1
On the Stability of Positive Linear Switched
Systems Under Arbitrary Switching Laws
Lior Fainshil, Michael Margaliot and Pavel Chigansky
Abstract—We consider n-dimensional positive linear switched
systems. A necessary condition for stability under arbitrary
switching is that every matrix in the convex hull of the matrices
defining the subsystems is Hurwitz. Several researchers conjec-
tured that for positive linear switched systems this condition is
also sufficient. Recently, Gurvits, Shorten, and Mason showed
that this conjecture is true for the case n = 2, but is not true
in general. Their results imply that there exists some minimal
integer n
p
such that the conjecture is true for all n < n
p
, but is
not true for n = n
p
. We show that n
p
= 3.
Index Terms—Switched systems, stability under arbitrary
switching law, positive linear systems, Metzler matrix.
I. INTRODUCTION
Consider the linear switched system:
˙
x(t) = A
σ(t)
x(t), x(0) = x
0
, (1)
where x(·) : R
+
→ R
n
, A
0
, A
1
∈ R
n×n
, and σ(·) : R
+
→
{0, 1} is a piecewise constant function of time, referred to as
the switching law. Roughly speaking, this models a system
that may switch between the two linear subsystems:
˙
x = A
0
x
and
˙
x = A
1
x.
Recall that a function α : [0, ∞) → [0, ∞) is said to
be of class K if it is continuous, strictly increasing, and
α(0) = 0. A function β : [0, ∞) × [0, ∞) → [0, ∞) is
said to be of class KL if β(·, t) is of class K for each fixed
t ≥ 0 and β(s, t) decreases to 0 as t → ∞ for each fixed
s ≥ 0. We say that (1) is globally uniformly asymptotically
stable (GUAS) if there exists a class KL function β such that
for any initial condition x(0) = x
0
and any switching law σ
the corresponding solution of (1) satisfies |x(t)| ≤ β(|x
0
|, t)
for all t ≥ 0. This implies in particular that lim
t→∞
x(t) = 0.
Denote A(k) := kA
0
+(1−k)A
1
. By the classic Lie-Trotter
product formula [1, Chapter 2]:
lim
n→∞
exp(kA
0
/n) exp((1 − k)A
1
/n)
n
= exp(A(k))
for any k ∈ [0, 1]. It follows from this that if A(k) is not
Hurwitz for some k ∈ [0, 1], then (1) is not GUAS. Thus,
from hereon we assume the following.
Assumption 1 Every matrix in
co{A
0
, A
1
} := {A(k) : k ∈ [0, 1]}
is Hurwitz.
Assumption 1 is a necessary (but not sufficient) condition for
GUAS of (1).
Recently, the problem of establishing conditions that guar-
antee GUAS of (1) has attracted considerable interest [2], [3],
LF (liorfainshil@gmail.com) and MM (michaelm@eng.tau.ac.il) are with
the School of EE-Systems, Tel Aviv University, Tel Aviv, Israel 69978;
PC (pchiga@mscc.huji.ac.il) is with Department of Statistics, The Hebrew
University, Jerusalem, Israel 91905.
[4], [5], [6], [7]. A natural idea, first suggested by Pyatnitskiy
and his colleagues [8], is to try and characterize the “most
unstable” switching law. If the corresponding trajectory is
asymptotically stable, then so are all the other trajectories.
Thus, the problem can be reduced to analyzing the behavior of
this single trajectory. The “most unstable” switching law can
be characterized using variational principles (see the survey
paper [9]).
Recall that a linear system
˙
x = Ax, with A ∈ R
n×n
, is
called positive if R
n
+
:= {x ∈ R
n
|x
i
≥ 0, i = 1, . . . , n} is an
invariant set of the dynamics, that is, if x(0) ∈ R
n
+
implies
that x(t) ∈ R
n
+
for all t ≥ 0. A necessary and sufficient
condition for this is that A is a Metzler matrix, that is, all the
non-diagonal elements of A are non-negative. Positive linear
systems play an important role in system and control theory
because in many physical systems the state-variables represent
quantities that can never attain negative values (see e.g. [10]).
If both A
0
and A
1
are Metzler and x
0
∈ R
n
+
, then we refer
to (1) as a positive linear switched system (PLS). PLSs were
used for modeling communication systems [11] and formation
flying [12] (see also [7]).
Mason and Shorten [13], and independently David Angeli,
posed the following conjecture.
Conjecture 1 A PLS that satisfies Assumption 1 is GUAS.
Recently, Gurvits, Shorten, and Mason [14] proved that this
conjecture is true for the case n = 2 (even when the number
of subsystems is arbitrary). This result was also proved using
the variational approach in [15].
Gurvits, Shorten, and Mason [14] also showed that Con-
jecture 1 is in general false. As noted in [14], this naturally
raises the following question. What is the minimal integer n
p
for which there exists a PLS that satisfies Assumption 1 but
is not GUAS?
In this paper, we solve this problem by presenting a specific
three-dimensional system for which Conjecture 1 is false.
Since n
p
> 2, this proves that n
p
= 3.
The remainder of this note is organized as follows. The next
section provides a brief review of the results in [14]. Section III
presents a specific three-dimensional counterexample to Con-
jecture 1. The final section concludes.
II. THE GURVITS ET AL. COUNTEREXAMPLE
In this section, we explain the construction in [14]. The first
step is an argument that allows transforming a linear switched
system with an invariant cone into a PLS.
Recall that a set Ω ∈ R
n
is called a closed convex cone if
for any y
1
, y
2
∈ Ω and any c
1
, c
2
≥ 0, c
1
y
1
+ c
2
y
2
∈ Ω.
The cone is said to be proper if its interior is non-empty,
and Ω ∩ (−Ω) = {0}.
Suppose that there exists a proper polyhedral convex cone
Ω =
n
k
X
i=1
c
i
z
i
: c
i
≥ 0
o
,
with z
i
∈ R
n
, that is an invariant set of (1). In other words,
if x
0
∈ Ω then
exp(A
i
t)x
0
∈ Ω, t ≥ 0, i = 0, 1.
2
It then follows [16] that there exists τ > 0 such that for
any x
0
∈ Ω:
(I + τA
i
)x
0
∈ Ω, i = 0, 1.
This implies in particular that
(I + τA
i
)z
j
∈ Ω, i = 0, 1.
By the definition of Ω, there exist c
i
pj
≥ 0 such that
(I + τA
i
)z
j
=
k
X
p=1
c
i
pj
z
p
,
i.e.
A
i
z
j
=
1
τ
(
k
X
p=1
c
i
pj
z
p
− z
j
).
In other words,
A
i
z
j
=
k
X
p=1
a
i
pj
z
p
,
with a
i
pj
≥ 0 if p 6= j. Thus, the matrices B
0
= {a
0
pj
}
k
p,j=1
and B
1
:= {a
1
pj
}
k
p,j=1
are Metzler. It can be shown that
every matrix in co{A
0
, A
1
} is Hurwitz if and only if ev-
ery matrix in co{B
0
, B
1
} is Hurwitz. Furthermore, the n-
dimensional switched system (1) is GUAS if and only if the k-
dimensional PLS
˙
x(t) = B
σ(t)
x(t) is GUAS.
Now fix two matrices A
0
and A
1
such that every matrix
in co{A
0
, A
1
} is Hurwitz, yet the switched system (1) is not
GUAS. Define
Z(t) := x(t)x
0
(t). (2)
Then
˙
Z = A
σ
Z + ZA
0
σ
, (3)
with Z(0) = x(0)x
0
(0). Since the original system is not
GUAS, it follows from (2) that so is the switched system (3)
(see also [17] for some related considerations). The system (3)
evolves on the cone of symmetric non-negative definite ma-
trices. It is possible to approximate this cone, to arbitrary
accuracy, using a proper polyhedral cone. It follows from
the arguments above that this yields a PLS that satisfies
Assumption 1 yet is not GUAS. This elegant construction
proves that Conjecture 1 is, in general, false.
Note that the dimension of the resulting PLS is k, where k is
the number of vertices defining the approximating polyhedral
cone. Intuitively, this suggests that constructing an explicit
counterexample using this scheme is non trivial and, further-
more, that the resulting PLS will have a very large dimension.
III. A THREE-DIMENSIONAL COUNTEREXAMPLE
From here on we consider the switched system (1) with n =
3,
A
0
=
−1 0 0
10 −1 0
0 0 −10
, A
1
=
−10 0 10
0 −10 0
0 10 −1
.
The eigenvalues of A
0
(A
1
) are {−1, −1, −10}
({−1, −10, −10}), so both matrices are Hurwitz. Also
both A
0
and A
1
are Metzler, so clearly every matrix
in co{A
0
, A
1
} is Metzler. We claim that this system provides
a counterexample to Conjecture 1.
We now show that every matrix in co{A
0
, A
1
} is Hurwitz.
This can be done directly by using the Routh-Hurwitz crite-
rion. We use a simpler approach, suggested to us by Leonid
Gurvits and also by one of the anonymous reviewers, and
based on an idea from [18, Chapter 2, Section 5, Exercise 5].
Fix some arbitrary k ∈ [0, 1]. We begin by showing
that A(k) is non-singular. If this is not true, then clearly k ∈
(0, 1) and there exists v ∈ R
n
\ {0} such that A(k)v = 0.
Using the definition of A(k) yields
A
−1
0
A
1
v =
k
k − 1
v,
i.e. A
−1
0
A
1
has a real and negative eigenvalue. However, the
characteristic polynomial of A
−1
0
A
1
is
p(s) = s
3
− 20.1s
2
+ 2s − 10,
and clearly it has no real negative roots. Thus we conclude
that A(k) is non-singular for all k ∈ [0, 1].
Since A(k) = kA
0
+ (1 − k)A
0
is a Metzler matrix,
exp(A(k)) has nonnegative entries. By the Perron-Frobenius
theorem (see e.g. [18]), the eigenvalue of exp(A(k)) with the
maximal absolute value is real and nonnegative. Consequently,
if λ(k) is the eigenvalue of A(k) with the largest real part,
then λ(k) is real. As both A
0
and A
1
are Hurwitz, λ(0) < 0
and λ(1) < 0. The function l 7→ λ(l) is continuous, and we
already know that A(k) is non-singular for all k ∈ (0, 1).
We conclude that λ(l) < 0 for all l ∈ [0, 1], so every
matrix in co{A
0
, A
1
} is Hurwitz. Thus our PLS satisfies
Assumption 1.
Finally, a calculation reveals that the matrix
exp(A
0
) exp(A
1
) has one real eigenvalue η ≈ 1.66879.
The corresponding eigenvector ζ satisfies ζ ∈ R
3
+
(this is
again a consequence of the Perron-Frobenius theorem). Thus,
for any integer j:
exp(A
0
) exp(A
1
)
j
ζ = η
j
ζ.
Since η > 1, this implies that the switched system admits
a trajectory x(t), with x(0) = ζ, corresponding to a peri-
odic switching law, such that x(t) is unbounded. Hence, the
switched system is clearly not GUAS.
Fig. 1 depicts the trajectory of the switched system ema-
nating from ζ, following
˙
x = A
1
x for t = 1 seconds (dashed
line), and then
˙
x = A
0
x for t = 1 seconds (solid line). The
final point is then exp(A
0
) exp(A
1
)ζ = ηζ.
Fig. 2 depicts the trajectory of the linear system
˙
x = A
1
x
emanating from x(0) = ζ. The trajectory is attracted to the
line c(1, 0, 0.9), c ∈ R, which is an eigenvector of A
1
, and
then converges, of course, to the origin.
Fig. 3 depicts the trajectory of the linear system
˙
x = A
0
x
emanating from µ := exp(A
1
)ζ. The trajectory is attracted
to the line c(0, 1, 0), c ∈ R, which is an eigenvector of A
0
,
and then converges, of course, to the origin. During the
transient, the trajectory moves away from the origin. Indeed,
3
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
x
•
ζ
•
ηζ
y
z
Fig. 1. A trajectory of the switched system with x(0) = ζ.
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
x
•
y
z
Fig. 2. Trajectory of
˙
x = A
1
x with x(0) = ζ.
a calculation shows that
| exp(A
0
)µ|
|µ|
≈ 2.747.
Summarizing, by properly combining the transient behav-
iors of the two linear subsystems, it is possible to obtain a
diverging trajectory of the switched system. This is of course
the standard mechanism yielding instability in a switched
system that satisfies Assumption 1. The fact that the system
is a PLS does not preclude the possibility of this mechanism
here. The interesting question is then why Conjecture 1 does
hold for the case n = 2. As noted above, two answers to this
question can be found in [14], [15].
IV. CONCLUSIONS
We considered an n-dimensional PLS satisfying the fol-
lowing condition: every matrix in co{A
0
, A
1
} is Hurwitz.
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
•
x
y
z
Fig. 3. Trajectory of
˙
x = A
0
x with x(0) = exp(A
1
)ζ.
This condition is necessary for GUAS, and several researchers
conjectured that for PLSs it is also sufficient for GUAS.
Recently, Gurvits, Shorten, and Mason showed that this
conjecture holds for n = 2, but is not true in general. In
other words, there exists some finite integer n
p
such that the
conjecture is true for all n < n
p
, but is not true for n = n
p
.
In this paper, we presented a three-dimensional counterex-
ample to this conjecture. Since n
p
> 2, this proves that n
p
=
3. This result provides another demonstration of the fact that
linear switched systems, even in relatively low dimensions,
may exhibit surprising and non trivial behavior. It also suggests
that new behaviors may emerge when we move from the planar
case n = 2 to the three-dimensional case n = 3. This is
perhaps not surprising, as linear switched systems behave, in
many respects, as nonlinear systems.
ACKNOWLEDGMENTS
We are grateful to Michael Branicky, Leonid Gurvits, and
Oliver Mason for helpful comments. We thank the anonymous
reviewers for their useful and constructive suggestions.
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