RESIDUAL COMPUTATION AND EVALUATION
USING DYNAMIC TIME WARPING
Francisco I. Gamero, David A. Llanos, Joan Colomer and Joaquim Mel¶endez
Grupo eXiT, University of Girona. Av. Lluis Santalo s/n E-17071-Girona (Spain)
fgamero, dllanosr, colomer, quimmelg@eia.udg.es
Abstract
This work describes the adaptation of Dynamic Time
Warping (DTW) algorithm in order to be used on-line
for residual computation and evaluation purpose. The
main motivation is to compensate modelling errors and
characteristic hybrid systems behavior by using DTW.
A laboratory plant has been used to test this approach.
Keywords: Fault diagnosis, residual computation
and evaluation, dynamic programming
1 INTRODUCTION
Fault Detection and Isolation (FDI) methods based on
analytical redundancy (Chow and Willsky 1984) are
widely used to diagnose systems whose mathematical
model is available. The task of FDI is typically accom-
plished in two steps, namely residual generation and
residual evaluation. A residual is a signal generated
from some computation based on measured variables.
It is ideally zero in the fault-free case and di®erent
from zero, in the faulty case. In practice the gener-
ated residuals are not identically zero, due to various
errors (measurement noises, modelling uncertainties).
Residual generation consists in designing fault indi-
cators satisfying speci¯cations such as sensitivity to
faults and robustness to disturbance, to modelling er-
rors and to noise. Residual generation has received
considerable attention in the literature during the last
decade. Commonly, residuals are analytical symp-
toms. In (Gertler 1988; Isermann 1993; Frank, Ding,
and KÄoppen-Seliger 2000; Blanke, Kinnaert, Lunze,
and Staroswiecki 2003) the most common techniques
are introduced : diagnostic observer, parity space, pa-
rameter estimation and structural analysis.
Residual evaluation (known also as decision proce-
dure) consists in translating the symptoms into in-
formation about the faults that may have occurred.
There are several methods of residual evaluation:
threshold logic (Jacques, Hamelin, and Aubrun 2003),
statistical decision theory (Basseville 2003), pattern
recognition, fuzzy decision making (Koscielny and
Syfert 2003), neural networks (Frank 1996).
A possible structure for fault detection is presented in
Figure 1. Residual r is obtained as a result of compar-
ison of the model output yM with real process measure
y.
The aim of this work is to adapt Dynamic Time Warp-
ing (DTW) to be used on-line in order to carry out the
residual computation and evaluation. Dynamic Time
Warping (DTW) algorithm is normally used to com-
pare and classify similar patterns by means of a mea-
sure of similarity. This approach is specially suitable
for those errors related with time distortions. There-
fore, it will be useful for distributed systems with com-
munication delays and for hybrid systems with on/o®
sensors or actuators. In this case, small modelling er-
rors can cause large residual mistakes due to abrupt
changes in speci¯c time instants.
In order to illustrate the proposed method a laboratory
plant has been used to test this approach.
System
Model of the
system
un
u2
u1
:
:
y
yM
r
Residual
generation
Figure 1: Diagram of residual generation.
This paper is organized as follows: In section 2, Dy-
namic Time Warping (DTW) algorithm is summa-
rized. In section 3, modi¯cation of DTW in order
to be applied on-line is explained. Section 4 presents
the residual evaluation approach applied. Section 5
presents the applications in a laboratory plant. Fi-
nally some conclusions and further work are given in
section 6.
.....