Conflict Resolution Algorithms for Fault Detection and Diagnosis

We present two approaches for conflict resolution between two fault detection schemes, detecting the same fault, via optimization with bounded adjustment of detection thresholds. In our first method, we assume initially that there is no conflict and optimize the thresholds of both schemes with respect to a partial cost function that penalizes false alarms and missed detections. Then we continuously update thresholds based on a comprehensive cost function that penalizes conflicts in addition to false alarms and missed detections. Our updates are bounded and controlled in such a way that the cost function always assumes the lowest possible cost as a function of thresholds. We make use of residual signals to minimize computational complexity. In our second method, we present a more general solution to the conflict resolution problem using a Markov Decision Process framework that generates an optimal policy for fault detection threshold. This method is computationally more complex but it is more general, does not require knowledge of residuals, and does not require initial optimization of the thresholds. We introduce an error signal that indicates failure in resolving the conflict using threshold updating in which case, a supervisor (human or computer) can be alerted and prompted to take a corrective action. We implemented our methods on a spacecraft attitude control thrustervalve system simulation with high noise. Our results show good performance and substantial reduction in conflicts under highly uncertain conditions. Nomenclature i a+ = Penalty weight for missed detection of fault by detection scheme i i a− = Penalty weight for false alarm of fault by detection scheme i i b = Binary flag indicating presence of fault detected by scheme i (depends on thresholds and input to the fault detection scheme). i v = Threshold value for fault detection in scheme i i v = Optimal value of threshold based on receiver operating characteristics of detection scheme i i v ) = Upper bound on threshold value based on penalties in the cost function i v ( = Lower bound on threshold value based on penalties in the cost function q = Penalty on conflict between detection schemes c = Binary flag representing presence of conflict J = Cost function ) ( i MD P = Probability of missed detection for scheme i ) ( i FA P = Probability of false alarm for scheme i i r = Residual signal for scheme i indicating difference between the output and the threshold i α = Primary parameter indicating the optimal amount by which threshold should change to resolve the conflict i β = Secondary parameter indicating the optimal amount by which threshold should change to resolve the conflict * Graduate Student, Aerospace Engineering, Ann Arbor, MI 48109, email: [email protected], AIAA student member. † Associate Professor, Aerospace Engineering, Ann Arbor, MI 48109, email: [email protected], Associate Fellow. ‡ Professor, Aerospace Engineering, Ann Arbor, MI 48109, email: [email protected], AIAA member. Infotech@Aerospace 2011 29 31 March 2011, St. Louis, Missouri AIAA 2011-1587 Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2 i f = Binary flags indicating whether or not the threshold of scheme i must be changed to resolve a conflict. i σ = Change in the cost function due to change in threshold to resolve a conflict e = Binary flag indicating failure of threshold change in resolving the conflict κ = Binary flag indicating oscillatory behavior of fault flags in both schemes i − φ = Penalty on false alarm for the MDP (Markov Decision Process) i + φ = Penalty on missed detection for the MDP φ = Penalty on conflict for the MDP S = Set of states for the MDP { } N s s s s S ,..., , , 3 2 1 = M = Set of actions for the MDP { } NOOP M , , , , 2 2 1 1 − + − + = μ μ μ μ ) ( i s R = Reward of state si in the MDP ) , , ( p k r i s s T μ = Probability of transitioning from state si to sp by executing action μr k