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Title: Indirect adaptive fuzzy control scheme based on observer for nonlinear systems_ A novel SPR-filter approach
Author: A. Boulkroune

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Neurocomputing 135 (2014) 378–387

Contents lists available at ScienceDirect

Neurocomputing
journal homepage: www.elsevier.com/locate/neucom

Indirect adaptive fuzzy control scheme based on observer
for nonlinear systems: A novel SPR-filter approach
A. Boulkroune a,n, N. Bounar a,b, M. M0 Saad c, M. Farza c
a

LAJ, Department of Automatic Control, University of Jijel, BP. 98, Ouled-Aissa, 18000 Jijel, Algeria
LCP, Department of Automatic Control, Ecole Nationale Polytechnique (ENP), 10, Av. Hassen Badi, BP182, Algiers, Algeria
c
GREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN, 6 Bd Maréchal Juin, 14050 Caen Cedex, France
b

art ic l e i nf o

a b s t r a c t

Article history:
Received 3 February 2013
Received in revised form
31 August 2013
Accepted 24 December 2013
Communicated by: H. Zhang
Available online 10 January 2014

In this paper, a novel fuzzy indirect adaptive controller based on observer for uncertain nonlinear
perturbed systems is proposed. A tracking-error observer is introduced to resolve the problem of the
unavailability of state variables. Adaptive fuzzy systems are employed to approximate the unknown
smooth nonlinear functions. The control system is augmented by a low-pass filter designed to meet a SPR
condition of a transfer function of the observation error dynamics. The SPR condition is used in the
Lyapunov stability analysis to construct the adaptation laws using only available measurements (i.e. the
output observation error and the output tracking error). The main contributions of this paper lie in the
following: (a) The SPR-filter approach used here avoids the filtering of the fuzzy basis functions. (b)
Unlike in the previous works, the stability analysis is rigorously proven by using a SPR-based Lypunov
approach. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of
the proposed controller.
& 2014 Elsevier B.V. All rights reserved.

Keywords:
Fuzzy system
Indirect adaptive control
Observer
SPR condition
Nonlinear system

1. Introduction
Fuzzy systems (FS) as well as neural networks (NN) have been
successfully applied to many control problems because they do not
need an accurate mathematical model of the system under control.
It is also known that these intelligent systems (i.e. FS and NN) can
approximate uniformly any nonlinear continuous function over a
compact set [1–3]. Recently, much attention has been focused on
adaptive neural/fuzzy control of nonlinear dynamical systems
[4–17]. With suitable coordinate transformation, a class of nonlinear dynamic systems can be transformed into the so-called
normal form [4–7], or into the cascade triangular forms, which are
classified as strict-feedback and pure-feedback forms [8–17]. For
the first class [4–7], fuzzy or neural adaptive controls were
developed in which uncertainties are well dealt with. The stability
of the closed-loop system has been only investigated using a
standard Lyapunov approach. For the second class [8–17], fuzzy or
neural adaptive backstepping controls were developed in which
both uncertainties and non-matching conditions are well dealt
with. The stability analysis of these control schemes has been
investigated using the backstepping concept and Lyapunov
approach.
n

Corresponding author.
E-mail addresses: boulkroune2002@yahoo.fr (A. Boulkroune),
bounar18@yahoo.fr (N. Bounar), msaad@greyc.ensicaen.fr (M. M0 Saad),
mfarza@greyc.ensicaen.fr (M. Farza).
0925-2312/$ - see front matter & 2014 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.neucom.2013.12.011

This last class of the nonlinear systems is beyond the scope of
this paper. We are interested here to the nonlinear systems having
the normal form or which can be transformed into this form.
Although the adaptive neural/fuzzy control systems designed in
the literature [4–7] for this class of systems are simple and can
give satisfactory results, they have certain limitations and shortcomings, namely: (1) the state vector of these systems is assumed
to be available for measurement. But this measurement requirement is more an exception than a rule in the engineering practice.
That is why observer-based controllers (or output-feedback controllers) are most used in practice. (2) These control systems can
only be applied to the nonlinear systems which can satisfy a harsh
requirement named matching condition, i.e. the nonlinearities
only appear in the same equation as the control in the state space
representation.
Based on state or tracking-error observer, direct and indirect
adaptive fuzzy control schemes have been developed in [18–24].
These schemes require strictly positive real (SPR) condition on the
observation error dynamics (i.e. the estimation error dynamics) so
that one can use Meyer–Kalman–Yakubovich (MKY) lemma in the
stability analysis. The original observation error dynamics, which
are not SPR in general, are augmented by a low-pass filter
designed to satisfy the SPR condition of a transfer function
associated with the Lyapunov stability analysis. However, according to [25], these schemes result in the filtering of the fuzzy basis
function (FBF) which is not generally suitable. Moreover, as stated
in [26], these observer-based fuzzy (direct or indirect) adaptive

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

controllers have not been derived rigorously in mathematics. In
fact, some comments on these control schemes have been made in
[26].
In this paper, a novel fuzzy indirect adaptive controller is
investigated for a class of uncertain perturbed monovariable nonlinear systems. The main difficulties are how to deal with
unknown nonlinear functions, to appropriately design an
observer-based indirect adaptive control using the SPR condition
and to compensate for the fuzzy approximation errors which
depend on the control input. In this paper, these difficulties can
be respectively solved by using the fuzzy systems for the function
approximation, by introducing an auxiliary error and by designing
a new robust dynamic compensator. The main contributions of
this paper lie in the following:

A novel fuzzy indirect adaptive output-feedback control



scheme based on SPR condition for nonlinear systems is
proposed.
Unlike in [18–24], there is no filtering of the FBF vectors in this
proposed adaptive control scheme.
By using the SPR condition and Lyapunov theory, the stability of
the closed-loop system is rigorously proven. Recall that all
previous observer-based fuzzy indirect and direct adaptive
control schemes [18–24] have not been derived rigorously in
mathematics, as stated in [26].

379

Assumption 3. There exists an unknown positive constant g0 such
that: 0 og 0 rjgðxÞj [26,27].
The following remarks allow to motivate the above assumptions with respect to the considered design framework:
Remark 1. Assumption 1 is usually required in the system theory,
e.g. see [18,22,25,26]. Assumption 2 is a standard assumption in
the adaptive control literature. The latter is the first to be made in
an adaptive control scheme and can be given explicitly or
implicitly.
Remark 2. Many practical systems can be expressed or transformed in the form (1) such as inverted pendulum system [3,26],
Duffing oscillator [3,26], Chua0 s chaotic circuit [28], mass–
springer–damper system [29], induction servo-motor system
[30], gyro system [31], Genesio-Tesi chaotic system [32], singlelink robot [33], atomic force microscope [34], autocatalysed
chemical reaction [35] and many others. Assumption 3 is not
restrictive as it is satisfied by all these practical systems. It
guarantees the controllability of the system (1).
Let us define the reference signal vector y r and the tracking
error vector as follows:
1Þ T
y r ¼ ½yr ; y_ r ; …; yðn
;
r

e ¼ y r x ¼ ½e1 ; e2 ; …; en T ¼ ½e; e_ ; …; eðn 1Þ T

By using the fact that y_ r ¼ Ay r þ ByðnÞ
r , we get
T
e_ ¼ Ae þ B½yðnÞ
r f ðxÞ gðxÞu dðtÞ ; e1 ¼ C e:

Based on the feedback linearization approach, when the functions
f ðxÞ and gðxÞ are known, dðtÞ ¼ 0 and the state x is available for
measurement, the so-called ideal controller can be chosen as
follows:

2. Problem formulation, preliminaries and fuzzy systems
2.1. Problem formulation and preliminaries
Consider the nth order nonlinear dynamical system of the
form:
xðnÞ ¼ f ðx; x_ ; …; xðn 1Þ Þ þ gðx; x_ ; …; xðn 1Þ Þu þ dðtÞ; y ¼ x

ð1Þ

Or equivalently of the form
x_ ¼ Ax þ B½f ðxÞ þ gðxÞu þ dðtÞ ; y ¼ C T x
with
2

0

60
6
6
A¼6
6⋮
6
40
0



1

0

0

1









0

0



0

0



0

3

07
7
7
⋮7
7;
7
15
0

2 3
2 3
1
0
607
607
6 7
6 7
6 7
6 7
7
6 7
B¼6
6 ⋮ 7; C ¼ 6 ⋮ 7;
6 7
6 7
0
4 5
405
1
0

ð4Þ

ð2Þ

ð3Þ

where uA R is the control input, x ¼ ½x1 ; x2 ; …; xn T ¼
½x; x_ ; …; xðn 1Þ T A Rn is the vector of unmeasured states and yA R
is the measured output. f ðxÞ and gðxÞ are unknown smooth
functions, d(t) is the external disturbance. Note that the pair
(A,B) is controllable and the pair (CT,A) is observable.
Design objective: Determine an output-feedback Determine an
output-feedback control law u to steer the system output y closes
to a reference signal yr, while ensuring that all involved signals in
the closed-loop system remain uniformly ultimately bounded
(UUB).
To facilitate control system design, the following usual
assumptions are presented and will be used in the subsequent
developments.
Assumption 1. There exists an unknown positive constant dn such
that |d(t)| rdn.
Assumption 2. The reference signals yr,y_ r , …,yrðn 1Þ , and yrðnÞ are
assumed to be continuous and bounded.

T
u ¼ un ¼ g 1 ðxÞ½ f ðxÞ þ yðnÞ
r þ K c e
T

ð5Þ

n

where K c ¼ ½kc1 ; kc2 ; …; kcn A R is the feedback-gain vector to be
selected such that the characteristic polynomial of A BK Tc is
strictly Hurwitz.
Substituting (5) into (4) yields
eðnÞ þ K Tc e ¼ eðnÞ þ kcn eðn 1Þ þ ⋯ þ kc1 e ¼ 0
Thus, it can be obtained that limt-1 eðtÞ ¼ 0. However, since the
functions f ðxÞ and gðxÞ are unknown and the state vector x is not
available for measurement, the ideal controller (5) cannot be
implemented. Thereafter, to overcome such problems, we will use:

adaptive fuzzy systems to approximate the unknown nonlinear
functions (f ðxÞ and gðxÞ),

an observer to estimate the tracking error vector. From the
estimate of the tracking-error vector, we can directly determine
the estimate of the state vector x.
Remark 3.
1. Note that the system (2) is written in the observability
canonical-form. This system is observable for any input: such
a feature is vital for the observer design.
2. The form of the system (2) is also called the Byrnes–Isidori
normal form [36]. This form is controllable, if gðxÞ a 0:
3. The controllability property of the pair (A,B) does not imply the
controllability of the nonlinear system (1) (or (2)). But, this
property guarantees the existence of a feedback-gain vector, Kc,
so that the characteristic polynomial of A BK Tc is strictly
Hurwitz.
4. Also, the observability property of the pair (CT,A) assures the
existence of an observer-gain vector, Ko, so that the characteristic polynomial of A KoCT is strictly Hurwitz.

380

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

Let us define the observation error vector as e~ ¼
½e~ 1 ; e~ 2 ; …; e~ n T ¼ e e^ . Subtracting (8) from (4), we get the
dynamics of the observation error as

Fuzzy Rules Base

x



Fuzzifier

e_~ ¼ Ao e~ þ B½v^ f ðxÞ gðxÞu dðtÞ ;

Defuzzifier

e~ 1 ¼ C T e~ :

with Ao ¼ A K o C , where v^ ¼
The unknown continuous nonlinear functions f ðxÞ and gðxÞ can
be approximated respectively, on the compact set Ωx , by the fuzzy
system (7) as follows:

Fuzzy Inference
Engine
Fig. 1. The basic configuration of a fuzzy logic system.

2.2. Description of the fuzzy logic system
The basic configuration of a fuzzy logic system consists of a
fuzzifier, some fuzzy IF–THEN rules, a fuzzy inference engine and a
defuzzifier, as shown in Fig. 1.
The fuzzy inference engine uses the fuzzy IF–THEN rules to
perform a mapping from an input vector x T ¼ ½x1 ; x2 ; …; xn A Rn to
an output f^ A R. The ith fuzzy rule is written as
i
RðiÞ : if x1 is Ai1 and … and xn is Ain then f^ is f

i

n
∑m
i ¼ 1 f ð∏j ¼ 1 μAi ðxj ÞÞ
j

n
∑m
i ¼ 1 ð∏j ¼ 1 μAi ðxj ÞÞ
j

T

¼ θ ψðxÞ

ð7Þ

where μAi ðxj Þ is the degree of membership of xj to Aij , m is the
j
1 2
m
number of fuzzy rules, θT ¼ ½f ; f ; …; f is the adjustable parameter vector (composed of consequent parameters), and
ψ T ¼ ½ψ 1 ψ 2 …ψ m with
ψ i ðxÞ ¼

ð∏nj¼ 1 μAi ðxj ÞÞ
j

n
∑m
i ¼ 1 ð∏j ¼ 1 μAi ðxj ÞÞ
j

being the fuzzy basis function (FBF). Throughout the paper, it is
assumed that the FBFs are selected so that there is always at least
n
one active rule [3], i.e.Σ m
i ¼ 1 ðΠ j ¼ 1 μAij ðxj Þ 4 0Þ.
It is worth noticing that the fuzzy system (7) is widely applied
in modeling, identification and control of nonlinear systems
because it has been proven by [3] that this simple fuzzy system
can approximate an arbitrary nonlinear smooth function f ðxÞ
defined on a compact operating space to an given accuracy. Of
particular importance, it is assumed that the FBFs ψðxÞ are properly
specified beforehand by designer. But, the consequent parameters
θ are determined by appropriate adaptation laws.

3. Observer-based fuzzy adaptive controller
Consider now the following observer for estimating the tracking error vector e:
e_^ ¼ Ac e^ þ K o e~ 1 ; e^ 1 ¼ C T e^ :

f^ ðx; θf Þ ¼ θTf ψ f ðxÞ;

ð10Þ

^ θg Þ ¼ θTg ψ g ðxÞ;
gðx;

ð11Þ

where ψ f ðxÞ and ψ g ðxÞ are FBF vectors fixed a priori by the
designer, and θf and θg are the adjustable parameter vector of
the fuzzy system.
The respective optimal values of θf and θg are defined as
"
#
ð12Þ
θn ¼ arg min sup jf ðxÞ f^ ðx; θ Þj
f

θf

ð8Þ

where Ac ¼ A BK Tc , e~ 1 ¼ e1 e^ 1 ¼ y^ y, e^ ¼ y r x^ , with x^ is the
estimate of the state vector x and e^ is the estimate of the tracking
error vector e. K o ¼ ½ko1 ; ko2 ; …; kon T A Rn is the observer-gain
vector to be selected such that the characteristic polynomial of
A K o C T is strictly Hurwitz, and the vector K c has been previously
defined.

x A Ωx

"

ð6Þ

where Ai1 ; Ai2 ; …; and Ain are fuzzy sets and fi is the fuzzy singleton
for the output in the ith rule. By using the singleton fuzzifier,
product inference, and center-average defuzzifier, the output of
the fuzzy system can be expressed as follows:
f^ ðxÞ ¼

ð9Þ
yrðnÞ þ K Tc e^ .

T

f

#

^ θg Þj
θg ¼ arg min sup jgðxÞ gðx;
n

θg

x A Ωx

ð13Þ

Notice that the quantities θnf and θng are introduced only for
analysis purposes, and their values are not needed when implementing the controller [37,38].
Define
θ~ f ¼ θf θnf ;

ð14Þ

θ~ g ¼ θg θng

ð15Þ

as the parameter estimation errors, and
εf ðxÞ ¼ f ðxÞ f^ ðx; θnf Þ;

ð16Þ

^ θng Þ
εg ðxÞ ¼ gðxÞ gðx;

ð17Þ

as the fuzzy approximation errors, where f^ ðx; θnf Þ ¼ θnf T ψ f ðxÞ and
^ θng Þ ¼ θgnT ψ g ðxÞ.
gðx;
As in the literature [3,26,39–43], we assume that those fuzzy
approximation errors are bounded for all 8 x A Ωx , i.e.
jεf ðxÞj r εf and jεg ðxÞj r εg ;

8 x A Ωx

ð18Þ

where εf and εg are unknown positive constants.
Since the state vector x is not available for measurement, the
fuzzy systems (10) and (11) used to approximate the unknown
functions (f ðxÞ and gðxÞ) are replaced by the following fuzzy
systems:
f^ ðx^ ; θf Þ ¼ θTf ψ f ðx^ Þ;

ð19Þ

^ x^ ; θg Þ ¼ θTg ψ g ðx^ Þ;


ð20Þ

where the vector x^ is the estimate of x.
From (14)–(17) and (19) and (20), we have
f ðxÞ ¼ f ðxÞ f^ ðx; θnf Þ þ f^ ðx; θnf Þ f^ ðx^ ; θnf Þ þ f^ ðx^ ; θnf Þ
¼ θfnT ψ f ðx^ Þ þεf ðxÞ þθnf T ½ψ f ðxÞ ψ f ðx^ Þ ;
¼ θfnT ψ f ðx^ Þ þwf ðx; x^ Þ:

ð21Þ

^ θng Þ þ gðx;
^ θng Þ gð
^ x^ ; θng Þ þ gð
^ x^ ; θng Þ
gðxÞ ¼ gðxÞ gðx;
¼ θgnT ψ g ðx^ Þ þ εg ðxÞ þ θngT ½ψ g ðxÞ ψ g ðx^ Þ ;
¼ θgnT ψ g ðx^ Þ þ wg ðx; x^ Þ:

ð22Þ

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

381

with wf ðx; x^ Þ ¼ εf ðxÞ þ θnf T ½ψ f ðxÞ ψ f ðx^ Þ and wg ðx; x^ Þ ¼ εg ðxÞ þ
θgnT ½ψ g ðxÞ ψ g ðx^ Þ are the approximation errors. Notice that
wf ðx; x^ Þ and wg ðx; x^ Þ have also upper bounds [3,26].
Substituting (21) and (22) into (9) yields

Remark 5. For example, one can easily show that for n ¼ 2, HðsÞ is
not SPR (because the real part of HðjωÞ can be negative, i.e. the
third condition in Remark 4 is not satisfied). One will note this
effect in the simulation examples given in Section 4.

T
T
e_~ ¼ Ao e~ þ B½v^ f^ ðx^ ; θf Þ gðx^ ; θg Þu þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g ðx^ Þu w1 ;

Let us define a novel error em1 , called the modified error, as
follows:

e~ 1 ¼ C e~ :
T

ð23Þ

where w1 ¼ wf ðx; x^ Þ þ wg ðx; x^ Þu þ dðtÞ, θ~ f ¼ θf θnf and θ~ g ¼ θg θng
are the parameter approximation errors.
The control input for the system (1) can be determined as


Tanhðδ2 θTg ψ g ðx^ ÞÞ
θTg ψ g ðx^ Þ Tanhðδ2 θTg ψ g ðx^ ÞÞ þ δ1

ð30Þ

where the error ea1 is called the auxiliary error, which is generated
by the following dynamics:
ea1 ¼ HðsÞðurf ur TðsÞ½θTf ψ f ðx^ Þ þ θTg ψ g ðx^ Þu þθTf ψ f ðx^ Þ þ θTg ψ g ðx^ ÞuÞ

ð θTf ψ f ðx^ Þ þ Satðv^ Þ þ ur Þ

ð24Þ

where δ1 ; and δ2 are strictly positive design constants, Tanh(.)
denotes the hyperbolic tangent function and Satð:Þ the usual
saturation function. ur is a dynamic adaptive control term which
will be designed later.
Substituting the control input (24) into (23) yields

ð31Þ
From (28), (30) and (31), the dynamics of the modified error em1
can be expressed as follows:
T
T
em1 ¼ HðsÞð ur þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g ðx^ Þu þ w3 Þ

T

e~ 1 ¼ C T e~ :

ð25Þ

where

where

Now, to facilitate the controller design and the stability analysis,
we make the following mild assumption:
Assumption 4. The following inequality holds

w2 ¼ w1 þu þ v^ Satðv^ Þ

jw3 j r a0 þ a1 juf j þ a2 juf j þ a3 jujþ a4 jΔv^ f j ¼ κ nT φ

and
δ1
u¼ T
ð θTf ψ f ðx^ Þ þ Satðv^ Þ þ ur Þ
θg ψ g ðx^ Þ Tanhðδ2 θTg ψ g ðx^ ÞÞ þ δ1

ð26Þ

Since only the output observation-error e~ 1 in (25) is measurable,
one will use the SPR-Lyapunov design approach to analyze the
stability of the observation error dynamics (25) and to generate
the adaptive laws to estimate the fuzzy parameter vectors and
unknown constants.
The dynamics (25) can be expressed in frequency domain using
the mixed notation (i.e. time–frequency)
T
T
e~ 1 ¼ HðsÞ½ ur þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g ðx^ Þu þ w2

ð27Þ
T

where s is the Laplace variable and HðsÞ ¼ C ðsI Ao Þ B is the
stable transfer function of (25). Note that this notation is very
common in the adaptive control literature such as in [32,41,
44–47]. It also refers to the convolution between the inverse
T
T
Laplace transform of H(s) and the term ½ ur þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g
ðx^ Þu þ w2 .
Now, since HðsÞ in (27) is not SPR in general, we introduce a low
pass filter TðsÞ such that HðsÞ ¼ HðsÞT 1 ðsÞ is SPR:
e~ 1 ¼ HðsÞð urf þ TðsÞ½θTf ψ f ðx^ Þ þ θTg ψ g ðx^ Þu θfnT ½TðsÞψ f ðx^ Þ
ð28Þ

e_ m ¼ Ao e m þB½ ur þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g ðx^ Þu þw3 ;
T

T

T

em1 ¼ C e m ;

ð34Þ
T

n n

n

n

where e m ¼ ½em1 ; em2 ; …; emn and (Ao A R ,B A R ,C A R ) is a
T
minimal state realization of HðsÞ ¼ HðsÞT 1 ðsÞ ¼ C ðsI Ao Þ 1 B,
T
with C ¼ ½1; 0; …; 0 .
Since HðsÞ is SPR, the following holds:
Ao P þ PAo ¼ Q o0
PB ¼ C

ð35Þ

ð29Þ

From (28), it is clear that the presence of the filtered terms urf and
TðsÞ½θTf ψ f ðx^ Þ þ θTg ψ g ðx^ Þu in the output observation-error dynamics
makes the control system design very difficult.
Remark 4. HðsÞ is SPR, with s ¼ s þ jω, if the following three
conditions are satisfied [48]:
(a) When s is real, HðsÞ is real.
(b) The poles of HðsÞ are not in the right half plane.
(c) For any real ω, the real part of HðjωÞ is positive, i.e.
Re½HðjωÞ Z 0:

T

T

where P ¼ P 4 0 and Q ¼ Q 4 0. Note that the matrix Eq. (35) and
the dynamics (34) will be used later in the stability analysis.
In order to dynamically compensate for the uncertain term w3 ,
the robust control term ur can be designed as follows:
"
#
Tanhðδ3 ur Þ
T
u_ r ¼ γ r ur þ γ r em1
κ
φje
j
ð36Þ
m1 ;
ur Tanhðδ3 ur Þ þ δ24
with
δ_ 4 ¼ γ δ sδ δ4 γ δ

and w2f ¼ TðsÞ½wf ðx; x^ Þ TðsÞ½wg ðx; x^ Þu

TðsÞ½dðtÞ þ TðsÞ½u þ TðsÞ½v^ Satðv^ Þ

where κ ¼ ½a0 ; a1 ; a2 ; a3 ; a4 is an unknown positive constant
vector,
uf ¼ TðsÞ½ u ,
uf ¼ TðsÞ½u ,
Δv^ f ¼ TðsÞ½v^ Satðv^ Þ
and
φ ¼ ½1; juf j; juf j; juj; jΔv^ f j T :
The state space realization of (33) is given by:

T

1

θngT ½TðsÞψ g ðx^ Þu þ w2f Þ

ð33Þ

T

n

with urf ¼ TðsÞ½ur ;

ð32Þ

w3 ¼ w2f θnf T ½TðsÞψ f ðx^ Þ θgnT ½TðsÞψ g ðx^ Þu þ θnf T ψ f ðx^ Þ þ θgnT ψ g ðx^ Þu:

e_~ ¼ Ao e~ þ B½ ur þ θ~ f ψ f ðx^ Þ þ θ~ g ψ g ðx^ Þu þ w2 ;
T

em1 ¼ e~ 1 þ ea1

δ4
κ T φjem1 j
ur Tanhðδ3 ur Þ þ δ24

κ_ ¼ γ κ sκ κ þ γ κ jem1 jφ

with κ i ð0Þ Z 0

with δ4 ð0Þ 4 0

ð37Þ
ð38Þ

where γ r ; γ δ ; γ κ , sκ , sδ and δ3 are positive design constants, κ A R5 is
the estimate of the unknown vector κ n .
The adaptation laws for θf and θg can be determined as follows:
θ_ f ¼ γ f sf θf γ f em1 ψ f ðx^ Þ

ð39Þ

θ_ g ¼ γ g sg θg γ g em1 uψ g ðx^ Þ

ð40Þ

where γ f ; γ g ; sf and sg are positive design constants.
Fig. 2 shows the scheme of the proposed fuzzy indirect
adaptive controller.

382

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387





e~1 +

yˆ +

-

y

y

Observer
Eq. (8)

Fuzzy adaptive
controller, Eqs. (24)
and (36-40)

em1

u

Plant
Eq. (1)

y

+
ea1
T

θgψ

g

θ Tf ψ

f

Estimator of

ur
ea1

u

Eq. (31)

Fig. 2. The scheme of the proposed fuzzy indirect adaptive controller based on observer.

The following theorem establishes the stability and performance properties of the closed-loop system with this proposed
controller.
Theorem 1. Consider the system (1) under Assumptions 1–4 and the
observer (8). Then, the proposed fuzzy adaptive controller, defined by
(24), (30), (31) and (36)–(40), guarantees the following nice
properties:
1. All signals in the closed-loop system are bounded, i.e.
e m ; em1 ; ea1 ; e~ 1 , e~ , e^ , e, θf ; θg ur , δ4 , κ and u A L1 ,
2. The output tracking error converges to the residual set
Ωe ¼ fe1 j je1 jr ηe , where ηe is a positive constant which will
be defined later.
Proof of Theorem 1. Let us consider the following Lyapunov-like
function:
1
1 ~T ~
1 ~T ~
1 2
1 2
1 T
V 1 ¼ e Tm Pe m þ
κ~ κ~
u þ
δ þ
θ θ þ
θ θg þ
2
2γ f f f 2γ g g
2γ r r 2γ δ 4 2γ κ

ð41Þ

where κ~ ¼ κ κn .
The time derivative of V 1 is
1
1
1 T
1 T
1
1
1
V_ 1 ¼ e Tm P e_ m þ e_ Tm Pe m þ θ~ f θ_ f þ θ~ g θ_ g þ ur u_ r þ δ4 δ_ 4 þ κ~ T κ_
2
2
γf
γg
γr
γδ
γκ
ð42Þ
Evaluating (42) along the trajectories (34) and (36)–(40) gives
1
T
T
V_ 1 r e Tm Q e m u2r sδ δ24 sf θ~ f θf sg θ~ g θg sκ κ~ T κ
2

ð43Þ

Since the following inequalities are valid:
sf 2 sf 2
T
sf θ~ f θf r θ~ f þ θnf ;
2
2
sg ~ 2 sg n 2
T
~
sg θ g θg r θ g j þ θg ; and
2
2
2 s 2
s







κ
κ
sκ κ~ T κ r κ~ þ κ n ;
2
2

where

π1 ¼

sf n 2 sg n 2 sκ n 2

θ þ θg þ κ :
2 f
2
2
Let μ1 ¼ min fλmin ðQ Þ=λmax ðPÞ; 2γ r ; 2γ δ sδ ; γ f sf ; γ g sg ; γ κ sκ g; hence
we can rewrite (44) as follows:
V_ 1 r μ1 V 1 þ π 1

ð45Þ

where λmin ðQ Þ denotes the smallest eigen-value of Q , and λmax ðPÞ
the largest eigen-value of P:
Multiplying (45) by eμ1 t leads to the following result:
d
ðV 1 eμ1 t Þ r π 1 eμ1 t
dt
Integrating (46) over [0,t] yields


π1
π 1 μ1 t
e
0 r V 1 ðtÞ r þ V 1 ð0Þ
μ1
μ1

ð46Þ

ð47Þ

From (47), therefore, the modified error e m , the robust term ur , the
adaptive parameter δ4 and the parameter errors (θ~ f ,θ~ g and κ~ )
are UUB.
With the help of (41), V 1 ð0Þ is defined as follows:
1
1 ~T
1 ~T
θ ð0Þθ~ f ð0Þ þ
θ ð0Þθ~ g ð0Þ
V 1 ð0Þ ¼ e Tm ð0ÞPe m ð0Þ þ
2
2γ f f
2γ g g
1 2
1 2
1 T
κ~ ð0Þκ~ ð0Þ
u ð0Þ þ
δ ð0Þ þ
þ
2γ r r
2γ δ 4
2γ κ
From (41) and (47), one has




1=2
2
π1
π 1 μ1 t
jje m jj r
þ V 1 ð0Þ
:
e
λmin ðPÞ μ1
μ1

ð48Þ

ð49Þ

Then, the solution e m exponentially converges to a bounded
region Ωem ¼ fe m j jje m jjr ηem g, with


2 π 1 1=2
:
ηem ¼
λmin ðPÞ μ1

we can rewrite (43) as follows:

sf 2 sg 2 sκ 2
1
V_ 1 r e Tm Q e m u2r sδ δ24 θ~ f θ~ g ~κ þ π 1
2
2
2
2
ð44Þ

From the boundedness of θ~ f ,θ~ g and κ~ , one can directly conclude
about the boundedness of θf ; θg and κ. The boundedness of the
control u follows that of ur , θf ; θg and Satðv^ Þ. From (31), since the
term
ðurf ur LðsÞ½θTf ψ f ðx^ Þ þ θTg ψ g ðx^ Þu þ θTf ψ f ðx^ Þ þ θTg ψ g ðx^ ÞuÞ
is
bounded and HðsÞ is a stable transfer function, one can easily
show that the auxiliary error ea1 is also bounded (i.e. it has an
upper bound). Now, in order to quantifier this upper bound, one

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

383

Table 1
Comparison between our control scheme and the previous works.
Comparaison Features of the proposed controller
Scheme in
[18]

Scheme in
[19]

Advantages/disadvantages

An indirect adaptive control has been proposed.
A linear observer has been used to estimate the tracking error
vector.
A projection algorithm has been incorporated in the adaptive
laws to avoid the parameters drift.
The problem of an possible singularity of the controller has
been solved by modifying this projection algorithm.
This scheme requires SPR condition on the observation error
dynamics
A sliding mode control term has been used to deal with the
fuzzy approximation errors.

Advantages
The tracking errors can theoretically converge to the origin.

A direct adaptive control has been designed.
A linear observer has been used to estimate the tracking error
vector.
A projection algorithm has been incorporated in the adaptive
laws to avoid the parameters drift.
This scheme requires SPR condition on the observation error
dynamics
A sliding mode control term has been used to compensate for
the fuzzy approximation errors.

Advantages
The tracking errors can theoretically converge to the origin.

Disadvantages
The filtering of the FBFs can make the dynamic order of the controller-observer
system very large.
Authors have designed a robust control term filtered vf instead of v. The
problem is how to obtain v from vf ?
The robust control term designed vf is not smooth.
The projection algorithm used requires the knowledge of the norm bound of
the uncertain optimal fuzzy parameters.
The stability analysis is questionable, according to [26].

Disadvantages
Authors have designed a robust control term filtered vf instead of v. The
problem is how to obtain v from vf ?
The robust control term designed vf is not smooth.
The projection algorithm used requires the knowledge of the norm bound of
the uncertain optimal fuzzy parameters.

Scheme in
[21]

Advantages:
Indirect and direct adaptive controllers have been designed.
The controller designed is very simple.
A linear observer has been used to estimate the tracking error
There are fewer parameters to be adapted.
vector.
The problem of a possible drift of the adaptive fuzzy parameters Disadvantages:
has not been treated.
Authors have designed the filtered control terms (ua1 and us1 ) instead of (ua and
The singularity problem of the controller has not been
us ). The question is how to obtain the control terms (ua and us ) from (ua1 and
discussed.
us1 ), respectively?
This scheme requires SPR condition on the observation error
Because the authors have not treated the singularity problem and parameters
dynamics.
drift, they assumed that the adjusted parameters never reach some known
An H 1 robust control term has been used to compensate for the
boundaries. Note that this assumption is not realistic in the practice.
fuzzy approximation errors.
The FBF filtering makes the dynamic order of the controller-observer system
very large.
Moreover, the stability analysis is questionable, according to [26].

Scheme in
[26]

Advantages:
A direct adaptive control has been designed.
An unified design frame-work for the high-gain observers has been proposed.
An unified observer has been used to estimate the tracking
This scheme does not require the SPR condition.
error vector.
The FBF filtering is no longer required.
To avoid the parameters drift, a s-modification term has been
incorporated in the adaptive laws.
Disadvantages:
it does not require the SPR condition on the observation error
The robust control term designed is not smooth.
dynamics.
The tracking errors cannot converge to zero. They are only UUB.
A sliding mode term has been used to compensate for the fuzzy
High-gain observers proposed are very sensitive to noise.
approximation errors.

Our
proposed
scheme

An indirect adaptive control has been designed.
Advantages:
An linear observer has been used to estimate the tracking error
Unlike in [18,21], there is no filtering of the FBF vectors.
vector.
Unlike in [18,19,21], by using the SPR condition and the Lyapunov theory, the
To avoid the parameters drift, a s-modification term has been
stability of the closed-loop is rigorously proven.
The dynamic adaptive compensator designed is smooth.
incorporated in the adaptive laws.
Unlike [26], the proposed observer is not an high-gain observer.
This scheme requires the SPR condition on the observation error
dynamics.
Disadvantages:
A dynamic adaptive compensator has been used to compensate
The tracking errors cannot converge to zero. They are only UUB.
for the fuzzy approximation errors together with other
uncertainties

Let us consider the following Lyapunov-like function:

makes the state space realization of the dynamics (31):
e_ a ¼ Ao e a þ B½ua ;
T

ea1 ¼ C e a ;

ð50Þ

with
ua ¼

urf ur LðsÞ½θTf ψ f ðx^ Þ þ θTg ψ g ðx^ Þu þ θTf ψ f ðx^ Þ þ θTg ψ g ðxÞu;

ð51Þ

where e a ¼ ½ea1 ; ea2 ; …; ean T and (Ao A Rn n ,B A Rn ,C A Rn ) is a miniT
mal state realization of HðsÞ ¼ HðsÞT 1 ðsÞ ¼ C ðsI Ao Þ 1 B with
T
C ¼ ½1; 0; …; 0 .

1
V 2 ¼ e Ta Pe a
2
Its time derivative is given by
1
1
V_ 2 ¼ e Ta P e_ a þ e_ Ta Pe a
2
2
1 T
¼ e a Q e a þ e Ta PBua
2
2 c 2 ε
1

1
r λmin ðQ Þ e a þ e a þ
2
2


ð52Þ

384

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

r μ2 V 2 þ π 2

ð53Þ

with c1 ¼ jjPBjj2 jua j2 ; μ2 ¼ ðλmin ðQ Þ ðc1 =εÞÞ=λmax ðPÞ and π 2 ¼ ε=2,
where ε is a small positive constant, and λmin ðQ Þ 4 ðc1 =εÞ.
From (52) and (53), one has

jje a jj r




1=2
2
π2
π 2 μ2 t
þ V 2 ð0Þ
:
e
λmin ðPÞ μ2
μ2

ð54Þ

Then, the solution e a exponentially converges to a bounded
region Ωea ¼ fe a j jje a jj r ηea g, with

ηea ¼

2 π2
λmin ðPÞ μ2

1=2
:

In the following, we present simulation results showing the
performances of the proposed controller applied to a Duffing
oscillator. This chaotic system can be described as [53]
(
x_ 1 ¼ x2
ð57Þ
x_ 2 ¼ p1 x2 p2 x1 p3 x31 þ q cos ðωtÞ
where x ¼ ½x1 ; x2 T ¼ ½x; x_ T is the state vector, p1 ; p2 ; p3 ; and q are
positive constants, t is the time variable, and ω is the frequency.
Depending on the choice of these constants, it is known that the
solutions of (57) exhibit periodic, almost periodic, and chaotic
behavior [53]. A typical chaotic behavior of the uncontrolled
Duffing equation can be obtained with
p1 ¼ 0:4; p2 ¼ 1:1; p3 ¼ 1; q ¼ 2:1 and ω ¼ 1:8

From (30) and the bounds of the errors e a and e m , one can get the
upper bound of the output tracking error e1 as follows:
je~ 1 j r jem1 j þjea1 j r ηe~

ð55Þ

where




2 π 2 1=2
2 π 1 1=2
þ
ηe~ ¼
λmin ðPÞ μ2
λmin ðPÞ μ1
From the observer dynamics (8), one can demonstrate that e^ 1 and
e^ are bounded and converge to a bounded adjustable region.
From (55) and because e^ 1 and e^ are bounded, then the output
tracking error remains in a compact set Ωe specified as:
Ωe ¼ fe1 j je1 j rηe g where ηe ¼ ηe~ þ ηe^ with ηe^ Z jje^ 1 jj. This ends
the proof. □
Remark 6. This paper considers only single-input single-output
(SISO) systems to simplify the representation. The extension of our
fundamental results to the following class of multivariable nonlinear uncertain systems is straightforward (that is why we
omitted the control design for this class of multivariable systems):

Then, the controlled Duffing equation can be written as follows:



0 1
0
x_ ¼

ðf ðxÞ þ gðxÞu þ dðtÞÞ
0 0
1
y ¼ ½ 1 0 x
ð58Þ
where f ðxÞ ¼ p1 x2 p2 x1 p3 x31 þ q cos ðωtÞ and gðxÞ ¼ 1. The
external disturbance dðtÞ is selected as a square wave having an
amplitude 7 1 with a period of 2πðsÞ:
The control objective is to force the system output y to track the
reference signal yr ¼ sin ðtÞ. It is worth noting that the function
f ðxÞ and the control gain gðxÞ are assumed here to be unknown by
the controller. In addition, the state vector is assumed to be nonmeasurable and only the system output y is measurable. In fact,
the model (58) is only required for simulation purposes.
The observer-gain vector and the feedback-gain vector
are selected respectively as follows:K o ¼ ½160; 6400 T and
K Tc ¼ ½144; 24 . The transfect function HðsÞ is given by
HðsÞ ¼

1
:
s2 þ 160s þ 6400

According to Remark 4, HðsÞ is not SPR because the real part of
HðjωÞ can be negative. One should choose a low-pass filter TðsÞ so
that HðsÞ ¼ HðsÞT 1 ðsÞ ¼ ð1=s2 þ 160s þ 6400ÞT 1 ðsÞ is SPR. This filter can be selected as follows:

x_ p1 ¼ xp2
x_ p2 ¼ xp3

x_ pnp ¼ f p ðx1 ; …; xm Þ þ g p ðx1 ; …; xm Þup þ dðtÞ

4.1. Example 1

for p ¼ 1; 2; …; m:

ð56Þ

where f p ðx1 ; …; xm Þ is the unknown nonlinear functions of the pth
subsystem, up is the input of the pth subsystem, g p ðx1 ; …; xm Þ a 0 is
the unknown positive control gain. x ¼ ½xT1 ; …; xTm T is the overall
state vector, and x1 ¼ ½xp1 ; …; xpnp T is the state vector of the pth
subsystem. Note that the model (56) can be used to describe a
relatively large class of multivariable nonlinear dynamical systems,
namely: the robotic manipulator used in [49], the induction motor
[50], the mass–spring–damper system in [51], the unified chaotic
systems [52], and many others.
Remark 7. Note that some closely related works have been
investigated in [18,19,21,26]. Unlike in [26], the stability of the
closed-loop system has been proven in [8,9,11] and in this paper
by using the SPR condition and the Lyapunov theory. A detailed
comparison between our present work and that of [18,19,21,26] is
summarized in Table 1.
4. Simulation results
Simulation studies are carried out to show the effectiveness of
the proposed controller. Two control problems are considered at
this end. The first one concerns a Duffing oscillator, while the
second one concerns an inverted pendulum system.

TðsÞ ¼

1
0:0255s þ 0:8197

From the expression of HðsÞ, we can find that


160 1
T
T

; B ¼ ½ 0:0255 0:8197 ; and C ¼ ½ 1
6400 0
Given

10

5

5

0 :



10

solving the matrix equation (35), one obtains the following
symmetric positive-definite matrix:


200:0312 5:0000

5:0000
0:1555
The design parameters are selected as γ f ¼ 103 ; γ g ¼ 0:5;
γ k ¼ 500; γ δ ¼ 10 5 ; γ r ¼ 2, sf ¼ 5 10 3 ; sg ¼ 10 3 ; sk ¼ 10 3 ;
sδ ¼ 10 7 , δ1 ¼ 10 2 , δ2 ¼ 1 and δ3 ¼ 5.
The fuzzy membership functions are defined for the variables
x^ 1 , x^ 2 as follows:



x^ j þ 0:5
1
1 þ Tanh
;
μA1 ðx^ j Þ ¼
j
2
0:4
!
!
2
1 x^ j
μA2 ðx^ j Þ ¼ exp
; and
j
2 0:6

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

385

2
2

Tracking of x

Tracking of x

1

0
1

0

-5

-10
-1
0

5
time(s)

10

10

0

5
time(s)

10

200
control u

0

1

Estimates of e and e

5
time(s)

300

2

20

0

-20

100
0
-100

-40
-200
-60

0

5
time(s)

10

-300

Fig. 3. Simulation results of Example 1: (a) the system output y ¼ x1 (solid line) and the reference signal yr (dotted line). (b) The state variable x2 (solid line) and the
reference signal y_ r (dotted line). (c) The tracking-errors estimates e^ 1 (dotted line) and e^ 2 (solid line). (d) The control signal u.
Table 2
IAE, ISE and IAU criteria for this proposed method and that of [26] (for Example 1).
Control method

IAE

ISE

IAU

The proposed fuzzy control method
The fuzzy control method of [26]

0.5271
2.02

0.3893
2.028

107.1
81.95

μA3 ðx^ j Þ ¼
j




x^ j 0:5
1
1 þ Tanh
2
0:4

The initial conditions are chosen as xð0Þ ¼ ½x1 ð0Þ; x2 ð0Þ T ¼ ½2; 2 T ,
e^ ð0Þ ¼ ½e^ 1 ð0Þ; e^ 2 ð0Þ T ¼ ½0; 1 T , δ4 ð0Þ ¼ 1:2 and κð0Þ ¼ ½ 0; 0; 0;
0; 0 T . The elements of θf ð0Þ are randomly selected in the interval
½ 1; 1 . As well, the elements of θg ð0Þ are randomly selected in
the interval ½0:4; 1:2 .
Fig. 3 shows the simulation results obtained by applying the
proposed fuzzy indirect adaptive controller. Fig. 3(a) and
(b) illustrates the tracking performances of the state variables.
The estimates (e^ 1 and e^ 2 ) of tracking errors are given in Fig. 3(c).
Fig. 3(d) shows the boundedness of the control signal u.
The fuzzy adaptive control method proposed in [26] is also
simulated and compared to represent the efficiency of our method
proposed in this paper. And to represent a qualitative comparison
between these two controllers, integral of absolute (output-tracking) error (IAE) integral of squared error (ISE) and Integral
Absolute input (IAU) criteria are calculated for each one in Table 2.
From Table 2, it is clear that the output-tracking performances
(IAE and ISE) obtained by applying the proposed fuzzy controller
are very good, in comparison with the fuzzy controller in [26].
However the IAU criterion is relatively increasing, because the
control effort in the proposed scheme is important in the beginning period.

4.2. Example 2
In this section, we present simulation results showing the
tracking performances of the proposed fuzzy adaptive controller
applied to an inverted pendulum system.
Let x1 ¼ θ be the angle of the pendulum with respect to the
_ The dynamic equations of such a system
vertical line and x2 ¼ θ.
are given by [3]
" #
" #
x_ 1
0 1 x1
0
¼
þ
ðf ðx1 ; x2 Þ þ gðx1 ; x2 Þu þ dðtÞÞ;
x_ 2
0 0 x2
1


y¼ 1

" #
x1
0
:
x2

ð59Þ

with
f ðx1 ; x2 Þ ¼

mlx2 sin x1 cos x1 ðM þ mÞG sin x1
;
ml cos 2 x1 ð4=3ÞlðM þ mÞ

gðx1 ; x2 Þ ¼

cos x1
;
ml cos 2 x1 ð4=3ÞlðM þ mÞ

where G is the acceleration due gravity, M is the mass of the cart,
m is the mass of the pole, l is the half-length of pole and u is the
applied force. It is assumed that the external disturbance dðtÞ is
a square wave having an amplitude 7 1 with a period of 2πðsÞ:
The system parameters are given as M ¼ 1 kg; m ¼ 0:1 kg;
l ¼ 0:5 m; G ¼ 9:8 m=s2 .
The control objective is to force the system output y to track the
reference signal yr ¼ sin ðtÞ. We assume that the functions f ðxÞ and
gðxÞ are completely unknown by the controller and only the
system output y is available for measurement. In fact, the model
(59) is only required for simulation purposes.

386

A. Boulkroune et al. / Neurocomputing 135 (2014) 378–387

2
2

0.5

Tracking of x

Tracking of x

1

1

0
-0.5

0
-2
-4
-6

-1
5
time(s)

-8

10

0

5
time(s)

10

0

5
time(s)

10

5
200
control u

0

1

Estimates of e and e

2

0

-5
-10
-15

0

-200
0

5
time(s)

10

Fig. 4. Simulation results of Example 2: (a) the system output y ¼ x1 (solid line) and the reference signal yr (dotted line). (b) The state variable x2 (solid line) and the
reference signal y_ r (dotted line). (c) The tracking-errors estimates e^ 1 (dotted line) and e^ 2 (solid line). (d) The control signal u.

Table 3
IAE, ISE and IAU criteria for this proposed method and that of [26] (for Example 2).
Control method
The proposed fuzzy control method
The fuzzy control method of [26]

IAE

ISE

IAU

0.3647
0.4798

0.02654
0.04118

109.1
105.1

The observer-gain vector K o , the feedback-gain vector K c , the
SPR filter TðsÞ and the matrix Q are all selected as in the previous
example.
The design parameters are selected as γ f ¼ 103 ; γ g ¼ 0:2; γ k ¼
500; γ δ ¼ 10 5 ; γ r ¼ 2, sf ¼ 8 10 3 ; sg ¼ 10 2 ; sk ¼ 10 3 ; sδ ¼
10 7 , δ1 ¼ 10 2 , δ2 ¼ 1 and δ3 ¼ 5.
The fuzzy membership functions are defined for the variables
x^ 1 , x^ 2 as follows:
μA1 ðx^ j Þ ¼

1
;
1 þ expð3ðx^ j þ0:25ÞÞ

μA3 ðx^ j Þ ¼

1
;
1 þ expð 3ðx^ j 0:25ÞÞ

j

j

μA2 ðx^ j Þ ¼ exp
j

2
1 x^ j
2 0:6

!!
;

and

The initial conditions are chosen as xð0Þ ¼ ½x1 ð0Þ; x2 ð0Þ T ¼ ½0:5; 0 T ,
e^ ð0Þ ¼ ½e^ 1 ð0Þ; e^ 2 ð0Þ T ¼ ½0; 1 T , δ4 ð0Þ ¼ 1:2 and κð0Þ ¼ ½0; 0; 0; 0; 0 T .
The elements of θf ð0Þ are randomly selected in the interval
½ 1; 1 . As well, the elements of θg ð0Þ are randomly selected in
the interval ½0:4; 1:2 .
The simulation results are depicted in Fig. 4. From this figure,
we can see that the system tracks its desired trajectories and the
control signal is bounded.
The fuzzy adaptive control method proposed in [26] is also
simulated and compared to represent the efficiency of our method
proposed in this paper. The results of these qualitative comparison
are illustrated in Table 3. It is clear from this table that the tracking
performances are better than those in the previous work [26].

5. Conclusion
In this paper, an observer-based fuzzy indirect adaptive controller for a class of SISO nonlinear systems has been presented. In
the controller designing, neither measurement of the system
states nor knowledge of the system nonlinearities is required.
Indeed, an observer has been constructed to estimate the tracking
error vector and an adaptive fuzzy system has been used to
approximate the system nonlinearities. Using the SPR condition
and Lyapunov theory, the stability of the closed-loop system has
been rigorously proven. Simulation results have been reported to
emphasize the performances of the proposed controller.
References
[1] K. Hornik, Multilayer feedforward networks are universal approximators,
Neural Netw. 2 (1989) 359–366.
[2] L.X. Wang, J.M. Mendel, Fuzzy basis functions, universal approximation, and
orthogonal least square learning, IEEE Trans. Neural Netw. 3 (5) (1992) 807–814.
[3] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis,
Prentice-Hall, Englewood Cliffs, NJ, 1994.
[4] Y.C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISO systems
via VSS and H1 approaches, IEEE Trans. Fuzzy Syst. 9 (2001) 278–292.
[5] B.S. Chen, C.H. Lee, Y.C. Chang, H1 Tracking design of uncertain nonlinear SISO
systems: adaptive fuzzy approach, IEEE Trans. Fuzzy Syst. 4 (1) (1996) 32–43.
[6] J.T. Spooner, K.V. Passino, Stable adaptive control using fuzzy systems and
neural networks, IEEE Trans. Fuzzy Syst. 4 (3) (1996) 339–359.
[7] C.Y. Sue, Y. Stepanenko, Adaptive control of a class of nonlinear systems with
fuzzy logic, IEEE Trans. Fuzzy Syst. 2 (4) (1994) 285–294.
[8] S. Tong, X. He, H. Zhang, A combined backstepping and small-gain approach to
robust adaptive fuzzy output feedback control, IEEE Trans. Fuzzy Syst. 17 (5)
(2009) 1059–1069.
[9] S. Tong, Y. Li, Observer-based fuzzy adaptive control for strict-feedback
nonlinear systems, Fuzzy Sets Syst. 160 (12) (2009) 1749–1764.
[10] S. Tong, C. Liu, Y. Li, Fuzzy adaptive decentralized output-feedback control for
large-scale nonlinear systems with dynamical uncertainties, IEEE Trans. Fuzzy
Syst. 18 (5) (2010) 845–861.
[11] M. Wang, X.P. Liu, P. Shi, Adaptive neural control of pure-feedback nonlinear
time-delay systems via dynamic surface technique, IEEE Trans. Syst. Man
Cybern. part B: Cybern. 41 (6) (2011) 1681–1692.


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