Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VI (2011), No. 3 (September), pp. 418-427
Uncertain Fractional Order Chaotic Systems Tracking Design via
Adaptive Hybrid Fuzzy Sliding Mode Control
T.C. Lin, C.H. Kuo, V.E. Balas
Tsung-Chih Lin
Feng-Chia University, 40724, Taichung, Taiwan
E-mail: tclin@fcu.edu.tw
Chia-Hao Kuo
Ph.D Program in Electrical and Communications Engineering
Feng-Chia University, Taichung, Taiwan
E-mail: peterqo022@hotmail.com
Valentina E. Balas
Aurel Vlaicu University of Arad, Romania
B-dul Revolutiei 77, 310130 Arad, Romania
E-mail: balas@drbalas.ro
Abstract: In this paper, in order to achieve tracking performance of un-
certain fractional order chaotic systems an adaptive hybrid fuzzy controller is
proposed. During the design procedure, a hybrid learning algorithm combining
sliding mode control and Lyapunov stability criterion is adopted to tune the
free parameters on line by output feedback control law and adaptive law. A
weighting factor, which can be adjusted by the trade-off between plant knowl-
edge and control knowledge, is adopted to sum together the control efforts
from indirect adaptive fuzzy controller and direct adaptive fuzzy controller.
To confirm effectiveness of the proposed control scheme, the fractional order
chaotic response system is fully illustrated to track the trajectory generated
from the fractional order chaotic drive system. The numerical results show
that tracking error and control effort can be made smaller and the proposed
hybrid intelligent control structure is more flexible during the design process.
Keywords: Fractional order chaotic systems; fuzzy logic control, adaptive
hybrid control.
1 Introduction in domain
Due mainly to its demonstrated applications in numerous seemingly diverse and widespread
fields of science and engineering, fractional calculus has gained considerable popularity and im-
portance during past three decades [1]- [2]. In control system, due to the fact that the theoretical
aspects are well established, fractional order controllers are successfully used to enhance the per-
formance of the feedback control loop. It is observed that the description of some systems is
more accurate when the fractional derivative is used. Nowadays, many fractional-order differ-
ential systems behave chaotically, such as the fractional-order Chua’s system [3], the fractional-
order Duffing system [4], the fractional-order system, the fractional-order Chen’s system [5], the
fractional-order cellular neural network [6], the fractional-order neural network [7]. The tracking
problem of fractional order chaotic systems is first investigated by Deng and Li [21] who carried
out tracking in case of the two fractional Lü systems. Afterwards, they studied chaos tracking
of the Chen system with a fractional order in a different manner [22]-[24].
Based on the universal approximation theorem, [9]- [20] (fuzzy logic controllers are general
enough to perform any nonlinear control actions) there is rapidly growing interest in systematic
Copyright c⃝ 2006-2011 by CCC Publications
Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy
Sliding Mode Control 419
design methodologies for a class of nonlinear systems using fuzzy adaptive control schemes. Like
the conventional adaptive control, the adaptive fuzzy control is classified into direct and indirect
fuzzy adaptive control categories [9], [17]- [19]. A direct adaptive fuzzy controller uses fuzzy logic
systems as controller in which linguistic fuzzy control rules can be directly incorporated into the
controller. On the other hand, an indirect adaptive fuzzy controller uses fuzzy descriptions to
model the plant in which fuzzy IF-THEN rules describing the plant can be directly incorporated
into the indirect fuzzy controller. Moreover, a hybrid adaptive fuzzy controller can be constructed
using a weighting factor to sum together the control efforts from indirect adaptive fuzzy controller
and direct adaptive fuzzy controller.
Although the concept of sliding mode control (SMC) and the theory of fractional order
system are well known, their integration, fractional sliding mode control, is an interesting filed
of research dwelt on this paper with some applications [8]. The motivation of this paper stands
on two driving forces: One, most systems in the reality display behavior characterized best in
time domain of fractional operators, the other, the uncertainties on the process dynamics can
appropriately be alleviated by utilizing SMC technique.
In this paper, by combining the approximate mathematical model, linguistic model descrip-
tion and linguistic control rules into a single adaptive fuzzy controller, an adaptive hybrid fuzzy
controller is proposed to achieve prescribed tracking performance of fractional order chaotic sys-
tems. A new adaptive hybrid fuzzy SMC algorithm incorporated Lyapunov stability criterion is
proposed so that not only the stability of adaptive fuzzy control system is guaranteed but also
the influence of the approximation error and external disturbance on the tracking error can be
attenuated to an arbitrarily prescribed level.
This paper is organized as follows: In section2, an introduction to fractional derivative and its
relation to the approximation solution will be addressed. Section 3 generally proposes adaptive
hybrid fuzzy SMC of uncertain fractional order systems in presence of uncertainty and its stability
analysis. In Section 4, application of the proposed method on fractional order expression chaotic
system is investigated. Finally, the simulation results and conclusion will be presented in Section
5.
2 Basic definition and preliminaries for fractional order systems
The concept of fractional calculus is popularly believed to have steamed from a question raised
in the year 1695 by Marquis de L’Hoptial to Gottfried Wilhelm Leibniz. It is a generalization
of integration and differentiation to non-integer order fundamental operator, denoted by aD
q
t ,
where a and t are the limits of the operator. This operator is a notation for taking both the
fractional integral and functional derivative in a single expression defined as [1]
aD
q
t =
dq
dtq
, q > 0
1 q = 0∫ a
t
(dτ)−q, q < 0
(1)
There are some basic definitions for the general fractional and the commonly used defini-
tions are Grunwald-Letnikov and Riemann-Liouville [1]. The Grunwald-Letnikov definition is
expressed as
aD
q
t f(t) = lim
h→0
[t−ah ]∑
j=0
(−1)j
(
a
b
)
f(t − jh) (2)
420 T.C. Lin, C.H. Kuo, V.E. Balas
where [.] is the integer part. The simplest and easiest definition is Riemann-Liouville defini-
tion given as
aD
q
t f(t) =
1
Γ(n − q)
dn
dtn
∫ t
0
f(τ)
(t − τ)q−n+1
dτ (3)
where n is the first integer which is not less q, i.e., n − 1 < q < n, and Γ is the Gamma
function.
The numerical simulation of a fractional differential equation is not simple as that of an
ordinary differential equation. In this paper, the algorithm which is an improved version of
Adams-Bashforth-Moulton algorithm to find an approximation for fractional order systems based
on predictor-correctors is given. Consider the following differential equation
aD
q
t y(t) = r(y(t), t), 0 ≤ t ≤ T and y
(k)(0) = y(k)o ,k = 0,1,2, ...,m − 1 (4)
where
aD
q
t y(t) =
1
Γ(m−q)
∫ t
0
f(m)(τ)
(t−τ)q−m+1 dτ, m − 1 < q < m
dm
dtm
y(t), q = m
(5)
and m is the first integer larger the q. The solution of the equation (4) is equivalent to
Volterra integral equation [1] described as
y(t) =
[q]−1∑
k=0
y
(k)
0
tk
k!
+
1
Γ(q)
∫ t
0
(t − λ)q−1r(y(λ),λ)dλ (6)
Let h=T/N, tn = nh, n=0,1,2,· · · N . Then (6) can be discretized as follows.
yh(tn+1) =
[q]−1∑
k=0
y
(k)
0
tkn+1
k!
+
hq
Γ(q + 2)
r(y
p
h(tn+1), tn+1) +
hq
Γ(q + 2)
n∑
j=0
aj,n+1r(yh(tj), tj) (7)
where predict value yph(tn+1) is determined by
y
p
h(tn+1) =
[q]−1∑
k=0
y
(k)
0
tkn+1
k!
+
hq
Γ(q)
n∑
j=0
bj,n+1r(yh(tj), tj) (8)
and
aj,n+1 =
nq+1 − (n − q)(n + 1)q, j = 0
(n − j + 2)q+1 + (n − j)q+1 − 2(n − j + 1)q+1 1 ≤ j ≤ n
1 j = n + 1
(9)
bj,n+1 =
hq
q
((n + 1 − j)q − (n − j)q) (10)
The approximation error is given as
max
j=0,1,2,···N
|y(tj) − yh(tj)| = O(hp) (11)
where p=min(2,1+q). Therefore, the numerical solution of a fraction order chaotic system
discussed in this paper can be obtained by applying the above mentioned algorithm.
Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy
Sliding Mode Control 421
3 Adaptive hybrid fuzzy sliding mode control of uncertain frac-
tional order chaotic systems
In this section, we study adaptive hybrid fuzzy tracking control of uncertain fractional order
chaotic systems, i.e., to force output trajectory which is obtained by the algorithm mentioned in
section 2 of the response system to track output trajectory of the drive system.
Consider a fractional order chaotic dynamic system
x(nq) = f(x,t) + g(x,t)u + d(t), y = x1 (12)
where x = [x1,x2, ...,xn]T = [x,x(q),x(2q), ...,x((n−1)q)]T is the state vector, f(x,t) and g(x,t)
are unknown but bounded nonlinear functions which express system dynamics, d(t) is the exter-
nal bounded disturbance, |d(t)| ≤ D, and u(t) is the control input. The control objective is to
force the system output y to follow a bounded reference signal yd which is the output trajectory
of a drive system, under the constraint that all signals involved must be bounded. To begin with,
the reference signal vector y
d
and the tracking error vector e will be defined as
y
d
=
[
yd,y
(q)
d , ...,y
((n−1)q)
d
]T
∈ Rn,
e = y
d
− x =
[
e,e(q),e(2q), ...,e((n−1)q)
]T
∈ Rn,e(iq) = y(iq)d − x
(iq) = y
(iq)
d − y
(iq)
In general, in the space of the error state a sliding surface is defined by
s(x,t) = −(ke) = −
(
k1e + k2e
(q) + ... + kn−1e
(n−2)q + e(n−1)q
)
(13)
where k = [k1,k2, ...,kn−1,1] in which the ki’s are all real and are chosen such that h(r) =∑n
i=1 kir
(i−1)q,kn = 1 is a Hurwitz polynomial where r is a Laplace operator. The tracking
problem will be considered as the state error vector e remaining on the sliding surface s(x,t) = 0
for all t ≥ 0. The sliding mode control process can be classified into two phases, the approaching
phase with s(x,t) ̸= 0 and the sliding phase with s(x,t)= 0 for initial error e(0) = 0. In order
to guarantee that the trajectory of the state error vector e will translate from the approaching
phase to the sliding phase, the sufficient condition
s(x,t)ṡ(x,t) ≤ −η > 0 (14)
must be satisfied. Two type of control law must be derived separately for those two phases
described above. In the sliding phase, it implies s(x,t) = 0 and s(q)(x,t) = 0. In order to force
the system dynamics to stay on the sliding surface, the equivalent control u can be derived as
follows:
If f(x,t) and g(x,t) are known and free of external disturbance, i.e., d(t)=0, taking the
derivative of the sliding surface with respective to time, we get
s(q) = −
(
n−1∑
i=1
cie
(iq) + e(nq)
)
= −
(
n−1∑
i=1
kie
(iq) + y
(nq)−y(nq)
d
)
= −
(
n−1∑
i=1
kie
(iq)−f(x,t)−g(x,t)ueq
)
− y(n)d = −
n−1∑
i=1
kie
(i) + f(x) + b(x)u(t) − x(n)d = 0 (15)
Therefore, the equivalent control can be obtained as
422 T.C. Lin, C.H. Kuo, V.E. Balas
u =
1
g(x,t)
(
n−1∑
i=1
kie
(iq) − f(x,t) + y(nq)d
)
(16)
On the contrary, in the approaching phase, s(x,t) ̸= 0, an approaching-type control uap must
be added in order satisfy the sufficient condition (4) and the complete sliding mode control will
be expressed as
u = u − uap, uap = ψhsgn(s) (17)
where ψh ≥ η > 0.
To obtain the sliding mode control (17), the system functions f(x,t),g(x,t) and switching
parameter ψh must be known in advance. However, f(x,t) and g(x,t) are unknown and external
disturbance, d(t) ̸= 0, the ideal control effort (16) cannot be implemented. We replace f(x,t),
g(x,t) and uap by the fuzzy logic system f(x|θf), g(y|θg) and h(s|θh) in specified form as [9], [17]-
[19], i.e.,
f(x|θf) = ξ
T (x)θf,g(x|θg) = ξ
T (x)θg, h(s|θh) = ∅
T (s)θh (18)
let |h(s|θh)| = D + ψh + ωmax when s(x,t) is outside the boundary layer. Here the fuzzy
basis functions ξ(x) and ∅(s) depend on the fuzzy membership functions and is supposed to be
fixed, while θf,θg and θh are adjusted by adaptive laws based on Lyapunov stability criterion.
Therefore, depending on plant knowledge and control knowledge, a hybrid adaptive fuzzy con-
troller can be constructed by incorporating both fuzzy description and fuzzy control rules using
a weighting factor α to combine the indirect adaptive fuzzy controller and the direct adaptive
fuzzy controller. Based on the trade-off between plant knowledge and control knowledge, the
weighting factor α ∈ [1,1] can be adjusted. Therefore, the total control effort can be expressed
as
uc = αui + (1 − α)ud (19)
where the direct adaptive fuzzy controller ud and the indirect adaptive fuzzy controller ui
are given as follows:
ud(x) = uD(x|θD) −
h(s|θh)
g(x,t)
and ui(x) =
1
g(x|θg)
[
n−1∑
i=1
kie
(iq) + y
(nq)
d − f(x, |θf) − h(s|θh)
]
(20)
where uD(x|θ) is obtained by fuzzy logic system specified as
uD(x|θD) = ξ
T (x)θD (21)
The optimal parameter estimations θ∗f,θ
∗
g, θ
∗
h and θ
∗
D are defined as
θ∗f = arg minθf ∈Ωf
[
sup
x∈Ωx
|f(x|θf) − f(x,t)|
]
, θ∗g = arg minθg∈Ωg
[
sup
x∈Ωx
|g(x|θg) − g(x,t)|
]
θ∗D = arg minθD∈ΩD
[
sup
x∈Ωx
|uD(x|θg) − u|
]
, θ∗h = arg minθh∈Ωh
[
sup
x∈Ωx
|h(s|θh) − uap|
]
Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy
Sliding Mode Control 423
where Ωf,Ωg,ΩD and Ωx are constraint sets of suitable bounds on θf,θg,θ
∗
h,θD and x respec-
tively and they are defined as Ωf = {θf|θf| ≤ Mf}, Ωg = {θg|θg| ≤ Mg}, ΩD = {θD|θD| ≤ MD},
Ωh = {θh|θh| ≤ Mh} and Ωx = {x||x| ≤ Mx}, where Mf,Mg,MD,Mh and are positive constants.
By using (20), (21), sliding surface equation (15) can be rewritten as
s(q) = ω + α
[
f(x|θ∗f) − f(x|θf)
]
+ α
[
g(x|θ∗g) − g(x|θg)
]
ui − (1 − α)h(s|θ∗h)
−αh(s|θh) − (1 − α)g(x) [uD(x|θ
∗
D) − uD(x|θD)] + αh(s|θ
∗
h) − αh(s|θ
∗
h) (22)
+(1 − α)h(s|θ∗h) − (1 − α)h(s|θ
∗
h) + d(t)
where the minimum approximation errors is defined as
ω = α
[
f(x) − f(x|θ∗f)
]
+ α
[
g(x) − g(x|θ∗g)
]
ui + (1 − α) [uD(x|θ∗) − uD] (23)
If θ̃f = θf − θ
∗
f , θ̃g = θg − θ
∗
g and, θ̃D = θD − θ
∗
D, we have
s(q) = −(1 − α)h(s|θ∗h) + ω − αθ̃
T
h ∅(s) − αθ̃
T
f ξ(x) − αθ̃
T
g ξ(x)ui
+(1 − α)g(x)θ̃
T
Dξ(x) − αh(s|θ
∗
h) − (1 − α)θ̃
T
h ∅ + d(t) (24)
Following the proceeding consideration, the following theorem can be obtained.
Theorem: Consider the fractional order SISO nonlinear chaotic system (12) with control
input (19), if the fuzzy-based adaptive laws are chosen as
θ
(q)
f = r1sξ(x), θ
(q)
g = r2sξ(x)ui, θ
(q)
D = r3s∅(s) and θ
(q)
h = −r4sg(x)ξ(x) (25)
where ri > 0, i = 1 ∼ 4. Then, the overall adaptive scheme guarantees the global stability
of the resulting closed-loop system in the sense that all signals involved are uniformly bounded
and the tracking error will converge to zero asymptotically.
Proof: In order to analyze the closed-loop stability, the Lyapunov function candidate is chosen
as
V =
1
2
s2 +
α
2r1
θ̃
T
f θ̃f +
α
2r2
θ̃
T
g θ̃g +
α
2r4
θ̃
T
h θ̃h +
(1 − α)
2r3
θ̃
T
Dθ̃D
(1 − α)
2r4
θ̃
T
h θ̃h (26)
Taking the derivative of the (26) with respect to time, we get
V (q) = ss(q) +
α
r1
θ̃
T
f θ̃
(q)
f +
α
r2
θ̃
T
g θ̃
(q)
g +
α
r4
θ̃
T
h θ̃
(q)
h +
(1 − α)
r3
θ̃
T
θ̃
q
+
(1 − α)
r4
θ̃
T
h θ̃
(q)
h
= −(1 − α)sh(s|θ∗h) + sω − αsθ̃
T
h ∅(s) − αsθ̃
T
h ξ(x) − αsθ̃
T
g ξ(x)ui + (1 − α)sg(x)θ̃
T
ξ(x)
−ash(s|θ∗h) − (1 − α)sθ̃
T
h ∅ + sd(t) +
α
r1
θ̃
T
f θ̃
(q)
f +
α
r2
θ̃
T
g θ̃
(q)
g +
α
r4
θ̃
T
h θ̃
(q)
h +
(1 − α)
r3
θ̃
T
θ̃
q
+
(1 − α)
r4
θ̃
T
h θ̃
(q)
h
≤
α
r1
θ̃
T
f
(
θ̃
(q)
f − r1sξ(x)
)
+
α
r2
θ̃
T
g
(
θ̃
(q)
g − r2sξ(x)ui
)
+
1
r4
θ̃
T
h
(
θ̃
(q)
h − r4s∅(s)
)
− αs(D + η)sgn(s)
+
1 − α
r3
θ̃
T
(
θ̃
(q)
+ r3sg(x)ξ(x)
)
− (1 − α)s(D + ψh)sgn(s) + sd(t) + sω (27)
From the robust compensator ua and the fuzzy-based adaptive laws are given (25), after
simple manipulation, we have
V (q) ≤ sω − sψhsgn(s) = sω − |s|ψh (28)
Using the corollary of Barbalat’s Lemma [16]-[19], we have limt→∞ |s(x,t)| = 0. Therefore,
limt→∞ |e(t)| = 0. The proof is completed.
424 T.C. Lin, C.H. Kuo, V.E. Balas
4 Simulation example
In this section, we will apply our adaptive hybrid fuzzy sliding mode controller to force
the fractional order chaotic gyro response system to track the trajectory of the fractional order
chaotic gyro drive system.
Example: The fractional order chaotic gyro drive and response systems are given as follows:
Drive System:
{
y
(q)
1 = y2
y
(q)
2 = −100
(
y1
4
)
+
y31
12
− 0.5y2 − 0.05y32 + sin(y1) + 35.5 sin(2t)y1 −
x31
6
+ d(t)
Response System:
{
x
(q)
1 = x2
x
(q)
2 = −100
(
x1
4
)
+
x31
12
− 0.7x2 − 0.08x32 + sin(x1) + 33 sin(2t)x1 −
x31
6
+ ∆f(x1,x2) + d(t) + uc(t)
where structured uncertainty ∆f(x1,x2) = −0.1sin(x1) and external disturbance d(t) =
0.2cos(πt). The main objective is to control the trajectories of the response system to track the
reference trajectories obtained from the drive system. The initial conditions of drive and response
systems are chosen as [y1(0),y2(0)]T = [1,−1]T and [x1(0),x2(0)]T = [1.6,0.8]T , respectively. For
q=0.95, α = 0.7 and all design constants are specified as k1 = k2 = 1,r1 = 150,r2 = 20,r3 =
1,r4 = 1 and step size h = 0.01. The phase portrait of the drive and response systems for free
of control input is given in Figure 1. It is obvious that the tracking performance is bad without
control effort supplied to response system.
Figure 1: Phase portrait of chaotic drive and response systems
The membership functions for xi i=1,2 are selected as follows:
µF i1
(xi) = exp
[
−0.5
(
xi−4
2
)2]
, µF i2(xi) = exp
[
−0.5
(
xi−2.7
2
)2]
, µF i3(xi) = exp
[
−0.5
(
xi−1.2
2
)2]
,
µF i4
(xi) = exp
[
−0.5
(
xi
2
)2], µF i5(xi) = exp[−0.5(xi+1.22 )2], µF i6(xi) = exp[−0.5(xi+2.72 )2],
µF i7
(xi) = exp
[
−0.5
(
xi+4
2
)2]
,
From the adaptive laws (25)-(28), the control effort of the response system can be obtained
as
uc = αui + (1 − α)ud (29)
Figure 2 shows the trajectories of the states xi,yi and x2,y2, respectively. Control effort
trajectory is given in Figure 3 and phase portrait, tracking performance, of the drive and response
Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy
Sliding Mode Control 425
Figure 2: The trajectories of the states xi,yi and x2,y2
Figure 3: Trajectory of the control effort
Figure 4: Phase portrait, tracking perfor-
mance, of the drive and response systems
systems is shown in Figure 4. Trajectory of the sliding surface is given is Figure 5. The maximum
value of V (q)(t) is -1.711e-4 which is always negative defined and consequently is stable.
In order to show the robustness of the proposed adaptive hybrid fuzzy sliding mode control,
the control effort is activated at 5 second. The phase portrait, tracking performance, of the drive
and response systems is given in Figure 6. Figure 7 shows the trajectories of the states xi,yi
and x2,y2 respectively. We can see that a fast tracking of drive and response is achieved as the
control effort is activated. Control effort trajectory is given in Figure 8. Trajectory of the sliding
surface is given is Figure 9. The maximum value of V (q)(t) is -1.732e-4 which is always negative
defined and consequently is stable.
Figure 5: Trajectory of the sliding surface
Figure 6: Phase portrait, tracking perfor-
mance, of the drive and response systems
426 T.C. Lin, C.H. Kuo, V.E. Balas
Figure 7: The trajectories of the states xi,yi and x2,y2
Figure 8: Trajectory of the control effort Figure 9: Trajectory of the sliding surface
5 Conclusions
A novel adaptive hybrid fuzzy sliding mode controller is proposed to achieve tracking perfor-
mance of fractional order chaotic systems in this paper. It is a flexible design methodology by the
trade-off between plant knowledge and control knowledge using a weighting factor ? adopted to
sum together the control effort from indirect adaptive fuzzy controller and direct adaptive fuzzy
controller. Based on the Lyapunov synthesis approach, free parameters of the adaptive fuzzy
controller can be tuned on line by output feedback control law and adaptive laws. The simulation
example, the output trajectory of the fractional order chaotic response system to tracking the
trajectory of the fractional order chaotic drive system, is given to demonstrate the effectiveness
of the proposed methodology.
Bibliography
[1] M. S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-order systems via active
sliding mode controller, Physica A, Vol. 387, pp. 57-70, 2008.
[2] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differ-
ential equations, North-Holland Math. Studies 204, Elsevier, Amsterdam, 2006.
[3] I. Petras, A note on the fractional-order Chua’s system, Chaos, Solitons & Fractals, 38 (1),
pp. 140-14,7 2008.
[4] X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing’s oscil-
lators, Chaos, Solitons & Fractals 26, pp. 1125-1133, 2005.
[5] J.G. Lu and G. Chen, A note on the fractional-order Chen system, Chaos, Solitons & Fractals
27, pp. 685-688, 2006.
Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy
Sliding Mode Control 427
[6] P. Arena and R. Caponetto, Bifurcation and chaos in non-integer order cellular neural net-
works, Int J Bifurcat Chaos 8 (7), pp. 1527-1539, 1998.
[7] Petras I., A note on the fractional-order cellular neural networks. In: Proceedings of the
IEEE world congress on computational intelligence, international joint conference on neural
networks, Vancouver, Canada; pp.16-21, 2006.
[8] S. H. Hosseninnia, R. Ghaderi, A. Ranjbar N., M. Mahmoudian, S. Momani, Sliding mode
synchronization of an uncertain fractional order chaotic system, Journal Computers & Math-
ematics with Applications, Vol. 59, pp. 1637-1643, 2010.
[9] L.A. Zadeh, Fuzzy logic, neural networks and soft computing. Commun. ACM 37 3, pp.
77-84, 1994.
[10] X. Z. Zhang; Y.N. Wang; X. F. Yuan, H? Robust T-S Fuzzy Design for Uncertain Non-
linear Systems with State Delays Based on Sliding Mode Control, International Journal of
Computers Communications & Control, 5(4):592-602, 2010.
[11] Balas, M.M., Balas, V.E., World Knowledge for Control Applications by Fuzzy-Interpolative
Systems, International Journal of Computers Communications and Control, Vol. 3, Supple-
ment: Suppl. S, pp. 28-32, 2008.
[12] L. X. Wang, and J. M. Mendel, Fuzzy basis function, universal approximation, and orthog-
onal least square learning, IEEE Trans. Neural Networks, vol. 3, no. 5, pp. 807-814, 1992.
[13] L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Engle-
wood Cliffs, NJ: Prentice-Hall, 1994.
[14] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional
order, Elec. Trans. Numer. Anal. 5, pp. 1-6, 1997.
[15] J. L. Castro, "Fuzzy logical controllers are universal approximators," IEEE Trans. Syst.,
Man, Cybern., 25, pp. 629-635, 1995.
[16] S. S. Ge, T. H. Lee, C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators,
World Scientific Publishing Co., Singapore, 1998.
[17] C. H. Wang, T. C. Lin, T. T.. Lee and H. L. Liu, "Adaptive Hybrid Intelligent Control for
Unknown Nonlinear Dynamical Systems", IEEE Transaction on Systems, Man, and Cyber-
netics Part B, 32 (5), October, pp. 583-597, 2002.
[18] C. H. Wang, H. L. Liu and T. C. Lin, "Direct Adaptive Fuzzy-Neural Control with Observer
and Supervisory Control for Unknown Nonlinear Systems", IEEE Transaction on Fuzzy Sys-
tems, 10(1) , pp. 39-49, 2002.
[19] T. C. Lin, C. H. Wang and H. L. Liu, "Observer-based Indirect Adaptive Fuzzy-Neural
Tracking Control for Nonlinear SISO Systems Using VSS and ", Fuzzy Sets and Systems,
143, pp.211-232, 2004.
[20] L. A. Zadeh, Knowledge Representation in Fuzzy Logic, IEEE Trans. Knowledge and Data
Engineering, 1 (1), pp. 89-100, 1989.