Many important engineering systems, such as transport-reaction processes and fluid flow systems, are characterized by spatial variations and are modeled by systems of Partial Differential Equations (PDEs). While the study of distributed parameter systems has been an area of active research within process control over the past few decades (see, for example, [1] and the references therein), the development of systematic methods for the diagnosis and handling of faults in distributed control systems has received only limited attention. This is an important problem given the vulnerability of automated industrial control systems to malfunctions in the control actuators, measurement sensors and process equipment, as well as the increasingly stringent requirements placed on safety and reliability in industrial process operation.

Recently, we developed in [2] and [3] a unified framework for the integration of model-based fault detection, isolation and control system reconfiguration for distributed processes modeled by nonlinear parabolic PDEs with control constraints and actuator faults. A key idea in these works is to tie the design of the fault detection and isolation (FDI) filters as well as the actuator reconfiguration logic, via singular perturbation techniques, to the intrinsic separation between the slow and fast eigenvalues of the differential operator of the infinite-dimensional system. This naturally leads to the derivation of explicit FDI thresholds and actuator reconfiguration rules that minimize false or missed alarms due to approximation errors when the low-order model-based architecture is implemented on the infinite-dimensional system. Practical implementation issues such as the presence of plant-model mismatch and the availability of measurements at a finite number of locations along the spatial domain were subsequently addressed in [4] and [5]. The central idea in these works is to shape the fault-free closed-loop behavior, via robust bounded feedback control, in a specific way that facilitates the derivation of FDI rules that are less sensitive to the uncertainty. Uniting robust control and FDI leads to an explicit characterization of the state space regions where FDI and actuator reconfiguration are feasible under uncertainty and constraints.

Beyond the problems of nonlinearities and model uncertainty, one of the key issues that needs to be accounted for in the design of monitoring and fault-tolerant control systems is the issue of measurement sampling. In practice, measurements of the process outputs are typically available from the sensors at discrete time instances and not continuously. The frequency at which the measurements are available is dictated by the sampling rate which is typically constrained by the inherent limitations on the data collection and processing capabilities of the measurement sensors. In some cases, constraints on the sampling rate may be imposed by the designer in order to limit the transfer of data over a bandwidth-limited communication channel that connects the sensor and the controller for the purpose of reducing network resource utilization. The limitations on the frequency of measurement availability imposes restrictions on the implementation of the feedback controller and can also erode the diagnostic and fault-tolerance capabilities of the fault-tolerant control architecture if not explicitly accounted for in the monitoring and control system design. Within the feedback control layer, for example, infrequent measurement sampling could result in substantial errors in the implemented control action leading to possible loss of stability or performance degradation. The lack of frequent measurements also limits our ability to accurately monitor the trajectory of the process variables rendering it difficult to evaluate the residuals or diagnose faults. At the control reconfiguration level, knowledge of the dependence of a given control configuration (i.e., the spatial placement of actuators and sensors) on the sampling rate is critical for identifying the appropriate backup configuration that should be activated following fault detection to preserve closed-loop. Unless the various components of the fault-tolerant control architecture are redesigned to account for the lack of continuous measurements, enforcing fault tolerance will be a difficult task.

Motivated by these considerations, we develop int his work a fault detection and fault-tolerant control structure for distributed processes modeled by parabolic PDEs with a limited number of measurements that are sampled at discrete time instances. The structure consists of a family of output feedback controllers, a fault detection filter that accounts for the discrete sampling of measurements and a switching law that orchestrates the transition from the faulty actuators to the healthy fall-backs following fault detection. The control, detection and reconfiguration components are designed on the basis of an approximate finite-dimensional system that captures the dominant dynamic modes of the PDE and is obtained using modal decomposition techniques. A key idea is to design a state observer that uses the available measurements to generate estimates of the states of the reduced-order system in the absence of faults, and to use these estimates both for controller implementation as well as fault detection. Fault detection is achieved by comparing the expected fault-free behavior of the reduced-order system with the actual process behavior, and using the discrepancy as a residual. Since the output measurements are available only at discrete time instances, a reduced-order model of the process is embedded with the state observer to provide it with estimates of the output measurements in between sampling instances when measurements are unavailable. The state of this model is then updated with the actual measurements whenever they become available from the sensors at discrete sampling times. By formulating the overall closed-loop system as a discrete jump system in which the model estimation error is re-set to zero at the sampling times, an explicit characterization of the minimum allowable sampling rate that guarantees both closed-loop stability and residual convergence in the absence of faults is obtained. The minimum sampling rate is characterized in terms of the model accuracy, the controller design parameters and the spatial placements of the actuators and sensors. This characterization leads to the derivation of (1) a time-varying threshold on the residual which can be used to detect faults under the given sampling constraint, and (2) an actuator reconfiguration law that determines the set of feasible fall-back actuators that preserve closed-loop stability under a given measurement sampling rate. Finally, the design and implementation of the proposed fault detection and fault-tolerant control architecture are demonstrated using a simulated model of a low-density polyethylene tubular reactor example.

References:

[1] Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Birkhauser, Boston, 2001.

[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 S. Ghantasala, ``Actuator fault isolation and reconfiguration in transport-reaction processes," AIChE J., 53:1518--1537, 2007.

[4] Ghantasala, S. and N. H. El-Farra, ``Robust Detection and Handling of Actuator Faults in Control of Uncertain Parabolic PDEs," Dyn. Cont. Dis. & Impl. Syst. - Series A, 14 (S2), 203-208, 2007.

[5] Ghantasala, S. and N. H. El-Farra, ``Detection, isolation and management of actuator faults in parabolic PDEs under uncertainty and constraints," Proceedings of 46th IEEE Conference on Decision and Control, pages 878-884, New Orleans, LA, 2007.