Lu Cao
1
Assistant Research Fellow
The State Key Laboratory of
Astronautic Dynamics (ADL),
China Xi’an Satellite Control Center,
Xi’an 710043, China
e-mails: caolu_space2015@163.com;
lu.cao2@mail.mcgill.ca
Xiaoqian Chen
Professor
College of Aerospace Science and Engineering,
National University of Defense Technology,
Changsha 410073, China
e-mail: chenxiaoqian@nudt.edu.cn
A Novel Input–Output
Linearization Minimum Sliding
Mode Error Feedback Control
for Synchronization of
FitzHugh–Nagumo Neurons
A novel input–output linearization minimum sliding mode error feedback control (I/OMS-
MEFC) is proposed for the synchronization between two uncoupled FitzHugh–Nagumo
(FHN) neurons with different ionic currents and external electrical stimulations. To esti-
mate and offset the system uncertainties and external disturbances, the concept of equiva-
lent control error is introduced, which is the key to utilization of I/OMSMEFC. A cost
function is formulated on the basis of the principle of minimum sliding mode covariance
constraint; then the equivalent control error is estimated and fed back. It is shown that
the proposed I/OMSMEFC can compensate various kinds of system uncertainties and
external disturbances. Meanwhile, it can reduce the steady-state error more than the con-
ventional sliding mode control (SMC). In addition, the sliding mode after the I/OMS-
MEFC will tend to be the ideal SMC, resulting in improved control performance and
quantity. Sufficient conditions are given based on the Lyapunov stability theorem and nu-
merical simulations are performed to verify the effectiveness of presented I/OMSMEFC
for the chaotic synchronization accurately. [DOI: 10.1115/1.4032074]
Keywords: synchronization, minimum sliding mode error feed back control, input–output
linearization, FitzHugh–Nagumo (FHN) neuron, different external electrical stimulations
1 Introduction
As we know, the chaotic phenomenon exists widely in nonlin-
ear science fields, which is a very complex dynamical nonlinear
system and exhibits some characteristic, such as excessive sensi-
tivity to initial conditions, ergodic property in some areas of state
space. That is, small variances in the initial condition can induce
large deviations in system states. Hence, chaotic system has the
widespread application values in many fields [1–4]. Specifically,
the chaos synchronization control exhibits tremendous utilization
potential and development, which have received great attentions.
Neural network is one of the complex dynamical networks with
strong backgrounds and various potential real-world applications
[5]. Therefore, it has been a fascinating and important subject of
many research fields [6]. During the last few years, there emerged
many results about the coupled FHN neuron system, which is usu-
ally utilized to study neural firings to investigate processing of
information in brain due to the simplicity. The FHN neuron model
was first derived as simplified model of the Hodglein–Huxley neu-
ron model by FitzHugh [7] and Nagumo et al. [8]. There are some
literatures [9,10] that have done many qualitative study of the
FHN model for its nonlinear phenomena. Especially, the synchro-
nization of chaotic neurons under external electric stimulations
(EES) is attracted many interests during the last decade. The pur-
pose of synchronization is to synchronize the states of slave sys-
tem identical or opposite to the states of master system [11]. With
the development of control theory, various control schemes have
been successfully applied to control and synchronization of cha-
otic neurons [12–16]. In Ref. [12], controlling chaos in FHN neu-
ron by the adaptive passive method was introduced.
Synchronization of two uncoupled FHN neurons in EES was
found in Ref. [13] where a nonlinear control technology was
addressed. A novel internal model control method was proposed
for the robust output synchronization [14]. Ambrosio and Aziz
investigated the synchronization and control of coupled
reaction–diffusion FHN systems in Ref. [15]. In addition, Yu
et al. discussed the synchronization of two coupled neurons in the
external electrical stimulation based on adaptive backstepping
SMC method in Ref. [16].
In the past, the chaotic synchronization was discussed under the
assumption of two FHN neurons had the same ionic current and
EES [13,16–18]. In this study, the chaotic synchronized problem
with the existence of system uncertainties and external disturban-
ces between two uncoupled FHN neurons is considered with
different ionic currents and external electrical stimulations. Due
to the existence of system uncertainties and external disturbances,
the SMC is a suitable control methodology. It is well-known that
the SMC is a robust nonlinear feedback control methodology, in
which the structure between the switching surfaces is changed to
achieve the desired performance. The excellent theoretical foun-
dation endows SMC with several advantages, such as rapid
response, low sensitivity to parameter variations and external dis-
turbances, and low computational cost [19]. In Refs. [20,21], a
simple type sliding mode was defined to design the controllers for
synchronization between two identical unified chaotic systems
where the driven system was assumed to have a single input. Syn-
chronization of two different uncertain chaotic systems with
unknown parameters was realized in Refs. [22,23]. However, for
the practical usage in engineering, the system uncertainties and
external disturbances are always time-varying, which hinders the
control precision of conventional SMC, even to inspire the chat-
tering phenomenon.
With a view to tackle the above limitation, the main goal of this
study is to introduce the I/OMSMEFC to accomplish chaotic syn-
chronization between two uncouple FHN neurons with high
1
Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the
J
OURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February
28, 2014; final manuscript received November 1, 2015; published online December
11, 2015. Assoc. Editor: Gabor Stepan.
Journal of Computational and Nonlinear Dynamics JULY 2016, Vol. 11 / 041011-1
Copyright
V
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2016 by ASME
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