IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2
for real-time scheduling as we can tune the threshold dynamics
according to the available resources. The second strategy
is inspired by [
19] where a time-triggered implementation
is studied. The idea here is to make the dynamics of the
clock variable used in [
19] state-dependent in order to adapt
transmissions to the system state. This strategy is compared
with [
28] on a van der Pol oscillator. We also show that the
approach is general enough to capture many of the event-
triggering laws available in the literature [
13], [16], [20], [28],
[34]. By doing so, we revisit and generalize these techniques
by relaxing the conditions on which they rely and by allowing
general holding functions. Indeed, the results of the paper
not only apply to zero-order-holds but to various holding
functions such as the model-based one proposed in [
15] for
linear systems. In addition, the proposed new techniques can
be used as starting points to address other event-based control
problems, such as the stabilization of systems with distributed
sensors and actuators ([
24]), or the coordination of cyber-
physical systems for instance ([21]). Note that a related study
has been independently proposed in [
26] where strategies
similar to those in [6] and [31] are investigated. Contrary to
[26], we ensure the existence of a uniform inter-execution
time for all our strategies, which is essential in practice
as explained above; furthermore we develop different event-
triggering policies and we revisit existing techniques.
The remainder of the paper is organized as follows.
Preliminaries are stated in Section II. Event-triggered control
systems are modeled as hybrid systems in Section III. The
analytical tools used to prove stability as well as the existence
of a minimum amount of time between any two transmissions
are stated in Section
IV. In Section V, we propose new
triggering rules and we show how several existing event-
triggering policies can be addressed with the proposed
approach. The event-triggering techniques are compared
on illustrative examples in Section
VI and conclusions are
proposed in Section VII. The proofs are given in the Appendix.
II. PRELIMINARIES
Let R := (−∞, ∞), R
≥0
:= [0, ∞), R
>0
:= (0, ∞),
Z
≥0
:= { 0, 1, 2, . . .}, Z
>0
:= {1, 2, . . .}. A function γ :
R
≥0
→ R
≥0
is of class K if it is continuous, zero at zero and
strictly increasing, and it is of class K
∞
if in addition it is
unbounded. The notation I denotes the identity mapping from
R
≥0
to R
≥0
. Let (x, y) ∈ R
n+m
, (x, y) stand for [x
T
, y
T
]
T
.
The distance of a vector x to a set A ⊂ R
n
is denoted by
|x|
A
:= inf{|x − y| : y ∈ A}.
The notation f
−1
(M) for a function f : R
n
→ R
m
and a set M ⊂ R
m
stands for {x : f(x) ∈ M}. When
f : R
n
→ R, f
−1
≤
(m) := {x : f(x) ≤ m} and
f
−1
≥
(m) := {x : f(x) ≥ m} for m ∈ R. The notation
| · | stands for the Euclidean norm. We use U
◦
(x; v) to
denote the generalized directional derivative of Clarke of a
locally Lipschitz function U at x in the direction v, i.e.
U
◦
(x; v) := lim sup
h→0
+
, y→x
(U(y + hv) − U(y))/h, which
reduces to the standard directional derivative h∇U (x), vi when
U is continuously differentiable; see [
4] for more detail. We
will use the following result which corresponds to Proposition
1.1 in [
18].
Lemma 1: Consider two continuously differentiable func-
tions U
1
: R
n
→ R and U
2
: R
n
→ R. Let A :=
{x : U
1
(x) > U
2
(x)}, B := {x : U
1
(x) < U
2
(x)} and
Γ := {x : U
1
(x) = U
2
(x)}. For any v ∈ R
n
, the function U :
x 7→ max{U
1
(x), U
2
(x)} satisfies U
◦
(x; v) = h∇U
1
(x), vi
for all x ∈ A, U
◦
(x; v) = h∇U
2
(x), vi for all x ∈ B, and
U
◦
(x; v) = max{h∇U
1
(x), vi , h∇U
2
(x), vi} for all x ∈ Γ.
The lemma below will be useful for lower bounding
Lyapunov functions. Its proof follows from the fact that
α(s
1
+ s
2
) ≤ α(2s
1
) + α(2s
2
) for any α ∈ K
∞
, s
1
, s
2
≥ 0.
Lemma 2: For any α
1
, α
2
∈ K
∞
, α
1
(s
1
) + α
2
(s
2
) ≥
α
(s
1
+ s
2
) for any s
1
, s
2
≥ 0 where α : s 7→
min{α
1
(s/2), α
2
(s/2)} ∈ K
∞
.
The following technical lemma is an extension of Lemma
4.3 in [12], which is convenient when designing Lyapunov
functions to investigate set stability.
Lemma 3: Let O ⊆ R
n
and A ⊂ O be non-empty and
compact sets and let V : O → R
≥0
be a continuous function
which is equal to zero only on A. There exist α
1
, α
2
∈ K
∞
such that
α
1
(|x|
A
) ≤ V (x) ≤ α
2
(|x|
A
) ∀x ∈ O.
(1)
When O is non-empty, closed and unbounded and V tends to
infinity as |x|
A
→ ∞, (
1) holds.
We recall the definition of the tangent cone to a set at a
given point and state a useful lemma.
Definition 1: [
8] The tangent cone to a set S ⊂ R
n
at a
point x ∈ R
n
, denoted T
S
(x), is the set of all vectors w ∈ R
n
for which there exist x
i
∈ S, τ
i
> 0 with x
i
→ x, τ
i
→ 0 as
i → ∞ such that w = lim
i→∞
(x
i
− x)/τ
i
.
Lemma 4: Let f : R
n
→ R
n
be continuous, C = {q :
V (x) ≥ W (x)} and D = {q : V (x) ≤ W (x)} for
some continuously differentiable functions V and W from
R
n
to R. For any x ∈ C ∩ D, f(x) ∈ T
C
(x) implies that
h∇V (x), f(x)i ≥ h∇W (x), f(x)i.
III. SYSTEM MODELS
We will write event-triggered control systems as hybrid
systems using the formalism of [
8], like in [5], [6], [26]. Thus,
we consider systems of the form
˙q = F (q)
for q ∈ C,
q
+
= G(q)
for q ∈ D, (2)
where q ∈ R
n
q
is the state and n
q
∈ Z
>0
. The sets C ⊂ R
n
q
and D ⊂ R
n
q
are closed and respectively denote the flow and
the jump sets. The vector fields F and G are assumed to be
continuous on C and D, respectively. The continuity of F and
G together with the closedness of the sets C and D ensure
that the system is ‘well-posed’, see Chapter 6 in [
8].
In event-triggered control, the transmissions occur whenever
a state-dependent criterion is satisfied. The hybrid model
(
2) is well-equipped to describe these systems. Indeed, a
transmission can be modeled as a jump of (2) that occurs
whenever the event-triggering condition is violated, which
corresponds to the fact that the state of system (2) enters in
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