(429b) Diagnosis And Management Of Actuator Faults In Uncertain Distributed Processes: Uniting Fdi And Robust Control

The development of systematic methods for the diagnosis and handling of faults in process systems is a central problem that has received considerable attention in both the academic and industrial circles. The significance of this problem stems in part from the vulnerability of automated processes to faults, as well as the increased emphasis placed on safety and reliability in the operation of industrial processes. Despite the extensive and growing body of literature on fault diagnosis and fault-tolerant control (FTC), most of the available methods have been developed for spatially homogeneous processes modeled by systems of ordinary differential equations. For spatially-distributed processes (e.g., transport-reaction systems) modeled by Partial Differential Equations (PDEs), existing results have focused either on the fault diagnosis task alone -- based on approximate linear process models (e.g., [1]) and without taking complexities such as constraints and measurement limitations into account -- or on the control reconfiguration strategy alone (e.g., [3]) under the assumptions that the faults are known and that complete state measurements are available.

The lack of a unified framework for the diagnosis and management of faults in spatially distributed processes limits the achievable reliability and fault-tolerance capabilities in the operation of transport-reaction processes. This has motivated a series of recent works on the design of integrated fault-diagnosis and fault-tolerant control architectures for distributed processes. In [2] a model-based fault-tolerant control architecture that brings together fault detection, feedback and supervisory control on the basis of an appropriate reduced-order model was developed. The fault detection filter is designed to replicate 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. Appropriate detection thresholds and controller reconfiguration criteria were derived for the implementation of the fault-tolerant control architecture on the distributed system to prevent false alarms due to model reduction errors. The proposed architecture was subsequently generalized in [4] to incorporate fault isolation capabilities in the diagnosis layer, and performance-based actuator reconfiguration rules in the supervisory control layer.

In addition to nonlinearities and spatial variations, another important issue that must be accounted for in the design of model-based fault diagnosis and fault-tolerant control systems is the presence of plant-model mismatch. Parabolic PDEs that model transport-reaction processes are inherently uncertain due to the presence of unknown or partially known process parameters as well as time-varying exogenous disturbances which, if not properly accounted for, can adversely affect all components of the fault-tolerant control system. Within the feedback control layer, for example, the uncertainty alters the stability regions of the nominal controllers and may even destabilize or degrade the performance of the closed-loop system if not accounted for explicitly in the controller design. Also, at the fault detection level, unless the filter is re-designed to achieve uncertainty decoupling (which is a difficult task for nonlinear systems), the residual will be sensitive to both the uncertainty and the faults, thus leading to possible false detection alarms that can trigger unnecessary control system reconfiguration that destabilizes the closed-loop system or significantly degrades its performance. Finally, at the supervisory control level, the actuator reconfiguration logic is based on the stability regions associated with the various control configurations. The size of these regions, however, will not remain the same under significant model uncertainty and, consequently, the nominal stability regions cannot be used as the basis for proper actuator reconfiguration under uncertainty.

In an effort to address some of these issues, a rule-based method has been developed in [5] for actuator fault detection in uncertain parabolic PDEs on the basis of an approximate low-order model. The method relies on shaping the healthy closed-loop behavior, via robust control, in a way that facilitates the derivation of fault detection rules that are less sensitive to the uncertainty. In many practical applications, however, it is important not only to detect that a fault has occurred, but also to isolate the source or location of the fault in order to be able to take appropriate corrective action. 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. Furthermore, it is important to address the implementation of the reduced-order model-based fault-tolerant control system on the distributed process to ensure its robustness against the approximation errors made in deriving the low-order model at the design stage. Finally, effective management of faults requires that the fault detection and isolation (FDI) tasks be tightly integrated with the control reconfiguration logic and that complexities due to nonlinear behavior, control constraints and uncertainty be explicitly accounted for.

Motivated by these considerations, we present in this work a methodology for the detection, isolation and management of actuator faults in distributed processes modeled by nonlinear parabolic PDEs with time-varying uncertain variables and actuator constraints. An integrated robust FDI-FTC architecture is designed on the basis of an approximate, finite-dimensional system that captures the dominant dynamic modes of the PDE. The key idea in the design is to use bounded robust control as a tool for enhancing actuator FDI. This is accomplished as follows. Initially, an invertible coordinate transformation, obtained with the aid of a judicious spatial placement of the control actuators, is used to transform the approximate system into an equivalent form where the evolution of each dominant mode is excited directly by only one actuator and decoupled from the rest. For each mode, a robustly stabilizing bounded feedback controller that achieves an arbitrary degree of asymptotic attenuation of the effect of uncertainty is then synthesized and its constrained stability region is explicitly characterized in terms of the constraints, actuator locations and the size of uncertainty. A key idea in the controller synthesis is to shape the healthy closed-loop response of each mode in a prescribed fashion that practically decouples the effects of uncertainty and other modes on its dynamics, thus allowing the derivation of performance-based FDI rules for each actuator. Uniting the tasks of constrained robust stabilization and FDI allows obtaining an explicit characterization of the state-space regions where robust FDI is feasible under uncertainty and constraints. Following FDI, a switching law is derived to orchestrate actuator reconfiguration in a way that preserves robust closed-loop stability. Precise FDI rules and control reconfiguration criteria that account for model reduction errors are derived for the implementation of the FDI-FTC structure on the distributed parameter system. A singular perturbation formulation is used to link these thresholds with the separation between the slow and fast eigenvalues of the spatial differential operator necessary for closed-loop stability. Finally, the implementation of the robust FDI-FTC architecture is illustrated through an application to a non-isothermal tubular reactor with recycle.

References:

[1] Demetriou, M. A., ``A model-based fault detection and diagnosis scheme for distributed parameter systems: A learning systems approach," ESAIM-Control Optimization and Calculus of Variations, 7:43--67, 2002.

[2] El-Farra, N. H., ``Integrated fault detection and fault-tolerant control architectures for distributed processes,'' Ind. Eng. Chem. Res., 45: 8338-8351, 2006.

[3] El-Farra, N. H. and P. D. Christofides, "Coordinating feedback and switching for control of spatially distributed processes," Comp. Chem. Eng., 28: 111--128, 2004.

[4] El-Farra, N. H. and S. Ghantasala, ``Actuator Fault Isolation and Reconfiguration in Transport-Reaction Processes,'' AIChE J., in press, 2007.

[5] Ghantasala, S. and N. H. El-Farra, ``Robust Detection and Handling of Actuator Faults in Control of Uncertain Parabolic PDEs,'' J. Dynamics of Continuous, Discrete and Impulsive Systems (Series A), in press, 2007.