Despite the significant, and growing, research work on control of particulate processes, the problem of designing and implementing fault-tolerant control systems for particulate processes has not received much attention. This is an important problem given the vulnerability of process control systems to faults (e.g., in the actuators, sensors or process equipment) and the detrimental effects that such faults can have on the process operating efficiency and, ultimately, on the final product quality. While an extensive body of literature exists on fault diagnosis of chemical processes, most methods have been developed for lumped parameter processes described by systems of ordinary differential equations. The dynamic models of particulate processes, however, 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.
One of the fundamental issues that arise in model-based control of particulate processes is model order reduction. Owing to their infinite-dimensional nature, population balance models cannot be used directly for the synthesis of practically implementable controllers. In the context of model-based fault-tolerant control, this issue impacts not only the controller synthesis but also the design of the fault diagnostic filters which need to be designed on the basis of similar, reduced-order models to be suitable for practical implementation. Owing to the approximation errors inherent in the reduced-order models, it is important that fault diagnosis filters be designed and implemented in a way that allows them to discriminate between approximation errors and the errors caused by the faults. Furthermore, the design of an effective fault-tolerant control system requires that the fault diagnosis filters be integrated with an appropriate controller reconfiguration strategy, and that complexities such as nonlinear behavior (e.g., owing to complex growth, nucleation, agglomeration and breakage mechanisms, and the Arrhenius dependence of nucleation laws on solute concentration in crystallizers), control constraints and limited measurements be explicitly accounted for in the fault-tolerant control system design.
In this paper, we develop a hierarchical fault-tolerant control architecture that integrates model-based fault detection, feedback and supervisory control for particulate processes described by population balance models with control constraints and actuator faults. The architecture is designed on the basis of appropriate reduced-order models that capture the dominant dynamics of the process and are obtained through application of the method of weighted residuals. Under full state feedback conditions, the architecture consists of a family of control configurations, together with a fault detection filter and a supervisor. For each configuration, a stabilizing feedback controller is obtained and its stability region is explicitly characterized in terms of the control constraints. A fault detection filter that replicates the dynamics of the fault-free, reduced-order model is designed, and its behavioral discrepancy from that of the actual process is used as a residual for fault detection. Appropriate fault detection criteria are derived for the implementation of the fault-tolerant control architecture on the particulate process to prevent false alarms resulting from the model reduction errors. The criteria is expressed in terms of optimized residual thresholds that capture the closeness of solutions between the fault-free reduced and full-order models. Finally, a switching law based on the stability regions of the constituent control configurations is derived to orchestrate the transition from the faulty actuators to a well-functioning fall-back configuration. The state feedback architecture is then extended to address the output feedback problem where measurements of the principal moments of the PSD, the process temperature and of the concentrations of the continuous-phase species are assumed to be limited. An appropriate state estimation scheme, based on the reduced-order model, is incorporated into the control structure and the effects of estimation errors are accounted for in the design of the feedback controller, the fault detection filter and the control reconfiguration logic. Finally, the proposed approach is applied to the problem of constrained, actuator fault-tolerant stabilization of an unstable steady-state of a continuous crystallizer.
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