Page 1

On H∞Control with Multiple Packet Dropouts:

Dealing with Repeated Scalar Nonlinearities

Hongli Dong, Zidong Wang, Jinling Liang and Huijun Gao

Abstract—This paper investigates the H∞ control problem

for a class of systems with repeated scalar nonlinearities and

multiple packet dropouts. The nonlinear system is described

by a discrete-time state equation involving repeated scalar

nonlinearities. The multiple packet-dropout phenomenon is

assumed to occur in both the senor-to-controller and the

controller-to-actuator channels, and the data missing law for

each individual sensor/actuator satisfies individual probabilistic

distribution in the interval [0 1]. An observer-based feedback

controller is designed to stochastically stabilize the closed-loop

control system and preserves a guaranteed H∞ performance.

Two examples are provided to show the applicability of the

proposed theoretical results.

I. INTRODUCTION

In the past few decades, the observer-based H∞ control

has gained a great deal of research attention due mainly to the

fact that this control can produce an estimate of the system

state and reduce the effect of the disturbance input on the

regulated output according to a prescribed level (see, e.g.,

[8], and references therein).

On the other hand, networked control systems (NCSs)

have been extensively studied and successfully applied in

a wide range of areas (see, e.g., [4], [9], [11], [12], and

references therein). However, most relevant literature has

been based on the implicit assumption that the packet dropout

problem occurs only in the channel from the sensor to the

controller. Another typical kind of packet dropouts, which

happen in the channel from the controller to the actuator,

has not yet been fully investigated [9]. Also, the probability

0 is usually used to account for an entire signal missing

and the probability 1 denotes the intactness, and all the

sensors or actuators are assumed to have the same missing

probability [6], [8]). Such a description, however, does have

its limitations since it cannot cover some practical cases, for

example, the case when only partial information is missing

This work was supported in part by the Engineering and Physical Sciences

Research Council (EPSRC) of the U.K. under Grant GR/S27658/01, in part

by the Royal Society of the U.K., in part by the Foundation for the Author

of National Excellent Doctoral Dissertation of China under Grant2007B4,

in part by the Heilongjiang Outstanding Youth Science Fund under Grant

JC200809, in part by the Postdoctoral Science Foundation of China, and in

part by the Alexander von Humboldt Foundation of Germany.

H. Dong is with the College of Electrical and Information Engineering,

Daqing Petroleum Institute, Daqing 163318, China, and is also with the

Space Control and Inertial Technology Research Center, Harbin Institute of

Technology, Harbin 150001, China.

Z. Wang is with the Department of Information Systems and Computing,

Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom.

(Email: Zidong.Wang@brunel.ac.uk)

J. Liang is with the Department of Mathematics, Southeast University,

Nanjing 210096, China.

H. Gao is with the Space Control and Inertial Technology Research

Center, Harbin Institute of Technology, Harbin 150001, China.

and the case when the individual sensor or actuator has

different missing probability [10].

Motivate by this, in this paper, the H∞control problem is

addressed for a class of nonlinear NCSs with random packet

dropouts in sensor-to-controller and controller-to-actuator

channels. The packet dropout phenomenon is assumed to be

random and could be different for individual sensor/actuator,

which is modeled by an individual random variable satisfying

certain probabilistic distribution on the interval [0 1]. The

repeated scalar nonlinearity [1], [3], [5], which typically

appears in recurrent neural networks, is employed to describe

the networked systems. The focus is on the design of an

observer-based controller, such that the closed-loop system

is stochastically stable and satisfies a prescribed disturbance

attenuation level.

II. PROBLEM FORMULATION

A. The Physical Plant

In this paper, we consider the discrete-time system with

repeated scalar nonlinearities described as following:

control input; zk∈ Rris the controlled output; yck∈ Rpis

the process output; wk∈ Rqis the disturbance input which

belongs to l2[0,∞); A,B1,B2,C1,C2,D1 and D2 are known

real matrices with appropriate dimensions. f is a nonlinear

function satisfying the following assumption as in [2].

Assumption 1: The nonlinear function f : R → R in sys-

tem (1) satisfies

xk+1= Af(xk)+B2uk+B1wk

zk=C1f(xk)+D1wk

yck=C2xk+D2wk

(1)

where xk∈ Rnrepresents the state vector; uk∈ Rmis the

∀a,b ∈ R

|f(a)+ f(b)| ≤ |a+b|.

(2)

In the sequel, for the vector x = [x1x2···xn]T, we denote

f(x)

△

= [f(x1) f(x2) ··· f(xn)]T.

B. The Controller

The dynamic observer-based control scheme for the sys-

tem (1) is described by

is the measured output, ˆ uk∈ Rmis the control input without

transmission missing, and L ∈ Rn×pand K ∈ Rm×nare the

observer and controller gains, respectively.

ˆ xk+1= Af(ˆ xk)+B2uk+L(yk− ˆ yk)

ˆ yk=C2ˆ xk

ˆ uk= Kˆ xk

(3)

where ˆ xk∈Rnis the state estimate of the system (1), yk∈Rp

Joint 48th IEEE Conference on Decision and Control and

28th Chinese Control Conference

Shanghai, P.R. China, December 16-18, 2009

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C. The Communication Links

In this paper, the measurement with multiple communica-

tion packet loss is described by

yk= Ξyck=

p

∑

i=1

αi(C2ixk+D2iwk),

(4)

where Ξ := diag{α1,...,αp} with αi (i = 1,...,p) being p

unrelated random variables which are also unrelated with wk.

It is assumed that αi has the probabilistic density function

qi(s) (i = 1,...,p) on the interval [0 1] with mathematical

expectation µiand variance σ2

i. C2iand D2iare defined by

C2i

:

= diag{0,··· ,0

? ?? ?

? ?? ?

i−1

,1,0,··· ,0

? ?? ?

,1,0,··· ,0

? ?? ?

p−i

}C2,

D2i

:

= diag{0,··· ,0

i−1

p−i

}D2.

αi could satisfy any discrete probabilistic distributions on

the interval [0 1], which includes the widely used Bernoulli

distribution as a special case. In the sequel, we denote¯Ξ =

E{Ξ}.

Similarly, the control input with multiple communication

packet loss is described by

uk= Ωˆ uk=

m

∑

j=1

βjKjˆ xk,

(5)

where Ω = diag{β1,...,βm} with βj(j = 1,...,m) being m

unrelated random variables and

Kj= diag{0,··· ,0

? ?? ?

j−1

,1,0,··· ,0

? ?? ?

m−j

}K.

It is assumed that βjhas the probabilistic density function

mj(s) on the interval [0 1] with mathematical expectation

ϑjand variance ξ2

j. We define¯Ω = E{Ω}.

D. The Closed-loop System

Letting the estimation error be

ek:= xk− ˆ xk,

(6)

the closed-loop system can be obtained as follows by sub-

stituting (3), (4) and (5) into (1) and (6)

or, in a compact form,

xk+1= Af(xk)+B2¯ΩKxk+B2(Ω−¯Ω)Kxk−B2¯ΩKek

−B2(Ω−¯Ω)Kek+B1wk

ek+1= A[f(xk)− f(ˆ xk)]+(LC2−L¯ΞC2)xk−L(Ξ−¯Ξ)C2xk

−LC2ek+?(B1−L¯ΞD2)−L(Ξ−¯Ξ)D2

?wk,

(7)

ςk+1=ˇAηk+¯Aςk+ψkˆAςk+¯Bwk

(8)

where

ςk

=

?

xT

k

eT

k

?T,ηk=?

,¯A =

fT(xk)

B2¯ΩK

LC2−L¯ΞC2

fT(xk)− fT(ˆ xk)

−B2¯ΩK

−LC2

?T

ˇA

=

?

A

0

0

A

??

?

ψk

=

?

?

B2(Ω−¯Ω)

0

0

L(Ξ−¯Ξ)

?

,

ˆA =

?

K

−K

0

−C2

?

(9)

¯B

=

B1

(B1−L¯ΞD2)−L(Ξ−¯Ξ)D2

?

.

Definition 1: [7] The solution ςk= 0 of the closed-loop

system in (8) as wk≡ 0 is said to be stochastically stable if,

for any ε > 0, there exists a δ > 0 such that E{||ςk||} < ε

whenever k ∈ I+and ||ς0|| < δ.

Assumption 2: [9] The matrix B2is of full column rank,

i.e., rank(B2) = m.

Remark 1: For the matrix B2 of full column rank, there

always exist two orthogonal matrices U ∈ Rn×nand V ∈

Rm×msuch that

?

U1 ∈ Rm×n

and

U2 ∈ R(n−m)×n,

diag{τ1,τ2,··· ,τm}, where τi (i = 1,2,··· ,m) are nonzero

singular values of B2.

Lemma 1: [9] For the matrix B2∈Rn×mwith full column

rank, if matrix P1is of the following structure:

˜B2=UB2V =

U1

U2

?

B2V =

?

Σ

0

?

,

(10)

where and

Σ =

P1=UT

?

P11

0

0

P22

?

U =UT

1P11U1+UT

2P22U2,

(11)

where P11∈ Rm×m> 0 and P22∈ R(n−m)×(n−m)> 0, and U1

and U2are defined in (10), then there exists a non-singular

matrix P ∈ Rm×msuch that B2P = P1B2.

Lemma 2: [7] If there exist a Lyapunov function V(ςk)

and a function Γ(r)∈OL satisfying the following conditions

V(0) = 0,

(12)

Γ(||ς||) ≤V(ς),

(13)

E?V(ςk+1)?−E{V(ςk)} < 0,

then the solution ςk=0 of the closed-loop system in (8) with

wk≡ 0 is stochastically stable.

Problem HCMDL (H∞ control with multiple data

losses):

For given the communication link parameters¯Ξ and¯Ω and

the scalar γ > 0, design the controller (3) for the system (1)

such that the closed-loop system satisfies the following two

performance requirements:

i) (stochastic stability) the closed-loop system in (8) is

stochastically stable in the sense of Definition 1;

ii) (H∞performance) under zero initial condition, the con-

trolled output zksatisfies ||z||E≤ γ||w||2, where

k ∈ I+,

(14)

?¯ z?E? E

??

∞

∑

k=0

zT

kzk

?

,

(15)

and ?·?2stands for the usual l2norm.

If the above two conditions are satisfied, the closed-loop

system is said to be stochastically stable with a guaranteed

H∞performance γ, and the problem HCMDL is solved.

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III. H∞CONTROL PERFORMANCE ANALYSIS

Theorem 1: Suppose that both the controller gain matrix

K and the observer gain matrix L are given. The closed-loop

system in (8) is stochastically stable with a guaranteed H∞

performance γ if there exist positive definite matrices P1, P2

and two scalars ρ1> 0, ρ2> 0 satisfying

?

Λ+Λ1

ΛT

Λ2

Λ3

2

?

< 0,

(16)

P1≤ ρ1I,

P2≤ ρ2I

(17)

where

Λ

=

2

?

B2¯ΩK

LC2−L¯ΞC2

B2¯ΩK

LC2−L¯ΞC2

−B2¯ΩK

−LC2

−B2¯ΩK

−LC2

?T?

?

P1

0

p

∑

i=1

?

0

P2

?

×

?

?

B2Kj

λmax(ATA), λ2= λmax(CT

?

−(B2¯ΩK)TP1B1−(LC2)TP2(B1−L¯ΞD2)

2BT

p

∑

i=1

(LC2−L¯ΞC2)TP2(B1−L¯ΞD2).

+

σ2

i¯CT

iP2¯Ci

+

2ρ1λ1I−P1

0

0

2ρ2λ1I−P2

?,

0

ρ2λ1I

p

∑

+

m

∑

j=1

ξ2

j¯BT

jP1¯Bj,

¯Bj

λ1

=

?

−B2Kj

¯Ci=?

−LC2i

0

?,

=

1C1),

?

i(LC2i)TP2(LD2i)+ˆΛ2

Λ1

=

ρ1λ1I+2λ2I

0

,

Λ2

=

(B2¯ΩK)TP1B1+

1P1B1+2(B1−L¯ΞD2)TP2(B1−L¯ΞD2)

i=1σ2

,

Λ3

=

+

σ2

i(LD2i)TP2(LD2i)+2DT

1D1−γ2I,

ˆΛ2

=

(18)

Proof: Assume wk≡ 0 and define the following Lya-

punov function candidate as:

Vk= xT

kP1xk+eT

kP2ek,

(19)

where P1and P2are solutions to (16)-(17).

E{∆Vk}

=

=

E{Vk+1|xk,...,x0,ek,...,e0}−Vk

E?xT

k+1P1xk+1+eT

k+1P2ek+1

?−xT

kP1xk

−eT

kP2ek

It follows from the definition (4) and (5) that

E{αi−µi}{αl−µl} =

?

?

σ2

i

0

i = l

i ?= l,

(i,l = 1,...,p),

E{βj−ϑj}{βq−ϑq} =

ξ2

j

j = q

j ?= q,

0

(j,q = 1,...,m),

(20)

and then we can obtain that

E?[B2(Ω−¯Ω)K(xk−ek)]TP1

=

∑

j=1

E{[−L(Ξ−¯Ξ)C2xk]TP2[−L(Ξ−¯Ξ)C2xk]}

p

∑

i=1

?B2(Ω−¯Ω)K(xk−ek)??

m

ξ2

j[B2Kj(xk−ek)]TP1[B2Kj(xk−ek)],

(21)

=

σ2

i[LC2ixk]TP2[LC2ixk].

then, we have

E{∆Vk} ≤ ςT

kΛςk.

By Schur complement, (16) implies that Λ < 0, hence

E{∆Vk} = ςT

kΛςk< 0

which satisfies (14). Taking Γ(xk,ek) = λmin(P1)x2

λmin(P2)e2

ksuch that Γ(xk,ek) ∈ OL, we obtain

λmin(P1)||xk||2+λmin(P2)||ek||2

≤

xT

k+

Γ(||xk,ek||)=

kP1xk+eT

kP2ek=V(xk,ek),

which satisfies (13). Considering V(0)=0, it follows readily

from Lemma 2 that the closed-loop system in (8) with wk≡0

is stochastically stable.

Next, the H∞ performance criterion for the closed-loop

system in (8) will be established. We denote

ˆΞ1:=

Af(xk)+B2¯ΩKxk+B2(Ω−¯Ω)Kxk−B2¯ΩKek

−B2(Ω−¯Ω)Kek+B1wk

A(f(xk)− f(ˆ xk))+(LC2−L¯ΞC2)xk−L(Ξ−¯Ξ)C2

×xk−LC2ek+((B1−L¯ΞD2)−L(Ξ−¯Ξ)D2)wk]T

ˆΞ2:=

Now, assuming zero initial conditions, we have:

E{Vk+1}−E{Vk}+E?zT

≤

wk

J

=

kzk

??

?−γ2E?wT

ςk

wk

,

kwk

?(22)

?

ςk

?T?

Λ+Λ1

ΛT

Λ2

Λ3

2

?

where Λ, Λ1, Λ2, Λ3come from (18). It follows from (16)

that J < 0, summing up (22) from zero to ∞ with respect to

k yields

∞

∑

k=0

E{?zk?2} < γ2

∞

∑

k=0

E{?wk?2}+E{V0}−E{V∞}.

Hence we can easy to conclude (15). The proof is complete.

IV. CONTROLLER DESIGN

Theorem 2: Consider the system (1). There exists a dy-

namic observer-based controller in the form of (3) such that

the closed-loop system in (8) is stochastically stable with a

guaranteed H∞performance γ, if there exist positive-definite

matrices P11∈ Rm×m, P22∈ R(n−m)×(n−m), P2∈ Rn×n, real

matrices Mj∈ Rm×n(j=1,...,m), N ∈ Rn×pand two scales

ρ1> 0, ρ2> 0 satisfying

?

P1≤ ρ1I,P2≤ ρ2I.

Π1

Π2

ΠT

Π3

2

?

< 0,

(23)

(24)

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Where

Π1= diag{−P1+3ρ1λ1I+2λ2I,−P2+3ρ2λ1I,

−γ2I+2DT

Π3= diag?−P1,−P2,−P1,−P2,−P1,−P2,−ˆP1,−ˆP2

ˆB = [ξ1MT

ˆC = [−σ1C21TNT,···,−σpC2pTNT]T,

ˆP1= diag{P1,··· ,P1

?

P1:= UT

1D1

?,

?,

Π2=

m

∑

j=1ϑjB2Mj

NC2−N¯ΞC2

m

∑

j=1ϑjB2Mj

NC2−N¯ΞC2

0

0

ˆB

ˆC

−

m

∑

j=1ϑjB2Mj

−NC2

m

∑

j=1ϑjB2Mj

−NC2

0

0

−ˆB

0

1BT

P1B1

P2B1−N¯ΞD2

−

0

0

P1B1

P2B1−N¯ΞD2

0

ˆD

mBT

,

2,···,ξmMT

2]T,

?? ?

m

}, ˆP2= diag{P2,··· ,P2

?

???

p

},

ˆD = [−σ1D21TNT,···,−σpD2pTNT]T,

1P11U1+UT

2P22U2.

(25)

Furthermore, the controller parameters are given by

K =

m

∑

j=1

VΣ−1P−1

11ΣVTMj,

L = P−1

2N.

(26)

Proof: Noticing

P1B2¯ΩK =

m

∑

j=1

ϑjP1B2Kj,

and applying the Schur complement to (16), we have

?

where

N1

m

∑

j=1ϑjP1B2Kj

N1

0

0

ˇB

ˇC

N1= P2LC2−P2L¯ΞC2,N2= P2B1−P2L¯ΞD2,

ˇB = [ξ1(P1B2K1)T,···,ξm(P1B2Km)T]T,

ˆC = [−σ1(P2LC21)T,···,−σp(P2LC2p)T]T,

ˇD = [−σ1(P2LD21)T,···,−σp(P2LD2p)T]T.

Π1

ˆΠ2

ˆΠT

Π3

2

?

< 0,

ˆΠ2=

m

∑

j=1ϑjP1B2Kj

−

m

∑

j=1ϑjP1B2Kj

−P2LC2

m

∑

j=1ϑjP1B2Kj

−P2LC2

0

0

−ˇB

0

P1B1

N2

−

0

0

P1B1

N2

0

ˇD

,

(27)

Since there exist P11> 0 and P22> 0 such that P1=

UT

1P11U1+UT

follows from Lemma 1 that there exists a non-singular matrix

P ∈ Rm×msuch that B2P = P1B2. Now let us calculate such

a matrix P from the relation B2P = P1B2as follows:

?

2P22U2, where U1and U2are defined in (10), it

P1UT

Σ

0

?

VT=UT

?

Σ

0

?

VTP,

i.e.

UT

?

P11

0

0

P22

??

Σ

0

?

VT=UT

?

Σ

0

?

VTP,

which implies that

P = (VT)−1Σ−1P11ΣVT.

(28)

Since B2P = P1B2, we define

Mj= PKj,

N = P2L,

(29)

and we can obtain (23) and (26) readily. The proof is now

complete.

Remark 2: As we can see from Theorem 2, in the pres-

ence of multiple random packet losses, the H∞ control

problem is solved for systems with repeated scalar nonlinear-

ities, and an observer-based feedback controller is designed

to stochastically stabilize the networked system and also

achieve the prescribed H∞disturbance rejection attenuation

level.

V. ILLUSTRATIVE EXAMPLES

In this section, two simulation examples are presented to

illustrate the usefulness and flexibility of the observer-based

controller design method developed in this paper.

Example 1: In this example, we are interested in design-

ing the observer-based controller for systems with repeated

scalar nonlinearities and multiple missing measurements. The

system data of (7) are given as follows:

A =

−1

0

0.5

−1

0

0.1

−1

0.2

0.3

00

0.6

0

0

0.3

1.2

1

0.3

0.1

0.6

0.5

−0.5

0.5

0.4

0.3

, B1=

0.1

0.2

0.6

0.8

0.4

1

0.1

0.2

1

0.2

1

0

1.9

−0.2

0

0.4

0.5

0.2

−1

−0.3

0.1

0.4

1

0.1

0

0.5

0.2

0.3

0.3

−0.1

,

,

B2=

, C1=

, D1=

−0.2

0

1

C2=

−0.4

0

,

D2=

0.1

0.2

0.4

.

Assuming that the probabilistic density functions of α1,

α2and α3in [0 1] are described by

q1(s1)=

0

s1= 0

s1= 0.5

s1= 1

s3= 0

s3= 0.5

s3= 1

0.2

0.8

,q2(s2) =

0.1

0.1

0.8

s2= 0

s2= 0.5

s2= 1

q3(s3)=

0

0.1

0.9

from which the expectations and variances can be easily

calculated as µ1= 0.9, µ2= 0.85, µ3= 0.95, σ1= 0.2,

σ2= 0.32 and σ3= 0.15. In the same way, we assume the

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probabilistic density functions of β1, β2and β3in [0 1] to

be

m1(s1)=

0

s1= 0

s1= 0.5

s1= 1

s3= 0

s3= 0.5

s3= 1

0.2

0.8

,m2(s2) =

0.05

0.15

0.8

s2= 0

s2= 0.5

s2= 1

m3(s3)=

0

0.4

0.6

from which we can calculate that ϑ1 = 0.9, ϑ2= 0.875,

ϑ3= 0.8, ξ1= 0.2, ξ2= 0.268 and ξ3= 0.245. Here, the

nonlinear function f(xk) = sin(xk) satisfies (2). By applying

Theorem 2, we can obtain an admissible solution as follows:

K

=

0.0044

−0.0031

−0.0247

−0.0500

−0.2554

0.0099

0.1395

0.4381

−0.1918

,

L

=

0.4538

0.0326

0.0401

0.9296

−0.0357

−0.0369

1.4553

0.2117

0.2550

.

For the purpose of simulation, we let the initial conditions

be x0= [1 0 0]T, ˆ x0= [0 0 0]T, and the disturbance input

be wk= [k−3k−2k−2]T. Fig. 1 displays the state responses

of the uncontrolled system, which are apparently unstable.

Fig. 2 shows the state simulation results of the closed-loop

system, from which we can see that the desired objective is

achieved.

Example 2: In this example, we aim to illustrate the

effectiveness of our results for different measurement missing

cases. Here

and the other system data of (7) is the same as in Example

1. We obtain an admissible solution as follows:

A =

0.4

0.1

0.5

−0.2

0.8

0.25

0.9

0.5

0

,

K

=

0.0044

−0.0026

−0.0248

−0.0518

−0.2622

0.0060

0.1429

0.4541

−0.1858

,

L

=

0.4517

0.0329

0.0398

0.9247

−0.0390

−0.0359

1.4496

0.2163

0.2523

,

for which the simulation result of the state responses is given

in Fig. 3 that confirms the realization of our design goal.

Next, let us consider the case when the multiple packet-

loss probability becomes higher. Take the probabilistic den-

sity functions of α1, α2and α3in [0 1] as

q1(s1)=

0.5

0

0.5

s1= 0

s1= 0.5

s1= 1

s3= 0

s3= 0.5

s3= 1

,q2(s2) =

0.4

0

0.6

s2= 0

s2= 0.5

s2= 1

q3(s3)=

0

0.5

0.5

and the probabilistic density functions of β1, β2and β3in

[0 1] as

m3(s3)=

By calculating the expectations and variances of the ran-

dom variables, we have arrived at the following solution:

L

=

Again, the simulation result of the state responses are de-

picted in Fig. 4. When the packet losses are more severe,

that is

corresponding to the case that 98% data are lost during the

communication, the simulation result of the state responses

are depicted in Fig. 5, which are apparently unstable. As we

can see from Figs. 3-5, when the packet losses are severer,

the dynamical behavior of the NCS takes longer to converge

or not converge to zero and, furthermore, the robustness of

the closed-loop system is rather degraded.

m1(s1)=

0.8

0.1

0.1

s1= 0

s1= 0.5

s1= 1

s3= 0

s3= 0.5

s3= 1

,m2(s2) =

0.2

0.1

0.7

s2= 0

s2= 0.5

s2= 1

0.2

0.1

0.7

.

K

=

−0.0335

−0.0401

−0.0009

−0.0603

−0.1444

−0.0068

0.2361

0.2679

−0.0907

,

,

0.4898

−0.0321

0.0072

1.1559

−0.1211

−0.0414

1.1026

−0.0294

0.0851

q1(s1) =

0.98

0

0.02

s1= 0

s1= 0.5

s1= 1

VI. CONCLUSIONS

This paper has solved the problem of observer-based H∞

control for systems with repeated scalar nonlinearities and

multiple missing measurements. The random communication

packet dropout have been allowed to occur, simultaneously,

in sensor-to-controller and controller-to-actuator channels,

and the missing probability for each sensor is governed by an

individual random variable satisfying a certain probabilistic

distribution in the interval [0 1]. In the presence of random

multiple missing measurements, an observer-based feedback

controller has been considered to stochastically stabilize the

networked system. Both the stability analysis and controller

design problems have been formulated. Simulations have

been carried out to show the effectiveness of the proposed

design scheme.

REFERENCES

[1] Y. Chu, Further results for systems with repeated scalar nonlinearities,

IEEE Trans. Automat. Control, Vol. 44, No. 12, pp. 2031-2035, 2001.

[2] Y. Chu and K. Glover, Bounds of the induced norm and model

reduction errors for systems with repeated scalar nonlinearities, IEEE

Trans. Automat. Control, Vol. 44, No. 3, pp. 4215-4226, 1999.

[3] Y. Chu and K. Glover, Stabilization and performance synthesis for sys-

tems with repeated scalar nonlinearities, IEEE Trans. Automat. Con-

trol, Vol. 44, No. 3, pp. 484-496, 1999.

[4] H. Gao and T. Chen, H∞estimation for uncertain systems with limited

communication capacity, IEEE Trans. Automat. Control, Vol. 52,

No. 11, pp. 2070-2084, 2007.

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