978-1-4799-5458-2 ©2014 IEEE 654
2014 2nd International Conference on Systems and Informatics (ICSAI 2014)
H-infinity Filtering for Networked Control Systems
with Packet Dropout using Delta Operator
Duanjin Zhang, Xue Liu
School of Information Engineering, Zhengzhou University
Zhengzhou 450001, China
E-mail: djzhang@zzu.edu.cn, liuxuejolin@163.com
Abstract—In this paper, we consider the H-infinity filtering for
networked control system with packet dropout. By using delta
operator approach, a new system model is proposed. For a given
packet loss rate, the filtering error system is modeled as an
asynchronous dynamic system. Sufficient conditions are obtained
to illustrate stability of the desired filtering error system in the
form of a number of some linear matrix inequalities, method of
designing H-infinity filter is also given. A numerical example
illustrates that the proposed approach is effective.
Keywords- networked control systems;
H
filtering; linear
matrix inequality; delta operator
I. INTRODUCTION
Networked control system (NCS) which is a
network-based system has attracted more and more attention in
the past decades. The NCS has a lot of good features so as low
cost and ease of maintenance and so on [1-3]. Filtering for NCS,
what is of super practical and research significance, has got lots
of attention. Compared with Kalman filtering, H
filtering
method has lots of advantages such as good robust performance
and high estimation precision. H
filtering is a very effective
approach in estimating system states [4,5]. H
filtering for
network-based system with packet dropout has been
extensively studied in [6]. However, majority existing works
we have learned in the previous discussion cannot give an
appropriate unified form of continuous and discrete time
systems.
At the same time, the study of delta operator has attracted
huge attention in the study of signal processing. By the
traditional shift operator approach, when the sampling is
extremely fast, majority of the poles will not ensure to set in
the stable scale. To solve the induced problem, Goodwin
established the delta operator approach for fast sampling [7].
When the sampling period is fast the delta operator can avoid
the problems like unstable and pathological induced by shift
operator approach. And when the sampling interval closes to
zero the model formulated by the delta operator is approximate
with the equivalent continuous model as well. [7–10] shown
that compared with the traditional shift operator approach the
characteristics of delta operator approach are described in the
following: (i) In delta operator systems the sampling period is
exact and explicit; (ii) In delta operator systems when sampling
period approaches to zero the parameters of the model will
close to the parameters in continuous system, the same to the
control results; (iii) When the sampling rates is extremely high,
the delta operator system will has wonderful numerical
properties. So the filtering problems can be studied in the
combination form via delta operator approach. Several progress
has achieved in NCS with delta operator approach [11, 12].
In the paper, a new method is obtained to solve the filtering
problems for the network-based system with both long-time
delay and packet-dropout using delta operator. We suppose the
time delays are longer than a sample period. When estimating
the system state, a mathematical expression of H
filter for
systems derived by delta operator is introduced. There has an
asynchronous dynamic system with two event rate constraints
modeled for the filtering error system. Using the stability
theory and giving a new stochastic Lyapunov functional for the
delta operator system, a sufficient condition for stable of the H
filter will be derived and expressed as several linear matrix
inequalities. And the numerical example will be illustrated to
confirm that the developed techniques are effective.
II. P
ROBLEM STATEMENT
Notation:
n
represents the n-dimensional Euclidean
space,
22nn
×
expresses the real matrices with order of 22nn× ,
is the identity matrix that of appropriate dimensions, the
superscript T expresses the matrix transpose,
|| ||⋅
denotes the
Euclidean vector norm or the induced matrix 2-norm as
appropriate.
0X > denotes that the matrix X is a real
symmetric positive definite.
λ
means the eigenvalue of a
symmetric matrix
. For a matrix
and two symmetric
matrices
A
,
C
. A symmetric matrix is represented as
*A
C
ªº
«»
¬¼
∗
is the entries implied by symmetry.
The definition of delta operator [7] is given as below:
()/ (), 0
()
[( ) ()]/ , 0
dx dt h
xt
xt h xt hh
δ
=
=
®
+− ≠
¯
and with h expresses the sampling period, the considered
system is continuous-time when
0h = and while 0h ≠ it is
discrete-time.
Consider the state equation of continuous-time system
expressed by the following:
12 3
() () () ()
() () () ()
=++
=+ +
xt Axt Bwt Cut
yt Dxt Dwt Du t
(1)