Filtering of Polytopic-Type Uncertain State-Delayed Noisy Systems

Eli Gershon

Holon Institute of Technology, HIT, Holon, Israel

−

Keywords:

Robust Estimation, Stochastic Systems, Finsler Lemma, Polytopic Uncertainties.

Abstract:

The problem of H

∞

state estimation is considered for uncertain polytopic retarded linear discrete-time stochas-

tic systems. We ﬁrst bring the solution of the estimation problem for the nominal case based on a previously

developed BRL for state-delayed stochastic systems. We then extend our solution to the robust uncertain

polytopic case where a vertex-dependent approach is applied. The latter is achieved via the application of a

modiﬁed version of the Finsler lemma. The use of this lemma enable us to derive a solution which is less

conservative comparing to the ”quadratic” solution where a single Lyapunov function is applied over all the

uncertain polytope. The solution obtained for the robust case is composed of a set of LMIs based on only two

tuning parameters. The theory presented is demonstrated by a numerical example.

1 INTRODUCTION

We address the problem of state-estimation of lin-

ear stochastic state-multiplicative systems which are

state-delayed and may contain large uncertainties. In

our study we make use of a general-type ﬁlter and

we ﬁrst bring the solution of the estimation problem

for nominal systems via a single LMI condition based

on a Bounded Real Lemma (BRL) for these systems

(Gershon and Shaked, 2013),(Fridman, 2014). Based

on the latter solution, we solve the estimation prob-

lem for polytopic-type uncertain systems where we

apply a special version of the Finsler lemma. This

lemma enable us to assign a different Lyapunov func-

tion to each vertex of the uncertain polytope, thus

greatly reducing the over-design which is inherent to

the quadratic solution, which, in turn, is based on as-

signing the same Lyapunov function over all the un-

certain polytope.

The ﬁeld of control and estimation of stochastic

state-multiplicative noisy systems has been greatly

developed since the onset of the H

∞

control theory

in the early 80‘s (see (Gershon and Shaked, 2013),

(Gershon et al., 2005) and the references therein).

Within a span of more than four decades, many ap-

proaches to the study of the various stochastic con-

trol and ﬁltering problems, including those that en-

sure a worst case performance bound in the H

∞

sense,

have been derived for both: delay-free systems (Ger-

shon et al., 2005), (Dragan and Stoica, 1998),(Dragan

and Morozan, 1997a), (Dragan and Morozan, 1997b),

(Dragan et al., 1992), (Hinriechsen and Pritchard,

1998), (Bouhtouri et al., 1999) and state-delayed, lin-

ear, stochastic systems (Gershon et al., 2007), (Ger-

shon and Shaked, 2011), (Verriest and Florchinger,

1995), (Mao, 1996), (Xu et al., 2005), (Chen et al.,

2005), (Gao and Chen, 2007), (Yue et al., 2009).

Delay-free systems with parameter uncertainties

that are modeled as white noise processes in a

linear setting have been treated in (Gershon and

Shaked, 2013), (Gershon et al., 2005), (Dragan and

Stoica, 1998) for both the continuous-time and the

discrete-time cases. Such models of uncertainty are

encountered in many areas of applications such as:

nuclear ﬁssion and heat transfer, population models

and immunology. In control theory, such models are

encountered in gain scheduling when the scheduling

parameters are corrupted with measurement noise.

The study of both continuous-time and discrete-

time retarded systems has been developed extensively

since the emergence of optimal control theory.

In the ﬁeld of control and estimation theory, the

input-output approach has been applied, in the last

two decades, to both continuous and discrete-time

retarded systems. The input-output approach is

based on the representation of the system’s delay

action by delay-free linear operators which allows

one to replace the underlying system with one that

possesses a norm-bounded uncertainty, and therefore

may be treated by the theory of norm bounded uncer-

tain, non-retarded systems with state-multiplicative

Gershon, E.

Filtering of Polytopic-Type Uncertain State-Delayed Noisy Systems.

DOI: 10.5220/0013715500003982

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 22nd International Conference on Informatics in Control, Automation and Robotics (ICINCO 2025) - Volume 1, pages 399-405

ISBN: 978-989-758-770-2; ISSN: 2184-2809

399

noise (Gershon et al., 2005). Based on the latter

approach, the stability and BRL issues of stochastic

state-multiplicative systems were obtained (Ger-

shon and Shaked, 2013), followed by the solution

of various control problems that include, among

others, state-feedback control, ﬁltering and static and

dynamic output-feedback control. We note that the

robust solution of both the state-feedback control and

ﬁltering problems for uncertain systems in (Gershon

and Shaked, 2013), rely on assigning a single Lya-

punov function over all the uncertainty polytope (the

so called ”quadratic” solution) thus leading to the

most conservative solution type (see (Gershon and

Shaked, 2013) and the references therein). Obviously,

the latter handicap can be relaxed by resorting to

vertex-dependent solution which is the merit of the

present work.

Similarly to the systems treated in (Gershon and

Shaked, 2013), in the present theory, our systems may

encounter a time-varying delay where the uncertain

stochastic parameters multiply both the delayed and

the non delayed states in the dynamics state-space

model of the systems as well as the non-delayed states

in the system measurement. We treat both the nomi-

nal case and the uncertain case where the system ma-

trices encounter polytopic type uncertainties. In the

latter case, we apply the Finsler lemma which leads

to a less conservative solution compared to the sim-

ple “quadratic” solution where the same decision vari-

ables are assigned to all the vertices of the polytope.

The paper is organized as follows: Following

the problem formulation in Section 2, we bring

a preliminary result in Section 3. Based on the

latter result, we bring the solution of the improved

robust vertex-dependant H

∞

ﬁlter in Section 4. In

Section 5 an example is given that demonstrates the

applicability and tractability of the various solution

methods derived in this work.

Notation: Throughout the paper the superscript

‘T ’ stands for matrix transposition, R

denotes the n

dimensional Euclidean space and R

n×m

is the set of

all n ×m real matrices. For a symmetric P ∈ R

n×n

P > 0 means that it is positive deﬁnite. We de-

note expectation by E {·} and we provide all spaces

, k ≥ 1 with the usual inner product < ·, · > and

with the standard Euclidean norm ||·||. By ||f (t)||

we denote the product of f

(t)R f (t). We denote by

(Ω, R

) the space of square-integrable R

−valued

functions on the probability space (Ω, F , P ), where

Ω is the sample space, F is a σ algebra of a sub-

sets of Ω called events and P is the probability mea-

sure on F . By (F

)

t>0

we denote an increasing

family of σ-algebras F

⊂ F . We also denote by

([0, ∞);R

) the space of nonanticipative stochas-

tic processes f (·) = ( f (t))

t∈[0,∞)

in R

with respect

to (F

)

t∈[0,∞)

satisfying

||f (·)||

= E{

∞

||f (t)||

dt} =

∞

E{||f (t)||

}dt < ∞,

and by C o{

Ω

, ...,

Ω

} the convex hull of the ma-

trices

Ω

, i = 1, ..., N. Stochastic differential equa-

tions will be interpreted to be of It ˆo type (Hinriechsen

and Pritchard, 1998).

2 PROBLEM FORMULATION

We consider the following nominal linear stochastic

retarded system:

dx =[A

x(t) + A

x(t −τ(t)) + B

w(t)]d t+

Hx(t −τ(t))dν(t)+Gx(t)dβ(t), x(θ)=0, θ≤ 0,

y(t) = C

x(t) + D

n(t) + Fx(t)dζ(t),

z(t) = Cx(t),

(1a-c)

where x(t) ∈ R

is the state vector, w(t) ∈

([0, ∞);R

) is an exogenous disturbance, y(t) ∈

is the measurement vector, n(t) ∈

([0, ∞);R

)

is an additive measurement noise, z(t) ∈ R

is the ob-

jective vector, A

, A

, B

, C, C

, D

and F, G, H are

time-invariant matrices of the appropriate dimension.

τ(t) is an unknown time-delay which satisﬁes:

0 ≤ τ(t) ≤ h,

τ(t) ≤ d < 1.

(2a,b)

The zero-mean real scalar Wiener process

β(t), ν(t), ζ(t) satisfy:

E{ζ(t)ζ(s)} = min(t, s), E {β(t)β(s)} =min(t, s),

E{ν(t)ν(s)} = min(t, s),

E{β(t)ν(s)}= 0, E{ν(t)ζ(t)}= 0, E{β(t)ζ(s)}= 0

In the uncertain case, we assume that the system

matrices in (1a-c), lie within the following polytope:

Ω

∆



A A

C C

F G H



(3)

which is described by the vertices:

Ω = C o{

Ω

, ...,

Ω

}, (4)

where

Ω

∆

(i)

(5)

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

400

and where N is the number of vertices. In other

words:

Ω =

∑

i=1

Ω

∑

i=1

= 1, f

≥ 0. (6)

We treat the following problem:

i) Robust H

∞

Filtering:

We consider the system of (1a-c) where the system

matrices lie within the polytope

Ω of (3). We consider

an estimator of the following general form:

d ˆx(t) = A

ˆx(t)dt + B

y(t),

ˆz(t) = C

ˆx(t).

(7a,b)

We denote

e(t) = x(t) − ˆx(t) and ¯z(t) = z(t) − ˆz(t), (8)

and consider the following cost function:

∆

= ||¯z(t)||

−γ

[||w(t)||

+ ||n(t)||

]. (9)

Given γ > 0 , we seek an estimate C

ˆx(t) of

Cx(t) over the inﬁnite time horizon [0, ∞) such

that J

given by (9) is negative for all nonzero

w(t) ∈

([0, ∞);R

), n(t) ∈

([0, ∞);R

3 PRELIMINARY RESULT

In this section we bring the solution of the continuous-

time robust quadratic ﬁltering problem of uncer-

tain polytopic systems with state-multiplicative noise.

The result in the sequel was obtained in (Gershon and

Shaked, 2013), Chapter 2, Theorem 2.12).

Denoting ξ

(t)

∆

= [x(t)

ˆx(t)

], ¯w

(t)

∆

[w(t)

n(t)

] we obtain the following augmented

system:

dξ(t) = [

ξ(t) +

B ¯w(t)]dt +

ξ(t −τ(t))d t

Hξ(t −τ(t))dν(t) +

Gξ(t)dβ(t) +

Fξ(t)dζ(t),

ξ(θ) = 0, over[−h 0],

˜z(t) =

Cξ(t),

(10)

where





B =



0 B





0 0



H =



H 0

0 0



G =



G 0

0 0



F =



0 0

F 0



C = [C

−C

(11)

where the system matrices lie within the polytope

Ω of (3). We obtain the following lemma:

Lemma 1. (Gershon and Shaked, 2013) Con-

sider the system of (1a-c) where the system matri-

ces lie within the polytope

Ω of (3). For a prescribed

scalar γ > 0 and tuning scalar ε

, there exists a ﬁlter

of the structure (7) that achieves (9) for all nonzero

w ∈

([0, ∞);R

), n ∈

([0, ∞);R

), if there exist

matrices

X > 0, Y > 0, K

, U,

M, Z, that satisfy

(13).







i,11

i,12

i,14

i,15

∗ −

0 0

i,25

∗ ∗ −ε

0 −ε

∗ ∗ ∗ −γ

i,45

∗ ∗ ∗ ∗ −ε

∗ ∗ ∗ ∗ ∗

(12)

i,16

i,17

i,18

0 0 0

i,29

0 0 0 0

−I

0 0 0

∗∗ −

0 0

∗ ∗ −

∗ ∗ ∗ −







< 0, (13)

∀i, i = 1, 2, ...., N, where

i,11

+ A

i,T

i,11a

+UC

+ K

+ A

i,T

i,11b

M +

1−d

i,11a

+ A

i,T

Y +C

i,T

+ K

i,11b

= YA

+UC

+ A

i,T

Y +C

i,T

i,12





−

i,14



Y B



Filtering of Polytopic-Type Uncertain State-Delayed Noisy Systems

401

i,15

= ε

i,T

X A

i,T

Y +C

i,T

+ K

i,T

X C

i,T

i,17



i,T

X G

i,T

X G

i,T



i,18



0 F

i,T

0 F

i,T



i,25

= ε

i,T

X A

i,T

X A

i,T

+ ε

i,29



i,T

X H

i,T

X H

i,T



i,45

= ε

i,14

(14)

and

i,16



i,T

−Z

−C

i,T



. In the latter case the ﬁlter

parameters can be extracted as follows:

= N

−T

−1

, B

= N

−T

U, C

= Z

−1

(15)

Proof: see (Gershon and Shaked, 2013), Section

2.6.1.

4 ROBUST h

∞

FILTERING

The solution of the ﬁltering problem for uncer-

tain systems has already appeared in (Gershon and

Shaked, 2013) for the most conservative case where

the same Lyapunov function is assigned all over the

uncertainty polytope. In this section we show that

relying on Lemma 1, one can manipulate the re-

sulting LMI condition of the latter lemma in such

a way that a vertex-dependent solution is obtained

for a minimal number of tuning parameters. Even-

tually, the quadratic solution can be derived from

our new vertex-dependent solution as a special case.

We consider the result of Lemma 1 and we denote

E =



−I



and J

= diag{E, E, E,

I, E, I

, E, E, E}. We then

consider the inequality of

< 0 where,

∆

= J







i,11

i,12

i,14

i,15

i,16

i,17

i,18

∗ −

0 0

i,25

0 0 0

i,29

∗ ∗

i,33

i,35

0 0 0 0

∗ ∗ ∗ −γ

i,45

0 0 0 0

∗ ∗ ∗ ∗

i,33

0 0 0 0

∗ ∗ ∗ ∗ ∗ −I

0 0 0

∗ ∗ ∗ ∗ ∗ ∗

i,33

0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗

i,33

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

i,33







(16)

= E

M = E

I =



−I



i,33

= −ε

diag{

, ∆

i,35

= −ε

i,11



−∆

∆

+ B

∆





−∆

∆

+ B

∆



(1−d)

i,12



∆



−

i,14



−

∆

−∆

+ B



i,15

= ε

i,T

∆

i,T

−∆

0 −A

i,T

∆

−∆

+ε

i,16



i,T

+∆

−2C

i,T

−∆



i,17



i,T

∆

0 0



i,18



0 F

i,T

0 0



i,25

= ε

i,T

∆

0 0

+ε

i,29



i,T

∆

0 0



and

i,45

= ε

i,14

(17)

Remark 1: As given in (Gershon and Shaked,

2013), Section 2.6.1, since Y X + N

M = I we chose

N = I and obtained that K

= −A

∆ where ∆ =Y −

We also have then that U = B

and that M

X = −∆.

We also denote:

∆

= diag{∆

, ∆

, 0

5n+2q

, ∆

, 0

n+r

, ∆

, 0

, ∆

∆

= diag{

, 0

5n+2q

, 0

n+r

, 0

∆









0 −A

 

0 0





0 0

−B



2n,2q

2q,2n



0 0

−A





0 0





0 0

−B



r,2n

r,2q



0 0



2n,2q



0 0



2n,2q

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

402



0 −A

 

−C



2n,6n

2n,r

2n,6n

2n,r

2n,6n

2q,2n

2q,r

2q,6n

2n,r

2n,6n

r,2n

r,6n

2n,r

2n,6n

2n,r

2n,6n

2n,r

2n,6n







and Φ







2n,6n+r

2n,2q

2n,6n+r

2n,2q

2n,6n+r

2q,2n

2q,2q

2q,2n

2q,6n+r

2n,6n+r

r,2n

r,2q

r,2n

r,6n+r

2n,2q

2n,6n+r

2n,2q

2n,6n+r



0 0



2n,2q

2n,6n+r







(18a-d)

where



0 0



, Φ



0 0





−B

0 0



, Φ



0 0



where 0

and 0

m,r

are the m ×m and m ×r matri-

ces of zeros, respectively.

We also deﬁne:

−

∆

−Φ

i,T

∆

−

−Φ

i,T

(19)

and build below the following requirement by apply-

ing the Finsler lemma

∆

+ Φ

T,i

∆

+ Φ

T,i

< 0 (20)

Denoting:

= diag{S

, S

, 0

5n+2q

, S

, 0

n+r

, S

, 0

, S

= diag{S

, 0

5n+2q

, S

, 0

n+r

, S

, 0

, S

, 0

= diag{ε

, S

, I

5n+2q

, H

, 0

n+r

, H

, 0

, H

= diag{H

, I

5n+2q

, H

, I

n+r

, H

, I

, H

, 0

where ε

is a positive tuning parameter. We thus

obtain the following theorem:

Theorem 1: Consider the system of (1a-c) where

the system matrices lie within the polytope

Ω

of (3). For a prescribed scalar γ > 0 and tun-

ing positive scalars ε

, ε

, there exists a ﬁlter of

the structure (7) that achieves (9) for all nonzero

w ∈

([0, ∞);R

), n ∈

([0, ∞);R

), if there

exist a matrix B

and positive deﬁnite matrices:

∆

, i = 1, ..., N, H

, H

, S

and S

that, for

i = 1, ..., N, solve the following set of inequalities:



Ω

∗ −2diag{

}



< 0, i = 1, ..., N (21)

where:

∆









∆



i,T

∆

i,T



 







i,11

i,12

M Ψ

i,14

i,15

i,16

i,17

i,18

∗ −

0 0 Ψ

i,25

0 0 0 Ψ

i,29

∗ ∗ X

∆

0 −ε

0 0 0 0

∗ ∗ ∗ −γ

I Ψ

i,45

0 0 0 0

∗ ∗ ∗ ∗ X

∆

0 0 0 0

∗ ∗ ∗ ∗ ∗ −I

0 0 0

∗ ∗ ∗ ∗ ∗ ∗ X

∆

0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ X

∆

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ X

∆







(22)

with:

I =



−I



, X

∆

= −ε

diag{

, ∆

i,11



−

+ B



+ [



−

+ B



(1−d)

i,12





−

i,14



−S

+ B



i,15

= ε

i,T

−

0 −A

i,T

−

+ ε

i,16

i,T

−2C

i,T

−

, Ψ

i,17



i,T

0 0



i,18



0 F

i,T

0 0



, Ψ

i,25

= ε

i,T

0 0

+ε

, Ψ

i,29



i,T

0 0



i,45

= ε

hΨ

i,14

(23)

Filtering of Polytopic-Type Uncertain State-Delayed Noisy Systems

403

and where we denoted

= A

and

= C

Ω







Ω



−A

 



2n,n

Ω



 



2n,n

Ω

hε



−B



2n,n

Ω



∆

−S



2n,n

Ω

r,n

2n,n



∆

−S



2n,n

4n,2n

4n,n



3n,n

∆

−S









where

Ω



∆

−S

−ε

∆

−S

+ ε



Ω





Ω



q,n

−ε



Ω



−ε



Ω





is the matrix that is obtained by deleting in the ma-

trix

∆

−

+ Φ

∆

the 9n + 2q + r columns that are

identically zero, and

Ω







−S

hε

3n,n

q,n

−B

q,n

2n,n



−S



2n,n

r,n

−S

3n,n

−S







is the matrix that is obtained by deleting the 10n+

r + 2q zero columns in

−

+ Φ

= diag{ε

, S

, H

and

= diag{H

, H

Proof: The result of (21) is obtained by applying the

Finsler lemma twice. We ﬁrst apply the latter to:

∆

+ Φ

i,T

∆

< 0, where

i,T

and obtain the requirement:



∆

+ Φ

i,T

∆

−

+ Φ

i,T

∆

∗ −2



< 0.

To the (1,1) block of the latter inequality we apply

again the Finsler lemma and readily obtain (21).

Remark 2: In the above, we consider S

∆

= A

and S

∆

= C

in the LMI of (21) to be decision vari-

ables.

5 EXAMPLE

We consider the system (1a,c) which is described by:



−2 ±a 0

1 −1



, A



−1 0

−0.5 −1



G = H =

√

0.1I

, with

= 0, D

= 0, C



1 2



F =



0.03 0.02



, D

= 0.01 and

= [0 0], C



−0.5 1



, d = 0, where a = 0.9.

We assume that the stochastic multiplicative noise

processes in both the delayed and the non delayed

states are uncorrelated. Considering the nomi-

nal case where a=0 and taking h = 0.1 and applying

the results of Theorem 2.10 in (Gershon and Shaked,

2013) for the near minimum attenuation level of γ =

0.136 is obtained.

Considering the uncertain case and applying the var-

ious solution methods, the results of Table 1 are ob-

tained. There, the quadratic H

∞

solution method was

calculated by assigning a single Lyapunov function

over the uncertainty polytope and is given in Lemma

1. We note that in the case where h = 0.8 the H

∞

attenuation level is γ = 41.16 whereas the quadratic

approach of Lemma 1 fails to solve the problem. We

note also the signiﬁcant reduction in the H

∞

system

norm, in the improved vertex-dependent case, com-

pared to the quadratic solution.

The resulting ﬁlter parameters for the robust case

are:



−25.8337 10.2329

−20.8500 0.4604



, B



−0.3239

−2.5225



= [24.6821 −14.0506].

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

404

Table 1: The results of the various solution methods in Ex-

ample 1. The nominal solution for the H

∞

case was obtained

in (Gershon and Shaked, 2013), Chapter 2, Theorem 2.10.

The ”Robust Vertex-dependent” (RVd) result refers to the

application of the Finsler lemma in the solution of the ro-

bust case [Theorem 1].

Solution Method γ

Nominal 0.136, (Theorem 2.10. )

Quadratic 0.61, (Lem. 1)

RVd 0.48, ε

= 0.001, ε

= 5,[Thm. 1]

6 CONCLUSIONS

In this paper the theory of robust linear H

∞

estima-

tion of state-multiplicative noisy systems is developed

and extended for state-delayed continuous-time un-

certain systems with multiplicative noise, that is en-

countered in both the dynamic and the measurement

matrices in the state space model of the system. Suf-

ﬁcient conditions are derived for the estimation prob-

lem of uncertain polytopic-type systems by applying

a vertex-dependent Lyapunov function, based on the

Finsler lemma. This approach enables us to apply a

unique Lyapunov function to each vertex of the uncer-

tain polytope. As shown in the example, the vertex-

dependent approach performs better the the solution

which is based on a single Lyapunov function over all

the uncertain polytope. We note also that our solution

depends only on two tuning parameters that can be

readily determined.

REFERENCES

Bouhtouri, A. E., Hinriechsen, D., and Pritchard, A. (1999).

∞

type control for discrete-time stochasic systems. In

Int. J. Robust Nonlinear Control, vol. 9, pp. 923-948.

Chen, W. H., Guan, Z., and Lu, X. (2005). Delay-dependent

exponential stability of uncertain stochastic systems

with multiple delays: an lmi approach. In Systems

and Control Letters, vol. 54, pp. 547-555.

Dragan, V. and Morozan, T. (1997a). Mixed input-output

optimization for time-varying ito systems with state

dependent noise. In Dynamics of Continuous, Discrete

and Impulsive Systems, vol. 3, pp. 317-333.

Dragan, V. and Morozan, T. (1997b). Mixed input-output

optimization for time-varying ito systems with state

dependent noise. In Dynamics of Continuous, Discrete

and Impulsive Systems,vol. 3, pp. 317-333.

Dragan, V., Morozan, T., and Halanay, A. (1992). Optimal

stability compensator for linear systems with state de-

pendent noise. In Stochastic Analysis and Application,

vol. 10, pp. 557-572.

Dragan, V. and Stoica, A. (1998). A γ attenuation prob-

lem for discrete-time time-varying stochastic systems

with multiplicative noise. In Reprint series of the In-

stitute of Mathematics of the Romanian Academy, vol.

10. 317-333.

Fridman, E. (2014). Introduction to Time Delay Systems:

Analysis and Control. Birkhauser, London, 1st edi-

tion.

Gao, H. and Chen, T. (2007). New results on stability of

discrete-time systems with time-varying state delay.

In IEEE Trans. on Automat. Contr., vol. 52(2), pp.

328-334.

Gershon, E. and Shaked, U. (2011). Robust h

∞

output-

feedback control of state-multiplicative stochastic sys-

tems with delay. In International Journal of Robust

and Nonlinear Control, vol. 21(11), pp. 1283-1296.

Gershon, E. and Shaked, U. (2013). Advanced Topics in

Control and Estimation of State-multiplicative Noisy

Systems. Lecture Notes in Control and Information

sciences, LNCIS, Springer, Vol. 439, London, 1st edi-

tion.

Gershon, E., Shaked, U., and Berman, N. (2007). H

∞

control

and estimation of state-multiplicative stochastic sys-

tems with delay. In IEEE Trans. on Automat. Contr.,

vol. 52(9), pp. 329-334.

Gershon, E., Shaked, U., and Yaesh, I. (2005). H

∞

con-

trol and Estimation of State-multiplicative linear sys-

tems. Lecture Notes in Control and Information sci-

ences, LNCIS, Springer, London, 1st edition.

Hinriechsen, D. and Pritchard, A. (1998). Stochastic h

∞

. In

SIAM Journal of Control Optimization, vol. 36(5), pp.

1504-1538.

Mao, X. (1996). Robustness of exponential stability of

stochastic differential delay equations. In IEEE Trans.

on Automat. Contr., vol. 41, pp. 442-447.

Verriest, E. I. and Florchinger, P. (1995). Stability of

stochastic systems with uncertain time delays. In Sys-

tems and Control Letters, vol. 24(1), pp. 41-47.

Xu, S., Lam, J., Mao, X., and Zou, Y. (2005). A new

lmi condition for delay-dependent robust stability of

stochastic time-delay systems. In Asian Journal of

Control, vol. 7, pp. 419-423.

Yue, D., Tian, E., and Zhang, Y. (2009). Stability analysis

of discrete systems with stochastic delay and its appli-

cations. In International Journal of Innovative Com-

puting, Information and Control, vol. 5(8), pp. 2391 -

2403.

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