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Signal Processing
Signal Processing 90 (2010) 2648–2654
0165-16
doi:10.1
� Cor
fax: +9
E-m
hnkar1@1 Te
journal homepage: www.elsevier.com/locate/sigpro
An LMI approach to robust optimal guaranteed cost control of 2-Ddiscrete systems described by the Roesser model
Amit Dhawan �, Haranath Kar 1
Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India
a r t i c l e i n f o
Article history:
Received 3 August 2009
Received in revised form
4 March 2010
Accepted 8 March 2010Available online 15 March 2010
Keywords:
2-D discrete systems
Guaranteed cost control
Roesser model
Linear matrix inequality
Robust stability
Lyapunov methods
84/$ - see front matter & 2010 Elsevier B.V. A
016/j.sigpro.2010.03.008
responding author. Tel.: +91 532 2271933;
1 532 2545341.
ail addresses: [email protected]
rediffmail.com (H. Kar).
l.: +91 532 2271815; fax: +91 532 2545341.
a b s t r a c t
The optimal guaranteed cost control problem via static-state feedback controller is
addressed in this paper for a class of two-dimensional (2-D) discrete systems described
by the Roesser model with norm-bounded uncertainties and a given quadratic cost
function. A novel linear matrix inequality (LMI) based criterion for the existence of
guaranteed cost controller is established. Furthermore, a convex optimization problem
with LMI constraints is formulated to select the optimal guaranteed cost controller
which minimizes the guaranteed cost of the closed-loop uncertain system.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
In recent years, two-dimensional (2-D) discrete sys-tems have found various applications in many areas suchas filtering, image processing, seismographic data proces-sing, thermal processes, gas absorption, water streamheating, etc. [1–4]. The stability properties of 2-D discretesystems described by the Roesser model [5] have beeninvestigated extensively [6–23]. Many publications relat-ing to 2-D Lyapunov equation with constant coefficientsfor the Roesser model have appeared in [6–9]. In [6], the2-D Lyapunov equation has been developed in terms ofenergy stored in the delays. It has been shown [8] that theexistence of the positive definite solutions to the 2-DLyapunov equation is, in general, only sufficient but notnecessary for 2-D stability. Based on the properties ofstrictly bounded real matrices, a necessary and sufficient
ll rights reserved.
(A. Dhawan),
condition has been developed [8] for the existence ofpositive definite solutions of the 2-D Lyapunov equation.Several algorithms for finding the positive definitesolution to the 2-D Lyapunov equations have beenpresented in [9]. The stability margin of 2-D discretesystems has been studied in [24–26]. In [27], the solutionsfor the H1 control and robust stabilization problems for2-D systems in Roesser model using the 2-D systembounded realness property have been presented. Thedesign methods for the H2 and mixed H2/HN control of2-D systems in Roesser model have been developed in [28].
The guaranteed cost control technique for 2-D un-certain systems aims to design a controller such that theclosed-loop system is asymptotically stable and theclosed-loop cost function value is not more than aspecified upper bound for all admissible uncertainties.Recently, a few results [29–32] have been obtained for theguaranteed cost control of 2-D discrete uncertain systemsdescribed by the Fornasini–Marchesini (FM) secondmodel. Optimal guaranteed cost control problem for 2-Ddiscrete uncertain systems is an important problem [32].However, to the best of the authors’ knowledge, theoptimal guaranteed cost control problem for 2-D discrete
ARTICLE IN PRESS
A. Dhawan, H. Kar / Signal Processing 90 (2010) 2648–2654 2649
uncertain systems represented by the Roesser modelwhich is structurally quite distinct from FM second modelhas not been addressed so far in the literature.
This paper, therefore, addresses the optimal guaran-teed cost control problem for 2-D discrete uncertainsystems described by the Roesser model with norm-bounded uncertainties. The paper is organized as follows.Section 2 deals with the problem formulation of robustguaranteed cost control for the uncertain 2-D discretesystem described by the Roesser model. Some usefulrelated results are also recalled in this section. In Section3, we relate the notion of cost matrix to the quadraticstability and an upper bound on the closed-loop costfunction. An LMI based sufficient condition for theexistence of guaranteed cost static-state feedback controllaws is established and the feasible solutions to this LMIare used to construct the guaranteed cost controllers.Further, a convex optimization problem is introduced tofind the optimal guaranteed cost controller which mini-mizes the upper bound on the closed-loop cost function.In Section 4, several illustrative examples are given toshow the potential of the proposed technique.
2. Problem formulation and preliminaries
The following notations are used throughout thepaper:
Rn
real vector space of dimension nRn�m
set of n�m real matrices0
null matrix or null vector of appropriate dimensionI
identity matrix of appropriate dimensionGT
transpose of matrix GG40
matrix G is positive definite symmetricGo0
matrix G is negative definite symmetriclmaxðGÞ
maximum eigenvalue of matrix G" # G¼ G1 � G2 direct sum, i.e., G¼G1 0
0 G2
This paper deals with the problem of optimal guaran-teed cost control for a class of 2-D discrete uncertainsystems represented by the Roesser model [5]. Specifi-cally, the system under consideration is given by
xhðiþ1, jÞ
xvði, jþ1Þ
" #¼ ðAþDAÞ
xhði, jÞ
xvði, jÞ
" #þðBþDBÞuði, jÞ, ð1aÞ
where xhði, jÞ 2 Rn and xvði, jÞ 2 Rm are the horizontal andvertical states, respectively, uði,jÞ 2 Rq is the control input.The matrices A 2 RðnþmÞ�ðnþmÞ and B 2 RðnþmÞ�q are knownconstant matrices representing the nominal plant. Thematrices DA and DB represent parameter uncertaintieswhich are assumed to be of the form
½DA DB� ¼ LFði,jÞ½M1 M2�: ð1bÞ
In the above equation, L 2 RðnþmÞ�g , M1 2 Rh�ðnþmÞ andM2 2 Rh�q can be regarded as known structural matricesof uncertainty and Fði,jÞ 2 Rg�h is an unknown matrixrepresenting parameter uncertainty which satisfies
JFði,jÞJr1: ð1cÞ
The matrices L and M1 (M2) characterize how theuncertain parameters in F(i,j) enter the state matrix A
(input matrix B). Observe that F(i,j) can always berestricted as (1c) by appropriately choosing L, M1 andM2. Therefore, without loss of generality, one can alwayschoose F(i,j) as in (1c).
It is assumed that the system (1a) has a finite set ofinitial conditions [6,17–19,22] i.e., there exist two positiveintegers k and l such that
xvði,0Þ ¼ 0, iZk; xhð0,jÞ ¼ 0, jZ l, ð1dÞ
and the initial conditions are arbitrary, but belong to theset
S¼ fxvði,0Þ,xhð0,jÞ : xvði,0Þ ¼ ZN1, xhð0,jÞ ¼ ZN2, NTt Nt o1 ðt ¼ 1,2Þg,
ð1eÞ
where Z is a given matrix. It may be mentioned that thestructure of initial conditions similar to (1e) has beenwidely adopted in guaranteed cost control for 2-Duncertain systems [29,31,32]. Observe that the vectorNt ðt¼ 1,2Þ can always be restricted as NT
t Nt o1 ðt¼ 1,2Þby appropriately choosing Z. In other words, there is noloss of generality by choosing initial conditions as in (1e).
Associated with the uncertain system (1a) is the costfunction:
J¼X1i ¼ 0
X1j ¼ 0
uT ði,jÞRuði,jÞþxT ði,jÞW1xði,jÞ, ð2aÞ
where
0oR¼ RT2 Rq�q, ð2bÞ
0oW1 ¼WT1 2 RðnþmÞ�ðnþmÞ, ð2cÞ
xði,jÞ ¼xhði,jÞ
xvði,jÞ
" #: ð2dÞ
Suppose the system state is available for feedback, thepurpose of this paper is to develop a procedure to design astatic-state feedback control law:
uði,jÞ ¼Kxði,jÞ, ð3Þ
for the system (1) and the cost function (2), such that theclosed-loop system
xhðiþ1,jÞ
xvði,jþ1Þ
" #¼ ðAþDAþBKþDBKÞ
xhði,jÞ
xvði,jÞ
" #ð4Þ
is asymptotically stable and the closed-loop cost function:
J¼X1i ¼ 0
X1j ¼ 0
xT ði,jÞW2xði,jÞ, ð5aÞ
where
W2 ¼W1þKT RK , ð5bÞ
satisfies Jr J� , where J� is some specified constant.
Definition 1. Consider the system (1) and cost function(2), if there exist a control law u� ði,jÞand a positive scalarJ* such that for all admissible uncertainties, the closed-loop system (4) is asymptotically stable and the closed-loop value of the cost function (5) satisfies Jr J� , then J* issaid to be a guaranteed cost and u� ði,jÞ is said to be aguaranteed cost control law for the uncertain system (1).
ARTICLE IN PRESS
A. Dhawan, H. Kar / Signal Processing 90 (2010) 2648–26542650
Now, we recall some useful related results on thestability of 2-D discrete uncertain systems described bythe Roesser model. As an extension of [8], we have thefollowing lemma.
Lemma 1. Anderson et al. [8]. The uncertain system (4) is
quadratically stable if there exists a positive definite block
diagonal matrix
P ¼ PT¼ Ph � Pv ¼
Ph 0
0 Pv
" #,
satisfying
C¼ ½AþDAþBKþDBK�T P½AþDAþBKþDBK ��Po0 for all JFði,jÞJr1,
ð6Þ
where Ph 2 Rn�n, Pv 2 Rm�m.
On the basis of the above lemma, we have the followingdefinition.
Definition 2. A state feedback controller uði,jÞ ¼Kxði,jÞ issaid to define a quadratic guaranteed cost controlassociated with a ðnþmÞ � ðnþmÞ positive definite sym-metric block diagonal cost matrix P for the system (4) andcost function (5) if there exist a ðnþmÞ � ðnþmÞ positivedefinite symmetric matrix W2 given by (5b) such that
½AþDAþBKþDBK �T P½AþDAþBKþDBK �
�PþW2o0 for all JFði,jÞJr1: ð7Þ
The following well-known lemmas are needed in the
proof of our main results.
Lemma 2. [29]. Let A 2 Rn�n, H 2 Rn�k, E 2 Rl�n andQ ¼Q T
2 Rn�n be given matrices. Then there exists a positive
definite matrix P such that
½AþHFE�T P½AþHFE��Q o0 ð8Þ
for all F satisfying FT FrI, if and only if there exists ascalar e40 such that
�P�1þeHHT A
AT e�1ET E�Q
" #o0: ð9Þ
Lemma 3. Boyd et al. [33]. For real matrices M, L, Q of
appropriate dimensions, where M=MT and Q ¼Q T 40, then
MþLT QLo0 if and only if
M LT
L �Q�1
" #o0 or equivalently
�Q�1 L
LT M
" #o0: ð10Þ
3. Main results
In the following, we aim to relate the notion of costmatrix to the quadratic stability and an upper bound onthe cost function (5).
Theorem 1. Suppose there exists a ðnþmÞ � ðnþmÞ positive
definite symmetric block diagonal cost matrix P for the
system (4) with initial conditions (1d), (1e) and cost function
(5) such that (7) holds. Then, (i) system (4) is quadratically
stable and (ii) the cost function satisfies the bound
Jr J� ¼ llmaxðZT PhZþÞklmaxðZ
T PvZÞ ð11Þ
for all admissible uncertainties.
Proof. Proof of (i) directly follows from Lemma 1 andDefinition 2.
To prove (ii), consider a quadratic 2-D Lyapunov
function [17,22]:
vðxði,jÞÞ ¼ vhðxhði,jÞÞþvvðxvði,jÞÞ ¼ xhT ði,jÞPhxhði,jÞþxvT ði,jÞPvxvði,jÞ: ð12Þ
Let Dvðxði,jÞÞ be defined as
Dvðxði,jÞÞ ¼ vhðxhðiþ1,jÞÞþvvðxvði,jþ1ÞÞ�vhðxhði,jÞÞ�vvðxvði,jÞÞ:
ð13Þ
Along the trajectory of the closed-loop system (4), we
obtain
Dvðxði,jÞÞ ¼ xT ði,jÞCxði,jÞ, ð14Þ
where xði,jÞ and C are defined in (2d) and (6), respectively.
Since P is a cost matrix, it follows from Definition 2 that
xT ði,jÞðCþW2Þxði,jÞo0: ð15Þ
From (14) and (15), we have
xT ði,jÞW2xði,jÞo�Dvðxði,jÞÞ: ð16Þ
Summing both sides of the above inequality over
i,j¼ 0-1 yields
Jo�X1i ¼ 0
X1j ¼ 0
Dvðxði,jÞÞ ¼�X1i ¼ 0
X1j ¼ 0
vhðxhðiþ1,jÞÞþvvðxvði,jþ1ÞÞ
�vhðxhði,jÞÞ�vvðxvði,jÞÞ ¼X1i ¼ 0
X1j ¼ 0
½xhT ði,jÞPhxhði,jÞ
�xhT ðiþ1,jÞPhxhðiþ1,jÞ�þX1i ¼ 0
X1j ¼ 0
½xvT ði,jÞPvxvði,jÞ
�xvT ði,jþ1ÞPvxvði,jþ1Þ� ¼X1j ¼ 0
xhT ð0,jÞPhxhð0,jÞ
þX1i ¼ 0
xvT ði,0ÞPvxvði,0Þ ¼Xl�1
j ¼ 0
xhT ð0,jÞPhxhð0,jÞ
þXk�1
i ¼ 0
xvT ði,0ÞPvxvði,0Þr llmaxðZT PhZÞþklmaxðZ
T PvZÞ,
ð17Þ
where use has been made of (5), (1d) and (1e) and the
relation limiþ j-1
xði,jÞ ¼ 0. This completes the proof of the
Theorem 1. &
Remark 1. The assumption (1e) on initial conditions hasbeen utilized in the proof of Theorem 1 to remove thedependence of the initial conditions on the bound of costfunction.
The following theorem establishes that the problem of
determining guaranteed cost control for system (4) and
the cost function (5) can be recast to an LMI feasibility
problem.
Theorem 2. Consider system (4) with initial conditions (1d),(1e) and cost function (5), then there exists a static-state
feedback controller uði,jÞ ¼Kxði,jÞ that solves the addressed
ARTICLE IN PRESS
A. Dhawan, H. Kar / Signal Processing 90 (2010) 2648–2654 2651
robust guaranteed cost control problem if there exist a
positive scalar e, a q� ðnþmÞ matrix U, a ðnþmÞ � ðnþmÞ
positive definite symmetric matrix S ¼ Sh � Sv such that the
following LMI is feasible:
�SþeLLT A 0 0 0
AT
�S SW1=21 UT R1=2 M1
0 WT=21 S �I 0 0
0 RT=2U 0 �I 0
0 MT
1 0 0 �eI
2666666664
3777777775o0, ð18Þ
where A ¼ASþBU, M1 ¼ SMT1þUT MT
2. In this situation, asuitable control law is given by
K ¼US�1: ð19Þ
Moreover, closed-loop cost function satisfies the bound
Jr J� ¼ llmaxðZT S�1
h ZÞþklmaxðZT S�1
v ZÞ: ð20Þ
Proof. Using (1b), (1c), (5b) and Lemma 2, (7) can berearranged as
�P�1þeLLT
ðAþBKÞ
ðAþBKÞT e�1ðM1þM2KÞT ðM1þM2KÞþW1þKT RK�P
" #o0:
ð21Þ
Premultiplying and postmultiplying (21) by the matrix
I 0
0 P�1
� �,
one obtains
�SþeLLT A
AT
�S
" #þ
0 0
0 e�1M1MT
1þSW1SþUT RU
" #o0,
ð22Þ
where
S ¼ P�1: ð23Þ
The equivalence of (22) and (18) follows trivially fromLemma 3. Using (23), the bound of the cost function canbe easily obtained from (11). This completes the proof ofTheorem 2. &
Remark 2. Note that (18) is linear in the variables e, U,and S which can be easily solved using Matlab LMIToolbox [33,34].
Theorem 2 provides a parameterized representation of a
set of guaranteed cost controllers (if they exist) in terms
of the feasible solutions to the LMI (18). This parameter-
ized representation can be exploited to design the
guaranteed cost controllers with some additional require-
ments. In particular, the optimal guaranteed cost control
law which minimizes the value of the guaranteed cost for
the closed-loop uncertain system can be determined by
solving a certain optimization problem. Based on Theorem
2, the design problem of the optimal guaranteed cost
controller can be formulated as follows.
Theorem 3. Consider system (4) with initial conditions (1d),(1e) and cost function (5), then there exists an optimal static-
state feedback controller uði,jÞ ¼Kxði,jÞ if the following
optimization problem
minimize ðlaþkbÞ
s:t:
ðiÞ ð18Þ,
ðiiÞ�aI ZT
Z �Sh
" #o0
ðiiiÞ�bI ZT
Z �Sv
" #o0
8>>>>>>><>>>>>>>:
ð24Þ
has a feasible solution a40, b40, e40, U 2 Rq�ðnþmÞ and0oS ¼ Sh � Sv ¼ ST
2 RðnþmÞ�ðnþmÞ. In this situation, anoptimal control law is K ¼US�1 which ensures theminimization of the guaranteed cost in (20).
Proof. By Theorem 2, the control law (19) constructed interms of any feasible solution e, U and S is a guaranteedcost controller of system (4). To obtain the optimum valueof the upper bound of guaranteed cost, the termslmaxðZ
T S�1h ZÞ and lmaxðZ
T S�1v ZÞ in (20) are changed to
lmaxðZT S�1
h ZÞoa3ZT S�1h ZoaI and lmaxðZ
T S�1v ZÞob3
ZT S�1v ZobI, respectively, which, in turn, implies the
constraints (ii) and (iii), in (24). Thus, the minimizationof ðlaþkbÞ implies the minimization of the guaranteedcost in (20). This completes the proof of Theorem 3. &
Remark 3. The optimization problem given by (24) is anLMI eigenvalue problem [33,34], which provides aprocedure to design optimal guaranteed cost controller.
Remark 4. Recently, in [32], the problem of designing anoptimal guaranteed cost controller for 2-D discreteuncertain systems described by FM second model isstudied and an upper bound on the cost function has beenobtained (see [32, Lemma 4]). Despite the fact that Roessermodel can be embedded into the FM second model [35,36],there are some structural differences between the systemconsidered in [32] and that of the present paper. From theproof of Theorem 1, it may be observed that the initialconditions (1d) and (1e) play key role in obtaining theupper bound of the cost function. A closer examination of[32, Eqs. {(1g), (1h)}] and {(1d), (1e)} reveals that theassumptions on initial conditions made in [32] do notmatch with those of the present paper. Further, the costfunction defined in [32] is not same as (2). Thus, one cansee that the 2-D uncertain system considered in this paperis an altogether different from that of [32]. Further, noapproximations of the form [32, Eq. (21)] have been used inthe proof of Theorem 1. Consequently, there is no directrelation between the results on cost bound given in [32, Eq.(14)] and (11). By exploiting the information of the systemunder consideration in a greater detail, the present paperestablishes the upper bound of the cost function in a moreefficient way as compared to that of [32]. Summarizing theabove, the presented approach for designing the optimalguaranteed cost controller is quite distinct from theapproach of [32].
4. Illustrative examples
Some examples illustrating the applicability ofTheorem 3 are in order.
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Example 1. Consider the 2-D discrete uncertain systemrepresented by (1) and (2) with
A¼1:02 �2:5
0:1 0
� �, B¼
0:1
0:2
� �, M1 ¼ ½0:005 �0:005�,
M2 ¼ 0:01, L¼1
6
� �,
W1 ¼0:0025 0
0 0:0025
� �, R¼ 0:25, Z ¼ 0:1, k¼ l¼ 2: ð25Þ
The 2-D characteristic polynomial of the present system
with DA=0 (i.e., in absence of uncertainties in the state
matrix) is given by
gðz1,z2Þ ¼ det1�1:02z1 2:5z1
�0:1z2 1
" #¼ 1�1:02z1þ0:25z1z2,
ð26Þ
where ‘det’ stands for determinant. For the solution
x(i, j)=0 of the underlying linear system (with DA=0) to
be 2-D asymptotically stable, one requires
gðz1,z2Þa0 8ðz1,z2Þ 2 U2 , ð27Þ
where U2¼ fðz1,z2Þ : jz1jr1,jz2jr1g. It is seen that (27) is
violated (choose, for instance, z1=0.99, z2=98/2475) for
the characteristic polynomial given in (26). Therefore, the
system under consideration is not 2-D asymptotically
stable. We wish to construct an optimal guaranteed cost
controller for this system. It is found using the LMI
toolbox in Matlab [33,34] that the optimization problem
(24) is feasible for the present example, and the optimal
solution is given by
S ¼0:6343 0
0 1:1025
� �, U ¼ ½�0:2478 �1:0264�,
e¼ 9:5605� 10�4, a¼ 0:0158, b¼ 0:0091: ð28Þ
While solving the optimization problem (24) for the
present example, using the function mincx and setting f-
radius saturation=109 in the LMI tool box, it is observed
that the optimal algorithm takes 25 iterations to reach to
the optimal solution (28).
Now, using Theorem 3, the optimal guaranteed cost
controller is found to be
uði,jÞ ¼ ½�0:3907 �0:9310�xði,jÞ ð29Þ
and the least upper bound of the corresponding closed-
loop cost function is J� ¼ 0:0497.
Example 2. In this example, we shall demonstrate theapplication of Theorem 3 to the control of thermalprocesses in chemical reactors, heat exchangers and pipefurnaces, which can be expressed in the following partialdifferential equation with time [1]:
@Tðx,tÞ
@x¼�
@Tðx,tÞ
@t�Tðx,tÞþuðx,tÞ, ð30Þ
with the initial conditions:
Tðx,0Þ ¼ f1ðxÞ, Tð0,tÞ ¼ f2ðtÞ, ð31Þ
where Tðx,tÞ is the temperature at space x 2 ½0,xf � and timet 2 ½0,1Þand uðx,tÞ is the input function. Taking
Tði,jÞ ¼ TðiDx,jDtÞ, uði,jÞ ¼ uðiDx,jDtÞ, ð32aÞ
@Tðx,tÞ
@t�
Tði,jþ1Þ�Tði,jÞ
Dt,
@Tðx,tÞ
@x�
Tði,jÞ�Tði�1,jÞ
Dx,
ð32bÞ
(30) can be expressed in the following form:
Tði,jþ1Þ ¼Dt
Dx
� �Tði�1,jÞþ 1�
Dt
Dx�Dt
� �þDtuði,jÞ: ð33Þ
Denote xhði,jÞ ¼ Tði�1,jÞ and xvði,jÞ ¼ Tði,jÞ. Now, (33) can bewritten in the setting of a 2-D Roesser model given by
xhðiþ1,jÞ
xvði,jþ1Þ
" #¼
0 1Dt
Dx
� �1�
Dt
Dx�Dt
� �24
35 xhði,jÞ
xvði,jÞ
" #þ
0
Dt
� �uði,jÞ:
ð34Þ
Next, consider the problem of optimal guaranteed cost
control of a system represented by (34) with
Dx¼10
101, Dt¼ 0:1, ð35Þ
and the initial conditions satisfy (1d) and (1e) with
k¼ l¼ 2, Z ¼ 0:1: ð36Þ
It is also assumed that the above system is subjected to
parameter uncertainties of the form (1b) and (1c) with
L¼0
1
� �, M1 ¼ ½0:007 �0:007�, M2 ¼ 0:02: ð37Þ
Associated with the uncertain system (34)–(37), the
cost function is given by (2) with
W1 ¼0:0064 0
0 0:0064
� �, R¼ 0:64: ð38Þ
We wish to design an optimal guaranteed cost con-
troller for the above system. The optimization problem
(24) is feasible for the present example and the optimal
solution is given by
S ¼6:7349 0
0 5:9657
� �, U ¼ ½�1:7137 �0:5824�,
e¼ 0:0652, a¼ 0:0015, b¼ 0:0017: ð39Þ
By Theorem 3, the optimal guaranteed cost controller
for this system is
uði,jÞ ¼ ½�0:2545 �0:0976�xði,jÞ, ð40Þ
and the least upper bound of the corresponding closed-
loop cost function is J*=0.0063.
Example 3. In this example, we shall illustrate theapplicability of Theorem 3 to the control of dynamicalprocesses in gas absorption, water steam heating andair drying, which are represented by the Darboux
ARTICLE IN PRESS
A. Dhawan, H. Kar / Signal Processing 90 (2010) 2648–2654 2653
equation [37–39]. Consider the Darboux equation [37–39]given by
@2sðx,tÞ
@x @t¼ a1
@sðx,tÞ
@tþa2
@sðx,tÞ
@xþa0sðx,tÞþbf ðx,tÞ, ð41Þ
with the initial conditions:
sðx,0Þ ¼ pðxÞ, sð0,tÞ ¼ qðtÞ, ð42Þ
where sðx,tÞ is an unknown function at space x 2 ½0,xf � andtime t 2 ½0,1Þ, a1, a2, a0 and b are real constants and f ðx,tÞis the input function.
Define
rðx,tÞ ¼@sðx,tÞ
@t�a2sðx,tÞ, ð43Þ
then (41) can be transformed into an equivalent system of
the form:
@rðx,tÞ
@x@sðx,tÞ
@t
2664
3775¼ a1 a1a2þa0
1 a2
" #rðx,tÞ
sðx,tÞ
" #þ
b
0
� �f ðx,tÞ, ð44Þ
with initial condition
rð0,tÞ ¼@sðx,tÞ
@t x ¼ 0
�a2sð0,tÞ ¼dqðtÞ
dt�a2qðtÞ9zðtÞ:
���� ð45Þ
Taking rði,jÞ ¼ rðiDx,jDtÞ9xhði,jÞ, sði,jÞ ¼ sðiDx,jDtÞ9xvði,jÞ,
f ðx,tÞ ¼ uði,jÞ and approximating the partial derivatives as
@rðx,tÞ
@x�
rðiþ1,jÞ�rði,jÞ
Dx,
@sðx,tÞ
@t�
sði,jþ1Þ�sði,jÞ
Dt, ð46Þ
(44) leads to
xhðiþ1,jÞ
xvði,jþ1Þ
" #¼ð1þa1DxÞ ða1a2þa0ÞDx
Dt ð1þa2DtÞ
" #xhði,jÞ
xvði,jÞ
" #þ
bDx
0
� �uði,jÞ,
ð47Þ
with the initial conditions
xhð0,jÞ ¼ zðjDtÞ, xvði,0Þ ¼ pðiDxÞ: ð48Þ
Now, consider the problem of optimal guaranteed cost
control of a system represented by (47) with
a0 ¼�5, a1 ¼ 1, a2 ¼�10, b¼ 1, Dx¼ 0:2, Dt¼ 0:1
ð49Þ
and the initial conditions (48) satisfy (1d) and (1e) with
k¼ l¼ 2, Z ¼ 0:12: ð50Þ
It is also assumed that the above system is subjected to
parameter uncertainties of the form (1b) and (1c) with
L¼1
9
� �, M1 ¼ ½0:009 �0:009�, M2 ¼ 0:04: ð51Þ
Associated with the uncertain system (47)–(51), the
cost function is given by (2) with
W1 ¼0:0044 0
0 0:0044
� �, R¼ 0:24: ð52Þ
We now apply Theorem 3 to find an optimal guaranteed
cost controller for the system under consideration. It is
checked that (24) is feasible for the present system and
the optimal solution is given by
S ¼0:1485 0
0 3:2068
� �, U ¼ ½�0:2693 �2:6715�,
e¼ 0:0275, a¼ 0:0970, b¼ 0:0045: ð53Þ
Moreover, the optimal guaranteed cost controller for
this system is
uði,jÞ ¼ ½�1:8135 �0:8331�xði,jÞ ð54Þ
and the least upper bound of the corresponding closed-
loop cost function is J*=0.2029.
5. Conclusions
A solution to the optimal guaranteed cost controlproblem via static-state feedback control laws for theuncertain 2-D discrete system described by the Roessermodel has been presented. The feasibility of a certain LMIhas proved to be the sufficient condition for the existenceof a guaranteed cost controller. A convex optimizationproblem has been formulated to select the optimalguaranteed cost controller which minimizes the upperbound on the closed-loop cost function. Finally, someexamples have been provided to illustrate the applic-ability of the proposed method.
Acknowledgment
The authors wish to thank the reviewers for theirconstructive comments and suggestions.
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