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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1
Remote Estimator Design for Time-Delay Neural
Networks Using Communication State Information
Yong Xu, Member, IEEE, Chang Liu, Renquan Lu , Senior Member, IEEE,and
Chun-Yi Su, Senior Member, IEEE
Abstract—This paper investigates the estimator design for the
neural networks, where distributed delays and imperfect mea-
surements are included. A randomly occurred neuron-dependent
nonlinearity is used to describe the uncertain measurements
disturbed by neurons. The measurements are transmitted over
multiple transmission channels, and Markov chains are intro-
duced to model packet dropouts of these channels. A one-to-
one map is constructed to transform m independent Markov
chains to an augmented one to facilitate system analysis. A new
variable called channel state is defined based on the cases of
packet dropouts, and the channel-state-dependent estimator is
designed to trade off between the number and the performance
of the estimator. Sufficient conditions are established to guarantee
that the augmented system is stochastically stable and satisfies the
strict ( Q, S, R)− γ −dissipativity. The estimator gains are derived
using linear matrix methods. Finally, an example is applied to
illustrate the effectiveness of the developed methods.
Index Terms— Dissipative estimator, distributed delays,
Markov chain, neural networks, packet dropouts.
I. INTRODUCTION
N
EURAL networks (NNs) have attracted intensive atten-
tion in the past several decades, since they are widely
applied in the fields of nonlinear system analysis and opti-
mization, industrial processes, channel equalization, and so
on [1]–[7]. Time-delay systems widely exist in our daily
life, and a number of researchers have paid attention to the
NNs with time delays to further extend the application of the
NNs [8]. In [9], the exponential mean-square stability was
analyzed for the stochastic NNs with state time delays. In [10],
the passive performance of the Markov jump NNs was studied
with time-varying delays and distributed delays. What is more,
Manuscript received July 18, 2017; revised October 29, 2017; accepted
January 3, 2018. This work was supported in part by the National Natural
Science Foundation of China under Grant 61503106 and Grant U1611262,
in part by the China National Funds for Distinguished Young Scientists under
Grant 61425009, and in part by the Fundamental Research Funds for the
Central Universities under Grant 2017FZA5010. The work of R. Lu was sup-
ported by the Guangdong Province Higher Vocational Colleges and Schools
Pearl River Scholar approved in 2015. The work of Y. Xu was supported by
the Guangdong Province Higher Vocational Colleges and Schools Pearl River
Young Scholar approved in 2017. (Corresponding author: Renquan Lu.)
Y. Xu, C. Liu, and R. Lu are with the Guangdong Key Laboratory of
IoT Information Technology, School of Automation, Guangdong University
of Technology, Guangzhou 510006, China (e-mail: xuyong809@163.com;
liuchangwm@163.com; rqlu@gdut.edu.cn).
C.-Y. Su is with the Guangdong Key Laboratory of IoT Information
Technology, School of Automation, Guangdong University of Technology,
Guangzhou 510006, China, on leave from the Department of Mechanical
Engineering, Concordia University, Montreal, H3G 1M8, Canada.
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2018.2793185
the NNs have been used to analyze the stability and design the
controller for nonlinear systems with time delays [11]. In [12],
the adaptive NNs were employed to handle the consensus
problem of the nonlinear multiagent sytems with time delays.
Thus, in order to further understand the properties of the
biological NNs and extend the application of the artificial NNs
in the smart control fields, studying on the NNs with time
delays is still a challenging problem.
The internal states of the NNs are complex (i.e., they possess
the properties of dynamics and nonlinearity), and always
cannot be measured directly. Thus, designing an estimator
based on the available measurements is requisite for the NNs.
What we have to point out is that the measurements are
always imperfect, bringing a lot of difficulties to design the
estimator, are mainly caused by two factors [13]. One is that
sensors are unavoidably disturbed by external noises, such
as Gaussian noises and energy bounded noises [14], [15].
The other is that measurements are influenced by the system
dynamics, such as the measurements of the NNs are disturbed
by the neurons [16]. Considering the second factor, a robust
filter was designed using the measurements with neuron-
dependent nonlinearities for the NNs with time delays [17].
This paper pays attention to randomly occurred neuron-
dependent nonlinearities, which are more general than the
existing ones, and how to use such measurements to design
an estimator deserves more attention.
In order to further promote the development of the NNs,
the obtained measurements are always transmitted over the
networks to realize remote state estimation. Thus, networked
systems possessing numerous advantages have been intro-
duced to the NNs. For the large scalar networked sys-
tems, the following facts make that transmission over the
multiple channels become an unavoidable issue [18], [19].
First, the capacity of the transmission channels is always
limited, being unavailable to transmit all of the information in
one packet [20]. Second, sensors are distributed in a large area,
it is unable to collect all sensor measurements and transmit
them together [21]. In [22], the linear quadratic controller was
designed for the multiple transmission channels with fading
property, and a sufficient and necessary condition for the
existence of the optimal controller was established. In [23],
the state and the output feedback controllers were designed
for the mutually dependent multiple transmission channels.
One drawback of the shared communication channels of the
networked control systems is packet dropouts, and there are
two reasons, which cause packet dropouts [24]–[28]. The first
one is that the shared communication channel is influenced
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