1 INTRODUCTION
In recent years, the estimation problem for networked
control systems and sensor networks has gained much
attention[1-2]. In networked control systems, random
measurement delays and packet dropouts are unavoidable
in data transmission due to the limited carrying capacity
and bandwidth. So, the research on filtering problems over
networks is significant[3-4].
For systems with stochastic measurement delays, a
suboptimal linear filter is proposed due to the Kalman-like
form[5]. Another suboptimal filter is designed by treating a
colored noise as a white noise[6]. The robust filters are
investigated in [7] and [8]. The filters dependent on
time-stamps and transmission probabilities are studied,
respectively[9]. For systems with packet dropouts, an
optimal linear filter in the linear minimum variance sense is
proposed in [10] based on the packet dropout model which
was first presented in [11]. Reference [12] extends the
results of [11] to deal with the case that unreliable
transmissions exist in both sensor and control channels. In a
very recent study [13], using a group of Bernoulli
distributed random variables, a new measurement model
which can describe the systems with random measurement
delays and multiple packet dropouts is proposed. Based on
the new model, the finite horizon optimal filter, predictor
and smoother are given by innovation analysis method.
However, the filter has high computational cost due to the
high dimension augmented state vector.
In this paper, for the measurement model proposed in [13],
we define a new augmented state vector which has a lower
dimension than the state vector used in [13]. Then, based on
the new augmented state vector, we derive the new
estimators including filter, predictor and smoother which
have the same accuracy with the filter proposed in [13].
This work is supported by National Nature Science Foundation under
Grant
NSFC-61174139.
2 PROBLEM DESCRIPTION
Consider the following linear discrete-time stochastic
system
(1) ()() ()()xt txt twt
ΦΓ
+= + (1)
() () () ()
tHtxtvt=+ (2)
where
()
n
xt R∈
is the system state at time
t , ()
m
wt R∈
and ( )
m
vt R∈ are correlated white noises with mean zeros
and cross-covariance matrices
T
E[ () ()]
w
wt w t Q= ,
T
E[ () ()]wt v t S= ,
T
E[ () ()]
v
vtv t Q= . ( )ĭ t ,()ī t ,()
t are
time-varying matrices with suitable dimensions. The initial
state
(0)x
is independent of
()wt
and ( )vt , and satisfies
that
0
E{ (0)}x
= and
T
000
E{[ (0) ][ (0) ] }xx P
μμ
−−=
,
where
E is the mathematical expectation, T is the
transpose operator.To reduce the effect of packet loss, we
assume that one packet at the sensor side is probably sent
several times repeatedly, and the estimator can only receive
a packet at each time instant. Then, based on the
assumptions, the following model is estibashied in [13].
001
1
0
0
() () () (1 ()) () ( 1)
(1 ( )) ( ) ( )
(1 ( )) ( 1)
d
kd
k
d
k
k
yt tzt t tzt
ttztd
tyt
ξξ
ξ
−
=
=
=+− −+
+− −
+− −
∏
∏
"
(3)
where ( )
i
t
ξ
, 1, 2, ,id= " are uncorrelated Bernoulli
distributed random variables with the probabilities
{() 1} ()
ii
Pt t
ξα
== and { ( ) 0} 1 ( )
ii
Pt t
ξα
==−
0()1
i
t
α
≤≤.
Our aim is to find the optimal linear estimators including
filter, predictor and smoother in the linear minimum
variance sense based on the received measurements
Optimal linear estimators for systems with multiple random measurement delays
and packet dropouts
Yu Luyang
1
, Ma Jing
1
, Sun Shuli
2
1. School of Mathematical Science, Heilongjiang University, Harbin 150080
E-mail: fds926@163.com
, E-mail: majing427@gmail.com
2. School of Electronic Engineering, Heilongjiang University, Harbin 150080
E-mail: sunsl@hlju.edu.cn
Abstract: For linear discrete-time stochastic systems with multiple random measurement delays and packet dropouts,
we give a new augmented method by introducing measurement outputs into the augmented state vector. Based on this
augmented state vector, the optimal linear estimators including filter, predictor and smoother are developed in the linear
minimum variance sense. They can reduce the computational burden compared with the augmented method in the
existing literature. A simulation example verifies their effectiveness.
Key Words: Random delay, Packet dropout, Optimal linear estimator, Linear minimum variance
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978-1-4799-3708-0/14/$31.00
c
2014 IEEE