Existence Conditions of Asymptotically Stable 2-D Feedback Control
Systems on the Basis of Block Matrix Diagonalization
Giido Izuta
Yonezawa Women’s Junior College, Yonezawa, Yamagata, Japan
Keywords:
2-D Systems, Asymptotic Stability, Feedback Control, Block Matrix Diagonalization.
Abstract:
This work is concerned with the existence of asymptotically stable 2-D (2-dimensional) systems by means of a
feedback control model represented by the system of partial difference equations and their Lagrange solutions.
Thus, the goal is to establish a controller thatprovides a feedback control system with state variables depending
solely on its Lagrange solution in the sense that the solution to the variable state is not a linear combination
of other Lagrange solutions. Roughly speaking, the results showed that, to achieve such a control system,
the controller has to diagonalize the block matrices of the matrices composing the system description model.
Finally, a numerical example is presented to show how the controller is designed in order to generate a stable
feedback control with given Lagrange solutions.
1 INTRODUCTION
Discrete 2-D control systems have been a subject
of research for over half a century now; and many
approaches have been proposed to study the stabil-
ity and design of feedback control systems. As far
as the mathematical model is concerned with, either
the Roesser system description (Givone and Roesser,
1972) or the FM model (Attasi, 1973; Fornasini and
Marchesini, 1978; Fornasini and Marchesini, 1980),
which are essentially equivalent to one another in the
sense that it is possible to transform the partial dif-
ference equations describing one model into the other
(Kaczorek, 1985), is taken as the system representa-
tion to handle this type of control systems. In addi-
tion, the formalism adopted has spread over a large
range of mathematical fields. Among those, the z−
transform showed to be a very effective strategy to in-
vestigate systems with single input and single output
corresponding to what is called the class of ‘delayless
systems’ in the ordinary 1-D system counterpart the-
ory; and a great deal of stability criteria and controller
design procedures were accomplished for 2-D con-
trol systems (Anderson and Jury, 1974; Bose, 1982;
Tzafestas, 1986; Lim, 1990).
During the time period around the turning of the
century, a remarkable shift in the paradigm from han-
dling those problems on the grounds of analytical
mathematics into the incorporation of computational
methods, which in control system theory is closely as-
sociated to the energy method framework, took place.
In fact, this approach relies basically on the Lyapunov
stability theory as well as the optimization algorithms,
and has undergone tremendous developments hand in
hand with the advances in computational techniques;
so that the formalization of the control system de-
sign based on the linear matrix inequalities gained
widespread popularity and established its status as a
standard procedure to handle not only the ordinary
1-D but also multi-input multi-output n-D systems
(Piekarski, 1977; Du and Xie, 2002; Pazke et al.,
2004; Izuta, 2007; Rogers et al., 2007). However,
this procedure has been pointed out to yield conserva-
tive controllers in the control theory sense. Also it is
worth noting that many other techniques have flour-
ished and contributed to the broadening of the field
(Jerri, 1996; Zerz, 2000; Cheng, 2003; Elaydi, 2005;
Izuta, 2012; Izuta, 2014a).
Unlike the theoretical accounts aforementioned,
this paper examines the existence conditions of
asymptotically stable 2-D feedback control systems
from the Lagrange method frame of reference, which
has shed light on the stability analysis problem (Izuta,
2010; Izuta, 2014b). In these reports, the Lagrange
method was used in conjunction with Jordan matrix
transformation in order to transform the original sys-
tem of partial difference equations into a system com-
posed by only diagonal matrices, which allow the
transformed system be solved analytically.
Motivated by these results, this paper aims to fig-
Izuta, G.
Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization.
DOI: 10.5220/0005975604630470
In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 463-470
ISBN: 978-989-758-198-4
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
463
ure out the conditions that the controller has to ful-
fill so as to form a feedback control system hav-
ing defined asymptotically stable Lagrange solutions.
Moreover, the kind of systems concerned with here
corresponds to the class of ‘systems with delays’ in
the 1-D case.
Finally, the paper is organized as follows: in sec-
tion 2, the 2-d control system and the concepts neces-
sary throughout the paper are presented; the asymp-
totic stability conditions are establish in section 3; a
numerical example is provided in 4; and some final
remarks are enunciated in section 5 .
2 PRELIMINARIES
In this section, the concepts and definitions used
throughout this paper are presented. Firstly, the 2-d
control system to be investigated is presented here.
Definition 2.1. Let the 2-d control system be given by
the following system of partial difference equations.
x
p
(i+ 1, j)
x
q
(i, j+ 1)
=
A
11
A
12
A
21
A
22
x
p
(i, j)
x
q
(i, j)
+
B
11
B
12
B
21
B
22
x
p
(i− δ
p
, j− σ
p
)
x
q
(i− δ
q
, j− σ
q
)
+
C
11
C
12
C
21
C
22
u
p
(i, j)
u
q
(i, j)
(1)
where x
p
(i, j) and x
q
(i, j) are vectors representing the
states of the system and given by
x
p
(i+ 1, j) =
def
x
1
(i+ 1, j)
.
.
.
x
n
(i+ 1, j)
x
q
(i, j+ 1) =
def
ˆx
1
(i, j+ 1)
.
.
.
ˆx
n
(i, j+ 1)
(2)
with the indices i and j (i, j ∈ Z) indicating that the
system is discrete and doubly indexed. In addition,
the inputs of the system u
p
(i, j) and u
q
(i, j) are and
are defined similarly to (2).
Assumption 2.2. Hereafter the matrices C
11
, C
12
,
C
21
, C
22
are all real valued non-singular matrices.
In order to simplify the notations, the following
definitions are adopted.
⋆(i, j) =
⋆
p
(i, j)
⋆
q
(i, j)
=
def
⋆
1
(i, j)
.
.
.
⋆
n
(i, j)
ˆ
⋆
1
(i, j)
.
.
.
ˆ
⋆
n
(i, j)
,
⋆ = x
or
⋆ = u
x(i+ 1, j+ 1) =
def
x
1
(i+ 1, j)
.
.
.
x
n
(i+ 1, j)
ˆx
1
(i, j+ 1)
.
.
.
ˆx
n
(i, j+ 1)
x(i− δ, j− σ) =
def
x
1
(i− δ
1
, j− σ
1
)
.
.
.
x
n
(i− δ
n
, j− σ
n
)
ˆx
1
(i−
ˆ
δ
1
, j−
ˆ
σ
1
)
.
.
.
ˆx
n
(i−
ˆ
δ
n
, j−
ˆ
σ
n
)
(3)
Thus, a compact notation for (1) is represented by
x(i+ 1, j+ 1) = Ax(i, j) + Bx(i− δ, j− σ)
+ Cu(i, j)
(4)
Remark 2.3. Small bold faced letters stand for vec-
tors whereas their non-bold faced counterparts mean
the variables. Capital letters describe either matrices
or block matrix.
Definition 2.4. A 2-d feedback control system is a
system accomplished from (1) by setting the inputs
variables be the states of the system itself. That is
to say
x(i+ 1, j+ 1) =
¯
Ax(i, j) +
¯
Bx(i− δ, j− σ)
(5)
in which
u(i, j) = −Fx(i, j) − Gx(i− δ, j− σ)
¯
A = A− CF
¯
B = B− CG
(6)
Remark 2.5. Under the notations of (1), (6) yields
¯
A
11
= A
11
− C
11
F
11
− C
12
F
21
¯
A
12
= A
12
− C
11
F
12
− C
12
F
22
¯
A
21
= A
21
− C
21
F
11
− C
22
F
21
¯
A
22
= A
22
− C
21
F
12
− C
22
F
22
¯
B
11
= B
11
− C
11
G
11
− C
12
G
21
¯
B
12
= B
12
− C
11
G
12
− C
12
G
22
¯
B
21
= B
21
− C
21
G
11
− C
22
G
21
¯
B
22
= B
22
− C
21
G
12
− C
22
G
22
(7)
In this paper, the system is asymptotically stable
if all the state variables vanish as the indices increase.
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
464
Definition 2.6. 2-D feedback control system (5) is
said to be asymptotically stable if all the solutions
x
1
(i, j), ..., ˆx
n
(i, j) fulfill the conditions described by
lim
(i+ j)→∞
| x
1
(i, j) |→ 0
.
.
.
lim
(i+ j)→∞
| ˆx
n
(i, j) |→ 0
(8)
Furthermore, an asymptotically stable Lagrange
solution is an analytic solution given by
Definition 2.7. A non-null asymptotically stable La-
grange solution to the state variable is the equation
x
⋆
(i, j) =
n
∑
k=1
I
k
α
i
k
β
j
k
+
n
∑
k=1
ˆ
I
k
ˆ
α
i
k
ˆ
β
j
k
(9)
where I
⋆
’s and
ˆ
I
⋆
’s are the initial values and non-null
real numbers α
∗
’s,
ˆ
α
∗
’s , β
⋆
’s and
ˆ
β
⋆
’s are such that
|α
∗
|, |
ˆ
α
∗
|, |β
∗
|, |
ˆ
β
∗
| < 1 .
Finally, this paper handles the following problem.
Problem 2.8. To establish conditions on the feedback
controller matrices for the feedbackcontrol system (5)
have asymptotically stable Lagrange solution given
by
x
1
(i, j)
.
.
.
x
t
(i, j)
=
I
1
α
i
1
β
j
1
.
.
.
I
n
α
i
n
β
j
n
ˆ
I
1
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
I
m
ˆ
α
i
m
ˆ
β
j
m
(10)
such that the numbers |α
r
| < 1, |
ˆ
α
r
| < 1 , |β
s
| < 1 and
|
ˆ
β
s
| < 1 , ∀ r, s.
Remark 2.9. The initial condition for (10) yield
x(0, 0) =
"
x
p
(0, 0)
x
q
(0, 0)
#
=
K
p
0
0 K
q
1
.
.
.
1
(11)
where
K
p
=
"
I
1
0
0 I
n
#
, K
q
=
"
ˆ
I
1
0
0
ˆ
I
n
#
(12)
3 RESULTS
Henceforth the solution to the problem is presented
gathered in a theorem. However, before doing it, a
very basic result is given for the sake of the compu-
tational procedures that are to be followed in the nu-
merical example.
Proposition 3.1. Let the controller be as in (6) and
(7). Then, their defining matrices are computed as
F
11
= C
−1
11
(A
11
−
¯
A
11
)−
C
−1
11
C
12
(C
−1
11
C
12
− C
−1
21
C
22
)
−1
×
C
−1
11
(A
11
−
¯
A
11
) − C
−1
21
(A
21
−
¯
A
21
)
F
21
= (C
−1
11
C
12
− C
−1
21
C
22
)
−1
×
C
−1
11
(A
11
−
¯
A
11
) − C
−1
21
(A
21
−
¯
A
21
)
F
12
= (C
−1
22
C
21
− C
−1
12
C
11
)
−1
×
C
−1
22
(A
22
−
¯
A
22
) − C
−1
12
(A
12
−
¯
A
12
)
F
22
= C
−1
22
(A
22
−
¯
A
22
)−
C
−1
22
C
21
(C
−1
22
C
21
− C
−1
12
C
11
)
−1
×
C
−1
22
(A
22
−
¯
A
22
) − C
−1
12
(A
12
−
¯
A
12
)
(13)
and
G
11
= C
−1
11
(B
11
−
¯
B
11
)−
C
−1
11
C
12
(C
−1
11
C
12
− C
−1
21
C
22
)
−1
×
C
−1
11
(B
11
−
¯
B
11
) − C
−1
21
(B
21
−
¯
B
21
)
G
21
= (C
−1
11
C
12
− C
−1
21
C
22
)
−1
×
C
−1
11
(B
11
−
¯
B
11
) − C
−1
21
(B
21
−
¯
B
21
)
G
12
= (C
−1
22
C
21
− C
−1
12
C
11
)
−1
×
C
−1
22
(B
22
−
¯
B
22
) − C
−1
12
(B
12
−
¯
B
12
)
G
22
= C
−1
22
(B
22
−
¯
B
22
)−
C
−1
22
C
21
(C
−1
22
C
21
− C
−1
12
C
11
)
−1
×
C
−1
22
(B
22
−
¯
B
22
) − C
−1
12
(B
12
−
¯
B
12
)
(14)
Proof. It follows straightforwardly from assumption
2.2.
Now, the main result reads.
Theorem 3.2. Let the 2-D feedback control system be
given by (5). If there exist diagonal matrices
¯
A
11
,
¯
A
12
,
¯
A
21
,
¯
A
22
,
¯
B
11
,
¯
B
12
,
¯
B
21
and
¯
B
22
in the sense of the
feedback controller (14) and such that the following
conditions all are satisfied then the control system is
asymptotically stable and its Lagrange solutions can
be established.
1. the diagonal entries of the diagonal matrices
¯
A
11
+
¯
B
11
¯
B
−1
21
¯
A
21
, and
¯
A
22
+
¯
B
22
¯
B
−1
12
¯
A
12
have all
non-null absolute values less then unit.
2. the diagonal entries of diagonal matrices
Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization
465
¯
A
−1
21
¯
B
21
α
−δ
1
1
0
.
.
.
0 α
−δ
n
n
(15)
and
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
β
−
ˆ
σ
n
n
¯
A
−1
12
¯
B
12
(16)
have all positive values less then unit.
Proof. Firstly, note that (5) can be rewritten as
x
p
(i+ 1, j) =
¯
A
11
x
p
(i, j) +
¯
A
12
x
q
(i, j)
+
¯
B
11
x
p
(i− δ, j− σ)+
¯
B
12
x
q
(i− δ, j− σ)
x
q
(i, j+ 1) =
¯
A
21
x
p
(i, j) +
¯
A
22
x
q
(i, j)
+
¯
B
21
x
p
(i− δ, j− σ)+
¯
B
22
x
q
(i− δ, j− σ)
(17)
Now substituting (10) into the first equation in
(17) leads to the subsystem described by
I
1
α
i+1
1
β
j
1
.
.
.
I
n
α
i+1
n
β
j
n
−
¯
A
11
I
1
α
i
1
β
j
1
.
.
.
I
n
α
i
n
β
j
n
−
¯
B
11
I
1
α
i−δ
1
1
β
j−σ
1
1
.
.
.
I
n
α
i−δ
n
n
β
j−σ
n
n
=
¯
A
12
ˆ
I
1
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
I
n
ˆ
α
i
n
ˆ
β
j
n
−
¯
B
12
ˆ
I
1
ˆ
α
i−
ˆ
δ
1
1
ˆ
β
j−
ˆ
σ
1
1
.
.
.
ˆ
I
n
ˆ
α
i−
ˆ
δ
n
n
ˆ
β
j−
ˆ
σ
n
n
(18)
whereas the second equation in (17) provides the set
of equations gathered as
ˆ
I
1
¯
α
i
1
¯
β
j+1
1
.
.
.
ˆ
I
n
¯
α
i
n
¯
β
j+1
n
−
¯
A
22
¯
I
1
¯
α
i
1
¯
β
j
1
.
.
.
¯
I
n
¯
α
i
n
¯
β
j
m
−
¯
B
22
ˆ
I
1
¯
α
i−
¯
δ
1
1
¯
β
j−
¯
σ
1
1
.
.
.
ˆ
I
n
¯
α
i−
¯
δ
n
n
¯
β
j−
¯
σ
n
n
=
¯
A
21
I
1
α
i
1
β
j
1
.
.
.
I
n
α
i
n
β
j
n
−
¯
B
21
I
1
α
i−δ
1
1
β
j−σ
1
1
.
.
.
I
n
α
i−δ
n
n
β
j−σ
n
n
(19)
Splitting the matrices further in order to single out
the terms related to the index, (18) yields
K
p
α
i+1
1
β
j
1
.
.
.
α
i+1
n
β
j
n
−
¯
A
11
K
p
α
i
1
β
j
1
.
.
.
α
i
n
β
j
n
−
¯
B
11
K
p
α
i−δ
1
1
β
j−σ
1
1
.
.
.
α
i−δ
n
n
β
j−σ
n
n
=
¯
A
12
K
q
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
α
i
n
ˆ
β
j
n
−
¯
B
12
K
q
ˆ
α
i−
ˆ
δ
1
1
ˆ
β
j−
ˆ
σ
1
1
.
.
.
ˆ
α
i−
ˆ
δ
n
n
ˆ
β
j−
ˆ
σ
n
n
(20)
on the other hand (19) produces
K
q
ˆ
α
i
1
ˆ
β
j+1
1
.
.
.
ˆ
α
i
n
ˆ
β
j+1
n
−
¯
A
22
K
q
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
α
i
n
ˆ
β
j
n
−
¯
B
22
K
q
ˆ
α
i−
ˆ
δ
1
1
ˆ
β
j−
ˆ
σ
1
1
.
.
.
ˆ
α
i−
ˆ
δ
n
n
ˆ
β
j−
ˆ
σ
n
n
=
¯
A
21
K
p
α
i
1
β
j
1
.
.
.
α
i
t
β
j
t
−
¯
B
21
K
p
α
i−δ
1
1
β
j−σ
1
1
.
.
.
α
i−δ
n
n
β
j−σ
n
n
(21)
Yet, equations (20) and (21) can be recasted as
K
p
α
1
0
.
.
.
α
n
−
¯
A
11
K
p
−
¯
B
11
K
p
α
−δ
1
1
β
−σ
1
1
0
.
.
.
0 α
−δ
n
n
β
−σ
n
n
×
α
i
1
β
j
1
.
.
.
α
i
n
β
j
n
=
¯
A
12
K
q
−
¯
B
12
K
q
×
ˆ
α
−
ˆ
δ
1
1
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
α
−
ˆ
δ
n
n
ˆ
β
−
ˆ
σ
n
n
×
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
α
i
n
ˆ
β
j
n
(22)
and
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
466
K
q
ˆ
β
1
0
.
.
.
ˆ
β
n
−
¯
A
22
K
q
−
¯
B
22
K
q
ˆ
α
−
ˆ
δ
1
1
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
α
−
ˆ
δ
n
n
ˆ
β
−
ˆ
σ
n
n
×
ˆ
α
i
1
ˆ
β
j
1
.
.
.
ˆ
α
i
n
ˆ
β
j
n
=
¯
A
21
K
p
−
¯
B
21
K
p
α
−δ
1
1
β
−σ
1
1
0
.
.
.
0 α
−δ
n
n
β
−σ
n
n
×
α
i
1
β
j
1
.
.
.
α
i
n
β
j
n
(23)
respectively. Note that these equations have terms de-
pending only on α’s and β’s on one side of the equal-
ity, and
ˆ
α’s and
ˆ
β’s on the other side. Furthermore
this equality has to hold for all values that the indices
assume. Since, in general, the pairs of α’s and β’s,
and the corresponding pairs
ˆ
α’s and
ˆ
β’s are not equal,
these equalities are valid only if the following expres-
sions are valid.
K
p
α
1
0
.
.
.
α
n
−
¯
A
11
K
p
−
¯
B
11
K
p
α
−δ
1
1
β
−σ
1
1
0
.
.
.
0 α
−δ
n
n
β
−σ
n
n
= 0
(24)
¯
A
12
K
q
−
¯
B
12
K
q
×
ˆ
α
−
ˆ
δ
1
1
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
α
−
ˆ
δ
n
n
ˆ
β
−
ˆ
σ
n
n
= 0
(25)
K
q
ˆ
β
1
0
.
.
.
ˆ
β
n
−
¯
A
22
K
q
−
¯
B
22
K
q
ˆ
α
−
ˆ
δ
1
1
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
α
−
ˆ
δ
n
n
ˆ
β
−
ˆ
σ
n
n
= 0
(26)
¯
A
21
K
p
−
¯
B
21
K
p
α
−δ
1
1
β
−σ
1
1
0
.
.
.
0 α
−δ
n
n
β
−σ
n
n
= 0
(27)
Hence, (24) and (27) yield
K
p
α
1
0
.
.
.
α
n
−
¯
A
11
K
p
−
¯
B
11
K
p
(
¯
B
21
K
p
)
−1
¯
A
21
K
p
= 0
(28)
taking into account the hypothesis of the theorem (28)
translates into
α
1
0
.
.
.
0 α
n
=
¯
A
11
+
¯
B
11
¯
B
−1
21
¯
A
21
(29)
As for β
⋆
(⋆ = 1, ··· , n), they are computed from
(27) , which with the hypothesis of the theorem lead
to
β
σ
1
1
0
.
.
.
0 β
σ
n
n
=
¯
A
−1
21
¯
B
21
α
−δ
1
1
0
.
.
.
0 α
−δ
n
n
(30)
Similarly, from (26) and (25)
K
q
ˆ
β
1
0
.
.
.
0
ˆ
β
n
−
¯
A
22
K
q
−
¯
B
22
K
q
(
¯
B
12
K
q
)
−1
¯
A
12
K
q
= 0
(31)
and it turns out that the hypothesis of the theorem
brings up
ˆ
β
1
0
.
.
.
ˆ
β
n
=
¯
A
22
+
¯
B
22
¯
B
−1
12
¯
A
12
(32)
In the same way,
ˆ
α
⋆
, ⋆ = 1, ··· , n are computed
from (25) according to
ˆ
α
ˆ
δ
1
1
0
.
.
.
0
ˆ
α
ˆ
σ
n
n
=
ˆ
β
−
ˆ
σ
1
1
0
.
.
.
0
ˆ
β
−
ˆ
σ
n
n
¯
A
−1
12
¯
B
12
(33)
Finally, (29), (30), (32), and (33) establish the
claim of the theorem.
Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization
467
In practical terms, the key to solving the problem
comes down to the determination of the diagonal ma-
trices. The following remark provides a directive to
roughly evaluate the diagonal matrices.
Remark 3.3. It is clear from (15) and (16) that the
diagonal entries of
¯
A
−1
21
¯
B
21
as well as
¯
A
−1
12
¯
B
12
must
have absolute values smaller than unit in order to be
able to establish Lagrange solutions. Thus, in prac-
tice, begin focusing on these diagonal matrices trying
to set the product at relatively small values; then set
each of the matrices individually. Once these matri-
ces are defined, handle (29) and (32) to define the re-
maining diagonal matrices. The controller matrices
are computed only after an estimate of these diagonal
matrices are obtained.
In the next section, a numerical example is pre-
sented to show how to make the calculations.
4 NUMERICAL EXAMPLE
Let the 2-D control system be given by
x
1
(i+ 1, j)
x
2
(i+ 1, j)
ˆx
1
(i, j + 1)
ˆx
2
(i, j + 1)
=
0.66 0.19 0.29 0.13
0.15 0.27 0.53 0.35
0.83 0.43 0.31 0.55
0.44 0.17 0.29 0.57
x
1
(i, j)
x
2
(i, j)
ˆx
1
(i, j)
ˆx
2
(i, j)
+
0.10 0.23 0.47 0.15
0.20 0.37 0.17 0.25
0.51 0.67 0.71 0.15
0.35 0.11 0.31 0.13
×
x
1
(i− δ
1
, j − σ
1
)
x
2
(i− δ
2
, j − σ
2
)
ˆx
1
(i−
ˆ
δ
1
, j −
ˆ
σ
1
)
ˆx
2
(i−
ˆ
δ
2
, j −
ˆ
σ
2
)
+
0.70 0.13 0.17 0.11
0.05 0.80 0.07 0.15
0.11 0.23 0.75 0.17
0.15 0.13 0.37 0.90
u
1
(i, j)
u
2
(i, j)
ˆu
1
(i, j)
ˆu
2
(i, j)
(34)
from which the composing matrices read
A
11
=
0.66 0.19
0.15 0.27
A
12
=
0.29 0.13
0.53 0.35
A
21
=
0.83 0.43
0.44 0.17
A
22
=
0.31 0.55
0.29 0.57
B
11
=
0.10 0.23
0.20 0.37
B
12
=
0.47 0.15
0.17 0.25
B
21
=
0.51 0.67
0.35 0.11
B
22
=
0.71 0.15
0.31 0.13
C
11
=
0.70 0.13
0.05 0.80
C
12
=
0.17 0.11
0.07 0.15
C
21
=
0.11 0.23
0.15 0.13
C
22
=
0.75 0.17
0.37 0.90
(35)
Here
¯
A
−1
21
¯
B
21
as well as
¯
A
−1
12
¯
B
12
are required to
satisfy
¯
A
−1
21
¯
B
21
=
"
0.0667 0.000
0.0000 0.1667
#
(36)
as well as
¯
A
−1
12
¯
B
12
=
"
0.0723 0.000
0.0000 0.2704
#
(37)
From these and keeping in mind the equations(29)
and (32), the diagonal matrices are set as
¯
A
11
=
0.11 0.00
0.00 0.67
¯
A
12
=
6.50 0.00
0.00 2.70
¯
A
21
=
1.50 0.00
0.00 3.00
¯
A
22
=
0.51 0.00
0.00 0.47
¯
B
11
=
0.05 0.00
0.00 0.01
¯
B
12
=
0.47 0.00
0.00 0.73
¯
B
21
=
0.10 0.00
0.00 0.50
¯
B
22
=
0.03 0.00
0.00 0.07
(38)
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
468
Thus (7) gives the controller matrices
F
11
=
0.9422 0.5315
0.0816 0.0607
F
21
=
−1.2451 1.3361
0.8320 −3.7911
F
22
=
0.4778 1.6626
1.5606 −0.1926
F
12
=
−9.4028 0.3825
0.9157 −3.0708
(39)
along with
G
11
=
−0.1045 0.1509
0.1804 0.5314
G
21
=
0.4637 0.9148
0.1896 −0.9113
G
22
=
0.8978 0.3677
−0.0055 −0.0334
G
12
=
−0.2451 0.2494
0.1503 −0.6415
(40)
Now taking into account expressions in (29) and
(30)
"
α
1
0
0 α
2
#
=
"
0.76 0.00
0.00 0.73
#
(41)
and
"
β
σ
1
1
0
0 β
σ
2
2
#
=
"
0.0667 0.0000
0.0000 0.1667
#
×
(0.76)
−δ
1
0.00
0.00 (0.73)
−δ
2
(42)
are computed. Note that as far as the diagonal values
of the resulting matrix on the right hand side of (42)
have absolute values less than unit, β’s compose La-
grange solutions to the control system. On the other
hand (32) and (33) reduce to
"
ˆ
β
1
0
ˆ
β
2
#
=
"
0.9249 0.0000
0.0000 0.7289
#
(43)
and
"
ˆ
α
−
ˆ
δ
1
1
0
0
ˆ
α
−
ˆ
σ
2
2
#
=
"
0.0723 0.0000
0.0000 0.2704
#
×
(0.9249)
ˆ
σ
1
0.0000
0.0000 (0.7289)
ˆ
σ
2
(44)
Analogous comments to (42) hold here for the exis-
tence of Lagrange solutions.
If (41), (42), (43) and (44) are not Lagrange, (36)
and (37) are set to different values; in general to
smaller values, and the computations are carried out
all over again.
5 FINAL REMARKS
This paper was concerned with the design of a feed-
back controller such that the overall system is asymp-
totically stable. The key point was the imposition of
a very specific Lagrange solution to then find out the
condition for the existence of such a system.
The results showed that a solution can be found if
the controller can diagonalize the matrix blocks of the
matrices composing the model, which is represented
by the set of partial difference equations. A numerical
example was presented to show how the mechanics of
the calculations work..
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