H∞ Filtering for Networked Markovian Jump Systems with Multiple
H∞ Filtering for Networked Markovian Jump Systems with Multiple Stochastic
Communication Delays
Hui Dong and Juntong Chen
Department of Automation, Zhejiang University of Technology, Hangzhou, 310032, P.R. China
Abstract: This paper is concerned with the H∞ filtering for a class of networked
Markovian jump systems with multiple communication delays. Due to the existence
of communication constraints, the measurement signal cannot arrive at the filter
completely on time, and the stochastic communication delays are considered in the
filter design. Firstly, a set of stochastic variables is introduced to model the
occurrence probabilities of the delays. Then based on the stochastic system approach,
a sufficient condition is obtained such that the filtering error system is stable in the
mean-square sense and with a prescribed H∞ disturbance attenuation level. The
optimal filter gain parameters can be determined by solving a convex optimization
problem. Finally, a simulation example is given to show the effectiveness of the
proposed filter design method.
Keywords: Markovian jump system, stochastic systems, networked systems,
communication delays, H∞ filtering.
1. Introduction
Analysis and synthesis of the networked systems has received much research
attention in the last decade due to the wide applications of networked systems into the
industrials, such that process control system, chemical systems, remote surgery.
Despite for the advantages brought by the networked systems, various challenging
problems occur, such as packet dropouts, communication delays, medium access
constraints. In the last a few years, the state estimation for networks systems has
become a hot research topic. Basically, there are state estimation algorithms, one is
the Kalman filtering and the other one is the H∞ filtering. For example, the authors in
[1] studied the remote state estimation for the networked systems with missing
observations. They addressed this problem under the discrete Kalman filtering
formulation, and also gave the statistical convergence properties of the estimation
error covariance, showing that the existence of a critical value for the arrival rate of
the observations. Many new results have also been reported based on the results in [1],
see [2-5] and the references therein. It is well known that the Kalman filter is only
applicable to the systems with a certain well-posed model, and the external noise is
required to be a stochastic process with known statistical property. In many
applications, these requirements may not hold. In this scenario, an alternative filtering
algorithm has been proposed, i.e., the H∞ filtering algorithm. The recent advances on
the H∞ filtering for networked systems with various networked issues are referred into
[6-10].
On another research front line, Markovian jump systems have attracted
significant attention from control society for many decades. This is due to the fact that
such systems are shown to be appropriate and convenient to model a large number of
practical systems that are subjected to abrupt variations in their structures, owing to
sudden environmental disturbances, abrupt variations in the operating point of a
nonlinear plant, and so on. Many works have been reported onto the analysis and H∞
filtering for Markovian jump systems [11-16]. To mention a few, the H∞ filtering for a
class of Markovian jump systems with time-varying transition probabilities has been
considered in [16], the authors showed that the filter gain parameter can be
determined by solving a set of linear matrix inequalities. Recently, the H∞ filtering for
Markovian jump systems with missing measurement and signal quantization is
investigated in [17], the corresponding filter synthesis problem is transformed into a
convex optimization problem that can be efficiently solved by using standard software
packages. It should be noted that the communication delay has not been considered in
their design of the filter. The communication delay is an important issue that occurs in
the networked control systems and it cannot be ignored. Though the H∞ filtering for
networked with stochastic delays in [18], the authors assumed that the measurement
signal is transmitted to the remote filter within a single packet. In reality, the
networked systems may distribute in a large area, incorporating the measurement into
one packet is impossible. Hence, the H∞ filtering for networked systems with multiple
communication delays deserves investigation, which motivates the present study.
In this paper, the H∞ filtering problem is addressed for a class of networked
Markovian jump systems with multiple communication delays. Firstly, a set of
stochastic variables are introduced to model the communication delays. Based on the
Laypunov stability theory and the stochastic system approach, a sufficient condition is
obtained such that the filtering error system is asymptotically stable in the
mean-square sense and achieves a prescribed H∞ performance level. Finally, a
simulation study is given to show the effectiveness of the proposed design.
n
Notations: the notations used in this paper is standard. The R denotes the
n-dimensional Euclidean space, l2 [0, ) is the space of summable sequences.
AT represents the transpose of the matrix A.
2. Problem formulation
The structure of the considered filtering system is shown in Fig. 1. The physical
plant under the consideration is:
x(k 1) A ( k ) x(k ) B ( k ) w(k )
z (k ) L ( k ) x(k )
(1)
where x(k ) R nx is the state, z (k ) R nz is the signal to be estimated, and
w(k ) R nw is the unknown disturbance
belonging to l2 [0, ) . A ( k ) , B ( k ) , and
L ( k ) are the constant matrices with appropriate dimensions. (k ) {1, 2,L , S} is
a discrete-time homogeneous Markov chain with following probabilistic transfer
matrix [ ij ] :
ij Prob{ (k 1) j | (k ) i} 0,
S
ij 1, i, j
j 1
w(k )
Plant
sensor 1
y1(k )
y1(k )
sensor 2
y2 (k )
y2 (k )
…
…
…
…
sensor m
ym (k )
(2)
Network
Filter
…
…
ym (k )
z (k )
z f (k )
e( k )
Fig.1. The structure of networked sytstems
Suppose there are m distributed sensors for the system and the p-th sensor
produces its measurement as
y p (k ) C p x(k ) D p w(k ) ( p 1, 2,
, m)
(3)
where y p (k ) R np , C p and D p
are the known constant matrix with appropriate
dimensions. Due to the existence of the communication delay, the filter may not be
able to use the current measurement signal to update its state. In this paper, we
assume that the largest delay in the p-th channel is N p time step. Then, at each time
instant k, the filter input, y p (tk ) , will be the most recent member of the transmitted
subset of
y
p
, y p (k N p ) .
(k ), y p (k 1),
In the real networked systems, the delay may occur in a stochastic way, hence, a
set of stochastic variables, p,s (k ) 0,1 , s 0,1,
, N p , is introduced such that
p,s (k ) 1 if y p (k s ) is the value used as the filter input at k. Due to the fact that at
each time instant, only one case may happen in our system and we have
Np
s 0
p,s
(k ) 1. In this paper, the probabilities E p,s (k ) 1 p ,s are assumed to
be known, which may be obtained by some stochastic analysis method. Clearly,
Np
s 0
p,s
1. The input of the filter can be described as
y p (k ) p,0 (k ) y p (k ) p,1 (k ) y p (k 1)
p , N p ( k ) y p (k N p )
(4)
Let N max {N p },
p 1,2, , m
T
X (k ) xT (k ) xT (k 1)
xT (k N ) ,
W (k ) wT (k ) wT (k 1)
wT (k N ) .
T
We have
Np
y p (k ) p,s (k ) C p Esp X (k ) Dp H spW (k )
(5)
s 0
where
Esp 0
0 I 0
0 , and Hsp 0
0 I 0
0 , in
which all elements are zeros except that the ( s +1)-th block is an identity matrix.
Let p (k ) [ p,0 (k ), p,1 (k ),
, p, N p (k )] . Then we have
1, 0, 0, , 0
0,1, 0, , 0
p (k )
0, 0, 0, ,1
(6)
Define (k ) [1 (k ), 2 (k ),
(k ) is N N1 N 2
, m ( k )] . The total number of possible realizations of
N m . Let
E (k ) [ E1,T1 ( k )
E2,T 2 ( k )
EmT ,m ( k ) ]T ,
H ( k ) [H1,T1 ( k )
H2,T 2 ( k )
HmT ,m ( k ) ]T .
One particular realization of (k ) means one sequence of p ( k ) , which specifies
one particular case of ( E ( k ) , H ( k ) ) . For presentation simplicity, we now define a
new variable l (k ) {1, 0}, l to model each possible case for all the measurement
signals, where {1, 2,
, N } , and N is the possible numbers. For example,
1 (k ) 1 if and only if p (k ) [1, 0,
p (k ) [1, 0,
N
, 0], p 1, 2,..., m 1, m ( k ) [0,1,
l (k ) 1, and
l 1
, 0], p 1, 2,..., m ; 2 (k ) 1 if and only if
N
l 1
l
, 0] ; and so on. It is easy to see that
1.
Based on the above discussions, the filter input signal is
y (k ) l (k ) CEl X (k ) DH lW (k )
N
(7)
l 1
where C diag{C1 , C2 ,
, Cm } , D diag{D1 , D2 ,
, Dm } . We propose the following
filter:
x f (k 1) Af x f (k ) B f y (k )
z f (k ) C f x f (k )
(8)
where x f (k ) R nx is the state of the filter and z f (k ) R nz is the estimate of z (k ) .
A f , B f , and C f are the filter parameters to be designed.
In order to derive the filtering error system, we rewrite state equation in (1) and
the estimation equation in (7) as
X (k 1) A ( k ) X (k ) B ( k )W (k )
z (k ) L ( k ) X (k )
where
(9)
A ( k )
A ( k )
I
n
0
0
L ( k ) L ( k )
Define
0
0
In
0
0
0
0
In
0
B ( k )
0
0
0 , B ( k ) 0
0
0
0
0
0
0
0
0
0
0
0
0
0 ,
0
0 .
0 0
(k ) X T (k ) xTf (k )
T
and
the
estimation
error
as
e(k ) z (k ) z f (k ) . Then the filtering error system is given by
N
(
k
1)
A
(
k
)
B
W
(
k
)
l (k ) l Aˆl (k ) BˆlW (k )
(k )
(k )
l 1
e( k ) L ( k )
(k )
(10)
where
A
A ( k ) ( k )
B f CE
0
Aˆl
B f CEl
B ( k )
0
, B ( k )
, L ( k ) L ( k )
Af
B f DH
C f ,
N
N
0
0
ˆ
B
E
E
H
l H l .
,
,
,
B DH
l
l l
0
f
l
l
1
l
1
System (10) is a complex stochastic systems, we need the following definitions
before presenting our main results.
Definition 1 [12]. System (10) with w(k ) 0 , is said to be stochastically stable
if the following inequality:
E{ (k ) | (0)}<
2
(11)
k 0
holds for any initial condition (0)={ (t0 ), (0)} .
Definition 2. For a given scalar 0 , system (10) is said to be asymptotically
stable in the mean-square sense and achieves a prescribed H∞ performance , if it is
asymptotically
stable
s 0
s 0
and
under
zero
initial
condition,
E{eT (s)e(s)} 2 wT (s)w(s) holds for all nonzero w(k ) l2[0, ) .
3. Main results
A sufficient condition is firstly presented to guarantee the stability of the filtering
error system.
Theorem 1. For given scalars i and , if there exist a positive-definite
matrix Pi such that the following inequalities
1
*
*
*
*
*
2
1 1
N N
Pi
*
*
*
0
Pi
*
*
*
0
0
0
Pi
*
*
*
*
0
0
3
0
0
0 ,
0
0
I
(12)
hold for all i , then the filtering error system is asymptotically stable in the
mean-square sense and achieves a prescribed H∞ performance level N 1 ,
where
P
1 i
0
0
, 2 Ai
2 I
Bi , l Aˆl
T
T
Bˆl , 3 Li
T
0 ,
1
s
Pi ij Pj .
j 1
Proof: We first consider the stability of the filtering error system (10)
with w(k ) 0 . To do this, we choose the following Lyapunov function for system
(10):
V (k ) T (k )P ( k ) (k )
(13)
Then, one sees that
E V (k 1) V (k )
T s
Ai (k ) ij Pj Ai (k ) T (k ) Pi (k )
j 1
T
N
s
N
E l (k ) Aˆl (k ) ij Pj l (k ) Aˆl (k )
j 1
l 1
l 1
where i (k ) i ( k ) i . On the other hand, we have
(14)
T
N
N
s
ˆ
E l (k ) Al (k ) ij Pj l (k ) Aˆl (k )
j 1
l 1
l 1
T s
N
2
l (k ) Aˆl (k ) ij Pj Aˆl (k )
l 1
j 1
E
N
N
s
T
Aˆl (k ) ij Pj Aˆt (k )
(
k
)
(
k
)
l
t
j 1
l 1 t 1,t l
N
T s
l Aˆl (k ) ij Pj Aˆl (k )
l 1
j 1
(15)
Hence,
E V (k 1) V (k )
T s
Ai (k ) ij Pj Ai (k ) T (k ) Pi (k )
j 1
(16)
N
T s
l Aˆl (k ) ij Pj Aˆl (k )
l 1
j 1
It is seen that the right hand side of (16) is negative under (12), and then
EV (k 1)| (k ) V (k ) . It is not difficult to find a scalar 0 1 , such that
EV (k 1)| (k ) V (k )
holds,
then
by
deduction,
we
have
E[V (k )| (0)] kV (0) , consequently,
h
E[ V (k )| (0)] （1+ +L + h）V ( k0 )
k 0
1 h 1
V ( k0 )
1
(17)
Let h , the following inequality is obtained:
E[ (k ) | (0)]
2
k 0
1
(1 )
V ( k0 )
(18)
where min min ( Pi ) . According to definition 1, the system (10) is stochastically
i
stable.
Now we consider the H∞ performance of the filtering error system (10). It
follows from the above analysis method that
E V (k 1) V (k ) eT (k )e(k ) 2W T (k )W (k )
T s
Ai (k ) BW
(
k
)
(k ) T (k ) Pi (k )
i
i
ij Pj Ai (k ) BW
j 1
T
N
l (k ) l Aˆl (k ) BˆlW (k )
l 1
E
s
N
P
ij j l (k ) l Aˆl (k ) BˆlW (k )
j 1
l 1
(19)
T
Li (k ) Li (k ) 2W T (k )W (k )
N
s
s
T (k ) 1 2 ij Pj T2 i i ij Pj Ti 3T3 (k )
i 1
j 1
j 1
T
where (k ) T (k ) W T (k ) . By using the schur complement [19], it is easy to
see
E V (k 1) V (k ) eT (k )e(k ) 2W T (k )W (k ) 0 .
(20)
Summing both sides of (16) from k 0 to k T gives
T
E V (k 1) V (0) eT (k )e(k ) 2W T (k )W (k ) 0 ,
k 0
which implies, by the zero initial condition and positiveness of
(21)
V (k 1) , that
T
E eT (k )e(k ) 2W T (k )W (k ) 0 .
k 0
(22)
Let T , then
E e
T
k 0
(k )e(k ) 2W T (k )W (k ) 2 wT (k ) w(k )
(23)
Hence, system (10) achieves a prescribed H∞ performance level as well. This
completes the proof.
It should be pointed out that theorem 1 can not be used to determine the filter
gain directly due to the co-existence of P and P1 . In the following theorem, we
present the filter gain design.
Theorem 2. For given scalars i and , if there exist a positive-definite
matrix Pi and a matrix G with appropriate dimensions, such that the following
inequalities
2
1
* Pˆi
* *
* *
* *
* *
1 1
0
Pˆ
i
N N
0
0
0
0
0
Pˆ
*
*
*
*
*
i
*
3
0
0 0
0
0
I
(24)
hold, then the filtering problem is solvable. Moreover, the filter gains are determined
by Af G3T AF , B f G3T BF , C f C F . where
1i
12
13 ˆ s
2 2 , i 2 , 3 , Pi ij Pj G GT ,
0
2
j 1
i
with
AT G E T BFT M
12 i 1 T
AF M
AiT G2 E T BFT
,
AFT
22 BiT G1 H T BFT M
BiT G2 H T BFT ,
EiT C T BFT M
0
1
i
EiT C T BFT
2
T
T T
, i Hi D BF M
0
LT
P
T , Pi 1i
*
CF
P2i
G1
, G
P3i
G3 M
1
3
HiT DT BFT ,
G2
, M I
G3
I .
Proof: By pre and post multiplying (12) with diag I , G T , G T , G T , I and its
N
transpose respectively, (12) is equivalent to
G
2
1
*
P
i
*
*
*
*
*
*
*
*
N N G 3
1 1G
0
Pi
*
*
0
0
*
0
0
0
Pi
*
*
*
On other
hand,
0
I
0
0
0
0
(25)
1
where
s
Pi G ij Pj G .
j 1
T
1
it
is easy to
see that
s
s
G ij Pj GT ij Pj G GT always holds for any matrix G . Then,
j 1
j 1
T
(25) holds if
G
2
1
*
Pˆi
*
*
*
*
*
*
*
*
N N G 3
1 1G
0
Pˆ
i
0
0
0
0
0
Pˆ
*
*
*
*
*
i
*
P1i
The above inequality is same as (24) with Pi
*
0
0 0
0
0
I
P2i
G1
, G
P3i
G3 M
(26)
G2
,
G3
AF G3T Af , BF G3T Bf , and C F C f . This completes the proof.
Remark 1. In order to obtain the minimum H∞ performance * , one can solve
the following optimization problem:
min
subject to (24) with = 2
(27)
and find the minimum H∞ performance * by * * ( N 1) .
4. Simulation example
Consider a satellite yaw angles control system with noise perturbation, which is
given by the following state-space representations ([20]):
1
0
0
0
0
0
1 0
0 J1
0
0
0 1 (t ) 0
0 2 (t ) 0
0 1 (t ) k
J 2 2 (t ) k
0
1
0
0
k
k
f
f
0 1 (t ) 0
0
0.1
1 2 (t ) 0
u (t ) w(t )
0
f 1 (t ) 1
f 2 (t ) 0
0.1
(28)
The satellite yaw angles control system consists of two rigid bodies jointed by a
flexible link. This link is modeled as a spring with torque constant k and viscous
damping f. 1 (t ) and 2 (t ) are the yaw angles for the main body and the
instrumentation module of the satellite. Moreover, 1 (t ) 1 (t ) and 2 (t ) 2 (t ) .
u (t ) is the control torque, while J 1 and J 2 are the moments of the main body and
the instrumentation module, respectively.
Choosing J1 J 2 1, k 0.3, f 0.004 and setting the sampling period T 0.1s ,
then with the controller u(k ) 103 0.1591 5.9343 0.0172 2.8604 x(k) , we
obtain the following discrete-time satellite yaw angles closed-loop system:
29.6580 0.0142 14.2960
0.2035
0.000005189
0.0012
0.01049
0.9871
0.0000
0.0944
w(k )
x(k 1)
x(k )
15.9256 592.9813 0.7168 285.8330
0.0001569
0.4451
0.0007
0.7980
0.0188
0.009843
where
x(k ) 1 (k ) 2 (k ) 1 (k ) 2 (k ) .
T
In real control systems, controller failure may occur. In this example, we only
consider a partial failure case, that is the real controller is taken as ur (k ) (k )u (k )
with (k ) {0.5,1} . The binary variable (k ) is assumed to be a two-state
0.6 0.4
Markovian process. The probabilistic transfer matrix is taken as
,
0.6 0.4
which will be generated stochastically in the simulation setup. Hence, such a satellite
yaw angles control system with partial controller failure can be modeled as the
discrete-time Markovian jump system (1).
Suppose the remote filter is designed to estimate the signal z (k ) Lx(k ) by the
measurement
y p (k ) C p x(k ) D p w(k )
of
the
satellite
system,
where
C1 1 0 0 0 , C2 0 1 0 0 , D1 0.3 , D2 0.2 , and L 1 1 0 0 .
The two measurements y1 and y 2 are transmitted to the remote filter through
communication channels with stochastic communication delays. The maximal time
delays are assumed to be one time step, and the occurrence probabilities are
1,0 0.2, 1,1 0.8, 2,0 0.4, and 2,1 0.6 . By solving the optimization problem
(17), we have * 3.1916 .
In the simulation setup, we choose the zero initial conditions, and the noise
signal is taked as w(k ) exp(-0.1k )*sin(0.8k ) . One sample trajectories of z (k ) and
z f ( k ) are shown in Fig. 2. The trajectory of the filtering error is shown in Fig. 3. By
100
simple calculation, we have
eT (k )e(k )
k 0
100
w
T
k 0
(k ) w(k ) 1.1622 * .
2
z
zf
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
10
20
30
40
50
Time(k)
60
70
80
90
100
Fig.2. Trajectories of z (k ) and z f ( k )
1.2
e
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
-0.4
0
10
20
30
40
50
Time(k)
60
70
80
90
100
Fig.3. Trajectory of e(k )
5. Conclusions
In this paper, the H∞ filtering for a class of networked Markovian jump system
with multiple stochastic communication delays has been investigated. A new
stochastic delay model is proposed and formulated. Based on the stochastic system
approach, a sufficient condition has been obtained such that the filtering error system
is asymptotically stable in the mean-square sense and achieves a prescribed H∞
performance level. A simulation study of the satellite yaw control system is given to
show the effectiveness of our results.
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