Positive Realization of Continuous Linear Systems with Order Bound
Kyungsup Kim and Jaecheol Ryou
Department of Computer Engineering, Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon, Korea
Keywords:
Positive Realization, Positive Linear System, Metzler Matrix, Polyhedra Cone.
Abstract:
This paper discusses the realization problem of a class of linear-invariant system, in which state variables, input
and output are restricted to be nonnegative to reflect physical constraints. This paper presents an efficient and
general algorithm of positive realization for positive continuous-time linear systems in the case of transfer
function with (multiple) real or complex poles. The solution of the corresponding problem for continuous-
time positive is deduced from the discrete-time case by a transformation. We deal with the positive realization
problem through convex cone analysis. We provide a simple general and unified construction method for the
positive realization of the transfer function, which has multiple poles, upper-bound and a sparse realization
matrix. We consider a sufficient condition of positive realization.
1 INTRODUCTION
This paper discuss the realization problem of a class
of linear-invariant system, in which state variables,
input and output are restricted to be nonnegative to
reflect physical constraints. The nonnegative con-
straints can be encountered in engineering, medicine
and economics (Brown, 1980) (Gersho and Gopinath,
1979), and (Benvenuti and Farina, 2001).
In the cases of discrete time, the powerful con-
structive tools of proper generators for general trans-
fer functions have been introduced a lot in the
last decade (Nagy and Matolcsi, 2003)(Nagy et al.,
2007)(Hadjicostis, 1999)(Nagy and Matolcsi, 2005).
Constructive efficient general methods to solve the
positive realization in close to minimal dimension
have mainly focused on the problems of discrete
systems (Anderson et al., 1996)(Benvenuti et al.,
1999)(Nagy et al., 2007). However, the positive real-
ization problem of the continuoustime case have been
studied less than that in the discrete time. We propose
a constructive efficient algorithm to solve the posi-
tive realization for some given positive system with
(possibly multiple) complex poles in continuous time
domain. First, we solve the general problem for the
positive realization of transfer function with complex
poles in the continuous time linear system. The posi-
tive realization problem can be derived by finding an
appropriate generator of a polyhedral cone interven-
ing reachability and observability. Because the posi-
tive realization is not unique, we can choose a proper
sparse realization matrices by selecting spanning vec-
tors from the cone generator. We also handle the pos-
itive realization of the transfer function with multiple
complex poles. The sufficient conditions for the pos-
itive realization of the transfer function with multiple
complex poles are given and analyzed. The format of
the paper is as follow. In Section 2, we introduce the
preliminary concepts for the analysis of the continu-
ous positive linear system. The positive realization
problems of the transfer function with simple poles
are discussed in Section. We consider the generalized
positive realization problem of the transfer function
with multiple complex or real poles in Section 4.
2 PRELIMINARY
The convex cone X = cone(X) denotes the smallest
convex cone of a set X, which consists of all finite
nonnegative linear combinations of elements of the
set X. The dual cone, X
∗
, of a cone X is defined by
X
∗
= { y|x
T
y ≥ 0,∀x ∈ X } . A convexcone X is said to
be a polyhedral cone if it is spanned by a finite num-
ber of vector set X = {x
1
,··· ,x
m
} with x
i
∈ R
n
and X
is called by a polyhedra generator of X . From now,
X is also denoted by the matrix with columns x
i
∈ X.
An extreme point of a convex cone is one which is
not a proper positive linear combination of any two
points of the set. A finite set X is said to be a frame of
the polyhedral cone X if the points of X are extreme
points in X and X spans X . A polyhedral cone is a
closed convex cone (Berman and Plemmons, 1994).
566
Kim K. and Ryou J..
Positive Realization of Continuous Linear Systems with Order Bound.
DOI: 10.5220/0003981405660569
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 566-569
ISBN: 978-989-8565-21-1
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

A matrix A ∈ R
n×n
is said to be a Metzler matrix if
all its off-diagonal elements are in R
+
. A matrix A is
a Metzler matrix if and only if there exists an α ∈ R
satisfying (A+ αI) ∈ R
n×n
+
(van den Hof, 1997)(Ben-
venuti and Farina, 1998). Consider a single-input, sin-
gle output linear time-invariant system ˙x = Ax + bu,
and y = cx where A ∈ R
n×n
, b ∈ R
n×1
and c ∈ R
1×n
.
The linear system is said to be a positive linear sys-
tem if for all x
0
∈ R
n
+
and for all u(t) ∈ R, we have
y(t) ∈ R for all t. A strictly proper rational function is
said to be positive realizable if there exist a matrix A
with nonnegative off-diagonal elements and nonneg-
ative b, c such that H(s) = c(sI − A)
−1
b. H(s) is the
set of strictly proper rational transfer functions. Such
a realization (A,b, c) is called the positive realization,
since it yields nonnegative state response whenever
initial states and inputs are nonnegative.
The necessary and sufficient condition of the pos-
itive realization has been introduced in (Ohta et al.,
1984). The problem of positive realization of a given
transfer function is reduced to finding an appropriate
polyhedral cone in the room sandwiched by the reach-
ability and observability cones. We summary the nec-
essary and sufficient condition of the existence of pos-
itive in the next theorem.
Theorem 2.1 ((Ohta et al., 1984)). Let (A, g,h) be
a minimal realization of H(s). Then H(s) is positive
realizable if and only if there exists a generator matrix
P such that a polyhedral cone P = cone(P) satisfies
1. exp(At)P ⊂ P for all t ≥ 0,
2. R ⊂ P ⊂ S .
where P ∈ R
n×m
, R is a reachable set and S is an
observable set.
Technically, it is difficult to find out a proper poly-
hedral cone P satisfying the condition of Theorem
2.1. The explicit construction methods of the poly-
hedral cone for discrete time case can be applied to
the continuous time version.
Lemma 2.1. Let P be a polyhedral cone in R
n
and
A ∈ R
n×n
. Then exp(At)P ⊂ P for any t ≥ 0 if and
only if (A+ λI)P ⊂ P for some λ ≥ 0.
3 SIMPLE POLE CASE
We consider a simple third-order asymptotically sta-
ble positive proper transfer function H(s) with partial
fractional form as
H(s) =
R
s− λ
0
+
β
1
s− λ
1
+
¯
β
1
s−
¯
λ
1
(1)
where λ
0
< 0, λ
1
is a complex pole with real(λ
1
) ≤
λ
0
,
¯
λ
1
is a complex conjugate of λ
1
and β
1
is complex
number. A minimal Jordan realization is given by
A =


λ
0
0 0
0 x y
0 −y x


, λ
0
⊕ A
1
, g =


1
1
0


(2)
h =

R c
2
c
3

,

R ˆc

where λ
0
is a real pole with λ
0
< 0, a complex pole
is λ
1
= x + iy with x < λ
0
. Here R > 0, c
2
and c
3
are
appropriate real values.
Definition 3.1. Let P
m
(ρ) for m ≥ 1 denote the set of
points in the complex plane that lie in the interior of
the regular polygon with m edges having one vertex
at point ρ and including a zero point 0. For m ≥ 3,
the polygon P
m
is defined by a subset in R
2
through
the following inequalities:
P
m
(ρ) =

(x,y)|rcos

(2k+ 1)π
m
− ϕ

≤ ρcos
π
m

,
(3)
for all k with 0 ≤ k ≤ m, where x = rcosϕ and y =
rsinϕ. When ρ = 1, it is simply denoted by P
m
.
Our constructive method is similar to the results in
(Benvenuti et al., 1999)(Nagy et al., 2007). A 2× m
matrix V consisting of the vertices of a polygon in the
complex plane is defined as
V =
"
1 cos
2π
m
cos
4π
m
··· cos
2π(m−1)
m
0 sin
2π
m
sin
4π
m
··· sin
2π(m−1)
m
#
. (4)
for a given m. A cone generator matrix P ∈ R
3×m
is
defined as
P =

e
V

,

p
1
p
2
··· p
m

(5)
where e represents an 1 × m vector with all entries
equal to 1 and p
i
’s are extreme points in a polyhedral
cone cone(P).
Theorem 3.1. Assume that the three dimensional
transfer function H(s) with a pair of strictly conju-
gate complex poles (λ
1
,
¯
λ
1
) and a real pole λ
0
with
λ
0
> real(λ
1
) has a realization such as (2) and the
extreme generator P is constructed as in (5). If we
have
−
π
2
+
π
m
≤ arg(λ
1
− λ
0
) ≤
π
2
−
π
m
, (6)
|β
1
| ≤
β
0
2
(7)
then there is a polyhedral generator P ∈ R
3×m
such
that cone(P) is exp(At)-invariant for any t > 0.
Theorem 3.2. Assume that the conditions of Theo-
rem 3.1 are satisfied. Then there is a sparse circu-
lar Toeplitz matrix A
+
with 3m elements such that
(ηI + A)P = PA
+
for a proper η > 0. Thus we get
a Metzler matrix A
∗
= A
+
− ηI.
PositiveRealizationofContinuousLinearSystemswithOrderBound
567

4 MULTIPLE POLES CASE
We consider the asymptotically stable transfer func-
tion H(z) being a positive linear system in the form
as
H(s) =
β
0
s− λ
0
+
r
∑
j=1
n
j
∑
i=1
β
(i)
j
(s− λ
j
)
i
(8)
where β
0
> 0 and H(s) has a non-negative impulse
response. An n
j
× n
j
Jordan form matrix A
i
, an n
j
× 1
matrix b
i
and an 1× n
j
matrix c
i
are defined ,respec-
tively, as follows: A
j
= J(λ
j
),
J(λ
j
) ,






λ
j
1 0 · ·· 0
0 λ
j
1 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 · ·· 1
0 0 0 · ·· λ
j






, b
j
= e
n
j
,






0
0
.
.
.
0
1






c
j
=
h
β
(n
j
)
j
β
(n
j
−1)
j
··· β
(1)
j
i
for 1 ≤ j ≤ r and A
0
= λ
0
where A⊕ B , diag(A, B)
and the basis vector e
k
has a 1 as its k-th component
and 0’s elsewhere. The transfer function H(z) has
a canonical minimal Jordan form realization (A,b,c)
such that A =
L
r
j=0
A
i
, b =

1 b
T
1
b
T
2
... b
T
r

T
and c =

1 c
1
c
2
... c
r

. First, let us consider
a real rational transfer function H
1
(s) with multiple
complex conjugate poles of the form
H
1
(s) =
n
∑
k=1

β
k
(s− λ
1
)
k
+
¯
β
k
(s−
¯
λ
1
)
k

(9)
where the pole λ
1
and coefficients β
i
are complex and
¯
β
i
is defined as the conjugate of β
i
. The transfer func-
tion H
1
(s) has a Jordan canonical form realization
such as (J(λ
1
) ⊕ J(
¯
λ
1
),

e
T
n
e
T
n

T
,

c
1
¯c
1

). Then
by using similarity transformation, we can obtain a
real Jordan form realization (J(λ
1
,w),
ˆ
b, ˆc) of H
1
(s)
such that J(λ
1
,w) ∈ R
2n×2n
,
ˆ
b ∈ R
2n
and ˆc
T
∈ R
2n
are given by
J(λ, w) ,






C I O ··· O
O C I · ·· O
O O C ·· · O
.
.
.
.
.
.
.
.
.
O O O · ·· C






,
ˆ
b =






0
.
.
.
0
1
0






ˆc =

˜
β
2n
˜
β
2n−1
···
˜
β
1

(10)
for any given number w and C = C(x,y) ,

x y
−y x

where the entries of c
2
are defined by
˜
β
2k
= 2Re(β
k
)
and
˜
β
2k−1
= 2Im(β
k
) for each k.
Theorem 4.1. Assume that a transfer function of the
form
H(s) =
β
0
z− λ
0
+
n
∑
k=1

β
k
(s− λ
1
)
k
+
¯
β
k
(s−
¯
λ
1
)
k

(11)
has a non-negative impulse response function where
β
0
> 0 and real(λ
1
) < λ
0
and the maximal order n of
the pole is larger than 1. Let us define a function r
m
asr
m
(z) = max{ˆr|(ˆrcosθ, ˆrsinθ) ∈ P
m
, ˆr > 0} with
respect to some z = rcosθ + irsinθ. Set z
1
=
η+λ
1
η+λ
0
and z
2
=
1
η+λ
0
. For sufficiently large η > 0, a suffi-
cient condition is given by
0 < w ≤

1−
|z
1
|
r
m
(z
1
)

r
m
(z
2
)
|z
2
|
(12)
|β
k
| ≤
β
0
(2w)
k−n
. (13)
Then, there exists a positive realization (A
+
,b
+
,c
+
)
of the transfer function H(s) that has the order mn for
all 1 ≤ k ≤ n.
Proof. We try to find a sufficient condition of the ex-
istence of a positive realization (A
+
,b
+
,c
+
) for the
given transfer function H(s). We obtain a real block
Jordan form realization (A,b,c) of the transfer func-
tion H(z) in equation (11) as
A = λ
0
⊕ J(λ
1
,w), b =

1
ˆ
b

(14)
c =

1 ˆc

where x = rcosθ, y = rsinθ and J(λ
1
,w),
ˆ
b and ˆc are
defined in equation (10). We use the fact that there
exists an η > 0 such that (ηI + A)-invariant cone P
with mn edges (i.e., (ηI + A)P ⊂ P ). We generalize
the concept of the cone generator introduced in (Ben-
venuti et al., 1999) for the case with multiple complex
poles. In order to formulate a polyhedral cone genera-
tor with (ηI + A)-invariant property, a block shift ma-
trix Z ∈ R
2n×2n
and a matrix V ∈ R
2n×m
are defined
as:
Z =






O O ··· O O
I O O O
O I · ·· O O
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
O O ··· I O






,
ˆ
V =






V
O
O
.
.
.
O






where I is an identity matrix, O is a zero matrix with
proper dimension and a 2× m matrix V is defined as
(4). A cone generator matrix P ∈ R
(2n+1)×mn
is de-
fined as P =

P
1
P
2
··· P
n

where P
k
is defined as
P
k
=

e
T
ϕ
k

for 1 ≤ k ≤ n − 1, e represents an m × 1
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
568

vector with all entries equal to 1 and ϕ
k
= Z
k−1
ˆ
V. The
columns of matrix P represent the extreme vertices of
a finite generated cone P in R
2n+1
(i.e., cone(P) = P )
and are positive independent. The polyhedral cone
P
k
is generated by P
k
, i.e., P
k
= cone(P
k
) for each
k. Note that (A + ηI)P
k
for each k has only 1-th, k-
th and k + 1-th block components. By the invariance
property,
(A+ηI)
λ
0
+η
P
k
∈ P is required. Set
˜
A =
(A+ηI)
λ
0
+η
.
The matrix
˜
A has eigenvalues {1, z
1
, ¯z
1
} with |z
1
| < 1.
Then the positive realization problem with respect to
˜
A is close related to that of the discrete time domain as
in (Benvenuti et al., 1999)(Nagy et al., 2007). Choose
{W
k
,W
k+1
} such that
(A+ ηI)
λ
0
+ η
P
k
= α
1
W
k
+ α
2
W
k+1
(15)
for each k where α
j
’s satisfy α
j
≥ 0, α
1
+ α
2
= 1 and
all the entries in the first row of W
k
∈ P
k
are equal
to 1. A sufficient condition for a feasible solution
(α
1
,α
2
) should satisfy two inequalities,
w|z
2
|
r
m
(z
2
)
≤ α
2
and
|z
1
|
r
m
(z
1
)
≤ α
1
, and an equality α
1
+ α
2
= 1 for a
given w. By rearranging the above conditions, we ob-
tain an inequality (12). From this result, we can see
that w and η are tunable parameters to get a positive
matrix. The polyhedral cone P is ηI+A-invariantun-
der the above condition in (12). We can prove that
R ⊂ P and P ⊂ S without difficulty.
Theorem 4.2. Assume that the conditions of Theo-
rem 4.1 are satisfied. Then there is a sparse circular
matrix A
+
with at most 3nm non-zero elements such
that (ηI + A)P = PA
+
for a proper η > 0. We note
W
k
∈ cone(P
k
) in Eq. (15). The columns of W
k
is pos-
itively linearly combined by the three vectors chosen
from P
k
similar to the process of Theorem 3.2. We can
verify that we can choose a sparse matrix A
+
such
that A
+
is defined by
A
+
=






T
1
εI 0 ·· · 0
0 T
2
εI · ·· 0
0 0 T
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 ··· T
2






(16)
with T
1
= T(
~
t
1
),T
2
= T(
~
t
2
) and T
3
= T(
~
t
3
) where
~
t
1
has at most three nonzero elements and
~
t
2
and
~
t
3
have
at most two nonzero elements. Finally, we get a sparse
Metzler matrix A
∗
= A
+
− ηI.
Some of theorems in paper were only mentioned
without detail proof due to page limit.
ACKNOWLEDGEMENTS
This research was supported by Next-Generation In-
formation Computing Development Program through
the National Research Foundation of Korea(NRF)
funded by the Ministry of Education, Science and
Technology (2011-0020516).
REFERENCES
Anderson, B., Deistler, M., Farina, L., and Benvenuti, L.
(1996). Nonnegative realization of a linear system
with nonnegative impulse response. Circuits and Sys-
tems I: Fundamental Theory and Applications, IEEE
Transactions on, 43(2):134 –142.
Benvenuti, L. and Farina, L. (1998). A note on minimal-
ity of positive realizations. Circuits and Systems I:
Fundamental Theory and Applications, IEEE Trans-
actions on, 45(6):676 –677.
Benvenuti, L. and Farina, L. (2001). The design of
fiber-optic filters. Lightwave Technology, Journal of,
19(9):1366 –1375.
Benvenuti, L., Farina, L., and Anderson, B. (1999). Filter-
ing through combination of positive filters. Circuits
and Systems I: Fundamental Theory and Applications,
IEEE Transactions on, 46(12):1431 –1440.
Berman, A. and Plemmons, R. J. (1994). Nonnegative Ma-
trices in the Mathematical Sciences. Society for In-
dustrial and Applied Mathematics.
Brown, R. F. (1980). Compartmental system analysis: State
of the art. Biomedical Engineering, IEEE Transac-
tions on, 27(1):1 –11.
Gersho, A. and Gopinath, B. (1979). Charge-routing net-
works. Circuits and Systems, IEEE Transactions on,
26(2):81 – 92.
Hadjicostis, C. (1999). Bounds on the size of minimal non-
negative realizations for discrete-time LTI systems.
Systems & Control Letters, 37:39–43.
Nagy, B. and Matolcsi, M. (2003). Algorithm for positive
realization of transfer functions. Circuits and Sys-
tems I: Fundamental Theory and Applications, IEEE
Transactions on, 50(5):699 – 702.
Nagy, B. and Matolcsi, M. (2005). Minimal positive real-
izations of transfer functions with nonnegative multi-
ple poles. Automatic Control, IEEE Transactions on,
50(9):1447 – 1450.
Nagy, B., Matolcsi, M., and Szilvasi, M. (2007). Order
bound for the realization of a combination of posi-
tive filters. Automatic Control, IEEE Transactions on,
52(4):724 –729.
Ohta, Y., Maeda, H., and Kodama, S. (1984). Reachabil-
ity, observability, and realizability of continuous-time
positive systems. SIAM J. on Control and Optimiza-
tion, 22(2):171–180.
van den Hof, J. (1997). Realization of continuous-time
positive linear systems. Systems & Control Letters,
31(4):243 – 253.
PositiveRealizationofContinuousLinearSystemswithOrderBound
569