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On the D-Stability of Linear and Nonlinear Positive Switched Systems

V. S. Bokharaie, O. Mason and F. Wirth

Abstract— We present a number of results on D-stability

of positive switched systems. Different classes of linear and

nonlinear positive switched systems are considered and simple

conditions for D-stability of each class are presented.

I. INTRODUCTION

Positive systems have found wide application in different

areas of science and engineering, for example, wireless

communications, economics, populations dynamics, systems

biology [1]. While the stability properties of Positive linear

time-invariant systems are thoroughly investigated and well

understood, the theory for nonlinear and time-varying sys-

tems needs more attention.

Recently, switched systems have attracted a lot of attention

[2] [3]. This has been primarily motivated by the fact that

many man-made systems and some physical systems can be

modelled within this framework. While major advances have

been made, many important questions that relate to their

behaviour still remain unanswered, even for linear switched

systems. Perhaps the most important of these relate to the

stability of such systems.

In this paper, we are concerned with the stability of

positive switched systems. A popular approach to this issue

is to exploit copositive Lyapunov functions to investigate

the stability of such systems. In particular, linear copositive

Lyapunov functions have been considered in [4], while

results on quadratic copositive Lyapunov functions were

presented in [5]. Linear copositive Lyapunov functions and

functionals have also been used to establish strong delay-

independent conditions for classes of linear and nonlinear

positive systems [6][7][8]. Within the class of switched

systems, most of the research in this ﬁeld to date has focused

on linear switched systems. Considering the importance of

nonlinear switched systems, there is a need to extend these

results to positive nonlinear switched systems.

One of the key properties of positive LTI systems is D-

stability, which we deﬁne formally below. In the recent paper

[9], this concept was extended to positive switched linear

systems and separate necessary and sufﬁcient conditions for

D-stability for this system class were presented. In this paper

we develop the work of [9] in two main directions. First,

we show that for irreducible linear systems, the separate

This work is supported by the Irish Higher Education Authority (HEA)

PRTLI Network Mathematics grant, by Science Foundation Ireland (SFI)

grant 08/RFP/ENE1417.

V. S. Bokharaie is with Hamilton Institute, National University of Ireland

Maynooth, Ireland vahid.bokharaie@nuim.ie

O. Mason is with Hamilton Institute, National University of Ireland

Maynooth, Ireland oliver.mason@nuim.ie

F. Wirth is with Institut f ¨

ur Mathematik, Universit¨

at W¨

urzburg, W¨

urzburg,

Germany wirth@mathematik.uni-wuerzburg.de

necessary and sufﬁcient conditions of [9] can be combined

to give a single necessary and sufﬁcient condition for D-

stability. Further, we show that positive switched systems

with commuting system matrices are D-stable. A major con-

tribution of the paper is to extend the sufﬁcient condition for

D-stability for linear switched systems to a class of nonlinear

positive switched systems. Speciﬁcally, we derive a condition

for D-stability for switched systems whose constituent vector

ﬁelds are homogeneous and cooperative. We also show, as

a corollary, that the result on commuting linear systems

extends to irreducible, cooperative homogeneous systems.

II. MATHEMATICAL BACKGROUND

A. Mathematical Notations

Throughout the paper, Rand Rndenote the ﬁeld of real

numbers and the vector space of all n-tuples of real numbers,

respectively. Rn×ndenotes the space of n×nmatrices with

real entries. For x∈Rnand i= 1, . . . , n ,xidenotes the ith

coordinate of x. Similarly, for A∈Rn×n,aij denotes the

(i, j)th entry of A. Also, for x∈Rn,diag(x)is the n×n

diagonal matrix in which dii =xi. For a diagonal matrix

D∈Rn×n, the notation D > 0indicates that dii >0for

i= 1, . . . , n.

In the interest of brevity, we shall slightly abuse notation

and refer to a system as being Globally Asymptotically

Stable, GAS for short, when the origin is a GAS equilibrium

of the system. Also, as we are dealing with positive systems

throughout, when we refer to a system as GAS, it is with

respect to initial conditions in Rn

+, where Rn

+is the set of

all vectors in Rnwith non-negative entries:

Rn

+:= {x∈Rn:xi≥0,1≤i≤n}.

The interior of Rn

+is given by

int(Rn

+) := {x∈Rn:xi>0,1≤i≤n}.

The boundary bd(Rn

+) := Rn

+\int(Rn

+).

For vectors x, y ∈Rn, we write: x≥yif xi≥yifor

1≤i≤n;x>yif x≥yand x6=y;xyif xi>

yi,1≤i≤n.

B. Cooperative Homogeneous Systems

Given an n-tuple r= (r1, . . . , rn)of positive real numbers

and λ > 0, the dilation map δr

λ(x) : Rn→Rnis given by

δr

λ(x)=(λr1x1, ..., λrnxn). If r= (1,...,1), then we will

have a standard dilation map. The vector ﬁeld f:Rn→Rn

is said to be homogeneous of degree τ≥0with respect to

δr

λ(x)if

∀x∈Rn, λ ≥0, f (δr

λ(x)) = λτδr

λ(f(x)).

Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary

ISBN 978-963-311-370-7

795

The following fact, which is known as Euler’s formula,

will be useful in Section IV.

Proposition 2.1 (Euler’s Formula): Let f:Rn→Rnbe

a homogeneous vector ﬁeld of degree τwith respect to the

dilation map δr

λ. Then for any a∈Rn:

∂f

∂x (a)diag(r)a=diag(r+τ∗)f(a)(1)

where τ∗:= (τ, · · · , τ )

We say that f:Rn→Rnis cooperative if the Jacobian

matrix ∂f

∂x (a)is Metzler for all a∈Rn

+[12].

Following the deﬁnition of [10], f:Rn→Rn, is said to

be irreducible if:

(i) for a∈int(Rn

+),∂f

∂x (a)is irreducible [11];

(ii) for a∈bd(Rn

+)\ {0}, either ∂ f

∂x (a)is irreducible or

fi(a)>0∀i:ai= 0.

The key fact for our later analysis is that cooperative

systems are monotone [12]. Formally, if f:Rn→Rn

is cooperative and we denote by x(t, x0)the solution of

˙x(t) = f(x(t)) satisfying x(0) = x0, then x0≤y0implies

x(t, x0)≤x(t, y0)for all t≥0. Moreover, as the origin is

automatically an equilibrium of a homogeneous system, it

follows that homogeneous cooperative systems are positive,

meaning that they are Rn

+invariant.

C. Commuting vector ﬁelds

We brieﬂy recall the deﬁnition of commuting nonlinear

vector ﬁelds.

Deﬁnition 2.1 (Commuting Nonlinear Vector ﬁelds): We

say that the vector ﬁelds f1and f2commute if

∂f1

∂x (a)f2(a) = ∂ f2

∂x (a)f1(a)(2)

for all a∈Rn.

III. D-STABILITY CONDITIONS FOR POSITIVE

LINEAR SWITCHED SYSTEMS

A key property of positive LTI systems is that a GAS

system is automatically D-stable [1]. The LTI system ˙x(t) =

Ax(t)is said to be D-stable if ˙x(t) = DAx(t)is GAS for

all diagonal matrices D > 0.

In this section, we consider an extension of the notion

of D-stability to positive switched linear systems. First, we

recall the deﬁnition introduced in [9].

Deﬁnition 3.1: Consider the switched system:

˙x(t) = A(t)x(t)A(t)∈ {A1, A2}(3)

in which A1and A2are Hurwitz Metzler matrices. We say

that (3) is D-stable, if the corresponding switched system

˙x(t) = A(t)x(t)A(t)∈ {D1A1, D2A2}(4)

is GAS for all diagonal D1>0and D2>0.

In [9], the following result, giving separate sufﬁcient

and necessary conditions for D-stability of positive linear

switched systems was presented.

Theorem 3.1: Let A1, A2∈Rn×nbe Metzler and Hur-

witz. Then:

(i) If there is some v0with A1v0,A2v0then

the system (3) is D-stable;

(ii) If (3) is D-stable then there exists some non-zero v > 0

with A1v≤0,A2v≤0.

It was highlighted in [9] that there is a clear gap between

the two conditions given above. In the next result, we show

that under the additional assumption of irreducibility, it is

possible to give a single necessary and sufﬁcient condition

for D-stability.

Theorem 3.2: Let A1, A2∈Rn×nbe Metzler, irreducible

and Hurwitz. Then the switched system (3) is D-stable if and

only if there exists a vector v0such that A1v≤0and

A2v≤0.

Proof:

Necessity:

Based on Theorem 3.1, we already know that if (3) is D-

stable, then there exists a v > 0such that A1v≤0and

A2v≤0. We shall show that if A1and A2are irreducible,

then any such vmust be strictly positive.

To this end, assume that v > 0,Aiv≤0for i= 1,2

and vis not strictly positive. Without loss of generality, we

assume that precisely the ﬁrst kelements of vare non-zero,

so vi>0for i= 1,· · · , k and vi= 0 for i=k+ 1,· · · , n.

Now we partition A1and A2as follows:

A1=A11 A12

A21 A22 and A2=A0

11 A0

12

A0

21 A0

22

In which A11 and A0

11 are k×k,A22 and A0

22 are (n−

k)×(n−k)and A21 and A0

21 are (n−k)×ksub-

matrices. Please note that A11, A0

11, A22 and A0

22 are Metzler

and A12, A0

12, A21 and A0

21 are element-wise non-negative.

We know that A1v≤0and since the last n−kelements

of vare zero, then we should have A21v0≤0, in which

v0= [v1,· · · , vk]T. Since we know A21 is a non-negative

matrix and v00, then the only way that this inequality can

hold is that A21 = 0 in which 0refers to a matrix with zero

entries and appropriate dimensions. Using the same method,

we can easily conclude that A0

21 = 0. This implies that both

A1and A2are reducible, which is a contradiction. Therefore,

vcannot have zero entries and we must have v0as

claimed.

Sufﬁciency:

Let σbe a given switching signal with switching instances

t0, t1, t2, ... and, as is standard in the literature on switched

systems, we assume that there is some τ > 0such that

tj+1 −tj≥τfor all j. We shall also write ij=σ(tj)for

j= 0,1, ....

For x0∈Rn

+, we denote by x(σ, t, x0)the solution of (3)

corresponding to the switched signal σand initial condition

x0.

Now note that for an irreducible Metzler matrix A,eAt

0for all t > 0. Consider for any such A, the system ˙x(t) =

Ax(t). Then for any solution x(t)of this system, y(t) =

Ax(t)also satisﬁes ˙y(t) = Ay(t). As eAt 0for all t > 0,

it immediately follows that if y(0) <0then we must have

y(t)0for all t > 0. In terms of the original system, this

means that Ax(0) <0implies that Ax(t)0for all t > 0.

V. S. Bokharaie et al. • On the D-Stability of Linear and Nonlinear Positive Switched Systems

796

This argument guarantees that there is some α < 1such that

for i= 1,2:

eAiτv≤αv. (5)

Further, as tj+1 −tj≥τfor all j, we can also conclude that

for i= 1,2and j= 0,1,2,3, ...

eAi(tj+1−tj)v≤eAi(τ)v. (6)

Now consider any time t > 0and assume that tKis the ﬁnal

switching instant before t. Then

x(σ, t, v) = eAiK(t−tK)eAiK−1(tK−tK−1)· · · eAi0(t1−t0)v.

(7)

It follows from (6) and (7) that x(σ, t, v)≤αKv. A little

thought (we can “lose” at most one power of αper switch)

shows that if we deﬁne Ntto be the largest integer less than

or equal to t

2τ, then for any switching signal (whether there

are ﬁnitely many switches or inﬁnitely many switches) we

must have

x(σ, t, v)≤αNtv

implying that x(σ, t, v)→0as t→ ∞.

Now let x0∈Rn

+be given. Choose λ > 0with x0≤λv.

It follows from eAit0for all t > 0and i= 1,2that

x(σ, t, x0)≤x(σ, t, λv) = λx(σ, t, v)

and hence x(σ, t, x0)→0also. The result now follows as it

is immediate that DiAiv < 0for i= 1,2, for any diagonal

matrices D1>0,D2>0.

It has been previously shown [13] that switched linear

systems with commuting system matrices are GAS. In the

following result, we show that for positive switched linear

systems, commutativity implies the stronger property of D-

stability.

Theorem 3.3: Let A1∈Rn×nand A2∈Rn×nbe Metzler

and Hurwitz. Further, assume that A1A2=A2A1. Then the

switched system (3) is D-stable under arbitrary switching.

Proof: Recall that for Metzler, Hurwitz matrices A1and A2,

A−1

1<0and A−1

2<0[14]. Now let w0in Rnbe given.

Then v=A−1

1A−1

2w0, and therefore:

A1v=A1A−1

1A−1

2w=A−1

2w0

and

A2v=A2A−1

1A−1

2w=A2A−1

2A−1

1w=A−1

1w0

Thus, we have v0such that A1v0and A2v0,

and it follows from Theorem (3.1) that the switched system

is D-stable.

IV. D-STABILITY CONDITIONS FOR POSITIVE

NONLINEAR SWITCHED SYSTEMS

In this section, we consider extensions of the previous re-

sults on D-stability to classes of nonlinear positive switched

systems; speciﬁcally, we consider switched systems deﬁned

by cooperative homogeneous vector ﬁelds.

Throughout this section, all vector ﬁelds are assumed to be

cooperative and homogeneous of degree 0with respect to a

ﬁxed dilation map δr

λ. Note that for the standard dilation map,

this amounts to assuming that f(λx) = λf(x)for all x∈Rn

and all λ > 0. Further, we shall assume that all vector

ﬁelds are continuous on Rnand C1on Rn\{0}. As noted in

[10], this ensures existence and uniqueness of solutions for

the associated autonomous system. As we are interested in

D-stability and stability under arbitrary switching, we also

assume that all constituent systems ˙x=fi(x)are GAS

throughout the section.

To begin, we state the following deﬁnition, which is a

natural extension of the Deﬁnition (3.1) for linear switched

systems.

Deﬁnition 4.1: Consider the switched system

˙x(t) = f(x(t), t); f(·, t)∈ {f1(·), . . . , fm(·)}(8)

We call this system D-stable if the associated switched

system

˙x(t) = f(x(t), t); f(·, t)∈ {D1f1(·), . . . , Dmfm(·)}(9)

is GAS for any diagonal matrices Djwith Dj>0for j=

1,· · · , m.

For the proof of the next result, we will need the following

proposition, which is Proposition 3.2.1 in [12].

Proposition 4.1: Let f:Rn→Rnbe cooperative. Then

if f(x0)≤0(f(x0)≥0) the trajectory x(t, x0)of ˙x=f(x)

with initial condition x0is nonincreasing (nondecreasing) for

t≥0.

We can now present and prove the main Theorem of this

section, which extends the sufﬁcient condition of Theorem

3.1 to cooperative homogeneous systems.

Theorem 4.1 (Main Theorem): Consider the switched

system (8). If there exists a v0such that fi(v)0

for all i∈ {1,2, ..., m}, then (8) is D-stable under arbitrary

switching.

Proof: We will prove the Theorem in a number of steps.

The ﬁrst step is to show Lyapunov stability of the system.

(i) Proof of Stability

Let an arbitrary switching signal σ: [0,∞)→ {1, . . . , m}

be given with switching instances 0 = t0, t1, t2,· · · . For

x0∈Rn

+, let x(σ, t, x0)denote the solution of (8) corre-

sponding to the initial condition x0and the switching signal

σ.

To begin with, from the homogeneity of the vector ﬁelds

fiit follows that for any λ > 0,fi(δr

λ(v)) 0for i=

1, . . . , m. Thus as each fiis cooperative, Proposition 4.1

implies that the trajectory starting from the initial condition

x0=δr

λ(v)is non-increasing for 0≤t<t1. In particular,

for 0≤t≤t1,x(σ, t, δr

λ(v)) < x(σ, 0, δr

λ(v)) = δr

λ(v).

At t=t1, we switch to a new vector ﬁeld, which starts

from initial condition equal to x(σ, t1, δr

λ(v)). Since this new

system is also cooperative it follows from fi(δr

λ(v)) 0for

i= 1, . . . , m and x(σ, t1, δr

λ(v)) < δr

λ(v)that

x(σ, t, δr

λ(v)) < δr

λ(v)

for t1≤t≤t2. Continuing in this way it is clear that the

trajectory x(σ, t, δr

λ(v)) < δr

λ(v)for all t≥0.

Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary

797

Let > 0be given. Then we can choose λ > 0so that

kδr

λ(v)k∞< . Now putting

δ= min

i(δr

λ(v))i

we see that if x0≥0and kx0k∞< δ, then x0≤δr

λ(v)and

the above argument guarantees that

kx(σ, t, x0)k∞<

for all t≥0. Note that our choice of δdoes not depend on

the switching signal σ.

(ii) Proof of Global Asymptotic Stability

We next show that the origin is uniformly attractive for

(8). To this end choose α > 0such that

fi(v) + αv 0

for 1≤i≤mand for i= 1, . . . , m, deﬁne gi:Rn→Rn

by

gi(y) = fi(y) + αdiag(r)y

and consider the switched system

˙y(t) = g(y(t), t)g(y, .)∈ {g1, . . . , gm}.(10)

Let x(σ, t, x0)be a solution of (8) with initial condition x0.

Then it can be veriﬁed by direct calculation that

y(t) = δr

α(x(t)) = (er1αtx1,· · · , ernαtxn)(11)

is a solution of (10) with y(0) = x0.

It follows from the argument given in part (i) that for any

x0there exists some λ > 0with y(σ, t, x0)≤δr

λ(v)for all

t≥0. This immediately implies that

x(σ, t, x0)→0

as t→ ∞.

(iii) Proof of D-stability

Let diagonal matrices D1, . . . , Dmbe given with Di>0

for i= 1,...m. Then it is simple to check that each vector

ﬁeld Difi(·)is cooperative and homogeneous of degree 0

with respect to δr

λ. Further,

Difi(v)0

for 1≤i≤m. It now follows from the previous arguments,

that (9) is GAS and hence (8) is D-stable as claimed.

Remark: While we have assumed that all vector ﬁelds

are homogeneous with respect to the same dilation map, this

does not appear to be necessary and extending the above

result in this and other directions is the work of ongoing

research.

Now, we present the result on commuting vector ﬁelds,

but before that, we need the following Theorem which is

Theorem 5.2 in [10].

Theorem 4.2: Let f:Rn→Rnbe cooperative and

irreducible. Further assume that fis homogeneous of degree

0with respect to δr

λ. Then there exists a γ∈Rand a vector

v0, such that f(v) = γdiag(r)vand ˙x=f(x)is GAS

if and only if γ < 0.

It is known that a switched system with GAS commuting

vector ﬁelds is GAS itself. Our next result, provides a

condition for D-stability of a class of commuting vector

ﬁelds.

Corollary 4.1: Consider the switched system:

˙x(t) = f(x(t), t)f(·, t)∈ {f1(·), f2(·)}.(12)

Assume that f1and f2commute and are irreducible. Then

the switched system (12) is D-stable under arbitrary switch-

ing.

Proof: Since f2is GAS, homogeneous, cooperative and

irreducible, then based on Theorem (4.2), we know that there

exists a v0such that

f2(v) = γdiag(r)v(13)

in which γ < 0is a scalar. Now, applying Euler’s formula

to f2and evaluating it at v, we have:

∂f2

∂x (v)diag(r)v=diag(r)f2(v)(14)

Substituting f2(v)in the right-hand side of (14) from (13),

we have:

∂f2

∂x (v)diag(r)v=diag(r)γdiag(r)v

⇒diag(r)−1∂f2

∂x (v)diag(r)v=γdiag(r)v

therefore diag(r)vis an eigenvector of diag(r)−1∂f2

∂x (v).

Since ∂f2

∂x (v)and therefore diag(r)−1∂f2

∂x (v)is irreducible

and Metzler, and since diag(r)v0, the Perron-Frobenius

Theorem for irreducible matrices [1] implies that γis the

right-most eigenvalue of diag(r)−1∂f2

∂x (v)and diag(r)vis

its unique eigenvector (up to scalar multiple).

On the other hand, by evaluating the commutativity equal-

ity at x=v, we have:

∂f1

∂x (v)f2(v) = ∂ f2

∂x (v)f1(v)

By applying (13) and Euler’s formula to the left-hand side

of the above equation, we have:

γdiag(r)f1(v) = ∂f2

∂x (v)f1(v)

⇒γf1(v) = diag(r)−1∂f2

∂x (v)f1(v)

Therefore, f1(v)is also an eigenvector corresponding to γ.

Since the eigenvector corresponding to this eigenvalue is

unique up to scalar multiple, then we should have:

f1(v) = κdiag(r)v

where κis a scalar. Since f1is GAS, homogeneous, coop-

erative and irreducible, then based on Theorem (4.2) κ < 0.

Thus f1(v)0and from (13) we know f2(v)0. It now

follows from Theorem 4.1 that the switched system (12) is

D-stable under arbitrary switching.

V. S. Bokharaie et al. • On the D-Stability of Linear and Nonlinear Positive Switched Systems

798

V. CONCLUSIONS

We have shown that the separate necessary sufﬁcient

conditions for D-stability for switched positive linear systems

previously presented in [9] can be combined into a single

necessary and sufﬁcient condition in the case of irreducible

systems. Further, we have shown that switched positive linear

systems with commuting system matrices are D-stable. A

simple extension of the concept of D-stability for switched

nonlinear positive systems has been considered and a sufﬁ-

cient condition for cooperative homogeneous (of degree 0)

switched systems to be D-stable has been derived. The result

on commuting system matrices has also been extended to

irreducible vector ﬁelds in this case.

VI. ACKNOWLEDGEMENTS

This work has been supported by Science Foundation

Ireland (SFI) grant 08/RFP/ENE1417 and by the Irish Higher

Education Authority (HEA) PRTLI Network Mathematics

grant.

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Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary

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