Journal of Signal and Information Processing, 2012, 3, 377-381
http://dx.doi.org/10.4236/jsip.2012.33049 Published Online August 2012 (http://www.SciRP.org/journal/jsip) 377
Robust Non-Fragile Control of 2-D Discrete Uncertain
Systems: An LMI Approach
Paramanand Sharma, Amit Dhawan
Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabd, India.
Received December 23rd, 2011; revised January 31st, 2012; accepted February 20th, 2012
ABSTRACT
This paper considers the problem of robust non-fragile control for a class of two-dimensional (2-D) discrete uncertain
systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under controller gain varia-
tions. The parameter uncertainty is assumed to be norm-bounded. The problem to be addressed is the design of
non-fragile robust controllers via state feedback such that the resulting closed-loop system is asymptotically stable for
all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such
controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. Fi-
nally, a numerical example is illustrated to show the contribution of the main resu lt.
Keywords: 2-D Discrete Systems; Fornasini-Marchesini Second Local State-Space Model; Non-Fragile Control;
Linear Matrix Inequality; Lyapuno v Methods
1. Introduction
In recent years, the research on two-dimensional (2-D)
discrete systems have received considerable attention,
since 2-D systems exist in many practical applications
such as image data processing, seismographic data proc-
essing, thermal processes, gas absorption, water stream
heating, river pollution modeling, etc. [1-5]. The stability
properties of 2-D discrete systems described by the For-
nasini-Marchesini second local state-space (FMSLSS)
model [6] have been studied in [7-15]. The asymptotic
stability conditions for linear FMSLSS model based on
2-D Lyapunov equation approach have been established
in [7-10]. Many publications related to stability analysis
of 2-D discrete systems employing various finite word-
length nonlinearities hav e appeared [10-1 4 ]. The prob lem
of robust stability analysis and stabilization of 2-D dis-
crete systems via the linear matrix inequality (LMI) ap-
proach has been considered in [15].
Recently, there has been a growing interest in the study
of robust non-fragile control problems. The aim of robust
non-fragile control is to design a robust controller for a
given uncertain system such that the controller is insensi-
tive to som e amount of error with re gard to its gain. Based
on this idea, many significant results have been obtained
for one-dimensional case [16-22]. Robust non-fragile
control for 2-D discrete uncertain systems in the FMSLSS
setting is an important prob lem.
This paper, therefore, deals with the problem of robust
non-fragile control for a class of 2-D discrete uncertain
systems described by the FMSLSS model. The paper is
organized as follows. In Section 2, we formulate the
problem of robust non-fragile control for a class of 2-D
discrete uncertain systems described by the FMSLSS
model and recall some useful results. The main result of
the paper is presented in Section 3. In Section 4, a nu-
merical example is given to illustrate the effectiveness of
the proposed method.
Notations denotes real vector space of dimension
n,
n
R
nm
R

is the set of nm

real matrices, 0 denotes null
matrix or null vector of appropriate dimension, I is the
identity matrix of appropriate dimension, the superscript T
stands for matrix transposition, () stands for
the matrix G is symmetric and positive (negative) definite,
and diag{···} stands for a block diagonal matrix.
0G0G
2. Problem Formulation and Preliminaries
This paper studies the problem of robust non-fragile con-
trol for a class of 2-D discrete uncertain systems de-
scribed by the FMSLSS model [6]. Specifically, the sys-
tem under consideration is given by



 
 
11
221 2
1, 11,
,1 1, ,1
ij ij
iji jij
 
,


xAAx
AAxBuBu
(1a)
where


,n ij Rx and


,m ij Ru are the state and

Robust Non-Fragile Control of 2-D Discrete Uncertain Systems: An LMI Approach
378
the control input, respectively. The matrices 12 , nn R

AA
A
and 12 are known constant matrices repre-
senting the nominal plant. The matrices 1 and 2
, nm R BB 

A
represent parameter uncertainties in the system matrices
which are assumed to be of the form





12
 1 ,ij2
A
ALF MM
, (1b)
where nk
R 
L
, 12 are known structural
matrices of uncertainty and is an un-
known matrix representing parameter uncertainty which
satisfies
, ln R MM F

,ijkl R




,
,,
valently, ,1
Tij ij
ij


FF I
For equi
mn
R 
.

(1c)
Suppose the system state is available for feedback, the
objective of this paper is to develop a procedure to de-
sign a non-fragile state feedback control law

,ij ijuKKx
, (2a)
where
K
is the nominal controller gain and

K
represents the control l er gain perturbati on of the form

,
kk k
ij
K
LF M, (2b)
with and
mg
kR
Lhn
kR

M

, being known constant
matrices, and
g
h
ij R


k an unknown uncertain
parameter matrix satisfying
F







1,
,1
ij
ij


Kx
x
,,
valently, ,
T
kk
k
ij ij
ij


FF
F

or equi

1, 1ijxA
P
1 ,
I

B
K
(2c)
for system (1) such that the resulting closed-loop system


11 11
22 22
 
 
BKA
ABK AB(3)
is asymptotically stable for all admissible uncertainties
and perturbation in controller gain.
Now, we recall the following lemmas, which are
needed in the proof of our main result. As an extension
of [7], one can easily arrive at the following lemma.
Lemma 2.1 [7] The system (3) is asymptotically stable
if there exists an positive definite symmetric ma-
trix such that nn


112 2112 2
00,
01
T




  







AA AAPAA AA
P
P

(4a)
for all admissible uncertainties (1b) and (2b) satisfying
(1c) and (2c), respectively, where
1112 22
1112 22
,,
,,
01. 
 

 

AABKAABK
AABKAABK
(4b)
Lemma 2.2 [23] Let H, E, F and M be real matrices
of appropriate dimension with M satisfying T

MM
then
0
TT T

MHFEEFH (5a)
for all F satisfying T

F
FI, if and only if there exists
a scalar 0

 such that
10 TT



MHH EE. (5b)
Lemma 2.3 [24] For real matrices M, L, Q of appro-
priate dimensions, where and
T
MM 0
T

QQ
then 0
T

MLQL if and only if
10
T






ML
LQ or equivalently 10
T






QL
LM . (6)
3. Main Result
In this section, we give a LMI-based sufficient condition
for the existence of non-fragile robust controllers in the
form of (2a) with the gain perturbation satisfying (2b)
and (2c), such that the resultin g closed-loop system (3) is
asymptotically stable for all admissible parameter uncer-
tainties and controller ga in variation s .
Theorem 3.1 Consider the system (1a) and the con-
troller gain perturbation

K
in (2b) and (2c). Then
the robust non-fragile control problem is solvable if
there exist positive scalars 1

and 2

, an mn

ma-
trix U, and an nn

positive definite symmetric matrix
S with a fixed 01


 such that the following LMI
holds:

222
T
kk


 

 
121111 22
11 1
22 2
121
2
2
000
00
010 0
00
000
0000
TT T
kk
TTT
k
TTk
k
k







 

 


T





0
0
T







S






LLBLLBASBUASBU
AS BUSSMSM
AS BUSSMSM
MSMS I
MS I
MS I
BLLB
. (7)

Robust Non-Fragile Control of 2-D Discrete Uncertain Systems: An LMI Approach 379
In this case, a state feedback controller chosen by
1


K
US (8)
will be such that, for all admissible un certainties (1b) and
(1c), and controller gain variations in (2b) and (2c), the
resulting closed-loop system (3) is asymptotically stable.
Proof: Applying Lemma 2.3, (4a) can be written as



 
11122
11
22
00
01
T
T











 


PAAAA
AA P
AA P

0
0
0
. (9)
Using (1b), (2b) and (4b), (9) can be represented as
 

 
 


 


1112 212
11 1
2
22
12
1
2
0,,0 0
000 0,0
00 0,0
01
0, ,000
000, 00
000, 00
TTT T
TT T
T
kk kkk kTTTT
kk k
TT TT
kk k
ij ij
ij
ij
ij ij
ij
ij






 


 
 


 


 


 








PABKABKLFM LFM
ABKPMF L
MF L
ABK P
BLFMBLFM
MF LB
MF LB
0.






(10)
Equation (10) can be rewritten as
 

 
  




1112 2
11 12 12
22
12
12
00,0 0,
00
01
,00 0
00 0,00
00
00,00
000 ,
T
T TT
T
kk
kk
kk
T
Tkk
kk
kk
ij ij
ij
ij
ij
ij





0


 


 
 


 













 









PABKABK

L
L
ABKPFM MM MF
ABK P
BL BLFM
FM
BL BL
MF
MF 00.
00
T






(11)
Using Lemma 2.2, (11) can be rearranged as



 
11211222 1122
11 1
1 11112112
11
22121 1222
0.
1
TTT TT
kk kk
TTT T
kk
TTT
kk
 
 


 

 
 
 
 

 
 

 
PLLBLLBBLLBA BKABK
A BKPMMMMMM
A BKMMPMMMM
1T


(12)
Premultiplying and postmultiplying (12) by


11
diag ,,

IP P, one obtains



 
121122211 22
11
22
11 1
1112 112
111
121 1222
0
01
00 0
00
0
TTT TT
kk kk
T
T
TTT
kk
TTT
kk
 


 

 


 















,


S
LLBLLBBLLBAS BUAS BU
AS BUS
AS BUS
SM MSSM MSSM MS
SM MSSM MSSM MS
(13)
where 1 
S
P and 1


K
US .
The equivalence of (13) and (7) follows trivially from
Lemma 2.3. This completes the proof of Theorem 3.1.
4. Numerical Example
As an illustration of Theorem 3.1, consider a 2-D discrete
uncertain system represented by (1) with

Robust Non-Fragile Control of 2-D Discrete Uncertain Systems: An LMI Approach
380
1
00
10



A
2
0.008
0




B
, , ,
, ,
2
0.1 1
00



A
0
1



L
1
0.002
0.002



B


1
M0.7 0.008,


20.5 0.02M.
Suppose the actual controller is with perturbations in
the form of (2b) and (2c) with parameters as
0.15
kL,


00.02
kM.
We wish to design a robust non-fragile state feedback
controller for this system such that the resulting closed-
loop system is asymptotically stable for all admissible
uncertainties and controller gain variations. By solving
LMI (7) using the Matlab LMI toolbox [24,25] with
0.7

, we obtain the following feasible solution:

12
2.48020.2958 , 7.1774,7.6242
0.295818.1199
70.82140.3 .






S
U

(14)
Therefore, by Theorem 3.1, we can see that the robust
non-fragile control problem is solvable. A desired state
feedback controller to solve this problem can be chosen
as


14.4699 117.8843K. (15)
5. Conclusion
In this paper, we have investigated the problem of robust
non-fragile control for a class of 2-D discrete uncertain
systems in the FMSLSS setting under state feedback gain
variations. Using the Lyapunov method, a criterion for
robust non-fragile control via state feedback is derived in
terms of LMI. Finally, a numerical example has been
presented to illustrate the effectiveness of the proposed
method.
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