(625f) Optimal Test Design Framework for Model-Based Active Fault Detection and Isolation

The complexity and widely variable operating envelope of
modern cyber-physical systems yields a need for advanced fault detection and
isolation (FDI) algorithms. Efficient and accurate FDI tests can reduce
maintenance cost, and improve system reliability and performance,^1,2
which are becoming increasingly significant in the aerospace and automotive
industries. Recent advances in the field of active FDI^3,4 allow for
better detectability and isolatability of faults by
implementing advanced control logic for the Built-In Tests deploying FDI.^5,6
In this presentation, we will outline a comprehensive workflow that improves
FDI robustness and accuracy through an optimal selection of input settings
prior to FDI test execution.

Figure 1. Outline of framework for improving active FDI test
designs

In the
proposed workflow, model-based, active fault detection and isolation is
accomplished by solving a set of optimization problems, constrained within the
system operating space. Figure 1 presents an overview of this framework for
test design selection. Models that accurately represent the system (especially
the steady state and transient system responses to faults) are used to generate
output forecasts with and without the anticipated fault(s) present. Modelling
error and uncertainty in the form of environment variability are minimized, in
terms of their impact on the FDI test. The improvement in fault identifiability
is accomplished with two different optimal test design criteria, both using the
output sensitivities, Q_i, of i-th outputs at a predetermined number of sampling points
with respect to the targeted faults and uncertainty, ξ. The first criterion designs tests
satisfying D-optimality, which minimizes the correlation of all the elements in
ξ.^7–9
The second approach maximizes the information with respect to the faults and
minimizes the impact of uncertainty, using the so-called Ds-optimality
criterion.¹⁰ Sensitivities are compiled into modified formulations
of the Fisher Information Matrix, H_ξ,
which is a function of the parameters representing faults and uncertainty in
the system model and the test design vector. As shown in Eq.(1), H_ξ is
partitioned into submatrices corresponding to (ξ₁…
ξ_f) and (ξ_f+1…
ξ_u) that contain
the correlations between faults (H_ff), uncertainty (H_uu),
and that between faults and uncertainty, (H_uf , H_fu):

where σ_ij
is the estimated measurement variance between the i-th and j-th outputs. The
optimal test design vector for each criterion is formulated for differential
algebraic equations describing dynamic systems as shown in Eqs.
(2) and (3) of Table 1.

Table 1. Mathematical formulation of
the FDI test optimization for steady-state and dynamic test designs

Execution of the optimal FDI tests designed, leads to data
that need to be analyzed in terms false alarm rates (FAR) and non-detection
rates (NDR). First, the space of uncertain parameters and boundaries is
explored to determine if similar output trajectories can be generated for
faulty and fault-free systems at the chosen design of admissible inputs. The
mathematical formulation of this search is based on practical identifiability
of model parameters (not shown in this abstract). Thereafter, thresholds are
designed to minimize the FARs and NDRs. We evaluate the residuals generated
from an FDI test through a positive definite function (e.g. normalized root
mean square). We iterate this calculation at various fault and uncertain parameter
values, and then compare the results with fault thresholds to determine if the
test satisfies FAR and NDR requirements. Threshold designs are analyzed in the
form of receiver operating characteristic (ROC) plots. An example of a ROC
curve is shown in Figure 2, where the optimal test design obtained using this
framework significantly improves the overall quality of the FDI test in one of
the systems studied.

Figure 2. Example ROC plot of a fouled heat exchanger at
various fault detection thresholds for particulate fouling at nominal and
optimal test designs.

Several case studies comparing the FDI effectiveness (in
terms of FAR and NDR) at nominal and optimal test designs will be presented.
The first case study will focus on built-in tests for a plate-fin heat
exchanger from an aircraft environmental control system (ECS) that is prone to
particulate fouling, which extends prior work on fouling quantification.⁹
The second case study will also focus on particulate fouling in the integrated
ECS, to illustrate the impact of increased system complexity of FDI, when
multiple sources of uncertainty are present. False alarms and non-detections at
nominal test conditions, and their improvement through the proposed framework
will be presented. Lastly, the application of the proposed framework on the
three-tank benchmark system will be presented. This analysis shows how the
design of an active FDI test using the proposed framework compares to more
computationally intensive methods proposed in the literature.

 
Acknowledgment

This work was sponsored by the UTC Institute for Advanced
Systems Engineering (UTC-IASE) of the University of Connecticut and the United
Technologies Corporation. Any opinions expressed herein are those of the
authors and do not represent those of the sponsor. Help and guidance by Modelon and Modelon-AB are
gratefully acknowledged.

 
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