Particulate processes are found in a number of important processing industries including agricultural, chemical, food, minerals, and pharmaceuticals. The realization that the end-product quality in these processes is critically dependent on the ability to manipulate the particle size distribution (PSD) has motivated significant research work on the design of model-based feedback control systems that achieve PSDs with desired characteristics (e.g., see [1] for a recent survey of results in this area). Despite the significant and growing body of literature on control of particulate processes, the problems of designing and implementing fault detection and fault-tolerant control systems for particulate processes have received limited attention. These are important problems given the fact that the erosion of control authority caused by control system malfunctions can directly impact product quality and lead to substantial production losses if such faults are not properly diagnosed and handled.

One of the central issues in the design of model-based fault diagnosis schemes for particulate processes is the infinite-dimensional nature of particulate process models which are typically obtained through the application of population, material and energy balances and consist of systems of nonlinear partial integro-differential equations that describe the evolution of the PSD, coupled with systems of nonlinear ordinary differential equations that describe the evolution of the state variables of the continuous phase. The infinite-dimensional nature of population balance models precludes their direct usage for the synthesis of practically implementable controllers or fault diagnostic filters which need to be designed on the basis of appropriate low-order models to be suitable for practical implementation. This requires that the fault diagnosis filters be designed and implemented in a way that allows discriminating between approximation errors and faults.

To address this problem, we recently developed in [2] a methodology for the detection and handling of actuator faults in particulate processes on the basis of appropriate reduced-order models that capture the dominant process dynamics. The fault detection task was addressed by means of a filter that simulates the behavior of the fault-free, reduced-order model and uses the discrepancy from the behavior of the actual process as a residual signal. Failure compensation, on the other hand, was accomplished through a switching mechanism that reconfigures the control system based on the stability regions of the constituent control configurations in a way that preserves closed-loop stability in the event of fault detection. Using regular perturbation theory, appropriate fault detection thresholds and control reconfiguration criteria that account for model reduction and state estimation errors were derived for the implementation of the control architecture on the particulate process. These results were subsequently generalized in [3] to address the problems of fault isolation and model uncertainty.

The implementation of the above schemes requires the availability of accurate measurements of the process outputs which are needed to implement the feedback controllers and monitor the closed-loop system for FDI and supervisory control purposes. In practice, exact measurements are often unavailable due to the presence of measurement noise, the occurrence of sensor malfunctions that degrade the accuracy of the measurements, or the presence of inherent limitations on the capabilities of the sensing devices as in discrete sensors that provide only a limited (i.e., qualitative) information about the state of the system. These practical limitations on the number and accuracy of the available measurements can seriously erode the diagnostic and fault-tolerance capabilities of the fault-tolerant control architecture, if not explicitly accounted for in the design of each layer. Within the feedback control layer, for example, measurement errors can degrade the stability and performance properties of the nominal controllers and may render the closed-loop system unstable unless the controller is designed with a sufficient robustness margin. At the fault diagnosis level, the presence of measurement errors limits our ability to accurately monitor the actual evolution of the process to determine if and when a fault can be declared. Unless the FDI rules are re-designed to discriminate between those errors and faults, the FDI scheme may lead to false or missed alarms that trigger unnecessary control system reconfiguration. The lack of full or accurate state measurements also limits the size of the stability regions as well as the supervisor's knowledge of where the system trajectory is relative to those regions. This in turn complicates the actuator reconfiguration task in the event of faults.

Motivated by these considerations, we develop in this work a robust fault diagnosis and fault-tolerant control structure for particulate processes described by population balance models with control constraints, time-varying uncertain variables, actuator faults and a limited number of output measurements with limited accuracy. To facilitate the controller synthesis and fault diagnosis tasks, a finite-dimensional system that approximates the dominant dynamic modes of the process is initially derived using the method of weighted residuals and then decomposed into an interconnection of subsystems where each subsystem is excited directly by only one manipulated input. A robustly stabilizing bounded output feedback controller is then designed for each subsystem by combining a bounded Lyapunov-based robust state feedback controller with a state estimation scheme that relies on the available measurements of the principal moments of the PSD and the continuous phase variables to provide estimates of the dominant process modes. The controller synthesis procedure facilitates the derivation of (1) an explicit characterization of the fault-free behavior of each subsystem in terms of a time-varying bound on the dissipation rate of the corresponding Lyapunov function that accounts for the uncertainty and measurement errors, and (2) an explicit characterization of the robust stability region where constraint satisfaction and robustness with respect to uncertainty and measurement errors are guaranteed.

Using the fault-free Lyapunov dissipation bounds as thresholds for FDI, the detection and isolation of faults in a given actuator is accomplished by monitoring the evolution of the dominant modes within the corresponding stability region and declaring a fault when the threshold is exceeded. A key feature of this threshold is that it is dependent on the achievable degree of asymptotic uncertainty attenuation and thus can be made small by properly tuning the robust controller. Robustness of the FDI scheme to measurement errors is ensured by confining the FDI region to an appropriate subset of the stability region, and enlarging the FDI thresholds appropriately. It is shown that these safeguards can be tuned by appropriate selection of the sensor configuration. The robust FDI scheme is integrated with a controller reconfiguration strategy that orchestrates the transition from the faulty actuators to a well-functioning fall-back configuration following FDI. Appropriate FDI criteria are derived for the implementation of the fault-tolerant control architecture on the particulate process to ensure its robustness with respect to model reduction errors. Using regular perturbations theory, the criteria are expressed in terms of residual thresholds that capture the closeness of solutions between the fault-free reduced and full-order models. Finally, the proposed approach is applied to the problem of robust fault-tolerant control of the Crystal Size Distribution in a continuous crystallizer with a fines trap.

References:

[1] Christofides, P. D., N. H. El-Farra, M. H. Li and P. Mhaskar, ``Model-Based Control of Particulate Processes,'' Chemical Engineering Science, 63:1156-1172, 2008.

[2] El-Farra, N. H. and A. Giridhar, ``Detection and Management of Actuator Faults in Controlled Particulate Processes Using Population Balance Models,'' Chemical Engineering Science, 63:1185-1204, 2008.

[3] Giridhar, A. and N. H. El-Farra, ``A Unified Framework for Robust Fault Detection, Isolation and Compensation in Uncertain Particulate Processes,'' Proceedings of American Control Conference, to appear, Seattle, WA, 2008.