Journal of Signal and Information Processing, 2011, 2, 292-300
doi:10.4236/jsip.2011.24042 Published Online November 2011 (http://www.SciRP.org/journal/jsip)
1
LMI Approach to Suboptimal Guaranteed Cost
Control for 2-D Discrete Uncertain Systems
Amit Dhawan, Haranath Kar
Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad, India.
Received September 28th, 2011; revised October 30th, 2011; accepted November 14th, 2011.
ABSTRACT
This paper studies the problem of the guaranteed cost control via static-state feedback controllers for a class of
two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS)
model with norm bound ed uncertain ties. A co nvex optimizatio n problem with linear matrix inequality (LMI) constra ints
is formulated to design the suboptimal guaranteed cost controller which ensures the quadratic stability of the
closed-loop system and minimizes the associated closed-loop cost function. Application of the proposed controller de-
sign method is illustrated with the help of one example.
Keywords: Linear Matrix Inequality, Lyapunov Methods, Robust Stability, 2-D Discrete Systems, Uncertain System s,
Fornasini-M ar chesi ni Second Local State- S p ace Mo del
1. Introduction
In the past few years, due to the rapid increase of a wide
variety of applications of two-dimensional (2-D) discrete
systems in many practical application domains such as
digital filtering, image and video processing, seismogr-
aphic data processing, thermal processes, gas absorption,
water stream heating, control systems etc. [1-10], there
has emerged a continuously growing interest in the sys-
tem theoretic problems of 2-D discrete systems. Many
authors have proposed and analyzed linear state-variable
models for 2-D discrete systems [11-14]. The more popu-
lar models are Roesser model [11], Fornasini-Marchesini
first model [13] and Fornasini-Marchesini second local
state-space (FMSLSS) model [14]. Many publications
relating to 2-D Lyapunov equation with constant coeffi-
cients for the Roesser model [11] have appeared [15-22].
The stability properties of 2-D discrete systems described
by the FM first model [13] have been investigated exten-
sively [23-29]. The stability analysis of 2-D discrete sys-
tems described by the FMSLSS model [14] has attracted
a great deal of interest and many significant results have
been obtained [22,30-44].
Due to assumptions in the modeling process and/or the
changing operating conditions of a real world system, it is
usually impossible for a mathematical model to describe
the real world system exactly. The problem of designing
robust controllers for 2-D uncertain systems has drawn the
attention of several researchers in recent years [39,40].
When controlling a system subject to parameter uncer-
tainty, it is also desirable to design a control system which
is not only stable but also guarantees an adequate level of
performance. One approach to this problem is the so-
called guaranteed cost control approach [45]. This appr-
oach has the advantage of providing an upper bound on a
given performance index and thus the system performance
degradation incurred by the uncertainties is guaranteed to
be less than this bound. Based on this idea, many signifi-
cant results have been proposed [42-51]. In [42-44], the
guaranteed cost control problem for 2-D discrete uncertain
systems in FMSLSS setting has been considered and a
robust controller design method has been established. The
approach of [42] does not provide a true linear matrix ine-
quality (LMI) based result which is not beneficial in terms
of numerical complexity. Subsequently, in [43], an LMI
based criterion for the existence of robust guaranteed cost
controller has been formulated. Robust suboptimal guar-
anteed cost control for 2-D discrete uncertain systems in
FMSLSS setting is an important problem.
In recent years, LMI has emerged as a powerful tool in
control design problems [52-58]. The introduction of
LMI in control theory has given a new direction in the
area of robust control problems. A widely accepted met-
hod for solving robust control problems now is to simply

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems293
reduce them to LMI problems. Since solving LMIs is a
convex optimization problem, such formulations offer a
numerically efficient means of attacking problems that
are difficult to solve analytically. These LMIs can be
solved effectively by employing the recently developed
Matlab LMI toolbox [53].
This paper, therefore, deals with the suboptimal guaran-
teed cost control problem for 2-D discrete uncertain syst-
ems described by FMSLSS model with norm-bounded
uncertainties. The paper is organized as follows. In Section
2, we formulate the problem of robust guaranteed cost
control for the uncertain 2-D discrete system described by
the FMSLSS model and recall some useful results. An
LMI based approach for the design of suboptimal guaran-
teed cost controller via static-state feedback is presented in
Section 3. In Section 4, an application of the presented
robust guaranteed cost controller design method is given.
Finally, some concluding remarks are given in Section 5.
2. Problem Formulation and Preliminaries
The following notations are used throughout the paper:
Rn real vector space of dimension n
Rnm set of n  m real matrices
0 null matrix or null vector of
appropriate dimension
I identity matrix of appropriate
dimension
GT transpose of matrix G
G > 0 matrix G is positive definite symmetric
G < 0 matrix G is negative definite symmetric
det (G) determinant of matrix G
max

(G) maximum eigenvalue of matrix G.
In this paper, we are concerned with the problem of
guaranteed cost control for 2-D discrete uncertain syst-
ems described by FMSLSS model [14]. The system un-
der consideration is given by




11
22
11
22
1, 11,
,1
1,
,1
ij ij
ij
ij
ij
 
 
 
 
xAAx
AAx
BBu
BBu




, (1a)


12

A
AA, (1b)
where and are the state and
control input, respectively. The matrices

,n ij Rx

,m ij Runn
kR

A and
k (k = 1, 2) are known constant matrices repre-
senting the nominal plant, k Ad k 
nm
R B
an
B
(k = 1, 2) are
real valued matrix functions representing parameter un-
certainties in the system model. The parameter uncertain-
ties under consideration are assumed to be norm-bounded
and of the form





12
,ij
A
BLF MM, (1c)
where


1

2
A
AA


1
 BBB
2
, (1d)


1111
MMM
2


2212
MMM
2
. (1e)
In the above,
L
, 1 and 2 can be regarded as
known structural matrices of uncertainty and
M M


,ijF is
an unknown matrix representing parameter uncertainty
which satisfies


,ij F1. (1f)
It may be mentioned that the uncertainty of (1c) satis-
fying (1f) has been widely adopted in robust control lit-
erature [38,39,42-44,59-62]. The matrices
L
and 1
(2) specify how the elements of the nominal matrices
A (B) are affected by the uncertain parameters in
M
M


,ijF. Note that


,ijF can always be restricted as
(1f) by appropriately selecting
L
, 1 and 2.
Therefore, there is no loss of generality in choosing
M M


,ijF as in (1f).
It is assumed that the system (1a) has a finite set of
initial conditions [22,34,36,38,43,44] i.e., there exist two
positive integers p and q such that


,0 ,i

0x ; , , (1g)
ip

0, j0xjq
and the initial conditions are arbitrary, but belong to the
set [42-44]





1
,0 ,0,:,0,
n
Si jRixxx MN



2
0,,1(1, 2)
T
kk
jkxMNNN , (1h)
where M is a given matrix.
Associated with the uncertain system (1) is the cost
function [43, 44]:


1
00
2
1
00
1, 1,
,1,1
T
ij
T
T
ij ij
ij
J
ij ij
ij ij









 




uRu
uRu
W
, (2a)
where



1,
,1
ij
ij
ij










x
x, (2b)
Tm
kk
R  0RR m (k =1, 2), (2c)
1
1
2







0
0
Q
WQ, (2d)
Tn
kk
R  0QQ n
objective of this paper is to develop a procedure to de-
(k =1, 2). (2e)
Suppose the system state is available for feedback, the

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems 294
sign a static-state feedback control law



,,ij ijuKx (3)
for the system (1) and the cost function (2), such that the
closed-loop system





 
111 1
222 2
1,
,1
ij
ij
 

  

BKABKx
ABKA BKx





(4)
is asymptotically stable and the closed-loop cost func
1, 1ij  xA
tion
J
is minimized where
2
00
T
ij ij
ij
J




 W, (5a)


. (5b)
Definition 2.1 A control law (3) is said to be an optimal
lobal asymptotic
st
2.1 [44] The 2-D discrete uncertain system (1) is
 
for all
11
2
22
T
T





0
0
QKRK
WQKRK
quadratic guaranteed cost control if it ensures the quad-
ratic stability of the closed-loop system (4) and mini-
mizes the closed-loop cost function (5).
As an extension of the result for the g
ability condition of 2-D discrete FMSLSS model given
in [14,30-33], one can easily arrive at the following
lemma.
Lemma
globally asymptotically stable if and only if
det
 
zz IALF MALF
2 1111 2120M

2
12
,, Uzz F, (6)
where
 
2
U= ,1,1zzz .
12 1 2
, :1,zFF
ion 2.2 [42-44] Consider the uncertain (1)
(7a)
where
(7b)
and
11
(7c)
12
1
Definit system
and cost function (2), then the static-state feedback con-
troller
 
,,ij ijuKx is said to define a quadratic
guaranteed cost control associated with cost matrix
Tnn
R  0PP if there exist a 2n  2n positive defi-
trix 2 W given by (5b) and an n  n
positive definite symmet matrix 1 P such that
2CL 0ΓW,
nite symmetric ma
ric


1122
1
112 2
1
T
CL  
 
 

 



0
0
ΓABKABKP
P
ABKABKPP
11 11AAAALFM,
22 22AAAALFΜ, (7d)
11 112
B
BBBLFM, (7e)
22 2222

 
B
BBBLFM. (7f)
The following lemmas are needed in t
m
2,44,51] Let
he proof of our
ain result.
Lemma 2.2 [4 nnnk ,RA,RH
ln
R


E
and Tn RQQ
tts a po matrix P such that

T
n be give
nite
n matrices. Then
here exissitive defi

0AHFE PAHFEQ (8)
for all F satisfying FT F  I, if and only if the

. (9)
Lemma 2.3 [52, 63] For real matrices M, L, Q of appro-
re exists a
scalar   0 such that
1



PH
1
T
TT





0
H A
AEEQ
priate dimensions, where T MM and T

0QQ ,
then T

0MLQL if an
T
ML
d only if
1 





0
LQ (10)
or equivalently
(11)
Lemma 2.4 [44] Suppose there exists a quadratic guar-
1
T





0
QL
LM .
anteed cost matrix T 0PP for the uncertain closed-
loop system (4) withditions (1g), (1h) and cost
function (5) such that (7) holds. Then, a) the uncertain
closed-loop system (4) is quadratically stable and b) the
cost function satisfies the bound
initial con



max
2Jpq  MP

(12)
3. Main Result
establish that the problem of deter-
TM
In this section, we
mining quadratic guaranteed cost control for system (1)
and cost function (2) can be recast to a convex optimiza-
tion problem. The main result may be stated as follows.
Theorem 3.1 Consider system (1) and cost function (2),
then there exists a suboptimal static-state feedback con-
troller


,iju =


,ijKx that solves the addressed
robust ed col problem if the following
optimization problem
minimize
guaranteost contr







(). 1,
().
T
i
ii




 

0
IM
MS
3
s.t. 

(14)
has a feasible solution 0

, mn R U,
nn
R

 0T SS andnn.
The constraint (13) is give
0
n by
R 
T
YY

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems
295

12
1/2 1/2
11 111
1/2 1/2
22
212
11 12
/2
1
/2
2
/2
1
/2
2
0
T
T

SA A
A
T
T
T
TT
T
T
T
T

























00 0 0
0
00
00
00
00
00 00 00 0
0 00 00
00
00 0 0000
000
000
00 0 0000
00 0 00 00
L
SQU R
Y M
SQU R
ASYM
LI
MM I
I QS
I
QS
I RU
I RU
, (13)
where
11 1
AASBU, (15a)
222
AASBU, (15b)
11 1121
TT
MSMUM,
T
(15c)
12 1222
TT
MSMUM .
T
(15d)
In this situation, a suboptimal control law is
K = which ensures the minimization of the upper
boun for the closed-loop uncertain system.
Proof: Using (5b) and (7b), matrix Inequality (7a) can
be expressed as



112 2
1
112 2
1
11
22
T
T
T
 
 


 









0
0
R0
0
0R
ABKABKP
P
ABKABKPP
QK K
QK K
1
US
d of (2)
, (16)
which, in view of (7c)-(7f), takes the form


(17)
Applying Lemma 2.2, (17) can be rearranged as



1 12211211222
1 12211211222
111
12 2
T
T
T
 








 


R0
0
0R
ABKA BKLFMMKMMK
PABKABKLFMMKMMK
PQ KK
PPQ KK
.


 




11
11
111 1111211121
2 212221121
22
11 211222
12 212221222
.
T
TT
T
TT
T
T
T






 
 






 


 

LL
Q
0
Q
PABK
ABKPKRKM MKM MK
ABKMMK MMK
ABK
MMKM MK
PPKRKMMK MMK

(18)
Premultiplying and postmultiplying (18) by the matrix
1/2
1/2 1
1/2 1















00
00
00
I
P
P
, one obtains
1








1 1
11
11 1
1111111 2111 21
11
2 212221121
1
22
11
11 2112 22
1 1
12 212221222
,
T
TT
T
TT
T
T
T

 
 



 
 



 
 








 


 
0
PLL ABKP
PABKPPQKRKM MKMMKP
PABKPMMK MMKP
ABKP
PMMK MMKP
PPPQKRKMMKMMKP

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems 296
which can be rewritten as

1
11
111
21211
2
11 12
11
221212
T
TT
TT
T
TT





 





1111
T



 
0
SLL A
AYSQSURUM
AMM
A
MM
SYSQSURU MM
M
(19)
where
1



S
P, (20)
S
and
1
1


YSP, (21)
1
A, 2 A, 11 M and 12 M
xpress
are de
Equation (19) can be eed as
The equivalence of (22) and (13) fo
Lemma 2.3. Using (20), the bound of t
(12) becomes
(23)
fined in (15).
Equation (22).
llows trivially from
he cost function


1
max
2T Jpq 

MSM .
Clearly, the upper bound (23) is not a convex function
1
in
S
and

. Hence, finding the minimum of this
per bound can not be considered as a convex optimiza-up
blemSince tion pro.

and are posi-
may obtainuboost con-
mi obtain
th mume ofteed cost,
to

1
max
T


MS M
ptimal guaranteed c

1
max
T
MS M. To
bound of guaran
in (23) is changed
tive, we
troller by
e opti
the term
inim
valu
a s
zing
th
1
 
e upper

M
max
T
 MS

1
max
T


MS M1T


MS MI
which, in turn, implies the constraint (ii) in (14). Thus,
the minimization of


 
st in (23).
implies the minimization
of the guaranteed co The optimality of the solu-
tion of the optimization problem (14) follows from the
convexity of the objective function and of the constraints.
This completes the proof of Theorem 3.1.
Remark 3.1 It should be pointed out that the optimi-
zation problem given by (14) is an LMI eigenvalue prob-
lem [52,53], which provides a procedure to design subo-
ptimal guaranteed cost controller.
4. Application to the Guaranteed Cost
Control of Dynamical Processes Described
by the Darboux Equation
In this section, we shall demonstrate
our proposed method (Theorem 3.1) in robust guaranteed
cost control of processes in the Darboux equation. It is
known that some dynamical processes in gas absorption,
water stream heating and air drying can be described by
the Darboux equation [3,7,8]:
the application of



  
2,sxts
a


120
,,
,,
xt sxt
aa sxtbfxt
xt tx


 
(24)
with the initial conditions




,0
s
xpx,

0,
s
tqt (25)
where


,
s
xt is an unknown function at space
0,
f
x
x

 

 and time


0,t

,1 a,2 a,0 a
eal constants and
and are b
r


,
f
xt is the input function.
Let



2
,
,,
sxt
rxt asxt
t


 (26)
then (24) can be transformed into a
first-order differential equation of the form:
n equivalent system of




 
112 0
2
,
,,
1, 0
,
rxt aaaa rxtb
x
f
xt
asxt
sxt
t


 















. (27)
It follows from (26) that


   
22
0
,d
0, 0,
x
sxt qt
rtast aqtzt
tdt



.
(28)
Taking

12
1/2 1/2
11111
1/2 1/2
21222
1
11 12
/2
1
/2
2
/2
1
2
T T
T T
T
TT
T
T
T





  00SA AL







 


















0 00
000 0
0000
0000000
00000 0
0000000
00 00000
00 00000
00000 00
AYMSQ UR
ASYM SQUR
IL
IMM
IQS
IQS
IRU
IR/2T











0
U
. (22)

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems297




,,rijri xj t, (29a)
 
,,
s
ijsixj t, (29b)



,,
f
xtuij (29c)
and applying the forward difference quotients for both
derivatives in (27), it is easy to verify that (27) can be ex-
pressed in the following form:
,
,

 
 




 
1120
2
,1
1
,1
00
00 ,1
1,1
0
1,, 1
00
rijri j
axaa ax
s
ijsij
rij
tat
sij
bxui juij
 

 

  

  




 



 

 
 



(30)
with the initial conditions
 
,0
s
ipix,


0,rjzjt.

(31)
By setting




,
,,
rij
ij
s
ij



x, (32)
(30) can be converted into the following FMSLSS
model:
(33)
with the initial conditions
,  (34)
Now, consider the problem of suboptimal guaranteed
cost control of a system represented by (33) with






 
2
1120
00
1, 11,
1
1
,1
00
0
1,,1
00
ij ij
tat
axaaaxij
bx
ui juij

 



  








xx
x
 

2
,0 apix ipix
 



x
 

0, zjt jqjt




x
0
1
15
a, (35a)
1
3
15 a , (35b)
2
1
3
a , (35c)
(35d)
(35e)
(35f)
and the initial conditions (34) satisfy (1g) and (1h) with
(36a)
2 b,
0.5x ,
0.9t
2pq,
0.01 0.05
0.006 0.001







M. (36b)
It is also assumed that the above system is subjected to
parameter uncertainties of the form (1c)-(f) with 1
0
1







L, (37a)


11 0.0005 0M, (37b)


12 0 0.005M, (37c)
(37d)
21 0M,
22 0.007

M. (37e)
Associated with the uncertain system (33)-(37), the
cost function is given by (2) with
1
0.09 0
00.09







Q, (38a)
2
0.9 0
00.9







Q, (38b)
12
0.0025

RR . (38c)
Applying Lemma 2.1, it is easy to verify that the
above system is unstable. We wish to construct a suitable
guaranteed cost controller for this system, such that the
corresponding cost bound is minimized. To this end, we
apply our proposed method (Theorem 3.1) to find the
suboptimal guaranteed cost controller. It is found using
the LMI toolbox in Matlab [53] that the optimization
problem (14) is feasible for the present example and the
optimal solution is given by
5.03810 4.53485
4.53485 7.41531









S, (39a)
1.22942 0.12961
0.12961 1.19500







Y, (39b)


0.35283 1.31762U, (39c)
11.01117


, (39d)
0.00121


. (39e)
By Theorem 3.1, the suboptimal guaranteed cost con-
troller for this system is





,0.19999 0.29999,uijijx, (40)
and the least upper bound of the corresponding closed-
loop cost function is
0.02682J

. (41)
5. Conclusions
In this paper, we have presented a method of designing a
suboptimal guaranteed cost controller via static-state

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems 298
feedback for a class of 2-D discrete systems described by
the FMSLSS model with norm bounded uncertainties. A
suboptimal guaranteed cost controller is obtained through
a convex optimization problem which can be solved by
using Matlab LMI Toolbox [53]. Application of pres-
ented controller design method is demonstrated through
processes described by a Darboux equation [3,7,8]. The
presented method can also be applied for the robust
guaranteed cost controller design for metal rolling con-
trol problem [4,9,10].
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