ArticlePDF Available

On the Stability of Switched Positive Linear Systems

Authors:

Abstract

It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching. In this note, we show that (1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; (2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and (3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note.
1
On the stability of switched positive linear
systems
L. Gurvits, R. Shorten and O. Mason
Abstract
It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices
is a necessary and sufficient condition for the asymptotic stability of the associated switched linear
system under arbitrary switching. In this paper we show that: (i) this conjecture is true for systems
constructed from a pair of second order Metzler matrices; (ii) the conjecture is true for systems
constructed from an arbitrary finite number of second order Metzler matrices; and (iii) the conjecture
is in general false for higher order systems. The implications of our results, both for the design of
switched positive linear systems, and for research directions that arise as a result of our work, are
discussed toward the end of the paper.
Key Words: Stability theory; Switched linear systems; Positive linear systems
I. INTRODUCTION
Positive dynamical systems are of fundamental importance to numerous applications in areas
such as Economics, Biology, Sociology and Communications. Historically, the theory of positive
linear time-invariant (LTI) systems has assumed a position of great importance in systems
theory and has been applied in the study of a wide variety of dynamic systems [1], [2], [3],
[4]. Recently, new studies in communication systems [5], formation flying [6], and other areas,
have highlighted the importance of switched (hybrid) positive linear systems (PLS). In the
L. Gurvits is with Los Alamos National Laboratory, USA - email: gurvits@lanl.gov
R. Shorten is with the Hamilton Institute, NUI Maynooth, Ireland, Corresponding Author - email: robert.shorten@nuim.ie
O. Mason is with the Hamilton Institute, NUI Maynooth, Ireland - email: oliver.mason@nuim.ie
DRAFT
last number of years, a considerable effort has been expended on gaining an understanding of
the properties of general switched linear systems [7], [8]. As is the case for general switched
systems, even though the main properties of positive LTI systems are well understood, many
basic questions relating to switched PLS remain unanswered. The most important of these
concerns their stability, and in this paper we present some initial results on the stability of
switched PLS.
Recently, it was conjectured by the authors of [9], and independently by David Angeli, that
the asymptotic stability of a positive switched linear system can be determined by testing
the Hurwitz-stability of an associated convex set of matrices. This conjecture was based on
preliminary results on the stability of positive switched linear systems and is both appealing
and plausible. Moreover, if it were true, it would have significant implications for the stability
theory of positive switched linear systems. In this paper, we shall extend some earlier work and
show that the above conjecture is true for some specific classes of positive systems. However,
one of the the major contributions of the paper is to construct a counterexample which proves
that, in general, the conjecture is false. However, this in turn gives rise to a number of open
questions for future research, some of which we discuss towards the end of the paper.
The layout of the paper is as follows. In Section 2 we present the mathematical background
and notation necessary to state the main results of the paper. Then in Section 3, we present
necessary and sufficient conditions for the uniform asymptotic stability of switched second
order positive linear systems. In Section 4 we show by means of an abstract construction that
the results derived in the preceding sections do not generalise to higher dimensional systems. In
Section 5, we demonstrate that these results also fail to generalise for the more restrictive case
of matrices with constant diagonals and we make some observations on the computation of the
joint Lyapunov exponent for positive switched systems in Section 6. Finally, our conclusions
are presented in Section 7.
II. M
ATHEMATICAL PRELIMINARIES
In this section we present a number of preliminary results that shall be needed later and
introduce the main notations used throughout the paper.
(i) Notation
Throughout, R denotes the field of real numbers, R
n
stands for the vector space of all n-tuples
of real numbers and R
n×n
is the space of n × n matrices with real entries. For x in R
n
, x
i
denotes the i
th
component of x, and the notation x 0 (x 0) means that x
i
> 0 (x
i
0)
for 1 i n. R
n
+
= {x R
n
: x 0} denotes the non-negative orthant in R
n
. Similarly, for
a matrix A in R
n×n
, a
ij
or A(i, j) denotes the element in the (i, j) position of A, and A 0
(A 0) means that a
ij
> 0(a
ij
0) for 1 i, j n. A B (A B) means that A B 0
(A B 0). We write A
T
for the transpose of A and exp(A) for the usual matrix exponential
of A R
n×n
.
For P in R
n×n
the notation P > 0 (P 0) means that the matrix P is positive (semi-)definite,
and P SD(n) denotes the cone of positive semi-definite matrices in R
n×n
. The spectral radius
of a matrix A is the maximum modulus of the eigenvalues of A and is denoted by ρ(A). Also
we shall denote the maximal real part of any eigenvalue of A by µ(A). If µ(A) < 0 (all the
eigenvalues of A are in the open left half plane) A is said to be Hurwitz or Hurwitz-stable.
Given a set of points, {x
1
, . . . , x
m
} in a finite-dimensional linear space V , we shall use
the notations CO(x
1
, . . . , x
m
) and Cone(x
1
, . . . , x
m
) to denote the convex hull and the cone
generated by x
1
, . . . , x
m
respectively. Formally:
CO(x
1
, . . . , x
m
) = {
m
X
i=1
α
i
x
i
: α
i
0, 1 i m, and
m
X
i=1
α
i
= 1};
Cone(x
1
, . . . , x
m
) = {
m
X
i=1
α
i
x
i
: α
i
0, 1 i m }.
A closed convex cone in R
n
is a set R
n
such that, for any x, y and any α, β 0,
αx + βy . A convex cone is said to be: solid if the interior of is non-empty; pointed
if (Ω) = {0}; polyhedral if = Cone(x
1
, . . . , x
m
) for some finite set {x
1
, . . . , x
m
}
of vectors in R
n
. We shall call a closed convex cone that is both solid and pointed, a proper
convex cone.
(ii) Positive LTI systems and Metzler matrices
The LTI system
Σ
A
: ˙x(t) = Ax(t), A R
n×n
, x(0) = x
0
is said to be positive if x
0
0 implies that x(t) 0 for all t 0. See [3] for a description of
the basic theory and several applications of positive linear systems. The system Σ
A
is positive
if and only if the off-diagonal entries of the matrix A are non-negative. Matrices of this form
are known as Metzler matrices. The next result concerning positive combinations of Metzler
Hurwitz matrices was pointed out in [10].
Lemma 2.1: Let A
1
, A
2
be Metzler and Hurwitz. Then A
1
+ γA
2
is Hurwitz for all γ > 0 if
and only if A
1
+ γA
2
is non-singular for all γ > 0.
(iii) Common Quadratic Lyapunov Functions and Stability
It is well known that the existence of a common quadratic Lyapunov function (CQLF) for
the family of stable LTI systems Σ
A
i
: ˙x = A
i
x i {1, . . . , k} is sufficient to guarantee
that the associated switched system Σ
S
: ˙x = A(t)x A(t) {A
1
, . . . , A
k
} is uniformly
asymptotically stable under arbitrary switching. Throughout the paper, when we speak of the
stability (uniform asymptotic stability) of a switched linear system, we mean stability (uniform
asymptotic stability) under arbitrary switching.
Note that any initial state x
0
R
n
can be written as x
0
= u v where u, v 0. Hence, for
linear systems, uniform asymptotic stability with respect to initial conditions in the positive
orthant is equivalent to uniform asymptotic stability with respect to arbitrary initial conditions
in R
n
. In particular, if a positive switched linear system fails to be uniformly asymptotically
stable (UAS) for initial conditions in the whole of R
n
, then it is also not UAS for initial
conditions in the positive orthant.
Formally checking for the existence of a CQLF amounts to looking for a single positive definite
matrix P = P
T
> 0 in R
n×n
satisfying the k Lyapunov inequalities A
T
i
P + P A
i
< 0 i
{1, . . . , k}. If such a P exists, then V (x) = x
T
P x defines a CQLF for the LTI systems Σ
A
i
.
While the existence of such a function is sufficient for the uniform asymptotic stability of
the associated switched system, it is in general not necessary for stability [8], and CQLF
existence can be a conservative condition for stability. However, recent work has established
a number of system classes for which this is not necessarily the case [11], [12]. The results
in these papers relate the existence of an unbounded solution to a switched linear system to
the Hurwitz-stability of the convex hull of a set of matrices and are based on the following
theorem.
Theorem 2.1: [13], [14] Let A
1
, A
2
R
n×n
be Hurwitz matrices. A sufficient condition for
the existence of an unstable switching signal for the system ˙x = A(t)x, A(t) {A
1
, A
2
}, is
that A
1
+ γA
2
has an eigenvalue with a positive real part for some positive γ.
Any trajectory of a positive system originating in the positive orthant will remain there as time
evolves. Consequently, to demonstrate the stability of such systems, one need not search for
a CQLF, but rather the existence of a copositive Lyapunov function. Formally, V (x) = x
T
P x
is a copositive CQLF if the symmetric matrix P R
n×n
is such that x
T
P x > 0 for x R
n
+
,
x 6= 0, and x
T
(A
T
i
P + P A
i
)x
T
< 0 i {1, . . . , k}, x 0, x 6= 0.
III. S
ECOND ORDER POSITIVE LINEAR SYSTEMS
In this section, we shall show that the conjecture in [9] is true for second order positive switched
linear systems. First, we recall the result of [11] which described necessary and sufficient
conditions for the existence of a CQLF for a pair of general second order LTI systems.
Theorem 3.1: Let A
1
, A
2
R
2×2
be Hurwitz. Then a necessary and sufficient condition for
Σ
A
1
, Σ
A
2
to have a CQLF is that the matrix products A
1
A
2
and A
1
A
1
2
have no negative
eigenvalues.
It is only necessary to check one of the products in the above theorem if the individual systems
Σ
A
1
, Σ
A
2
are positive systems.
Lemma 3.1: Let A
1
, A
2
R
2×2
be Hurwitz and Metzler. Then the product A
1
A
2
has no
negative eigenvalue.
Proof: As A
1
, A
2
are both Hurwitz, the determinant of A
1
A
2
must be positive. Also, the
diagonal entries of A
1
A
2
must both be positive. Hence the trace of A
1
A
2
is positive. It now
follows easily that the product A
1
A
2
cannot have any negative eigenvalues.
Combining Theorem 3.1 and Lemma 3.1 yields the following result.
Theorem 3.2: Let A
1
, A
2
R
2×2
be Hurwitz and Metzler. Then the following statements are
equivalent:
(a) Σ
A
1
and Σ
A
2
have a CQLF;
(b) Σ
A
1
and Σ
A
2
have a common copositive quadratic Lyapunov function;
(c) The switched system ˙x = A(t)x, A(t) {A
1
, A
2
} is uniformly asymptotically stable;
(d) The matrix product A
1
A
1
2
has no negative eigenvalues.
Proof : (a) (d): From Lemma 3.1 it follows that the matrix product A
1
A
2
cannot have a
negative eigenvalue. Hence, the equivalence of (a) and (d) follows from Theorem 3.1.
(b) (d): If A
1
A
1
2
has no negative eigenvalues, then Σ
A
1
and Σ
A
2
have a CQLF. Thus, they
certainly have a copositive common quadratic Lyapunov function. Conversely, suppose that
A
1
A
1
2
has a negative eigenvalue. It follows that A
1
+ γ
0
A
2
has a real, non-negative eigenvalue
for some γ
0
> 0. Since, A
1
+ γ
0
A
2
= N α
0
I, where N 0, it follows that the eigenvector
corresponding to this eigenvalue is the Perron eigenvector of N and consequently lies in the
positive orthant [2]. It follows that a copositive Lyapunov function cannot exist.
(c) (d): Suppose that A
1
A
1
2
has a negative eigenvalue; namely, A
1
+ γA
2
is non-Hurwitz
for some γ > 0. It now follows from Theorem 2.1 that there exists some switching signal for
which the switched system Σ
S
: ˙x = A(t)x A(t) {A
1
, A
2
} is not uniformly asymptotically
stable. This proves that (c) implies (d). Conversely, if A
1
A
1
2
has no negative eigenvalues, then
Σ
A
1
, Σ
A
2
have a CQLF and the associated switched systems is uniformly asymptotically stable.
This completes the proof.
The equivalence of (c) and (d) in the previous theorem naturally gives rise to the following
question. Given a finite set {A
1
, . . . , A
k
} of Metzler, Hurwitz matrices in R
2×2
, does the
Hurwitz stability of CO(A
1
, ..., A
k
) imply the uniform asymptotic stability of the associated
switched system? This is indeed the case and follows from the following theorem, which can
be thought of as an edge theorem for positive systems. This theorem extends a result presented
recently in [15] by removing the restrictive assumption that the diagonal entries of all the
system matrices are equal to 1.
Theorem 3.3: Let A
1
, . . . , A
k
be Hurwitz, Metzler matrices in R
2×2
. Then the positive switched
linear system,
˙x = A(t)x A(t) {A
1
, . . . , A
k
}, (1)
is uniformly asymptotically stable if and only if each of the switched linear systems,
˙x = A(t)x A(t) {A
i
, A
j
}, (2)
for 1 i < j k is uniformly asymptotically stable.
Outline of Proof
(a) First, we show that R
2
+
, can be partitioned into a finite collection of wedges, Sec
j
, 1
j m such that, for 1 j m there exists a quadratic form x
T
P
j
x, which is non-
increasing along each trajectory of (1) within Sec
j
. Formally, for x Sec
j
and 1 i k,
x
T
(A
T
i
P
j
+ P
j
A
i
)x 0.
(b) Using level sets of the quadratic forms in (a), we show that the system (1) has uniformly
bounded trajectories.
(c) Finally, we show that for sufficiently small ǫ > 0 the same conclusion will hold if we
replace each system matrix A
i
with A
i
+ ǫI. This then establishes the uniform asymptotic
stability of the system (1).
Proof : It is immediate that if the system (1) is uniformly asymptotically stable (for arbitrary
switching), then each of the systems (2) is also.
Now suppose that for each i, j with 1 i < j k, the system (2) is uniformly asymptotically
stable. We can assume without loss of generality that for all a > 0 and 1 i < j k, the
matrix A
i
aA
j
is not zero. Let R
2
+
be the nonnegative orthant in R
2
. For any vector x R
2
,
Cone(A
1
x, . . . , A
k
x) =
1i<jk
Cone(A
i
x, A
j
x).
Moreover, as the switched system (2) is uniformly asymptotically stable for 1 i < j k
and the system matrices are Metzler, Cone(A
i
x, A
j
x) R
2
+
= {0} for all 1 i < j k and
nonzero x R
2
+
. Therefore Cone(A
1
x, . . . , A
k
x) R
2
+
= {0}.
For a nonzero vector x R
2
define arg(x), the argument of x in the usual way, viewing
x as a complex number. Let (l(x), u(x)), 1 l(x), u(x) k be a pair of integers such that
arg(A
l(x)
x) arg(A
i
x) arg(A
u(x)
x). Then clearly
Cone(A
1
x, . . . , A
k
x) = Cone(A
l(x)
, A
u(x)
).
For 1 i, j k define
D
(i,j)
= {y R
2
+
, y 6= 0 : Cone(A
1
y, . . . , A
k
y) = Cone(A
i
y, A
j
y)}
Here (i, j) is a pair of integers, not necessarily ordered and possibly equal and arg(A
i
y)
arg(A
j
y). It now follows that R
2
+
{0} =
1i,jk
D
(i,j)
. Note that D
(i,j)
{0} is a closed
cone, not necessarily convex and that if x D
(i,j)
and arg(A
i
x) < arg(A
m
x) < arg(A
j
x) for
m 6= i, j then x belongs to the interior of D
(i,j)
.
Consider the set Symp = {ˆa =: (a, 1 a)
T
: 0 a 1}, and define d
(i,j)
= Symp D
(i,j)
.
We shall write ˆa <
ˆ
b if and only if a < b. The sets d
(i,j)
are closed and their (finite) union
is equal to Symp. Moreover, the only way for x Symp not to lie in the interior of some
d
(i,j)
is if there exists b > 0, 1 l 6= m k such that A
l
x = bA
m
x. As we assumed that for
all a > 0, 1 i < j k the matrix A
i
aA
j
is not zero, it follows that there exists a finite
subset Si ng =: {0 ˆa
1
< ... < ˆa
q
1} such that all vectors in Symp Sing belong to the
interior of some d
(i,j)
.
It now follows that Symp can be partitioned into a finite family of closed intervals, each of
them contained in some d
(i,j)
. This in turn defines a partition of R
2
+
{0} into finitely many
closed cones/wedges Sec
j
, 1 j m, each of which is contained in some D
(L(j),U (j))
. We
shall label the rays which define this partition r
1
, . . . , r
m+1
where r
1
is the y-axis, r
m+1
is the
x-axis and the rays are enumerated in the clockwise direction.
Now, by assumption, the switched system ˙x = A(t)x A(t) {A
L(j)
, A
U(j)
} is uniformly
asymptotically stable for all 1 j m. Thus, it follows from Theorem 3.2 that, for 1
j m, there exist quadratic forms x
T
P
j
x, P
j
= P
T
j
> 0, such that A
T
L(j)
P
j
+ P
j
A
L(j)
< 0,
A
T
U(j)
P
j
+ P
j
A
U(j)
< 0. As Sec
j
D
(L(j),U (j))
, it follows that x
T
P
j
A
i
x 0 for all x Sec
j
and all i with 1 i k.
Now, choose a point T
1
= (0, y)
T
, y > 0 and consider the level curve of x
T
P
1
x which passes
through T
1
. This curve intersects the second ray r
2
at some point T
2
and the level curve of
x
T
P
2
x going through T
2
intersects the third ray r
3
at some point T
3
. We can continue this
process until we reach some point T
m+1
on the x-axis. This gives us a domain bounded by the
y-axis, the chain of ellipsoidal arcs defined above, and the x-axis. This domain is an invariant set
for (1), which implies that the trajectories of the system (1) are uniformly bounded. The same
conclusion will hold if we replace the system matrices A
1
, . . . , A
k
with {A
1
+ ǫI, .., A
k
+ ǫI}
for some small enough positive ǫ. This implies that the original system (1) is in fact uniformly
asymptotically stable and completes the proof of the theorem.
IV. HIGHER DIMENSIONAL SYSTEMS
Motivated by results such as those described in the previous section, a number of authors have
recently formulated the following conjecture.
Conjecture 1: Let A
1
, . . . , A
k
be a finite family of Hurwitz, Metzler matrices in R
n×n
. Then
the following statements are equivalent:
(i) All matrices in the convex hull, CO(A
1
, . . . , A
k
), are Hurwitz;
(ii) The switched linear system, ˙x = A(t)x A(t) {A
1
, . . . , A
k
}, is uniformly asymptoti-
cally stable.
In the remainder of this section, we shall present a counterexample to Conjecture 1, based on
arguments first developed by Gurvits in [16](which extended the results in [17], [18]).
Lemma 4.1: Let A
1
, . . . , A
k
be a finite family of matrices in R
n×n
. Assume that there exists a
proper polyhedral convex cone in R
n
such that exp(A
i
t)(Ω) for all t 0 and 1 i k.
Then there is some integer N n and a family of Metzler matrices A
M
1
, . . . A
M
k
in R
N×N
such that:
(i) All matrices in CO(A
1
, . . . , A
k
) are Hurwitz if and only if CO(A
M
1
, . . . , A
M
k
) consists
entirely of Hurwitz matrices;
(ii) The switched linear system ˙x = A(t)x, A(t) {A
1
, . . . , A
k
} is uniformly asymptotically
stable if and only if the positive switched linear system ˙x = A(t)x, A(t) {A
M
1
, . . . , A
M
k
}
is uniformly asymptotically stable.
Proof:
As is polyhedral, solid and pointed, we can assume without loss of generality that there exist
vectors z
1
, . . . , z
N
in R
n
, with N n, such that = Cone(z
1
, . . . , z
N
). Also, (see Theorem
8 in [19]) for 1 i k, exp(A
i
t)(Ω) for all t 0 if and only if there is some τ > 0
such that (I + τ A
i
)(Ω) .
Define a linear operator Φ : R
N
R
n
by Φ(e
i
) = z
i
for 1 i N where e
1
, . . . , e
N
is the
standard basis of R
N
. We shall now show how to construct Metzler matrices A
M
i
R
N×N
satisfying the requirements of the lemma.
First, we note the following readily verifiable facts:
(i) For any trajectory,x(t) =
P
1iN
α
i
(t)z
i
, α
i
(t) 0, in , lim
t→∞
x(t) = 0 if and only
if lim
t→∞
α
i
(t) = 0 for 1 i N;
(ii) For each i {1, . . . , k} and q {1, . . . , N}, we can write (non-uniquely)
A
i
(z
q
) =
N
X
p=1
a
pq
z
p
where a
pq
0 if p 6= q.
In this way, we can associate a Metzler matrix, A
M
i
= (a
pq
: 1 p, q N) in R
N×N
with
each of the system matrices A
i
in R
n×n
.
(iii) By construction, ΦA
M
i
= A
i
Φ and Φ(exp(A
M
i
t)) = (exp(A
i
t))Φ for all t 0. Hence,
A
M
i
is Hurwitz if and only if A
i
is Hurwitz for 1 i k.
From points (i) and (iii) above we can conclude that all matrices in the convex hull CO(A
1
, ..., A
k
)
are Hurwitz if and only if all matrices in the convex hull CO(A
M
1
, ..., A
M
k
) are Hurwitz. More-
over, the switched linear system ˙x = A(t)x,, A(t) {A
1
, . . . , A
k
} is uniformly asymptotically
stable if and only if the positive switched linear system ˙x = A(t)x, A(t) {A
M
1
, . . . , A
M
k
} is
uniformly asymptotically stable. This proves the lemma.
It follows from Lemma 4.1 that if Conjecture 1 was true, then the same statement would also
hold for switched linear systems having an invariant proper polyhedral convex cone.
Given a matrix A R
n×n
, define the linear operator
ˆ
A, on the space of n × n real symmetric
matrices, by
ˆ
A(X) = A
T
X + XA. It is a straightforward exercise to verify that if x
1
(t) and
x
2
(t) are solutions of the system ˙x = A
T
x with initial conditions x
1
(0) = x
1
, x
2
(0) = x
2
, then
x
1
(t)x
2
(t)
T
+ x
2
(t)x
1
(t)
T
is a solution of the linear system
˙
X =
ˆ
A(X), with initial conditions
x
1
x
T
2
+ x
2
x
T
1
. The following result follows easily by combining this observation with standard
facts about the existence and uniqueness of solutions to linear systems.
Lemma 4.2: Consider a family, {A
1
, . . . , A
k
}, of matrices in R
n×n
. Then:
(i) CO(A
1
, . . . , A
k
) consists entirely of Hurwitz stable matrices if and only if all of the
operators in CO(
ˆ
A
1
, . . . ,
ˆ
A
k
) are Hurwitz stable;
(ii) The cone, P SD(n), of positive semi-definite matrices in R
n×n
is an invariant cone for
the switched system
˙
X =
ˆ
A(t, X)
ˆ
A(t, X) {
ˆ
A
1
(X), . . . ,
ˆ
A
k
(X)};
(iii) The system
˙
X =
ˆ
A(t, X),
ˆ
A(t, X) {
ˆ
A
1
(X), . . . ,
ˆ
A
k
(X)} is uniformly asymptotically
stable if and only if the system ˙x = A(t)x, A(t) {A
1
, . . . , A
k
} is uniformly asymptoti-
cally stable.
The Counterexample
To begin, consider the following two matrices in R
2×2
A
1
=
0 1
1 0
, A
2
=
0 a
b 0
where a > b 0. Then, for some t
1
, t
2
> 0 the spectral radius ρ((exp(A
1
t
1
)(exp(A
2
t
2
)) > 1. In
fact, if we take a = 2, b = 1 then this is true with t
1
= 1, t
2
= 3/2. By continuity of eigenvalues,
if we choose ǫ > 0 sufficiently small, we can ensure that ρ((exp((A
1
ǫI)t
1
))(exp((A
2
ǫI)t
2
))) > 1. Hence, the switched linear system associated with the system matrices A
1
ǫI,
A
2
ǫI is unstable and moreover, all matrices in the convex hull CO({A
1
ǫI, A
2
ǫI}) are
Hurwitz.
The above remarks establish the existence of Hurwitz matrices B
1
, B
2
in R
2×2
such that all
matrices in CO(B
1
, B
2
) are Hurwitz and the switched linear system ˙x = A(t)x, A(t)
{B
1
, B
2
} is unstable.
Next, consider the Lyapunov operators,
ˆ
B
1
,
ˆ
B
2
on the symmetric 2× 2 real matrices. It follows
from Lemma 4.2 that CO(
ˆ
B
1
,
ˆ
B
2
) consists entirely of Hurwitz stable operators and that the
switched linear system associated with
ˆ
B
1
,
ˆ
B
2
is unstable, and leaves the proper (not polyhedral)
cone P SD(2) invariant. Formally, exp(
ˆ
B
i
t)(P SD(2)) P SD(2) for i = 1, 2, and all t 0.
From examining the power series expansion of exp(
ˆ
B
i
t), it follows that for any ǫ > 0, there
exists τ > 0 and two linear operators
i
, i = 1, 2 such that (τI +
ˆ
B
i
+
i
)(P SD(2)) P SD(2)
with ||
i
|| < ǫ for i = 1, 2. Combining this fact with standard results on the existence of
polyhedral approximations of arbitrary proper cones in finite dimensions (see Theorem 20.4
in [20]), we can conclude that for any ǫ > 0, there exists a proper polyhedral cone P H
ǫ
P SD(2), and two linear operators δ
i
, i = 1, 2 such that (τI +
ˆ
B
i
+
i
+ δ
i
)(P H
ǫ
) P H
ǫ
with ||
i
||, ||δ
i
|| < ǫ for i = 1, 2.
Recall that CO(
ˆ
B
1
,
ˆ
B
2
) consists entirely of Hurwitz-stable operators and that the switched
linear system associated with
ˆ
B
1
,
ˆ
B
2
is unstable. For ǫ > 0, define the linear operators B
i,ǫ
=
ˆ
A
i
+
i
+ δ
i
for i = 1, 2. By choosing ǫ > 0 sufficiently small, we can ensure that all operators
in CO(B
1
, B
2
) are Hurwitz-stable and that the switched linear system ˙x = A(t)x, A(t)
{B
1
, B
2
} is unstable. Moreover, this switched linear system leaves the proper, polyhedral
cone P H
ǫ
invariant.
Thus, the statement of Conjecture 1 is not true for switched linear systems with an invariant
proper, polyhedral cone and hence, it follows from Lemma 4.1 that Conjecture 1 itself is
also false. However, on examining the proof of Lemma 4.1, we see that the dimension of the
counterexample is determined by the number of generators of the polyhedral approximation
P H
ǫ
, and this may be very large.
V. M
ATRICES WITH CONSTANT DIAGONALS
In the recent paper [15], it was shown that for Metzler, Hurwitz matrices A
1
, . . . , A
k
in R
2×2
all
of whose diagonal entries are equal to -1 (A
j
(i, i) = 1 for i = 1, 2, j = 1, . . . , k), the Hurwitz-
stability of all matrices in CO(A
1
, . . . , A
k
) is equivalent to the uniform asymptotic stability of
the associated switched linear system. Motivated by this result, Mehmet Akar recently asked
if a counterexample to Conjecture 1 exists for the more restrictive system class satisfying:
A
j
(i, i) = 1 for 1 i n, 1 j k. We shall now show that such a counterexample
does indeed exist. Note that it is enough to provide a counterexample such that each matrix
A
j
, 1 j k has a constant diagonal in the sense that there are real numbers c
1
, . . . , c
k
such
that A
j
(i, i) = c
j
for i = 1, . . . , n, j = 1, . . . , k.
Given a Metzler matrix A in R
n×n
, let A(l, l) = m in
1in
A(i, i), and define the 2 × 2 blocks:
B
i,j
=
A(i, j) 0
0 A(i, j)
if i 6= j
B
i,i
=
A(l , l) A(i, i) A(l, l)
A(i, i) A(l, l) A(l, l)
.
Let Lift(A) R
2n×2n
be the block matrix whose (i, j) block is B
i,j
. Next define the linear
operator F R
n×2n
by F (x
1
, ..., x
2n
) = (y
1
, ..., y
n
),, where y
i
= x
2i1
+ x
2i
for i = 1, . . . , n.
It is straightforward to check that for any Metzler A R
n×n
, Lift(A) R
2n×2n
is Metzler,
has a constant diagonal, and F (Lift(A)) = AF . The next lemma now follows readily from
the previous equation.
Lemma 5.1: Consider a set of Metzler matrices A
1
, ..., A
k
in R
n×n
. Then the following state-
ments hold ;
(i) The convex hull CO(A
1
, ..., A
k
) is Hurwitz iff the convex hull CO(Lift(A
1
), ..., Lif t(A
k
))
is Hurwitz .
(ii) The switched system ˙x = A(t)x, A(t) {A
1
, ..., A
k
} is uniformly asymptotically sta-
ble iff the switched system ˙x = A(t)x, A(t) {Lif t(A
1
), ..., Lif t(A
k
)} is uniformly
asymptotically stable .
In the last section, we proved that there exists a positive integer n and a pair of Metzler,
Hurwitz matrices A
1
, A
2
in R
n×n
which violate Conjecture 1. Now consider one such pair
{A
1
, A
2
} and lift it to the pair {Lift(A
1
), Lift(A
2
)}. It now follows, using Lemma 5.1, that
the pair Lif t(A
1
), Lif t(A
2
) provides the required counterexample.
VI. T
HE JOINT LYAPUNOV EXPONENT
In this section, we shall make some simple observations concerning the computation of the
joint Lyapunov exponent, which is a continuous-time analogue of the joint spectral radius.
Definition 6.1: Let S be a compact subset of R
n×n
. The joint Lyapunov exponent of the
associated continuous-time switched linear system Σ
S
, JLE(S) is defined as
JLE(S) = inf{λ : a matrix norm ||.|| : ||exp(At)|| e
λt
for A S, t 0}
Notice that uniform asymptotic stability of the switched linear system Σ
S
is equivalent to the
inequality JLE(S) < 0. A relatively straightforward modification of the proof of Theorem 3.3
yields the following result.
Theorem 6.1: (i) Let S R
2×2
be a compact set of Metzler matrices. Then
JLE(S) = max
A,BS
JLE({A, B});
(ii) JLE(S) = max
MCO(S)
µ(M), where CO(S) is the convex hull of S and µ(M) is the
maximal real part of the eigenvalues of M;
(iii) Let S = {A
1
, ..., A
k
} be a finite set of 2 × 2 Metzler matrices . Then the joint Lyapunov
exponent JLE(S) can be computed in O(k
2
) arithmetic operations .
VII. C
ONCLUSIONS
In this paper we have presented a counterexample to a recent conjecture presented in [9],
and formulated independently by David Angeli, concerning the uniform asymptotic stability of
switched positive linear systems. In particular, we have shown that the stability of a positive
switched linear system is not in general equivalent to the Hurwitz stability of the convex hull
of its system matrices. Furthermore, we have also shown that this conjecture fails for the more
restrictive case where the system matrices are required to have constant diagonals. While this
conjecture is now known to be false, the lowest dimension for which it fails is still not known.
Thus it may be true for other, low-dimensional classes of positive systems. Also, it is not
known how large the set of counterexamples is, which means that the conjecture may be true
for significant sub-classes of switched positive linear systems.
Acknowledgements: The first author thanks David Angeli for numerous e-mail communications
on the subject of this paper. This work was partially supported by Science Foundation Ireland
(SFI) grant 03/RP1/I382, SFI grant 04/IN1/I478, the European Union funded research training
network Multi-Agent Control, HPRN-CT-1999-00107
1
and by the Enterprise Ireland grant
SC/2000/084/Y. Neither the European Union or Enterprise Ireland is responsible for any use
of data appearing in this publication.
R
EFERENCES
[1] D. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley, 1979.
1
This work is the sole responsibility of the authors and does not reflect the European Union’s opinion
[2] A. Berman and R. Plemmons, Non-negative matrices in the mathematical sciences. SIAM Classics in Applied
Mathematics, 1994.
[3] L. Farina and S. Rinaldi, Positive linear systems. Wiley Interscience Series, 2000.
[4] T. Haveliwala and S. Kamvar, “The second eigenvalue of the Google matrix, Stanford University, Tech. Rep., March
2003.
[5] R. Shorten, D. Leith, J. Foy, and R. Kilduff, “’towards an analysis and design framework for congestion control in
communication networks, in Proceedings of the 12th Yale workshop on adaptive and learning systems, 2003.
[6] A. Jadbabaie, J. Lin, and A. S. Morse, “Co-ordination of groups of mobile autonomous agents using nearest neighbour
rules, IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 988–1001, 2003.
[7] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems, IEEE Control Systems
Magazine, vol. 19, no. 5, pp. 59–70, 1999.
[8] W. P. Dayawansa and C. F. Martin, A converse Lyapunov theorem for a class of dynamical systems which undergo
switching, IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 751–760, 1999.
[9] O. Mason and R. Shorten, A conjecture on the existence of common quadratic Lyapunov functions for positive linear
systems, in Proceedings of American Control Conference, 2003.
[10] R. Horn and C. Johnson, Topics in matrix analysis. Cambridge University Press, 1991.
[11] R. N. Shorten and K. Narendra, “Necessary and sufficient conditions for the existence of a common quadratic Lyapunov
function for a finite number of stable second order linear time-invariant systems, International Journal of Adaptive
Control and Signal Processing, vol. 16, p. 709.
[12] R. N. Shorten and K. S. Narendra, “On common quadratic lyapunov functions for pairs of stable LTI systems whose
system matrices are in companion form, IEEE Transactions on Automatic Control, vol. 48, no. 4, pp. 618–621, 2003.
[13] R. Shorten and K. Narendra, A Sufficient Condition for the Existence of a Common Lyapunov Function for Two Second
Order Systems: Part 1. Center for Systems Science, Yale University, Tech. Rep., 1997.
[14] R. Shorten, F.
´
O Cairbre, and P. Curran, “On the dynamic instability of a class of switching systems, in Proceedings
of IFAC conference on Artificial Intelligence in Real Time Control, 2000.
[15] M. Akar et al, “Conditions on the stability of a class of second-order switched systems, IEEE Transactions on Automatic
Control, vol. 51, no. 2, pp. 338–340, 2006.
[16] L. Gurvits, “What is the finiteness conjecture for linear continuous time inclusions?” in Proceedings of the IEEE
Conference on Decision and Control, Maui, USA, 2003.
[17] ——, “Controllabilities and stabilities of switched systems (with applications to the quantum systems), in in Proc. of
MTNS-2002, 2002.
[18] ——, “Stability of discrete linear inclusions, Linear Algebra and its Applications, vol. 231, pp. 47–85, 1995.
[19] H. Schneider and M. Vidyasagar, “Cross-positive matrices, SIAM Journal on Numerical Analysis, vol. 7, no. 4, pp.
508–519, 1970.
[20] R. T. Rockafellar, Convex Analysis. Princeton University Press, 1970.
... It must be remarked that the nonparametric approach to the inverse problem, that we follow in this paper, is different from the one followed in identification or realization of nonnegative and linear systems, see [2] for a survey, and for instance [1], [12], [17], [23], [24], [26]. ...
... where the last identity follows from Equation (17). ...
Preprint
Full-text available
We pose the problem of approximating optimally a given nonnegative signal with the scalar autoconvolution of a nonnegative signal. The I-divergence is chosen as the optimality criterion being well suited to incorporate nonnegativity constraints. After proving the existence of an optimal approximation we derive an iterative descent algorithm of the alternating minimization type to find a minimizer. The algorithm is based on the lifting technique developed by Csisz\'ar and Tusn\'adi and exploits the optimality properties of the related minimization problems in the larger space. We study the asymptotic behavior of the iterative algorithm and prove, among other results, that its limit points are Kuhn-Tucker points of the original minimization problem. Numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm.
... Internally positive, or simply positive, dynamic systems are those whose state and output trajectory solutions have non-negative components for all time under any given nonnegative initial conditions and any given input with non-negative components for all time. See, for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and references therein. The concept of externally positive systems refers to the situation when all the components of the output trajectory are non-negative for all time under zero initial conditions and any given input with non-negative components for all time. ...
... The stability and stabilization properties have been also widely studied in the background literature. See, for instance [10][11][12][13][14][15][16], and some references there in, including the problems of finite-time stabilization, stabilization of switched positive systems and stability and stabilization of fractional positive models. However, it can be pointed out that the positivity of the solution is not an intrinsic property to a dynamic system, as it could be for instance, the eigenvalues of a time-invariant linear system, which are not dependent on the particular state space description. ...
Article
Full-text available
The property of external positivity of dynamic systems is commonly defined as the non-negativity of the output for all time under zero initial conditions and any given non-negative input for all time. This paper investigates the extension of that property for a structured class of initial conditions of a single-input single-output (SISO) linear dynamic system which can include, in general, certain negative initial conditions. The above class of initial conditions is characterized analytically based on the structure of the transfer function. The basic study is performed in the delay-free case, but extensions are then given for systems subject to a finite number of internal and external, in general incommensurate, point delays and for the closed-loop dynamic systems which incorporate a feedback compensator. The formulation relies on calculating the output based on the impulse responses by considering the relation of the mentioned sets of structured initial conditions with the zero-state response which allows to keep the non-negativity of the zero-input response and that of the total response provided the non-negativity for all time of the zero-state response.
... Assumption 3. The spectral radius ρ(f ) of a function f (·) has a lower bound that ρ(f ) ≥ C ≥ 0, where a spectral radius is the maximum modulus of eigenvalues (Gurvits, Shorten, and Mason 2007) ...
... Assumption 6. The spectral radius ρ(f ) of a function f (·) has a lower bound that ρ(f ) ≥ C ≥ 0, where a spectral radius is the maximum modulus of eigenvalues (Gurvits, Shorten, and Mason 2007), i.e., ρ(f ) = max 1≤i≤∞ √ λ i . ...
Preprint
Adversarial attacks by generating examples which are almost indistinguishable from natural examples, pose a serious threat to learning models. Defending against adversarial attacks is a critical element for a reliable learning system. Support vector machine (SVM) is a classical yet still important learning algorithm even in the current deep learning era. Although a wide range of researches have been done in recent years to improve the adversarial robustness of learning models, but most of them are limited to deep neural networks (DNNs) and the work for kernel SVM is still vacant. In this paper, we aim at kernel SVM and propose adv-SVM to improve its adversarial robustness via adversarial training, which has been demonstrated to be the most promising defense techniques. To the best of our knowledge, this is the first work that devotes to the fast and scalable adversarial training of kernel SVM. Specifically, we first build connection of perturbations of samples between original and kernel spaces, and then give a reduced and equivalent formulation of adversarial training of kernel SVM based on the connection. Next, doubly stochastic gradients (DSG) based on two unbiased stochastic approximations (i.e., one is on training points and another is on random features) are applied to update the solution of our objective function. Finally, we prove that our algorithm optimized by DSG converges to the optimal solution at the rate of O(1/t) under the constant and diminishing stepsizes. Comprehensive experimental results show that our adversarial training algorithm enjoys robustness against various attacks and meanwhile has the similar efficiency and scalability with classical DSG algorithm.
... The estimations of stability radii of switched linear systems and periodically switched linear systems were introduced in the paper (Nguyen Khoa Son et al., 2020, Do Duc Thuan et al., 2019. The stability radii of the positive linear system proposed by Son-Hinrichsen (see Nguyen Khoa Son et al., 1996) has a real stability radius equal to the complex stability radius, while the switched positive linear system interested by many authors and given conditions stable (see Blanchini et al., 2015, Ding et al., 2011, Gurvits et al., 2007, Le Van Ngoc et al., 2020, Sun., 2016, have studied robust stability in (Le Van Ngoc et al., 2020). However, with an inevitable limitation, the formula of stability radius has not been fully studied. ...
Article
The constrained stabilization problem of switched positive linear systems (SPLS) with bounded inputs and states is investigated via the set-theoretic framework of polyhedral copositive Lyapunov functions (PCLFs). It is shown that the existence of a common PCLF is proved to be necessary and sufficient for the stabilizability of an SPLS. As a primary contribution of this paper, we propose a PCLF-based approach for stabilization with a larger estimate of the domain of attraction for the constrained SPLSs. The analysis problems are converted into optimization problems whose constraints become linear matrix inequalities when a few variables are fixed. Finally, a turbofan engine model is employed to demonstrate the potential and effectiveness of the theoretical conclusions.
Article
A new design of a non-parametric adaptive approximate model based on Differential Neural Networks (DNNs) applied for a class of non-negative environmental systems with an uncertain mathematical model is the primary outcome of this study. The approximate model uses an extended state formulation that gathers the dynamics of the DNN and a state projector (pDNN). Implementing a non-differentiable projection operator ensures the positiveness of the identifier states. The extended form allows producing continuous dynamics for the projected model. The design of the learning laws for the weight adjustment of the continuous projected DNN considered the application of a controlled Lyapunov-like function. The stability analysis based on the proposed Lyapunov-like function leads to the characterization of the ultimate boundedness property for the identification error. Applying the Attractive Ellipsoid Method (AEM) yields to analyze the convergence quality of the designed approximate model. The solution to the specific optimization problem using the AEM with matrix inequalities constraints allows us to find the parameters of the considered DNN that minimizes the ultimate bound. The evaluation of two numerical examples confirmed the ability of the proposed pDNN to approximate the positive model in the presence of bounded noises and perturbations in the measured data. The first example corresponds to a catalytic ozonation system that can be used to decompose toxic and recalcitrant contaminants. The second one describes the bacteria growth in aerobic batch regime biodegrading simple organic matter mixture.
Article
In this paper, we deal with the finite-time stability of positive switched linear time-delay systems. By constructing a class of linear time-varying copositive Lyapunov functionals, we present new explicit criteria in terms of solvable linear inequalities for the finite-time stability of positive switched linear time-delay systems under arbitrary switching and average dwell-time switching. As an important application, we apply the method to finite-time stability of linear time-varying systems with time delay.
Article
In this paper, the global asymptotic stability (GAS) of continuous-time and discrete-time nonlinear impulsive switched positive systems (NISPS) are studied. For continuous-time and discrete-time NISPS, switching signals and impulse signals coexist. For both of these systems, using the multiple max-separable Lyapunov function method and average dwell-time (ADT) method, some sufficient conditions on GAS are given. Based on these, the GAS criteria are also given for continuous-time and discrete-time linear impulsive switched positive systems (LISPS). From our criteria, the stability of the systems can be judged directly from the characteristics of the system functions, switching signals and impulse signals of the systems. Finally, simulation examples verify the validity of the results.
Article
This article investigates the robust exponential stabilization of a class of switched positive systems with uncertainties by co-designing controllers for subsystems and dwell time switching strategy. The uncertainties refer to interval uncertainties. One of the distinguishing features is that none of the forced individual subsystems is assumed to be stabilized. The other feature is that the stabilization for the unforced switched system can not be solvable by designing dwell time switching signal. First, for switched positive systems without uncertainties, a type of multiple time-varying co-positive Lyapunov functions is used, and the computable sufficient conditions on the state feedback controller for each subsystem are derived in the framework of dwell time strategy. Moreover, the exponential stabilization problem is solved by confining the lower and upper bounds of the dwell time, restricting the upper bound of derivative of considered Lyapunov function of the active subsystem, and decreasing the Lyapunov function values at successive switching instants of the overall switched system. Then, the results are extended to the robust exponential stabilization case for switched positive systems with uncertainties, and sufficient conditions are given in the same framework of the dwell time for that of the forced switched systems by further designing state feedback controllers. Finally, illustration examples are given to illustrate the effectiveness of the proposed methods.
Article
This paper deals with stabilising uncertain Switched Positive Linear Systems (SPLSs) with a bumpless transfer controller. The contribution of this study is to improve switching controller robustness at the switching instances and also guarantee the L1–gain performance of the positive systems. Positive systems are widespread; most of the biological, economic systems, disease, etc., have nonnegative variables belonging to this class of systems. Positive Systems modelling encounters different types of uncertainties; therefore, stability analysis and stabilisation of uncertain SPLSs are crucial in analysing positive systems. The stability of the SPLSs can be guarantee by using multiple copositive Lyapunov functions with the Dwell Time (DT) strategy. The stability of the SPLS has been investigated under synchronous and asynchronous switching modes. The so-called asynchronous switching means that the switches between the candidate controllers and system modes are asynchronous. Also, the L1–gain performance of the system is derived. Finally, two illustrative examples are provided to demonstrate the proposed scheme’s effectiveness compared with the foremost existing solutions.
Conference Paper
Full-text available
The problem of shaping the kinetic and potential energy of a mechanical system by feedback is cast in a differential geometric framework. The nature of the set of solutions to the potential energy shaping problem is described. The kinetic energy shaping problem is posed in (1) an affine differential geometric framework and (2) a manner where the geometric integrability theory for partial differential equations can be applied.
Article
In this paper, necessary and sufficient conditions are derived for the existence of a common quadra-tic Lyapunov function for a finite number of stable second order linear time-invariant systems. Copyright © 2002 John Wiley & Sons, Ltd.
Book
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.
Article
There can be few books on mathematical mechanics as famous as this, a work that forms a comprehensive account of all the classical results of analytical dynamics.
Article
We present a simplied model of a network of TCP-like sources that compete for a shared bandwidth. We show that: (i) networks of communicating devices operating AIMD congestion control algorithms may be modelled as a positive linear system; (ii) that such networks possess a unique stationary point; and (iii) that this stationary point is globally exponentially stable. Using these results we establish conditions for the fair co-existence of trac in networks employing heterogeneous AIMD algorithms. A new protocol for operation over high-speed links is proposed and its dynamic properties discussed as a positive linear system.