# LMI Based Fuzzy Control of a Wing Doubled Fractional-Order Chaos

Authors: Bin Wang; Yuzhu Wang; Hongbo Cao; Delan Zhu
Journal: Journal of Control Science and Engineering
 (2015)

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## Abstract

This paper investigates a new wing doubled fractional-order chaos and its control. Firstly, a new fractional-order chaos is proposed, replacing linear termx in the second equation by its absolute value; a new improved system is got, which can make the wing of the original system doubled. Then, circuit diagram is presented for the proposed fractional-order chaos. Furthermore, based on fractional-order stability theory and T-S fuzzy model, a more practical stability condition for fuzzy control of the proposed fractional-order chaos is given as s set of linear matrix inequality (LMI) and the strict mathematical norms of LMI are presented. Finally, numerical simulations are given to verify the effectiveness of the proposed theoretical results.


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## Body

## 1. Introduction

Fractional calculus has the same history as integer calculus, which has appeared 300 years ago. Nevertheless, it has not been widely used in actual project until recent decades [1–4]. Recently, people find that many actual systems can be well described with the help of fractional calculus, especially for memory and hereditary properties of various materials and processes [5–7]. For example, it can be better described with fractional calculus of power system [8], memristor system [9], physical system [10], chemical system [11], and so on.It has been widely verified that chaos is universal in integer-order chaos. For the advantages of fractional-order chaotic systems in secure communication and signal processing [12–14], many new fractional-order chaotic systems have been proposed and analyzed. For example, Chen et al. proposed a new fractional-order chaotic system and the circuit synchronization of the system was implemented which was very important in theory and practice [15]. Jia et al. studied the chaotic features of the fractional-order Lorenz system and the circuit implementation was presented [16]. A new fractional-order hyperchaotic system based on the Lorenz system was presented and the fractional Hopf bifurcation was investigated in [17]. A new three-dimensional King Cobra face shaped fractional-order chaotic system was designed and the multiscale synchronization of two identical fractional-order King Cobra chaotic systems was derived through feedback control in [18]. In particular, since the potential merits of fractional-order chaos in secure communication and many other fields, fractional-order chaos control and synchronization has become a hot topic.Many studies indicated that integer-order chaos could be well controlled. Recently, many scholars tried to investigate the control methods for fractional-order chaotic systems. Until now, some control strategies have been designed for the control of fractional-order chaotic systems such as sliding mode control [19], finite time control [20], and pinning control [21], among many others. Fuzzy control is well known as an effective control strategy, and it has attracted more and more scholar’s attention. Linear matrix inequality (LMI), as a very important and classic tool, has been widely used in the fuzzy control and synchronization of integer-order chaos. For example, in [22], by employing LMI method, a new fuzzy controller based on T-S fuzzy model is designed for chaos synchronization of two Rikitake generator systems. In [23], a T-S fuzzy receding horizon H-infinity synchronization (TSFRHHS) approach is proposed and a novel set of LMI conditions are given. And the scheme is applied to synchronize Lorenz meteorological chaos. In [24], a robust static output feedback controller is derived in strict LMI terms for discrete-time T-S fuzzy systems. However, this control method has been mostly used in integer-order systems up to now. As we all know, the controllability and stability region of fractional-order systems are different with integer-order systems. Can LMI be applied to the fuzzy control of the wing doubled fractional-order chaos? If so, what are the special application conditions and strict mathematical norms? There is almost no relevant literature. Research in this area should be meaningful and challenging.In light of the above analysis, there are three advantages which make our research attractive. Firstly, a new wing doubled fractional-order chaos is proposed and its experimental circuit simulation is presented. Secondly, based on fractional-order stability theory, a more practical stability condition for fuzzy control of the proposed fractional-order chaos is proposed and the strict mathematical norms of LMI are presented. Finally, numerical simulations are given to verify the effectiveness of the proposed theoretical results.The paper is organized as follows. In Section2, the fractional-order calculus and stability theory are presented. Section 3 introduces the system description and circuit implementation. In Section 4, the fuzzy controller design and numerical simulations are presented. Conclusions are drawn in Section 5.


## 2. Preliminaries

### 2.1. Fractional-Order Calculus

There are two commonly used definitions of fractional calculus operator: Riemann-Liouville definition and Caputo definition, which are listed here for clarity.Definition 1.
The Riemann-Liouville fractional integral is described as(1)Jt0qfx=1Γq∫t0tt-τq-1fτdτ,where q∈R+, f:R→R, Jt0q denotes q order integral operator, Γ· means the Gamma function, and Γq=∫0∞tq-1e-tdt.Definition 2.
The Riemann-Liouville fractional derivative is given by(2)Dt0qft=Dt0nJt0n-qft=dndtn1Γn-q∫t0tt-τfτdτ,n-1<q<n,dndtnft,q=n,where Dt0q denotes q order Riemann-Liouville derivative operator.Definition 3.
The Caputo derivative can be written by(3)Dt0q0cft=Jt0n-qDt0nft=1Γn-q∫t0tt-τn-q-1fnτdτ,n-1<q<n,dndtnft,q=n,where Dt0q0c denotes q order Caputo derivative operator, which can be simplified as Dq.The Caputo derivative has a popular application in engineering and is adopted in this paper.


### 2.2. Stability in Fractional-Order Systems

Theorem 4 (see [25]).
Consider the following linear fractional-order system:(4)Dqx=Ax,x0=x0,where A∈Rn×n, x∈Rn, and q=[q1,q2,…,qi,…,qn](0<qi≤1). System (4) is asymptotically stable if and only if argλi>qπ/2 is satisfied for all eigenvalues λi of matrix A. Besides, this system is stable if and only if arg(λi)≥qπ/2 is satisfied for all eigenvalues λi of matrix A and those critical eigenvalues, which satisfy condition arg(λi)=qπ/2, have geometric multiplicity one.


## 2.1. Fractional-Order Calculus

There are two commonly used definitions of fractional calculus operator: Riemann-Liouville definition and Caputo definition, which are listed here for clarity.Definition 1.
The Riemann-Liouville fractional integral is described as(1)Jt0qfx=1Γq∫t0tt-τq-1fτdτ,where q∈R+, f:R→R, Jt0q denotes q order integral operator, Γ· means the Gamma function, and Γq=∫0∞tq-1e-tdt.Definition 2.
The Riemann-Liouville fractional derivative is given by(2)Dt0qft=Dt0nJt0n-qft=dndtn1Γn-q∫t0tt-τfτdτ,n-1<q<n,dndtnft,q=n,where Dt0q denotes q order Riemann-Liouville derivative operator.Definition 3.
The Caputo derivative can be written by(3)Dt0q0cft=Jt0n-qDt0nft=1Γn-q∫t0tt-τn-q-1fnτdτ,n-1<q<n,dndtnft,q=n,where Dt0q0c denotes q order Caputo derivative operator, which can be simplified as Dq.The Caputo derivative has a popular application in engineering and is adopted in this paper.


## 2.2. Stability in Fractional-Order Systems

Theorem 4 (see [25]).
Consider the following linear fractional-order system:(4)Dqx=Ax,x0=x0,where A∈Rn×n, x∈Rn, and q=[q1,q2,…,qi,…,qn](0<qi≤1). System (4) is asymptotically stable if and only if argλi>qπ/2 is satisfied for all eigenvalues λi of matrix A. Besides, this system is stable if and only if arg(λi)≥qπ/2 is satisfied for all eigenvalues λi of matrix A and those critical eigenvalues, which satisfy condition arg(λi)=qπ/2, have geometric multiplicity one.


## 3. New Wing Doubled System and Circuit Simulation

### 3.1. System Model

A new three-dimensional fractional-order chaos is written as(5)dqxdtq=2811x-yz,dqydtq=-10y+xz+kx,dqzdtq=-4z+xy,where x, y, and z are the state variables, k is a real number, here, k=0.2, and q is the fractional-order. In order to study the chaotic characteristics of system (5), firstly, it is easy to get the Jacobian matrix:(6)λ1=-10.0000,λ2=2.5455,λ3=-4.0000J0=2811-z-yz+0.2-10xyx-4.Besides, the dynamic characteristics of the system are related to the eigenvalues. Eigenvalues are determined by the Jacobian matrix at equilibrium points. System (5) has five equilibrium points; the corresponding eigenvalues are given as in Table 1.Table 1
Equilibrium points
Their corresponding eigenvalues
E
1(0, 0, 0)
λ
1
=
-
10.0000, λ2=2.5455, λ3=-4.0000
E
2(−6.4512, 3.1909, −5.1462)
λ
1
=
1.1552
+
5.3713
i, λ2=1.1552-5.3713i, λ3=-13.7650
E
3(−6.2004, −3.1909, −4.9462)
λ
1
=
-
11.4914, λ2=0.0368, λ3=0
E
4(6.2004, 3.1909, 4.9462)
λ
1
=
1.0344
+
5.3349
i, λ2=1.0344-5.3349i, λ3=-13.5233
E
5(6.4512, −3.1909, −5.1462)
λ
1
=
1.1552
+
5.3713
i, λ2=1.1552-5.3713i,  λ3=-13.7650According to Theorem4, nonstable region is determined by the following conditions:(7)qπ2≥mini⁡arg⁡λi=maxi⁡arctan⁡Im⁡λiRe⁡λi=arctan⁡5.33491.0344=1.3793⟶q>0.8781.Therefore, the necessary conditions for generating chaos of system (5) is q>0.8781, and this paper takes q=0.9. The phase trajectories of system (5) are shown in Figure 1 with the initial value (x(0),y(0),z(0))=(20,0.6,0.8). We can clearly see that the system has two-wing chaotic attractor.Figure 1
Phase trajectories of fractional-order chaotic system (5).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
zNow we replace the linear termx in the second equation of system (5) by its absolute value x; a new improved system can be got:(8)dqxdtq=2811x-yz,dqydtq=-10y+xz+kx,dqzdtq=-4z+xy.Note that system (8) has certain symmetry. Removing the absolute value of the second equation, each equation contains the corresponding linear term and the cross product term of the other two state variables. Figure 2 shows the phase trajectories of system (8) when k=0.2; in this case, we can see that system (8) has four-wing structure and four chaotic attractors, which realize the wing doubled system (5).Figure 2
Phase trajectories of wing doubled fractional-order chaotic system (8).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
z


### 3.2. Circuit Simulation

Fractional circuit is generally expressed as follows:(9)Fs=R1sR1C1+1+R2sR2C2+1+⋯RnsRnCn+1,where n is the unit number of basic integration circuit.According to the literature [26], when the fractional-order is 0.9, (9) can be written as(10)Fs=R1sR1C1+1+R2sR2C2+1+R3sR3C3+1,where R1=62.84 MΩ, R2=250 kΩ, R3=2.5 kΩ, C1=1.232 μF, C2=1.835 μF, and C3=1.1 μF. The unit circuit of (10) is shown in Figure 3.Figure 3
The unit circuit of1/s0.9.System (5) is rewritten as the following corresponding circuit equation:(11)d0.9xdt0.9=R3R2R4C0x-R3R1R4C0yzd0.9ydt0.9=-R7R8R5R15R6C0y+R5R14R6C0xz+R5R14R6C0xd0.9zdt0.9=-R11R18R9R17R10C0z+R9R16R10C0xy.The designed overall circuit diagram is shown in Figure4. Each channel of the circuit is formed by resistors, capacitors, operational amplifiers, and multiplier. There are three channels, representing the three states of system (5). The circuit parameters of system (5) are designed as follows: R1=700 kΩ, R2=275 kΩ, R3=14 kΩ, R4=R6=20 kΩ, R5=7 kΩ, R10=25 kΩ, R7=R8=R11=R17=R18=10 kΩ, R13=700 kΩ, R14=350 kΩ, R15=35 kΩ, and R16=40 kΩ; C0 is the unit circuit of Figure 3.Figure 4
Circuit diagram to realize the new fractional-order chaotic system (5).Figure5 shows the simulation results of the fractional-order system (5); we can see that the circuit simulation results in Figures 5(a)–5(c) are in line with the phase diagram in Figures 1(a)–1(c). It indicates that the proposed fractional-order system (5) can be realized by circuit simulation, and the circuit diagram is valid and practical.Figure 5
Circuit simulation results of the new fractional-order chaotic system (5).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
zSimilarly, the designed circuit diagram of system (8) is shown in Figure 6, and the corresponding simulation results are presented in Figure 7. We can also see that the circuit simulation results in Figures 7(a)–7(c) are in line with the phase diagram in Figures 2(a)–2(c). It indicates that the wing doubled system (8) can be easily achieved by circuit simulation. Therefore, the circuit diagram of system (8) is valid and practical.Figure 6
Circuit diagram to realize the new wing doubled fractional-order chaotic system (8).Figure 7
Circuit simulation results of the new wing doubled fractional-order chaotic system (8).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
z


## 3.1. System Model

A new three-dimensional fractional-order chaos is written as(5)dqxdtq=2811x-yz,dqydtq=-10y+xz+kx,dqzdtq=-4z+xy,where x, y, and z are the state variables, k is a real number, here, k=0.2, and q is the fractional-order. In order to study the chaotic characteristics of system (5), firstly, it is easy to get the Jacobian matrix:(6)λ1=-10.0000,λ2=2.5455,λ3=-4.0000J0=2811-z-yz+0.2-10xyx-4.Besides, the dynamic characteristics of the system are related to the eigenvalues. Eigenvalues are determined by the Jacobian matrix at equilibrium points. System (5) has five equilibrium points; the corresponding eigenvalues are given as in Table 1.Table 1
Equilibrium points
Their corresponding eigenvalues
E
1(0, 0, 0)
λ
1
=
-
10.0000, λ2=2.5455, λ3=-4.0000
E
2(−6.4512, 3.1909, −5.1462)
λ
1
=
1.1552
+
5.3713
i, λ2=1.1552-5.3713i, λ3=-13.7650
E
3(−6.2004, −3.1909, −4.9462)
λ
1
=
-
11.4914, λ2=0.0368, λ3=0
E
4(6.2004, 3.1909, 4.9462)
λ
1
=
1.0344
+
5.3349
i, λ2=1.0344-5.3349i, λ3=-13.5233
E
5(6.4512, −3.1909, −5.1462)
λ
1
=
1.1552
+
5.3713
i, λ2=1.1552-5.3713i,  λ3=-13.7650According to Theorem4, nonstable region is determined by the following conditions:(7)qπ2≥mini⁡arg⁡λi=maxi⁡arctan⁡Im⁡λiRe⁡λi=arctan⁡5.33491.0344=1.3793⟶q>0.8781.Therefore, the necessary conditions for generating chaos of system (5) is q>0.8781, and this paper takes q=0.9. The phase trajectories of system (5) are shown in Figure 1 with the initial value (x(0),y(0),z(0))=(20,0.6,0.8). We can clearly see that the system has two-wing chaotic attractor.Figure 1
Phase trajectories of fractional-order chaotic system (5).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
zNow we replace the linear termx in the second equation of system (5) by its absolute value x; a new improved system can be got:(8)dqxdtq=2811x-yz,dqydtq=-10y+xz+kx,dqzdtq=-4z+xy.Note that system (8) has certain symmetry. Removing the absolute value of the second equation, each equation contains the corresponding linear term and the cross product term of the other two state variables. Figure 2 shows the phase trajectories of system (8) when k=0.2; in this case, we can see that system (8) has four-wing structure and four chaotic attractors, which realize the wing doubled system (5).Figure 2
Phase trajectories of wing doubled fractional-order chaotic system (8).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
z


## 3.2. Circuit Simulation

Fractional circuit is generally expressed as follows:(9)Fs=R1sR1C1+1+R2sR2C2+1+⋯RnsRnCn+1,where n is the unit number of basic integration circuit.According to the literature [26], when the fractional-order is 0.9, (9) can be written as(10)Fs=R1sR1C1+1+R2sR2C2+1+R3sR3C3+1,where R1=62.84 MΩ, R2=250 kΩ, R3=2.5 kΩ, C1=1.232 μF, C2=1.835 μF, and C3=1.1 μF. The unit circuit of (10) is shown in Figure 3.Figure 3
The unit circuit of1/s0.9.System (5) is rewritten as the following corresponding circuit equation:(11)d0.9xdt0.9=R3R2R4C0x-R3R1R4C0yzd0.9ydt0.9=-R7R8R5R15R6C0y+R5R14R6C0xz+R5R14R6C0xd0.9zdt0.9=-R11R18R9R17R10C0z+R9R16R10C0xy.The designed overall circuit diagram is shown in Figure4. Each channel of the circuit is formed by resistors, capacitors, operational amplifiers, and multiplier. There are three channels, representing the three states of system (5). The circuit parameters of system (5) are designed as follows: R1=700 kΩ, R2=275 kΩ, R3=14 kΩ, R4=R6=20 kΩ, R5=7 kΩ, R10=25 kΩ, R7=R8=R11=R17=R18=10 kΩ, R13=700 kΩ, R14=350 kΩ, R15=35 kΩ, and R16=40 kΩ; C0 is the unit circuit of Figure 3.Figure 4
Circuit diagram to realize the new fractional-order chaotic system (5).Figure5 shows the simulation results of the fractional-order system (5); we can see that the circuit simulation results in Figures 5(a)–5(c) are in line with the phase diagram in Figures 1(a)–1(c). It indicates that the proposed fractional-order system (5) can be realized by circuit simulation, and the circuit diagram is valid and practical.Figure 5
Circuit simulation results of the new fractional-order chaotic system (5).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
zSimilarly, the designed circuit diagram of system (8) is shown in Figure 6, and the corresponding simulation results are presented in Figure 7. We can also see that the circuit simulation results in Figures 7(a)–7(c) are in line with the phase diagram in Figures 2(a)–2(c). It indicates that the wing doubled system (8) can be easily achieved by circuit simulation. Therefore, the circuit diagram of system (8) is valid and practical.Figure 6
Circuit diagram to realize the new wing doubled fractional-order chaotic system (8).Figure 7
Circuit simulation results of the new wing doubled fractional-order chaotic system (8).
(a)
x
-
y
(b)
x
-
z
(c)
y
-
z


## 4. Controller Design and Numerical Simulations

### 4.1. T-S Fuzzy Model

The fractional-order T-S fuzzy model is given in the following form:RuleRi is as follows: IF z1(t) is Mi1 and ⋯ and zn(t) is Min(12)THENdqxdtq=Aixt+Biuti=1,2,…,r,where Mij(j=1,2,…,r) is the fuzzy set and r is the number of IF-THEN rules, x(t)∈Rn is the state vector, Ai∈Rn×n, z(t)=[z1(t),z2(t),…,zn(t)] are the premise variables, and q is the fractional-order; u(t) is control input.By adopting single point fuzzification, product inference, and weighted average defuzzification, the final output of the fractional-order T-S fuzzy model is inferred as follows:(13)dqxdtq=∑i=1rhiztAixt+∑i=1rhiztBiut,where(14)hizt=∏j=1nMijzjt∑i=1r∏j=1nMijzjt≥0,∑i=1rhizt=1with Mijzj(t) being the grade of membership of zj(t) in Mij. hi(z(t)) can be regarded as the normalized weights of the IF-THEN rules.


### 4.2. T-S Fuzzy Controller Design Based on LMI

RuleRi is as follows: IF z1(t) is Mi1 and ⋯ and zn(t) is Min(15)THENut=Kixti=1,2,…,r.The inferred output of the PDC controller is expressed in the following form:(16)ut=∑j=1rhjztKjxt,where Kj represents the feedback gain.By substituting (16) into (13), one has(17)dqxdtq=∑i=1rhiztAixt+∑i=1rhiztBi∑j=1rhjztKjxt.In order to simplify (17), it is modified as follows: (18)∑i=1rhiAi=h1A1+h2A2+⋯+hrAr.Considering∑i=1rhi(z(t))=1 in (17), (18) can be written as(19)∑i=1rhiAi=h1h1+h2+⋯+hrA1+⋯+hrh1+h2+⋯+hrAr=h12+h1h2+⋯+h1hrA1+⋯+hrh1+hrh2+⋯+hr2Ar=∑i=1r∑j=1rhihjAi.For formula (17), ∑i=1rhiBi∑j=1rhjKj can be written as(20)∑i=1rhiBi∑j=1rhjKj=h1B1+h2B2+⋯+hrBr·h1K1+h2K2+⋯+hrKr=h1B1h1K1+⋯+hrKr+⋯+hrBrh1K1+h2K2+⋯+hrKr=h12B1K1+h22B2K2+⋯+hi2BiKi+⋯+hr2BrKr+h1h2K2+h3K3+⋯+hrKrB1+h2h3K3+⋯+hrKrB2+⋯+hihi+1Ki+1+⋯+hrKrBi+⋯+hr-1hrKrBr-1+h2h1K1B2+h3h2K2+h1K1·B3+⋯+hjhj-1Kj-1+hj-2Kj-2+⋯+h1K1·Bj+⋯+hrhr-1Kr-1+hr-2Kr-2+⋯+h1K1·Br=∑i=1rhi2BiKj+∑i<jrhihjKjBi+∑i>jrhihjKjBi=∑i=1r∑j=1rhihjBiKj.Submitting (19) and (20) into (17), one gets(21)dqxdtq=∑i=1r∑j=1rhizthjztAi+BiKjxt.Before utilizing the T-S fuzzy model into fractional-order systems (5) and (8), make the following assumptions.Assumption 5.
According to the boundedness of the system, the scope of the system state can be defined as(22)Ω≡xt∈R∣xtn≤δ,xit∈-di,di,di>0i=1,2.Assumption 6.
The feedback control can be applied to every state of the system. The PDC approach together with the T-S fuzzy model can be adopted to control systems (5) and (8).Based on the above assumptions, we can use T-S fuzzy model to describe system (5). Suppose that x1(t)∈[-d1,d1], x2(t)∈[-d2,d2], where d1=15 and d2=10.Thus, the T-S fuzzy model can be presented as follows:R1: IF x1(t) is M1(x1(t)), THEN dqx/dtq=A1x(t)+B1u(t),R2: IF x1(t) is M2(x1(t)), THEN dqx/dtq=A2x(t)+B2u(t),R3: IF x2(t) is M3(x2(t)), THEN dqx/dtq=A3x(t)+B3u(t),R4: IF x2(t) is M4(x2(t)), THEN dqx/dtq=A4x(t)+B4u(t),wherex(t)=[x1(t),x2(t),x3(t)]T, u(t)=[u1(t),u2(t),u3(t)]T,(23)A1=2811000.2-1015015-4,A2=2811000.2-10-150-15-4,A3=28110-100.2-100100-4,A4=28110100.2-100-100-4,B1=B2=B3=B4=I3×3Iis  a  unit  matrix.The membership functions of fuzzy sets are taken as follows:(24)M1x1t=121+x1td1,M2x1t=121-x1td1,M3x2t=121+x2td2,M4x2t=121-x2td2.Thus, the fractional-order system (5) based on T-S fuzzy model can be expressed as(25)dqxdtq=∑i=14hiztAixt+Biut.According to the PDC method, the controller is given as follows:R1: IF x1(t) is M1(x1(t)), THEN u(t)=K1x(t),R2: IF x1(t) is M2(x1(t)), THEN u(t)=K2x(t),R3: IF x2(t) is M3(x2(t)), THEN u(t)=K3x(t),R4: IF x2(t) is M4(x2(t)), THEN u(t)=K4x(t).The overall controller is obtained as(26)ut=∑j=14hjztKjxt.Submitting (26) into (25), one can get the fractional-order system:(27)dqxdtq=∑i=14∑j=14hizthjztAi+BiKjxt.Similarly, for wing doubled system (8), d1=30 and d2=20. One has(28)A1=2811000-1030030-4,A2=2811000-10-300-30-4,A3=28110-200-100200-4,A4=28110200-100-200-4.To stabilize the fractional-order system (27), the following lemma and theorem are given.Lemma 7 (see [27]).
For the arbitrary state variablex=[x1,x2,…,xn]T, if there exists a symmetric positive definite real matrix P satisfying J=xTP(dqx/dtq)≤0 (xTP(dqx/dtq) is called J function), then the fractional-order system (27) is globally asymptotically stable. Condition J=xTPdqx/dtq≤0 can be equally written as J0=xTPdqx/dtq+dqx/dtqTPx≤0.Theorem 8.
If there exist positive definite matricesP, E, and F as well as a positive constant η, suppose that λi(i=1,2,3,4) and ξi(i=1,2,3,4) are the eigenvalues of matrices PEETP and FTF, respectively. If λi(i=1,2,3,4)>0, ξi(i=1,2,3,4)>0, and we select the controller gain matrix Ki(i=1,2,3,4) which satisfies the following inequalities, the fractional-order system (27) is globally asymptotically stable:(29)GiiTP+PGii+ηPEETP+η-1FTF<0i,j=1,2,3,4,(30)GijTP+PGij+ηPEETP+η-1FTF<0i<j<4,where Gii=Ai+BiKi and Gij=(Ai+BiKj)+(Aj+BjKi)/2.Proof.
By Lemma7, select J0=xTPdqx/dtq+dqx/dtqTPx as J function of system (27):(31)J0=xTPdqxdtq+dqxdtqTPx=xTP∑i=14∑j=14hihjAi+BiKjx+∑i=14∑j=14hihjAi+BiKjTPx=∑i=14∑j=14hihjxTAi+BiKjTP+PAi+BiKjx=∑i=14hi2xTAi+BiKiTP+PAi+BiKix+∑i<j4hihjxTAi+BiKjTP+PAi+BiKjx+∑i>j4hihjxTAi+BiKjTP+PAi+BiKjx.
According to∑i=14hi2+2∑i<j4hihj=1 in (14), so there is(32)∑i<j4hihjxTAi+BiKjTP+PAi+BiKjx+∑i>j4hihjxTAi+BiKjTP+PAi+BiKjx=∑i<j4hihjxTAi+BiKj+Aj+BjKiTP+PAi+BiKj+Aj+BjKix=2∑i<j4hihjxTAi+BiKj+Aj+BjKi2TP+PAi+BiKj+Aj+BjKi2x.
SelectingGii=Ai+BiKi, Gij=(Ai+BiKj)+(Aj+BjKi)/2 and under conditions (29) and (30), one gets (33)J0=∑i=14hi2xTGiiTP+PGiix+2∑i<j4hihjxTGijTP+PGij<x-∑i=14hi2xTηPEETP+η-1FTFx+2∑i<j4hihjxTηPEETP+η-1FTFx.
According to Theorem4, the eigenvalues of the matrices PEETP and FTF are λi(i=1,2,3,4)>0 and ξi(i=1,2,3,4)>0, respectively, so the matrices PEETP and FTF both are positive definite matrix; that is, PEETP+FTF>0; since η>0, one gets(34)ηPEETP+η-1FTF>0.
Considering (34) for (33), one can get(35)J0<-∑i=14hi2xTηPEETP+η-1FTFx+2∑i<j4hihjxTηPEETP+η-1FTFx<0.
Therefore, when inequalities (29) and (30) hold, one has(36)J0=xTPdqxdtq+dqxdtqTPx≤0.
This shows that the fractional system (27) is globally asymptotically stable. This completes the proof.According to Schur complement theorem, inequalities (29) and (30) can be transformed into the problem of solving linear matrix inequalities (LMIs).From (29), we can obtain (37)AiTP+KiTBiP+PAi+PBiKi+ηPEETP+η-1FTF<0.MultiplyingP-1 both in the left and in the right in (37), one gets (38)P-1AiT+P-1KiTBi+AiP-1+BiKiP-1+ηEET+η-1P-1FTFP-1<0.In (38), selecting Q=P-1 and Mi=KiP-1, we can obtain(39)QAiT+AiQ+MiTBiT+BiMi+ηEET+η-1QFTFQ<0.The controller gain isKi=MiQ-1.From (30), we can obtain(40)12AiTP+KjTBiTP+AjTP+KiTBjTP+PAi+PBiKj+PAj+PBjKi+ηPEETP+η-1FTF<0.MultiplyingP-1 both in the left and in the right in (40), we obtain(41)12P-1AiT+P-1KjTBiT+P-1AjT+P-1KiTBjT+AiP-1+BiKjP-1+AjP-1+BjKiP-1+ηEET+η-1P-1FTFP-1<0.In (41), selecting Q=P-1, Mi=KiP-1, and Mj=KjP-1, we can obtain(42)12QAiT+AiQ+MiTBjT+BjMi+QAjT+AjQ+MjTBiT+BiMj+ηEET+η-1QFTFQ<0.The controller gain isKi=MiQ-1.By Schur complement theorem, (39) can be written as(43)QAiT+MiTBiT+AiQ+BiMi+ηEETQFTFQ-η-1I<0.Similarly, (42) can be written as(44)12QAiT+AiTQ+MiTBjT+BjMi+QAjT+AjTQ+MjTBiT+BiMj+ηEETQFTFQ-η-1I<0Q>0.For a given positive numberη>0, we can use Matlab LMI toolbox to solve the linear matrix inequalities (43) and (44) to obtain the positive definite matrix P and the controller gain Ki(i=1,2,3,4).


### 4.3. Simulation Results

For fractional-order system (5), take d1=15, d2=10, and η=10000. According to Theorem 4, select positive definite matrices:(45)E=100000000;010000000;001000000;F=100;010;001;100;010;001;100;010;001.Through Matlab 7.0 LMI toolbox, the positive definite matrixP and the feedback controller gain Ki(i=1,2,3,4) can be obtained:(46)P=diag⁡0.00066,0.00066,0.00066,K1=-9.2609-0.05830.1021-0.14173.2846-22.0523-0.1021-7.9477-2.7154,K2=-9.2609-0.12170.0497-0.07833.284611.4239-0.049718.5761-2.7154,K3=-9.2609-0.1278-0.2628-0.07223.2846-0.00170.26280.0017-2.7154,K4=-9.2609-0.1234-0.1296-0.07663.2846-0.00680.12960.0068-2.7154.Substituting the value ofKi(i=1,2,3,4) into (26), the desired feedback controller u(t) can be got. Figure 8 shows the state trajectories of system (5) without controller; the system states are unstable. Figure 9 illustrates the state trajectories of closed-loop system (5) with controller (26), and the states of system (5) globally converge to the equilibrium point which implies the effectiveness of the designed controller.Figure 8
State trajectories of system (5) without controller.
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State trajectories of closed-loop system (5) with controller (26).
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y
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t
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z
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tSimilarly, for wing doubled system (8), select d1=30, d2=20, and η=10000. According to Theorem 4, take positive definite matrices:(47)E=100000000;010000000;001000000;F=100;010;001;100;010;001;100;010;001.Through Matlab LMI toolbox, the positive definite matrixP and the feedback controller gain Ki(i=1,2,3,4) can be obtained:(48)P=diag⁡0.00066,0.00066,0.00066K1=-9.26090.00560.0005-0.00563.2846-30.4410-0.0005-29.5590-2.7154,K2=-9.2609-0.00910.00020.00913.284630.1005-0.000229.8995-2.7154,K3=-9.2609-0.0000-0.78190.00003.2846-0.00000.78190.0000-2.7154,K4=-9.2609-0.00000.02550.00003.2846-0.0000-0.02550.0000-2.7154.Substituting the value ofKi(i=1,2,3,4) into (26), the controller can be got. Figure 10 shows the state trajectories of system (8) without controller; the system states are unstable. Figure 11 illustrates the state trajectories of system (8) with controller (26). We can see that the controller can be applied to the more complicated wing doubled chaotic system (8) with short time and small overshoot, which shows the speed ability and robustness of the control method.Figure 10
State trajectories of system (8) without controller.
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State trajectories of closed-loop system (8) with controller (26).
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## 4.1. T-S Fuzzy Model

The fractional-order T-S fuzzy model is given in the following form:RuleRi is as follows: IF z1(t) is Mi1 and ⋯ and zn(t) is Min(12)THENdqxdtq=Aixt+Biuti=1,2,…,r,where Mij(j=1,2,…,r) is the fuzzy set and r is the number of IF-THEN rules, x(t)∈Rn is the state vector, Ai∈Rn×n, z(t)=[z1(t),z2(t),…,zn(t)] are the premise variables, and q is the fractional-order; u(t) is control input.By adopting single point fuzzification, product inference, and weighted average defuzzification, the final output of the fractional-order T-S fuzzy model is inferred as follows:(13)dqxdtq=∑i=1rhiztAixt+∑i=1rhiztBiut,where(14)hizt=∏j=1nMijzjt∑i=1r∏j=1nMijzjt≥0,∑i=1rhizt=1with Mijzj(t) being the grade of membership of zj(t) in Mij. hi(z(t)) can be regarded as the normalized weights of the IF-THEN rules.


## 4.2. T-S Fuzzy Controller Design Based on LMI

RuleRi is as follows: IF z1(t) is Mi1 and ⋯ and zn(t) is Min(15)THENut=Kixti=1,2,…,r.The inferred output of the PDC controller is expressed in the following form:(16)ut=∑j=1rhjztKjxt,where Kj represents the feedback gain.By substituting (16) into (13), one has(17)dqxdtq=∑i=1rhiztAixt+∑i=1rhiztBi∑j=1rhjztKjxt.In order to simplify (17), it is modified as follows: (18)∑i=1rhiAi=h1A1+h2A2+⋯+hrAr.Considering∑i=1rhi(z(t))=1 in (17), (18) can be written as(19)∑i=1rhiAi=h1h1+h2+⋯+hrA1+⋯+hrh1+h2+⋯+hrAr=h12+h1h2+⋯+h1hrA1+⋯+hrh1+hrh2+⋯+hr2Ar=∑i=1r∑j=1rhihjAi.For formula (17), ∑i=1rhiBi∑j=1rhjKj can be written as(20)∑i=1rhiBi∑j=1rhjKj=h1B1+h2B2+⋯+hrBr·h1K1+h2K2+⋯+hrKr=h1B1h1K1+⋯+hrKr+⋯+hrBrh1K1+h2K2+⋯+hrKr=h12B1K1+h22B2K2+⋯+hi2BiKi+⋯+hr2BrKr+h1h2K2+h3K3+⋯+hrKrB1+h2h3K3+⋯+hrKrB2+⋯+hihi+1Ki+1+⋯+hrKrBi+⋯+hr-1hrKrBr-1+h2h1K1B2+h3h2K2+h1K1·B3+⋯+hjhj-1Kj-1+hj-2Kj-2+⋯+h1K1·Bj+⋯+hrhr-1Kr-1+hr-2Kr-2+⋯+h1K1·Br=∑i=1rhi2BiKj+∑i<jrhihjKjBi+∑i>jrhihjKjBi=∑i=1r∑j=1rhihjBiKj.Submitting (19) and (20) into (17), one gets(21)dqxdtq=∑i=1r∑j=1rhizthjztAi+BiKjxt.Before utilizing the T-S fuzzy model into fractional-order systems (5) and (8), make the following assumptions.Assumption 5.
According to the boundedness of the system, the scope of the system state can be defined as(22)Ω≡xt∈R∣xtn≤δ,xit∈-di,di,di>0i=1,2.Assumption 6.
The feedback control can be applied to every state of the system. The PDC approach together with the T-S fuzzy model can be adopted to control systems (5) and (8).Based on the above assumptions, we can use T-S fuzzy model to describe system (5). Suppose that x1(t)∈[-d1,d1], x2(t)∈[-d2,d2], where d1=15 and d2=10.Thus, the T-S fuzzy model can be presented as follows:R1: IF x1(t) is M1(x1(t)), THEN dqx/dtq=A1x(t)+B1u(t),R2: IF x1(t) is M2(x1(t)), THEN dqx/dtq=A2x(t)+B2u(t),R3: IF x2(t) is M3(x2(t)), THEN dqx/dtq=A3x(t)+B3u(t),R4: IF x2(t) is M4(x2(t)), THEN dqx/dtq=A4x(t)+B4u(t),wherex(t)=[x1(t),x2(t),x3(t)]T, u(t)=[u1(t),u2(t),u3(t)]T,(23)A1=2811000.2-1015015-4,A2=2811000.2-10-150-15-4,A3=28110-100.2-100100-4,A4=28110100.2-100-100-4,B1=B2=B3=B4=I3×3Iis  a  unit  matrix.The membership functions of fuzzy sets are taken as follows:(24)M1x1t=121+x1td1,M2x1t=121-x1td1,M3x2t=121+x2td2,M4x2t=121-x2td2.Thus, the fractional-order system (5) based on T-S fuzzy model can be expressed as(25)dqxdtq=∑i=14hiztAixt+Biut.According to the PDC method, the controller is given as follows:R1: IF x1(t) is M1(x1(t)), THEN u(t)=K1x(t),R2: IF x1(t) is M2(x1(t)), THEN u(t)=K2x(t),R3: IF x2(t) is M3(x2(t)), THEN u(t)=K3x(t),R4: IF x2(t) is M4(x2(t)), THEN u(t)=K4x(t).The overall controller is obtained as(26)ut=∑j=14hjztKjxt.Submitting (26) into (25), one can get the fractional-order system:(27)dqxdtq=∑i=14∑j=14hizthjztAi+BiKjxt.Similarly, for wing doubled system (8), d1=30 and d2=20. One has(28)A1=2811000-1030030-4,A2=2811000-10-300-30-4,A3=28110-200-100200-4,A4=28110200-100-200-4.To stabilize the fractional-order system (27), the following lemma and theorem are given.Lemma 7 (see [27]).
For the arbitrary state variablex=[x1,x2,…,xn]T, if there exists a symmetric positive definite real matrix P satisfying J=xTP(dqx/dtq)≤0 (xTP(dqx/dtq) is called J function), then the fractional-order system (27) is globally asymptotically stable. Condition J=xTPdqx/dtq≤0 can be equally written as J0=xTPdqx/dtq+dqx/dtqTPx≤0.Theorem 8.
If there exist positive definite matricesP, E, and F as well as a positive constant η, suppose that λi(i=1,2,3,4) and ξi(i=1,2,3,4) are the eigenvalues of matrices PEETP and FTF, respectively. If λi(i=1,2,3,4)>0, ξi(i=1,2,3,4)>0, and we select the controller gain matrix Ki(i=1,2,3,4) which satisfies the following inequalities, the fractional-order system (27) is globally asymptotically stable:(29)GiiTP+PGii+ηPEETP+η-1FTF<0i,j=1,2,3,4,(30)GijTP+PGij+ηPEETP+η-1FTF<0i<j<4,where Gii=Ai+BiKi and Gij=(Ai+BiKj)+(Aj+BjKi)/2.Proof.
By Lemma7, select J0=xTPdqx/dtq+dqx/dtqTPx as J function of system (27):(31)J0=xTPdqxdtq+dqxdtqTPx=xTP∑i=14∑j=14hihjAi+BiKjx+∑i=14∑j=14hihjAi+BiKjTPx=∑i=14∑j=14hihjxTAi+BiKjTP+PAi+BiKjx=∑i=14hi2xTAi+BiKiTP+PAi+BiKix+∑i<j4hihjxTAi+BiKjTP+PAi+BiKjx+∑i>j4hihjxTAi+BiKjTP+PAi+BiKjx.
According to∑i=14hi2+2∑i<j4hihj=1 in (14), so there is(32)∑i<j4hihjxTAi+BiKjTP+PAi+BiKjx+∑i>j4hihjxTAi+BiKjTP+PAi+BiKjx=∑i<j4hihjxTAi+BiKj+Aj+BjKiTP+PAi+BiKj+Aj+BjKix=2∑i<j4hihjxTAi+BiKj+Aj+BjKi2TP+PAi+BiKj+Aj+BjKi2x.
SelectingGii=Ai+BiKi, Gij=(Ai+BiKj)+(Aj+BjKi)/2 and under conditions (29) and (30), one gets (33)J0=∑i=14hi2xTGiiTP+PGiix+2∑i<j4hihjxTGijTP+PGij<x-∑i=14hi2xTηPEETP+η-1FTFx+2∑i<j4hihjxTηPEETP+η-1FTFx.
According to Theorem4, the eigenvalues of the matrices PEETP and FTF are λi(i=1,2,3,4)>0 and ξi(i=1,2,3,4)>0, respectively, so the matrices PEETP and FTF both are positive definite matrix; that is, PEETP+FTF>0; since η>0, one gets(34)ηPEETP+η-1FTF>0.
Considering (34) for (33), one can get(35)J0<-∑i=14hi2xTηPEETP+η-1FTFx+2∑i<j4hihjxTηPEETP+η-1FTFx<0.
Therefore, when inequalities (29) and (30) hold, one has(36)J0=xTPdqxdtq+dqxdtqTPx≤0.
This shows that the fractional system (27) is globally asymptotically stable. This completes the proof.According to Schur complement theorem, inequalities (29) and (30) can be transformed into the problem of solving linear matrix inequalities (LMIs).From (29), we can obtain (37)AiTP+KiTBiP+PAi+PBiKi+ηPEETP+η-1FTF<0.MultiplyingP-1 both in the left and in the right in (37), one gets (38)P-1AiT+P-1KiTBi+AiP-1+BiKiP-1+ηEET+η-1P-1FTFP-1<0.In (38), selecting Q=P-1 and Mi=KiP-1, we can obtain(39)QAiT+AiQ+MiTBiT+BiMi+ηEET+η-1QFTFQ<0.The controller gain isKi=MiQ-1.From (30), we can obtain(40)12AiTP+KjTBiTP+AjTP+KiTBjTP+PAi+PBiKj+PAj+PBjKi+ηPEETP+η-1FTF<0.MultiplyingP-1 both in the left and in the right in (40), we obtain(41)12P-1AiT+P-1KjTBiT+P-1AjT+P-1KiTBjT+AiP-1+BiKjP-1+AjP-1+BjKiP-1+ηEET+η-1P-1FTFP-1<0.In (41), selecting Q=P-1, Mi=KiP-1, and Mj=KjP-1, we can obtain(42)12QAiT+AiQ+MiTBjT+BjMi+QAjT+AjQ+MjTBiT+BiMj+ηEET+η-1QFTFQ<0.The controller gain isKi=MiQ-1.By Schur complement theorem, (39) can be written as(43)QAiT+MiTBiT+AiQ+BiMi+ηEETQFTFQ-η-1I<0.Similarly, (42) can be written as(44)12QAiT+AiTQ+MiTBjT+BjMi+QAjT+AjTQ+MjTBiT+BiMj+ηEETQFTFQ-η-1I<0Q>0.For a given positive numberη>0, we can use Matlab LMI toolbox to solve the linear matrix inequalities (43) and (44) to obtain the positive definite matrix P and the controller gain Ki(i=1,2,3,4).


## 4.3. Simulation Results

For fractional-order system (5), take d1=15, d2=10, and η=10000. According to Theorem 4, select positive definite matrices:(45)E=100000000;010000000;001000000;F=100;010;001;100;010;001;100;010;001.Through Matlab 7.0 LMI toolbox, the positive definite matrixP and the feedback controller gain Ki(i=1,2,3,4) can be obtained:(46)P=diag⁡0.00066,0.00066,0.00066,K1=-9.2609-0.05830.1021-0.14173.2846-22.0523-0.1021-7.9477-2.7154,K2=-9.2609-0.12170.0497-0.07833.284611.4239-0.049718.5761-2.7154,K3=-9.2609-0.1278-0.2628-0.07223.2846-0.00170.26280.0017-2.7154,K4=-9.2609-0.1234-0.1296-0.07663.2846-0.00680.12960.0068-2.7154.Substituting the value ofKi(i=1,2,3,4) into (26), the desired feedback controller u(t) can be got. Figure 8 shows the state trajectories of system (5) without controller; the system states are unstable. Figure 9 illustrates the state trajectories of closed-loop system (5) with controller (26), and the states of system (5) globally converge to the equilibrium point which implies the effectiveness of the designed controller.Figure 8
State trajectories of system (5) without controller.
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State trajectories of closed-loop system (5) with controller (26).
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y
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z
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tSimilarly, for wing doubled system (8), select d1=30, d2=20, and η=10000. According to Theorem 4, take positive definite matrices:(47)E=100000000;010000000;001000000;F=100;010;001;100;010;001;100;010;001.Through Matlab LMI toolbox, the positive definite matrixP and the feedback controller gain Ki(i=1,2,3,4) can be obtained:(48)P=diag⁡0.00066,0.00066,0.00066K1=-9.26090.00560.0005-0.00563.2846-30.4410-0.0005-29.5590-2.7154,K2=-9.2609-0.00910.00020.00913.284630.1005-0.000229.8995-2.7154,K3=-9.2609-0.0000-0.78190.00003.2846-0.00000.78190.0000-2.7154,K4=-9.2609-0.00000.02550.00003.2846-0.0000-0.02550.0000-2.7154.Substituting the value ofKi(i=1,2,3,4) into (26), the controller can be got. Figure 10 shows the state trajectories of system (8) without controller; the system states are unstable. Figure 11 illustrates the state trajectories of system (8) with controller (26). We can see that the controller can be applied to the more complicated wing doubled chaotic system (8) with short time and small overshoot, which shows the speed ability and robustness of the control method.Figure 10
State trajectories of system (8) without controller.
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State trajectories of closed-loop system (8) with controller (26).
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## 5. Conclusions

A new three-dimensional fractional-order chaos was proposed in this paper. When the linear termx in the second equation of the system was replaced by its absolute value, the new system can make the wing of the original system doubled, and the circuit diagram was implemented. Based on the T-S fuzzy model and fractional-order stability theory, a more practical stability condition for fuzzy control of the proposed wing doubled fractional-order chaos was given as a set of LMI and the strict mathematical derivation was presented. Numerical simulation results were consistent with theoretical results.More and better methods for the control of fractional-order chaotic systems should be studied. The authors will continue the study of application condition of integer-order chaotic theory into fractional-order chaotic systems. There will be focus on the stability of fractional-order chaotic systems.



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*Source: 604257-2015-07-05.xml*