Chaos, Solitons and Fractals 167 (2023) 113092
Available online 7 January 2023
0960-0779/© 2023 Elsevier Ltd. All rights reserved.
An optimal robust fuzzy adaptive integral sliding mode controller based
upon a multi-objective grey wolf optimization algorithm for a nonlinear
uncertain chaotic system
M.J. Mahmoodabadi
Department of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran
A R T I C L E I N F O
Keywords:
Robust control
Integral sliding mode control
Multi-objective grey wolf optimization
Fuzzy logic systems
Uncertain chaotic systems
Duffing-Holmes oscillator
A B S T R A C T
This paper presents a novel robust fuzzy adaptive integral sliding mode controller designed via a multi-objective
grey wolf optimization algorithm for a nonlinear uncertain chaotic system. To this end, proper integral surfaces
are defined and a sliding mode stabilizer is designed to converge the system errors to zero. Next, the gradient
descent approach is employed to tune the design gains of the integral sliding mode controller. Then, fuzzy rules
are employed to regulate the coefficients of the proposed robust control approach. In order to acquire the proper
parameters of the proposed controller and avoid trial-and-error processes, a multi-objective grey wolf optimi­
zation algorithm is employed to enhance the performance of the proposed controller. The challenging case study
of a Duffing-Holmes oscillator, as a nonlinear autonomous uncertain chaotic system without stable equilibrium
points, is considered to assess the behavior of the suggests optimal robust fuzzy adaptive integral sliding mode
controller. The results of this study are compared with the outcomes of a distinguished work in the literature.
Lastly, the discussions elucidate the efficiency of the proposed controller with respect to uncertain chaotic
nonlinear systems in terms of optimal control inputs and minimum tracking error.
1. Introduction
In recent years, chaos, as an agent of instability and a source of vi­
bration generation, has been broadly studied [1–5]. In this way, Duffing-
Holmes oscillator, as a nonlinear uncertain chaotic system, has attracted
attentions of many scientists to challenge new ideas of control schemes.
To name but a few, Thenozhi and Concha [6] presented a backstepping-
based adaptive control design for bi-stable Duffing-Holmes oscillators.
The notion of virtual stabilizability for dynamical systems as well as the
virtual stabilizability of uncertain Duffing–Holmes control systems were
investigated by Sun [7]. A variable structure control design for a class of
uncertain chaotic systems subject to sector nonlinear inputs was
considered by Yan [8]. Karami-Mollaee and Barambones [9] studied
concerns the control of uncertain fractional-order multi-inputs chaotic
systems, using fractional dynamic sliding mode controller. Chang [10]
proposed an adaptive single neural controller for a class of uncertain
nonlinear chaotic systems subject to a nonlinear input. A simple robust
controller was developed for digital secure communication between two
different chaotic systems (Duffing-Holmes oscillator and chaotic gyro)
with uncertainties, external disturbances, and unknown parameters in a
finite time by Zirkohi [11]. A novel sliding mode nonlinear proportional-
integral control scheme was proposed for driving a class of time-variant
chaotic systems with uncertainty to arbitrarily desired trajectory with
high accuracy by Dong-Chuan et al. [12]. The mentioned literature re­
views clearly depict that a wide range of various control approaches
have been successfully utilized by researchers for stabilization or tra­
jectory tracking of complicated nonlinear chaotic dynamical systems.
However, employing the integral sliding mode control automatically
regulated by the adaptation laws and fuzzy rules which are optimized by
a multi-objective meta-heuristic algorithm is a novel idea that is
implemented for the Duffing-Holmes nonlinear uncertain chaotic oscil­
lator by this work.
Due to its robustness for handling structural and unstructural un­
certainties, sliding mode control is a popular nonlinear approach in
many fields of sciences and technologies such as robotics, mechanical
and electrical systems [13–19]. In order to enhance the abilities of the
sliding mode control, the outstanding attempts have been made by re­
searchers in the recent years as follows: Chu et al. [20] extended a
continuous terminal sliding mode controller based upon meta-cognitive
fuzzy neural networks for an active power filter. To assure the control­
ler’s performance, an integral-type terminal sliding surface was utilized
to force the tracking error to zero value in a finite time. Cuong et al. [21]
E-mail address: mahmoodabadi@sirjantech.ac.ir.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals
journal homepage: www.elsevier.com/locate/chaos
https://doi.org/10.1016/j.chaos.2022.113092
Received 15 March 2022; Received in revised form 23 December 2022; Accepted 28 December 2022

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
2
enhanced the core of the sliding mode control via combining with
fractional calculus and adaptive techniques. Indeed, an adaptive robust
control system for rubber-tired gantry cranes was constructed under
parametric variations and unknown winds. Fan and Wang [22] guar­
anteed the reachability of formulated surfaces by designing an event-
triggered sliding mode control law through a Lemma introduced to
deal with the caused difficulty. Song et al. [23] developed a sliding mode
control scheme for a class of stochastic hyperbolic partial differential
equation systems by employing a state observer for regarding unavai­
lable system states. Guo [24] proposed an adaptive sliding mode control
approach, compared it with its classical version, and confirmed that the
proposed method is superior in terms of chattering, overshoot and
response time. Wang et al. [25] proposed a design of iterative learning
control with an adaptive sliding mode approach in order to address the
problem of unknown periodic parameters for multi-input multi-output
nonlinear systems. In addition, a universal barrier Lyapunov function
was proposed to solve the constraint requirements under the time
domain and the iteration domain. Yang and Ding [26] employed an
event-triggered sliding mode control method for the discrete-time two-
dimensional systems shown by the Roesser model with time delays.
Further, the Lyapunov function method was used to drive the state
trajectories of the resultant closed-loop system into a bounded region
and maintain there for the subsequent time. Abdollahzadeh and
Esmailifar [27] utilized two sliding mode control approaches for the
rotational and translational control of a chaser spacecraft in the close
vicinity docking phase with a target subjected to external disturbances.
By analyzing the control gains via the Lyapunov function, the robustness
of the closed-loop system in the presence of external disturbances,
measurement noises and uncertainties was guaranteed. Ammar et al.
[28] proposed an estimation method for induction motors based upon a
coupling of sliding mode direct torque control with sliding mode flux
and speed observers. A sliding mode speed/flux observer was designed
to augment the control performances via using a sensorless derivative-
integral-algorithm to gain a precise estimation and enhance the reli­
ability of the system and decrease the cost of usage sensors. Kang et al.
[29] designed a second-order sliding mode controller with a propor­
tional surface and extended the state observers for a quad-rotor un­
manned aerial vehicle. Roy and Roy [30] compared the performance of
the conventional sliding mode control with fractional-order one via
applying both controllers for position handling of a ferromagnetic ball
against gravity in a magnetic levitation system. The results elucidated
that the fractional-order sliding mode control performs better than its
conventional version in the aspects of the speed of response, tracking
accuracy, chattering and control effort.
It could be evident from the above literature review that to provide
the best feasible performance of any controller, specially the sliding
mode controller, its coefficients must be properly regulated by using a
suitable mechanism such as fuzzy logic systems. Further, as a brief re­
view of implementation of the fuzzy logic systems, Liu et al. [31] pre­
sented a robust adaptive control strategy based upon the coupling of a
fuzzy logic system with sliding mode approach. In fact, a fuzzy logic
system was employed to approximate the unknown functions, and the
approximation errors were removed by utilizing an adaptive algorithm.
Zheng et al. [32] designed a double fuzzy robust scheme based upon the
combination of sliding mode control and fuzzy logic systems. The su­
periority of the proposed controller was demonstrated by its application
on a complex robot system with disturbances and comparing the results
with the outcomes of other existing approaches. Elsisi et al. [33] pro­
posed a hybrid control method based upon the fuzzy logic and nonlinear
sliding mode control in order to manage the energy of the distributed
controllable loads in a smart grid. The proposed controller handles the
system nonlinearities and enhances the damping characteristics of the
response against the uncertainties of the parameters, while its gains
were optimized via the imperialist competitive algorithm. Kutlu et al.
[34] proposed an intuitionistic fuzzy approach to find the parameters of
an adaptive integral sliding mode control technique for robust handling
of nonlinear systems. Liu et al. [35] developed a general fuzzy robust
controller via substituting an element of the sliding mode law by the
output of a type-2 fuzzy system to augment the anti-interference ability
of a power-line inspection robot for gaining motion balance control.
In this study, the rules of the fuzzy logic systems are chosen via
intelligent optimization algorithms to minimize the tracking error and
control effort, simultaneously. In this respect, particle swarm optimi­
zation [36–38], genetic algorithm [39–41], artificial bee colony
[42–44], ant colony optimization [45–47] as smart optimization algo­
rithms have been employed in the literature. However, hear, grey wolf
optimization, which is based upon the leadership hierarchy and hunting
mechanism of grey wolves in nature, as a new meta heuristic idea would
be used to select the control gains. Indeed, in the algorithm, three main
steps of hunting, which involve searching for prey, encircling prey, and
attacking prey, have been implemented to perform optimization. This
optimization algorithm has been successfully used by a number of re­
searchers to optimize stochastic problems [48–51], such as
manufacturing systems [52], an adaptive fuzzy logic controller [53], a
nonlinear model predictive controller [54], scheduling of workflows in
cloud computing environments [55] and feedstock selection of biomass
gasification [56]. More specifically, regarding the implementation of the
grey wolf optimization on sliding mode control, the following research
studies have been conducted. Zhou et al. [57] proposed a hybrid grey
wolf optimization algorithm to optimize the scheme of the sliding mode
control based on an extreme learning machine to address uncertainties
and external interference. Two-link manipulator simulation results
illustrated that the proposed sliding mode control scheme could gain
high-precision and high-speed tracking and could suppress chatterings.
Roy and Ghoshal [58] expanded the grey wolf-second order sliding
mode control to stabilize an inchworm robot manipulator. The control
capability was augmented and the chattering was diminished. Lately,
Rahmani et al. [59] presented a sliding mode method based on the
extended grey wolf optimizer to control a 2-DOF robot manipulator.
Two proportional derivative and sliding mode control approaches were
utilized and combined to benefit from the advantages of the both
controllers.
This paper has endeavored substantially to advance authors’ previ­
ous study [60] by means of employing integral sliding mode control and
designing its parameters via gradient descent based adaptation laws.
Besides, by using the linguistic variables, singleton fuzzifier, Mamdani
product inference engine and center average defuzzifier, an efficient
fuzzy logic based estimator is introduced to tune the controller gains.
Consequently, a multi-objective grey wolf optimization algorithm is
utilized to acquire the parameters of the fuzzy logic system to diminish
the tracking error and control inputs. The Duffing-Holmes oscillator as a
challenging case study was regarded to evaluate the performance of the
proposed controller. Based on the above statements, the novel contri­
butions of this research work could be summarized as follows:
• Designing an integral sliding mode controller for trajectory tracking
defined for nonlinear uncertain chaotic systems.
• Providing an automatic tuning scheme based on the gradient descent
approach for the control parameters.
• Introducing a fuzzy logic based system for online regulating of the
coefficients appeared in the adaptation laws.
• Presenting a Pareto frontier of non-commensurable objective func­
tions in the multi-criteria optimum design for the suggested control.
• Employing the optimized robust fuzzy adaptive integral sliding mode
controller for a nonlinear Duffing-Holmes oscillator as an autono­
mous uncertain chaotic system without stable equilibrium points.
The rest of this paper is organized as follows: Section 2 presents the
dynamical equations of the Duffing-Holmes uncertain chaotic system.
Section 3 provides the structure of the integral sliding mode control
employed in this study. The considered fuzzy logic system is designed in
Section 4. Section 5 provides the architecture of the multi-objective grey
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
3
wolf optimization algorithm. The results and discussions on the results
of the proposed optimal robust fuzzy adaptive controller are illustrated
in Section 6. Lastly, Section 7 presents the conclusions and potential
future works.
2. Duffing-Holmes uncertain chaotic system
The Duffing-Holmes system as a nonlinear dynamical oscillator can
show chaotic behaviors at certain conditions. Since it is a reduced form
of lots of practical models, it has great potential for applications in
several fields of studies [61]. A common form of its dynamical equations
could be written as follows [62].
ẋ1(t) = x2(t) (1a)
ẋ2(t) = − c1x1(t) − c2x2(t) − c3x3
1(t) − c4cos(ωt) + g(x, t) + D(x, t) + u(x, t)
(1b)
where ci, i = 1, 2, 3, 4 are the system constants and ω denotes the fre­
quency of the oscillation. Let consider the system uncertainty as g(x, t),
the external disturbance as D(x, t), and the control input as u(x, t). It
should be noted that there are positive upper bounds for the system
dynamics and uncertainty as fu
and gu
; i.e.
⃒
⃒ − c1x1(t) − c2x2(t) −
c3x3
1(t) − c4cos(ωt)
⃒
⃒ ≤ fu
and |g(x, t) | ≤ gu
. Moreover, maximum value
dm
is regarded for the absolute value of the external disturbance; i.e.
|D(t) | ≤ dm
. The control goal is defined in such a way that the output of
the system tracks the desired input considered as yd(t). In order to
illustrate the nonlinear and chaotic behavior of the above described
plant, the phase plane diagrams of the states are depicted in Fig. 1 for
different values of the frequency ω = 0.5, 1 and 1.5 as well as the
following characteristics: x1(0) = x2(0) = 0, c1 = − 1, c2 = 1, c3 = 1
and c4 = − 1, while g(x, t) = D(x, t) = u(x, t) = 0.
3. Integral sliding mode control
The sliding mode control as a variable structure control method
switches from one continuous structure to another based upon the
present position in the state space. Moreover, the adjustment of the
dynamics is conducted by the implementation of a discontinuous control
signal [63]. In the sliding mode control, the final trajectory is not inside
of one control structure, and the real trajectory will result from sliding
along the boundaries of the control structure.
By regarding a nonlinear system in the general state space, the
following equation can be written.
ẋ = f(x, u, t) (2)
where, xϵRn
is the state vector, uϵRm
represents the control input vector,
n is the order of the system, and m stands for the number of inputs. Here,
an integral sliding surface is introduced as follows.
s(e, t) =
{
e : HT
e = 0
}
(3)
where, HϵRn
shows the coefficients or the slope of the sliding surface.
Moreover, tracking error e, its derivative and integral for the Duffing
Holmes system are defined as follows.
e = xoutput − xd = x1 − xd, ė = x2 − ẋd,
∫
edt =
∫
x1dt −
∫
xddt (4)
Integral sliding surface s is defined by calculating the following
scalar equation.
s(e, t) =
(
d
dt
+ ∂1 + ∂2
∫
dt
)n− 1
e = 0 (5)
where, ∂1 and ∂2 are strictly positive constants, and n denotes the order
of the system. Therefore, sliding surface s for the second order Duffing
Holmes system could be formulated as follows.
s(e, t) =
(
x2 − ẋd
)
+ ∂1(x1 − xd) + ∂2
( ∫
x1dt −
∫
xddt
)
(6)
The second-order tracking problem is now being substituted by a
first-order stabilization problem, where scalar s must be kept at zero via
a governing reaching condition. By regarding the Lyapunov function as
V(x) = 1
2
s2
, the reaching condition would be guaranteed via the
following relation [64].
V̇(x) < − ηs (7)
The sliding mode of the system response is inclined to chatter along
s = 0. By regarding Eq. (9), the convergence and existence condition is
re-written as follows.
sṡ ≤ − ηs (8)
The existence of a non-switching region is guaranteed by Eq. (8).
Furthermore, η is a strictly positive constant chosen based on the
knowledge of disturbances and system dynamics [65]. The sliding
condition mentioned in Eq. (10) is always satisfied by having:
uSMC = ueq − κ sgn(s) (9)
where, ueq represents the equivalent control effort that would be found
by solving equation ṡ = 0. Moreover, κ stands for a design parameter.
Since function sign (sgn) makes the extreme frequency chattering in the
sliding mode control, and a discontinuity happens in the controller, a
thin boundary layer around the sliding surface is utilized through
substituting function sign by the saturation function formulized as fol­
lows.
sat
(s
φ
)
=
⎧
⎪
⎨
⎪
⎩
sgn
(s
φ
)
if ∣
s
φ
∣ ≥ 1
(s
φ
)
if ∣
s
φ
∣ < 1
(10)
Fig. 1. Phase portraits for the states of the Doffing-Holmes uncertain chaotic oscillator for different values of system frequency.
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
4
4. Gradient descent based adaptation laws
The adaptation laws, as automatic tuning methods for the controller
parameters, employ algorithms based on the identified or estimated
process model [66,67]. In this way, the gradient descent scheme as a
minimization technique is broadly utilized to adapt control gains [68].
Indeed, it minimizes functions by iteratively moving in the direction of
steepest descent via the negative of the gradient [69].
γ̇1 = − δ1
∂sṡ
∂γ1
= − δ1
∂sṡ
∂u
∂u
∂γ1
=
⎧
⎪
⎨
⎪
⎩
− δ1sė if ∣
s
φ
∣ ≥ 1
− δ1s
(
ė +
κ
φ
e
)
if ∣
s
φ
∣ < 1
(11)
γ̇2 = − δ2
∂sṡ
∂γ2
= − δ2
∂sṡ
∂u
∂u
∂γ2
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
− δ2se if ∣
s
φ
∣ ≥ 1
− δ2s
(
e +
κ
φ
∫
edt
)
if ∣
s
φ
∣ < 1
(12)
κ̇ = − δ3
∂sṡ
∂κ
= − δ3
∂sṡ
∂u
∂u
∂κ
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
− δ3s sgn
(s
φ
)
if ∣
s
φ
∣ ≥ 1
− δ3
s2
φ
if ∣
s
φ
∣ < 1
(13)
5. Fuzzy system
In order to regulate the coefficients of the adaptation laws (δ1,δ2 and
δ3), a fuzzy logic system is utilized in this study. Let consider that fuzzy
set Bl
in the structure of the fuzzy rule base is normal with center yl
.
Further, if the center average defuzzifier, product inference engine,
singleton fuzzifier are employed to construct the fuzzy system, then the
intendance variables would be calculated by the following formulation
[70]:
δf
i (h) =
∑
M
l=1
yi
lμAi
l
(h)
∑
M
l=1
μAi
l
(h)
(14)
where, δ
f
i (h)ϵV⊂R is the output of the fuzzy system, hϵU⊂Rn
denotes its
input, μAi
l represents the membership function of the inputs, yi
l is the
center of the output membership function, l is the rule number, and M
shows the maximum number of the rules. For fuzzification of adaptation
parameters δ1, δ2 and δ3 the input variables are respectively regarded as
e,
∫
edt and ė. Each fuzzy system has three rules according to Table 1,
while the input membership function are depicted in Fig. 2 and the
center of the output membership functions (yi
l) would be determined via
the optimization process based on the multi-objective grey wolf algo­
rithm. Finally, the coefficients would be computed by the following
equations:
δi(h) = δc
i (h) + δp
i (h) × δf
i (h) (15)
where δc
i (h) and δ
p
i (h) correspondingly represent the constant and
product parameters that would be determined by employing the opti­
mization algorithm.
6. Multi-objective grey wolf optimization
The grey wolf optimization (GWO) algorithm was initially intro­
duced by Mirjalili et al. [71]. Indeed, grey wolves as members of Cani­
dae family generally prefer to live in a pack and in a group size of 5–12
on average. The leaders, which are responsible for making decisions
about hunting, time to wake, sleeping place, are called alphas. Inter­
estingly, the alpha is not essentially the most powerful member of the
pack but is the best in the aspect of managing the pack. Beta in the
second level in the hierarchy of the grey wolves helps the alpha in
decision-making or other pack activities. The beta wolf is most likely the
best candidate when one of the alpha wolves dies or becomes very old.
Omega as the lowest ranking grey wolf contains the last wolves which
are permitted to eat that must yield to all the other dominant wolves. If a
wolf is not alpha, beta, or omega, it is called delta. Delta wolves must
yield to alphas and betas; however, they dominate the omega. Based
upon the four divisions mentioned above, the fittest solution is regarded
as the alpha (α) wolf for mathematically modeling the social hierarchy of
wolves. Accordingly, the second solution is named beta (β), wolves and
Table 1
Fuzzy rules for the parameters of the adaptation laws.
Premise Conclusion
If ė is μA1
1 Then δ1 is B1
1
If ė is μA1
2 Then δ1 is B2
1
If ė is μA1
3 Then δ1 is B3
1
If e is μA2
1 Then δ2 is B1
2
If e is μA2
2 Then δ2 is B2
2
If e is μA2
3 Then δ2 is B3
2
If
∫
e dt is μA3
1 Then δ3 is B1
3
If
∫
e dt is μA3
2 Then δ3 is B2
3
If
∫
e dt is μA3
3 Then δ3 is B3
3
Fig. 2. Input membership functions of the fuzzy systems designed for the control parameters.
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
5
the third solution is called delta (δ) wolves. Lastly, the rest of the
candidate solutions are presumed to be omega (ω) wolves.
The following equations are used to mathematically model the
encircling behavior of wolves to hunt the prey [52]:
D
→
= ∣C
→
X
→
p(t) − Y
→
(t)∣ (16)
Y
→
(t + 1) = Y
→
p(t) − A
→
D
→
(17)
where, t presents the current iteration, A
→
and C
→
represent coefficient
vectors calculated through Eqs. (18) and (19). Y
→
is the position vector of
a grey wolf, and Y
→
p represents the position vector of the prey.
A
→
= 2 a
→ r
→
1 − a
→ (18)
C
→
= 2 r
→
2 (19)
where, vector a
→ is linearly decreased from 2 to 0 over the iteration. r
→
1
and r
→
2 are random vectors in the range of [0, 1].
The following equations are utilized for each search agent to simu­
late the hunting and acquire promising regions of the search space:
D
→
α = ∣C
→
1 X
→
α − Y
→
∣ (20)
D
→
β = ∣C
→
2 X
→
β − Y
→
∣ (21)
D
→
δ = ∣C
→
3 X
→
δ − Y
→
∣ (22)
Y
→
1 = Y
→
α − A
→
1 D
→
α (23)
Y
→
2 = Y
→
β − A
→
2 D
→
β (24)
Y
→
3 = Y
→
δ − A
→
3 D
→
δ (25)
Y
→
(t + 1) =
Y
→
1 + Y
→
2 + Y
→
3
3
(26)
where subscript indexes α, β and δ respectively indicate the parameters
related to the alpha, beta and delta wolves. Components A
→
and C
→
guarantee the exploration ability of the algorithm. In this regard, A
→
is
assigned random values >1 or less than − 1 to help the search agent to
diverge from the prey. Further, vector C
→
has components as random
values in range [0, 2] and plays an important role in emphasizing (C > 1)
or deemphasizing (C < 1) the effect of prey for defining the distance in
Eq. (15). Parameters a and A are linearly diminished over the iteration,
while C is not. To guarantee the exploration, parameter C is required to
provide random values at all times from the initial to final iterations,
increasingly. In the case of guaranteeing the exploitation, it begins when
∣A∣ < 1, and consequently, the next position of a search agent would be
placed within its current position and the position of the prey. When
∣A∣ > 1, the search agents tend to diverge from the prey.
In order to implement the GWO as a multi-objective algorithm, two
strategies, i.e. archive saving and leader selection, should be considered
in the architecture of the algorithm [72]. The archive is a storage unit
which can save or retrieve the non-dominated solutions of the Pareto
front. The size of the archive should be controlled when a solution wants
to enter the archive while the archive is full [72]. On the other hand, the
leader selection strategy would be utilized in order to select alpha, beta
and delta solutions from the archive as the leaders in the hunting pro­
cess. In this paper, the least crowded area in the search space is regar­
ded, and its non-dominated solutions are considered as the alpha
wolves. This selection is conducted by the roulette-wheel approach for
each segment by applying the following probability.
Pi =
c
Ni
(27)
where, Ni indicates the number of Pareto optimal solutions gained in the
ith segment, and c denotes a constant number greater than one. The
objective functions for the optimization process of this challenging
problem are introduced as follows.
ϕ1(ζ) =
∫ T
0
|e(t) |dt +
∫ T
0
|ė(t) |dt (28)
ϕ2(ζ) =
∫ T
0
|u(t) |dt (29)
7. Results and discussion
For the first case, in order to simulate the behavior of the considered
Duffing-Holmes system handled by the introduced robust fuzzy adaptive
integral sliding mode controller, the initial conditions are regarded as
x1(0) = 0.2 and x2(0) = 0 and the desired trajectory is identified as
yd(t) = sin(1.1t) while other constants are fixed at c1 = − 1,c2 = 0.25,
c3 = 1,c4 = − 0.3,ω = 1. Moreover, the uncertainty term is considered
as g(x, t) = 0.1sin(t)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
x2
1 + x2
2
√
, and the external disturbance is defined as
D(t) = 0.1 sin(t). The proper values of the control gains are determined
by applying the suggested optimization technique based on the objective
functions presented in Eqs. (28) and (29). As it is illustrated in Fig. 3, the
multi-objective GWO algorithm performs properly in designing the
robust fuzzy sliding controller, and a uniform scattered Pareto front is
Fig. 3. Flowchart of the grey wolf optimization algorithm.
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
6
obtained from this algorithm. In the following, one of the optimum
points illustrated in the Pareto front should be selected, and the corre­
sponding optimum variables would be utilized to simulate the system
behavior. Although all of the solutions displayed in Fig. 4 are non-
dominated, points A and C are the best solutions with respect to the
first and second objective functions respectively. Among the non-
dominated solutions of the Pareto front, Point B could be chosen as a
trade-off optimum answer for simulation of the states. In this way, the
numerical values of the design variables corresponding to point B,
represented in Table 2, are implemented to solve the dynamical equa­
tions of the Duffing-Holmes based on the robust adaptive fuzzy integral
sliding mode scheme. The found results are depicted in Figs. 5 through 8
for the trajectory tracking of the first state, trajectory tracking of the
second state, tracking error, control input, sliding surface, phase plane
and control parameters.
In Figs. 5 and 6, the performance of the suggested control strategy is
compared with that of an optimal adaptive robust PID (OARPID) scheme
proposed in Reference [60]. As it could be seen from these graphs, the
controller of this research work can track the desired trajectory more
accurately and faster. Moreover, Fig. 7 (a) represents that the introduced
optimal robust fuzzy adaptive integral sliding mode strategy is able to
stable the Duffing-Holmes system at about 1 s. Fig. 7 (b) illustrates the
control effort utilized for stabilization of the system states while its
maximum value is about 5.7 (N). Fig. 7 (c) displays that the designed
integral sliding surface converges to zero at about 2 s. Finally, the phase
plane diagram proves the stability of the system in Fig. 7 (d), while the
variations of the adaptive control parameters are shown in Fig. 8.
For the second case, the optimal robust fuzzy adaptive integral
sliding mode controller is utilized to handle the Duffing-Holmes system
of the first case for various values of the initial conditions. In this regard,
the initial conditions represented in Table 3 are regarded, and the sys­
tem states are shown in Fig. 9. These simulation results clearly depict the
capability of the suggested strategy to stable the introduced nonlinear
uncertain chaotic system having different conditions.
For the third case, the proposed control algorithm is verified through
different characteristics for the considered chaotic Duffing-Holmes sys­
tem. In this way, the initial conditions are regarded as x1(0) = 1 and
x2(0) = − 5 and the desired output is identified as yd(t) = 0, while other
constants are fixed at c1 = − 1, c2 = 0.073, c3 = 1, c4 = − 3.97, ω =
0.68. Moreover, the uncertainty term is considered as g(x, t) =
0.5cos(5πt), and the external disturbance is defined as D(x, t) = 0.2sin
(x1) + 2x1x2 + 0.1u(t). Furthermore, the optimum values of the control
gains found for the above mentioned conditions are regarded as δc
1 =
41.99,δp
1 = − 50.00,δc
2 = 4.8,δp
2 = 0.27,δc
3 = 3.89,δp
3 = 4.95,γ1(0) =
11.84, γ2(0) = 0.98, κ(0) = 50.04. Besides, the performance of the
introduced method is compared with three previously published
schemes, i.e. terminal sliding mode control based linear matrix
inequality (TSMCLMI) [73], sliding mode control with a nonlinear
disturbance observer (SMCNDO) [74] and robust adaptive sliding mode
control (RASMC) [75] to emphasize the contribution of the proposed
method. The obtained results for this case are illustrated in Figs. 10 and
11 that indicate the superiority of the optimum robust fuzzy adaptive
integral sliding mode controller in comparison with the TSMCLMI,
SMCNDO and RASMC with respect to the settling time and overshoot
values of the system states. With more details, the method of this work
displays setting time 0.33 (s) without any overshoots, while the
Fig. 4. Pareto front of the optimal robust fuzzy sliding controller found by the
multi-objective grey wolf optimization algorithm.
Table 2
Optimum values of the control gains corresponding to
the non-dominated point B illustrated in the Pareto front.
Control gains Optimum value
δc
1 61.71
δp
1
− 12.68
δc
2 8.0
δp
2
− 2.0
δc
3 1.33
δp
3
1.81
γ1(0) 15.0
γ2(0) 2.0
κ(0) 14.17
Fig. 5. Comparison between trajectory tracking for the first state obtained by the proposed controller and OARPID [60].
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
7
TSMCLMI, SMCNDO and RASMC methods respectively demonstrate
setting times 0.85 (s), 0.55 (s) and 0.39 (s) for state x1(t). Besides, the
TSMCLMI technique represents an overshoot value equal to 0.24 for this
system state. Further, the approach suggested by this research handles
state x2(t) at 0.38 (s), while the TSMCLMI, SMCNDO and RASMC
method control this state at 0.92 (s), 0.73 (s) and 0.47 (s), correspond­
ingly. Moreover, the TSMCLMI controller depicts overshoot value equal
to 0.45 for state x2(t).
8. Conclusions and future works
In this study, a novel optimal fuzzy adaptive integral sliding mode
controller was proposed to stabilize a class of uncertain chaotic
nonlinear systems. At first, an integral sliding surface was formulated,
and the control effort was successfully designed by using the Lyapunov
stability theory. In order to adaptively tune the parameters of the sug­
gested robust controller, the gradient descent was employed. Moreover,
fuzzy logic systems were implemented to acquire the parameters of the
adaptation laws. Then, multi-objective grey wolf optimization algorithm
was used to find the output membership functions of the fuzzy logic
system. In order to evaluate the performance of the proposed controller,
the Duffing-Holmes oscillator as a challenging case study was consid­
ered. The obtained results were compared with the outcomes of a
distinguished work in the literature that illustrated the efficiency of the
proposed optimal robust controller with respect to uncertain chaotic
nonlinear systems in terms of optimal control inputs and minimum
tracking error. As future studies, the following extensions of this
research study are suggested:
Fig. 6. Comparison between trajectory tracking for the second state obtained by the proposed controller and OARPID [60].
Fig. 7. Diagrams of the optimal robust fuzzy sliding control (a) tracking errors, (b) control input, (c) sliding surfaces and (d) phase plane.
Fig. 8. Control parameters of proposed optimal robust fuzzy adaptive integral
sliding mode technique.
Table 3
Initial conditions for the states of the Duffing-Holmes system.
Case x1(0) x2(0)
Condition 1 − 0.2 − 1
Condition 2 0 − 1
Condition 3 0.2 − 1
Condition 4 − 0.2 0
Condition 5 0 0
Condition 6 0.2 0
Condition 7 − 0.2 +1
Condition 8 0 +1
Condition 9 0.2 +1
M.J. Mahmoodabadi

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena 167 (2023) 113092
8
(1) Employing the introduced optimal fuzzy adaptive robust
controller for autonomous chaotic systems with/without stable
equilibrium points.
(2) Applying fraction-order calculus to enhance the performance of
the suggested controller.
(3) Acquiring better optimal solutions for the parameters of the
controller by other smart multi-objective optimization
algorithms.
(4) Regarding several different objective functions (design criteria)
and solving by many-objective optimization algorithms to
improve the performance of the controller.
(5) Utilizing the proposed control strategy to handle the fractional-
order nonlinear uncertain systems.
CRediT authorship contribution statement
Conceptualization, simulations, original draft preparation, review
and editing, M. J. Mahmoodabadi.
Declaration of competing interest
The authors have declared no conflict of interest.
Data availability
Data will be made available on request.
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