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

Conference Paper

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

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.
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 xRnand i= 1, . . . , n ,xidenotes the ith
coordinate of x. Similarly, for ARn×n,aij denotes the
(i, j)th entry of A. Also, for xRn,diag(x)is the n×n
diagonal matrix in which dii =xi. For a diagonal matrix
DRn×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
+:= {xRn:xi0,1in}.
The interior of Rn
+is given by
int(Rn
+) := {xRn:xi>0,1in}.
The boundary bd(Rn
+) := Rn
+\int(Rn
+).
For vectors x, y Rn, we write: xyif xiyifor
1in;x>yif xyand x6=y;xyif xi>
yi,1in.
B. Cooperative Homogeneous Systems
Given an n-tuple r= (r1, . . . , rn)of positive real numbers
and λ > 0, the dilation map δr
λ(x) : RnRnis given by
δr
λ(x)=(λr1x1, ..., λrnxn). If r= (1,...,1), then we will
have a standard dilation map. The vector field f:RnRn
is said to be homogeneous of degree τ0with respect to
δr
λ(x)if
xRn, 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:RnRnbe
a homogeneous vector field of degree τwith respect to the
dilation map δr
λ. Then for any aRn:
∂f
∂x (a)diag(r)a=diag(r+τ)f(a)(1)
where τ:= (τ, · · · , τ )
We say that f:RnRnis cooperative if the Jacobian
matrix ∂f
∂x (a)is Metzler for all aRn
+[12].
Following the definition of [10], f:RnRn, is said to
be irreducible if:
(i) for aint(Rn
+),∂f
∂x (a)is irreducible [11];
(ii) for abd(Rn
+)\ {0}, either f
∂x (a)is irreducible or
fi(a)>0i:ai= 0.
The key fact for our later analysis is that cooperative
systems are monotone [12]. Formally, if f:RnRn
is cooperative and we denote by x(t, x0)the solution of
˙x(t) = f(x(t)) satisfying x(0) = x0, then x0y0implies
x(t, x0)x(t, y0)for all t0. 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 aRn.
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, A2Rn×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 A1v0,A2v0.
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, A2Rn×nbe Metzler, irreducible
and Hurwitz. Then the switched system (3) is D-stable if and
only if there exists a vector v0such that A1v0and
A2v0.
Proof:
Necessity:
Based on Theorem 3.1, we already know that if (3) is D-
stable, then there exists a v > 0such that A1v0and
A2v0. We shall show that if A1and A2are irreducible,
then any such vmust be strictly positive.
To this end, assume that v > 0,Aiv0for 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)×(nk)and A21 and A0
21 are (nk)×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 A1v0and since the last nkelements
of vare zero, then we should have A21v00, 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 tjτfor all j. We shall also write ij=σ(tj)for
j= 0,1, ....
For x0Rn
+, 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:
eAiτvαv. (5)
Further, as tj+1 tjτfor all j, we can also conclude that
for i= 1,2and j= 0,1,2,3, ...
eAi(tj+1tj)veAi(τ)v. (6)
Now consider any time t > 0and assume that tKis the final
switching instant before t. Then
x(σ, t, v) = eAiK(ttK)eAiK1(tKtK1)· · · eAi0(t1t0)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 x0Rn
+be given. Choose λ > 0with x0λv.
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 A1Rn×nand A2Rn×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,
A1
1<0and A1
2<0[14]. Now let w0in Rnbe given.
Then v=A1
1A1
2w0, and therefore:
A1v=A1A1
1A1
2w=A1
2w0
and
A2v=A2A1
1A1
2w=A2A1
2A1
1w=A1
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 xRn
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:RnRnbe 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
t0.
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
x0Rn
+, 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 0t<t1. In particular,
for 0tt1,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 t1tt2. Continuing in this way it is clear that the
trajectory x(σ, t, δr
λ(v)) < δr
λ(v)for all t0.
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 x00and kx0k< δ, then x0δr
λ(v)and
the above argument guarantees that
kx(σ, t, x0)k< 
for all t0. 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 1imand for i= 1, . . . , m, define gi:RnRn
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
t0. 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 1im. 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:RnRnbe cooperative and
irreducible. Further assume that fis homogeneous of degree
0with respect to δr
λ. Then there exists a γRand 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)1f2
∂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
799
... To date, relatively little has been written on positive switched systems subject to uncertainty. The works of [MBS09], [BMW10a] and [BMW10a] are of this nature and are the basis for some of the results presented in this chapter and the next. ...
... To date, relatively little has been written on positive switched systems subject to uncertainty. The works of [MBS09], [BMW10a] and [BMW10a] are of this nature and are the basis for some of the results presented in this chapter and the next. ...
... Looking at Theorems 5.4.4 and 5.4.7, it can be easily seen that there is a gap between the sufficient and necessary D-stability conditions given in these theorems. In the next result, which appears in [BMW10a], we show that under the extra assumption of irreducibility, it is possible to give a single necessary and sufficient condition for D-stability in linear switched systems. ...
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Although in recent years much attention has been paid to positive systems in general, there are still many areas that are left untouched. One of these areas, is the stability analysis of positive systems under any form of uncertainty. In this book, we study three broad classes of positive systems subject to different forms of uncertainty: nonlinear, switched and time-delay positive systems. Our focus is on positive systems which are monotone. Naturally, monotonicity methods play a key role in obtaining our results. As an application of our theoretical work on positive systems, we study a class of epidemiological systems with time-varying parameters. Most of the work done so far in epidemiology has been focused on models with time-independent parameters. Based on some of the recent results in this area, we describe the epidemiological model as a switched system and present some results on stability properties of the disease-free state of the epidemiological model.
... Monotonicity means that the ordering of initial states is preserved. This is a very powerful property as it allows us to obtain results concerning the asymptotic behaviour of the system, see for example [5], [4]. Monotone dynamical systems have long been studied, and the theory in its modern form was developed by M.W. Hirsch in a series of papers called "Systems of differential equations that are competitive or cooperative". ...
... It is worth noting that monotonicity can be used to establish conditions for asymptotic stability [4], [5], [10]. These papers consider nonlinear and switched systems. ...
... However, nonlinear PSSs are seldom investigated. Recently, [34] discussed the D-stability of nonlinear PSSs without delays. On this ground, we will study the stability property of nonlinear PSSs with delays in the paper. ...
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1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions 7. Totally positive matrices.
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