigated in the case of the fractional-order α satisfying 0oαo1 and 1 rαo2. Firstly, the uncertain fractional-order chaotic system is described by the so-called fractional-order T–S fuzzy model, and then the fuzzy state feedback controller is correspondingly designed. Sufficient conditions are derived for the stability of the closed loop systems in the aforementioned two cases. These criteria are expressed in terms of linear matrix inequalities (LMIs), and therefore the feedback gain matrices can be formulated into the solvability of the relevant LMIs. The proposed controller overcomes defects in traditional control techniques and is easy to implement. Finally, two numerical examples are presented to demonstrate the effectiveness and the feasibility of the robust stabilizing controller and the robust stability criteria.

References [1] I. Podlubny, Fractional Differential Equations, Academic, New York, 1999. [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Hackensack, NJ, 2001. [3] J. Shen, J.D. Cao, J.Q. Lu, Consensus of fractional-order systems with nonuniform input and communication delays, Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 226 (2) (2012) 132–412. [4] J. Yu, C. Hu, H. Jiang, α-stability and α-synchronization for fractional-order neural networks, Neural Netw. 35 (2012) 82–87. [5] T.T. Hartley, C.F. Lorenzo, H.K. Qammer, Chaos in a fractional order Chua's system, IEEE Trans. Circuits Syst. I 42 (1995) 485–490. [6] Y. Yu, H. Li, S. Wang, J. Yu, Dynamic analysis of a fractional-order Lorenz chaotic system, Chaos Solitons Fractals 42 (2009) 1181–1189. [7] C. Li, G. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Physica A 341 (2004) 55–61. [8] D. Varsha, B. Sachin, Chaos in fractional ordered Liu system, Comput. Math. Appl. 59 (2010) 1117–1127. [9] X. Huang, Z. Zhao, Z. Wang, Y. Li, Chaos and hyperchaos in fractional-order cellular neural networks, Neurocomputing 94 (2012) 13–21. [10] E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. [11] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (6) (1992) 421–428. [12] Z.Y. Yan, Controlling hyperchaos in the new hyperchaotic Chen system, Appl. Math. Comput. 168 (2) (2005) 1239–1250. [13] M.T. Yassen, Adaptive control and synchronization of a modified Chuas circuit system, Appl. Math. Comput. 135 (2003) 113–128. [14] J. Lu, J. Cao, Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos 15 (4) (2005) 043901. [15] S.S. Ge, C. Wang, T.H. Lee, Adaptive backstepping control of a class of chaotic systems, Int. J. Bifurc. Chaos 10 (2000) 1149–1156. [16] T. Yang, L.O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44 (1997) 976–988. [17] K. Konishi, M. Hirai, H. Kokame, Sliding mode control for a class of chaotic systems, Phys. Lett. A 245 (6) (1998) 511–517. [18] Z. Wang, X. Huang, H. Shen, Control of an uncertain fractional order economic system via adaptive sliding mode, Neurocomputing 83 (2012) 83–88.

X. Huang et al. / Journal of the Franklin Institute 351 (2014) 5480–5493

5493

[19] M.R. Faieghi, Control of an uncertain fractional-order Liu system via fuzzy fractional-order sliding mode control, J. Vib. Control 18 (9) (2011) 1366–1374. [20] S. Dadras, H.R. Momeni, Control of a fractional-order economical system via sliding mode, Physica A 389 (12) (2010) 2434–2442. [21] C.G. Li, G.R. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (3) (2004) 549–554. [22] A.E. Matouk, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A 373 (2009) 2166–2173. [23] W.M. Ahmad, A.M. Harb, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos Solitons Fractals 18 (2003) 693–701. [24] J.-H. Kim, C.-H. Hyun, E. Kim, M. Park, Adaptive synchronization of uncertain chaotic systems based on T–S fuzzy model, IEEE Trans. Fuzzy Syst. 15 (3) (2007) 359–369. [25] X.W. Liu, X. Gao, Delayed fuzzy controller design for hyperchaos with application to hyperchaotic Chen's system, Int. J. Mod. Phys. C 18 (7) (2007) 1095–1105. [26] C. Hu, H. Jiang, Z. Teng, General impulsive control of chaotic systems based on a T–S fuzzy model, Fuzzy Sets Syst. 174 (2011) 66–82. [27] Y. Liu, S.W. Zhao, J.Q. Lu, A new fuzzy impulsive control of chaotic systems based on T–S fuzzy model, IEEE Trans. Fuzzy Syst. 19 (2) (2011) 393–398. [28] Y. Zheng, Y. Nian, D. Wang, Controlling fractional order chaotic systems based on Takagi–Sugeno fuzzy model and adaptive adjustment mechanism, Phys. Lett. A 375 (2010) 125–129. [29] J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0oαo1 case, IEEE Trans. Autom. Control 55 (1) (2010) 152–158. [30] J.G. Lu, G.R. Chen, Robust stability and stabilization of fractional-order interval systems: an LMI approach, IEEE Trans. Autom. Control 54 (6) (2009) 1294–1299. [31] X. Song, Z. Wang, Dynamic output feedback control for fractional-order systems, Asian J. Control 15 (2013) 834–848. [32] B. Zhang, J. Lam, S. Xu, Z. Shu, Robust stabilization of uncertain T–S fuzzy time-delay systems with exponential estimates, Fuzzy Sets Syst. 16 (12) (2009) 979–984. [33] D. Matgnon, Stability result on fractional differential equations with applications to control processing, in: Proceedings of IMACS, IEEE-SMC, Lille, France, July 1996, pp. 963–968. [34] M. Chilali, P. Gahinet, P. Apkarian, Robust pole placement in LMI regions, IEEE Trans. Autom. Control 44 (12) (1999) 2257–2270. [35] S.Y. Xu, B. Song, J.W. Lu, J. Lam, Robust stability of uncertain discrete-time singular fuzzy systems, Fuzzy Sets Syst. 158 (2007) 2306–2316. [36] S. Boyd, L. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [37] Y.W. Wang, Z.H. Guan, H.O. Wang, LMI-based fuzzy stability and synchronization of Chen's system, Phys. Lett. A 320 (2–3) (2003) 154–159. [38] X. Huang, J. Cao, Y. Lu, Takagi–Sugeno fuzzy-model-based control of hyperchaotic Chen system with normbounded uncertainties, Proc. IMechE Part I: J. Syst. Control Eng. 224 (3) (2010) 223–234.