A methodology is presented for the design of integrated, model-based fault diagnosis and reconfigurable control systems for transport-reaction processes modeled by nonlinear parabolic partial differential equations (PDEs) with control constraints and actuator faults. The methodology brings together nonlinear feedback control, fault detection and isolation (FDI), and performance-based supervisory switching between multiple actuator configurations. Using an approximate, finite-dimensional model that captures the PDE's dominant dynamic modes, a stabilizing nonlinear feedback controller is initially designed for each actuator configuration, and its stability region is explicitly characterized in terms of the control constraints and actuator locations. To facilitate the fault diagnosis task, the locations of the control actuators are chosen in a way that ensures that the evolution of each dominant mode, in appropriately chosen coordinates, is excited by only one actuator. Then, a set of dedicated FDI filters, each replicating the fault-free behavior of a given state of the approximate system, are constructed. The choice of actuator locations ensures that the residual of each filter is sensitive to faults in only one actuator and decoupled from the rest, thus, allowing complete fault isolation. Finally, a set of switching rules are derived to orchestrate switching from the faulty actuators to healthy fallbacks in a way that preserves closed-loop stability and minimizes the closed-loop performance deterioration resulting from actuator faults. Precise FDI thresholds and control reconfiguration criteria that account for model reduction errors are derived to prevent false alarms when the reduced order model-based fault-tolerant control structure is implemented on the process. A singular perturbation formulation is used to link these thresholds with the degree of separation between the slow and fast eigenvalues of the spatial differential operator. The developed methodology is successfully applied to the problem of constrained, actuator fault-tolerant stabilization of an unstable steadystate of a representative diffusion-reaction process.
SUMMARYThis work presents an integrated fault detection and fault-tolerant control architecture for spatially distributed systems described by highly dissipative systems of nonlinear partial differential equations with actuator faults and sampled measurements. The architecture consists of a family of nonlinear feedback controllers, observer-based fault detection filters that account for the discrete measurement sampling, and a switching law that reconfigures the control actuators following fault detection. An approximate finitedimensional model that captures the dominant dynamics of the infinite-dimensional system is embedded in the control system to provide the controller and fault detection filter with estimates of the measured output between sampling instances. The model state is then updated using the actual measurements whenever they become available from the sensors. By analyzing the behavior of the estimation error between sampling times and exploiting the stability properties of the compensated model, a sufficient condition for the stability of the sampled-data nonlinear closed-loop system is derived in terms of the sampling rate, the model accuracy, the controller design parameters, and the spatial placement of the control actuators. This characterization is used as the basis for deriving appropriate rules for fault detection and actuator reconfiguration. Singular perturbation techniques are used to analyze the implementation of the developed architecture on the infinite-dimensional system. The results are demonstrated through an application to the problem of stabilizing the zero solution of the Kuramoto-Sivashinsky equation.
This paper develops a robust fault detection and isolation (FDI) and fault-tolerant control (FTC) structure for distributed processes modeled by nonlinear parabolic partial differential equations (PDEs) with control constraints, time-varying uncertain variables, and a finite number of sensors that transmit their data over a communication network. The network imposes limitations on the accuracy of the output measurements used for diagnosis and control purposes that need to be accounted for in the design methodology. To facilitate the controller synthesis and fault diagnosis tasks, a finite-dimensional system that captures the dominant dynamic modes of the PDE is initially derived and transformed into a form where each dominant mode is excited directly by only one actuator. A robustly stabilizing bounded output feedback controller is then designed for each dominant mode by combining a bounded Lyapunov-based robust state feedback controller with a state estimation scheme that relies on the available output measurements to provide estimates of the dominant modes. The controller synthesis procedure facilitates the derivation of: (1) an explicit characterization of the fault-free behavior of each mode in terms of a time-varying bound on the dissipation rate of the corresponding Lyapunov function, which accounts for the uncertainty and networkinduced 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 are accomplished by monitoring the evolution of the dominant modes within the stability region and declaring a fault when the threshold is breached. The effects of network-induced measurement errors are mitigated by confining the FDI region to an appropriate subset of the stability region and enlarging the FDI residual thresholds appropriately. It is shown that these safeguards can be tightened or relaxed by proper selection of the sensor spatial configuration. Finally, the implementation of the networked FDI-FTC architecture on the infinite-dimensional system is discussed and the proposed methodology is demonstrated using a diffusion-reaction process example.
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