Robust Vertex-dependant H
∞
Filtering of Stochastic Discrete-time
Systems with Delay
Eli Gershon
1,2
and Uri Shaked
2
1
Holon Institute of Technology, HIT, Holon, Israel
2
School of Electrical Engineering, Tel-Aviv University, Tel Aviv, Israel
Keywords:
Robust Filtering, Discrete Stochastic Systems, Polytopic Uncertainties.
Abstract:
Linear, state delayed, discrete-time systems with stochastic uncertainties in their state-space model are con-
sidered. The problems of robust vertex-dependant polytopic H
∞
filtering is solved, for the stationary case, via
an input-output approach by which the system is replaced by a nonretarded system with deterministic norm-
bounded uncertainties. A vertex-dependent solution is obtained by applying a modified version of the Finsler
lemma. In this problem, a cost function is defined which is the expected value of the standard H
∞
performance
index with respect to the uncertain parameters.
1 INTRODUCTION
We address the problem of robust polytopic
H
∞
filtering of state-delayed, discrete-time, state-
multiplicative linear systems via the input-output ap-
proach based on the robust vertex-dependant stability
and Bounded Real Lemma (BRL) of these systems,
which are developed here. The multiplicative noise
appears in the system model in both the delayed and
the non delayed states of the system.
The stability and control of stochastic delayed sys-
tems of various types (i.e constant time-delay, slow
and fast varying delay) have been a central issue
in the theory of stochastic state-multiplicative sys-
tems over the last decade (Boukas and Liu, 2002),
(Mao, 1996), (Verriest and Florchinger, 1995), (S.Xu
and Chen, 2002), (Chen et al., 2005), (Gershon and
Shaked, 2011). The results that have been obtained
for the stability of deterministic retarded systems,
since the 90’s, have been extended also to the stochas-
tic case, mainly for continuous-time systems. In the
continuous-time stochastic setting, for example, the
Lyapunov-Krasovskii (L-K) approach is applied in
(Xu et al., 2005) and (Chen et al., 2005), to sys-
tems with constant delays, and stability criteria are
derived for cases with norm-bounded uncertainties.
The H
∞
state-feedback control for systems with time-
varying delay is treated in (S.Xu and Chen, 2002) for
restricted LKFs that provide delay-independent, rate
dependent results. Also (Boukas and Liu, 2002) con-
siders H
∞
control (both state and output feedback) and
estimation of time delay systems.
In the discrete-time setting, the mean square expo-
nential stability and the control and filtering problems
of these systems were treated by several groups (Xu et
al., 2004), (Gao and Chen, 2007), (Yue et al., 2009).
In (Xu et al., 2004), the state-feedback control prob-
lem solution is solved for norm-bounded uncertain
systems, for the restrictive case where the same multi-
plicative noise sequence multiplies both the states and
the input of the system. The solution there is delay-
dependent.
The filtering problem of the discrete-time retarded
stochastic systems was already solved in (Gershon
and Shaked, 2013). The point of view that was
taken in the latter work is similar to the one taken
for the solution of both the continuous-time state-
feedback control and filtering problems in (Gershon
and Shaked, 2011). This solution is based on the
input-output approach of (Kao and Lincoln, 2004),
(Fridman and Shaked, 2006) that was developed for
deterministic systems. The latter approach is based
on the representation of the system’s delay action by
linear operators, with no delay, which in turn allows
one to replace the underlying system with an equiv-
alent one which possesses a norm-bounded uncer-
tainty, and therefore may be treated by the theory of
norm bounded uncertain, non-retarded systems with
state-multiplicative noise (Gershon et al., 2005).
In the robust polytopic filtering case, the solution
in (Gershon and Shaked, 2013) draws on the solution
of the nominal uncertainty-free system. Basically, it
Gershon, E. and Shaked, U.
Robust Vertex-dependant H
∞
Filtering of Stochastic Discrete-time Systems with Delay.
DOI: 10.5220/0006432706210628
In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 621-628
ISBN: 978-989-758-263-9
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
621
applies the same Lyapunov function over the whole
uncertainty polytope, leading to a considerably con-
servative solution [i.e the quadratic solution]. In the
present paper we extend the results achieved in (Ger-
shon and Shaked, 2013) for the robust case by apply-
ing a different approach which make use of the Finsler
lemma. Unlike the quadratic solution, in this work we
assign a different Lyapunov function to each vertex of
the uncertainty polytope thus allowing for a possibly
less conservative solution. In order to derive the latter
solution we first develop a vertex-dependent stability
condition followed by a vertex-dependent BRL solu-
tion.
In our system we allow for a time-varying delay
where the uncertain stochastic parameters multiply
both the delayed and the non delayed states in the state
space model of the system. This paper is organized
as follows: Based on the input-output approach, the
solution of the robust stability issue is brought in Sec-
tion 3 and newly developed for the vertex-dependent
case, followed by the solution of the robust BRL in
Section 4. The robust vertex-dependent filtering prob-
lem is treated in Section 5, for a general-type filter,
resulting in a less conservative solution, compared to
the already solved quadratic one.
Notation. Throughout the paper the superscript ‘T ’
stands for matrix transposition, R
n
denotes the n di-
mensional Euclidean space, R
n×m
is the set of all
n × m real matrices, N is the set of natural num-
bers and the notation P > 0, (respectively, P ≥ 0) for
P ∈ R
n×n
means that P is symmetric and positive
definite (respectively, semi-definite). We denote by
L
2
(Ω, R
n
) the space of square-integrable R
n
− valued
functions on the probability space (Ω, F , P ), where
Ω is the sample space, F is a σ algebra of a subset
of Ω called events and P is the probability measure
on F . By (F
k
)
k∈N
we denote an increasing family
of σ-algebras F
k
⊂ F . We also denote by
˜
l
2
(N ; R
n
)
the n-dimensional space of nonanticipative stochas-
tic processes { f
k
}
k∈N
with respect to (F
k
)
k∈N
where
f
k
∈ L
2
(Ω, R
n
). On the latter space the following l
2
-
norm is defined:
||{ f
k
}||
2
˜
l
2
= E{
∑
∞
0
|| f
k
||
2
} =
∑
∞
0
E{|| f
k
||
2
} < ∞,
{ f
k
} ∈
˜
l
2
(N ; R
n
),
(1)
where ||·|| is the standard Euclidean norm. We denote
by Tr{·} the trace of a matrix and by δ
i j
the Kronecker
delta function. Throughout the manuscript we refer
to the notation of exponential l
2
stability, or internal
stability, in the sense of (Bouhtouri et al., 1999) (see
Definition 2.1, page 927, there).
2 PROBLEM FORMULATION
We consider the following linear retarded system:
x
k+1
= (A
0
+ Dν
k
)x
k
+ (A
1
+ Fµ
k
)x
k−τ(k)
+B
1
w
k
x
l
= 0, l ≤ 0,
y
k
= C
2
x
k
+ D
21
n
k
(2a,c)
with the objective vector
z
k
= C
1
x
k
, (3)
where x
k
∈ R
n
is the system state vector, w
k
∈ R
q
is
the exogenous disturbance signal, n
k
∈ R
p
is the the
measurement noise signal, y
k
∈ R
m
is the measured
output and z
k
∈ R
r
is the state combination (objective
function signal) to be regulated and where the time
delay bound is denoted by h. The variables {µ
k
} and
{ν
k
} are zero-mean real scalar white-noise sequences
that satisfy:
E{ν
k
ν
j
} = δ
k j
, E{µ
k
µ
j
} = δ
k j
E{µ
k
ν
j
} = 0, ∀k, j ≥ 0.
The matrices in (2a,c), (3) are constant matrices of
appropriate dimensions.
We treat the following two problems:
i) H
∞
Filtering.
We consider the system of (2a,b) and (3) where and
consider the estimator of the following general form:
ˆx
k+1
= A
c
ˆx
k
+ B
c
y
k
,
ˆz
k
= C
c
ˆx
k
.
(4)
We denote
e
k
= x
k
− ˆx
k
, and ¯z
k
= z
k
− ˆz
k
, (5)
and we consider the following cost function:
J
F
∆
= ||¯z
k
||
2
˜
l
2
− γ
2
[||w
k
||
2
˜
l
2
+ ||n
k+1
||
2
˜
l
2
]. (6)
Given γ > 0 , we seek an estimate C
c
ˆx
k
of C
1
x
k
over the infinite time horizon [0, ∞) such that J
F
given by (6) is negative for all nonzero w
k
, n
k
where
w
k
∈
˜
l
2
F
k
([0, ∞);R
q
), n
k
∈
˜
l
2
F
k
([0, ∞];R
p
).
i) Robust Filtering. In the robust stochastic H
∞
esti-
mation problem treated here, we assume that the sys-
tem parameters lie within the following polytope:
¯
Ω
∆
=
A
0
A
1
B
1
C
1
C
2
D
21
D F
, (7)
which is described by the vertices:
¯
Ω = C o{
¯
Ω
1
,
¯
Ω
2
, ...,
¯
Ω
N
}, (8)
where
¯
Ω
i
∆
=
h
A
(i)
0
A
(i)
1
B
(i)
1
C
(i)
1
C
(i)
2
D
(i)
21
D
(i)
F
(i)
i
(9)
and where N is the number of vertices. In other words:
¯
Ω =
N
∑
i=1
¯
Ω
i
f
i
,
N
∑
i=1
f
i
= 1 , f
i
≥ 0. (10)
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
622
3 ROBUST MEAN-SQUARE
EXPONENTIAL STABILITY
In order to solve the above two problems we bring
first the stability result for the retarded, stochastic,
discrete-time system that was already derived in
(Gershon and Shaked, 2013). Considering the system
of (2a) with B
1
= 0, we obtain the following theorem,
which is brought here for the sake of completeness :
Theorem 1. (Gershon and Shaked, 2013) The ex-
ponential stability in the mean square sense of the sys-
tem (2a) with B
1
= 0, is guaranteed if there exist n × n
matrices Q > 0, R
1
> 0, R
2
> 0 and M that satisfy the
following inequality:
¯
Γ
∆
=
¯
Γ
1,1
¯
Γ
1,2
0 0
¯
Γ
1,5
∗ −Q Q(A
1
− M) QM 0
∗ ∗
¯
Γ
3,3
0
¯
Γ
3,5
∗ ∗ ∗ −R
2
−hM
T
R
2
∗ ∗ ∗ ∗ −R
2
< 0.
where
¯
Γ
1,1
= −Q + D
T
(Q + h
2
R
2
)D+R
1
,
¯
Γ
1,2
= (A
0
+M)
T
Q,
¯
Γ
1,5
= h(A
T
0
+ M
T
)R
2
− R
2
h,
¯
Γ
3,3
= −R
1
+F
T
(Q+h
2
R
2
)F,
¯
Γ
3,5
= h(A
T
1
− M
T
)R
2
.
(11a-e)
We note that inequality (11a) is bilinear in the deci-
sion variables because of the terms QM and R
2
M. In
order to remain in the linear domain, we can define
Q
M
= QM and choose R
2
= εQ where ε is a posi-
tive tuning scalar. The resulting LMI can be found in
(Gershon and Shaked, 2013).
In the polytopic uncertain case we obtain two re-
sults, the first of which is the quadratic solution which
appears in (Gershon and Shaked, 2013) and is based
on assigning the same Lyapunov function over the all
uncertainty polytope. A new result is obtained here
by applying a vertex-dependent Lyapunov function.
Using Schur’s complement, (11a) can be written as:
Ψ + ΦQΦ
T
< 0, (12)
with
Ψ
∆
=
Ψ
1,1
0 0 h(A
T
0
+ M
T
)R
2
− R
2
h
∗ Ψ
2,2
0 h(A
T
1
− M
T
)R
2
∗ ∗ −R
2
−hM
T
R
2
∗ ∗ ∗ −R
2
,
Φ
∆
=
A
T
0
+ M
T
A
T
1
− M
T
M
T
0
, Ψ
1,1
=−Q+D
T
(Q + h
2
R
2
)D+R
1
and
Ψ
2,2
= −R
1
+F
T
(Q+h
2
R
2
)F.
(13a-d)
The following result is thus obtained:
Lemma 1. Inequality (12) is satisfied iff there exist
matrices: 0 < Q ∈ R
n×n
, G ∈ R
n×4n
, M ∈ R
n×n
and
H ∈ R
n×n
that satisfy the following inequality
Ω
∆
=
Ψ + G
T
Φ
T
+ ΦG −G
T
+ ΦH
−G + H
T
Φ
T
−H − H
T
+ Q
< 0.
(14)
Proof. Substituting G = 0 and H = Q in (14), inequal-
ity (12) is obtained. To show that (14) leads to (12) we
consider
I Φ
0 I
Ω
I 0
φ
T
I
=
Ψ + ΦQΦ
T
−G
T
− ΦH
T
+ ΦQ
−G − HΦ
T
+ QΦ
T
−H − H
T
+ Q
.
Inequality (12) thus follows from the fact that the
(1,1) matrix block of the latter matrix is the left side
of (12).
Taking H = G[I
n
0 0 0]
T
, R
2
= ε
r
H where ε
r
> 0 is
a scalar tuning parameter and denoting M
H
= H
T
M,
we note that in (14) the system matrices, excluding
D and F, do not multiply Q. It is thus possible to
choose vertex dependent Q
(i)
while keeping H and G
constant. We thus arrive at the following result:
Corollary 1. The exponential stability in the mean
square sense of the system (2a) where B
1
= 0 and
where the system matrices lie within the polytope
¯
Ω
of (7) is guaranteed if there exist matrices 0 < Q
j
∈
R
n×n
, ∀ j = 1, ...N, 0 < R
1
∈ R
n×n
, M
H
∈ R
n×n
,
G ∈ R
n×4n
and a tuning scalar ε
r
> 0 that satisfy the
following set of inequalities:
Ω
j
=
Ψ
j
+ G
T
Φ
j,T
+ Φ
j
G −G
T
+ Φ
j
H
−G + H
T
Φ
j,T
−H − H
T
+ Q
j
< 0,
(15)
∀ j, j = 1, 2, ...., N, where H ∈ R
n×n
= G[I
n
0 0 0]
T
Ψ
j
∆
=
Ψ
j,1,1
0 0 Ψ
j,1,4
∗ Ψ
j,2,2
0 hε
r
(A
j,T
1
H − M
T
H
)
∗ ∗ −ε
r
H −hε
r
M
T
H
∗ ∗ ∗ −ε
r
H
,
Φ
j
H =
A
j,T
0
H + M
T
H
A
j,T
1
H − M
T
H
M
T
H
0
,
(16)
and where
Ψ
j,1,1
= −Q
j
+ D
T
(Q
j
+ h
2
ε
r
H)D+R
1
,
Ψ
j,1,4
= hε
r
(A
j,T
0
H + M
T
H
) − ε
r
hH,
Ψ
j,2,2
= −R
1
+F
T
(Q
j
+ h
2
ε
r
H)F.
Robust Vertex-dependant H
∞
Filtering of Stochastic Discrete-time Systems with Delay
623
4 ROBUST BOUNDED REAL
LEMMA
Based on the stability result of Corollary 1, the fol-
lowing result is readily obtained where we consider
the system (2a) with z
k
= C
1
x
k
and the following in-
dex of performance:
J
B
= E{x
T
k+1
Qx
k+1
} − x
T
k
Qx
k
+ z
T
k
z
k
− γ
2
w
T
k
w
k
.
Theorem 2 (Gershon and Shaked, 2013). Consider
the system (2a) and (3). The system is exponen-
tially stable in the mean square sense and, for a pre-
scribed scalar γ > 0 and a given scalar tuning parame-
ter ε
b
> 0, the requirement of J
B
< 0 is achieves for all
nonzero w ∈
˜
l
2
F
k
([0, ∞);R
q
), if there exist n ×n matri-
ces Q > 0, R
1
> 0 and a n × n matrix Q
m
that satisfy
˜
Γ < 0 where
˜
Γ =
˜
Γ
11
˜
Γ
12
0 0
˜
Γ
15
0 C
T
1
∗ −Q
˜
Γ
23
Q
m
0 QB
1
0
∗ ∗
˜
Γ
33
0
˜
Γ
35
0 0
∗ ∗ ∗ −ε
b
Q −hε
b
Q
T
m
0 0
∗ ∗ ∗ ∗ −ε
b
Q ε
b
hQB
1
0
∗ ∗ ∗ ∗ ∗ −γ
2
I
q
0
∗ ∗ ∗ ∗ ∗ ∗ −I
r
(17)
where
˜
Γ
11
= −Q + D
T
Q[1 + ε
b
h
2
]D + R
1
,
˜
Γ
12
= A
T
0
Q + Q
T
m
,
˜
Γ
15
= ε
b
h[A
T
0
Q + Q
T
m
] − ε
b
hQ,
˜
Γ
23
= QA
1
− Q
m
,
˜
Γ
33
= −R
1
+ (1 + ε
b
h
2
)F
T
QF,
˜
Γ
35
= ε
b
h[A
T
1
Q − Q
T
m
].
Similarly to the stability condition for the uncer-
tain case of Section 3 we obtain two results for the
robust BRL solution. The first one, which is referred
to as the quadratic solution, is simply obtained by as-
signing the same Lyapunov function over the all un-
certainty polytope and thus is solved similarly to the
robust quadratic condition . A new, possibly less con-
servative condition is obtained by applying the fol-
lowing vertex-dependent Lyapunov function:
Ψ
0 C
T
1
0 0
0 0
hR
2
B
1
0
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
−γ
2
I
q
0
∗ I
r
+
Φ
B
T
1
0
Q
Φ
T
B
1
0
∆
=
ˆ
Ψ +
ˆ
ΦQ
ˆ
Φ
T
< 0,
where Ψ and Φ are given in (13a,b).
Following the derivation of the LMI of Corollary
1, the following result is readily derived:
ˆ
Ψ +
ˆ
G
T
ˆ
Φ
T
+
ˆ
Φ
ˆ
G −
ˆ
G
T
+
ˆ
ΦH
−
ˆ
G + H
T
ˆ
Φ
T
−H − H
T
+ Q
< 0, (18)
where now
ˆ
G ∈ R
n×4n+q+r
and H ∈ R
n×n
. We thus
arrive at the following result for the uncertain case,
taking H =
ˆ
G[I
n
0 0 0]
T
, R
2
= ε
r
H, M
H
= H
T
M:
Corollary 2. Consider the system (2a) and (3)
where the system matrices lie within the polytope
¯
Ω
of (7). The system is exponentially stable in the mean
square sense and, for a prescribed γ > 0 and given
tuning parameter ε
r
, the requirement of J
B
< 0 is
achieved for all nonzero w ∈
˜
l
2
F
k
([0, ∞);R
q
), if there
exist 0 < Q ∈ R
n×n
, 0 < R
1
∈ R
n×n
, M
H
∈ R
n×n
,
and
ˆ
G ∈ R
n×4n+q+r
, that satisfy the following set of
LMIs:
ˆ
Ψ
j
+
ˆ
G
T
ˆ
Φ
j,T
+
ˆ
Φ
j
ˆ
G −
ˆ
G
T
+
ˆ
Φ
j
H
−
ˆ
G + H
T
ˆ
Φ
j,T
−H − H
T
+ Q
j
< 0,
(19)
where H ∈ R
n×n
=
ˆ
G[I
n
0 0 0]
T
ˆ
Ψ
j
=
Ψ
j
0 C
j,T
1
0 0
0 0
hε
r
B
j
1
0
∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
−γ
2
I
q
0
∗ I
,
ˆ
Φ
j
H =
Φ
j
H
B
j,T
1
H
0
, ∀ j , j = 1, 2, ...., N,
where Ψ
j
and Φ
j
H are given in (16).
5 DELAYED FILTERING
In this section we address the filtering problem of the
delayed state-multiplicative noisy system. We start
with the nominal case and then we bring the quadratic
solution of the uncertain polytopic case. We consider
the system of (2a-c) and (3) and the general type filter
of (4). Denoting ξ
T
k
∆
= [x
T
k
ˆx
T
k
], ¯w
T
k
∆
= [w
T
k
n
T
k
] we
obtain the following augmented system:
ξ
k+1
=
˜
A
0
ξ
k
+
˜
B ¯w
k
+
˜
A
1
ξ
k−τ(k)
+
˜
Dξ
k
ν
k
+
˜
Fξ
k−τ(k)
µ
k
,
˜z
k
=
˜
Cξ
k
, ξ
l
= 0, l ≤ 0
(20)
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
624
where
˜
A
0
=
A
0
0
B
c
C
2
A
c
,
˜
B =
B
1
0
0 B
c
D
21
,
˜
A
1
=
A
1
0
0 0
,
˜
D =
D 0
0 0
,
˜
C
T
=
C
T
1
−C
T
c
(21)
and
˜
F =
F 0
0 0
. Using the BRL result of Section
4 we obtain the inequality condition
˜
ϒ < 0 where
˜
ϒ =
˜
ϒ
11
˜
ϒ
12
0 0
˜
ϒ
15
0
˜
C
T
˜
ϒ
18
∗ −
˜
Q
˜
ϒ
23
˜
Q
M
0
˜
Q
˜
B 0 0
∗ ∗
˜
ϒ
33
0
˜
ϒ
35
0 0 0
∗ ∗ ∗ −ε
f
˜
Q
˜
ϒ
45
0 0 0
∗ ∗ ∗ ∗ −ε
f
˜
Q
˜
ϒ
56
0 0
∗ ∗ ∗ ∗ ∗ −γ
2
I 0 0
∗ ∗ ∗ ∗ ∗ ∗ −I
r
0
∗ ∗ ∗ ∗ ∗ ∗ ∗ −
˜
Q
(22)
where
˜
ϒ
11
= −
˜
Q +
˜
R
1
,
˜
ϒ
12
=
˜
A
T
0
˜
Q +
˜
Q
T
M
,
˜
ϒ
15
= ε
f
h[
˜
A
T
0
˜
Q +
˜
Q
T
M
] − ε
f
h
˜
Q,
˜
ϒ
18
=
˜
D
T
˜
Q
p
1 + ε
f
h
2
,
˜
ϒ
23
=
˜
Q
˜
A
1
−
˜
Q
M
,
˜
ϒ
33
= −
˜
R
1
+ (1 + ε
f
h
2
)
˜
F
T
˜
Q
˜
F,
˜
ϒ
35
= ε
f
h[
˜
A
T
1
˜
Q −
˜
Q
T
M
],
˜
ϒ
45
= −hε
f
˜
Q
T
M
,
˜
ϒ
56
= hε
f
˜
Q
˜
B.
Defining
˜
P =
˜
Q
−1
, denoting the following parti-
tions
˜
P =
X M
T
M T
,
˜
Q =
Y N
T
N W
,
and J =
X
−1
Y
0 N
, we multiply (22) by
ˆ
J = diag{
˜
PJ,
˜
PJ,
˜
PJ,
˜
PJ,
˜
PJ, I, I,
˜
PJ}
from the right and by
ˆ
J
T
, from the left. We obtain,
denoting
¯
R
p
= J
T
˜
P
˜
R
1
˜
PJ,
ˆ
ϒ
11
ˆ
ϒ
12
0 0
ˆ
ϒ
15
∗ −J
T
˜
PJ
ˆ
ϒ
23
J
T
˜
P
˜
Q
M
˜
PJ 0
∗ ∗ −
¯
R
p
0
ˆ
ϒ
35
∗ ∗ ∗ −ε
f
J
T
˜
PJ
ˆ
ϒ
45
∗ ∗ ∗ ∗ −ε
f
J
T
˜
PJ
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
0 J
T
˜
P
˜
C
T
¯
εJ
T
˜
P
˜
D
T
J 0
J
T
˜
B 0 0 0
0 0 0
¯
εJ
T
˜
P
˜
F
T
J
0 0 0 0
hε
b
J
T
˜
B 0 0 0
−γ
2
I
q+p
0 0 0
∗ −I
r
0 0
∗ ∗ −J
T
˜
PJ 0
∗ ∗ ∗ −J
T
˜
PJ
< 0,
(23)
where
ˆ
ϒ
11
= −J
T
˜
PJ +
¯
R
p
,
ˆ
ϒ
12
= J
T
˜
P
˜
A
T
0
J + J
T
˜
P
˜
Q
T
M
˜
PJ,
ˆ
ϒ
15
= ε
f
h[J
T
˜
P
˜
A
T
0
J + J
T
˜
P
˜
Q
T
M
˜
PJ] − ε
f
hJ
T
˜
PJ,
ˆ
ϒ
23
= J
T
˜
A
1
˜
PJ − J
T
˜
P
˜
Q
M
˜
PJ,
ˆ
ϒ
35
= ε
f
h[J
T
˜
P
˜
A
T
1
J − J
T
˜
P
˜
Q
T
M
˜
PJ],
ˆ
ϒ
45
= −hε
f
J
T
˜
P
˜
Q
T
M
˜
PJ,
¯
ε
2
= 1 + ε
f
h
2
.
Denoting
¯
X = X
−1
and
¯
X
y
=
¯
X
¯
X
¯
X Y
,
˜
P
M
= J
T
˜
P
˜
Q
T
M
˜
PJ,
we obtain:
−
¯
X
y
+
¯
R
p
Ψ
12
0 0 Ψ
15
∗ −
¯
X
y
Ψ
23
˜
P
M
0
∗ ∗ −
¯
R
p
0 Ψ
35
∗ ∗ ∗ −ε
f
¯
X
y
Ψ
45
∗ ∗ ∗ ∗ −ε
f
¯
X
y
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
Robust Vertex-dependant H
∞
Filtering of Stochastic Discrete-time Systems with Delay
625
0 J
T
˜
P
˜
C
T
¯
εJ
T
˜
P
˜
D
T
J 0
J
T
˜
B 0 0 0
0 0 0
¯
εJ
T
˜
P
˜
F
T
J
0 0 0 0
hε
b
J
T
˜
B 0 0 0
−γ
2
I
q+p
0 0 0
∗ −I 0 0
∗ ∗ −
¯
X
y
0
∗ ∗ ∗ −
¯
X
y
< 0,
(24)
where
Ψ
12
= J
T
˜
P
˜
A
T
0
J +
˜
P
M
,
Ψ
15
= ε
f
h[J
T
˜
P
˜
A
T
0
J +
˜
P
M
] − ε
f
h
¯
X
¯
X
¯
X Y
,
Ψ
23
= J
T
˜
A
1
˜
PJ −
˜
P
M
,
Ψ
35
= ε
f
h[J
T
˜
P
˜
A
T
1
J −
˜
P
M
],
Ψ
45
= −hε
f
˜
P
M
,
¯
ε
2
= 1 + ε
f
h
2
.
Carrying out the various multiplications and denoting
K
0
= N
T
A
c
M
¯
X, U = N
T
B
c
and Z = C
c
M
¯
X, we obtain
the following result:
−
¯
X
y
+
¯
R
p
˜
Ψ
12
0 0
˜
Ψ
15
∗ −
¯
X
y
˜
Ψ
23
˜
P
M
0
∗ ∗ −
¯
R
p
0
˜
Ψ
35
∗ ∗ ∗ −ε
f
¯
X
y
−hε
f
˜
P
M
∗ ∗ ∗ ∗ −ε
f
¯
X
y
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
0
˜
Ψ
17
˜
Ψ
18
0
˜
Ψ
26
0 0 0
0 0 0
˜
Ψ
39
0 0 0 0
˜
Ψ
56
0 0 0
−γ
2
I
q+p
0 0 0
∗ −I
r
0 0
∗ ∗ −
¯
X
y
0
∗ ∗ ∗ −
¯
X
y
< 0, (25)
where
˜
Ψ
12
=
A
T
0
¯
X A
T
0
Y +C
T
2
U
T
+ K
T
0
A
T
0
¯
X A
T
0
Y +C
T
2
U
T
+
˜
P
M
,
˜
Ψ
15
= ε
f
h
A
T
0
¯
X A
T
0
Y +C
T
2
U
T
+ K
T
0
A
T
0
¯
X A
T
0
Y +C
T
2
U
T
+ε
f
h
˜
P
M
− ε
f
h
¯
X
¯
X
¯
X Y
,
˜
Ψ
17
=
C
T
1
− Z
T
C
T
1
,
˜
Ψ
18
=
D
T
¯
X D
T
Y
D
T
¯
X D
T
Y
,
˜
Ψ
23
=
¯
XA
1
¯
XA
1
YA
1
YA
1
−
˜
P
M
,
˜
Ψ
26
=
¯
XB
1
0
Y B
1
UD
21
,
˜
Ψ
35
= ε
f
h
A
T
1
¯
X A
T
1
Y
A
T
1
¯
X A
T
1
Y
− ε
f
h
˜
P
M
,
˜
Ψ
39
=
F
T
¯
X F
T
Y
F
T
¯
X F
T
Y
,
˜
Ψ
56
= hε
f
¯
XB
1
0
Y B
1
UD
21
,
¯
ε
2
= 1 + ε
f
h
2
.
We thus arrive at the following theorem:
Theorem 3. Consider the system of (2a-c) and
(3). For a prescribed scalar γ > 0 and a positive tun-
ing scalar ε
f
, there exists a filter of the structure (4)
that achieves J
F
< 0, where J
F
is given in (6), for
all nonzero w ∈
˜
l
2
([0, ∞);R
q
), n ∈
˜
l
2
([0, ∞);R
p
), if
there exist n × n matrices
¯
X > 0, Y > 0, 2n × 2n ma-
trix
¯
R
p
> 0, n × n matrices K
0
and U, 2n × 2n matrix
˜
P
M
and a n × l matrix Z, that satisfy (25). In the latter
case the filter parameters can be extracted using the
following equations:
A
c
= N
−T
K
0
¯
XM
−1
, B
c
= N
−T
U, C
c
= Z
¯
XM
−1
.
(26)
Noting that XY − M
T
N = I, the filter matrix parame-
ters A
c
, B
c
, and C
c
can be readily found, without any
loss of generality, by a singular value decomposition
of I − XY.
In the uncertain case a robust filter is obtained by
either applying the quadratic solution (based on the
quadratic BRL) or applying the following vertex de-
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
626
pendent approach: Starting with (25), we define
Γ = diag{E, E, E, E, E, I, I, E, E}, E =
I −I
0 I
.
(27)
We then multiply (25) by Γ and Γ
t
, from the left and
the right respectively. We obtain then:
−V
i
+
ˆ
R
i
p
¯
ψ
i
12
0 0
¯
ψ
i
15
∗ −V
i
¯
ψ
i
23
ˆ
P
i
M
0
∗ ∗ −
ˆ
R
i
p
0 hε
f
¯
ψ
i,T
23
∗ ∗ ∗ −ε
f
V
i
−hε
f
ˆ
P
i
M
∗ ∗ ∗ ∗ −ε
f
V
i
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
0
−Z
T
C
i,T
1
¯
ψ
18
0
−∆B
i
1
−UD
i
21
Y
i
B
i
1
UD
i
21
0 0 0
0 0 0
¯
ψ
39
0 0 0 0
hε
f
−∆B
i
1
−UD
i
21
Y
i
B
i
1
UD
i
21
0 0 0
−γ
2
I
q+p
0 0 0
∗ −I 0 0
∗ ∗ −V
i
0
∗ ∗ ∗ −V
i
< 0
(28)
where
V
i
=
∆ −∆
−∆ Y
i
, ∆ =Y
i
−
¯
X
i
,
ˆ
R
i
p
=E
¯
R
i
p
E
T
,
¯
ψ
i
12
=
−K
T
0
K
T
0
−A
i,T
0
∆−C
i,T
2
U
T
A
i,T
0
Y
i
+C
i,T
2
U
T
+
ˆ
P
i
M
,
¯
ψ
i
15
= hε
f
(
¯
ψ
i
12
−V
i
),
¯
ψ
18
=
0 0
−D
i,T
1
∆ D
i,T
1
Y
i
,
¯
ψ
i
23
=
0 −∆A
i
1
0 Y
i
A
i
1
−
ˆ
P
i
M
,
¯
ψ
39
=
0 0
−F
i,T
∆ F
i,T
Y
i
,
ˆ
P
i
M
= E
¯
P
i
M
E
T
, .
The latter can be written as
Ψ
i
(∆,Y
i
, K
o
,U,
ˆ
P
i
M
Z) + ∆
i
Y
+ ∆
i,T
Y
< 0
where ∆
Y
+ ∆
T
Y
is the part that includes products of Y
i
with A
i
0
, A
i
1
, B
i
1
, D
i
1
and F
i
.
We readily find that ∆
Y
=
0 0 0 0
0
Y
i
0 A
i
0
0
0
Y
i
0 A
i
1
0
0 0 0 0
0 0 0 0
ε
f
h
0
Y
i
0 A
i
0
0 ε
f
h
0
Y
i
0 A
i
1
0
0 0 0 0
0 0 0 0
0
Y
i
0 D
i
0 0 0
0 0
0
Y
i
0 F
i
0
0 0 0 0 0
0
0
Y
i
B
i
1
0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 ε
f
h
0
Y
i
B
i
1
0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
(29)
In the case where there is no uncertainty in F and
D we find that ∆
i
Y
∆
= Θ
i
α
i,T
=
0
0
Y
i
0
0
hε
f
0
Y
i
0
0
0
0
× α
i,T
, α
i,T
=
0 A
i
0
T
0
0 A
i
1
T
0
0
B
i
1
0
T
0
0
0
(30)
In this case we have
Ψ
i
+ Θ
i
α
i,T
+ α
i
Θ
i,T
< 0. (31)
By Finsler’s method the latter is equivalent to the fol-
lowing inequality:
Ψ
i
+ Gα
i,T
+α
i
G
T
Θ
i
−G+α
i
H
∗ −H−H
T
< 0. (32)
The dimension of the new constant matrix G are iden-
tical to those of Θ
i
. The constant decision matrix H
is a square matrix. In the case of (30) it is a n× n ma-
trix. In We thus arrive at the following possibly less
conservative result:
Robust Vertex-dependant H
∞
Filtering of Stochastic Discrete-time Systems with Delay
627
Theorem 4. Consider the system of (2a-c) and (3)
where the system matrices lie within the polytope
¯
Ω
of (7). For a prescribed scalar γ > 0 and a positive
tuning scalar ε
f
, there exists a filter of the structure
(4) that achieves J
F
< 0, where J
F
is given in (6), for
all nonzero w ∈
˜
l
2
([0, ∞);R
q
), n ∈
˜
l
2
([0, ∞);R
p
), if
there exist n × n matrices
¯
X > 0, Y > 0, 2n × 2n ma-
trix
¯
R
p
> 0, n × n matrices K
0
and U, 2n × 2n matrix
˜
P
M
and a n × l matrix Z, and matrices H and G that
satisfy (31). In the latter case the filter parameters can
be extracted using (26) as explained in Theorem 3.
6 CONCLUSIONS
In this paper the theory of linear H
∞
filtering of state-
multiplicative noisy discrete-time delayed systems,
is extended to the robust polytopic vertex-dependant
case. Delay dependent analysis and synthesis meth-
ods are developed for the robust case which are based
on the input-output approach. Sufficient conditions
are thus derived for the robust stability of the system
and the existence of a solution to the corresponding
robust BRL. Based on the robust vertex-dependant
BRL derivation, the robust filtering problem is for-
mulated and solved.
An inherent overdesign is admitted to our solu-
tion due to the use of the bounded operators which
enable us to transform the retarded system to a norm-
bounded one. Some additional overdesign is also ad-
mitted in our solution due to the special structure im-
posed on R
2
.
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