273190 Fault Detection and Accommodation in Particulate Processes Using Multi-Rate Sampled and Delayed Measurements
Fault detection and accommodation are fundamental problems at the interface of process control and operations. Agricultural, chemical, food, mineral, and pharmaceutical process industries place a huge importance on product quality which may deteriorate when faults are not properly diagnosed and handled once they occur. Many of the underlying processes in these industries are classified as particulate processes which, in addition to the continuous phase, also include a dispersed phase composed of particulates. The dispersed phase is described using a particle size distribution which is of significance in controlling the process since it links the product quality indices to product characteristics. Although there have been numerous studies on control of particulate processes, results on fault detection and accommodation are limited at present. Fundamental issues that arise in the design of model-based fault-tolerant control systems for particulate processes include the infinite-dimensional nature of the process models, as the complex nonlinear and uncertain dynamics of particulate processes. Efforts to address these problems were initiated in ,  where a unified framework for the robust detection, isolation and handling of control actuator faults in particulate processes was developed using low-order models.
In addition to the presence of nonlinearities and uncertainty, there is a host of practical implementation issues that need to be accounted for in the design of the fault diagnosis and fault-tolerant control systems. Among these issues is the delayed arrival of discrete multi-rate measurements from the sensors. In particulate processes, the measurements of the discrete (e.g., principal moments of the particle size distribution) and continuous phase variables (e.g., solute concentration and temperature in a crystallizer) are typically available from the sensors at discrete time instances and are delayed. The control system may also make use of multiple outputs which may be subject to different sampling rates. For instance, the dispersed phase properties may be collected using light scattering techniques whereas properties of the solute concentration in the continuous phase may be obtained from a refractometer. The frequency and times at which the measurements are received by the control system are constrained by the inherent limitations on the data collection, processing and transmission capabilities of the measurement sensors. These limit the controller implementation as well as accuracy of process monitoring. Ignoring these factors in process monitoring and controller design may erode the fault diagnosis and fault handling capabilities of the fault-tolerant control system.
Motivated by these considerations, we develop in this work a model-based framework for component fault detection and accommodation in particulate processes modeled by population balance equations with delayed, discretely-sampled, multi-rate sensor measurements. Initially, model reduction techniques are used to derive an approximate finite-dimensional system that captures the dominant dynamics of the particulate process. An observer-based output feedback controller is then designed based on this system to stabilize the fault-free process. To compensate for the lack of continuous measurements, an inter-sample model predictor is included within the control system to provide the observer with an estimate of the process outputs when measurements are unavailable from the sensors. The model state is then updated when measurements are received at discrete times. Because of the dissimilar sampling rates of the process outputs, this update is performed using different combinations of outputs at a given sampling time depending on which measurements are available from the sensors. To compensate for the measurement delays, the control system includes a propagation unit that uses the low-order model together with the past values of the control input to calculate an estimate of the current output from the delayed measurements received. This estimate is then used to update the inter-sample model predictor which, together with the controller, generates the control signal for the process.
Specific conditions for fault-free closed-loop stability are obtained by formulating the closed-loop system as a hybrid discrete-continuous system that exhibits variable update patterns generated by the dissimilar sensor sampling rates. This condition is expressed in terms of the plant-model mismatch, the different sampling rates, the controller and observer design parameters, and the size of the measurement delay. The characteristic fault-free closed-loop behavior obtained from this analysis is used as the basis for deriving appropriate rules for fault detection and accommodation. The idea is to use the state observer as a fault detection filter and compare its output with the estimate of the current plant output generated by the propagation unit at the sampling times. The discrepancy is used as a residual and compared against a time-varying alarm threshold obtained from the stability analysis to determine the fault or health status of the process components. Once a fault is detected its accommodated by adjusting the controller and observer design parameters to preserve stability and minimize performance deterioration. Finally, the results are illustrated using a simulated model of a continuous crystallizer.
 El-Farra, N. H. and A. Giridhar, "Detection and Management of Actuator Faults in Controlled Particulate Processes Using Population Balance Models," Chem. Eng. Sci., 63:1185-1204, 2008.
 Giridhar, A. and N. H. El-Farra, "A Unified Framework for Robust Fault Detection, Isolation and Compensation in Uncertain Particulate Processes," Chem. Eng. Sci., 64:2963-2977, 2009.
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