Decentralized event-triggered estimation of nonlinear systems

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Decentralized event-triggered estimation of nonlinear systems
E. Petria, R. Postoyana, D. Astolfib, D. Neˇsi´cc, W.P.M.H. Heemelsd
aUniversit´e de Lorraine, CNRS, CRAN, F-54000 Nancy, France.
bUniversit´e Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, F-69100, Villeurbanne, France.
cDepartment of Electrical and Electronic Engineering, The University of Melbourne, Parkville, 3010 Victoria, Australia.
dDepartment of Mechanical Engineering, Eindhoven University of Technology, The Netherlands.
Abstract
We investigate the scenario where a perturbed nonlinear system transmits its output measurements to a remote observer
via a packet-based communication network. The sensors are grouped into Nnodes and each of these nodes decides when
its measured data is transmitted over the network independently. The objective is to design both the observer and the
local transmission policies in order to obtain accurate state estimates, while only sporadically using the communication
network. In particular, given a general nonlinear observer designed in continuous-time satisfying an input-to-state
stability property, we explain how to systematically design a dynamic event-triggering rule for each sensor node that
avoids the use of a copy of the observer, thereby keeping local calculation simple. We prove the practical convergence
property of the estimation error to the origin and we show that there exists a uniform strictly positive minimum inter-
event time for each local triggering rule under mild conditions on the plant. The efficiency of the proposed techniques
is illustrated on a numerical case study of a flexible robotic arm.
1. Introduction
While digital networks exhibit a range of benefits for
control applications in terms of ease of installation, main-
tenance and reduced weight and volume, they also require
adapted control theoretical tools to cope with the induced
communication constraints (e.g., sampling, delays, packet
drops, scheduling, quantization), see e.g., [1,2]. In this
work, we concentrate on the state estimation of nonlin-
ear systems over a digital channel and we focus on the
effect of sampling. In particular, we consider state esti-
mation where the plant is nonlinear, perturbed and com-
municates its measurements over a digital network to a
remote observer, whose goal is to estimate the plant state.
The communication schedule is very important to guaran-
tee good estimation performance. An option is to generate
the transmission instants based on time, in which case we
talk of time-triggered strategies for which various results
are available in the literature, see, e.g., [3–7]. However,
this paradigm may generate (significantly) more transmis-
sions over the network than necessary to fulfill the esti-
mation task, thereby leading to a waste of the network
resources. As a potential and promising solution, one can
‹This work was funded by Lorraine Universit´e d’Excellence LUE,
HANDY project ANR-18-CE40-0010-02, the France Australian col-
laboration project IRP-ARS CNRS and the Australian Research
Council under the Discovery Project DP200101303.
Email addresses: elena.petri@univ-lorraine.fr (E. Petri),
romain.postoyan@univ-lorraine.fr (R. Postoyan),
daniele.astolfi@univ-lyon1.fr (D. Astolfi),
dnesic@unimelb.edu.au (D. Neˇsi´c), m.heemels@tue.nl (W.P.M.H.
Heemels)
use event-triggered transmissions to overcome this draw-
back, see e.g., [8] and the references therein. In this case,
an event-based triggering rule monitors the plant measure-
ment and/or the observer state and decides when an out-
put transmission is needed.
Various event-triggered techniques are available in the
literature for estimation, see, e.g., [9–24]. Numerous pa-
pers propose to implement a copy of the observer within
the sensor and then use its information to define the trans-
mission instants, see e.g., [9–15]. A possible drawback with
this technique is that it may require significant computa-
tion capabilities on the sensors, especially in the case of
large-scale systems, or highly nonlinear dynamics, which
may be unavailable. Another solution is to follow an event-
triggered strategy, which is only based on a static condition
involving the measured output and its past transmitted
value(s) see, e.g., [16–23]. Consequently, it is not necessary
to implement a copy of the observer in the sensors and thus
the sensors are not required to have significant computa-
tion capabilities. However, such static triggering rules may
generate a lot of transmissions and the results in [16–23]
only apply to specific classes of systems and a centralized
scenario, where all sensors communicate simultaneously
over the network, with the exception of [16,19]. An alter-
native are self-triggering policies, see e.g., [25,26], where
the observer requests a new output measurement when it
needs it to perform the estimation. However, the available
results only apply to specific classes of systems. Moreover,
self-triggering rules typically generate more transmissions
than event-triggered ones.
In this paper, we adopt a dynamic event-triggered ap-
Preprint submitted to Automatica October 10, 2022
arXiv:2210.03368v1 [eess.SY] 7 Oct 2022
proach based only on the measured output and the last
transmitted output value. This strategy keeps monitoring
the plant output, and thereby may lead to less transmis-
sions compared to a self-triggering approach. Moreover, it
does not require a copy of the observer, which simplifies
the implementation and requires less computation capabil-
ity on the sensor. The main novelties are, first, the design
of a new triggering rule, which involves an auxiliary scalar
variable for each sensor node, that has several benefits as
explained in the sequel. Second, the proposed results ap-
ply to general, perturbed nonlinear systems contrary to
the vast majority of works in the literature, which con-
centrates on specific classes of systems, see e.g., [10–24].
Third, the triggering strategies are decentralized. Indeed,
we consider the scenario with Nsensor nodes, where each
node decides independently when to transmit its local data
to the observer via a digital network. Consequently, each
sensor node has its own triggering rule.
Our design is following an emulation-based approach in
the sense that in the first step the observer is designed ig-
noring the effects of the communication network. In par-
ticular, we assume that the observer has been synthesized
in continuous-time in such a way that it satisfies an input-
to-state stability property, that holds for many observer
design techniques of the literature, see e.g., [27,28] and
the references therein. In the second step, we take the
network into account and propose a new hybrid model us-
ing the formalism of [29,30]. We then design a dynamic
triggering rule for each sensor node to approximately pre-
serve the original properties of the observer. In particu-
lar, we ensure that the estimation error system satisfies
a global practical stability property and we show that, in
some particular cases, it is possible to recover the same
decay rate for the Lyapunov function along the solutions
as in the absence of the communication network. Note
that, we do not guarantee an asymptotic stability prop-
erty, but a practical one in general, which is a consequence
of the absence of a copy of the observer in the triggering
mechanism as we explain later (see Remark 3). As al-
ready stated, the triggering rules are dynamic in the sense
that they involve a local scalar auxiliary variable, which
essentially filters an absolute threshold type condition, see
e.g., [20–23]. This is a new in the context of estimation,
to the best of the authors’ knowledge, and is inspired by
related event-triggering control techniques [31–33]. In ad-
dition, our design of the triggering rules rely on very mild
knowledge of the observer properties; only some qualita-
tive knowledge is needed on the gains appearing in the
input-to-state stability dissipativity property, which is as-
sumed to hold for the state estimation error system, as
will be explained in more detail below.
Compared to [16–19], we do not consider a stochastic
setting and discrete-time plants, but deterministic (non-
linear) continuous-time systems, which raise the issue of
potential Zeno phenomena. Moreover, in our work we
propose a new triggering rule, which filters the absolute
threshold rule proposed in e.g., [20–23] and, as a result,
typically leads to less transmissions, as illustrated on a
numerical robot example in this paper. The closest work
is [24] where a similar triggering rule is presented, but only
for polynomial systems and for a centralized approach (one
communication sensor node only). In contrary, our results
essentially only rely on an input-to-state stability assump-
tion of the estimation error system, which is commonly
satisfied [27]. Moreover we consider the more challenging
case of a decentralized set-up, we provide in-depth charac-
terizations of the domains of the solutions and we provide
various extensions for scenarios where the outputs are af-
fected by additive noise, and where the plant input is also
transmitted over the network (see Section 7). Compared
to our preliminary version of this work [34], here we con-
sider nonlinear systems, instead of linear time-invariant
ones, and the transmission strategy is decentralized, and
not centralized as in [34]. Moreover, the plant is affected
by unknown disturbances and we prove the completeness
of maximal solutions for the overall system.
The remainder of the paper is organized as follows. Pre-
liminaries are stated in Section 2. The problem setting, the
assumption on the observer and the problem statement are
presented in Section 3. The proposed triggering rule and
the overall hybrid system model are given in Section 4.
In Section 5we analyze the stability properties of the pro-
posed event-triggered observer. In Section 6we derive var-
ious properties of the solutions domains (completeness of
maximal solutions, existence of a minimum time between
any two transmissions of each sensor node as well as a
condition that allows transmissions to stop). Some gen-
eralizations and extensions are presented in Section 7and
a numerical case study on a flexible joint robotic arm is
reported in Section 8. Finally, Section 9concludes the
paper. Two technical lemmas are given in the Appendix.
2. Preliminaries
The notation Rstands for the set of real numbers
and Rě0:“ r0,`8q. We use Zto denote the set
of integers, Zě0:“ t0,1,2, ...uand Zą0:“ t1,2, ...u.
For a vector xPRn,|x|denotes its Euclidean norm.
For a matrix APRnˆm,}A}stands for its 2-induced
norm. For any signal v:Rě0ÑRnv, with nvPZą0,
}v}rt1,t2s:“ess suptPrt1,t2s|vptq|. Given a real, symmetric
matrix P, its maximum (minimum) eigenvalue is denoted
λmaxpPq pλmin pPqq. The notation INstands for the iden-
tity matrix of dimension NPZą0, while 0NˆMstands for
the null matrix of dimension NˆM, with N, M PZą0. We
consider class-K,K8,KL functions as defined in [29]. We
model hybrid systems in the formalism of [29,30], namely
H:"9x“Fpx, uq,px, uq P C,
x`PGpx, uq,px, uq P D,(1)
where CĎRnxˆRnuis the flow set, DĎRnxˆRnu
is the jump set, Fis the flow map and Gis the jump
map. Solutions to system (1) are defined on hybrid time
2
domains. A set EĂRě0ˆZě0is a compact hybrid time
domain if E“ŤJ´1
j“0prtj, tj`1s, jqfor some finite sequence
of times 0 “t0ďt1ď. . . ďtJand it is a hybrid time
domain if for all pT, Jq P E,EX pr0, T s ˆ t0,1, . . . , Juq is a
compact hybrid time domain. Given a hybrid time domain
E, supjE:“suptjPZě0:DtPRě0such that pt, jq P Eu.
A hybrid signal x: dom xÑRnxis called a hybrid arc
if xp¨, jqis locally absolutely continuous for each j. Given
a set UĎRnu,LUis the set of all functions from Rě0
to Uthat are Lebesgue measurable and locally essentially
bounded. We consider the notion of solution proposed in
[30]. Hence, a hybrid arc xis a solution to Hfor a given
input uPLU, if
•for all jPNsuch that Ij:“ tt| pt, jq P dom xu
has nonempty interior, 9xpt, jq P Fpxpt, jq, uptqq and
pxpt, jq, uptqq P Cfor almost all tPIj;
•for all pt, jq P dom xsuch that pt, j `1q P dom x,
pxpt, jq, uptqq P Dand xpt, j `1q P Gpxpt, jq, uptqq.
A solution xto Hfor a given input uPLUis maximal, if
there does not exist another solution ˜xto Hfor the same
input usuch that dom xis a proper subset of dom ˜xand
xpt, jq “ ˜xpt, jqfor all pt, jq P dom x. Moreover, a maximal
solution xto Hfor a given input uPLUis complete, if
dom xis unbounded.
3. Problem statement
3.1. Setting
Consider the nonlinear system
9
x“fppx, u, vq, y “hpxq,(2)
where xptq P Rnxis the state to be estimated by the ob-
server, uptq P Rnuis the measured input, yptq P Rnyis the
output measured by sensors, and vptq P Rnvis an unmea-
sured disturbance input at time tPRě0with nx, nyPZą0,
and nu, nvPZě0. The inputs uand vto (2) are such that
uPLUand vPLVfor some sets UĎRnuand VĎRnv.
The vector field fp:RnxˆRnuˆRnvÑRnxis locally
Lipschitz in its first argument and continuous in the others
and h:RnxÑRnyis continuously differentiable.
We follow an emulation-based design in the sense that
a continuous-time observer for system (2) is first designed
ignoring the packet-based nature of the communication
network. Afterwards, we will consider the network and
design a triggering rule to decide when the output data
need to be transmitted to the observer in order to approx-
imately preserve its original properties. In particular, we
assume the availability of a continuous-time observer for
system (2) of the form
9
z“fopz, u, y, ˆyq,
ˆx“ψpzq,ˆy“hpˆxq,(3)
where zptq P Rnzis the observer state, with nzěnx,
ˆxptq P Rnxis the state estimate, ˆyptqis the output esti-
mate at time tPRě0. The vector field fo:RnzˆRnuˆ
RnyˆRnyÑRnzis continuous, and ψ:RnzÑRnxad-
mits a right inverse ψ´Rof ψ, i.e., x“ψpψ´Rpxqq for any
xPRnx. Often z“ˆx, but this does not necessary have
to be the case, like in Kalman filters, which involve extra
variables that can be stacked in vector z. Observer (3)
has a general structure and can be designed using several
observer design procedures, including Luenberger-like ob-
servers and Kalman filters, see e.g., [3,27], [28, Section
IV] and the references therein. The precise assumption
we make on observer (3) is stated later in this section.
For simplicity, we do not consider in this work the case of
reduced-order observers (see e.g., [35]), but we believe that
similar derivations could be developed in this scenario. We
also adopt the following assumption.
Assumption 1. The observer has access to the input u
at any time instant. l
Assumption 1is reasonable in many control applications
such as, for example, when the control input is generated
on the observer side. It is worth noting that, when the
observer does not know the input u, meaning that As-
sumption 1is not satisfied, the input ucan be included
in the unknown disturbance input vin (2) and the results
presented in the sequel apply, as long as Assumption 2
presented later holds. Furthermore, in the case where the
input uis transmitted from the plant to the observer via
a digital network, we explain in Section 7.3 how to define
a triggering rule for uso that the forthcoming results hold
mutatis mutandis.
We investigate the scenario where the output mea-
surements of system (2) are transmitted to observer (3)
via a digital channel, as depicted in Figure 1. In par-
ticular, we consider the setup where the sensors are
grouped into Nnodes, where NP t1, . . . , nyuand we
write, after re-ordering (if necessary), y“ py1, . . . , yNq “
ph1pxq, . . . , hNpxqq with yiPRnyi,nyiP t1, . . . , nyuand
ny1`. . . `nyN“ny. Each sensor node decides when its
output measurement needs to be transmitted to the ob-
server over the network, independently of the other sensor
nodes. Hence several nodes are allowed to communicate
at the same time instant.
In this setting, the observer does not know y, but its
networked version ¯y:“ p¯y1,...,¯yNq P Rny. Each ¯yiP
Rnyi, with iP t1, . . . , Nu, is generated with a zero-order-
hold device between two successive transmission instants,
i.e., in terms of the hybrid systems notation of Section 2,
9
¯yi“0 (4)
and, when a transmission of node ioccurs the correspond-
ing output yiis transmitted, considering an ideal sampler,
hence
¯y`
i“yi,(5)
3
9
x“fppx, u, vq
y“hpxq
Plant
u
vy
Node 1
Node N
Sensor 1
Sensor N
.
.
.
y1
yN
ETM 1
ETM N
Network
¯y1
¯yN
9z“fopz, u, ¯y, ˆyq
ˆx“ψpzq
ˆy“hpˆxq
Observer
¯y
u
ˆx
Figure 1: Block diagram representing the system architecture (ETM: Event-Triggering Mechanism)
otherwise, when another node generates a transmission the
last received value is kept constant, i.e.
¯y`
i“¯yi.(6)
It is worth noting that the zero-order-hold is just a choice
we make to generate the output sampled version ¯yifor all
iP t1, . . . , Nubetween transmission times. Other options
are for example the first-order-hold and the model-based
holding function [36].
Since the output yis transmitted over the network, ob-
server (3) does not have access to the exact measurement
output y, but its networked version ¯y. As a result, the
observer equations in (3) become
9z“fopz, u, ¯y, ˆyq,
ˆx“ψpzq,ˆy“hpˆxq.(7)
We define the network-induced error for each sensor node
ei:“¯yi´yiPRnyi, with iP t1, . . . , Nu, and the concate-
nated vector e:“ pe1, . . . , eNq “ ¯y´yPRny. We obtain,
in view of (2) and (7),
9
z“fopz, u, y `e, ˆyq “ fopz, u, hpxq ` e, hpψpzqqq.(8)
The dynamics of variable ei, for iP t1, . . . , N u, between
two successive transmission instants is, in view of (2) and
(4) and since hiis (continuously) differentiable,
9ei“9
¯yi´9yi“ ´Bhipxq
Bxfppx, u, vq “:gipx, u, vq.(9)
Furthermore, at each transmission instant of the i-th sen-
sor node, we have
e`
i“0,(10)
in view of (5), while, for jP t1, . . . , N uwith j‰i,
e`
j“ej.(11)
3.2. Assumption on the observer
Inspired by [27], we require observer (3) to satisfy the
next input-to-state stability property.
Assumption 2. There exist α,α,α,γ1, . . . , γN, θ PK8,
V:RnxˆRnzÑRě0continuously differentiable, such
that for all xPRnx,zPRnz,uPU,vPV,ePRny,
ˆyPRny,
αp|x´ψpzq|q ď Vpx, zq ď αp|ψ´Rpxq ´ z|q (12)
x∇Vpx, zq,pfppx, u, vq, fopz, u, y `e, ˆyqqy ď
´αpVpx, zqq `
N
ř
i“1
γip|ei|q ` θp|v|q.(13)
l
Assumption 2implies that (3) is a global asymptotic ob-
server when v“0 for system (2) in the sense that (12) and
(13) guarantee that, in this case, for any initial condition
xp0q P Rnx,zp0q P Rnzand any input pu, vq P LUˆ t0u,
the corresponding (maximal) solution xand zto (2) and
(3), if complete1, satisfy xptq ´ ˆxptq Ñ 0 as tÑ `8,
where ˆxptq “ ψpzptqq. More precisely, Assumption 2
implies that the estimation error system x´ˆxsatisfies
an input-to-state stability property [37] with respect to
both the network-induced errors ei, which act as additive
measurement noises in (13), and to the unknown distur-
bance input v. In other words, there exist βPKL and
γPK8such that, for any input uPLUand any distur-
bance vPLVthe corresponding solutions xand zto (2)
and (3) respectively, for all tě0 satisfy |ˆxptq ´ xptq| ď
βp|ψ´Rpxp0qq ´ zp0q|, tq ` γp
N
ř
i“1
}ei}r0,ts` }v}r0,tsq.Hence,
Assumption 2is a robustness property of the observer with
respect to measurement noises, which is independent of the
network.
1Completeness of maximal solution will be ensured in Section 6.1
4
摘要：

Decentralizedevent-triggeredestimationofnonlinearsystemsE.Petria,R.Postoyana,D.Astol b,D.Nesicc,W.P.M.H.HeemelsdaUniversitedeLorraine,CNRS,CRAN,F-54000Nancy,France.bUniversiteClaudeBernardLyon1,CNRS,LAGEPPUMR5007,F-69100,Villeurbanne,France.cDepartmentofElectricalandElectronicEngineering,TheUniv...

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