An LMI approach to quantized H∞ control of uncertain linear systems with networkinduced delays

Mauricio Zapateiro
Universidade Tecnológica Federal do Paraná, Cornelio Procopio, Brazil
ABSTRACT This paper deals with a convex optimization approach to the problem of robust networkbased H_{∞} control for linear systems connected over a common digital communication network with normbounded parameter uncertainties. 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 LyapunovKrasovskii functional and a suitable change of variables, new required sufficient conditions are established in terms of delayrangedependent linear matrix inequalities for the existence of the desired networkbased 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 admissible norm bounded uncertainties. One example is utilized to illustrate the design procedure proposed in this paper.

Article: Robust H8 Output Feedback Control for a Class of Networked Control Systems with Long TimeDelay
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ABSTRACT: This paper studies the robust H∞ output feedback control problem for a class of uncertain networked control systems (NCSs) with long timedelay. Firstly, the NCS model with a random long timedelay is transformed into a discretetime system model with uncertain parameters. And then, a sufficient condition for the solvability of the robust H∞ dynamic output feedback control problem is proved and presented by a proper Lyapunov function and linear matrix inequality. The systems under action of the given controller are robust and satisfy H∞ performance. Finally, a numerical example is provided to demonstrate the validity of the proposed method.01/2011;
Page 1
An LMI Approach to Quantized H∞Control of Uncertain Linear
Systems with NetworkInduced Delays
Hamid Reza Karimi, Ningsu Luo, Mauricio Zapateiro
Abstract—This paper deals with a convex optimization ap
proach to the problem of robust networkbased H∞ control
for linear systems connected over a common digital com
munication network with normbounded 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 LyapunovKrasovskii functional and a suitable change of
variables, new required sufficient conditions are established in
terms of delayrangedependent linear matrix inequalities for
the existence of the desired networkbased 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 DPI200806699C0201 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.N4898Grimstad,
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 LyapunovRazumikhin 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 networkbased 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 delaydependent linear matrix inequalities (LMIs) for the
existence of the desired networkbased 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 ndimensional 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
2norm 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 continuoustime 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 networkbased 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 LeibnizNewton 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 closedloop 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 networkbased 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
closedloop 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 positivedefinite
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 LyapunovKrasovskii 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 LeibnizNewton 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 LeibnizNewton 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 L2gain 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 networkbased
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 networkbased quantized controller in the
form of (7)(8) such that the closedloop 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 statespace 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 interiorpint 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 networkinduced 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)