Abstract. Spectral properties of nonnegative and Metzler matrices are considered. The conditions for existence of Metzler spectrum in dynamical systems have been established. An electric RL and GC ladder-network is presented as an example of dynamical Metzler system. The suitable conditions for parameters of these electrical networks are formulated. Numerical calculations were done in MATLAB

W Mitkowski

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BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL SCIENCES
Vol. 56, No. 4, 2008
Dynamical properties of Metzler systems
W. MITKOWSKI∗
Department of Automatics, AGH University of Science and Technology, 30 Mickiewicza Ave, 30-059 Kraków, Poland
Abstract. Spectral properties of nonnegative and Metzler matrices are considered. The conditions for existence of Metzler spectrum in
dynamical systems have been established. An electric RL and GC ladder-network is presented as an example of dynamical Metzler system.
The suitable conditions for parameters of these electrical networks are formulated. Numerical calculations were done in MATLAB.
Key words: Metzler matrices, ladder and ring electrical networks, nonnegative matrices, positive linear systems.
1. Introduction
Analysis of properties of positive matrices was considered by
many authors [1–3]. These matrices are connected with pos-
itive discrete dynamical systems. Metzler matrices are con-
nected with positive continuous-time dynamical systems. The
essence of positive dynamical systems resides in spectrum of
the state matrices. The form of state matrices depends on the
basis of co-ordinate system. Depending on the basis we obtain
suitable positive-cone systems [4].
The paper is organized as follows: in Section 2 and 3
spectral properties of nonnegative and Metzler matrices are
considered. In Section 4 Metzler dynamical systems is in-
vestigated. The conditions for positivity of electrical ladder
network of RL and GC type are established in Section 5.
Concluding remarks are given in Section 6.
2. Spectral properties of nonnegative matrix
The real matrix A = [aij ] ∈ Rn×n is nonnegative matrix
if and only if aij ≥ 0. If A is nonnegative matrix we will
write A ≥ 0. Vector x with nonnegative real components is
called nonnegative vector. In this case we will write x ≥ 0. If
aij > 0, then matrix A = [aij ] is positive and we will write
A > 0.
Let λ(A) be the spectrum of the square matrix A. Let
λi(A) ∈ λ(A) be an eigenvalue of A. Denote by ρ(A) =
maxi |λi(A)| the spectral radius of matrix A and denote by
α(A) = maxi Reλi(A) the growth constant of A.
Remark 1. Let ρ(A) > 0. For any square real matrix
A ≥ 0 there exists a real number λmax(A) ∈ λ(A) such
that λmax = ρ(A). The eigenvalue λmax = ρ is a simple root
of characteristic polynomial of matrix A ≥ 0 with eigenvec-
tor w > 0. If matrix A ≥ 0 has k eigenvalues λmax = ρ,
λ1,λ2, . . . , λk−1 such that |λi| = ρ, then these eigenvalues
are various roots of equation λk − ρk = 0. See Theorem of
Frobenius [1].
Spectral properties of nonnegative square matrix are de-
fined by roots of characteristic polynomial. For A ≥ 0 (see
Remark 1) we have polynomial in the following form:
W (λ, A) = det[λI − A] = (λ − λmax)f(λ)
and f(λmax) 6= 0,
(1)
where roots λi of polynomial f(λ) satisfy the condition
{
|λi| ≤ ρ = λmax
|λk| = ρ are diffrent roots and λk 6= λmax
(2)
For A = [aij ] ∈ Rn×n degree of polynomial (1) is n. The
conditions (1) and (2) define the spectrum λ(A) of nonnega-
tive square matrix A (see Fig. 1 for n = 6).
Fig. 1. Spectrum of nonnegative matrix
Now consider the following problem. Polynomial W̃ (λ) is
given. We look for nonnegative matrix A such that W (λ, A) =
W̃ (λ). Solution of this problem is shown in the following ex-
ample.
Example 1. Following roots of polynomial W̃ (λ) = λ3 −
0.3333λ2 − 0.1953λ− 0.4714 are given
λmax = 1, λ1,2 = −0.3333± 0.6002 j,
j2 = −1, |λ1,2| = 0.4714.
(3)
∗e-mail: mitkowsk@agh.edu.pl
309
W. Mitkowski
It is clear, that W (λ, F ) = W̃ (λ), where
F =



0 1 0
0 0 1
0.4714 0.1953 0.3333



≥ 0 (4)
is nonnegative Frobenius matrix. For eigenvalue λmax = 1 =
ρ(F ) we have eigenvector w=[0.5774 0.5774 0.5774]T >0.
Example 2. Consider a discrete-time linear system described
by the equations
x(k + 1) = Ax(k), x(0) ∈ Rn ⇔ x(i) = Aix(0)
k = 0, 1, 2, . . . i = 0, 1, 2, . . .
(5)
If A is nonnegative matrix, then equations (5) generate
nonnegative dynamical system. If A ≥ 0 and x(0) ≥ 0 then
x(i) ≥ 0 for i = 0, 1, 2, . . .. System (5) is asymptotically
stable if and only if ρ(A) ∈ [0, 1).
Remark 2. Let η ∈ R and A = [aij ] ≥ 0. Real number η is
greater than eigenvalue λmax = ρ(A) of nonnegative matrix
A (λmax(A) < η) if and only if all principal minors of the
matrix ηI − A are greater than zero, i.e. Mi[η I − A] > 0,
i = 1, 2, . . . n, where M1[η I−A] = η−a11, M2[η I−A] =
det
[
η − a11 −a12
−a21 η − a22
]
, . . . , Mn[ηI−A] = det[ηI−A]
(see [1, p. 349]).
3. Spectral properties of Metzler matrix
The real square matrix M = [mij ] ∈ Rn×n is called the Met-
zler matrix if its all off-diagonal entries are nonnegative, i.e.
mij ≥ 0, i 6= j.
Remark 3. Every nonnegative matrix is the Metzler matrix.
Let γ = mini mii. For every Metzler matrix M there exists
a real number η ≥ γ such that matrix ηI +M = A is nonneg-
ative matrix. Let s ∈ λ(A). Then s − η = λ ∈ λ(M). Thus
spectrum λ(A) is copy of spectrum λ(M) and is shifted (see
Fig. 2) [5].
Fig. 2. Spectrum of matrices A and M
We consider real square n × n Frobenius matrix F ≥ 0.
The characteristic polynomial of F ≥ 0 is given by the fol-
lowing equation:
W (s, F ) = det[sI−F ] = sn+an−1sn−1+. . .+a1s+a0 (6)
where ai ≤ 0 for i = 0, 1, 2, . . . , n − 1.
Let smax = ρ(F ). Let s = λ + η, where η > smax. Thus
from polynomial (6) we obtain new polynomial of variable λ
W̃ (λ) = λn + bn−1λ
n−1 + . . . + b1λ + b0. (7)
It is obvious that M = F −η I is the Metzler matrix such
that W (λ, M) = det[λ I−M ] = W̃ (λ) and additionally M is
asymptotically stable, i.e. Reλ i(M) < 0, i = 1, 2, 3, . . . , n.
Example 3. Consider the characteristic polynomial of positive
Frobenius matrix F given in Eq. (4),
W (s, F ) = s3 − 0.3333s2 − 0.1953s− 0.4714. (8)
Let s = λ + η, where η = 1.1 > ρ(F ) = 1. Thus we have
W̃ (λ) = λ3 + 2.9667 λ2 + 2.7014 λ + 0.2415. (9)
The matrix M = F − η I ,
M =



−1.1 1 0
0 −1.1 1
0.4714 0.1953 −0.7667



(10)
is the Metzler matrix such that W (λ, M) = det[λ I − M ] =
W̃ (λ) and λmax = −0.1, λ1,2 = −1.4334 ± 0.6002 j.
For the eigenvalue λmax = −0.1 we have the eigenvector
w = [0.5774 0.5774 0.5774]T > 0.
4. Metzler dynamical systems
Let us consider a continuous-time linear system described by
the equations
ẋ(t) = Ax(t), x(0) ∈ Rn
t ≥ 0 ⇔ x(t) = eA tx(0).
(11)
If A is the Metzler matrix, then equations (11) generate
the Metzler dynamical system. It has been shown [2, 6] that
eAt ≥ 0 if and only if A ∈ Rn×n is the Metzler matrix. If A
is the Metzler matrix and x(0) ≥ 0 then x(t) ≥ 0 for t ≥ 0
(see [6]).
Denote by α(A) = maxi Reλi(A) the growth constant
of A. For any Metzler matrix M there exists a real number
λmax ∈ λ(M) such that λmax = α(M), where α(M) =
maxi Reλi(M), i = 1, 2, ..., n, is the growth constant of M
(see [9]). The number λmax(M) is a single eigenvalue of the
Metzler matrix M (see Fig. 2) and α(M) = λmax(M).
The Metzler matrix M is asymptotically stable if and only
if α(M) < 0. Let
W (λ, M) = det[λ I − M ]
= λn + bn−1λ
n−1 + . . . + b1λ + b0.
(12)
In this case α(M) < 0 if and only if bi > 0 for
i = 0, 1, 2, . . . , n − 1 [7]; see also [5].
It is well know that the system (11) is asymptotically sta-
ble if and only if Reλi(A) < 0, i = 1, . . . , n. When A is the
Metzler matrix then α(A) = λmax(A) and Reλi(A) < 0 if
and only if Mi[−A] > 0, i = 1, 2, . . . , n (see Remark 2 with
η = 0). We can notice that Mi[−A] > 0, i = 1, 2, ..., n if
and only if M1[A] < 0, M2[A] > 0, . . . , (−1)nMn[A] > 0
[1]; see also [8].
310 Bull. Pol. Ac.: Tech. 56(4) 2008
Dynamical properties of Metzler systems
Consider linear transformation of variables x(t) ∈ Rn giv-
en by following equations with matrix P such that detP 6= 0
x(t) = Pz(t), z(t) = P−1x(t). (13)
Thus from (11) we obtain
ż(t) = P−1APz(t). (14)
It is clear that the spectrum λ(P−1AP ) = λ(A). If A
is the Metzler matrix then the system (11) or (14) has spec-
trum of the Metzler matrix. Structure of the matrix P−1AP
depends on chosen base of space Rn (columns of matrix P ).
Example 4. Consider the electrical network shown in Fig. 3.
Parameters of the network R > 0, L and C > 0 are known.
The system shown in Fig. 3 is described by equation
LC ẍ1(t) + RC ẋ1(t) + x1(t) = 0. (15)
Fig. 3. Electrical network of RLC type
If R > 2
√
L/C then the system (15) has a spectrum of
the Metzler matrix. Let x(t) = [x1(t) x2(t)]
T . From (15) we
obtain equation (11) with
A =
[
0 1
−1/LC −R/L
]
. (16)
Let L = 1, C = 1/6, R = 5 and
T =
[
1 1
−2 −3
]
, T−1 =
[
3 1
−2 −1
]
,
Q =
[
cosϕ − sin ϕ
sinϕ cosϕ
]
, detQ = 1, Q−1 = QT .
(17)
Thus T−1AT = diag (−2, −3) and
M = Q−1T−1ATQ =
[
cos2 ϕ − 3 −0.5 sin2ϕ
−0.5 sin2ϕ − cos2 ϕ − 2
]
.
(18)
If sin 2ϕ ≤ 0, then M given in (18) is the Metzler
matrix. Consequently M is the Metzler matrix for ϕ ∈
[
−π
2
+ π
]
+ kπ, k = 0, ±1, ±2, . . . (see [9]).
5. Electrical ladder network of RL and GC type
We consider the electrical ladder network of the RL and GC
type shown in Fig. 4. The parameters of the network L > 0,
G > 0, R > 0 and C > 0 are known [10, 11], see also [12].
Fig. 4. Electric ladder network of RL and GC type
Let u(t) = 0. The eigenvalues of ladder network we can
obtain from following equation (see [11] and [10])
LCλ2 + (LG + RC)λ + RG + 2(1 − cosϕk) = 0, (19)
where
ϕk =
2k − 1
2n + 1
π, k = 1, 2, . . . , n. (20)
If |LG − RC| > 2
√
2 LC(1 − cos ϕn), then ladder net-
work has a spectrum of the Metzler matrix.
Let u(t) = xn(t). In this case the electrical network shown
in Fig. 4 is the ring network of RL and GC type [11]. The
eigenvalues of the ring network we obtain from (19) for
ϕk =
2k
n
π, k = 1, 2, . . . , n. (21)
If |LG − RC| > 2
√
2 LC(1 − cos ϕ[n/2]), then the ring
network has a spectrum of the Metzler matrix.
Fig. 5. Ladder network of RL and GC type
The eigenvalues of the ladder network shown in Fig. 5
with u(t) = 0 we can obtain from equation (19), where
ϕk =
k
n + 1
π, k = 1, 2, . . . , n. (22)
If |LG − RC| > 2
√
2 LC(1 − cos ϕn), then the ladder
network shown in Fig. 5 has a spectrum of the Metzler ma-
trix.
6. Concluding remarks
In this paper a spectral properties (see (2) and Remark 3) of
finite dimensional continuous-time Metzler systems are con-
sidered. The considerations have been illustrated by examples
of electrical ladder networks of RL and GC type. The con-
ditions for positivity of the electrical ladder network of RL
and GC type are given in Section 5 (see also Example 4 for
electrical network of RLC type given in Fig. 3).
The positive ladder networks can be applied in approxi-
mation of some positive distributed parameters systems [13].
Acknowledgements. This work was supported by Ministry of
Science and Higher Education in Poland in the years 2008–
2011 as a research project No N N514 414034.
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Dynamical properties of Metzler systems

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Dynamical properties of Metzler systems

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