A Tutorial on Positive Systems and Large Scale Control

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A Tutorial on Positive Systems and Large Scale Control
Anders Rantzer 1and Maria Elena Valcher 2
Abstract— In this tutorial paper we first present some founda-
tional results regarding the theory of positive systems. In partic-
ular, we present fundamental results regarding stability, positive
realization and positive stabilization by means of state-feedback.
Special attention is also paid to the system performance in
terms of disturbance attenuation. Under the asymptotic stability
assumption, such performance can be measured in terms of Lp-
gain of the positive system. In the second part of the paper we
propose some recent results about control synthesis by linear
programming and semidefinite programming, under the positiv-
ity requirement on the resulting controlled system. These results
highlight the value of positivity when dealing with large scale
systems. Indeed, stability properties for these systems can be
verified by resorting to linear (copositive) or diagonal Lyapunov
functions that scale linearly with the system dimension, and
such linear functions can be used also to design stabilizing
feedback control laws. In addition, stabilization problems with
disturbance attenuation performance can be easily solved by
imposing special structures on the state feedback matrices. This
is extremely valuable when dealing with large scale systems for
which state feedback matrices are typically sparse, and their
structure is a priori imposed by practical requirements.
I. INT ROD UC TI ON A ND M OTIVATING EXAMPLES
In its broadest meaning, a positive system is simply a sys-
tem whose describing variables can only take positive, or at
least nonnegative, values. The class of natural and technolog-
ical phenomena that can be naturally described by means of a
positive system is far larger than one would expect at a first
glance. Indeed, lots of physical quantities (concentrations,
population levels, buffer sizes, queue lengths, charge levels,
light intensity levels, prices, production quantities, etc.) are
naturally constrained to be nonnegative, and any accurate
mathematical model accounting for their dynamic evolution
has to necessarily incorporate this constraint. Mathematical
models with the positivity constraint on their describing
variables have therefore flourished in several research areas,
e.g., biology, ecology, physiology and pharmacology [17],
[19], [37], [38], [39], [44], biomolecular and biochemical
modelling [12], [13], [22], thermodynamics [12], [38], epi-
demiology [1], [40], [53], traffic and congestion modelling
[12], [67], power systems [77], filtering and charge routing
networks [6], [7], [10], econometrics [55], etc. In addition,
since probabilities are nonnegative quantities, Markov chains
[66], Hidden Markov chains and other probabilistic models
represent special cases of positive systems.
In this tutorial paper we mainly focus our attention on
positive linear state-space models, on which most of the
1A. Rantzer is with the Automatic Control LTH, Lund University, Box
118, SE-221 00 Lund, Sweden, e-mail: rantzer@control.lth.se
2M.E. Valcher is with the Dipartimento di Ingegneria dell’Informazione
Universit`
a di Padova, via Gradenigo 6B, 35131 Padova, Italy, e-mail:
meme@dei.unipd.it.
research efforts in the last 40 years have focused, and hence
in the following by a positive system we will always mean a
positive linear state-space model. The merit of stimulating a
systematic study of this class of systems, and hence of initi-
ating what is nowadays known as “Positive System Theory”,
must be credited to David Luenberger, who published in
1979 a fundamental book [50] whose Chapter 6 was entirely
devoted to positive linear systems (while Chapter 7 addressed
Markov chains). Not unexpectedly, the first aspects that were
investigated were asymptotic behavior (in particular, simple
and asymptotic stability) and equilibrium points. Research
on these topics heavily relies on (and benefits from) the
well-know “Positive Matrix Theory” developed by Perron
and Frobenius [35], [57] as well as by Karpelevich [46].
When moving a step further into classic System Theory prob-
lems, however, new mathematical tools had to be developed,
mainly relying on graph theory and cone theory. In fact, in
the nineties there was a long stream of research focusing
on positive controllability and reachability [16], [19], [28],
[32], [56], [64], [72], [73], observability of positive systems
and their observers [4], [20], and positive realization [3], [9],
[29], [30], [51], [56], [75].
The first part of this survey aims at recalling the fundamen-
tal results obtained about the stability and the state-feedback
stabilization of positive systems, as well as on the positive
realization problem. The interested reader is referred to the
surveys [8], [9], [31] and to the fundamental books [33], [45]
for a more complete account on these subjects, as well as for
a more detailed list of references. Other subjects of intense
research were robust stability [41], [42], [69] and positive
stabilization [2], [21], [36], [63].
More recently, the interest has focused on quite different
control problems, in particular, robust positive stabilization
and system performance [2], [15], [25], the bounded real
lemma [70] and the Kalman-Yakubovic-Popov lemma [60]
for positive systems, decentralized and distributed control
[26], [58], large scale positive systems and scalable control
[27], [59]. These results have highlighted the deep impact of
Positive System Theory in the study of large scale systems.
Indeed, the use of nonnegative matrices for the study of large
scale systems has long been recognized (see for example
the surveys [54] and [65]). However, while criteria based on
nonnegative matrices tend to be conservative when applied
to generic linear systems, they are tight in the context of
positive systems. There are two main reasons why positive
systems are more tractable than general ones in a large scale
setting:
1) Stability and performance analysis of standard linear
systems require Lyapunov functions with a number

of parameters that grows quadratically with the state
dimension. This is not feasible when dealing with large
scale systems. All the results for positive systems in this
paper are based on Lyapunov functions with only linear
growth. This makes a huge difference when the system
size grows.
2) Standard linear system theory can only handle feedback
laws taking the form u=Kx, where the matrix K
is dense. Again this leads to quadratic growth when
the number of inputs is proportional to the number of
states. For positive systems, sparse matrices Kcan be
optimized with standard tools. This makes it possible to
keep the complexity linear and manageable even when
the input dimension grows with the state dimension. For
positive systems, sparse matrices Kcan be optimized
with standard tools, and this keeps the complexity linear
and manageable.
During the the past decade, there has been a rapid growth of
large scale control applications triggered by new information
technology and internet-based services. This has triggered
renewed interest in scalable control paradigms and positive
systems [59], [76]. For example, hundreds of thousands
of papers have been devoted to the analysis of consensus
dynamics, which is a special form of positive system (see,
in this respect, [71], [74] that specifically address the con-
sensus problem under the positivity constraint on the agents’
description).
In the second part of this paper, we will put particu-
lar emphasis on scalable control synthesis methods based
on input-output gains. This is a problem formulation that
has been standard practice in general multi-variable control
theory for more than twenty years [78], but the results for
positive systems are much more recent.
A recurrent example used to illustrate the results will be
a simple model of a buffer network, i.e., a graph structure
whose nodes behave as buffers. They are subject both to local
inflow/outflow and to exchange with neighboring nodes. In
formal terms, given a directed graph (V,E), where Vis a set
of nodes and Eis a set of edges, we will consider state-space
models whose ith state variables are described by
˙xi=aixi+X
(i,j)∈E
uij −X
(j,i)∈E
uji +wii∈ V.(1)
See Figure 1 for an illustration. Here xirepresents the buffer
content of node iand the input uij the flow from node jto
node i. The term aixidescribes natural decay (or growth)
of the buffer content, while widescribes the effect of local
production or consumption.
In detail, the paper is organized as follows. Section II
will introduce the fundamental notation and the (minimal)
required background material. Positive systems and stabil-
ity properties will be investigated in Section III, together
with the positive system performance in terms of Lp-gains,
p∈(0,+∞], upon assuming the positive system as an
input-output map driven by an external disturbance. Positive
realization will be the subject of Section IV, while positive
Fig. 1. Positive systems are commonly used to model dynamics of buffer
networks (1). Each state represents the content of a buffer. Content can be
transferred from one buffer to another via the network links. The content
of a buffer can also change as a result of local production or consumption.
state-feedback stabilization will be investigated in Section
V. Control synthesis (with norm constraints) by linear pro-
gramming and semidefinite programming will be investigated
in Sections VI and VII, thus providing evidence for our
previous claim regarding the pros of positivity when dealing
with large scale systems. This aspect will be further explored
in Section VIII, where some results about large scale H∞
optimal control will be presented. Section IX will provide
some extensions of the previous results to the general class
of monotone systems, while bilinear positive systems will be
addressed in Section X.
To conclude this Introduction, we would like to remark
that an extensive research activity has been devoted to
classes of systems that represent natural extensions of the
class of positive systems here considered. In particular,
we mention positive systems with delay, positive switched
systems, positive two-dimensional (2D) systems, positive
fractional systems, and monotone systems. While we will
briefly consider monotone systems at the end of this tutorial,
space constraints prevent us from addressing the other topics.
II. NOTATI ON A ND BACK GRO UN D MATE RI AL
Given p∈Z, p > 0,we set [1, p] := {1,2,...,p}. We
denote by eithe ith canonical vector in Rn, and by 1n
and 0nthe n-dimensional vectors with all entries equal to 1
and 0, respectively. Given A∈Rn×n, we denote by σ(A)
the spectrum of A, i.e., the set of its eigenvalues, and by
ρ(A) := max{|λ|:λ∈σ(A)}its spectral radius.Ais
Schur if λ∈σ(A)implies |λ|<1(namely ρ(A)<1), and
it is Hurwitz if λ∈σ(A)implies Re(λ)<0. The (i, j)th
entry of a matrix Awill be denoted by aij , and the ith entry
of a vector vby vi. Given nreal numbers d1, d2,...,dn, we
denote by diag{d1, d2, . . . , dn}the n×ndiagonal matrix
whose (i, i)th entry is di. Given a matrix M,M⊤denotes
its transpose, while M∗its conjugate transpose.
Given two positive integers mand n, a sparsity structure
Sin Rm×nis a set of matrices in Rm×nwhose nonzero

pattern is constrained, i.e.
S={K∈Rm×n|kij = 0 for (i, j )6∈ E},(2)
for some given subset Eof [1, m]×[1, n].
R+is the semiring of nonnegative real numbers. A matrix
(in particular, a vector) Awith entries in R+is called
nonnegative, and if so we adopt the notation A≥0. If,
in addition, Ahas at least one positive entry, the matrix
is positive (A > 0), while if all its entries are positive, it
is strictly positive (A≫0). Given a nonnegative matrix
A∈Rn×n
+, Perron-Frobenius Theorem [52] ensures that
ρ(A)is always an eigenvalue of A, corresponding to a
positive eigenvector. A Metzler matrix is a real square matrix,
whose off-diagonal entries are nonnegative.
A set K ⊂ Rnis said to be a cone provided that for every
x∈ K and every α∈R, α > 0,the vector αxbelongs
to K. If the cone K ⊂ Rncontains an open ball of Rn,
then it is said to be solid; if K ∩ −K ={0},then Kis
said to be pointed. A cone is convex if x1,x2∈ K implies
αx1+ (1 −α)x2∈ K for every α∈R,0≤α≤1, and
it is closed if it is a closed set of Rn. A closed, convex,
solid and pointed cone is a proper cone. A cone is said to
be polyhedral if it coincides with the set of all nonnegative
linear combinations of a finite family of vectors, i.e. K=
Cone(v1,...,vN) := {PN
i=1 αivi:αi∈R, αi≥0},for
some v1,...,vN∈Rn.
A symmetric matrix P=P⊤∈Rn×nis said to be
positive definite (positive semidefinite) and if so we adopt
the notation P≻0(P0) if for every vector x∈R\ {0}
we have x⊤Px>0(x⊤Px≥0). A symmetric matrix
P=P⊤∈Rn×nis said to be negative definite (negative
semidefinite) and if so we adopt the notation P≺0(P0)
if −Pis positive definite (positive semidefinite).
Given a vector x∈Rn
+, for any p∈(0,+∞)we define its
vector p-norm as |x|p:= (Pn
i=1 xp
i)
1
p, while the vector ∞-
norm is |x|∞:= maxi∈[1,n]|xi|. Given a matrix M∈Rl×m
and p∈(0,+∞], we define the induced matrix p-norm as
kMkp−ind := sup
x:|x|p=1
|Mx|p.
Given a function f:R+→Rn, for every p∈(0,+∞)the
Lp-norm of fis defined as
kfkLp:= Z+∞
0
|f(t)|p
pdt
1
p
,
while for p= +∞we have
kfkL∞:= ess supt≥0|f(t)|∞.
We denote by Lk
pthe set of functions f:R+→Rkhaving
finite Lp-norm. If Gis an operator from Lm
pto Lr
p,p∈
(0,+∞],its Lp-gain is defined as
kGkLp−Lp:= sup
w:kwkLp=1
kGwkLp.
Given a proper rational matrix G(s)∈R(s)r×m, with no
poles in the right half-plane, we define its H∞-norm as
kGkH∞:= sup
ω
kG(iω)k2−ind .
It is well known [78, Chapter 4] that if Gis an operator
from Lm
2to Lr
2, associated with a linear time-invariant
asymptotically stable state-space model of transfer function
G(s), then kGkL2−L2=kGkH∞.
III. POS IT IV E SY ST EM S AN D STAB IL IT Y
In general terms, a positive system is a system whose
describing variables are constrained to take positive (or at
least nonnegative) values. Most of the literature on positive
systems, however, has focused on the specific class of (pos-
itive) linear state-space models described by the following
equations
x(t+ 1) = Ax(t) + Bu(t),(3a)
y(t) = Cx(t) + Du(t), t ∈Z+,(3b)
in the discrete-time case, and by the following equations
˙
x(t) = Ax(t) + Bu(t),(4a)
y(t) = Cx(t) + Du(t), t ∈R+,(4b)
in the continuous-time case. In these equations xrepresents
the n-dimensional state variable, uthe m-dimensional input
variable and ythe r-dimensional output variable.
For systems (3) and (4) two notions of positivity have been
defined [33], [45], [50]:
•internal positivity: for every nonnegative initial con-
dition x(0) and every nonnegative input u(t), t ≥0,
the associated state and output evolutions x(t)and
y(t), t ≥0,remain nonnegative at every time instant
t;
•external positivity: assuming zero initial condition x(0),
for every nonnegative input u(t), t ≥0,the associated
output evolution y(t), t ≥0,remains nonnegative at
every time instant t.
In the discrete-time case internal positivity is equivalent
to the fact that A, B, C and Dare nonnegative matrices,
while in the continuous-time case to the fact that Ais a
Metzler matrix, while B, C and Dare nonnegative matrices.
On the other hand, external positivity is equivalent to the
nonnegativity of the system impulse response, which is
expressed, in the discrete-time case, as
g(t) = (D, t = 0,
CAt−1B, t ∈Z+, t ≥1,(5)
and in the continuous-time case as
g(t) = CeAt Bδ−1(t) + Dδ(t),(6)
where δ(t)is the Dirac impulse, while δ−1(t)is the unitary
step function. It is well known [33], [45], [50] that internal
positivity ensures external positivity, while the converse is
not true, and we will come back to this topic in Section IV.
In the following, by a positive system we will always mean
an internally positive system. The first important property
that has been investigated for this class of systems is, of
course, stability in its various forms.

Definition 1: Systems (3) and (4) are said to be
•asymptotically stable (equivalently, exponentially sta-
ble) if, for every positive initial state x(0), the corre-
sponding unforced state evolution converges to zero as
tgoes to +∞;
•simply stable if, for every positive initial state x(0), the
corresponding unforced state evolution remains (posi-
tive and) bounded.
It turns out that the fact that initial conditions are confined
to belong to the positive orthant does not affect the standard
characterizations of asymptotic stability and simple stability
for linear state-space models. Indeed, [33], [45], [50] system
(3) (system (4)) is asymptotically stable if and only if Ais
a positive Schur matrix (a Metzler Hurwitz matrix). On the
other hand, system (3) (system (4)) is simply stable if and
only if λ∈σ(A)implies |λ| ≤ 1and when |λ|= 1 then λ
is a simple root of the minimal annihilating polynomial of A
(λ∈σ(A)implies Re(λ)≤0and when Re(λ) = 0 then λis
a simple root of the minimal annihilating polynomial of A).
In the following we will focus our attention on asymptotic
stability, since this is the property one typically wants a
positive system to exhibit.
In addition to the standard characterizations available for
Schur or Hurwitz matrices, positivity brings novel characteri-
zations that we will comment on in some detail. Specifically,
we have the following two propositions.
Proposition 1: [33], [45] For a positive matrix A∈Rn×n,
the following properties are equivalent:
(1.1) Ais Schur;
(1.2) There exists ξ≫0such that Aξ≪ξ;
(1.3) There exists z≫0such that z⊤A≪z⊤;
(1.4) There exists a diagonal P≻0such that A⊤P A −
P≺0;
(1.5) (In−A)−1exists and has nonnegative entries.
Proposition 2: [33], [43], [45]1For a Metzler matrix A∈
Rn×n, the following properties are equivalent:
(2.1) Ais Hurwitz;
(2.2) There exists a ξ≫0such that Aξ≪0;
(2.3) There exists a z≫0such that z⊤A≪0;
(2.4) There exists a diagonal P≻0such that A⊤P+
P A ≺0;
(2.5) −A−1exists and has nonnegative entries.
Conditions (1.2), (1.3) and (1.4) (as well as (2.2), (2.3)
and (2.4)) in the above propositions have nice interpretations
in terms of Lyapunov functions. Condition (1.4) in Propo-
sition 1 corresponds to saying that for positive matrices the
Schur property corresponds to the existence of a diagonal
quadratic Lyapunov function, i.e., V(x) = x⊤Pxwhere
P=P⊤≻0is diagonal, satisfying ∆V(x) = V(Ax)−
V(x)<0for every x6= 0. Similarly, condition (2.4) in
Proposition 2 corresponds to saying that for Metzler matrices
the Hurwitz property corresponds to the existence of a
1The characterizations in [43] were actually derived for M-matrices. A
matrix Mis an M-marix if and only if −Mis Metzler Hurwitz.
Fig. 2. Level curves of Lyapunov functions corresponding to the con-
ditions (1.2), (1.3) and (1.4) in Proposition 1 or (2.2), (2.3) and (2.4) in
Proposition 2.
diagonal quadratic Lyapunov function , i.e., V(x) = x⊤Px,
with P=P⊤≻0diagonal, such that ˙
V(x) = ∇V(x)Ax<
0for every x6= 0.
To comment on the meaning of conditions (1.2) and (2.2),
let us introduce a different kind of Lyapunov function that is
very well-known and extensively used in the literature about
positive systems because of its simplicity.
Definition 2: A function V:Rn→Ris said to be
copositive if V(x)>0for every x>0. A function
V:Rn→Ris said to be a linear copositive function if
V(x) = z⊤x, for some z≫0.
Condition (1.3) in Proposition 1 and (2.3) in Proposition 2
correspond to saying that a positive matrix (a Metzler matrix)
is Schur (Hurwitz) if and only if it admits a linear copositive
Lyapunov function, i.e., there exists V(x) = z⊤x, with z≫
0, such that ∆V(x)<0(˙
V(x)<0) for every x6= 0.
Finally, conditions (1.2) and (2.2) correspond to the exis-
tence of another special type of Lyapunov function, defined
as
V(x) = max
i∈[1,n]
xi
ξi
,
that exhibits rectangular level curves. The level curves of
these three Lyapunov functions are illustrated in Figure 2.
Asymptotic stability is concerned with the long term
behavior of the unforced state trajectories of a system, and
this is a fundamental property one needs to obtain, possibly
by resorting to a state-feedback control action (see Section
V). On the other hand, if the input uacting on the system
represents a disturbance, it is natural to evaluate the system
performance in terms of its capability to contain the effects
of the disturbance uon the output y. So, if we focus only on
the forced dynamics of system (4), by assuming x(0) = 0,
and regard the system as an input-output operator Gmapping
input trajectories into output trajectories, it is natural to
investigate the input-output performance of the system in
terms of its Lp-gain. The Lp-gain of the system is defined
as
kGkLp−Lp= sup
u:kukLp=1
kg∗ukLp,
where ∗represents the convolution product and g(t)the
system impulse response (see (5) and (6)). In the rest of
this section we will focus on the continuous-time case and
investigate the Lp-gain of an asymptotically stable positive

system. To this end, we introduce the transfer matrix
G(s) := C(sIn−A)−1B+D∈Rr×m(s),(7)
of the system (4) (or, equivalently, of system (3)). Under the
asymptotic stability assumption on the system, namely upon
assuming that Ais Metzler Hurwitz, quite remarkable results
have been obtained [15], [25], [26], [58], [59].
Proposition 3: [15], [58] Suppose that Gis the input-
output operator of an asymptotically stable positive system.
Then for p= 1,2and +∞we have
kGkLp−Lp=kG(0)kp−ind .(8)
In particular, if G1and G2have transfer functions G(s)and
G(s)⊤respectively, then
kG1kL1−L1=kG2kL∞−L∞.(9)
Moreover, if the system is SISO, i.e. r=m= 1, then
kGkLp−Lp=G(0),∀p∈[1,+∞].
Proposition 4: [15], [27] Given the positive system (4)
and any γ > 0, the following facts are equivalent:
(4.1) Ais Hurwitz and kGkL∞−L∞< γ;
(4.2) Ais Hurwitz and G(0)1m≪γ1r;
(4.3) There exists ξ≫0,ξ∈Rn, such that
A B
C D ξ
1m≪0
γ1r.(10)
Consequently, for an asymptotically stable positive system
(4) the L∞-gain can be found by solving a linear program:
kGkL∞−L∞= min{γ:(10) holds for some ξ≫0,ξ∈Rn}.
Obviously Proposition 4 can be combined with (9) to also
compute the L1-induced gain of a positive system by linear
programming. However, if the L2-induced gain, also known
as the H∞-norm, of a positive MIMO system is of interest,
then different methods are needed. The following theorem
from [60] is a generalization of a result in [70].
Proposition 5: Given the positive system (4), assume that
A∈Rn×nis Hurwitz, and the pair (A, B)is controllable.
Also, suppose that M=M⊤∈R(n+m)×(n+m)is a
symmetric matrix with all nonnegative entries, except for
the last mdiagonal elements. Then the following statements
are equivalent:
(5.1) For every ω∈[0,∞)one has
(iωIn−A)−1B
Im∗
M(iωIn−A)−1B
Im0;
(5.2) −A−1B
Im⊤
M−A−1B
Im0;
(5.3) There exists a diagonal P0such that
M+A⊤P+P A P B
B⊤P00;
(5.4) There exist x,p≥0,u≫0such that Ax+Bu≤
0and x⊤u⊤M+p⊤A B≤0.
If all inequalities are replaced by strict ones, then the equiv-
alences hold even without the controllability assumption.
In particular, Proposition 5 with
M=C⊤C C⊤D
D⊤C D⊤D−γ2I
can be used to test if G(s) = C(sI −A)−1B+Dhas H∞-
norm smaller than γ.
IV. POS IT IV E RE AL IZ ATION
One of the most challenging problems investigated in the
context of positive systems is surely the positive realization
problem [3], [9], [29], [30], [51], [56], [75], which can be
stated as follows:
Given a proper rational matrix G(s)∈R(s)r×m, under
what conditions it admits a positive realization, namely it can
be identified with the transfer matrix of a (continuous-time
or discrete-time) positive state-space model?
Most of the literature on this subject has focused on
the discrete-time case, so in the rest of the section we
will consider this specific case. Under this assumption, the
positive realization problem becomes:
Given a proper rational matrix G(s)∈R(s)r×m, under
what conditions there exist N∈Z, N > 0,and nonnegative
matrices A∈RN×N
+, B ∈RN×r
+, C ∈Rr×N
+and D∈
Rr×m
+such that G(s) = C(sIN−A)−1B+D?
Clearly, a necessary condition for the existence of a
solution to the positive realization problem is that G(s)
is the transfer matrix of an externally positive system,
which means that the Markov coefficients Giof G(s), i.e.,
G(s) = P+∞
i=0 Gis−i, are nonnegative matrices. However,
as previously remarked, external positivity does not ensure
internal positivity, and hence the nonnegativity of the Markov
coefficients is not a sufficient condition. It is also worth
mentioning that verification of external positivity for a given
rational transfer function has proved to be NP-hard (see [14]
for the discrete-time problem and [5] for the continuous-time
one).
On the other hand, the positive realization problem is typi-
cally posed in contexts related, for instance, to compartmen-
tal systems in pharmacokinetics or biological applications,
where external positivity is intrinsically guaranteed by the
nature of the system under investigation. So, oftentimes it
makes sense to assume that the nonnegativity of the Markov
coefficients can be taken for granted.
Note that as the matrix Din every state-space realization
is determined by G0, the positive realization problem can
always be restricted to strictly proper rational matrices.
Finally, G(s)admits a positive realization if and only if
all its entries gij (s), i ∈[1, r], j ∈[1, m],admit a positive
realization. So, in the following we will restrict our attention
to strictly proper scalar transfer functions.
The path to the problem solution started with three mile-
stone contributions [48], [51], [56] in the early eighties, that

characterized the existence of a positive realization in terms
of the existence of a proper polyhedral cone that enjoys
special properties. More specifically.
Theorem 1: Given a strictly proper rational function
G(s)∈R(s), let Σ = (F, g, H )be a minimal realization
of G(s)and let nbe the dimension of Σ. Then G(s)has
a positive realization if and only if there exists a proper
polyhedral cone K ⊂ Rnsuch that
1) Kis F-invariant, namely FK ⊆ K;
2) g∈ K;
3) K ⊆ {x∈Rn:H F k−1x≥0,∀k∈Z, k ≥1}.
If such a cone exists and K= Cone(v1,...,vN), for some
v1,...,vN∈Rn, then there exists a positive realization of
dimension N.
Based on this fundamental theoretical characterization,
that however lacks of practical feasibility, several important
results have been obtained. But, the final fundamental steps
toward the solution of this difficult problem were taken only
15 years later in [3].
Theorem 2: [3] Let G(s)∈R(s)be a strictly proper
rational function, with nonnegative Markov coefficients. If
G(s)has a single pole of maximum modulus which is
positive real and has arbitrary multiplicity, then G(s)admits
a (discrete-time) positive realization.
Theorem 3: [3] Let G(s)∈R(s)be a strictly proper
rational function, with nonnegative Markov coefficients, and
set
ρ:= max{|λ|:λ∈Cis a pole of G(s)}.
If ρ > 0,limi→+∞inf Gi
ρi>0, and each pole λof G(s)
having modulus ρsatisfies the following conditions:
1) it is simple;
2) there exists k∈Z, k > 0,such that λk=ρk;
then G(s)admits a (discrete-time) positive realization.
The general (scalar) case, when poles of maximum mod-
ulus are not simple, was later addressed in [30]. To solve
the problem, a procedure has been proposed that is based on
the evaluation of the poles of maximum modulus of a finite
sequence of strictly proper rational functions, whose Markov
coefficients are suitable subsets of the Markov coefficients of
the original G(s). The procedure not only makes it possible
to answer the question of whether a positive realization exists
or not, but when a positive realization exists it shows how to
explicitly derive one. We refer the interested reader to [30]
and to the survey [9].
Once the problem of determining whether a given strictly
proper transfer function G(s)∈R(s)admits a positive
realization has been solved, the next natural question is:
What is the minimal size that a positive realization of G(s)
may exhibit? By referring to the characterization given in
Theorem 1, this amounts to looking for the polyhedral proper
cone K ⊂ Rnfor which the number Nof generating vectors
v1,...,vNis minimal. As a matter of fact, to the best
of our knowledge this is still an open problem, and no
major advancements have been obtained recently, so that
the surveys [8], [9] still provide the state of the art in
this subject. Apart from specific results, obtained for strictly
proper rational functions of specific (McMillan) degrees or
poles endowed with special properties (see, e.g., [11]), the
main result available provides a lower bound on the size of
a minimal positive realization. This result is based on the
well-known Karpelevich Theorem [46], [52] that completely
characterizes the regions of the complex plane where the
eigenvalues of a positive matrix Awith spectral radius ρ
can be located. Specifically, by the symbol Θρ
nwe denote
the region of the complex plane where the eigenvalues of
an n×nnonnegative matrix with spectral radius ρlie. The
following result holds.
Proposition 6: Consider a strictly proper rational function
G(s), with nonnegative impulse response and minimal real-
ization of order n, and assume that it has a real dominant
pole ρ > 0. Then the minimal order of a positive realization
of G(s)is not less than max{n, N }, where Nis the minimal
positive integer such that every pole pof G(s)satisfies
p∈Θρ
N. Moreover in every minimal positive realization the
state matrix has ρas nonnegative real dominant eigenvalue.
For additional results on the minimal positive realization
problem, the interested reader is referred to [8], [9] and
references therein.
V. S TATE-F EE DBA CK S TABI LI ZATI ON
As previously mentioned, asymptotic stability is a fun-
damental property one needs to ensure. In general, when
dealing with positive systems, the goal of achieving stability
by means of a state-feedback law u(t) = Kx(t), where u
is now regarded as a control input, cannot be pursued at
the cost of losing the positivity of the resulting feedback
system. So, the standard stabilization problem is replaced by
the problem of making the resulting state-space system both
positive and asymptotically stable 2. In this context, it is also
possible to easily introduce constraints on the state-feedback
matrices, and hence to pose the positive stabilization problem
by assuming that the matrix Kbelongs to some sparsity
structure Sin Rm×n, rather than being an arbitrary matrix
in Rm×n.
Positive stabilization problem: Given a sparsity structure
S ⊂ Rm×nas in (2), determine if there exists a state
feedback matrix K∈ S such that the resulting feedback
state-space model
x(t+ 1) = (A+BK)x(t), t ∈Z+,(11)
in discrete-time or
˙
x(t) = (A+BK)x(t), t ∈R+,(12)
2For the sake of simplicity, in this section we consider only strictly proper
positive systems, namely we assume D= 0. Consequently, the output
equation is not affected by the state-feedback control and hence we omit
it. In the general case, the output equation of the resulting feedback state-
space model would be y(t) = (C+DK)x(t), and hence we would need
to include also the constraint C+DK ≥0. This will be considered in the
following sections.

in continuous-time, is positive and asymptotically stable.
Let us focus, again, on the continuous-time case. In
this context the positive stabilization problem amounts to
determining, if it exists, a matrix K∈ S such that A+BK
is Metzler and Hurwitz. This problem was first investigated3
in [36], and the existence of a solution was expressed in terms
of the solvability of a family of Linear Matrix Inequalities
(LMIs). We provide here the result given in Theorem 1 of
[36], suitably rephrased in order to make a comparison with
the subsequent characterizations more immediate.
Theorem 4: Given a continuous-time system (12) and
a sparsity structure S ⊂ Rm×n, the following facts are
equivalent:
(4.1) The positive stabilization problem has a solution;
(4.2) There exist Y∈ S and a positive diagonal matrix
X, such that AX +BY is Metzler and
(AX +BY )⊤+AX +BY ≺0.(13)
When so, a solution to the stabilization problem is
obtained as K=Y X−1.
This problem was later investigated in [2], where the
problem solution was converted into a Linear Programming
(LP) problem.
Theorem 5: Given a continuous-time system (12) and
a sparsity structure S ⊂ Rm×n, the following facts are
equivalent:
(5.1) The positive stabilization problem has a solution;
(5.1) There exist Y∈ S and a positive diagonal matrix
X, such that AX +BY is Metzler and
(AX +BY )1n≪0.(14)
When so, a solution to the stabilization problem is
obtained as K=Y X−1.
It is worth noticing that Theorems 4 and 5 make use of two
of the characterizations of Metzler Hurwitz matrices obtained
in Proposition 2. Indeed, the former makes use of the fact
that if A+BK is Metzler Hurwitz then it admits a diagonal
quadratic Lyapunov function, while the latter of the fact that
it admits a linear copositive Lyapunov function.
From a computational point of view, the solution in terms
of LP, even if equivalent from a theoretical viewpoint, is
preferable due to its lower computational complexity. Even
more, it is prone to be easily extended to cope with robust
stabilization in the presence of polytopic uncertainties, sta-
bilization with restricted sign controls and stabilization with
bounded controls [2].
Alternative approaches to the positive stabilization prob-
lem have been proposed in [63] and [15]. The charac-
terization derived in [63] is based on the construction of
certain polytopes and on verifying whether a selection of
3In fact, all the results about positive stabilization reported in this section
were obtained without imposing any constraint on K, namely for S=
Rm×n, but their adaption to the case when Kis constrained to belong to
some sparsity structure Sis immediate.
their vertices can be used to construct a stabilizing state-
feedback matrix. On the other hand, in [15] the problem of
achieving by means of a state-feedback not only positivity
and stability, but also certain L1and L∞performances, has
been investigated. Also in this case, necessary and sufficient
conditions for the existence of a solution have been expressed
as LPs. We will address this aspect in Sections VI and VII.
Finally, the previous stabilization problem together with
other versions of the positive stabilization problem have been
investigated in [21] by focusing on the special case of single-
input continuous-time positive systems.
VI. CO NT ROL S YN TH ES IS B Y LI NE AR P ROGRAMMING
We consider, now, a more general version of the state-
space models introduced in Section III, since it includes
both the control input and the external disturbance. As in
the previous section, we will focus our attention on the
continuous-time case and hence on the model:
˙
x(t) = Ax(t) + Bu(t) + Ew(t),(15a)
y(t) = Cx(t) + Du(t) + Fw(t), t ∈R+,(15b)
where xrepresents the n-dimensional state variable, u
the m-dimensional control variable, wthe q-dimensional
disturbance, and ythe r-dimensional output variable. Note
that we will not introduce any positivity assumption on the
real matrices A, B, C, D , E and F. Indeed, we will impose
the positivity on the resulting feedback system, but not on
the original one. By referring to the previous state-space
model (15), in the next sections we will extend the previous
stabilization methods in order to optimize the performance of
the resulting feedback system in terms of input-output gains.
All the results will be illustrated by examples dealing with
buffer networks.
If we introduce the state-feedback law u(t) = Kx(t), the
resulting feedback system becomes:
˙
x(t) = (A+BK)x(t) + Ew(t),(16a)
y(t) = (C+DK)x(t) + Fw(t), t ∈R+.(16b)
Note that the only input acting on system (16) is the
disturbance w. We denote by
GK(s) := (C+DK)[sI −(A+BK)]−1E+F∈Rr×q(s)
(17)
the transfer matrix of system (16). If we assume zero initial
condition, i.e., x(0) = 0, such a system can be seen as an
operator GKfrom the input disturbance wto the output y.
A fundamental problem in the literature on robust control
is the minimization of gain from disturbance to error. The
purpose of this section and the next one is to demonstrate
how this can be done by extending the ideas of Section V. We
start with the optimization of L∞-gain, which can be reduced
to linear programming along the lines of Proposition 4 and
Theorem 5.
Theorem 6: [15] Consider a continuous-time system (15),
a sparsity structure Sand a positive real number γ. If Eand

x1
x2
x3
x4
Fig. 3. A directed graph representing the buffer network in Example 1.
The arrow from node 2to node 1corresponds to the term κ12x2, which
indicates a flow from buffer 2to buffer 1proportional to the buffer level
x2.
Fare nonnegative matrices, then the following conditions
are equivalent:
(6.1) There exists a matrix K∈ S such that C+DK
is nonnegative, A+BK is Metzler Hurwitz and
kGKkL∞−L∞< γ.
(6.2) There exist a positive diagonal matrix Xand a
matrix Y∈ S such that ¯
A:= AX +BY is Metzler,
¯
C:= CX +DY is nonnegative and
¯
A1n+E1q≪0
¯
C1n+F1q≪γ1n.
If Xand Ysatisfy (6.2), then (6.1) holds for K=Y X−1.
Example 1: Consider the buffer network as described in
(1), with one input uij =κij xjfor every edge in the directed
graph illustrated in Figure 3:















˙x1=−x1−κ31x1+κ12 x2+w1
˙x2=−2x2−κ12x2−κ32 x2+κ23x3+w2
˙x3=−3x3+κ31x1+κ32 x2−κ23x3−κ43 x3+κ34x4
+w3
˙x4=−4x4+κ43x3−κ34 x4+w4.
Our problem is to find (nonnegative) feedback gains κij, to
minimize the influence of the disturbances wion states and
inputs uij . To do this, notice that the buffer network can be
described as in (16a) for
A= diag{−1,−2,−3,−4}(18)
B=



−1 1 0 0 0 0
0−1−1 1 0 0
1 0 1 −1−1 1
0 0 0 0 1 −1




(19)
K= diag κ31,κ12
κ32,κ23
κ43, κ34 (20)
E=I4(21)
and hence the transfer matrix from wto (x,u), where uis
the vector u=u31 u12 u32 u23 u43 u34⊤, can be
expressed as
GK(s) = I4
K[sI4−(A+BK)]−1.
This amounts to assuming y=x
uand hence
C=In
0D=0
ImF= 0.(22)
Hence Theorem 6 can be used to minimize the L∞-gain of
the input-output operator from wto y. With
X= diag{ξ1, ξ2, ξ3, ξ4}
Y= diag µ31,µ12
µ32,µ23
µ43, µ34 
the linear programming problem becomes to minimize γ
subject to the constraints
−ξ1−µ31 +µ12 + 1 ≤0
−2ξ2−µ12 −µ32 +µ23 + 1 ≤0
−3ξ3+µ31 +µ32 −µ23 −µ43 +µ34 + 1 ≤0
−4ξ4+µ43 −µ34 + 1 ≤0
0≤ξi≤γ
0≤µij ≤γ
where, to ensure the Metzler property of A+BK, we
imposed that also the ξi’s are nonnegative.
VII. SYN TH ES IS B Y SE MI DE FIN IT E PRO GR AM MI NG
This section and the next one are devoted to minimization
of the L2-gain, a subject which in the literature on robust
control is known as H∞-optimization. There is a very
rich mathematical theory associated with H∞-norm, and
corresponding control synthesis methods are either based on
Riccati equations or on semidefinite programming. It should
therefore come as no surprise that Theorem 6 has an analogue
for L2-gain optimization, which uses semidefinite rather than
linear programming.
Theorem 7: [70] Consider a continuous-time system (15),
a sparsity structure Sand a positive real number γ. If Eand
Fare nonnegative matrices, then the following conditions
are equivalent:
(7.1) There exists a matrix K∈ S such that C+DK
is nonnegative, A+BK is Metzler Hurwitz and
kGKkH∞< γ.
(7.2) There exist a positive diagonal matrix Xand a
matrix Y∈ S such that ¯
A:= AX +BY is Metzler,
¯
C:= CX +DY is nonnegative and
¯
C⊤¯
C+¯
A+¯
A⊤¯
C⊤F+E
F⊤¯
C+E⊤F⊤F−γ2I≺0.(23)
If Xand Ysatisfy (7.2), then (7.1) holds for K=Y X−1.
At first sight, it may look like condition (23) is poorly
scalable, since it involves all problem data in a single
inequality. However, for sparse matrices, the following result
makes it possible to break the inequality into pieces and
hence to apply distributed algorithms.
Theorem 8: [60] An n×n, n ≥3,symmetric Metzler
matrix with mnonzero entries above the diagonal is negative

semidefinite if and only if it can be written as a sum of m
negative semidefinite matrices, each of which has only four
nonzero entries.
Example 2: We consider the same example as in the
previous section, with just a different norm. Hence, for
A, B, C, D , E and Fgiven by (18)-(19) and (21)-(22), we
want to find a matrix Kas in (20) making A+BK Metzler
and minimizing
kGKkH∞=


I
K[sI −(A+BK)]−1


H∞
.
Applying Theorem 7, and keeping in mind the expressions
of the matrices A, B, C, D , E, F and K, equation (23) takes
the form
X2+Y⊤Y+AX +BY + (AX +BY )⊤I
I−γ2I≺0.
Equivalently
X2+Y⊤Y+AX +BY + (AX +BY )⊤+γ−2I≺0
and more explicitly




d1µ12 µ31 0
µ12 d2µ23 +µ32 0
µ31 µ23 +µ32 d3µ34 +µ43
0 0 µ34 +µ43 d4




≺0
where ξi≥0,µij ≥0and the diagonal elements are
d1=ξ2
1+γ−2−2ξ1−2µ31
d2=ξ2
2+γ−2−4ξ2−2µ12 −2µ32
d3=ξ2
3+γ−2−6ξ3−2µ23 −2µ43
d4=ξ2
4+γ−2−8ξ4−2µ34.
Given that H∞-optimal controllers can also be computed
by standard Riccati equations, it is natural to compare
the gains achievable by the two methods. This allows us
to determine how restrictive the demand for closed loop
positivity is. As it will be shown in the next section, there is
a large class of buffer network control problems for which
the positivity condition makes no difference at all in terms
of achievable performance.
VIII. LA RG E SCA LE H∞O PT IM AL CONTRO L
Consider, first, the following problem motivated by distur-
bance rejection in buffer networks:
Given a directed graph with nodes Vand edges E, con-
sider the buffer network whose ith node updates according
to the following equation
˙xi=aixi+X
(i,j)∈E
(uij −uji ) + wii∈ V.(24)
Find feedback gains κij such that by assuming uij =κij xj
A+BK is Metzler and the H∞-norm of the transfer function
from the disturbance wto the controlled output (x,u), where
uis the input vector (whose specific definition depends on
the network structure), is minimized. Suppose that all nodes
are stable, i.e. ai<0. Then we will see below that an optimal
control law is given by
uij =xi/ai−xj/aj.(25)
The closed loop system from wto xis a positive system
and the control law is decentralized in the sense that control
action on the edge (i, j)is entirely determined by the states
in node iand node j. The formula for the optimal control
law follows from the following result of [49].
Theorem 9: Suppose that Ais symmetric (in particular,
diagonal) and Hurwitz. Consider
min
K


I
K[sI −(A+BK)]−1


H∞
where minimization is done over all Ksuch that A+BK is
Hurwitz. Then the minimum is attained by K∗=B⊤A−1.
The minimal value of the norm is pk(A2+BB⊤)−1k.
The theorem identifies a rare but important class of
systems for which decentralized controllers are known to
achieve the same H∞-performance as the best centralized
ones. Moreover, the combination of (1) and the control
law (25) gives a closed loop positive system, so this is a
case where closed loop positivity can be attained without
sacrificing performance.
It is interesting to compare Theorem 9 with Theorem 7. It
is straightforward to verify using completion of squares that
X=−A,Y=−B⊤solves (23) whenever a solution exists.
However, this solution does not necessarily satisfy the other
constraints of Theorem 7, so there is only a partial overlap
between the two theorems.
Combining (1) with the control law (25) is insufficient for
many practical applications. One reason is that proportional
controllers like (25) are unable to remove static errors in
presence of constant disturbances. Motivated by this fact, an
extension to controllers with integral action was derived in
[62].
IX. EX TE NS IO NS TO MONOT ON E SY STEMS
All real control systems come with nonlinearities, and
buffer networks are no exception. On the contrary, proper
handling of (nonlinear) buffer capacity constraints is often a
main concern. It is therefore interesting to note that positive
systems have natural extensions to nonlinear “monotone”
systems [68]. Just like positivity, also monotonicity is a term
that has different meanings in different contexts, but here
we use it to denote a system ˙
x=f(x)that preserves a
partial ordering of the states (see Figure 4). In other words, a
dynamical system is said to be monotone if its linearizations
are positive systems. For example, if (24) is connected to the
saturated control law
uij = sat(xi/ai−xj/aj),(26)
where sat(x) = min{max{x, −1},1}, then the closed loop
system becomes
˙xi=aixi+ 2 X
(i,j)∈E
sat(xi/ai−xj/aj) + wii∈ V

x(0)
x(1)
x(2)
x(3)
y(0)
y(1)
y(2)
y(3)
Fig. 4. A monotone dynamical system is a system that preserves a partial
ordering of the states. For example, the system is monotone if, given two of
its trajectories, say y(t)and x(t), t ≥0, then y(t)−x(t)is a nonnegative
vector for t > 0whenever this is true for t= 0.
x1
x2
t= 0
t= 1
t= 2
t= 3
Fig. 5. The main idea behind Theorem 10 is very simple. All trajectories
starting in the yellow box at t= 0 will be confined, due to monotonicity,
to smaller and smaller rectangles as t→ ∞. The boundaries of these
rectangles can be used to define a Lyapunov function of the form V(x) =
max{V1(x1),...,Vn(xn)}.
which is a monotone system. Many attractive features of pos-
itive systems generalize to monotone systems. The existence
of simpler and more scalable stability certificates is one of
them, as illustrated by the following result from [24].
Theorem 10: Consider a monotone system ˙
x=f(x)
with a globally asymptotically stable equilibrium at x= 0.
Suppose that fis locally Lipschitz and that the system leaves
the compact set X⊂Rn
+invariant. Then there exist strictly
increasing functions Vk:R+→R+for k= 1,...,n such
that V(x) = max{V1(x1),...,Vn(xn)}satisfies
d
dtV(x) = −V(x)
for all the state trajectories x(t), t ∈R+,included in X.
The proof idea for Theorem 10 is illustrated in Figure 5.
A control system of the form
˙
x=f(x,u)
with f∈C1is said to be a convex monotone system if fis
convex, while ∂f/∂xis Metzler and ∂f/∂uis nonnegative
at every point (x,u). This non-linear generalization of linear
positive systems retains another important property of posi-
tive systems, namely convexity in the dependence on initial
conditions [61]:
Theorem 11: If ˙
x=f(x,u),x(0) = a, is a convex-
monotone system with a unique solution x(t) = φt(a,u),
then each component of φt(a,u)is a convex function of
(a,u).
The convexity property is of course very useful in the
numerical computation of optimal trajectories and can some-
times also yield analytical results. An interesting instance
will be studied in the next section.
X. BIL IN EA R PO SI TI VE S YS TE MS
Before concluding the paper, we will briefly discuss sys-
tems of the form
˙
x(t) = A+X
i
ui(t)Di!x(t),(27)
where Ais Metzler, while D1,...,Dmare diagonal matri-
ces. Such models are useful in the study of combination ther-
apies of diseases such as HIV and cancer, where Adescribes
the mutation dynamics without drugs, while D1,...,Dm
model the effects of drug doses u1(t),...,um(t)on the
mutant concentrations xk(t). We assume that all variables
ui(t)take values in the same set U.
For fixed values of ui, the model (27) is a linear positive
system. However, due to the multiplication of uand x,
the state generally has a nonlinear dependence on u. Such
systems are generally very complicated to analyze, but in
this case the positivity properties help a lot.
Theorem 12: [18], [61] Given a Metzler matrix A, let x(t)
be the solution of
˙
x(t) = A+
m
X
i=1
ui(t)Di!x(t)
where x(0) = a>0and D1,...,Dmare diagonal matrices.
Then log xk(t)is a convex function of (a,u).
The theorem follows from the fact that the change of
variables zk(t) = log xk(t)yields a convex monotone
system:
˙zk(t) = X
k,j
akj exp(zj−zk) + X
i
ui(t)Di
k.
The same transformation can also be used to address several
other synthesis problems for this system. For example, for a
bilinear state-space model
˙
x(t) = A+
m
X
i=1
ui(t)Di!x(t) + Bw(t)(28a)
y(t) = Cx(t), t ∈R+,(28b)

where Ais Metzler, Band Care nonnegative, D1,...,Dm
are diagonal, and wis a disturbance, [23] proved that the
square of the H2-norm of the input-output map from wto
yis a convex function of uand also that the H∞-norm is
a convex function of u. Moreover, and this is important for
large scale problems, not only convexity but also sparsity
can be exploited, just like we did earlier in Section VI. This
is illustrated by the following formulation of conditions for
minimization of the L∞-gain (L1-optimal control):
Theorem 13: [79] Consider the positive bilinear system
(28), and assume that Ais Metzler, Cis nonnegative and
D1,...,Dmare diagonal. Let γbe a positive real number,
then the following conditions are equivalent:
(13.1) There exist u1,...,um∈Usuch that
kGukL∞−L∞< γ, where Guis the operator
corresponding to the linear and time-invariant
input-output map defined by (28) once the input
variables ui(t)take the constant values ui.
(13.2) There exist u1,...,um∈Uand z1,...,znsuch
that Pjclj ezj≤γand
X
j
akj ezj−zk+X
i
uiDi
k+e−zk≤0
for all kand l.
All previous inequalities are convex in (z,u)and if Aand
Care sparse, each inequality only involves a small number
of terms.
XI. CO NC LU SI ON S
In this tutorial paper we have first presented some foun-
dational results regarding stability, Lp-gain, state-feedback
stabilization and positive realization of positive systems.
Subsequently, we have addressed the state-feedback stabi-
lization with norm constraints and imposed sparsity structure
on the feedback gain matrix. This highlighted how the
techniques available for positive systems scale well with
dimension. As a result, positive systems have a noteworthy
advantage over standard systems when dealing with large
scale systems. Extensions of these results to the classes of
monotone systems and bilinear positive systems have also
been briefly discussed.
While a number of system theoretic problems for positive
systems have reached complete maturity, we believe that
there are still several exciting open problems that deserve
attention. In this tutorial we have highlighted what, in our
opinion, is one of the most promising currently, namely the
exploitation of positivity properties when dealing with large
scale systems. In spite of the long history, it is clear that a
large number of important and useful results still remain to be
discovered. For example, extensions have recently been made
to time-varying systems [47], not to mention the exciting new
results on differentially positive (non-linear) systems [34].
Together with an ever growing number of applications where
positivity plays a major role, this suggests that many of the
most important results are yet to come.
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