In , a hierarchical fault-tolerant control architecture that integrates fault detection and control system reconfiguration for spatially distributed processes described by nonlinear parabolic PDEs with control constraints and control actuator faults was developed. The architecture integrates model-based fault detection, spatially distributed feedback and supervisory control to orchestrate switching between different actuator configurations in the event of faults. The various components are designed on the basis of appropriate reduced-order models that capture the dominant dynamics of the distributed process. The fault detection filter replicates the dynamics of the fault-free, reduced-order model and uses its behavioral discrepancy from that of the actual process as a residual for fault detection. Owing to the inherent approximation errors in the reduced-order model, appropriate fault detection thresholds and controller reconfiguration criteria are derived for the implementation of the fault-tolerant control architecture on the distributed system to prevent false alarms.
Since the diagnostic filter in  is designed to only detect faults, a residual exceeding the specified threshold indicates that some fault has occurred in one or more actuator of the active control configuration but does not pinpoint the location of the fault. This necessitates that the supervisor shut down all the actuators of the current configuration upon fault detection, including possibly healthy actuators, and switch to an appropriate fall back configuration whose entire set of actuators are well functioning to ensure fault-tolerance. To avoid the unnecessary shut down of healthy actuators, a fault-isolation scheme that identifies the faulty actuators within the active set needs to be incorporated into the fault-tolerant control architecture. The ability to distinguish between faults in different actuators depends to a large extent on the structure of the input operator which describes the channels through which the different actuators affect the process evolution. For spatially distributed processes, this structure depends on the actuator locations which provide the designer with an additional degree of freedom that can be exploited to guide the design of an easy-to-implement fault-isolation scheme.
In this paper, we focus on the development and integration of a model-based fault-isolation scheme within the fault-tolerant control architecture introduced in  for distributed processes modeled by nonlinear parabolic PDEs. The central idea is to select the actuator locations in a manner that gives the input operator a specific structure conducive to easy fault-isolation via a bank of dedicated fault-isolation filters. Initially, model reduction techniques are used to obtain a finite-dimensional system that captures the evolution of the slow eigenmodes of the PDE system. The actuator locations are then chosen such that the evolution of only one of the slow modes is excited by all the actuators, while the rest are each decoupled from (at least) one actuator. Next, a set of modal observers, each replicating the fault-free behavior of a given slow mode using measurements of the other modes, is constructed and their behavioral discrepancies from those of the actual slow modes are used as residuals. The specific way in which the actuators influence each mode ensures that the residual of each filter is insensitive to (at least) one actuator and can therefore be used to discern the fault or health status of that actuator at any given time. The immediate result of this is the generation of a unique pattern of residuals for each actuator fault, thus allowing complete actuator fault-isolation. Owing to the inherent approximation errors in the reduced-order model used for the design of the fault-isolation filters, appropriate fault detection and isolation criteria are derived for the implementation of the fault-tolerant control architecture on the distributed system to prevent false alarms. The criteria is expressed in terms of residual thresholds that capture the closeness of solutions between the fault-free reduced and full-order models. A singular perturbations formulation is used to link these thresholds with the extent of separation between the slow and fast eigenvalues of the spatial differential operator. Generalizations of this scheme that address the output feedback control problem and allow the isolation of multiple faults simultaneously are also discussed. Finally, the integrated fault detection, isolation and fault-tolerant control architecture is applied to the problem of actuator fault-tolerant stabilization of an unstable steady-state of a tubular reactor with recycle.
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