(268b) Actuator Fault Detection and Reconfiguration in Particulate Processes with Measurement Sampling Constraints

The development of systematic control methods for shaping the particle size distribution in particulate processes is a fundamental problem whose practical significance encompasses a wide range 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 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 impact product quality and lead to substantial production losses if such faults are not properly diagnosed and handled.

To address this problem, we recently developed in [2] a methodology for the detection and handling of actuator faults in single-input particulate processes on the basis of appropriate low-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.

Beyond the problems of nonlinearities and model uncertainty, a key issue that needs to be accounted for in the design of fault-tolerant control systems is the issue of measurement sampling. In practice, measurements of the principal moments of the particle size distribution (e.g., using light scattering techniques) and measurements of the continuous phase variables (e.g., solute concentration in a crystallizer) 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.

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 on the sampling rate is critical for identifying the appropriate backup configuration that should be activated following fault detection to preserve closed-loop stability. 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 difficult.

Motivated by these considerations, we develop int his work a fault detection and fault-tolerant control structure for particulate processes modeled by population balance equations with control actuator faults and 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 process dynamics and is obtained using the method of weighted residuals. A key idea is to design a state observer that uses the available measurements of the principal moments of the PSD and the continuous phase variables 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 and 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 between sampling instances when measurements are unavailable. The state of this model is then updated using the actual measurements whenever they become available from the sensors. 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 control configuration. 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 configurations 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 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,'' Chem. Eng. Sci., 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,'' Chem. Eng. Sci., 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,'' Chem. Eng. Sci., in press, 2009.