(284c) A Tractable Method for Closed-Loop Active Fault Diagnosis of Stochastic Linear Systems

Reliable fault detection and isolation in increasingly complex engineering systems is critical for ensuring stability and reliability of system operation in the face of anomalies such as sensor and actuator malfunctions, system failures, and drifts in system parameters. Various sources of system uncertainty, including incomplete knowledge of system faults, exogenous disturbances, and measurement noise, increase the challenges of the diagnosis problem. Methods for fault diagnosis can be broadly categorized as being data- or model-based [1]. The latter category generally seeks to detect the system fault status based on measurements obtained during operation, commonly referred to as passive fault diagnosis. The reliability of passive fault diagnosis can, however, be adversely impacted by system uncertainties and the presence of feedback controllers that can mask the impact of faults on system outputs. This observation motivates active fault diagnosis (AFD), which involves designing an input sequence that, when applied to the system, facilitates ‘correct’ (with high probability) fault diagnosis in the presence of uncertainty [2].

When the system uncertainty is deterministic and bounded, AFD methods generally aim to design input sequences that are robust to worst-case realizations of the uncertainty. Most of these methods, primarily developed for linear time-invariant systems, separate all reachable sets of the fault models to guarantee diagnosis [3, 4, 5]. With probabilistic descriptions of system uncertainty, the AFD problem is more naturally addressed in a stochastic setting. Probabilistic AFD formulations utilize distributional information on the uncertainties, potentially reducing the conservatism commonly observed in worst-case deterministic AFD. An early approach to open-loop (offline) AFD for linear time-variant systems with additive stochastic disturbances and measurement noise is developed in [6], which relies on minimizing an upper bound on the probability of misclassification of fault models that is defined in terms of the Bhattacharyya coefficient [7].

In this talk, we present a computationally efficient method for closed-loop AFD of multiple faults in stochastic linear systems with uncertain initial conditions. The proposed AFD method relies on computing an optimal open-loop input sequence that is applied to the system in a receding-horizon fashion by updating the input design problem online based on the system measurements. A key challenge in this approach is ensuring the online solution time is significantly shorter than the sampling time. This challenge is addressed by defining the finite-horizon AFD problem as maximizing the pairwise Bhattacharyya distances (a statistical distance measure) between the predicted output probability distributions, subject to system constraints represented by a convex polytope. The main advantage of the proposed closed-loop AFD method is that the global solution can be determined efficiently through exhaustive enumeration of a small number of vertices. This approach contrasts with formulations that directly minimize the nonconvex error bound, proposed in [6], that require solving a nonconvex problem either to some local optimum or to its global optimum using algorithms such as [8] with (possibly) excessive computational requirements. Furthermore, the computational complexity of the proposed method is independent of the number of fault models and the number of states in each model. We demonstrate the performance of the closed-loop AFD method on a benchmark stochastic fault diagnosis problem with five models.

References

[1] V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S. N. Kavuri, “A review of process fault detection and diagnosis. Part I: Quantitative model-based methods,” Computers & Chemical Engineering, vol. 27, no. 3, pp. 293–311, 2003.

[2] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, second edition. Springer, 2006.

[3] S. L. Campbell and R. Nikoukhah, Auxiliary Signal Design for Failure Detection. Princeton University Press, 2004.

[4] M. Sampath, S. Lafortune, and D. Teneketzis, “Active diagnosis of discrete-event systems,” IEEE Transactions on Automatic Control, vol. 43, no. 7, pp. 908–929, 1998.

[5] A. E. Ashari, R. Nikoukhah, and S. L. Campbell, “Auxiliary signal design for robust active fault detection of linear discrete-time systems,” Automatica, vol. 47, no. 9, pp. 1887–1895, 2011.

[6] L. Blackmore and B. Williams, “Finite horizon control design for optimal discrimination between several models,” in Proceedings of the IEEE Conference on Decision and Control, San Diego, CA, pp. 1147–1152, 2006.

[7] T. Kailath, “The divergence and Bhattacharyya distance measures in signal selection,” IEEE Transactions on Communication Technology, vol. 15, no. 1, pp. 52–60, 1967.

[8] M. Tawarmalani and N. V. Sahinidis, “A polyhedral branch-and-cut approach to global optimization,” Mathematical Programming, vol. 103, no. 2, pp. 225–249, 2005.