Common Diagonal Stability of Second Order Interval Systems
Bengi Yıldız
1
, Taner B
¨
uy
¨
ukk
¨
oro
˘
glu
2
and Vakif Dzhafarov
2
1
Department of Mathematics, Faculty of Science and Letters, Bilecik Seyh Edebali University,
Gulumbe Campus, 11210 Bilecik, Turkey
2
Department of Mathematics, Faculty of Science, Anadolu University, 26470 Eskisehir, Turkey
Keywords:
Hurwitz Diagonal Stability, Schur Diagonal Stability, Common Diagonal Stability, Interval Matrices.
Abstract:
In this paper for second order interval systems we obtain necessary and sufficient conditions for the existence
of a common diagonal solutions to the Lyapunov (Stein) inequality. Hurwitz and Schur cases are considered
separately. One necessary and sufficient condition is given for n ×n interval family of Z-matrices. The
obtained results also give diagonal solution in the case of existence.
1 INTRODUCTION
Consider n×n real matrix A = (a
i j
). If all eigenvalues
of A lie in the open left half plane (open unit disc) A is
said to be Hurwitz (Schur) stable. Necessary and suf-
ficient condition for Hurwitz (Schur) stability is the
existence of a symmetric positive definite matrix P
(i.e. P > 0) such that
A
T
P + PA < 0 (A
T
PA −P < 0) (1)
where B < 0 means −B > 0. If in (1) the matrix P
can be chosen to be positive diagonal then A is called
Hurwitz (Schur) diagonally stable.
Diagonal stability problem have many applica-
tions (see (Arcat and Sontag, 2006; Johnson, 1974;
Ziolko, 1990; Kaszkurewicz and Bhaya, 2000)).
The existence of diagonal type solutions are con-
sidered in the works (Khalil, 1982; Kaszkurewicz
and Bhaya, 2000; Mason and Shorten, 2006; Pastra-
vanu and Voicu, 2006; Dzhafarov and B
¨
uy
¨
ukk
¨
oro
˘
glu,
2006; B
¨
uy
¨
ukk
¨
oro
˘
glu, 2012) and references therein.
For a single 2 ×2 real matrix
A =

a
1
a
2
a
3
a
4

(2)
algebraic characterization of Hurwitz and Schur diag-
onal stability are available.
Fact 1.1 ((Cross, 1978; Kaszkurewicz and Bhaya,
2000)). The matrix (2) is Hurwitz diagonally stable
if and only if a
1
< 0, a
4
< 0 and a
1
a
4
−a
2
a
3
> 0.
Fact 1.2 ((Mills et al., 1978; Kaszkurewicz and
Bhaya, 2000)). The matrix (2) is Schur diagonally
stable if and only if |a
1
a
4
−a
2
a
3
| < 1, |a
1
+ a
4
| <
1 + a
1
a
4
−a
2
a
3
and |a
1
−a
4
| < 1 −(a
1
a
4
−a
2
a
3
).
Consider an 2 ×2 interval family
A =

a
1
a
2
a
3
a
4

: a
i
∈ [a
−
i
,a
+
i
], i = 1, 2,3,4

.
(3)
There are many important results for stability of in-
terval systems (see for example (Kharitonov, 1978;
Barmish, 1994; Deng et al., 1999)). Define the fol-
lowing 4-dimensional box
Q = [a
−
1
,a
+
1
] ×···×[a
−
4
,a
+
4
].
Without loss of generality all 2 ×2 positive diagonal
matrices may be normalized to have the form D =
diag(λ,1) where λ > 0.
Definition 1.1. The family (3) is robust Hurwitz
(Schur) diagonally stable if every matrix in (3) is Hur-
witz (Schur) diagonally stable, i.e. for every a =
(a
1
,a
2
,a
3
,a
4
) ∈ Q there exist λ > 0 such that
A
T
D + DA < 0 (A
T
DA −D < 0)
where
A =

a
1
a
2
a
3
a
4

, D =

λ 0
0 1

.
Robust Hurwitz (Schur) diagonal stability of the
family (3) can be easily tested due to multilinearity
of the diagonal stability conditions (see Fact 1.1 and
Fact 1.2).
Recall that a function f : Q → R is said to be mul-
tilinear if it is affine-linear with respect to each com-
ponent of a ∈Q. The following theorem expresses the
well-known property of a scalar multilinear function
defined on a box.
223
Yildiz B., Büyükköro
˘
glu T. and Dzhafarov V..
Common Diagonal Stability of Second Order Interval Systems.
DOI: 10.5220/0005025902230227
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 223-227
ISBN: 978-989-758-039-0
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

Theorem 1.1 ((Barmish, 1994, page 245)). Suppose
that Q is a box with extreme points a
i
, f : Q → R is
multilinear. Then both the maximum and the mini-
mum are attained at extreme points of Q. That is
max
a∈Q
f (a) = max
i
f (a
i
), min
a∈Q
f (a) = min
i
f (a
i
).
2 HURWITZ CASE
In this section for the family (3) we give a necessary
and sufficient condition for the existence of a common
diagonal solution to the Lyapunov inequality, i.e. the
existence of D = diag(λ
∗
,1) with λ
∗
> 0 such that
A
T
D + DA < 0
for all a ∈ Q = [a
−
1
,a
+
1
] ×···×[a
−
4
,a
+
4
]. Necessary
condition for the existence of a common diagonal so-
lution is the robust diagonal stability.
Proposition 2.1. The family (3) is robust Hurwitz di-
agonally stable if and only if
a
+
1
< 0, a
+
4
< 0, and a
+
1
a
+
4
−max{a
2
a
3
} > 0 (4)
where the maximum is calculated over extreme points
a
−
2
, a
+
2
, a
−
3
and a
+
3
.
Proof. By Theorem 1.1 for every a =
(a
1
,a
2
,a
3
,a
4
) ∈ Q
a
1
< 0, a
4
< 0, a
1
a
4
−a
2
a
3
> 0
or
maxa
1
< 0, max a
4
< 0, min
{
a
1
a
4
−a
2
a
3
}
> 0
or
a
+
1
< 0, a
+
4
< 0, min
{
a
1
a
4
}
+ min
{
−a
2
a
3
}
> 0.
Obviously min
{
a
1
·a
4
}
= a
+
1
·a
+
4
, min
{
−a
2
·a
3
}
=
−max
{
a
2
·a
3
}
and (4) follows. The maximum in (4)
is calculated over extreme points by Theorem 1.1.
Now we proceed to the necessary and sufficient
condition for the existence of common diagonal solu-
tions. Assume that the family (3) is robust Hurwitz
diagonally stable, that is (4) is satisfied and we are
looking for condition for the existence of common di-
agonal solution.
The existence of a common D = diag(λ
∗
,1) (λ
∗
>
0) means that
A
T
D + DA =

2a
1
λ
∗
a
2
λ
∗
+ a
3
a
2
λ
∗
+ a
3
2a
4

< 0
or
2a
1
λ
∗
< 0, 4a
1
a
4
λ
∗
> (a
2
λ
∗
+ a
3
)
2
(5)
for all a = (a
1
,a
2
,a
3
,a
4
) ∈ Q. The first condition of
(5) is satisfied automatically since by (4), a
+
1
< 0. The
second condition is equivalent to the following
min
(a
1
,a
4
)
(4a
1
a
4
)λ
∗
> max
(a
2
,a
3
)
(a
2
λ
∗
+ a
3
)
2
or
(4a
+
1
a
+
4
)λ
∗
> max

(a
−
2
λ
∗
+ a
−
3
)
2
,(a
+
2
λ
∗
+ a
+
3
)
2

or
(4a
+
1
a
+
4
)λ
∗
> (a
−
2
λ
∗
+ a
−
3
)
2
,
(4a
+
1
a
+
4
)λ
∗
> (a
+
2
λ
∗
+ a
+
3
)
2
or
(a
−
2
)
2
λ
2
∗
+ (2a
−
2
a
−
3
−4a
+
1
a
+
4
)λ
∗
+ (a
−
3
)
2
< 0,
(a
+
2
)
2
λ
2
∗
+ (2a
+
2
a
+
3
−4a
+
1
a
+
4
)λ
∗
+ (a
+
3
)
2
< 0.
(6)
Consider the function
f (x) = (a
−
2
)
2
x
2
+ (2a
−
2
a
−
3
−4a
+
1
a
+
4
)x + (a
−
3
)
2
(x ≥0)
which corresponds to the first condition in (6). Since
f (0) ≥ 0 and the family (3) is robust Hurwitz diag-
onally stable there exists a positive solution interval
(α
1
,α
2
) of the inequality f (x) < 0. For example, if
a
−
2
6= 0 then
α
1
=
(2a
+
1
a
+
4
−a
−
2
a
−
3
) −
p
(a
−
2
a
−
3
−2a
+
1
a
+
4
)
2
−(a
−
2
a
−
3
)
2
(a
−
2
)
2
α
2
=
(2a
+
1
a
+
4
−a
−
2
a
−
3
) +
p
(a
−
2
a
−
3
−2a
+
1
a
+
4
)
2
−(a
−
2
a
−
3
)
2
(a
−
2
)
2
(the discriminant ∆ = (a
−
2
a
−
3
−2a
+
1
a
+
4
)
2
−(a
−
2
a
−
3
)
2
is
positive by the robust Hurwitz diagonal stability of
(3)).
Analogously the exists an open interval (β
1
,β
2
)
which corresponds to the second condition in (6). If
a
+
2
6= 0 then
β
1
=
(2a
+
1
a
+
4
−a
+
2
a
+
3
) −
p
(a
+
2
a
+
3
−2a
+
1
a
+
4
)
2
−(a
+
2
a
+
3
)
2
(a
+
2
)
2
β
2
=
(2a
+
1
a
+
4
−a
+
2
a
+
3
) +
p
(a
+
2
a
+
3
−2a
+
1
a
+
4
)
2
−(a
+
2
a
+
3
)
2
(a
+
2
)
2
(the discriminant is positive).
Now we give the main result of this section.
Theorem 2.2. Assume that the family (3) is robust
Hurwitz diagonally stable. There exists a common di-
agonal solution to the Lyapunov inequality if and only
if the intervals (α
1
,α
2
) and (β
1
,β
2
) have nonempty
intersection, i.e.
max
{
α
1
,β
1
}
< min
{
α
2
,β
2
}
.
In this case for every λ ∈(α
1
,α
2
)∩(β
1
,β
2
) the matrix
D = diag(λ, 1) is a common solution.
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
224

Example 2.1. Consider the family

[−3,−2] [1, 2]
[−5,−4] −1

. (7)
Is there a common diagonal solution D = diag(λ,1)?
The family (7) is robust Hurwitz diagonally stable by
Proposition 2.1, since a
+
1
= −2 < 0, a
+
4
= −1 < 0,
a
+
1
a
+
4
−max{a
2
a
3
} = (−2) ·(−1) −(−4) = 6 > 0.
Corresponding to (6) inequalities are
x
2
−18x + 25 < 0, 4x
2
−24x + 16 < 0
and α
1
= 9 −2
√
14, α
2
= 9 +2
√
14, β
1
= 3 −
√
5 and
β
2
= 3 +
√
5. (α
1
,α
2
) ∩(β
1
,β
2
) = (9 −2
√
14,3 +
√
5). For every λ ∈ (9 −2
√
14,3 +
√
5) the matrix
D = diag(λ, 1) is a common diagonal solution.
Example 2.2. Consider the family

[−2,−1] [0,1]
−1 [−3,−2]

.
The family is robust Hurwitz diagonally stable by
Proposition 2.1, since a
+
1
= −1 < 0, a
+
4
= −2 <
0, a
+
1
a
+
4
−max{a
2
a
3
} = (−1) · (−2) − 0 = 2 > 0.
Inequalities, corresponding to (6) are −8x + 1 < 0
and x
2
−10x + 1 < 0 with common solution interval
(1/8,5 +
√
24), every λ from this interval gives com-
mon diagonal solution.
3 SCHUR CASE
Here we give a necessary and sufficient condition
for the existence of common diagonal solution in the
Schur case, i.e. the existence of D = diag(λ
∗
,1) with
λ
∗
> 0 such that
A
T
DA −D < 0
for all a ∈ Q = [a
−
1
,a
+
1
] ×···×[a
−
4
,a
+
4
].
In order to have a common diagonal solution a
family must be robust diagonally stable. From Fact
1.2 we obtain
Proposition 3.1. The family (3) is robust Schur diag-
onally stable, i.e. every member is Schur diagonally
stable if and only if the following six conditions are
satisfied
1 + a
2
a
3
−a
1
a
4
> 0,
1 + a
1
a
4
−a
2
a
3
> 0,
1 + a
1
a
4
−a
1
−a
4
−a
2
a
3
> 0,
1 + a
1
+ a
4
+ a
1
a
4
−a
2
a
3
> 0,
1 + a
4
+ a
2
a
3
−a
1
−a
1
a
4
> 0,
1 + a
1
+ a
2
a
3
−a
4
−a
1
a
4
> 0
(8)
for all (a
1
,a
2
,a
3
,a
4
) ∈ Q.
These conditions can be easily checked through
the extremal points of Q by using multilinearity of the
left-hand sides of (8) and Theorem 1.1.
Assume that the family (3) is robust Schur di-
agonally stable. Again, the existence of a common
D = diag(λ
∗
,1) (λ
∗
> 0) means that
A
T
DA −D =

λ
∗
(a
2
1
−1) + a
2
3
λ
∗
a
1
a
2
+ a
3
a
4
λ
∗
a
1
a
2
+ a
3
a
4
λ
∗
a
2
2
+ a
2
4
−1

< 0
or
λ
∗
(a
2
1
−1) + a
2
3
< 0,

λ
∗
(a
2
1
−1) + a
2
3

λ
∗
a
2
2
+ a
2
4
−1

−
(λ
∗
a
1
a
2
+ a
3
a
4
)
2
> 0
(9)
for all (a
1
,a
2
,a
3
,a
4
) ∈ Q. From the robust
Schur diagonal stability it follows that |a
1
| < 1
((Kaszkurewicz and Bhaya, 2000, page 78)). There-
fore the first condition of (9) gives λ
∗
·min(1 −a
2
1
) >
max(a
2
3
), which in turn gives λ
∗
> α = (max a
2
3
)/(1−
max(a
2
1
)). The second condition gives
(a
2
2
)λ
2
∗
−

a
2
3
a
2
2
+ (a
2
4
−1)(a
2
1
−1)−
2a
1
a
2
a
3
a
4
]λ
∗
+ a
2
3
< 0.
Consider the function
g(x) = (a
2
2
)x
2
−

a
2
3
a
2
2
+(a
2
4
−1)(a
2
1
−1)−
2a
1
a
2
a
3
a
4
]x + a
2
3
(x ≥ 0).
To avoid a division by zero, without loss of generality
assume that 0 6∈ [a
−
2
,a
+
2
]. From g(0) ≥ 0 and robust
Schur stability of (3) it follows that there exist pos-
itive, continuous root functions r
1
(a
1
,a
2
,a
3
,a
4
) and
r
2
(a
1
,a
2
,a
3
,a
4
) such that the inequality g(x) < 0 is
satisfied for all x ∈ (r
1
,r
2
). The continuous functions
r
i
(i = 1,2) can be written explicitly by using the dis-
criminant which is positive by the robust Schur diag-
onal stability.
Finally, we arrive at the main result of this section.
Theorem 3.2. Assume that the family (3) is given and
0 6∈ [a
−
2
,a
+
2
]. Let (3) be robust Schur diagonally sta-
ble. There exits a common Schur diagonal solution if
and only if the following two conditions are satisfied:
i)
γ
1
:= max
(a
1
,a
2
,a
3
,a
4
)
r
1
(a
1
,a
2
,a
3
,a
4
) < γ
2
:=
min
(a
1
,a
2
,a
3
,a
4
)
r
2
(a
1
,a
2
,a
3
,a
4
)
ii) (α, ∞) ∩(γ
1
,γ
2
) 6=
/
0
CommonDiagonalStabilityofSecondOrderIntervalSystems
225

In this case for every λ ∈ (α, ∞) ∩(γ
1
,γ
2
) the ma-
trix D = diag(λ,1) is a common solution to the Stein
inequality.
Example 3.1. Consider the following interval family





0,
1
2
 
1
3
,
1
2


−
1
10
,
1
10

1
2




.
This family is robust Schur diagonally stable by
Proposition 3.1 and Theorem 1.1:
min
a∈Q
(1 + a
2
a
3
−a
1
a
4
) = 0.7,
min
a∈Q
(1 + a
1
a
4
−a
2
a
3
) = 0.95,
min
a∈Q
(1 + a
1
a
4
−a
1
−a
4
−a
2
a
3
) = 0.2,
min
a∈Q
(1 + a
1
+ a
4
+ a
1
a
4
−a
2
a
3
) = 1.45,
min
a∈Q
(1 + a
4
+ a
2
a
3
−a
1
−a
1
a
4
) = 0.7,
min
a∈Q
(1 + a
1
+ a
2
a
3
−a
4
−a
1
a
4
) = 0.45
where Q =

0,
1
2

×

1
3
,
1
2

×

−
1
10
,
1
10

×
1
2
.
The maximization of the left root function
r
1
(a
1
,a
2
,a
3
,a
4
) of g(x) over Q gives γ
1
= 0.019,
and the minimization of the rgiht root function
r
2
(a
1
,a
2
,a
3
,a
4
) over Q gives γ
2
= 2.141. Since
α = 0.0134, for every λ ∈ (0.019,2.141) the matrix
D = diag(λ, 1) is a common diagonal solution.
4 Z-MATRICES
In this section for real n ×n interval Z-matrices we
give necessary and sufficient condition for the exis-
tence of a common diagonal solution to the Lyapunov
inequality.
A real n×n matrix A = (a
i j
) is said to be Z-matrix
if a
i j
≥ 0 for all i 6= j. The following properties of
Z-matrices are well-known (see (Horn and Johnson,
1991)):
Assume that A is a Z-matrix. Then:
1) If A is Hurwitz stable then it is diagonally stable
2) The spectral abscissa ρ(A) = max
i
Reλ
i
(A), i.e.
the maximum of the real parts of eigenvalues of
A is a real eigenvalue of A. Therefore A is Hur-
witz stable if and only if every real eigenvalue is
negative.
Assume that the following interval Z-matrix fam-
ily is given
A =
n
(a
i j
) : a
i j
∈ [a
−
i j
,a
+
i j
]
o
(10)
with a
−
i j
≥ 0 for all i 6= j. Denote by U the right-end
matrix of the family A: U = (a
+
i j
).
Theorem 4.1. Let the family A (10) be given. There
exists a common diagonal solution to the Lyapunov
inequality if and only if the matrix U is Hurwitz stable
(equivalently Hurwitz diagonally stable).
Proof. The implication ⇒ follows from the inclusion
U ∈ A .
⇐) : Since U is Hurwitz diagonally stable there exists
a positive diagonal D
∗
such that
U
T
D
∗
+ D
∗
U < 0 (11)
Take an arbitrary A ∈ A . Then
A
T
D
∗
+ D
∗
A ≤U
T
D
∗
+ D
∗
U
where the symbol “≤” means componentwise in-
equality. The matrices A
T
D
∗
+D
∗
A and U
T
D
∗
+D
∗
U
are Z-matrices. Take sufficiently large α > 0 such that
componentwisely
A
T
D
∗
+ D
∗
A + αI ≥ 0, U
T
D
∗
+ D
∗
U +αI ≥ 0.
Then from
A
T
D
∗
+ D
∗
A + αI ≤U
T
D
∗
+ D
∗
U +αI
and (Bernstein, 2005, page 160, Fact 4.11.7) it fol-
lows that
σ(A
T
D
∗
+ D
∗
A + αI) ≤ σ(U
T
D
∗
+ D
∗
U +αI)
where σ(·) denote the spectral radius. For a non-
negative (componentwise) matrix B, σ(B) = λ
max
(B)
where λ
max
(B) denotes the greatest real eigenvalue of
B. Therefore
λ
max
(A
T
D
∗
+D
∗
A +αI) ≤ λ
max
(U
T
D
∗
+D
∗
U + αI).
(12)
On the other hand for a symmetric matrix C
λ
max
(C + αI) = λ
max
(C) +α,
therefore from (11) and (12) we obtain
λ
max
(A
T
D
∗
+ D
∗
A) ≤ λ
max
(U
T
D
∗
+ D
∗
U) < 0.
Since A ∈A is arbitrary, the last inequality means that
the family A has a common diagonal solution D
∗
to
the Lyapunov inequality (recall that D
∗
is a positive
diagonal solution of (11)).
The matrix D
∗
can be evaluated by different ways:
by direct solution of (11), or by the linear matrix in-
equality (LMI) techniques, or by the algorithm from
(Khalil, 1982), or by the Perron-Frobenius theory.
Example 4.1. Consider the following interval Z-
matrix family
A =


[−8,−6] [1,5] [1,2]
[2,3] [−9,−8] [1,3]
[2,4] [2,5] [−9,−8]


.
ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics
226

Here
U =


−6 5 2
3 −8 3
4 5 −8


which is Hurwitz stable. The matrix D
∗
=
(0.249,0.308, 0.183) evaluated by LMI technique is
solution of (11) and a common solution for the family
A.
5 CONCLUSIONS
In this paper we consider common diagonal Lyapunov
function problem for interval systems. For second or-
der interval systems we obtain necessary and suffi-
cient conditions for the existence of common diagonal
solutions to Lyapunov and Stein inequalities. Neces-
sary and sufficient condition is given for n×n interval
Z-matrix family. The obtained results also give diag-
onal solutions in the case of existence.
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