(12e) Closed-Loop Active Fault Diagnosis for Uncertain Nonlinear Systems

Fault detection and diagnosis allows for maintaining stable, reliable, and profitable operation of complex systems in the presence of component malfunctions, drifting system parameters, and other abnormal events [1]. However, nonlinear dynamics, system uncertainty, disturbances, and measurement noise often render fault diagnosis a challenging task in many applications. The problem of fault diagnosis is further complicated by the increasingly stringent requirements on the high-performance operation of practical systems, which necessitate retaining the system within admissible operational constraints during fault diagnosis. To address these challenges, a wide range of methods have been developed including residual- and observer-based [2], set-based [3], and data-based [1] approaches. The vast majority of these methods are passive, where the status of the system is deduced by comparing measurements with model predictions or historical data. However, system uncertainties and/or corrective actions of the control system can often obscure faulty behavior to an extent that impairs reliable diagnosis of faults. In these cases, auxiliary input signals can be applied to the system to enhance the detectability and isolability of faults in the system measurements [4].

This notion has led to the development of active fault diagnosis (AFD) methods in which an input sequence is designed to diagnose the system faults within a given time window in the presence of uncertainty. Worst-case “robust” AFD methods have been reported for designing input sequences that account for the worst-case values of the uncertainty. These approaches, largely developed for linear systems, look to minimize the energy of the input sequence while guaranteeing separation of the reachable output sets for all model hypotheses [4,5]. When probabilistic descriptions of the system uncertainty are available, a more natural approach is to account for probability distribution of the model outputs in place of deterministic sets. By accounting for distributional information, the input sequences can be designed in a (possibly) less conservative manner when compared to worst-case approaches. Recently, a model predictive control approach with active model discrimination is presented to account for not only the uncertainty but also the presence of control in active fault diagnosis [6].

This talk addresses the problem of AFD for nonlinear systems subject to stochastic disturbances and measurement noise as well as probabilistic model uncertainty in parameters and initial conditions (representing our incomplete knowledge of the system). The talk further discusses the challenges of integrating the AFD and control problems for improved diagnosability without adversely affecting the average control performance. The natural choice of the AFD objective function is the Bayes risk, which represents the probability of misclassifying the model; however, the Bayes risk in known to be an intractable optimization criterion as discussed in [7]. Therefore, we instead minimize an upper bound on the Bayes risk that can be defined in terms of the Bhattacharyya coefficient, which has a closed-form solution for Gaussian random variables [8]. Input and state constraints are enforced to guarantee safe system operation that is minimally intrusive during the fault diagnosis period. State constraints are enforced as chance constraintsto provide flexibility and minimal conservatism. A key challenge in solving this stochastic AFD problem is propagation of uncertainties through the nonlinear system dynamics since there is no closed-form expression for the probability distribution of the predicted state evolution. This work explores two methods for efficient uncertainty propagation (including linearization and the unscented transform) to determine the evolution of the moments of the predicted states.

The proposed AFD method can be used to compute an input trajectory once offline or it can be implemented in an online (i.e., receding-horizon) fashion. Online AFD allows for re-planning the inputs (i.e., fault diagnosis experiment) at each measurement sampling time based on new system observations/measurements. This online procedure involves solving a nonlinear state estimation problem to recursively update the model probabilities using Bayes’ rule. The effectiveness of the proposed tractable approach for offline and online AFD is demonstrated on a continuous bioreactor with multiple fault scenarios including operational and structural changes to the process. Simulation results indicate that the designed input trajectories achieve a correct fault diagnosis in all considered cases despite the presence of uncertainty. For this case, online AFD shows superior performance compared to an offline implementation and the AFD optimization problem can be solved sufficiently fast for real-time implementation. When integrating control and AFD, the simulation results show that AFD can be performed with minimal loss of control performance and that adding AFD to the predictive control algorithm improves average control performance for certain fault scenarios.

References

[1] L. H. Chiang, E. L. Russell, R. D. Braatz. Fault Detection and Diagnosis in Industrial Systems. Springer, 2000.

[2] R. J. Patton, J. Chen. Observer-based fault detection and isolation: Robustness and applications. Control Engineering Practice, 5, 671–682, 1997.

[3] R. Nikoukhah. Guaranteed active failure detection and isolation for linear dynamical systems. Automatica, 34, 1345–1358, 1998.

[4] A. E. Ashari, R. Nikoukhah, S. L. Campbell. Auxiliary signal design for robust active fault detection of linear discrete-time systems. Automatica, 47, 1887–1895, 2011.

[5] J. K. Scott, R. Findeisen, R. D. Braatz, D. M. Raimondo. Input design for guaranteed fault diagnosis using zonotopes. Automatica, 50, 1580–1589, 2014.

[6] Heirung, T. A. N., Mesbah, A. Stochastic nonlinear model predictive control with active model discrimination: a closed-loop fault diagnosis application. In Proceedings of the IFAC World Congress, 2017, Toulouse.

[7] T. Cover, P. Hart. Nearest neighbor pattern classification. IEEE Transactions on InformationTheory, 13, 21–27, 1967.

[8] L. Blackmore, B. Williams. Finite horizon control design for optimal discrimination between several models. In Proceedings of the 45th IEEE Conference on Decision and Control, pp 1147–1152, 2006, San Diego.