Fault tolerant synchronization of chaotic heavy symmetric gyroscope systems versus external disturbances via lyapunov rule based fuzzy control

ISA Transactions 51 (2012) 50–64
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ISA Transactions
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Fault tolerant synchronization of chaotic heavy symmetric gyroscope systems
versus external disturbances via Lyapunov rule-based fuzzy control
Faezeh Farivara,∗
, Mahdi Aliyari Shoorehdelib
a
Department of Mechatronics Engineering, Science and Research Branch, Islamic Azad University, P.O. Box: 14515/775, Tehran, Iran
b
Faculty of Electrical Engineering, Department of Mechatronics Engineering, K. N. Toosi University of Technology, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 12 January 2011
Received in revised form
25 June 2011
Accepted 27 July 2011
Available online 24 August 2011
Keywords:
Fault tolerant
Synchronization
Chaotic gyroscope
Nonlinear control
Lyapunov rule-based fuzzy control
a b s t r a c t
In this paper, fault tolerant synchronization of chaotic gyroscope systems versus external disturbances
via Lyapunov rule-based fuzzy control is investigated. Taking the general nature of faults in the slave
system into account, a new synchronization scheme, namely, fault tolerant synchronization, is proposed,
by which the synchronization can be achieved no matter whether the faults and disturbances occur or not.
By making use of a slave observer and a Lyapunov rule-based fuzzy control, fault tolerant synchronization
can be achieved. Two techniques are considered as control methods: classic Lyapunov-based control and
Lyapunov rule-based fuzzy control. On the basis of Lyapunov stability theory and fuzzy rules, the nonlinear
controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The
fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics
of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. Two proposed
methods are compared. The Lyapunov rule-based fuzzy control can compensate for the actuator faults
and disturbances occurring in the slave system. Numerical simulation results demonstrate the validity
and feasibility of the proposed method for fault tolerant synchronization.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Since the synchronization of chaotic dynamical systems was
observed by Pecora and Carroll [1] in 1990, chaos synchronization
has become a topic of great interest. Chaos synchronization has
many potential applications in laser physics, chemical reactors,
secure communication, biomedicine, and so on [2–4]. The aim
of chaos synchronization is to make two systems oscillate in
a synchronized manner. Given a chaotic system considered as
the master system, and another identical system considered as
the slave system, the dynamical behaviors of these two systems
may be identical after a transient time when the slave system is
driven by a control input. But despite the amount of theoretical
and experimental results already obtained, chaos synchronization
seems a difficult task, particularly if it is considered that [5]: (1) due
to the sensitive dependence of chaos on the initial conditions, it is
almost impossible to reproduce the same starting conditions; (2) in
matching the master and slave systems exactly, even infinitesimal
parametric variations of any model will eventually result in the
divergence of orbits starting nearby each other; and (3) parametric
∗ Corresponding author. Tel.: +98 9121463845; fax: +98 21 88788281.
E-mail addresses: F.Farivar@srbiau.ac.ir, Faezeh_Farivar84@yahoo.com
(F. Farivar), Aliyari@eetd.kntu.ac.i (M.A. Shoorehdeli).
differences between chaotic systems yield different attractors.
Many methods of chaos synchronization have been presented [6–
8].
On the other hand, the dynamics of a gyro presents a very in-
teresting nonlinear problem in classical mechanics. The gyro has
attributes of great utility to navigational, aeronautical and space
engineering [9]. Gyros for sensing angular motion are used in
airplane automatic pilots, rocket vehicle launch/guidance, space
vehicle attitude systems, ship’s gyrocompasses and submarine in-
ertial auto-navigators. The concept of chaotic motion in a gyro
was first presented in 1981 by Leipnik and Newton [10], show-
ing the existence of two strange attractors. In the past few years,
gyros have been found with rich phenomena which are of bene-
fit for the understanding of gyro systems. Different kinds of gyros
with linear/nonlinear damping are investigated for predicting
the dynamic responses, such as periodic and chaotic motions
[11–13]. Some methods have been presented for synchronizing
two identical/non-identical nonlinear gyro systems, such as active
control [14], fuzzy sliding mode control [15], neural sliding mode
control [16,17] etc.
Fault diagnosis and isolation (FDI) and fault tolerant control
(FTC) have been active research topics during the past two decades,
with a view to increasing the safety and reliability of dynami-
cal systems [18,19]. In the literature, faults normally occur at two
places: the actuators and sensors. Actuator faults are faults that
act on the system, resulting in deviation of the process variables.
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.07.002

F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 51
The result is that the command (control) signal sent to this device
has no effect. Sensor faults are faults that act on the sensors that
measure the system variables, and do not directly affect the pro-
cess. The source of these faults could be wear and tear of the sen-
sor leading to inaccurate readings, or a total failure of the sensor.
Sensor faults could also arise from drift/poor calibration/extreme
process conditions. These faults will affect the process if the output
measurements are used to generate the input control signal [20].
FTC is an area of research that aims to increase applicability by
specifically designing control algorithms capable of maintaining
stability and performance despite the occurrence of faults. FTC
structures have also been studied where a model-matching strat-
egy is used to design linear [21] and nonlinear [22] controllers,
etc. [23,24]. Observer-based methods are commonly used as a basis
for FDI schemes. Residual generation approaches, using linear ob-
servers, have been widely used, where the difference between the
system output and observer output is processed to form so-called
residuals. Ideally, these will be zero during fault-free operation but
will give a specific response when a certain fault occurs. Residual
generation techniques have been demonstrated in [25,26].
But when a fault happens in chaotic systems, the conventional
methods for synchronizing the chaotic systems are invalid. In order
to increase the safety and the reliability of synchronization systems
when a fault arises, fault tolerant synchronization is essential to
guarantee the synchronization of chaotic systems. The objective
of fault tolerant synchronization (FTS) is to design a fault tolerant
controller to achieve the synchronization no matter whether a fault
occurs or not. To the best of the authors’ knowledge, the problem
of FTS is still an open problem and seldom researched [27,28].
Various approaches for achieving chaos synchronization using
fuzzy systems have been proposed [29]. Fuzzy set theory was first
presented by Zadeh [30], while fuzzy logic control (FLC) schemes
have been widely developed for almost 40 years, and have been
successfully deployed in many applications [31]. Furthermore,
adaptive fuzzy controllers have been used to control and synchro-
nize chaotic systems [32,33].
Until now, there has been no investigation on the subject of
fault tolerant synchronization of chaotic gyroscope systems. Two
notable papers are studied, namely [27,28]. Ref. [27] considers
Lur’e systems using time-delay feedback control. By making use
of an observer-based fault estimator and a modified time-delay
feedback controller, the fault tolerant master–slave synchroniza-
tion is formulated so as to allow discussion of the global asymp-
totic stability of the error system and the bound of the energy
gain from fault to state, and fault estimation error vectors. Some
delay-dependent criteria are derived in order to analyze the syn-
chronization error system, and on the basis of the analysis re-
sults, a sufficient condition for the existence of such a master–slave
synchronization scheme and a solution for the controller and fault-
estimator gain matrices are obtained in terms of linear matrix
inequalities (LMI). Also, a Chua circuit is used to illustrate the ef-
fectiveness of the proposed method.
Ref. [28] presents an offline and data-based procedure for fault
tolerant synchronization. In Ref. [28], to investigate the stability
of the error system and facilitate the design of the fuzzy sampled-
data controller, a Takagi–Sugeno (TS) fuzzy model is employed to
represent the chaotic system dynamics. A parameter-dependent
Lyapunov–Krasovksii functional and a relaxed stabilization tech-
nique are considered and the stability conditions based on the lin-
ear matrix inequality (LMI) are obtained in order to achieve fault
tolerant synchronization of the chaotic systems.
Motivation and overview of this study.
Fault tolerant control (FTC) has been an active research topic
during the past two decades with a view to increasing the safety
and reliability of dynamical systems. FTC is an area of research that
aims to increase applicability by specifically designing control al-
gorithms capable of maintaining stability and performance despite
the occurrence of faults.
On the other hand, dynamic chaos is a very interesting nonlin-
ear effect which has been intensively studied during the last three
decades. Chaotic phenomena can be found in many scientific and
engineering fields such as biological systems, electronic circuits,
power converters, chemical systems, and so on. A chaotic system
has complex dynamical behaviors that possess some special fea-
tures, such as excessive sensitivity to initial conditions, broad spec-
tra of the Fourier transform, bounded and fractal properties of the
motion in the phase space, etc. Chaos in control systems and con-
trolling chaos in dynamical systems have both attracted great in-
terest in recent years.
But when a fault happens in chaotic systems, the conventional
methods for synchronizing the chaotic systems are almost invalid.
In order to increase the safety and the reliability of synchroniza-
tion systems when a fault happens, fault tolerant synchronization
(FTS) is essential to guarantee the synchronization of chaotic sys-
tems. The objective of fault tolerant synchronization is to design
a fault tolerant controller to achieve the synchronization no mat-
ter whether a fault arises or not. FTS is capable of maintaining sta-
bility and performance despite the occurrence of faults in chaotic
systems.
Moreover, since the gyro has been utilized to describe modes
in navigational, aeronautical or space engineering, the chaos syn-
chronization procedure for chaotic gyroscope systems is important
and this study may have practical applications in the future.
Therefore, FTS of chaotic gyroscope systems versus external
disturbances via Lyapunov rule-based fuzzy control is investigated
in this study. Since fuzzy control schemes have been successfully
deployed in many applications, we choose this control method
for this study. In this study, on the basis of Lyapunov stability
theory and fuzzy rules, the nonlinear controller and some generic
sufficient conditions for global asymptotic synchronization are
obtained. The fuzzy rules are directly constructed subject to a
common Lyapunov function such that the error dynamics of two
identical chaotic motions of symmetric gyros satisfy stability in the
Lyapunov sense.
The goal of this paper is to synchronize two chaotic heavy sym-
metric gyroscope systems versus actuator faults and external dis-
turbances. To achieve this goal, Lyapunov rule-based fuzzy control
is applied as the active fault tolerant control. Also, an observer is
designed, assuming that the nonlinear term satisfies a Lipschitz
condition. In addition, the results of this paper may be extended
to synchronize many classes of nonlinear chaotic systems.
The aforementioned Refs. [27,28] have a different viewpoint on
the subject of FTS to ours. In our work, this subject is considered
as an online procedure and the fault occurs in the slave system.
We attempt to design a convenient and robust controller. Also,
the controller is easy to implement. On the other hand, since
the gyro has been utilized to describe modes in navigational,
aeronautical and space engineering, the FTS procedure for chaotic
gyroscope systems is important and this study may have practical
applications in the future.
This paper is organized as follows. In Section 2, the dynamics
of a heavy symmetric gyroscope system and the chaos synchro-
nization problem are explained. In Section 3.1, fault tolerant
control and synchronization are introduced. The scheme of fault
tolerant synchronization is presented in Section 3.2. In Section 3.3,
classic Lyapunov-based control is designed for chaos synchroniza-
tion. Then, Lyapunov rule-based fuzzy control is designed for chaos
synchronization in 3.4. In Section 3.5, various forms of slave system
as regards actuator faults and external disturbances are presented.
An appropriate observer is designed, and also residuals are gener-
ated in this section. In Section 3.6, the stability of FTS via Lyapunov

52 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64
rule-based control is presented. A theorem concerning the state es-
timator, actuator faults and external disturbances is presented and
proven to verify the stability of the system under these conditions.
Finally, simulations are presented in Section 4, to show the effec-
tiveness of the proposed FTS methods for gyroscope systems versus
disturbances. At the end, the paper is concluded in Section 5.
2. System description and the synchronization problem
The dynamics of a heavy symmetric gyroscope with linear-
plus-cubic damping of angle θ mounted on a vibrating base can
be described in terms of Euler’s angles θ, φ and ψ. The vibration
dynamics of the base can be described in terms of the multiple
harmonic motion
∑n
k=1 Ak sin ωkt. sin x. Let the state variables be
x = [θ ˙θ ˙φ]T
; then the dynamic equations can be obtained
as [11]
˙x1 = x2
˙x2 = −
(βφ − βψ cos x1)(Bψ − βφ cos x1)
I2
1
sin3
x1
−
c1
I1
x2 +
Mg l
I1
sin x1 −
Mg A
I1
sin ωt sin x1
˙x3 = −2x2x3
cos x1
sin x1
+
βψ
I1 sin x1
x2
(1)
where I1 and I3 are the polar and equatorial moments of inertia
of the typical gyroscope, Mg is the gravity force, and l is the
distance between the center of gravity and O. It is clear that
coordinates θ and φ are cyclic, which provides the conjugate
momenta. The momentum integrals are βφ and βψ . In order to
simplify the following procedure, two nonlinear functions are
defined as follows:
p(x1, x2) = −
(βφ − βψ cos x1)(Bψ − βφ cos x1)
I2
1
sin3
x1
−
c1
I1
x2 +
Mg l
I1
sin x1 −
Mg A
I1
sin ωt sin x1 (2)
q(x1, x2, x3) = −2x2x3
cos x1
sin x1
+
βψ
I1 sin x1
x2. (3)
The gyroscope system, Eq. (1), performs complex dynamics and
has been extensively studied by Ge [11], with specific values set
as follows: βφ = 2, βψ = 5, l = 0.25, I1 = 1, Mg = 4, c1 =
0.5, ω = 2, A = 50.
In the numeric simulational, the dynamic behavior gyroscope
system, Eq. (1), exhibits a chaotic motion as shown in Fig. 3(a) with
initial conditions (x1, x2, x3) = (1, −1, 1).
Consider two coupled, chaotic heavy symmetric gyro systems,
where the master and slave systems are denoted by x and y,
respectively. The master system is shown in Eq. (1) and the slave
system is presented as follows:
˙y1 = y2
˙y2 = p(y1, y2) + u1(t)
˙y3 = q(y1, y2, y3) + u2(t)
(4)
where u1, u2 ∈ R are the control inputs attached to the nonlinear
gyroscope system.
The control problem considered in this study is that of different
initial conditions of the master and slave systems to be synchro-
nized by designing appropriate input controls.
We define synchronization errors between the master and slave
systems as follows:

e1 = y1 − x1
e2 = y2 − x2
e3 = y3 − x3.
(5)
The synchronization error dynamics can be obtained as follows:

˙e1(t) = e2(t)
˙e2(t) = p(y1, y2) − p(x1, x2) + u1(t)
˙e3(t) = q(y1, y2, y3) − q(x1, x2, x3) + u2(t).
(6)
The objective of the current control problem is to design the
appropriate control inputs u1(t) and u2(t) such that for any initial
conditions of the master and slave systems, the synchronization
errors converge to zero such that the resulting synchronization
error vector satisfies Eq. (7):
lim
t→∞
‖y(t) − x(t)‖ → 0 (7)
where ‖·‖ is the Euclidean norm of a vector.
3. Fault tolerant synchronization of chaotic gyroscopes versus
disturbances
3.1. Fault tolerant control and synchronization
As previous mentioned, fault diagnosis and isolation (FDI), and
fault tolerant control (FTC) have been active research topics during
the past two decades with a view to increasing the safety and
reliability of dynamical systems [18,19]. Faults normally occur in
two places: the actuators and the sensors. Actuator faults are faults
that act on the system, resulting in the deviation of the process
variables. The result is that the command (control) signal sent
to this device has no effect. Sensor faults are faults that act on
the sensors that measure the system variables, and they do not
directly affect the process. The source of these faults could be wear
and tear of the sensor leading to inaccurate readings, or a total
failure of the sensor. Sensor faults could also arise from drift/poor
calibration/extreme process conditions. These faults will affect the
process if the output measurements are used to generate the input
control signal [20].
FTC is an area of research that aims to increase applicability by
specifically designing control algorithms capable of maintaining
stability and performance despite the occurrence of faults. When
faults happen in chaotic systems, the conventional methods for
synchronizing the chaotic systems are invalid. In order to increase
the safety and the reliability of synchronization systems when a
fault happens, fault tolerant synchronization (FTS) is essential to
guarantee the synchronization of chaotic systems. The objective
of fault tolerant synchronization (FTS) is to design a fault tolerant
controller to achieve the synchronization no matter whether a fault
occurs or not. To the best of the authors’ knowledge, the problem
of FTS is still an open problem and seldom researched [27,28].
In this study, the actuator faults are considered as intermittent
faults and ramp faults. These occur in the slave system. Also,
the external disturbances happen in the slave system, as shown
in Fig. 1. The proposed method is the active fault tolerant
synchronization. The disturbances are assumed to be bounded as
follows: |d(t)| ≤ γ , where γ is a positive constant value. Also,
the actuator faults assumed to be bounded as follows: |fa(t)| ≤ β,
where β is a positive constant value which is the maximum power
of actuator faults happening in the slave system. The actuator faults
and external disturbances considered in this study are shown in the
simulation section.
3.2. Description of fault tolerant synchronization of chaotic gyro-
scopes versus disturbances via classic Lyapunov-based control and
Lyapunov rule-based fuzzy control
The scheme of fault tolerant synchronization of chaotic gyro-
scope systems versus disturbances via Lyapunov rule-based fuzzy
control is shown in Fig. 1. First, Lyapunov-based control (classic
nonlinear control) is designed for chaos synchronization of chaotic
gyroscope systems in Section 3.3. Second, Lyapunov rule-based

Fig. 1. The scheme of fault tolerant synchronization of chaotic gyroscopes versus disturbances.
fuzzy control (classic intelligent control) is designed for chaos syn-
chronization of chaotic gyroscope systems in Section 3.4. Then, it is
assumed that the states of slave systems are not available. An ap-
propriate observer is designed for the linear part of the slave sys-
tem in Section 3.5. Therefore, Lyapunov rule-based fuzzy control is
designed for fault tolerant synchronization of the master and slave
systems, considering the actuator faults, disturbances, and a state
estimator. A theorem and its proof for the stability of fault tolerant
synchronization of chaotic gyroscope systems under the aforemen-
tioned conditions are presented at the end of Section 3.6.
3.3. Classic Lyapunov-based control for synchronization of chaotic
gyroscopes
In this section, classic nonlinear control based on Lyapunov sta-
bility theory is designed for chaos synchronization of chaotic gyro-
scope systems.
To solve the chaos synchronization problem presented in
Eq. (6), define a Lyapunov function as follows:
V =
1
2

e2
1 + e2
2 + e2
3

. (8)
We differentiate Eq. (8) with respect to time:
˙V = e1 ˙e1 + e2 ˙e2 + e3 ˙e3. (9)
We substitute Eq. (6) into Eq. (9):
˙V = e2 (e1 + ˙e2)
  
A
+ e3 ˙e3

B
. (10)
The corresponding requirement for Lyapunov stability is [34]
˙V < 0. (11)
If A < 0 and B < 0, then the Lyapunov stability conditions
will be satisfied. The following cases will satisfy all the stability
conditions. This discussion is separately presented for A and B
terms.
Case 1 for A:
If e2 > 0, then e1 + ˙e2 < 0. (12)
Substituting Eq. (6) into Eq. (12), we obtain
u1(t) < −e1 − p(y1, y2) + p(x1, x2). (13)
A solution of Eq. (13) is considered:
u∗
1−A1
(t) = −e1 − p(y1, y2) + p(x1, x2) − λ (14)
where λ is a positive constant value.
Case 2 for A:
If e2 < 0, then e1 + ˙e2 > 0. (15)
u1(t) > −e1 − p(y1, y2) + p(x1, x2). (16)
u∗
1−A2
(t) = −e1 − p(y1, y2) + p(x1, x2) + λ (17)
Case 3 for A:
For e1 > 0 and e2 ∈ zero,
e2 (e1 + ˙e2) = −η |e2| (18)
where η is a positive constant value. Substituting Eq. (6) into
Eq. (18), we simplify as follows:
u1(t) < −ηsgn(e2) − p(y1, y2) + p(x1, x2). (19)
u∗
1−A3
(t) = −ηsgn(e2) − p(y1, y2) + p(x1, x2) − λ (20)
Case 4 for A:
For e1 < 0 and e2 ∈ zero,
e2 (e1 + ˙e2) = −η |e2| (21)
u1(t) > −ηsgn(e2) − p(y1, y2) + p(x1, x2). (22)
u∗
1−A4
(t) = −ηsgn(e2) − p(y1, y2) + p(x1, x2) + λ (23)
If the system satisfies the conditions of Cases 1–4 then A < 0
and the error state will be asymptotically driven to zero.
Case 5 for B:
If e3 > 0, then ˙e3 < 0. (24)
u2(t) < −q(y1, y2, y3) + q(x1, x2, x3). (25)

u∗
2−B1
(t) = −q(y1, y2, y3) + q(x1, x2, x3) − λ (26)
Case 6 for B:
If e3 < 0, then ˙e3 > 0. (27)
u2(t) > −q(y1, y2, y3) + q(x1, x2, x3). (28)
u∗
2−B2
(t) = −q(y1, y2, y3) + q(x1, x2, x3) + λ (29)
If the system satisfies the conditions of Cases 5 and 6 then B < 0
and the error state will be asymptotically driven to zero. Therefore,
the Lyapunov-based control is as follows:
u1(t) =



−e1 − p(y1, y2) + p(x1, x2) − λ, e2 > 0
−e1 − p(y1, y2) + p(x1, x2) + λ, e2 < 0
−ηsgn(e2) − p(y1, y2) + p(x1, x2) − λ,
e1 > 0, e2 ∈ zero
−ηsgn(e2) − p(y1, y2) + p(x1, x2) + λ,
e1 < 0, e2 ∈ zero
(30)
u2(t) =

−q(y1, y2, y3) + q(x1, x2, x3) − λ, e3 > 0
−q(y1, y2, y3) + q(x1, x2, x3) + λ, e3 < 0
(31)
where λ and η are positive constant values.
3.4. Lyapunov rule-based fuzzy control for synchronization of chaotic
gyroscopes
A set of the fuzzy linguistic rules based on expert knowledge
are applied to design the control law for fuzzy logic control. To
overcome the trial-and-error tuning of the membership functions
and rule base, the fuzzy rules are directly defined such that
the error dynamics satisfies stability in the Lyapunov sense. The
basic fuzzy logic system is composed of five function blocks as
follows [35]. (1) A rule base contains a number of fuzzy if–then
rules. (2) A database defines the membership functions for the
fuzzy sets used in the fuzzy rules. (3) A decision-making unit
performs the inference operations on the rules. (4) A fuzzification
interface transforms the crisp inputs into degrees of match with a
linguistic value. (5) A defuzzification interface transforms the fuzzy
results for the inference into a crisp output.
The fuzzy rule base consists of a collection of fuzzy if–then rules
expressed in the form of if a is A then b is B, where a and b denote
linguistic variables, and A and B represent linguistic values that are
characterized by membership functions. All of the fuzzy rules can
be used to construct the fuzzy associated memory.
In this study, the two FLC are designed as follows. The signal
(e1, e2) in Eq. (6) is the antecedent part of the proposed FLC1 for
designing the control input u1 that will be used in the consequent
part of the proposed FLC1:
u1 = FLC1(e1, e2). (32)
The signal e3 in Eq. (6) is the antecedent part of the proposed
FLC2 for designing the control input u2 that will be used in the
consequent part of the proposed FLC2:
u2 = FLC2(e3) (33)
where the FLCs accomplish the objective of stabilizing the error
dynamics Eq. (6). The ith if–then rule of the fuzzy rule base of the
FLC1 is of the following form:
Rule i: if e1 is X1 and e2 is X2 then u1−Li ≡ f1−i(e1, e2) (34)
Table 1
Rule table for FLC1.
Rule Antecedent Consequent
e1 e2 u1−Li
1 P P u1−L1 = −e1 − p(y1, y2) + p(x1, x2) − λ
2 P Z u1−L2 = −ηsgn(e2)−p(y1, y2)+p(x1, x2)−λ
3 P N u1−L3 = −e1 − p(y1, y2) + p(x1, x2) + λ
4 Z P u1−L4 = −e1 − p(y1, y2) + p(x1, x2) − λ
5 Z Z u1−L5 = zero
6 Z N u1−L6 = −e1 − p(y1, y2) + p(x1, x2) + λ
7 N P u1−L7 = −e1 − p(y1, y2) + p(x1, x2) − λ
8 N Z u1−L8 = −ηsgn(e2)−p(y1, y2)+p(x1, x2)+λ
9 N N u1−L9 = −e1 − p(y1, y2) + p(x1, x2) + λ
Table 2
e3 u2−Li
1 P u2−L1 = −q(y1, y2, y3)+q(x1, x2, x3)−λ
2 Z u2−L2 = zero
3 N u2−L3 = −q(y1, y2, y3)+q(x1, x2, x3)+λ
where X1 and X2 are the input fuzzy sets, and u1−Li is the out-
put which is the analytical function f1−i(.) of the input variables
(e1, e2). For given input values of the process variables, their de-
grees of membership µxi, i = 1, 2, . . . , n, called rule-antecedent
weights, are calculated. The centroid defuzzifier evaluates the out-
put of all rules as follows:
u1 =
n∑
i=1,µ̸=0
µ1−i.u1−Li
n∑
i=1,µ̸=0
µ1−i
, µ1−i = µX1(e1)µX2(e2). (35)
u2 =
n∑
i=1,µ̸=0
µX (e3).u2−Li
n∑
i=1,µ̸=0
µX (e3)
. (36)
u2−Li is the output which is the analytical function f2−i(.) of
the input variable (e3). Its degrees of membership µx, called rule-
antecedent weights, are calculated to give the input values of the
process variables.
Table 1 lists the fuzzy rule base in which the input variables
in the antecedent parts of the rules are e1 and e2, and the output
variable in the consequent is u1−Li. Table 2 lists the fuzzy rule
base in which the input variable in the antecedent parts of the
rules is e3, and the output variable in the consequent is u2−Li. We
use P, Z and N as input fuzzy sets representing ‘positive’, ‘zero’
and ‘negative’, respectively. The Gaussian membership function is
considered. The combination of the two input variables (e1, e2)
forms n = 9 heuristic rules in Table 1 and each rule belongs to one
of the three fuzzy sets P, Z and N. The input variable (e3) forms
n = 3 heuristic rules in Table 2 and each rule belongs to one of the
three fuzzy sets P, Z and N. The rules in Tables 1 and 2 are read as,
taking rule 1 in Table 1 as an example, ‘rule 1: if input 1 e1 is P and
input 2 e2 is P, then the output is u1−L1’.
and the error state will be asymptotically driven to zero. In order
to achieve this result, we will design u1−L by using the rules in
Table 1 for FLC1. According to the stability analysis method [34],
only those fuzzy subsystems that correspond to each rule need to
be considered. From Table 1, it can be seen that (e1, e2) belong to
fuzzy sets P, Z and N. The heuristic rules in Table 1 are divided
into five parts in order to discuss the stability of each subsystem.
For rules 1, 4 and 7 in Table 1, the error state e2 is positive, and

u1−L1 = u1−L4 = u1−L7 = u∗
1−A1. For rules 3, 6 and 9 in Table 1, the
error state e2 is negative, and u1−L3 = u1−L6 = u1−L9 = u∗
1−A2. For
rule 2 in Table 1, the error state e1 is positive and e2 is zero, and
u1−L2 = u∗
1−A3. For rule 8 in Table 1, the error state e1 is negative
and e2 is zero, and u1−L8 = u∗
1−A4. For rule 5 in Table 1, the error
states e1 and e2 are zeros. This condition is included in the other
rules, and we define u1−L5 = 0.
If the system satisfies the conditions of Cases 5 and 6 then B < 0
and the error state will be asymptotically driven to zero. In order to
achieve this result, we will design u2−L by using the rules of Table 1
for FLC2.
According to the stability analysis method [34], only the fuzzy
subsystems that correspond to each rule need to be considered.
From Table 2, it can be seen that e3 belongs to fuzzy sets P, Z and N.
The heuristic rules in Table 2 are divided into three parts in order to
discuss the stability of each subsystem. For rules 1, the error state
e3 is positive, and u2−L1 = u∗
2−B1. For rules 3, the error state e3 is
negative, and u2−L3 = u∗
2−B2. For rule 2 in Table 2, the error state e3
is zero. This condition is included in the other rules, and we define
u2−L2 = 0.
Therefore, all of the rules in FLC1 and FLC2 can lead to
Lyapunov stable subsystems under the same Lyapunov function,
Eq. (8). Furthermore, the closed-loop rule-based system Eq. (6) is
asymptotically stable for each derivative of the Lyapunov function
that satisfies ˙V < 0 in Tables 1 and 2. That is, the error states
guarantee convergence to zero and the system of two nonlinear
gyros is synchronized.
3.5. Design of an observer and residuals
Consider the slave system when the actuator faults and external
disturbances are added to the system. It is represented as follows:



˙y1 = y2
˙y2 = p(y1, y2) + u1(t) + Fa1
(t) + d1(t)
= −
c1
I1
y2 + p(y1, 0) + u1(t) + Fa1
(t) + d1(t)
˙y3 = q(y1, y2, y3) + u2(t) + Fa2
(t) + d2(t)
Yout = C

y1 y2 y3
T
(37)
where C = I3×3. Fa1
and Fa2
are actuator faults. d1 and d2 are
external disturbances. The slave system is regarded as consisting
of two parts: the linear and nonlinear parts. The linear parts of Eq.
(30) are as follows:
A =



0 1 0
0 −
c1
I1
0
0 0 0


 ; B =

0
1
1

; C =

1 0 0
0 1 0
0 0 1

. (38)
Notice that the pair (A, C) is observable. A Luenberger observer
is designed for the linear part of Eq. (38):

˙ˆY = A ˆY + Bu + L(Yout − ˆYout)
Yout = C ˆY
(39)
where ˆY = [ˆY1
ˆY2
ˆY3]T
. The observer gain L is selected such
that the eigenvalues of A0 = A − LC are located in the left half-
plane. We define the estimated errors as follows:
eo = Yout − ˆYout. (40)
Hence the dynamics of observer error is
˙eo = AoCeo. (41)
Since Ao is stable, for any given positive-definite matrix Qo,
there exists a unique positive-definite Po such that the Lyapunov
equation is established [36]:
AT
o Po + PoAo = −Qo. (42)
In this study, L is calculated from pole-placement method.
A residual is defined as follows:
R(t) = Yout(t) − ˆYout(t). (43)
From Fig. 4, the estimated synchronization error is defined as



es1
= ˆY1 − x1
es2
= ˆY2 − x2
es3
= ˆY3 − x3.
(44)
Notice that Eq. (44) will be used for designing the input controls
via Lyapunov rule-based fuzzy control. These are considered as
input variables for FLC1 and FLC2.
3.6. Stability of fault tolerant synchronization of chaotic gyroscope
systems
In this section, a theorem is proven for the stability of fault
tolerant synchronization of chaotic gyroscope systems, concerning
the state estimator, actuator faults and external disturbances via
Lyapunov rule-based fuzzy control.
Theorem. Consider the master and the slave chaotic gyroscope sys-
tems presented by Eqs. (1) and (37). Actuator faults and disturbances
are occurring in the slave system. The states of the slave system are
unavailable and the state estimator presented by Eq. (39) is utilized
to observe the states of the slave system. The estimated synchroniza-
tion errors are represented by Eq. (44). Then, FTS is achieved by two
Lyapunov rule-based fuzzy controls as follows:
u1 = FLC1(es1, es2) and u2 = FLC2(es3)
where es1, es2, and es3 presented by Eq. (44) are the antecedent parts of
the FLCs and the control inputs (u1 and u2) are the outputs of the FLCs.
Proof. The two FLCs accomplish the objective of stabilizing the
error dynamics of Eq. (36).
The ith if–then rule of the fuzzy rule base of the FLC1 and FLC2
are of the following forms:
Rule i: if es1 is X1 and es2 is X2 then u1−Li ≡ f1−i(es1, es2) (45)
Rule i: if es3 is X1 then u2−Li ≡ f2−i(es3) (46)
where Xi are the input fuzzy sets, u1−Li and u2−Li are the outputs
which are the analytical functions f1−i(.) and f2−i(.) of the input
variables (es1, es2) and (es3).
In the ‘‘3.6.1: Consequent parts of the FLCs’’ subsection, u1−Li
and u2−Li are obtained.
For given input values of the process variables, their degrees of
membership µxi, i = 1, 2, . . . , n, called rule-antecedent weights,
are calculated. The centroid defuzzifier evaluates the output of all
rules as follows:
u1 =
n∑
i=1,µ̸=0
µ1−i.u1−Li
n∑
i=1,µ̸=0
µ1−i
, µ1−i = µX1(es1)µX2(es2) (47)
u2 =
n∑
i=1,µ̸=0
µX (es3).u2−Li
n∑
i=1,µ̸=0
µX (es3)
. (48)
Table 3 lists the fuzzy rule base in which the input variables in
the antecedent parts of the rules are es1 and es2, and the output
variable in the consequent is u1−Li. Table 4 lists the fuzzy rule
base in which the input variable in the antecedent part of the

Fig. 2a. Actuator faults: intermittent faults.
Fig. 2b. Actuator faults: ramp faults.
Fig. 2c. External disturbances.
rules is es3, and the output variable in the consequent is u2−Li. We
use P, Z and N as input fuzzy sets representing ‘positive’, ‘zero’
and ‘negative’, respectively. The Gaussian membership function is
considered. Notice that λ = γ +β+k, where k is a positive constant
value.
Therefore, all of the rules in the FLC1 and FLC2 can lead to
Lyapunov stable subsystems under the same Lyapunov function as
follows:
V =
1
2

e2
s1 + e2
s2 + e2
s3

. (49)
The closed-loop rule-based system is asymptotically stable for
each derivative of the Lyapunov function that satisfies ˙V < 0 in
Tables 3 and 4.
Notice that according to the Lyapunov equation presented by
Eq. (42), the estimated errors presented by Eq. (40) will converge
to zero.
Therefore, it is guaranteed that the error states converge to zero
and fault tolerant synchronization is achieved.
3.6.1. Consequent parts of the FLCs
In this subsection, u1−Li and u2−Li are obtained.

(a) Master: X. (b) Slave: Y. (c) Estimated slave: Yhat.
Fig. 3. Classic Lyapunov-based control: time responses of the master, slave and estimated slave systems, when disturbances and intermittent actuator faults occur.
(a) Synchronization error: y − x. (b) Estimation error: y − yhat. (c) Error: yhat − x.
Fig. 4. Classic Lyapunov-based control: synchronization error, estimation error and estimated synchronization error, when disturbances and intermittent actuator faults
occur.
Table 3
es1 es2 u1−Li
1 P P u1−L1 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ
2 P Z u1−L2 = −ηsgn(es2)−p(ˆY1, ˆY2)+p(x1, x2)−λ
3 P N u1−L3 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ
4 Z P u1−L4 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ
5 Z Z u1−L5 = zero
6 Z N u1−L6 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ
7 N P u1−L7 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ
8 N Z u1−L8 = −ηsgn(es2)−p(ˆY1, ˆY2)+p(x1, x2)+λ
9 N N u1−L9 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ
As presented by Eq. (44), the estimated errors are as follows:



es1
= ˆY1 − x1
es2
= ˆY2 − x2
es3
= ˆY3 − x3.
(50)
Table 4
e3 u2−Li
1 P u2−L1 = −q(ˆY1, ˆY2, ˆY3)+q(x1, x2, x3)−λ
2 Z u2−L2 = zero
3 N u2−L3 = −q(ˆY1, ˆY2, ˆY3)+q(x1, x2, x3)+λ
The dynamics of Eq. (50) is obtained from Eqs. (1) and (37):



˙es1(t) = es2(t)
˙es2(t) = p(ˆY1, ˆY2) − p(x1, x2) + u1(t) + Fa1
(t) + d1(t)
˙es3(t) = q(ˆY1, ˆY2, ˆY3) − q(x1, x2, x3) + u2(t) + Fa2
(t) + d2(t).
(51)
We define a Lyapunov function as follows:
V =
1
2

e2
s1 + e2
s2 + e2
s3

. (52)
We differentiate Eq. (52) with respect to time:
˙V = es1 ˙es1 + es2 ˙es2 + es3 ˙es3. (53)

Fig. 5. Classic Lyapunov-based control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and intermittent actuator faults occur.
Fig. 6. Classic Lyapunov-based control: time responses of the master, slave and estimated slave systems, when disturbances and ramp actuator faults occur.
We substitute Eq. (51) into Eq. (53):
˙V = es2 (es1 + ˙es2)
  
A
+ es3 ˙es3
  
B
. (54)
The corresponding requirement for Lyapunov stability is [34]
˙V < 0. (55)
If A < 0 and B < 0, then the Lyapunov stability will be satisfied.
The following cases will satisfy all the stability conditions. This
discussion is separately presented for A and B terms.
Case 1 for A:
If es2 > 0, then es1 + ˙es2 < 0. (56)
u1(t) < −es1 − p(ˆY1, ˆY2) + p(x1, x2) − Fa1
(t) − d1(t). (57)
u∗
1−A1
(t) = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ (58)
where λ = γ + β + k and k is a positive constant value.
Case 2 for A:
If e2 < 0, then es1 + ˙es2 > 0. (59)
u1(t) > −es1 − p(ˆY1, ˆY2) + p(x1, x2) − Fa1
(t) − d1(t). (60)
u∗
1−A2
(t) = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ (61)
Case 3 for A:
For es1 > 0 and es2 ∈ zero,

Fig. 7. Classic Lyapunov-based control: synchronization error, estimation error and estimated synchronization error, when disturbances and ramp actuator faults occur.
Fig. 8. Classic Lyapunov-based control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and ramp actuator faults occur.
es2 (es1 + ˙es2) = −η |es2| (62)
u1(t) < −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − Fa1
(t) − d1(t). (63)
u∗
1−A3
(t) = −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − λ (64)
Case 4 for A:
For es1 < 0 and es2 ∈ zero,
es2 (es1 + ˙es2) = −η |es2| (65)
where η is a positive constant value. Substituting Eq. (51) into Eq.
(65), we simplify as follows:
u1(t) > −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − Fa1
(t) − d1(t). (66)
u∗
1−A4
(t) = −ηsgn(e2) − p(ˆY1, ˆY2) + p(x1, x2) + λ (67)
Case 5 for B:
If es3 > 0, then ˙es3 < 0. (68)
u2(t) < −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − Fa2
(t) − d2(t). (69)
u∗
2−B1
(t) = −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − λ (70)
Case 6 for B:
If es3 < 0, then ˙es3 > 0. (71)

Fig. 9. Lyapunov rule-based fuzzy control: time responses of the master, slave and estimated slave systems, when disturbances and intermittent actuator faults occur.
Fig. 10. Lyapunov rule-based fuzzy control: synchronization error, estimation error and estimated synchronization error, when disturbances and intermittent actuator
faults occur.

Fig. 11. Lyapunov rule-based fuzzy control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and intermittent actuator faults occur.
Fig. 12. Lyapunov rule-based fuzzy control: time responses of the master, slave and estimated slave systems, when disturbances and ramp actuator faults occur.
u2(t) > −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − Fa2
(t) − d2(t). (72)
u∗
2−B2
(t) = −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) + λ (73)
If the system satisfies the conditions of Cases 5, 6 then B < 0
In Table 3, for rules 1, 4 and 7 in Table 1, the error state es2 is
positive, and u1−L1 = u1−L4 = u1−L7 = u∗
1−A1. For rules 3, 6 and
9 in Table 3, the error state es2 is negative, and u1−L3 = u1−L6 =
u1−L9 = u∗
1−A2. For rule 2 in Table 3, the error state es1 is positive
and es2 is zero, and u1−L2 = u∗
1−A3. For rule 8 in Table 3, the error
state es1 is negative and es2 is zero, and u1−L8 = u∗
1−A4. For rule 5
in Table 3, the error states es1 and es2 are zeros. This condition is
included in the other rules, and we define u1−L5 = 0. In Table 4, for
rules 1, the error state es3 is positive, and u2−L1 = u∗
2−B1. For rules
3, the error state es3 is negative, and u2−L3 = u∗
2−B2. For rule 2 in
Table 4, the error state es3 is zero. This condition is included in the
other rules, and we define u2−L2 = 0.
4. Simulation results
In this section, numerical simulations are given to demonstrate
the FTS of the chaotic gyros versus disturbances via Lyapunov-
based control and Lyapunov rule-based fuzzy control.
The utilized actuator faults and disturbances are introduced in
Section 3.1. d1 is attached between 3 < t < 4 and 7 < t < 8. d2 is
attached between 1 < t < 2 and 5 < t < 6. The actuator faults in
this study are shown in Figs. 2a–2c.
The parameters of the nonlinear chaotic gyroscope systems are
specified in Section 2. The initial conditions of the master and
slave systems are defined as follows: [x1(0)x2(0)x3(0)] = [1 −
1 1], [y1(0)y2(0)y3(0)] = [2 − 2 − 4]. The initial conditions of the
observer are defined as follows: [yO1(0)yO2(0)yO3(0)] = [1.5 1 2].
4.1. Simulation results for Lyapunov-based control
Figs. 3–5 are related to Lyapunov-based control when intermit-
tent actuator faults and disturbances occur. The time responses
of the master, slave and estimated slave systems are shown in
Fig. 3((a)–(c)). The synchronization error, estimation error and es-
timated synchronization error are shown in Fig. 4((a)–(c)). The er-
rors illustrated in Fig. 4 converge asymptotically to zero. Notice

Fig. 13. Lyapunov rule-based fuzzy control: synchronization error, estimation error and estimated synchronization error, when disturbances and ramp actuator faults occur.
that the residuals are the estimation errors shown in Fig. 4. Fig. 5
shows the control inputs obtained via Lyapunov-based control.
Figs. 6–8 are related to Lyapunov-based control when ramp
actuator faults and disturbances occur. The time responses of
the master, slave and estimated slave systems are shown in
Fig. 6((a)–(c)). The synchronization error, estimation error and
estimated synchronization error are shown in Fig. 7((a)–(c)).
Obviously, the errors illustrated in Fig. 7 converge asymptotically
to zero. Notice that the residuals are the estimation errors shown in
Fig. 7. Fig. 8 shows the control inputs obtained via Lyapunov rule-
based fuzzy control.
As shown in these simulation results (Figs. 3–8), the Lyapunov-
based control method is not robust. Also, it cannot be useful for
fault tolerant synchronization of chaotic gyroscope systems. The
synchronization errors (Figs. 4 and 7) are bounded, but they do not
converge to zero.
4.2. Simulation results for Lyapunov rule-based fuzzy control
Figs. 9–11 are related to Lyapunov rule-based fuzzy control
when intermittent actuator faults and disturbances occur. The
time responses of the master, slave and estimated slave sys-
tems are shown in Fig. 9((a)–(c)). The synchronization error, es-
timation error and estimated synchronization error are shown in
Fig. 10((a)–(c)). The errors illustrated in Fig. 10 converge asymp-
totically to zero. Notice that the residuals are the estimation er-
rors shown in Fig. 10. Fig. 11 shows the control inputs obtained via
Lyapunov rule-based fuzzy control.
Figs. 12–14 are related to Lyapunov-based control when ramp
actuator faults and disturbances occur. The time responses of
the master, slave and estimated slave systems are shown in
Fig. 12((a)–(c)). The synchronization error, estimation error and
estimated synchronization error are shown in Fig. 13((a)–(c)).
Obviously, the errors illustrated in Fig. 13 converge asymptotically
to zero. Notice that the residuals are the estimation errors shown
in Fig. 13. Fig. 14 shows the control inputs obtained via Lyapunov
rule-based fuzzy control.
The simulation results for FTS via Lyapunov rule-based fuzzy
control show good performances and confirm that the master
and slave systems achieve the synchronized states, when actuator
faults and disturbances occur.
From the simulation results, it is shown that the Lyapunov
rule-based fuzzy control is better than the classic Lyapunov-based

Fig. 14. Lyapunov rule-based fuzzy control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and ramp actuator faults occur.
control. It is robust and more capable than the classic control for
FTS.
Also, these results demonstrate that by using Lyapunov rule-
based fuzzy control, the synchronization error states are regulated
to zero asymptotically. It is clear that the proposed method
(Lyapunov rule-based fuzzy control) is capable of achieving FTS,
when actuator faults and disturbances occur.
5. Conclusion
The fault tolerant synchronization of chaotic gyroscope systems
via Lyapunov rule-based fuzzy control has been addressed. Two
techniques are compared: classic Lyapunov-based control and
Lyapunov rule-based fuzzy control. In this study, a slave observer
is designed to estimate the states of the slave system with respect
to actuator faults and disturbances. Then, classic Lyapunov-based
control and Lyapunov rule-based fuzzy control are applied in
the fault tolerant synchronization. The global asymptotic stability
of the synchronization errors is investigated. Finally, simulation
results show the effectiveness of the proposed method. Also,
the Lyapunov rule-based fuzzy control can compensate for the
actuator faults and disturbances occurring in the slave system.
The proposed method is considered as an online procedure and
the faults occur in the slave system. We tried to design a convenient
and robust controller. Also, the controller is easy to implement.
Since the gyro has been utilized to describe modes in nav-
igational, aeronautical and space engineering, the fault tolerant
synchronization procedure in this study may have practical appli-
cations in the future.
We will try to extend the proposed method to fault tolerant
synchronization of many classes of nonlinear chaotic systems.
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