The problems of on-line fault detection and handling in dynamic process systems have been the subject of considerable research interest over the past few decades in both the academic and industrial circles. Traditionally, most of the research work in this area has focused on the monitoring of purely continuous dynamic processes described by systems of ordinary differential equations using data-based or model-based approaches. Yet, there are many examples in the chemical process industries where the dynamical properties of the process depend rather on an intricate interaction between discrete and continuous variables, and are more appropriately modeled by intervals of piecewise continuous dynamics interspersed by discrete transitions. Process systems wherein the overall behavior is shaped by the interactions of continuous dynamics with discrete events are referred to as hybrid systems, and are characterized by a switched multi-modal structure in which discrete events trigger autonomous or controlled transitions between a collection of continuous dynamical subsystems or modes. The continuous dynamics in each mode typically arise from the underlying physical laws such as mass, momentum, and energy conservation, and are typically modeled by continuous-time differential equations. The discrete events, on the other hand, can be the result of inherent physico-chemical discontinuities in the continuous dynamics, transitions between different operating regimes, the use of discrete actuators and sensors in the control system, or the use of logic-based switching for supervisory and safety control.

Over the past two decades, the challenges posed by the combined discrete-continuous interactions in hybrid systems, together with the abundance of practical applications where such interactions arise, have motivated significant research work on the design of supervisory and control schemes for several classes of hybrid systems (see, for example, [1], [2], [3]). More recently, the fusion of hybrid system tools with advances in nonlinear process control has led to the formulation and solution of several practical control problems for nonlinear hybrid systems [4]. Compared with the efforts on control of hybrid systems, the problems of fault detection and monitoring have received limited attention. Existing results in this direction have focused mainly on the related problems of state estimation and observer design for switched linear systems (see, for example, [5]). For the majority of chemical processes, however, the underlying continuous dynamics are inherently nonlinear and cannot be monitored effectively using linear approaches. The lack of systematic methods for model-based fault detection in nonlinear hybrid process systems can significantly limit the achievable process performance and control quality in those processes.

One of the key issues in the design of model-based fault detection schemes for hybrid systems is handling the switched dynamics which requires the design of a hybrid observer that reconstructs both the continuous and discrete variables and ensures residual convergence under switching. Another key consideration is the ability of the fault detection scheme to discriminate effectively between process and/or control system faults on the one hand, and the discrete transitions that take place between the continuous modes on the other. Failure to distinguish between faults and mode transitions can lead to false or missed alarms and subsequent instability or deterioration in the overall process performance. The monitoring problem for a hybrid system is further complicated by the presence of uncertainty in both the continuous dynamics and the discrete events governing the transitions between them. Uncertainty in the continuous dynamics arises typically due to the presence of unknown or partially known process parameters as well as time-varying exogenous disturbances which if not properly accounted for can adversely affect the implementation of the monitoring and control systems. Uncertainty in the mode transitions, on the other hand, stems from the lack of a priori knowledge of either the timing or the sequence of transitions between the constituent modes. For example, in processes undergoing autonomous transitions due to intrinsic physico-chemical discontinuities (e.g., phase changes, flow reversals and transitions, saturation of control valves), the timing and sequence of switches are determined by the evolution of the process and cannot be determined beforehand. An effective hybrid monitoring scheme must therefore be robust with respect to model uncertainty and have the ability to decouple the effects of faults from mode transitions from disturbances.

Motivated by these considerations, we focus in this contribution on the problem of fault detection in a class of hybrid process systems modeled by switched nonlinear systems with uncertain continuous dynamics and uncertain mode transitions. A robust hybrid monitoring scheme that discriminates between faults, mode transitions and uncertainty is developed using tools from unknown input observer theory and results from switched system stability. The monitoring scheme consists of (1) a family of robust fault detection filters that detect the faults within the continuous modes, (2) a family of mode observers that locate the active operating mode at any given time, and (3) a supervisor that switches synchronously between the fault detection filters as the process transitions from one mode to another.

Each fault detection filter in this architecture relies on the available process measurements to estimate the fault-free behavior of a given mode and uses the discrepancy from the actual behavior as a residual signal. The filters are designed using the unknown input observer principle to ensure that the estimation errors, which are used as residual signals, are decoupled from the model uncertainty and disturbances. Conditions for the convergence of the fault-free residuals under switching are also derived to ensure that the residuals are insensitive to the mode transitions. In this manner, the residual of each filter becomes sensitive only to the faults within a given mode. The ability of the switched fault detection scheme to detect faults depends on the ability of the supervisor to correctly identify which mode is active at any given time to ensure that the corresponding filter is activated. For this purpose, a set of robust state observers that recreate the expected dynamic behavior of each mode are constructed. The observers are designed to ensure that the observation errors, which are used as residual signals, are decoupled from the model uncertainty and disturbances, as well as the faults within each mode, and are thus sensitive only to the mode location. By running the observers in parallel with the process, a unique pattern of residuals is obtained where at any given time only the observer estimating the active mode's states returns a converging residual, while the rest do not. Once a mode transition is detected and the active mode is identified, the supervisor switches to the corresponding fault detection filter to monitor the active mode. Finally, the design and implementation of the monitoring scheme are demonstrated using a chemical process example.

References:

[1] Yamalidou, E.C. and J. Kantor, ``Modeling and optimal control of discrete-event chemical processes using Petri nets," Comp. Chem. Eng., 15:503-519, 1990.

[2] Bemporad, A. and M. Morari, ``Control of systems integrating logic, dynamics and constraints," Automatica, 35:407-427, 1999.

[3] Engell, S., S. Kowalewski, C. Schulz and O. Stursberg, ``Continuous-Discrete Interactions in Chemical Processing Plants," Proc. IEEE, 88:1050-1068, 2000.

[4] Christofides, P. D. and N. H. El-Farra. Control of Nonlinear and Hybrid Process Systems: Designs for Uncertainty, Constraints and Time-Delays, Springer-Verlag, Berlin, Germany, 2005.

[5] Alessandri, A., M. Baglietto, G. Battistelli, ``Luenberger observers for switching discrete-time linear systems,'' Int. J. Contr., 80:1931-1943, 2007.