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International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems
Vol. 20, No. 3 (2012) 339−353
© World Scientific Publishing Company
DOI: 10.1142/S0218488512500171
339
ADAPTIVE FUZZY-BASED TRACKING CONTROL FOR A CLASS OF
STRICT-FEEDBACK SISO NONLINEAR TIME-DELAY SYSTEMS
WITHOUT BACKSTEPPING
HASAN A. YOUSEF*, ‡, MOHAMED HAMDY†, § and MUHAMMAD SHAFIQ*
*Department of Electrical and Computer Engineering, College of Engineering,
Sultan Qaboos University, Sultanate of Oman, Muscat 123, PO Box 33 †Department of Industrial Electronics and Control Engineering,
Faculty of Electronic Engineering, Menofia University, Menof-32952, Egypt ‡hyousef@squ.edu.om
§mhamdy72@hotmail.com
Received 22 August 2010
Revised 12 April 2012
In this paper, an adaptive fuzzy tracking control is presented for a class of SISO nonlinear strict
feedback systems with unknown time delays. The proposed algorithm does not use the backstepping scheme rather it converts the strict-feedback time-delayed system to the normal form. The Mamdani-
type fuzzy system is employed to approximate on-line the lumped uncertain system nonlinearity.
The developed controller guarantees uniform ultimate boundedness of all signals in the closed-loop system. The designed control law is independent of the time delays and has a simple form with only
one adaptive parameter vector needed to be updated on-line. As a result, the proposed control
algorithm is considerably simpler than the previous ones based on backstepping. The Lyapunov stability of the fuzzy system parameters and the filtered tracking error is employed to guarantee
semiglobal uniform boundedness for the closed loop system. Simulation results are presented to
verify the effectiveness of the proposed approach.
Keywords: Adaptive fuzzy control; strict-feedback nonlinear systems; time-delay; stability.
1. Introduction
Systems with delays frequently appear in engineering applications. Typical examples of
time-delay systems are electrical networks, chemical processes, rolling mill systems,
teleportation systems, underwater vehicles and so on. The existence of time delays may
degrade the control performance and make the stabilization problem more difficult. So
far, the stability analysis and robust control for these dynamic time-delay systems have
attracted a number of researchers over the past years; see Ref. 1, and 2 for example and
the references therein. Stability analysis and synthesis of time-delay systems are
important issues addressed by many authors and for which surveys can be found in many
articles.3-7
Recently, by combining Lyapunov–Krasovskii functional and backstepping tech-
nique, an adaptive neural tracking control scheme was proposed in Ref. 8 for a class of
strict-feedback nonlinear time-delay systems with unknown virtual control coefficients.
H. A. Yousef, M. Hamdy & M. Shafiq 340
The suggested controller guarantees the uniform ultimate boundedness of the adaptive
closed-loop system, while the output tracking is achieved. Further improvements were
given in Refs. 9 and 10. Robust stabilization methods were presented via the
approximation capability of neural network.11-13
In Refs. 10, 14 and 15, observer-based fuzzy adaptive control for strict-feedback
nonlinear systems and fuzzy adaptive observer backstepping control for MIMO nonlinear
systems are developed. In Ref. 16, the adaptive H∞ control was addressed via
backstepping and neural networks technique. The observer-based adaptive neural
controller was designed for a special class of nonlinear time-delay systems.17 However,
these adaptive neural control methods8-17
require a large number of neural weights to be
adapted online simultaneously. This makes the learning time unacceptably large. In view
of this disadvantage, several adaptive fuzzy control schemes have been developed in
Refs. 18−25 for nonlinear delay-free systems. The advantage of these control schemes is
that the proposed controllers require much less parameters to be updated online. The T-S
fuzzy model combines the flexibility of fuzzy logic theory and rigorous mathematical
analysis tools into a unified framework. Linear models are employed for the consequent
parts which makes it simple to use conventional linear system theory for the analysis. The
advantage of T-S fuzzy modeling is its greater accuracy, dimensionality, and also the
simplification of the structure of nonlinear systems. It provides an effective
representation of complex nonlinear systems. Stability analysis and control synthesis of
T-S fuzzy delayed systems were proposed in Refs. 26−33. An approximation-based
adaptive control has also been addressed for nonlinear systems with time-delay. An
attempt for using Mamdani-type fuzzy logic systems to design an adaptive fuzzy tracking
controller by using the backstepping technique and Lyapunov–Krasovskii functional
appeared in Ref. 29. Most of the previous adaptive fuzzy control algorithms for strict-
feedback nonlinear systems were based on the backstepping scheme, which makes the
control law and stability analysis very complicated. The main advantage of the results
proposed in Ref. 29 is that the developed fuzzy controllers contain less adaptation laws.
The adaptive fuzzy controllers based on backstepping design method have some
drawbacks. First, the determination of virtual control terms and their time derivatives
requires tedious and complex analysis. As pointed out in Ref. 33, for the relatively simple
application to dc motor control, the regression matrix in the adaptive backstepping
control almost covers an entire page.34 The complexity exhibits an exponential increase
as the order of the controlled system, i.e., n, grows. However, this complexity must be
avoided for practical implementation. Recently, in Ref. 35, an attempt for using an
adaptive neural control for SISO nonlinear systems without backstepping is proposed.
Based on the above observation, a novel systematic design procedure is developed in
this paper to synthesize a stable adaptive fuzzy controller for a class of strict feedback
nonlinear time-delay systems without backstepping. Fuzzy logic systems are employed to
approximate the lumped uncertain system nonlinearity, and then, the adaptive law of
adjustable parameters is obtained. The universal approximation property of fuzzy systems
developed in Ref. 36 is utilized to generate estimation for the lumped uncertain
parameters appeared in the control law. The main advantage of the proposed method is
that for an nth order strict-feedback nonlinear system, only one parameter vector is
needed to be estimated on-line and therefore, the computation burden is significantly
reduced and the algorithm is easily realized in practice.
Adaptive Fuzzy-Based Tracking Control 341
The paper is organized as follows. The system under investigation and its
transformation to normal form is outlined in Sec. 2. The adaptive fuzzy control design is
presented in Sec. 3. The main result is presented in Sec. 4. Simulation results are provided
in Sec. 5, with conclusions given in Sec. 6.
2. Problem Formulation
Consider the SISO nonlinear time-delay dynamic systems in the following strict feedback
form:
, , , ...., ,
) ) ,
i i i i i i i i i
n n n n n n n n
x g x x f x h x t i n
x g x u t f x h x t
y x
τ
τ+= + + − = −
= + + −
=
ɺ
ɺ
(1)
where [ , ,....., ]Ti ix x x x= , [ , ,....., ]T nnx x x x R= ∈ ∈∈∈∈R
n is the system state vector, u ∈R
u ∈∈∈∈R and y ∈R denote system control input and output, respectively. The functions
,i i ig f h are unknown smooth functions and τi is an unknown time delay of the
state variables, i = 1,… , n.
It is assumed that (1) has measurable state vector x. With
regard to controllability, the following assumption must be made.
Assumption 1. For 1 ≤ i ≤ n, the signs of ( )i ig x are known. Without loss of generality, it
is assumed that ( )i ig x > .
The control objectives are to design an adaptive fuzzy tracking controller for the
system (1) such that the system output y tracks a desired reference signal yd and to assure
the boundedness of all signals in the closed-loop system.
The first step in the design is to convert the original system (1) into the normal form
with respect to the newly defined state variables [ , ,....., ]Tnz z z= .
Let yz∆=1 and z z g x f d
∆= = + +ɺ where ( ( ))i i i id h x t τ= − . The time derivative of z2
is derived as
( )
( )( ) ( )
( ) ( ) ( ( ))j
f g dz x x x g x x t
x x x
f g dx f g x d g f g g x g d x t
x x x
a x b x x l x t∆
τ
τ
τ
∂ ∂ ∂= + + + −
∂ ∂ ∂
∂ ∂ ∂= + + + + + + + −
∂ ∂ ∂
= + + −
ɺ ɺ ɺ ɺɺ
ɺ
(2)
where ( ) (( ))( )f g
a x x f g x g fx x
∂ ∂= + + +
∂ ∂, ( )b x g g=
and
( ( )) (( )) ( )j
f g dl x t x d g d x t
x x xτ τ
∂ ∂ ∂− = + + + −
∂ ∂ ∂ɺ
H. A. Yousef, M. Hamdy & M. Shafiq 342
Again, let ( ) ( ) ( )z a x b x x l x= + + and time derivative is induced by
( )
( )( ) ( ) ( )
( ) ( ) ( ( ))
j j j j
j j jj j j
j j j j j jj j jj j
j
a b lz x x x b x x t
x x x
a b lx f g x d b f g x d x t
x x x
a x b x x l x t∆
τ
τ
τ
= = =
+= =
∂ ∂ ∂= + + + −
∂ ∂ ∂
∂ ∂ ∂= + + + + + + + −
∂ ∂ ∂
= + + −
∑ ∑ ∑
∑ ∑
ɺ ɺ ɺ ɺɺ
ɺ
(3)
where ( ) ( )( )j j jj jj
a ba x x f g x b f
x x+
=
∂ ∂= + + +
∂ ∂∑
, ( )b x b g g g g= = and
( ) ( ) j jj j jj j
a b ll x x d x b d
x x x= =
∂ ∂ ∂= + + +
∂ ∂ ∂∑ ∑ ɺ
In general, by induction, if we define a f= , b g= and l d= , the following is
satisfied for i = 2,..,n:
( ) ( ) ( ( ))
( ) ( ) ( ( ))
i i i i i i i i i
i i i i i i i i i
z a x b x x l x t
z a x b x x l x t
∆ τ
τ− − − − − − −
+ −
= + + −
= + + −ɺ
(4)
where
( ) ( )( )
( )
( ( )) (( ) )
ii i
i i i j j j i ij jj
i
i i i i j
j
ii i i
i i i i j j i ij j jj
a ba x x f g x b f
x x
b x b g g
a b ll x t x d x b d
x x xτ
−− −
+ −=
−=
−− − −
−=
∂ ∂= + + +
∂ ∂
= =
∂ ∂ ∂− = + + +
∂ ∂ ∂
∑
∏
∑ ɺ
(5)
As a result, the strict-feedback time-delayed system (1) can be represented as the
following normal form with respect to the new state variables iz :
( ) ( ) ( )
i i
n d
z z
z a x b x u l x
y z
+=
= + +
=
ɺ
ɺ
(6)
where ( ) ( ), ( ) ( ), ( ) ( ) and [ ( ) ( )]n n d n d d n na x a x b x b x l x l x x x t x tτ τ= = = = − −⋯
with
( )di i ix x t τ= − . It should be noted that the functions ( ), ( ), and ( )da x b x l x are unknown
functions of x and time delays.
Assumption 2. The unknown function ( )dl x satisfies the following
( ) ( ) ( ( ))n n
d i i di i i i i
i i
l x x x tρ φ ρ φ τ= =
= = −∑ ∑
where ( )i dixφ are known positive function with ( ( ( ))n
i i
i
x t Cφφ=
≤∑ and iρ are unknown
constants.
Adaptive Fuzzy-Based Tracking Control 343
From Assumption 1, it is noted that a constant b > exists such that ( ) , nb x b x R≥ ∀ ∈ .
This assumption poses a controllable condition on system (6).
3. Brief Description of Fuzzy Approximator
The fuzzy system considered in this paper has a center-average defuzzifier, product
inference and singleton fuzzifier. This type of fuzzy logic system is given by:
( ( ))
( )
( ( ))
i
i
nM
iFi
nM
iFi
q x
q x
x
µ
µ
= =
= =
=∑ ∏
∑ ∏
ℓ
ℓ
ℓ
ℓ
ℓ
(7)
where M is the number of IF-THEN rules in the fuzzy rule base. The If-Then rules take
the following form for , ,...,M=ℓ :
: If is and is and and Then isn nR x F x F x is F q Gℓ ℓ ℓ ℓ ℓ⋯
where ℓ
iF and ℓG are the fuzzy sets with membership functions iF
µ ℓand
Gµ ℓ
respectively, and q is the linguistic variables which can be considered as output of the
fuzzy logic system. The parameter q ℓ is the point at which ( )G
qµ ℓ
ℓ
achieves its
maximum value and we assume that ( )G
qµ =ℓ
ℓ . Equation (7) can be rewritten as
( ) ( )Tq x xψ ζ= (8)
where [ , ,...., ]TMψ ψ ψ ψ= is a parameter vector, and ( ) [ ( ), ( ),...., ( )]M Tx x x xζ ζ ζ ζ= is a
regressive vector with the regressor ( )xζ ℓ defined as fuzzy basis function (FBF) of the
form
( )
( )
( ( ))
i
i
n
iFi
nM
iFi
x
x
x
µ
ζ
µ
=
= =
=∏
∑ ∏
ℓ
ℓ
ℓ
ℓ
Property. For any given real continuous function ( )inxβ on a compact set inx
Ω and an
arbitrary small numberreε > , there exists a fuzzy system ˆ( )inxβ in the form of (8) and an
optimal parameter vector *ψ such that *ˆsup ( ) ( )in xin
in in rex
x xΩ
β β ψ ε∈
− < .
4. Adaptive Fuzzy Controller Design
In this section an adaptive fuzzy tracking controller for system (1), or equivalently (6), is
designed. First the ideal controller is determined assuming the functions ( ),a x ( )b x and
( )dl x are known. Then the adaptive control law is designed using the approximation
capability of fuzzy logic system.
H. A. Yousef, M. Hamdy & M. Shafiq 344
4.1. Ideal controller
It is assumed that the desired output trajectory and its derivatives ( )[ , ,..., ]n Td d d dY y y y −= ɺ
are measurable and bounded and ( )ndy
− denotes the (n-1)th derivative of dy , with respect
to time. Define the error vector e as
[ , ,...., ]Td ne z Y e e e= − =
(9)
and the filtered tracking error se as
( ) [ ] n T
s
de e e
dtλ Λ−= + =
(10)
where [ , ( ) ,..., ( ) ]n n Tn nΛ λ λ λ− −= − − , and λ > 0 is a positive constant to be specified by
the designer.
Assumption 3. The desired trajectory vector dy is continuous, available, and
satisfies ( )[ , ]nT T nd d dy Y y R += ∈ .
From (6) and (10), one obtains the following differential equation for the tracking
error
( ) ( ) ( )n
s i i di
i
e a x b x u x vρ φ=
= + + +∑ɺ (11)
where ( )[ ] nTdv e yΛ= − .
Using triangular inequality, we have
( ( ) ( ) ) ( )n
Ts s s s i di
i
e e e a x b x u v e xρ ρ φ=
≤ + + + +∑ɺ (12)
with Tnρ ρ ρ= …
Though the functions ( )i dixφ are known, it could not be used for the construction of
control law as dix is not available due to the unknown time delays. To tackle this
problem, let us introduce the following Lyapunov-Krasovskii functional
( ( ))
i
tn
U i i
i t
V x d
τ
φ λ λ= −
= ∑ ∫
The time derivative of UV is
( ( ( )) ( ))n
U i i i di
i
V x t xφ φ=
= −∑ɺ (13)
in which the non-positive term can be used to compensate for the terms with unknown
time delay.
Lemma 1. Considering that (6) satisfies Assumption 1, the filtered tracking error se is
uniformly ultimately bounded if the ideal control law is designed as
* *
*
( )
( )( )
( )
s ad
ad
u ke u w
a w vu w
b w
= − −
+=
(14)
where k > 0 is a design parameter and [ ]Tw x v= .
Adaptive Fuzzy-Based Tracking Control 345
Proof. Consider the Lyaunov function
se s UV e V= +
(15)
Using (13), the time derivative of (15) along (11) is
( ( ( )) ( ))s
n
e s s i i i di
i
V e e x t xφ φ=
= + −∑ɺ ɺ .
(16)
Substitution of (12) and (14) in (16) yields
( )( ( ) ( )( )) ( ) ( ( ( )) ( ))
( )s
n nT
e s s s i di i i i di
i i
a x vV e a x b x ke v e x x t x
b xρ ρ φ φ φ
= =
+≤ + − − + + + + −∑ ∑ɺ
which reduces to
s
Te s sV kbe e Cφρ ρ≤ − + +ɺ
( )
se s s
CV e e
φδ≤ − − − +ɺ (17)
where ( )Tkbδ ρ ρ= − > . Therefore, for all s
Ce
φ≥ , Eq. (15) becomes ( )se sV eδ≤ − −ɺ
which concludes that se is uniformly ultimately bounded.
4.2. Adaptive control law
Using the fuzzy system of (8), the fuzzy-logic based control input is designed as follows:
ˆ ( )
ˆˆ ( ) ( )
s ad
Tad
u ke u w
u w wψ ζ
= − −
= (18)
where ψ is the estimation of the parameter vector and can be updated as follows:
ˆ ˆ ˆ( ( ) ( ) )s sw e eψ γ ζ σ ψ ψ= −ɺ
(19)
In (19) γ is positive learning rate and the switching function ˆ( )σ ψ chosen as (see
Ref. 38)
ˆ, if ˆ( )
, otherwise
Cζψ
ψψ ε
εσ ψ
>
=
(20)
with ψε being a design constant and Cζζ ≤ .Then ˆ ψψ ε≤ .
It should be noted that the switching function (20) is employed to retain the learned
information of FBF and to prevent the loss of information if ψε is chosen sufficiently
large such that *ψψ ε< while guaranteeing the boundedness of ψ .
Remark. One parameter vector ψ that contains M adjustable parameters is required to
synthesis the control law (18) where M is the number of fuzzy sets. It is worth-noting to
mention that some adaptive fuzzy controllers for non-delayed systems which require less
number of adjustable parameters have appeared recently.38-40
H. A. Yousef, M. Hamdy & M. Shafiq 346
5. Main Result
In this section, the boundedness of the closed-loop signals is proved using the Lyapunov
function approach.
Theorem 1. For the adaptive system comprising (1) under Assumption 1, the controller
(18) and update law (19), the filtered error se is uniformly ultimately bounded.
Proof. Let us consider the following Lyapunov function candidate:
ˆ ˆs
TeV V ψ ψ
γ= + (21)
The time derivative of (21) is given by
ˆ ˆ ˆ( ( ) ( ) )
s
Te in s sV V x e eψ ζ σ ψ ψ= + −ɺ ɺ
(22)
where
( ( ) ( ) ))
s
Te s sV e a x b x u v e Cφρ ρ≤ + + + +ɺ (23)
Using (18) in (23) to get
ˆ( ( ) ( )( ) ))
s
T Te s s sV e a x b x ke v e Cφψ ζ ρ ρ≤ + − − + + +ɺ
Now equation (22) becomes
* * ˆ ˆ ˆ( ) ( ) ( ) ( ( ) )T T T T
s s s ad s s sV b x ke be b x e u e e e Cφψ ζ ψ ζ ψ ζ σ ψ ψ ρ ρ≤ − + + − + − + +ɺ ɶ
(24)
where TTT ψψψ ˆ~ * −= . Equivalently, (24) can be put in the following form
ˆ ˆ ˆ( ) ( ( ) )T
s s s re sV bke e C b e C C e Cφ ψ ζ ζρ ρ ε ψ σ ψ ψ≤ − + + + + − −ɶɺ (25)
where T C Cψ ζψ ζ <ɶ
ɶ . Now if ˆ ψψ ε> , then according to (20), the term
ˆ ˆ( ( ) )Cζσ ψ ψ − will be a positive constant Cσ and (25) becomes
ˆ( ) ( )s s s re s
CV e e b e C C e C C
φψ ζ σψδ ε≤ − − − + + + −ɶ
ɺ (26)
Therefore, for all ˆ( ) )s re
s
Ce b C C C C
e
φψ ζ σψε≥ + + −ɶ
, Eq. (26) will be ( ) sV eδ≤ − −ɺ
and hence the filtered error se is uniformly ultimately bounded.
6. Simulation Results
In this section, we demonstrate the effectiveness of the proposed adaptive fuzzy control
algorithm using the following illustrative examples.
Example 1. In this example, we consider a second order time–delay system of the form:9
.( ) ( )
( cos( )) . ( ) sin ( )
xx x x x e x t
x x x u x x x t x t
y x
τ
τ τ
−= + + + −
= + + + − −
=
ɺ
ɺ (27)
Adaptive Fuzzy-Based Tracking Control 347
In this example, secτ τ= = and the desired reference signal is chosen as
. (sin sin( . )dy t t= + .6
The above system is simulated under the proposed adaptive fuzzy control given by
(18) and (19) where [ , , ]Tinx x x v= and dv e yλ= − ɺɺ . In this case, seven Gaussian
membership functions with centers evenly spaced between [−1.5, 1.5] for each variable , , ,i inW x i∈ = are chosen as follows:
4
)5.1(5.0
4
)1(5.0
4
)5.0(5.0
4
)(5.0
4
)5.0(5.0
4
)1(5.0
4
)5.1(5.0
2
7
2
6
2
5
2
4
2
3
2
2
2
1
,,
,,,,
−−−−−−
−+−+−+−
===
====
i
i
i
i
i
i
i
i
i
i
i
i
i
i
W
F
W
F
W
F
W
F
W
F
W
F
W
F
eee
eeee
µµµ
µµµµ
(28)
The FBFs used to generate the approximation of )(ˆinad xu take the following form:
( ) ( )
( ) , ,i i
T
i iF Fi i
i
W W
WD D
µ µζ = =
=
∏ ∏1 7…
where ( )i
iFi
D Wµ= =
=∑∏ ℓ
ℓ
.
The controller parameters are selected as ; ; and k λ γ= = = and zero initial
conditions for the model dynamics as well as for the vector ˆ.ψ
Simulation results are
depicted in Figs. 1 and 2.
0 2 4 6 8 10 12 14 16 18 20-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ou
tpu
t y a
nd
re
fere
nce
sig
nal yd
sec
y
yd
Fig. 1. Tracking performance.
H. A. Yousef, M. Hamdy & M. Shafiq 348
0 2 4 6 8 10 12 14 16 18 20-2
0
2
4
6
8
10
12
co
ntr
ol in
pu
t
sec
Fig. 2. Control signal.
Example 2. In this example, a two-stage chemical reactor with delayed recycle streams is
simulated with the proposed controller. The reactor model is given by:41
. . sin( ) ( ),
. ( ) . sin( ) ( ),
,
x x x t x t
x u x x t t x t
y x
τ
τ τ
= − + + −
= − + − + −
=
ɺ
ɺ (29)
where the delay times are secτ τ= = . The desired reference signal is assumed
asdy = . For the vector [ , , ]Tinx x x v= seven Gaussian membership functions of the form
(28) are used. Simulation results are shown in Figs. 3 and 4 for ; . ; and k λ γ= = =
and initial conditions of [ ] ˆ( ) , and (0)=0x ψ= − .
0 5 10 15-2
-1.5
-1
-0.5
0
0.5
ou
tpu
t y
sec
Fig. 3. Tracking performance.
Adaptive Fuzzy-Based Tracking Control 349
0 5 10 15-15
-10
-5
0
5
10
15
co
ntr
ol in
pu
t
sec
Fig. 4. Control signal.
Example 3. Consider a controlled Brusselator model with both external disturbances and
existence of time delays of the form:33
.
.
( ) ( , ) ( ( )),
( cos( )) ( , ) ( ( )),
,
x C D x x x d t x h x t
x Dx x x x u d t x h x t
y x
τ
τ
= − + + + + −
= − + + + + −
=
(30)
where x1 and x2 denote the concentrations of the reaction intermediates, C, D > 0 are
parameters which describe the supply of “reservoir” chemicals, u is the control input. The
terms ( , )d t x and ( , )d t x represent external disturbances which come from the modeling
errors and other types of unknown nonlinearities in the practical chemical reactions. The
functions ( ( ))h x t τ− and ( ( ))h x t τ− are unknown time-delayed functions.
In simulation, we choose the same parameters, disturbances and delay functions as
given in Ref. 33, namely,
, , ( ) , ( ) . sin( ), . cos( . ), . ( ) sin ( )C D h x x h x x x d x t d x x t= = = = = = +
and secτ τ= = . The reference trajectory is selected as sin( . ) . sin( . ).dy t t= + +
Seven Gaussian membership functions of the form (28) are used for the
vector [ , , ] .Tinx x x v= Simulation results are shown in Figs. 5 and 6 for k = 150;
λ = 2.5; and γ = 4000 and initial conditions of x(0) = [2.5, 1] and ˆ(0) = 0y .
Simulation results of the above three examples demonstrate the effectiveness of the
proposed adaptive fuzzy controller in achieving good tracking performance and smooth
control signal.
H. A. Yousef, M. Hamdy & M. Shafiq 350
0 2 4 6 8 10 12 14 16 18 201.5
2
2.5
3
3.5
4
4.5
ou
tpu
t y a
nd
re
fere
nce
sig
na
l yd
sec
y
yd
Fig. 5. Tracking performance.
0 2 4 6 8 10 12 14 16 18 20-400
-300
-200
-100
0
100
200
con
tro
l in
pu
t
sec Fig. 6. Control signal for Example 3.
7. Conclusion
Adaptive fuzzy control has been developed for a class of SISO strict-feedback nonlinear
systems with unknown time-delays. The key aspect of the proposed method is that the
state-feedback control problem of the strict-feedback system can be viewed as the output-
feedback control problem of the affine system in the normal form; this results in avoiding
backstepping in the controller design. It is also demonstrated that control law and
stability analysis is considerably simpler than the previous backstepping based
algorithms. In the developed control algorithm, fuzzy logic systems of Mamdani-type are
utilized to approximate on-line the unknown nonlinear functions. The proposed control
Adaptive Fuzzy-Based Tracking Control 351
scheme guarantees the semi-global boundedness of all the signals in the closed-loop
system and good tracking performance. Moreover, the suggested adaptive fuzzy
controller is simple. This makes the design scheme easier to be implemented in practical
applications. Extensive simulation examples are provided to validate the effectiveness of
the proposed controller.
Acknowledgements
The authors would like to thank the Editor and the anonymous reviewers for their
inspiring encouragement and constructive comments, which have contributed much to the
improvement of the clarity and presentation of this paper. The first and third authors
acknowledge the support of Sultan Qaboos University.
References
1. K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems (Birkhäuser, Berlin,
Germany, 2003).
2. S. L. Niculescu, Delay Effects on Stability: A Robust Control Approach (Springer-Verlag, New
York, 2001).
3. C.-Y. Chen, J.-W. Lin, W.-I. Lee, and C.-W. Chen, Fuzzy control for an oceanic structure: A
case study in time-delay TLP system, J. Vibration and Control 16 (2010) 147–160.
4. Y.-J. Liu, R. Wang, and C. L. P. Chen, Robust adaptive fuzzy controller design for a class of
uncertain nonlinear time-delay systems, Int. J. Uncertainty, Fuzziness and Knowledge-Based
Systems 19 (2011) 329–360.
5. K. Yeh, C.-Y. Chen and C.-W. Chen, Robustness design of time-delay fuzzy systems using
fuzzy Lyapunov method, Applied Math. Computation (2008) 568–577.
6. M. Kchaou, A. Toumi and M. Souissid, Delay-dependent H∞ resilient output fuzzy control for
nonlinear discrete-time systems with time-delay, Int. J. Uncertainty, Fuzziness and Knowledge-
Based Systems 19 (2011) 229–250.
7. C. W. Chen, W. L. Chiang and F. H. Hsiao, Stability analysis of T-S fuzzy models for
nonlinear multiple time-delay interconnected systems, Math. Computer in Simulation 66 (2004)
523–537.
8. S. S. Ge, F. Hong, and T. Lee, Adaptive neural network control of nonlinear systems with
unknown time delays, IEEE Trans. Automatic Control 48 (2003) 2004–2010.
9. C.-W. Chen, Stability analysis and robustness design of nonlinear systems: An NN-based
approach, Applied Soft Computing (2011) 2735–2742.
10. Y.-J. Liu, C. L. P. Chen, G.-X. Wen and S.-C. Tong, Adaptive neural output feedback
controller design with reduced-order observer for a class of uncertain nonlinear SISO systems,
IEEE Trans. Neural Networks 22 (2011) 1328–1334.
11. D. W. C. Ho, J. M. Li and Y. G. Niu, Adaptive neural control for a class of nonlinearly
parametric time-delay systems, IEEE Trans. Neural Networks 16 (2005) 625–635.
12. C.-W. Chen, Modeling and control for nonlinear structural systems via a NN-based approach,
Expert Systems with Application (2009) 4765–4772.
13. Y.-J. Liu, C. L. P. Chen, G.-X. Wen and S.-C. Tong, Adaptive neural output feedback tracking
control for a class of uncertain discrete-time nonlinear systems, IEEE Trans. Neural Networks
22 (2011)1162–1167.
14. S.-C. Tong and Y.-M. Li, Observer-based fuzzy adaptive control for strict-feedback nonlinear
systems, Fuzzy Sets and Systems 160 (2009) 1749–1764.
H. A. Yousef, M. Hamdy & M. Shafiq 352
15. S.-C. Tong , C.-Y. Li and Y.-M. Li, Fuzzy adaptive observer backstepping control for MIMO
nonlinear systems, Fuzzy Sets and Systems 160 (2009) 2755–2775.
16. Y. Niu, J. Lam and X. Wang, Adaptive H∞ control using backstepping and neural networks,
J. Dynam. Syst., Meas. Control 127 (2005) 313–326.
17. C. C. Hua, X. P. Guan, and P. Shi, Robust output feedback tracking control for time-delay
nonlinear systems using neural network, IEEE Trans. Neural Networks 18 (2007) 495–505.
18. H. Yousef, M. Hamdy and E. El-Madbouly, Robust Adaptive fuzzy semi-decentralized control
for a class of large-scale nonlinear systems using input-output linearization concept, Int. J.
Robust and Nonlinear Control 20 (2010) 27–40.
19. H. Yousef, M. Hamdy, E. El-Madbouly, and D. Eteim, Adaptive fuzzy decentralized control
for interconnected MIMO nonlinear subsystems, Automatica 45 (2009) 456–462.
20. C.-W. Chen, K. Yeh and K. F.-R. Liu, Adaptive fuzzy sliding mode control for seismically
excited bridges with lead rubber bearing isolation, Int. J. Uncertainty, Fuzziness and
Knowledge-Based Systems 17 (2009) 705–727.
21. J.-Y. Chen, Design of adaptive fuzzy sliding mode control for nonlinear systems, Int. J.
Uncertainty, Fuzziness and Knowledge-Based Systems 7 (1999) 463–474.
22. C.-W. Chen, M. H. L. Wang and J.-W. Lin, Managing target the cash balance in construction
firms using a fuzzy regression approach, Int. J. Uncertainty, Fuzziness and Knowledge-Based
Systems 17 (2009) 667–684.
23. C.-W. Chen, Modeling and fuzzy PDC control and its application to an oscillatory TLP
structure, Math. Problems in Engineering − an Open Access Journal (2010). 24. Y.-J. Liu, W. Wang, S.-C. Tong and Y.-S. Liu, Robust adaptive tracking control for nonlinear
systems based on bounds of fuzzy approximation parameters, IEEE Trans. Systems, Man, and
Cybern. A 40 (2010) 170–184.
25. C.-W. Chen, Stability conditions of fuzzy systems and its application to structural and
mechanical systems, Advances in Engineering Software (2006) 624–629.
26. B. Chen, X. P. Liu, and S. C. Tong, New delay-dependent stabilization conditions of T-S fuzzy
systems with constant delay, Fuzzy Sets and Systems 158 (2007) 2209–2224.
27. F.-H. Hsiao, W.-L. Chiang, C.-W. Chen, S.-D. Xu and S.-L. Wu, Application and robustness
design of fuzzy controller for resonant and chaotic systems with external disturbance, Int. J.
Uncertainty, Fuzziness and Knowledge-Based Systems 13 (2005) 281–296.
28. C.-W. Chen, The stability of an oceanic structure with T-S fuzzy models, Math. and Computers
in Simulation (2009) 402–426.
29. C. Lin, Q. G. Wang, T. H. Lee, Y. He, and B. Chen, Observer-based H∞ fuzzy control design
for T-S fuzzy systems with state delays, Automatica 44 (2008) 868–874.
30. C. S. Ting, An observer-based approach to controlling time-delay chaotic systems via Takagi-
Sugeno fuzzy model, Information Sciences 177 (2007) 4314–4328.
31. C.-W. Chen, C.-L. Lin, C.-H. Tsai, C.-Y. Chen and K. Yeh, A novel delay-dependent criteria
for time-delay T-S fuzzy systems using fuzzy Lyapunov method, Int. J. Artificial Intelligence
Tools 16 (2007) 545–552.
32. B. Zhang, J. Lam, S. Xu, and Z. Shu, Robust stabilization of uncertain T-S fuzzy time-delay
systems with exponential estimates, Fuzzy Set and Systems 160 (2009) 1720–1737.
33. M. Wang, B. Chen, X. P. Liu, and P. Shi, Adaptive fuzzy tracking control for a class of
perturbed strict-feedback nonlinear time-delay systems, Fuzzy Sets and Systems 159 (2008)
949–967.
34. D. M. Dawson, J. J. Carroll, and M. Schneider, Integrator backstepping control of a brush dc
motor turning arobotic load, IEEE Trans. Control Systems Technology 2 (1994) 233–244.
35. J.-H. Park, S.-H. Kim and C.-J. Moon, adaptive neural control for strict-feedback nonlinear
systems without backstepping, IEEE Trans. Neural Network 20 (2009) 1204–1209.
Adaptive Fuzzy-Based Tracking Control 353
36. L,-X. Wang, Fuzzy Systems and Control: Design and Stability Analysis (Prentice-Hall,
Englewood Cliffs, NJ, 1994).
37. P. A. Ioannou and J. Sun, Robust Adaptive Control (Prentice-Hall, Englewood Clkiffs, NJ,
1996).
38. Y.-J. Liu, S.-C. Tong, and W. Wang, Adaptive fuzzy output tracking control for a class of
uncertain nonlinear systems, Fuzzy Sets and Systems 160 (2009) 2727–2754.
39. Y.-J. Liu, S.-C. Tong and T.-S. Li, Observer-based adaptive fuzzy tracking control for a class
of uncertain nonlinear MIMO systems, Fuzzy Sets and Systems 164 (2009) 25–44.
40. Y.-J. Liu and W. Wang, Adaptive fuzzy control for a class of uncertain non-affine nonlinear
Systems, Information Sciences 177 (2007) 3901–3917.
41. Y. Li, S. Qiang, X. Zhuang, and O. Kaynak, Robust and adaptive backstepping control for
nonlinear systems using RBF neural networks, IEEE Trans. Neural Network 15 (2004) 693–
701.