Page 1

An LMI Approach to Quantized H∞Control of Uncertain Linear

Systems with Network-Induced Delays

Hamid Reza Karimi, Ningsu Luo, Mauricio Zapateiro

Abstract—This paper deals with a convex optimization ap-

proach to the problem of robust network-based H∞ control

for linear systems connected over a common digital com-

munication network with norm-bounded parameter uncer-

tainties. Firstly, we investigate the effect of both the output

quantization levels and the network conditions under static

quantizers. Secondly, by introducing a descriptor technique,

using Lyapunov-Krasovskii functional and a suitable change of

variables, new required sufficient conditions are established in

terms of delay-range-dependent linear matrix inequalities for

the existence of the desired network-based quantized controllers

with simultaneous consideration of network induced delays

and measurement quantization. The explicit expression of the

controllers is derived to satisfy both asymptotic stability and

a prescribed level of disturbance attenuation for all admissi-

ble norm bounded uncertainties. One example is utilized to

illustrate the design procedure proposed in this paper.

I. INTRODUCTION

Networked control systems (NCS) in which control and

communication issues are combined together, and all the de-

lays and limitations of the communication channels between

sensors, actuators, and controllers are taken into account has

become an enabling technology for many military, commer-

cial and industrial applications. The study of NCSs is an

interdisciplinary research area, combining both network and

control theory. That is, in order to guarantee the stability and

performance of an NCS, analysis and design tools based on

both network and control parameters are needed. Modeling,

analysis, and design of NCSs have received increasing atten-

tion in recent years (see [1], [2], [8] and [21]).

However, due to network bandwidth restriction, the in-

sertion of communication network in the feedback control

loop inevitably leads to communication delays and makes

the analysis and design of NCSs complex. Communication

delays can deteriorate the performance of NCSs and even can

destabilize the systems when they are not considered in the

design of NCSs. So far, a variety of efforts have been devoted

This work has been partially funded by Norwegian Centre for Offshore

Wind Energy (NORCOWE), by the European Union (European Regional

Development Fund) and the Ministry of Science and Innovation of Spain

through the coordinated research projects DPI2008-06699-C02-01 and by

the Government of Catalonia (Spain) through SGR523. M. Zapateiro is

grateful to the grant of Juan de la Cierva program of the Ministry of Science

and Innovation (Spain).

Hamid Reza Karimiiswith

andScience, UniversityofAgder,

hamid.r.karimi@uia.no

Ningsu Luo is with the Institute of Informatics and Applications, Uni-

versity of Girona, Girona, Spain. ningsu.luo@udg.edu

MauricioZapateiroiswiththe

ematics III, Universitat Politcnica de Catalunya, Barcelona, Spain.

mauricio.zapateiro@upc.edu

the Facultyof Technology

Norway.N-4898Grimstad,

Departmentof AppliedMath-

to analyzing NCSs with communication delays (see, e.g., [3],

[4],[15]-[19] and the references therein). Specifically, [1] and

[20] analyzed the stability of NCSs and obtained stability

regions using a hybrid systems technique. [10] presented

linear matrix inequality (LMI) conditions for obtaining max-

imum allowable delay bounds, which guarantee the stability

of NCSs. Based on Lyapunov-Razumikhin function method,

[19] presented conditions on the admissible bounds of data

packet loss and delays for NCSs in terms of LMIs. Based

on stochastic control theory, optimal controller design of

NCSs with stochastic network delays was investigated in (

[7], [12]). For other control schemes, we refer readers to

the survey ([14]). Recently, the problem of output feedback

control for networked control systems (NCSs) with limited

communication capacity was studied by Tian et al. in [13].

In this paper, we contribute to the further development

of a convex optimization approach to the problem of ro-

bust network-based H∞control for uncertain linear systems

connected over a common digital communication network.

Here, We consider the case where quantizers are static and

the parameter uncertainties are norm bounded. Firstly, we

propose a new model to investigate the effect of both the out-

put quantization levels and the network conditions. Secondly,

by introducing a descriptor technique, using Lyapunov-

Krasovskii functional and a suitable change of variables,

new required sufficient conditions are established in terms

of delay-dependent linear matrix inequalities (LMIs) for the

existence of the desired network-based quantized controllers

with simultaneous consideration of network induced delays

and measurement quantization. The explicit expression of

the controllers is derived to satisfy both asymptotic stability

and a prescribed level of disturbance attenuation for all ad-

missible norm bounded uncertainties. A numerical example

is provided to illustrate the effectiveness of the approach

presented in this paper.

The notations used throughout the paper are fairly stan-

dard. Inand 0nrepresent, respectively, n by n identity matrix

and n by n zero matrix; the superscript T stands for matrix

transposition; ℜndenotes the n-dimensional Euclidean space;

ℜn×mis the set of all real m by n matrices. The matricesˆI

and˜I are defined, respectively, asˆI := [I,0] and˜I := [0,I].

?.? refers to the Euclidean vector norm or the induced matrix

2-norm and diag{···} represents a block diagonal matrix.

λmin(A) and λmax(A) denote, respectively, the smallest and

largest eigenvalue of the square matrix A. The operator

sym{A} denotes A+ATand [.] is the operation of round. The

notation P > 0 means that P is real symmetric and positive

definite and the symbol ∗ denotes the elements below the

Page 2

main diagonal of a symmetric block matrix.

II. SYSTEM DESCRIPTION

Consider the following continuous-time system with time-

varying structured uncertainties:

˙ x(t) = (A+∆A(t))x(t)+Du(t)+(B+∆B(t))w(t),

y(t) =Cx(t),

z(t) = Gx(t),

(1)

(2)

(3)

where x(t) ∈ ℜnis the state vector, y(t) ∈ ℜmis the mea-

sured output, considered as the control input; w(t) ∈ ℜl

and z(t) ∈ ℜrare the disturbance and the signal to be

estimated, respectively. The coefficient matrices A,B,C,G

are real matrices with appropriate dimensions. The time-

varying structured uncertainties ∆A(t) and ∆B(t) are said to

be admissible if the following form holds

?∆A(t)

∆B(t)?= M1F(t)?La

Lb

?,

(4)

where La,Lbare constant matrices with appropriate dimen-

sions; and F(t) is an unknown, real, and possibly time-

varying matrix with Lebesgue measurable elements, and its

Euclidean norm satisfies

?F(t)? ≤ 1, ∀t.

(5)

We are interested in investigating the stability property

of systems when the observer undergoes quantization and

delays. This kind of problem arises in scenarios in which

a finite bandwidth channel lies in the feedback loop and

introduces a delay.

In this paper, a quantizer means a piecewise constant

function q : ℜp→ Q, where Q is a finite subset of ℜl. We

will use quantized measurements of the form

qµ(z) := µq

?z

µ

?

=

µM∆,

−µM∆,

µ∆

?

z

µ> (M+0.5)∆

z

µ< −(M+0.5)∆

???z

z

µ

?

,

µ

??? ≤ (M+0.5)∆

(6)

where µ > 0 and the range of this quantizer is µM and the

quantization error is µ∆ ([11]).

The problem considered here is to design the signal u(t) by

a network-based quantized controller of a general structure

described by

˙ xf(t) = Afxf(t)+Bfµ1kq1

?y(ikh)

k,ik+1h+ηsc

µ1k

?

(7)

u(t) =Cfxf(t), t ∈ [ikh+ηsc

k+1)

(8)

where xf(t) is the controller state vector, µ1kq1

quantized plant output with ikh as the sampling instant of the

sensor and h as the sampling period, u(t) is the control signal

and Af,Bf,Cf are appropriately dimensioned matrices to be

designed. ηsc

k

denotes the transmission delay from sensor

to the controller. When considering the network conditions

from the controller to the plant output, the quantized output

signal can be expressed as

?u(jkh)

?y(ikh)

µ1k

?

is the

µ2kq2

µ2k

?

.

(9)

Define η1(t) =t −ikh−η1mfor t ∈ [ikh+ηsc

and η2(t) = t − jkh−η2m for t ∈ [jkh+ηca

with a natural assumption on the network induced delays as

follows

k,ik+1h+ηsc

k, jk+1h+ηca

k+1)

k+1)

η1m≤ ηsc

η2m≤ ηca

k≤ η1M

k≤ η2M

(10)

(11)

where constants ηimand ηiM, i = 1,2, denote the minimum

and maximum delays, respectively. ηca

sion delay from the controller to the actuator. Then, from

(10)-(11) we have

kdenotes the transmis-

0 ≤ ηi(t) ≤ ¯ ηi

(12)

where ¯ ηi:= ηiM−ηim. We assume that the values in both

sets {i1,i2,i3,···} and {j1, j2, j3,···} are ordered as follows

ik+1>ikand jk+1> jk, which means that there is no wrong

packet sequences in the network, and satisfy the following

conditions, respectively,

(ik+1−ik)h+ηsc

(jk+1− jk)h+ηca

k< η1M

k< η2M

(13)

(14)

Furthermore, it is noting that there are n − 1 continuous

packets dropped or lost if ik+1−ik= n(n ≥ 2) ([18]).

Replacing ikh and jkh in the quantized plant and controller

outputs with t−η1m−η1(t) and t−η2m−η2(t) , respectively,

in (7) and (9), we obtain

˙ xf(t) = Afxf(t)+Bfµ1kq1

?Cx(t −η1m−η1(t))

µ1k

?

= Afxf(t)+BfCx(t −η1m−η1(t))+Bfδ1(t)

(15)

and, for t ∈ [jkh+ηca

k, jk+1h+ηca

k+1) ,

µ2kq2

?u(jkh)

µ2k

?

=Cfxf(t −η2m−η2(t))+δ2(t)

(16)

where

δ1(t) = µ1kq1

?Cx(t −η1m−η1(t))

µ1k

?

−Cx(t −η1m−η1(t))

(17)

and

δ2(t) = µ2kq2

?Cfxf(t −η2m−η2(t))

µ2k

?

−Cfxf(t −η2m−η2(t))

(18)

By connecting the plant (1)-(3) and the controller (7)-(8) and

from the Leibniz-Newton formula, i.e.

X(t −η1m−η1(t)) = X(t −η1m)

−

?t−η1m

t−η1m−η1(t)

˙X(s) ds

(19)

we obtain the following closed-loop system as

˙X(t) = (¯A+∆¯A(t))X(t)+¯D1X(t −η2m)

Page 3

−¯D1

?t−η2m

?t−η1m

t−η2m−η2(t)

˙X(s) ds+¯B1X(t −η1m)

−¯B1

t−η1m−η1(t)

˙X(s) ds+¯B2δ1(t)+¯D2δ2(t)

+(¯B3+∆¯B3(t))w(t)

(20)

and

z(t) =¯C1X(t)

(21)

where X(t) = [x(t)T,xf(t)T]Tand

¯A =

?A

0

0

Af

?

DCf

0

,∆¯A(t) =

?∆A(t)

?0

?

0

00

?

,¯B1=

?

00

0

BfC

?

,

¯D1=

?0

0

?

,¯B2=

Bf

?

,¯C1=?G

,¯D2=

?D

0

?

,¯B3=

?B

0

?

,

∆¯B3=

?∆B(t)

0

0?

Finally, the problem of robust network-based H∞control for

uncertain linear systems with both the output quantization

levels and the network conditions can be expressed as below.

Problem: Given system (1)-(3), design the controller (7)-

(8) such that the augmented system (20)-(21) from w(t)

to z(t) is asymptotically stable with a prescribed H∞ per-

formance γ, that is ?z(t)?2

conditions for all admissible uncertain parameters.

2< γ2?w(t)?2

2under zero initial

III. H∞PERFORMANCE ANALYSIS

In this section, we investigate the problem of H∞perfor-

mance analysis for nominal system (1)-(3) with no uncer-

tainties and exactly known controller matrices. Specifically,

we will be concerned with the conditions under which the

closed-loop system with finite delay components is asymp-

totically stable from w(t) to z(t) with an H∞ performance

γ.

Theorem 1. Given the positive constants γ,∆i and

the matrices Af,Bf,Cf, if there exist positive-definite

matrices P1,R1,R2,S1,S2,Q1,Q2,Z1,Z2,T1,T2 and matrices

P2,P3,H1,H2,U1,U2,Ni,j(i = 1,2,··· ,4; j = 1,2,··· ,10) of

appropriate dimensions such that the following LMIs hold

Πη1mχ1

η2mχ2

η1Mχ3

∗−η1mT1

0

∗∗−η2mT2

∗∗∗−η1MQ1

∗∗∗

η2Mχ4

0

0

0

−η2MQ2

0

0

∗

< 0 (22)

?Hi

i,1,NT

Ui

Zi

∗

?

≥ 0 (i = 1,2)

(23)

with

Π = ΠT= [Πi,j]i,j=1,2,···,11, ˜Ni= N1,i+ N2,i+ N3,i+ N4,i ,

?P1

diag?R1+S1,∑2

Π1,2= PT

¯B1

χi = [NT

i,2,··· ,NT

i,10,0]T

(i = 1,2,··· ,4),

P=

0

P2

P3

?

and Π1,1=sym

?

PT

?0

I

¯A

−I

?

??

+ sym?˜N1ˆI?

¯B1

+∑2

i=1¯ ηiHi+

i=1ηiMQi+2¯ ηiZi+ηimTi

−N1,1+ˆI˜NT

,

?0

?

2, Π1,3=U1−PT

?0

?

−N1,1+

ˆI˜NT

3, Π1,4= −N3,1+ˆI˜NT

4, Π1,5= PT

?0

¯D1

?

− N2,1+ˆI˜NT

5,

Π1,6 = −N4,1 +ˆI˜NT

6,

Π1,7 = U2 − PT

?0

?0

0?T,

ˆITCTCˆI − NT

1,6,Π2,7 = −NT

Π2,10

= −NT

¯D1

?

+ˆI˜NT

7,

Π1,8

= PT

?0

?0

¯B2

?

?

+ ˆI˜NT

8,

Π1,9

= PT

¯D2

?

+ ˆI˜NT

9,

Π1,10 = PT

¯B3

+ˆI˜NT

10,

Π1,11 =

?¯C1

1

M2

Π2,2 =

−R1− R2− sym{N1,2},

Π2,4 = −N1,2− NT

Π2,8

= −NT

Π3,3

=−¯ η−1

Π3,5= −N2,3, Π3,6= −N4,3, Π4,4 = −R2− sym{N3,4},

Π4,5= −N2,4− NT

Π4,8 = −NT

S2−S1−sym?N2,5

Π5,7= −NT

2,7−

M2

2k

Π5,10= −NT

Π6,8

= −NT

Π6,9

= −NT

Π7,7 = −¯ η−1

M2

2k

Π10,10=−γ2I, Π11,11=−I and other elements Πi,jfor j ≥i

are equal to zero. Then, system (22)-(23) is asymptotically

stable with the H∞performance level γ > 0.

Proof. Firstly, we represent (20) in an equivalent descrip-

tor model form as

Π2,3 = −

∆2

1µ2

1k

1,3,

1,7,

1,10,

1,4,Π2,6 = −N4,2− NT

Π2,9

= −NT

∆2

1

M2

1k

1,8,

1,9,

1Z1

+

1µ2

ˆITCTCˆI,Π3,4

=−N3,3,

3,5,Π4,6= −N4,4− NT

3,8, Π4,9 = −NT

?+

2µ2

2,10, Π6,6= −S2− sym?N4,6

2Z2+

2µ2

3,6, Π4,7= −NT

3,10, Π5,5 =

fCf˜I, Π5,6= −N4,5−NT

3,7,

3,9, Π4,10 = −NT

˜ITCT

∆2

2µ2

fCf˜I,Π5,8= −NT

2

M2

2k

2,6,

∆2

2

˜ITCT

2,8, Π5,9= −NT

?, Π6,7= −NT

2,9,

4,7,

4,10,

4,8,

4,9,

Π6,10

= −NT

∆2

2

˜ITCT

fCf˜I, Π8,8 = Π9,9 = −I ,

˙X(t) = ξ(t),

0 = −ξ(t)+¯AX(t)+¯B1X(t −η1m)+¯D1X(t −η2m)

−¯B1

+¯B2δ1(t)+¯D2δ2(t)+¯B3w(t)

?t−η1m

t−η1m−η1(t)˙X(s) ds−¯D1

?t−η2m

t−η2m−η2(t)˙X(s) ds

(24)

Define the Lyapunov-Krasovskii functional [9]

V(t) =

5

∑

i=1

Vi(t)

(25)

where

V1(t) = X(t)TP1X(t) :=?X(t)T

ξ(t)T?TP

?X(t)

ξ(t)

?

,

V2(t) =

?t

?t−η1m

t−η1m

X(s)TR1X(s) ds

+

t−η1M

X(s)TR2X(s) ds

V3(t) =

?t

?t−η2m

t−η2m

X(s)TS1X(s) ds

+

t−η2M

X(s)TS2X(s) ds

Page 4

Vi+3(t) =

?t

?−ηim

?−ηim

−ηiM

?t

?t

?t

t+θξ(s)TQiξ(s) ds dθ

+2

−ηiM

t+θξ(s)TZiξ(s) ds dθ

+

−ηiM

t+θξ(s)TTiξ(s) ds dθ

with i = 1,2 and T = diag{I,0}. Differentiating V1(t) in t

we obtain

˙V1(t) = 2X(t)TP1˙X(t) = 2?X(t)T

= 2?X(t)T

?0

??t−η1m

ξ(t)T?PT

I

−I

?˙X(t)

0

?

ξ(t)T?PT

??0

¯A

??X(t)

ξ(t)

?

+

¯B1

?

X(t −η1m)+

?0

¯D1

?

??t−η2m

X(t −η2m)

−

?0

¯B1

t−η1m−η1(t)ξ(s) ds−

?0

¯D1

t−η2m−η2(t)ξ(s) ds

+

?0

¯B2

?

δ1(t)+

?0

¯D2

?

δ2(t)+

?0

¯B3

?

w(t)}

(26)

By Lemma 1 (in Appendix), it is clear that

−2?X(t)T

?t−η1m

ξ(t)T?PT

?0

¯B1

??t−η1m

t−η1m−η1(t)ξ(s) ds

≤

t−η1M

ξ(s)TZ1ξ(s) ds+ ¯ η1

?X(t)

ξ(t)

?T

H1

?X(t)

ξ(t)

?

+2

?X(t)

ξ(t)

?T

(U1−PT

?0

¯B1

?

)

?t−η1m

t−η1m−η1(t)ξ(s) ds

(27)

and, similarly,

−2?X(t)T

?t−η2m

ξ(t)T?PT

?0

¯D1

??t−η2m

t−η2m−η2(t)ξ(s) ds

≤

t−η2M

ξ(s)TZ2ξ(s) ds+ ¯ η2

?X(t)

ξ(t)

?T

H2

?X(t)

ξ(t)

?

+2

?X(t)

ξ(t)

?T

(U2−PT

?0

¯D1

?

)

?t−η2m

t−η2m−η2(t)ξ(s) ds

(28)

Differentiating other Lyapunov terms in (25) give

˙V2(t)+˙V3(t) = X(t)T(R1+S1)X(t)−X(t −η1m)T

×(R1−R2)X(t −η1m)−X(t −η2M)TS2X(t −η2M)

−X(t −η1M)TR2X(t −η1M)

−X(t −η2m)T(S1−S2)X(t −η2m)

(29)

and, using Jensen’s Inequality in Lemma 2 (in Appendix),

one gets, for i = 1,2,

˙Vi+3(t) ≤ ξ(t)T(ηiMQi+2¯ ηiZi+ηimTi)ξ(t)

−

?t

t−ηiM

ξ(s)TQiξ(s) ds−

?t−ηim

t−ηiM

ξ(s)TZiξ(s) ds

−¯ η−1

i

?t−ηim

t−ηim−ηi(t)ξ(s)TdsZi

?t−ηim

t−ηim−ηi(t)ξ(s) ds

−

?t

t−ηim

ξ(s)TTiξ(s) ds.

(30)

Moreover, from the Leibniz-Newton formula, the follow-

ing equations hold for any matrices {Ni}10

dimensions, for i = 1,2,:

i=1with appropriate

2ν(t)TTi(X(t)−X(t −ηim)−

?t

t−ηim

ξ(s) ds) = 0(31)

2ν(t)TTi+2(X(t)−X(t −ηiM)−

?t

t−ηiM

ξ(s) ds) = 0(32)

where

?t−η1m

state vector. According to the property of the quantizers

qi(.) and using the Leibniz-Newton formula, we readily

obtain

ν(t)

:=

col{X(t),ξ(t),X(t −η1m),

t−η1m−η1(t)ξ(s)

η2M),?t−η2m

ds,X(t − η1M),X(t − η2m),X(t −

t−η2m−η2(t)ξ(s) ds,δ1(t),δ2(t),w(t)} is an augmented

0 ≤ −δi(t)Tδi(t)+

∆2

iµ2

i

M2

ik

(x(t −ηim)

−

?t−ηim

t−ηim−ηi(t)˙ x(s) ds)TCTC(x(t −ηim)

−

?t−ηim

t−ηim−ηi(t)˙ x(s) ds).

(33)

Now, to establish the H∞ performance measure for the

system (1)-(3), assume zero initial condition, then we have

V(t)|t=0= 0. Consider the index J∞ in the form J∞=

?∞

?∞

From (26)-(30), (34) and adding the left and right sides of

equations (31)-(32) and (33), respectively, into˙V(t), we get

0[z(t)Tz(t)−γ2w(t)Tw(t)] dt , then along the solution of

(1) for any nonzero w(t) there holds

J∞≤

0

[e(t)Te(t)−γ2w(t)Tw(t)+˙V(t)] dt

(34)

J∞≤

?∞

0

ν(t)TΣν(t) dt

(35)

where

η1Mχ3Q−1

J∞< 0 which means that the L2-gain from the disturbance

w(t) to the controlled output z(t) is less than γ . By applying

Schur complements, we find that Σ < 0 is equivalent to

(22). ⊳

Σ := Π + η1Mχ1T−1

1χT

1

χT

Now,

1

+ η2Mχ2T−1

if

Σ < 0,

2

χT

2

+

3+ η2Mχ4Q−1

2χT

4.then

Page 5

IV. ROBUST H∞CONTROL DESIGN

In this section we investigate the robust network-based

H∞control design problem for system (1)-(3) with the norm

bounded uncertainty parameters defined in (4).

Theorem 2. Consider system (1)-(3) with the quantizer

given in (6). Given positive constants ε,γ

there exist a network-based quantized controller in the

form of (7)-(8) such that the closed-loop system (20)-

(21) is asymptotically stable with an H∞ disturbance

attenuation level γ if there exist the scalar ρ > 0, positive-

definite matrices P1,R1,R2,S1,S2,Q1,Q2,Z1,Z2,T1,T2 and

matrices

ˆP1,ˆP2,P22,¯P11,W1,W2,W3,H1,H2,U1,U2,Ni,j(i =

1,2,··· ,4; j = 1,2,··· ,10) of appropriate dimensions and

satisfying (23) and the LMI

˜ΠΓT

d

∗−ρI

∗∗

and ∆i,

ρΓT

0

−ρI

e

< 0(36)

with

˜Π :=

ˆΠ

∗

∗

∗

∗

η1mχ1

−η1mT1

∗

∗

∗

η2mχ2

0

−η2mT2

∗

∗

η1Mχ3

0

0

−η1MQ1

∗

η2Mχ4

0

0

0

−η2MQ2

Γd

?La

and P2=

=

?εMT

1P2

Lb

P11

P22

MT

0?

and

1P2

with ˆΠ =ˆΠT= [ˆΠi,j]i,j=1,2,···,11,

···

00?,

Γe

=

0

···

?P11

0

P22

?

ˆΠ1,1:= sym{

ε

?PT

?PT

2

∑

i=1

11A

PT

11A

PT

W1

W1

W1

W1

11A

?

P1−εPT

2

11A

?

−PT

2

}

?

+

2

∑

i=1

¯ ηiHi+diag

?

R1+S1,

ηiMQi+2¯ ηiZi+ηimTi

+sym?˜N1ˆI?,

− N1,1

ˆΠ1,2

=(εˆIT

+

˜IT)

?W2C

?W2C

0

0

W2C

?

0

0

+

ˆIT˜NT

2,

ˆΠ1,3 = U − (εˆIT+˜IT)

W2C

?

+ˆIT˜NT

3,

ˆΠ1,5 =

(εˆIT+˜IT)

?0

DW3

DW3

+ˆI˜NT

0

?

?

− N2,1+ˆI˜NT

5,

ˆΠ1,7 = U2− (εˆIT+

˜IT)

other elementsˆΠi,j are equal to their counterpart elements

in the matrix Π. Moreover, if the above conditions are

feasible, desired controller gain matrices are given by

Cf= (¯PT

?0

DW3

DW3

0

7, ˆΠ1,8= (εˆIT+˜IT)

?W2

W2

?

+ˆIT˜NT

8and

11)−1W3and

?Af

Bf

?= (PT

0?VT, P11 = V

22)−1?W1

W2

?

where DT= U?ˆD

UˆDˆP1ˆD−1UTwith the unitary matrices U,V and a diagonal

matrixˆD with positive diagonal elements in decreasing order.

?ˆP1

0

ˆP2

0

?

VT,

¯P11 =

Proof. The state-space matrices ¯A and ¯B3 in (22) are

replaced with¯A+M1F(t)Laand¯B3+M1F(t)Lb, respectively.

By considering P3= εP2with DTP11=¯P11DTand introduc-

ing change of variables

?W1

W2

?= PT

W3=¯PT

22

?Af

Bf

?

(37)

11Cf

(38)

then the inequality (22) is equivalent to the following con-

dition:

˜Π+sym?ΓT

By Lemma 3 (in Appendix), a necessary and sufficient

condition for (39) is that there exists a scalar ρ such that

˜Π+ρ−1ΓT

dF(t)Γe

?< 0(39)

dΓd+ρΓT

eΓe< 0(40)

then, applying Schur complements, we find that (40) is

equivalent to (36). ⊳

Remark 2. In Theorem 2, the results are expressed within

the framework of LMIs, which can be easily computed by

the interior-pint method.

V. NUMERICAL RESULTS

Consider an uncertain linear system with the system

matrices given by

A =

?−1

?0.1

0.4

0.10.2

?

?

,B =

?1

?0

1

?

,C =

?1

1

?

,G =?11?,

La=

0

0.10.05

,Lb=

0.1

?

,M1= I,F(t) = csin(t)

where |c| ≤ 1. It is assumed that the network-induced delay

bounds are given by η1m=η2m=20ms,η1M=η2M=200ms.

In addition, the quantizer parameters in (5) are assumed to

be ∆1= ∆2= 0.1,M1= M2= 5 and the sampling period

h = 20ms. By using the convex problem in Theorem 2

with a constant ε = 0.1, we obtain the minimum guaranteed

performance in terms of the feasibility of (23) and (36) as

γ = 0.45.

05 1015

−2

−1

0

1

2

3

4

5

6

7

Time (s)

z(t)

c=0

c=0.3

c=0.6

Fig. 1. Controlled output signals for the plant.

With the initial conditions x(0) = [1,1]Tand xf(0) =

[0,0]T, and an exogenous disturbance input as a unit step

function within [0,1], then the controlled output signal z(t)