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Lyapunov event-triggered control: a new
event strategy based on the control
N. Marchand ∗,∗∗ J.J. Martinez ∗,∗∗ S. Durand ∗∗∗
J.F. Guerrero-Castellanos ∗∗∗∗
∗CNRS, GIPSA-lab, 11 rue des math´ematiques, 38402 Grenoble,
France
∗∗ Univ. Grenoble Alpes, GIPSA-lab, F-38402 Grenoble, France
nicolas.marchand@gipsa-lab.fr
∗∗∗ UMI LAFMIA, CNRS - CINVESTAV, Mexico D.F., Mexico,
sylvain@durandchamontin.fr
∗∗∗∗ Autonomous University of Puebla (BUAP), Faculty of Electronics,
Puebla, Mexico, fguerrero@ece.buap.mx
Abstract: Event-triggered control is a sampling strategy that updates the control value only
when some events occur. An event is usually generated by an event-function that indicates
if the control signal must be updated or not. If one excepts self-triggered implementation,
event-triggered control requires the evaluation of the event function at each time instant.
Unfortunately, in the literature of nonlinear system event-based control, computing the event
function is more resource consuming than computing the control itself. Moreover, it requires the
knowledge of a Lyapunov function that is not necessarily available. The purpose of this paper
is to propose for affine nonlinear systems a new strategy for the choice of the event function
that only requires the computation of the control. This reduces the complexity of computing
the event and avoids to know the Lyapunov function.
Keywords: Nonlinear control, event-triggered control, Lyapunov, general formula for control
1. INTRODUCTION
In the past years, a new approach for sampling controlled
systems appeared. It is usually called event-based con-
trol and consists in updating the control upon an event
depending on the system itself, that is the state or the
output, instead of depending upon the time as for classical
periodic sampling. The idea was first developed by the
seminal works of ˚
Astr¨om and Bernhardsson [2002] and
˚
Arz´en [1999]. In the first contribution, it was shown that
with less computations, event-based control can achieve
the same performance when in the second, the first event
driven PID was proposed. The development of event-based
control strategies started with level-crossing approaches
like in [Velasco et al., 2009, 2003] or [Durand and Marc-
hand, 2009, Durand et al., 2011]. Event-based control now
usually relies on a triggering algorithm as represented in
figure 1.
System Sensor
Event-triggered
algorithm
Control
dx
dt =f(x, u)x(t)
u(t)
T(x)
Network Network
Fig. 1. Event-based control [Seuret et al., 2013]
As in [Marchand et al., 2013], the triggering algorithm
usually takes the form of an event function e:X ×
X → Rthat indicates if one needs (e≤0) or not
(e > 0) to update the control value. Xrepresents the
state space. The event function etakes the current state
xas input and a memory mof xlast time ebecame
negative (defined later on). In the Periodic Event-triggered
Control scheme (PEC), a periodic sampling is given and
the event function indicates the control must be updated
at the next sampling instant [Heemels et al., 2013, 2011]
whereas in the Continuous Event-triggered Control scheme
(CEC), the control function is updated instantaneously
after the zeroing of the event function [Anta and Tabuada,
2010, Tabuada, 2007, Anta and Tabuada, 2009, Marchand
et al., 2013]. If the first approach has the advantage to
guaranty a minimal sampling period (the control can not
be updated more often than the a priori given period) it is
not very suitable for general nonlinear systems of the form
˙x=f(x, u) where the application of the classical periodic
sampling is known to be delicate. The PEC strategy relies
either on the existence of a Lipschitz stabilizing control
law and an ISS-CLF, that is a Control Lyapunov Function
such that ∂V
∂x f(x, k(x+ε)) ≤ −α(||x||) + β(||ε||) where
αand βare two functions of class K∞and ε=m−x
denotes the measurement error. The control is updated
each time e(x, m) := σα(||x||)−β(||ε||)≤0 ensuring that
way the strict decrease of the CLF with 0 < σ < 1. Or,
as in [Marchand et al., 2013] for affine nonlinear systems
of the form ˙x=fa(x) + ga(x)u, the event function is
defined relative to the derivative of the CLF as e(x, m) :=
σ∂V
∂x (fa(x) + ga(x)k(x)) −∂ V
∂x (fa(x) + ga(x)k(m)) where
here again 0 ≤σ < 1 is a tuning parameter.
9th IFAC Symposium on Nonlinear Control Systems
The International Federation of Automatic Control
September 4-6, 2013. Toulouse, France
978-3-902823-47-2/2013 © IFAC 324 10.3182/20130904-3-FR-2041.00129
The triggering algorithm defined by the event function e
is associated to a static feedback law k:X → U that
derives from a continuous time analysis where Udenotes
the control space.
In all the existing continuous event-triggered control
strategies developed for nonlinear systems, computing the
event requires to compute a function depending upon the
current state value, upon the value of the state at the
previous event instant and upon the control value. If the
data sent over the network is lower with this approach, the
complexity of the computations is much higher than with a
classical continuous time implementation. For instance, for
linear system with a quadratic CLF, eis a quadratic poly-
nomial in xand m. Moreover, in all the event strategies
mentioned above, the knowledge of the Lyapunov function
is required to compute the event function.
The aim of this paper is to give some answers to the two
above limitations. In a first step, we assume that a control
stabilizing the system exists. Based on this assumption, an
event function is proposed. It only requires the knowledge
of the control to guaranty stability and non zero inter-
execution time. In a second step, we propose a new event
function based on the sole control for the general event-
triggered feedback formula proposed in [Marchand et al.,
2013] and derived from Sontag’s general formula (see
[Clarke et al., 1997] for its more general formulation). To
our knowledge, event functions based on the control has
never been proposed for nonlinear systems in the literature
before.
2. PRELIMINARIES
In this paper, we focus on affine in the control dynamical
systems defined by:
˙x=f(x) + g(x)u(1)
where x∈ X ⊂ Rn,u∈ U ⊂ Rp, and fand gare smooth
functions with fvanishing at the origin. For sake of
simplicity, we only consider in this paper null stabilization
with initial time instant t0= 0. If system (1) admits an
asymptotic stabilizing feedback k:X → U then there
exists a Control Lyapunov Function V:X → R, that is a
smooth function, positive definite and such that :
˙
V=∂V
∂x [f(x) + g(x)k(x)] (2)
It is worth noting that if kis assumed to be smooth, then
Vis known to exist and to be as smooth as k. In the
present paper, only the smoothness of Vis required which
is a little bit less restrictive than the one of k.
In the following, let us use the following notations. Let
a(x) := ∂V
∂x f(x)
and
b(x) := ∂V
∂x g(x)
Let us define the sets:
Sb:= {x∈ X| ||b(x)|| 6= 0}
Sa:= {x∈ X| a(x)≥0}
where ||·|| stand for the 2-norm. Let ||·||1and ||·||∞denote
respectively the 2-norm and the infinite norm. For any
set, let the superscript “∗” be added when considering
the set without {0}. Let videnote for any vector vits
ith component. Let B(δ) denote the ball or radius δ > 0
centred at the origin.
We recall here the definition of semi-uniform Minimum
Sampling Interval (MSI) event-triggered control:
Definition 1. [Marchand et al., 2013] An event-triggered
feedback (k, e) is said to be semi-uniformly MSI iff for all
δ > 0, and all x0in the ball of radius δcentered at the
origin B(δ), the inter-execution times, that is the duration
between two successive events, can be below bounded by
some τ(δ)>0.
It is known that a nonlinear system of the form (1) with
a semi-uniformly MSI event-based feedback (k, e), the
solution of (1) starting in x0∈ X at t= 0 is defined
for all positive tas the solution of the differential system:
˙x=f(x) + g(x)k(m) (3)
m=xif e(x, m)≤0, x6= 0
˙m= 0 elsewhere (4)
with: x(0) = x0and m(0) = x(0) (5)
3. AN EVENT FUNCTION BASED ON THE
CONTROL
In this section, we assume first that a stabilizing feedback
exists for system (1):
Hypothesis 2. There exists a feedback k:X → U and
a smooth Lyapunov function V:X → R+such that
a(x) + b(x)k(x)<0 for all x6= 0
This assumption implies in particularly that Sa⊂ Sb. In
addition, we assume:
Hypothesis 3. There exists a strictly positive function ω:
X → R+such that the feedback can be written as k(x) =
−bT(x)ω(x).
Hypothesis 3 is directly linked to the existence of a smooth
feedback. Indeed, coupled to hypothesis 2, it guaranties
that:
a(x)−b(x)bT(x)ω(x)<0 (6)
which is exactly inequality (11) of [Marchand et al., 2013]
that is known to ensure a smooth feedback using Sontag’s
general formula. Since ωis assumed to be positive, it also
gives for all ρ≥1:
a(x)−ρb(x)bT(x)ω(x)<0 (7)
With the above assumptions, our first result is the follow-
ing:
Theorem 4. Assume Hypotheses 2 and 3 are satisfied for
some feedback law k, then the event-triggered feedback
(κ, e) defined by (8)-(9) is semi-uniformly MSI for any
ρ≥1.
κ(x) := (1 + √2)k(x) (8)
e(x, m) := ||k(x)||2−ρp(1 + √2) ||k(m)−k(x)||2
∞on Sa
1 otherwise (9)
Recall that pdenotes the control vector size. Note
also that, right after an event, e(x, m) = ||k(x)||2=
IFAC NOLCOS 2013
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325
ω(x)2||b(x)||2>0 for all x∈ Sa. This follows from
inequality (6). Note also that (9) does not require the
knowledge of the Lyapunov function.
Proof of Theorem 4 :
The proof is divided into two steps. First, we prove that
the CLF Vis strictly decreasing except eventually at the
event instants. In a second step, we prove that the event-
based feedback (k, e) is semi-uniformly MSI, that is for
any ball of initial conditions, there exists a minimal inter-
execution time depending only upon the ball radius. Note
that the proof can be restricted to S∗
bsince outside this
set, a(x)<0 for all non zero x.
Step 1: Vis decreasing - Let us take β:= 1 + √2. Then
one can notice that βis the solution of
β2−2β−1 = 0 (10)
Recall also that pis defined as the size of the control vector.
Let us assume that e(x, m)>0, that is for all i:
||k(x)||2> ρpβ [ki(m)−ki(x)]2≥pβ [ki(m)−ki(x)]2(11)
because ρ≥1. Consider first the right hand side of this
equation and rewrite it using hypothesis 3 as follows for
any β:
β2b(x)bT(x)ω(x)2−(β2−1)b(x)bT(x)ω(x)2
=b(x)bT(x)ω(x)2=||k(x)||2
Since Vis a CLF for k, for all non zero xwe have a(x) +
b(x)k(x)<0 which can be written as:
a(x)−b(x)bT(x)ω(x)<0
Since ω(x) is assumed strictly positive for all x, it follows
for any β≥1:
−(β2−1)b(x)bT(x)ω2(x)≤ −(β2−1)ω(x)a(x)
which finally gives:
β2b(x)bT(x)ω(x)2−(β2−1)ω(x)a(x)> pβ [ki(m)−ki(x)]2
Multiplying left and right hand side by bi(x)2and sum-
ming for all i, it gives using (10):
b(x)b(x)Tβ2b(x)b(x)Tω(x)2−2βa(x)ω(x)> pβ X
i∈{1,...,p}
bi(x)2[ki(m)−ki(x)]2
First looking at the left hand side of this last inequality,
it follows:
a(x)2+b(x)b(x)Tβ2b(x)b(x)Tω(x)2−2βa(x)ω(x)
=a(x)−βb(x)b(x)Tω(x)2
≥b(x)b(x)Tβ2b(x)b(x)Tω(x)2−2βa(x)ω(x)
The right hand side of the inequality gives using the
inequality between the 1−norm and the 2−norm:
pβ X
i∈{1,...,p}
bi(x)2[ki(m)−ki(x)]2
≥β
X
i∈{1,...,p}|bi(x)||ki(m)−ki(x)|
2
≥β[b(x) (k(m)−k(x))]2
Combining this two last inequalities gives:
[a(x) + βb(x)k(x)]2> β [b(x) (k(m)−k(x))]2
Applying κ(·) := βk(·) as control, this last inequality
becomes
|a(x) + b(x)κ(x)|>|b(x) (κ(m)−κ(x))|
This ensures that the CLF Vis strictly decreasing along
the event-triggered trajectory of the system:
˙
V=a(x) + b(x)κ(m)
=a(x) + b(x)κ(x) + b(x) (κ(m)−κ(x)) <0
Step 2: (k, e)is uniformly MSI - The proof is identical
to [Marchand et al., 2013]. It relies on the smoothness of
k. We will not detail the proof here but we will just give
the spirit of it. The proof consists in showing that when
an event occurs, the next event can not happen instanta-
neously right after due to continuity considerations. This
follows from the fact that just after an event, that is just
after the control update, the event function can be lower
bounded by some bound as follows:
e(x, m) = ||k(x)||2≥sup
ζ∈Sa||k(ζ)||2
| {z }
=: χ(ϑm)>0
Considering ksmooth, it will necessary take a non zero
time τ(ϑm) for eto vanish again. ♣
4. THE GENERAL FORMULA FOR FEEDBACK
STABILIZATION WITH AN EVENT FUNCTION
BASED ON THE CONTROL
We focus now on the general formula for event-based
control given in [Marchand et al., 2013] that ensures the
existence of an event-based feedback having the semi-
uniform MSI property that we recall here:
Theorem 5. [Marchand et al., 2013] If there exists a CLF
Vfor system (1), then the event-based feedback (k, e)
defined below is semi-uniformly MSI, smooth on X\{0},
and such that:
∂V
∂x f(x) + ∂V
∂x g(x)k(m)<0, x ∈ X\ {0}(12)
where mis defined in (4) and
ki(x) := −bi(x)δi(x)γ(x) (13)
e(x, m) := −a(x)−b(x)k(m)
−σqa(x)2+θ(x)b(x)∆(x)b(x)T(14)
where
•x→∆(x) := diag(δ1(x), δ2(x), . . . , δp(x)) is a
smooth function of X∗to Rp×pstrictly positive on
S∗
b
•x→θ(x) is a smooth function of Xto Rpositive
definite on Sb
•σis a control parameter in [0,1[,
•γ:X → Ris defined by:
γ(x) :=
a(x) + pa(x)2+θ(x)b(x)∆(x)b(x)T
b(x)∆(x)b(x)Tif x∈ Sb
0 if x /∈ Sb
(15)
The main problem of the control law (13-14) resides 1) in
the complexity that requires the evaluation of the event
function, 2) in the knowledge of the Lyapunov function
that is required to compute (14). As before, the aim of the
following theorem is therefore to propose a simpler event
function for the general formula.
Theorem 6. If there exists a CLF Vfor system (1), then
the event-based feedback (k, e) is semi-uniformly MSI,
IFAC NOLCOS 2013
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326
smooth on X\{0}, and such that (12) is satisfied with
kas in (13) and edefined by:
e(x, m) := (inf
i∈{1,...,p}θ(x)δi(x)−p[ki(m)−ki(x)]2on Sa
1 otherwise (16)
Note that each time an event occurs (that is, each time
evanishes), right after the update of the control value,
the event function becomes strictly positive: e(x, m) =
θ(x)δi(x)>0.
Proof of Theorem 6 :
The proof is divided in two steps as for Theorem 4. In the
first one, we prove that the CLF Vis strictly decreasing
and in a second step, we establish that the event-based
feedback (k, e) is semi-uniformly MSI.
Step 1: Vis decreasing - If e(x, m)>0, then we have for
all i∈ {1, . . . , p}:
θ(x)δi(x)> p [ki(m)−ki(x)]2
and therefore
bi(x)2θ(x)δi(x)≥pbi(x)2[ki(m)−ki(x)]2
Note that this last inequality is strict everywhere b(x)
does not vanish, that is on Sb:= {x∈ X| ||b(x)|| 6= 0}.
Summing this last inequality for all iand writing this sum
in a matrix form, it gives:
θ(x)b(x)∆(x)b(x)T≥pX
i∈{1,...,p}
b2
i(x) [ki(m)−ki(x)]2
Since for all non zero x∈ X\S, one knows that a(x)<0,
it follows:
a2(x) + θ(x)b(x)∆(x)b(x)T> p X
i∈{1,...,p}
b2
i(x) [ki(m)−ki(x)]2
It gives using the inequality between the 2-norm and the
1-norm:
a2(x) + θ(x)b(x)∆(x)b(x)T>
X
i∈{1,...,p}
bi(x) [ki(m)−ki(x)]
2
This last inequality finally gives:
−qa(x)2+θ(x)b(x)∆(x)b(x)T+b(x) (k(m)−k(x)) <0
Since, along the trajectory of the system, one has:
˙
V=a(x) + b(x)k(m)
=a(x) + b(x)k(x) + b(x) (k(m)−k(x))
=−qa(x)2+θ(x)b(x)∆(x)b(x)T+b(x) (k(m)−k(x))
it follows that, as long as e(x, m)>0, the time derivative
of the CLF Vremains strictly negative.
Each time an event occurs (that is, each time evanishes),
right after the update of the control value, e(x, m) =
θ(x)δi(x)>0. Therefore, the CLF can vanish only “punc-
tually” at the instants when evanishes. At all other in-
stants, the CLF is strictly decreasing.
Step 2: (k, e)is uniformly MSI - The proof is identical to
the one in [Marchand et al., 2013]. ♣
5. THE LINEAR CASE
Consider a linear system given by:
˙x=Ax +Bu (17)
and P, the solution of the Ricatti equation:
P A +ATP−2εP B BTP=−Q(18)
where Qis a symmetric definite positive. Taking V=
xTP x as CLF, it follows
a(x) = xT(P A +ATP)x
b(x)=2xTP B
Taking ∆(x) = Inand θ(x) = ε2b(x)b(x)T−2εa(x), the
feedback law (13) becomes using (18):
k(x) = −εb(x)T=−2εBTP
| {z }
=:K
x(19)
Rewriting θ(x), one has:
θ(x) = ε2b(x)∆(x)b(x)T−2εa(x)
= 2εxTQx
Therefore, applying Theorem 6 gives the event function
e(x, m) := 2εxTQx −psup
i∈{1,...,p}
[ki(m)−ki(x)]2
Applying this event-triggered strategy therefore requires
to compute the current control value k(x) (event if not
applied) and the corresponding Lyapunov function. At the
opposite, applying Theorem 4 only requires the computa-
tion of k(x) and gives for ρ= 1:
e(x, m) := ||k(x)||2−p(1 + √2) sup
i∈{1,...,p}
[ki(m)−ki(x)]2
If this event function is positive, it follows for all i∈
{1, . . . , p}:
p(1 + √2)(ki(x)−ki(m))2< k(x)Tk(x) (20)
This last inequality can be further transformed:
p(1 + √2) ||k(x)−k(m)||2
∞<||k(x)||2(21)
Introducing the classical inequalities between norms, if the
following inequality holds, then (21) also holds:
(1 + √2) ||k(x)−k(m)||2<||k(x)||2(22)
Let Υ denote the principal square root of KTK. Since
KTKis symmetric semi-definite positive, Υ is uniquely
defined and symmetric semi-definite positive [Koeber and
Schfer, 2006]. Inequality (22) becomes the following linear
matrix inequality:
(x−m)TΥTΥ(x−m)<1
1 + √2xΥTΥx(23)
Note that deriving the event function from (21) instead of
(22) can be computationally less consuming.
6. NUMERICAL EXAMPLE
We consider here the nonlinear system proposed in [Anta
and Tabuada, 2008, Marchand et al., 2013]:
˙x1=−x3
1+x1x2
2
˙x2=x1x2
2−x2
1x2+u(24)
This system is known to admit V(x) = 1
2x2
1+1
2x2
2as CLF
with a(x) = −x4
1+x1x3
2,b(x) = x2. Therefore, using
Theorem 6 gives a first event-triggered control law for
instance taking σ= 0.9, ∆ = Iand θ(x) = ||b||2.
Defining ωas follows:
ω(x) = x1x2+1
2x2
1+1
2x2
2
it is also known that the control k(x) := −b(x)Tω(x)
globally asymptotically stabilizes system (24). Therefore,
IFAC NOLCOS 2013
September 4-6, 2013. Toulouse, France
327
Hypotheses 2 and 3 are satisfied. Hence, Theorem 4 gives
a second event-based control.
In order to test the proposed event-triggered control, we
compared them with the event-triggered discontinuous
control proposed [Marchand et al., 2013] with an event
function based on the time derivative of the CLF and with
[Anta and Tabuada, 2008] with an event function based on
the ISS property of the system. The result is presented in
Figure 2. The new event strategy looks parsimonious like
the general formula was. The impact of ρin Theorem 4 on
the performance seems to be limited. Increasing it seems to
reduce the number of event. The simulation where carried
out taking ρ= 10.
0 5 10 15 20 25 30 35 40
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Time evolution of CLFs
Marchand et al. (2013)
Anta&Tabuada (2009)
Theorem 6
Theorem 4 with ρ=10
Fig. 2. System (24) under different event-triggered control
strategies. The circles emphasize events, that is an
update of the control law.
7. CONCLUSION
In this paper, we have proposed two different event-
triggered feedback laws for control-affine nonlinear sys-
tems. Contrary to the feedback law existing in the litera-
ture, computing the proposed feedback and event function
is of the same complexity level as computing the sole
feedback. Moreover, the triggering strategies are not based
on the Lyapunov function that is no more required (at least
for Theorem 4).
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