On the Asymptotic Stability Analysis of a Certain Type of Discrete-time
3-D Linear Systems
Guido Izuta
Department of Social Information, Yonezawa Women’s College, 6-15-1 Toori Machi, Yonezawa, Yamagata, Japan
Keywords:
3-D Systems, Asymptotic Stability Analysis, Lagrange Method, Partial Difference Equations.
Abstract:
This work is concerned with the analysis of 3-d (3-dimensional) systems. The aim is to establish conditions
that guarantee the asymptotic stability of these kinds of systems. To accomplish it, the Lagrange candidate
solutions method for partial difference equations is adopted here. We show that the systems are asymptotically
stable if the entries of the matrices of their state space descriptions yield a solution in the Lagrange solution
sense. Furthermore, the particular cases in which the matrices can be turned into a diagonal matrix by means
of the canonical transformation is studied in order to figure out the role of the eigenvalues on the stability
conditions.
1 INTRODUCTION
Multidimensional linear systems theory grew out of
theoretical and practical research on systems that can
be mathematically modeled and described by partial
difference and differential equations; and the field,
which is an interdisciplinary area that shares common
grounds with engineering, mathematics, physics and
other sciences, has been developing ever since with
each section owing its particular tools and techniques
to handle the problems (see for example (Bose, 1982;
Tzafestas, 1986; Wall, 1987; Lim, 1990; Leondes,
1995; Jerri, 1996; Zerz, 2000; Du and Xie, 2002;
Matsuo and Hasegawa, 2003; Cheng, 2003; Rosen-
thal and Gilliam, 2003; Wood, 2004; Elaydi, 2005;
Russell and Cohn, 2013)).
The investigations of multidimensional linear sys-
tems in engineering date back to the 1960s when a
certain kind of electrical system network was mod-
eled by as a set of partial differential equations in
two independent variables, namely as 2-d linear sys-
tem (Kasami, 1960; Ansell, 1964). In the 1970s, the
state space description (Roesser, 1972; Attasi, 1973;
Roesser, 1975; Marchesini, 1978) and the matrix
polynomial (Juri, 1978; Bose, 1982) approaches were
established and these frameworks prevailed until the
1990’s when the energy method matured as a fruitful
formalism to the analysis and design of multidimen-
sional control systems (W. S. Lu, 1992). Eventually,
this reasoning evolved into linear matrix inequalities
that have been applied to 2-d discrete linear multidi-
mensional systems (Izuta, 2007a; Izuta, 2007b; Izuta,
2007c).
More recently, unlike the techniques so far, the
authors pursued a solution to 2-d discrete linear con-
trol systems on grounds of the Lagrange method for
solving partial difference equations from the con-
troller design standpoint (Izuta, 2010a; Izuta, 2010b).
Briefly, the key contribution of these studies was to
show how to obtain an explicit solution to the system
of partial difference equations based on the Lagrange
method when a solution to them exists.
Motivated by these works, in this paper we are
concerned with the asymptotic stability analysis of
discrete-time 3-d linear systems in the scope of La-
grange solutions to partial difference equations. The
state space description of the system consists of two
matrices on its right hand side. One of them is related
to the current states and the other one corresponds to
the states with smaller values indices, which in the
ordinary 1-d systems theory, these kinds of states are
often called systems with state delays.
Taking these into consideration, the aims of this
work are: (1) to extend the previous result to 3-d sys-
tems; (2) to establish conditions on the entries of the
matrix of the state space description in order to guar-
antee the asymptotic stability; and (3) to shed some
light on the link between the eigenvalues and the La-
grange solutions when the matrix composing the sys-
tem description can be transformed into a diagonal
matrix by means of the canonical transformation.
Finally, the paper is organized as follows: in sec-
665
Izuta G..
On the Asymptotic Stability Analysis of a Certain Type of Discrete-time 3-D Linear Systems.
DOI: 10.5220/0005043306650670
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 665-670
ISBN: 978-989-758-039-0
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

tion 2, the discrete-time 3-d linear system and the as-
sumptions are presented; the asymptotic stability con-
ditions are establish in section 3; and some final re-
marks are enunciated in section 4 .
2 PRELIMINARIES
In this section we provide the definitions and concepts
that define the scope as well as the problem to be in-
vestigated in the work. To begin with, the system is
described in the following definition.
Definition 1. The 3-d system is described by the sys-
tem of partial difference equations given by


x
1
(i + 1, j, k)
x
2
(i, j + 1, k)
x
3
(i, j, k + 1)


=


a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33




x
1
(i, j, k)
x
2
(i, j, k)
x
3
(i, j, k)


+


b
11
b
12
b
13
b
21
b
22
b
23
b
31
b
32
b
33




x
1d
x
2d
x
3d


in which


x
1d
x
2d
x
3d


=
.
.


x
1
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
x
2
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
x
3
(i − ∆
1
, j − ∆
2
, k − ∆
3
)


(1)
and i, j, k, ∆
?
∈ Z (∀?), x
∗
(i, j, j), x
∗
(i, j, j) ∈ ℜ, ∀∗,
are the states of the system, a
∗?
and b
∗?
, ∀∗ ? are real
valued entries of the matrices.
Remark 1. Ruling out the terms related to ∆
?
, ∀?, the
state space notation (1) reduces to one introduced by
Roesser (Roesser, 1975).
Remark 2. Hereafter for the sake of compact nota-
tions and when convenient we reference to the first
matrix on the right hand side of (13) as matrix A
and the second one as matrix B. Moreover, in some
cases we abusively reference to system (1) as x
+1
=
Ax + Bx
∆
.
Next, we state the definition of asymptotic stabil-
ity and the kind of solution searched for in this in-
vestigation. The statement of asymptotic stability is
conceptually the same as in (Jerri, 1996).
Definition 2. System (1) is asymptotically stable if its
solutions satisfy
lim
(i+ j+k)→∞
| x
1
(i, j, k) |→ 0
lim
(i+ j+k)→∞
| x
2
(i, j, j) |→ 0
lim
(i+ j+k)→∞
| x
3
(i, j, j) |→ 0
(2)
Furthermore, if there exist 0 6= |α
1
|, |β
1
|, |γ
1
|, |α
2
|,
|β
2
|, |γ
2
|, |α
3
|, |β
3
|, |γ
3
| < 1 such that the so called
Lagrange solution candidates, which are given by
x
1
(i, j, k) = α
i
1
β
j
1
γ
k
1
x
2
(i, j, k) = α
i
2
β
j
2
γ
k
2
x
3
(i, j, k) = α
i
3
β
j
3
γ
k
3
(3)
can be established, then
lim
(i+ j+k)→∞
(| x
1
(i, j, k) | + | x
2
(i, j, k) |
+ | x
3
(i, j, k) |) → 0
(4)
holds for i, j ≥ 0 and the symbol | ∗ | meaning the
conventional vector norm of ∗; and consequently, the
system is asymptotically stable.
Remark 3. Basically, (4) means that as the indices
increase the sum of absolute values of the states van-
ish.
Finally, the aim of this investigation is to establish
the conditions that the system has to satisfy in order to
be asymptotically stable in the sense of the Lagrange
solution method.
3 RESULTS
Here the system is analyzed and asymptotic stability
conditions are pursued. First, the existence of a solu-
tion is given by the conditions stated in the following
theorem.
Theorem 1. Consider system (1). Then there exists
an asymptotically stable solution on the grounds of
the Lagrange solutions if either or both of the follow-
ing conditions hold.
• Condition 1:
α
1
= α
2
= α
3
.
= α
β
1
= β
2
= β
3
.
= β
δ
1
= δ
2
= δ
3
.
= β
(5)
• Condition 2:
α
∆
1
+1
1
β
∆
2
1
γ
∆
3
1
− a
11
α
∆
1
1
β
∆
2
1
γ
∆
3
1
− b
11
= 0
α
∆
1
2
β
∆
2
+1
2
γ
∆
3
2
− a
22
α
∆
1
2
β
∆
2
2
γ
∆
3
2
− b
22
= 0
α
∆
1
3
β
∆
2
3
γ
∆
3
+1
3
− a
33
α
∆
1
3
β
∆
2
3
γ
∆
3
3
− b
33
= 0
(6)
and
a
12
α
∆
1
2
β
∆
2
2
γ
∆
3
2
+ b
12
= 0
a
21
α
∆
1
1
β
∆
2
1
γ
∆
3
1
+ b
21
= 0
a
31
α
∆
1
1
β
∆
2
1
γ
∆
3
1
+ b
31
= 0
(7)
and
a
13
α
∆
1
3
β
∆
2
3
γ
∆
3
3
+ b
13
= 0
a
23
α
∆
1
3
β
∆
2
3
γ
∆
3
3
+ b
23
= 0
a
32
α
∆
1
2
β
∆
2
2
γ
∆
3
2
+ b
32
= 0
(8)
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
666

Proof. To begin with, rewrite the equations of system
(1) as
x
1
(i + 1, j, k) = a
11
x
1
(i, j, j) + a
12
x
2
(i, j, k)
+a
13
x
3
(i, j, k)
+b
11
x
1
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
12
x
2
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
13
x
3
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
for the first entry of the matrix of the system and
x
2
(i, j + 1, k) = a
21
x
1
(i, j, j) + a
22
x
2
(i, j, k)
+a
23
x
3
(i, j, k)
+b
21
x
1
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
22
x
2
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
23
x
3
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
(9)
for the second one. Finally,
x
3
(i, j, k + 1) = a
31
x
1
(i, j, j) + a
32
x
2
(i, j, k)
+a
33
x
3
(i, j, k)
+b
31
x
1
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
32
x
2
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
+b
33
x
3
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
for the last one. Now substitute (3) into it to come up
with the equations
α
i−∆
1
1
β
j−∆
2
1
γ
k−∆
3
1
× (α
∆
1
+1
1
β
∆
2
1
γ
∆
3
1
−a
11
α
∆
1
1
β
∆
2
1
γ
∆
3
1
− b
11
) = α
i−∆
1
2
β
j−∆
2
2
γ
k−∆
3
2
×(a
12
α
∆
1
2
β
∆
2
2
γ
∆
3
2
+ b
12
) + α
i−∆
1
3
β
j−∆
2
3
γ
k−∆
3
3
×(a
13
α
∆
1
3
β
∆
2
3
γ
∆
3
3
+ b
13
)
(10)
and
α
i−∆
1
2
β
j−∆
2
2
γ
k−∆
3
2
× (α
∆
1
2
β
∆
2
+1
2
γ
∆
3
2
−a
22
α
∆
1
2
β
∆
2
2
γ
∆
3
2
− b
22
) = α
i−∆
1
1
β
j−∆
2
1
γ
k−∆
3
1
×(a
21
α
∆
1
1
β
∆
2
1
γ
∆
3
1
+ b
21
) + α
i−∆
1
3
β
j−∆
2
3
γ
k−∆
3
3
×(a
23
α
∆
1
3
β
∆
2
3
γ
∆
3
3
+ b
23
)
(11)
and finally
α
i−∆
1
3
β
j−∆
2
3
γ
k−∆
3
3
× (α
∆
1
3
β
∆
2
3
γ
∆
3
+1
3
−a
33
α
∆
1
3
β
∆
2
3
γ
∆
3
3
− b
33
) = α
i−∆
1
1
β
j−∆
2
1
γ
k−∆
3
1
×(a
31
α
∆
1
1
β
∆
2
1
γ
∆
3
1
+ b
31
) + α
i−∆
1
2
β
j−∆
2
2
γ
k−∆
3
2
×(a
32
α
∆
1
2
β
∆
2
2
γ
∆
3
2
+ b
32
)
(12)
From these equations we have that the left hand
terms depend on the indices i, j and k whereas the
expressions on the right are independent from them.
Moreover since the equalities must hold for all values
of the indices, we have that in order to have an asymp-
totically stable system, the claim of the theorem must
hold.
Theorem 2. Consider system (1) and let
¯
A
?
.
=
∑
t=1,···,3
a
?t
, ? = 1, · · · , 3
¯
B
∗
.
=
∑
s=1,···,3
b
∗s
, ∗ = 1, · · · , 3
ˆ
C
?∗
.
=
¯
A
?
×
¯
B
∗
(13)
Then there exists a Lagrange solution on the
grounds of the equalities (5) if either of the following
conditions are fulfilled.
• Condition 1: there exists an α, |α| < 1 subject to
m
α
< α < M
α
m
α
.
= max
n
ˆ
C
21
+
ˆ
C
12
−
¯
B
1
¯
B
2
,
ˆ
C
13
−
ˆ
C
31
−
¯
B
1
¯
B
3
o
M
α
.
= max
n
ˆ
C
21
+
ˆ
C
12
+
¯
B
1
¯
B
2
,
ˆ
C
13
−
ˆ
C
31
+
¯
B
1
¯
B
3
o
(14)
• Condition 2: there exists a β, |β| < 1 subject to
m
β
< β < M
β
m
β
.
= max
n
ˆ
C
21
−
ˆ
C
12
−
¯
B
2
¯
B
1
,
ˆ
C
23
−
ˆ
C
32
−
¯
B
2
¯
B
3
o
M
β
.
= max
n
ˆ
C
21
−
ˆ
C
12
+
¯
B
2
¯
B
1
,
ˆ
C
23
−
ˆ
C
32
+
¯
B
2
¯
B
3
o
(15)
• Condition 3: there exists a γ, |γ| < 1 subject to
m
γ
< γ < M
γ
m
γ
.
= max
n
ˆ
C
31
−
ˆ
C
13
−
¯
B
3
¯
B
1
,
ˆ
C
32
−
ˆ
C
23
−
¯
B
3
¯
B
2
o
M
γ
.
= max
n
ˆ
C
31
−
ˆ
C
13
+
¯
B
3
¯
B
1
,
ˆ
C
32
−
ˆ
C
23
+
¯
B
3
¯
B
2
o
(16)
Proof. Since (5) holds, we can write equations (10)
through (12) as
(α −
¯
A
1
)α
∆
1
β
∆
2
γ
∆
3
=
¯
B
1
(β −
¯
A
2
)α
∆
1
β
∆
2
γ
∆
3
=
¯
B
2
(γ −
¯
A
3
)α
∆
1
β
∆
2
γ
∆
3
=
¯
B
3
(17)
from which we get the equations
α
¯
B
2
− β
¯
B
1
=
ˆ
C
12
−
ˆ
C
21
α
¯
B
3
− γ
¯
B
1
=
ˆ
C
13
−
ˆ
C
31
β
¯
B
3
− γ
¯
B
2
=
ˆ
C
23
−
ˆ
C
32
(18)
from which the solutions α, β and γ can not be
uniquely established since an equivalent matrix rep-
resentation of (18) given by


¯
B
2
−
¯
B
1
0
¯
B
3
0 −
¯
B
1
0
¯
B
3
−
¯
B
2




α
β
γ


=


ˆ
C
12
−
ˆ
C
21
ˆ
C
13
−
ˆ
C
31
ˆ
C
23
−
ˆ
C
32


(19)
has a singular matrix on its left hand side. Thus work-
ing on the first equation of (18), we have that
β =
(α−
¯
A
1
)
¯
B
2
+
ˆ
C
21
¯
B
1
(20)
and since we must have |β| < 1 to have an asymptoti-
cally stable solution, the right side of (20) yields
−
¯
B
1
+
ˆ
C
21
+
ˆ
C
12
¯
B
2
< α <
¯
B
1
+
ˆ
C
21
+
ˆ
C
12
¯
B
2
(21)
OntheAsymptoticStabilityAnalysisofaCertainTypeofDiscrete-time3-DLinearSystems
667

Furthermore, plugging (20) into the third equation
in (18) leads to
γ =
(α−
¯
A
1
)
¯
B
3
+
ˆ
C
31
¯
B
1
(22)
which is part of an asymptotically stable solution if
|γ| < 1, which is equivalent to the condition
−
¯
B
1
−
ˆ
C
31
+
ˆ
C
13
¯
B
3
< α <
¯
B
1
+
ˆ
C
31
+
ˆ
C
13
¯
B
3
(23)
It is straightforward that (21) and (23) reduce to (15).
In order to show (16), note that the first two equa-
tions of (18) provide
α =
(β−
¯
A
2
)
¯
B
1
+
¯
A
1
¯
B
2
¯
B
2
(24)
and
γ =
(β−
¯
A
2
)
¯
B
3
+
¯
A
3
¯
B
2
¯
B
2
(25)
which under the conditions |α| < 1 and |γ| < 1 pro-
duce the desired result.
Likewise, to establish (17), consider the second
and first equations of (18), which render
α =
(γ−
¯
A
3
)
¯
B
1
+
¯
A
1
¯
B
3
¯
B
3
(26)
and
β =
(γ−
¯
A
3
)
¯
B
2
+
ˆ
C
23
¯
B
3
(27)
and the claim is fulfilled by taking into account |α| <
1 and |β| < 1.
Remark 4. Theorem 3 implies that we will have an
asymptotically stable system if the matrices of the
state space description provide the Lagrange solu-
tions according to one of the conditions established
in the proof.
Next we examine the conditions under which it is
possible to establish a Lagrange solution to system (1)
Theorem 3. There exists a Lagrange solution on the
grounds of the equalities (6) if
b
21
a
21
=
b
31
a
31
,
b
12
a
12
=
b
32
a
32
,
b
13
a
13
=
b
23
a
23
(28)
along with |α
1
|, |β
2
| and |γ
3
|, which are represented
respectively by
| a
11
−
a
21
b
11
b
21
| = | a
11
−
a
31
b
11
b
31
| < 1
| a
22
−
a
12
b
22
b
12
| = | a
22
−
a
32
b
22
b
32
| < 1
| a
33
−
a
13
b
33
b
13
| = | a
33
−
a
23
b
33
b
23
| < 1
(29)
hold.
Proof. Since equations (6) through (8) translate into
(α
1
− a
11
)α
∆
1
1
β
∆
2
1
γ
∆
3
1
= b
11
(β
2
− a
22
)α
∆
1
2
β
∆
2
2
γ
∆
3
2
= b
22
(γ
3
− a
11
)α
∆
1
3
β
∆
2
3
γ
∆
3
3
= b
33
α
∆
1
1
β
∆
2
1
γ
∆
3
1
=
−b
21
a
21
=
−b
31
a
31
α
∆
1
2
β
∆
2
2
γ
∆
3
2
=
−b
12
a
12
=
−b
32
a
32
α
∆
1
3
β
∆
2
3
γ
∆
3
3
=
−b
13
a
13
=
−b
23
a
23
(30)
and the claims follow straightforwardly by recalling
the conditions that the Lagrange solution must satisfy
to guarantee the asymptotic stability of the system.
Remark 5. The conditions (28) in theorem 3 say that
(6) is met only if the system is described by some very
particular kinds of matrices. In general, systems sat-
isfying these conditions are not common.
In what follows, the framework developed so far
is used to get some insights into the role of the eigen-
values of the matrices in the system stability.
3.1 Analysis of Diagonalizable Systems
In the sequel we examine systems which have diag-
onalizable matrices in its state space system descrip-
tion, and pursue the conditions on the eigenvalues to
have asymptotically stable systems.
Theorem 4. Consider system (1) and let it be com-
posed by a diagonalizable matrix B by means of the
canonical transformation (Gantmatcher, 1959; Suda,
1978). And let matrix
˜
A be the matrix obtained from
the canonical transformation of B applied on A. Also,
assume that the Lagrange solution candidate is com-
posed by ρ
?
, σ
?
and φ
?
, for ? = 1, ··· , 3. Then the
system is asymptotically stable in the sense of the La-
grange solution method if the eigenvalues of matrix B
defined by λ
1
, λ
2
and λ
3
satisfy either of the condi-
tions.
• Condition 1:
−1 −
˜
A
∗
< λ
∗
< 1 −
˜
A
∗
∗ = 1, · · · , 3
˜
A
∗
= sum of row entries of
˜
A
(31)
for ρ
1
= ρ
2
= ρ
3
.
= ρ, σ
1
= σ
2
= σ
3
.
= σ and φ
1
=
φ
2
= φ
3
.
= φ
• Condition 2: assuming the same reasoning of con-
dition 2 in theorem 1, the matrix
˜
A must also be a
diagonal matrix and the Lagrange solutions can
be established such that the eigenvalues of B can
be written as
λ
1
= ρ
∆
1
1
σ
∆
2
1
φ
∆
3
1
(ρ
1
− ˜a
11
)
λ
2
= ρ
∆
1
2
σ
∆
2
2
φ
∆
3
2
(σ
2
− ˜a
22
)
λ
3
= ρ
∆
1
3
σ
∆
2
3
φ
∆
3
3
(φ
3
− ˜a
33
)
(32)
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668

Proof. Since matrix B is diagonalizable by means of
canonical transformation, there exists a matrix T such
that T × B × T
−1
, so that (1) writes
T x
+1
= TAT
−1
(T x) + T BT
−1
(T x
∆
)
(33)
where T BT
−1
turns into the diagonal matrix D with
eigenvalues λ
1
, λ
2
and λ
3
of B as entries. Thus defin-
ing vector z as z
.
= T z, (33) is described by
z
+1
=
˜
Az + Dz
∆
(34)
which is a loose representation of the equations
z
1
(i + 1, j, k) = ˜a
11
z
1
(i, j, j) + ˜a
12
z
2
(i, j, k)
+ ˜a
13
z
3
(i, j, k)
+λ
1
z
1
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
z
2
(i, j + 1, k) = ˜a
21
z
1
(i, j, j) + ˜a
22
x
2
(i, j, k)
+ ˜a
23
z
3
(i, j, k)
+λ
2
z
2
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
z
3
(i, j, k + 1) = ˜a
31
z
1
(i, j, j) + ˜a
32
z
2
(i, j, k)
+ ˜a
33
z
3
(i, j, k)
+λ
3
z
3
(i − ∆
1
, j − ∆
2
, k − ∆
3
)
(35)
and letting the Lagrange solution candidates as
z
?
= ρ
i
?
σ
j
?
φ
k
?
? = 1, · · · , 3
(36)
and proceeding as in theorem 1, one concludes that
similar conditions as (5) - (8) as long as the corre-
sponding terms are suitably changed.
Now with ρ
1
= ρ
2
= ρ
3
.
= ρ, σ
1
= σ
2
= σ
3
.
= σ
and φ
1
= φ
2
= φ
3
.
= φ, the equations to be satisfied
are
ρ
∆
1
σ
∆
2
φ
∆
3
[ρ − (
˜
A
1
+ λ
1
)] = 0
ρ
∆
1
σ
∆
2
φ
∆
3
[σ − (
˜
A
2
+ λ
2
)] = 0
ρ
∆
1
σ
∆
2
φ
∆
3
[φ − (
˜
A
3
+ λ
3
)] = 0
(37)
from which (31) follows.
On the other hand, focusing on the equivalent con-
dition to (6), one realizes that to fulfill it we must have
ρ
∆
1
1
σ
∆
2
1
φ
∆
3
1
[ρ
1
− ˜a
11
] − λ
1
= 0
ρ
∆
1
2
σ
∆
2
2
φ
∆
3
2
[σ
2
− ˜a
22
] − λ
2
= 0
ρ
∆
1
3
σ
∆
2
3
φ
∆
3
3
[φ
3
− ˜a
11
] − λ
3
= 0
˜a
12
= ˜a
13
= ˜a
21
= ˜a
23
= ˜a
31
= ˜a
32
= 0
(38)
from which one reaches to (32).
Remark 6. Theorem 4 says that one can figure out
whether the system is asymptotically stable by just
checking out values of the eigenvalues of the trans-
formed system since condition 2, which imposes that
both matrices be diagonal, is quite unlikely to be sat-
isfied in general.
Finally, we investigate the systems with diagonal-
izable matrix A. The results are packed up in the fol-
lowing theorem.
Theorem 5. Let system (1) have diagonalizable ma-
trix B by means of the canonical transformation, and
matrix
˜
B be the result of the canonical transformation
of A applied on B. Then based on the Lagrange so-
lution candidates ρ
?
, σ
?
and φ
?
, for ? = 1, · · · , 3, the
eigenvalues of matrix A given by λ
1
, λ
2
and λ
3
satisfy
either of the conditions in order to assure the asymp-
totic stability.
• Condition 1: ρ
1
= ρ
2
= ρ
3
= ρ, σ
1
= σ
2
= σ
3
=
σ and φ
1
= φ
2
= φ
3
= φ. Furthermore, all the
sets ranging from maximum to minimum values as
defined by
max{λ
1
− (λ
2
+ 1)
˜
B
1
˜
B
2
, λ
1
− (λ
3
+ 1)
˜
B
1
˜
B
3
}
min{λ
1
− (λ
2
− 1)
˜
B
1
˜
B
2
, λ
1
− (λ
3
− 1)
˜
B
1
˜
B
3
}
(39)
and
max{λ
2
− (λ
1
+ 1)
˜
B
2
˜
B
1
, λ
2
− (λ
3
+ 1)
˜
B
2
˜
B
3
}
min{λ
2
− (λ
1
− 1)
˜
B
2
˜
B
1
, λ
2
− (λ
3
− 1)
˜
B
2
˜
B
3
}
(40)
and
max{λ
3
− (λ
1
+ 1)
˜
B
3
˜
B
1
, λ
3
− (λ
2
+ 1)
˜
B
3
˜
B
2
}
min{λ
3
− (λ
1
− 1)
˜
B
3
˜
B
1
, λ
3
− (λ
2
− 1)
˜
B
3
˜
B
2
}
(41)
contain at least one non-null element whose abso-
lute value is less than unity.
• Condition 2: proceeding similarly to condition 2
in the previous theorem 4, matrix
˜
A must be diag-
onal and there must exist Lagrange solutions such
that the eigenvalues of matrix B are to satisfy
λ
1
= ρ
1
−
˜a
11
ρ
∆
1
1
σ
∆
2
1
φ
∆
3
1
λ
2
= σ
2
−
˜a
22
ρ
∆
1
2
σ
∆
2
2
φ
∆
3
2
λ
3
= φ
3
−
˜a
33
ρ
∆
1
3
σ
∆
2
3
φ
∆
3
3
(42)
Proof. The steps to show the claims are close to the
ones describe in the previous theorems.
Remark 7. Theorem 5 requires the computation of
the maximums and minimums in order to establish the
range of testing. In this sense, whenever possible, the
conditions of theorem 4 are a bit easier to check.
4 FINAL REMARKS
In this paper we studied the asymptotic stability of
discrete time 3-d linear systems. Conditions to assure
the stability were established and collected in 5 theo-
rems. Theorem 1 gives the existence condition of the
solutions. In practice, to check whether the system
OntheAsymptoticStabilityAnalysisofaCertainTypeofDiscrete-time3-DLinearSystems
669

is asymptotically stable, one basically has to test ei-
ther theorem 2 or theorem 4 or theorem 5. Theorem 2
allows us to decide it by making the computations di-
rectly on the entries of the matrices, whereas the oth-
ers require the application of the canonical transfor-
mation of the matrices. The striking point of the use
of canonical transformation is that it yield some infor-
mation on the eigenvalues of the matrices, which pro-
vides deeper insights into the relationships between
the matrix structure and asymptotic stability.
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