Expected First Occurrence Time of Uncertain Future Events in
One-Dimensional Linear Systems
David E. Acu
˜
na-Ureta
1
, Diego I. Fuentealba-Secul
1
and Marcos E. Orchard
2
1
Department of Mechanical and Metallurgical Engineering, School of Engineering,
Pontificia Universidad Cat
´
olica de Chile, Av. Vicu
˜
na Mackenna 4860, Santiago, Chile
david.acuna@uc.cl
difuente@uc.cl
2
Department of Electrical Engineering, Faculty of Mathematical and Physical Sciences,
Universidad de Chile, Av. Tupper 2007, Santiago, Chile
morchard@ing.uchile.cl
ABSTRACT
The rapid advancement of machine learning algorithms has
significantly enhanced tools for monitoring system health,
making data-driven approaches predominant in Prognostics
and Health Management (PHM). In contrast, model-based
approaches have seen modest progress, as they are often con-
strained by the need for prior knowledge of specific governing
equations, limiting their applicability to a wide range of prob-
lems. Recently, rigorous theoretical foundations have been
established to extend dynamical systems theory by incorpo-
rating prognosis of uncertain events. This article leverages
this formal framework to introduce and demonstrate a fun-
damental mathematical result for one-dimensional linear dy-
namical systems. The presented theorem offers an analyti-
cal expression for approximating the expected time at which
an event will first occur in the future. Unlike typical thresh-
olds, this event is triggered by a hazard zone, defined as an
uncertain event likelihood function over the system’s state
space. Applications of this theorem can be found in imple-
menting real-time prognostic frameworks, where it is crucial
to quickly estimate the magnitude of impending failures. Em-
phasis is placed on minimizing computational burden to facil-
itate prognostic decision-making.
David Acu
˜
na-Ureta et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution 3.0 United States License, which
permits unrestricted use, distribution, and reproduction in any medium, pro-
vided the original author and source are credited.
NOMENCLATURE
k
p
Natural number denoting the present time.
k Natural number denoting any time in the
future so that k > k
p
.
X
k
Random system state at time k.
X
k
p
+1:k
Random system state trajectory between k
p
and k: X
k
p
+1:k
= {X
k
p
+1
, X
k
p
+2
, . . . , X
k
}.
X
k
Domain of X
k
: X
k
= R.
X
k
p
+1:k
Domain of X
k
p
+1:k
: X
k
p
+1:k
= R
k−k
p
.
E Qualitative description of an event.
E
k
Binary random variable denoting the
occurrence or not the event E at time k.
τ
E
Random variable depicting the first future
occurrence time of the event E.
P(·) Probability mass function.
p(·) Probability density function.
E{·} Expectation operator.
δ
x
(·) Dirac delta distribution located at x.
1
A
(·) Indicator function of an arbitrary set A.
N(µ, σ
2
) Gaussian distribution with mean µ and
standard deviation σ.
Φ(·) Cumulative distribution function of a
standard Gaussian distribution.
erf(·) Error function.
e
l
(·) Elementary symmetric polynomial of
degree l.
1. INTRODUCTION
The event prognostic problem is undoubtedly the most chal-
lenging regarding system health monitoring for different rea-
sons (Vachtsevanos & Zahiri, 2022). Frequently, the failure
data available is rather scarce since, normally, and under a
reliability engineering approach, preventive maintenance is
carried out periodically to ensure a failure situation is rarely
reached. On the other hand, the monitored systems tend to
1
