A control system design for chemical processes needs to account for the inherent complexity of the process exhibited in the form of nonlinear behavior, operational issues such as constraints and uncertainty, as well as to safeguard against eventualities including (but not limited to) faults in the control actuators. The nonlinear behavior exhibited by most chemical processes, together with the presence of constraints on the operating conditions (dictated by performance considerations or due to limited capacity of control actuators), the presence of modeling uncertainty and disturbances, and the unavailability of all the process states as measurements has motivated significant research work in the area of nonlinear control focusing on these issues (see, e.g., [1,2] for excellent reviews of results, and the references therein).

The stability guarantees provided by a controller may no longer hold in the presence of faults in the control actuators that prevent the implementation of the control action prescribed by the control law, and these faults can have substantial negative ramifications owing to the interconnected nature of processes [3]. One approach to maintain closed-loop stability would be to use all available (and redundant) control actuators so that even if one of the control actuators fail, the rest can maintain closed-loop stability (the reliable control approach). The use of redundant control actuators, however, incurs (possibly) preventable operation and maintenance costs. These economic considerations dictate the use of only as many control loops as is required at a time and to achieve tolerance with respect to faults through control-loop reconfiguration in the event of failure of the primary control configuration. Controller reconfiguration has been utilized to achieve fault-tolerance in the context of aerospace engineering applications, and in the context of chemical process control, based on the assumption of a linear process description. More recently, reconfiguration has been utilized to achieve fault-tolerant control for nonlinear systems (see, for example, [4]) where the main idea is as follows: first, backup control configurations are identified and an explicit characterization of the stability region associated with each control configuration is obtained using Lyapunov-based bounded robust nonlinear controllers. Switching laws that determine, on the basis of the stability regions, which of the available backup control configurations can preserve closed-loop stability in the events of faults in the primary control configuration are subsequently devised. The approach in [4] addresses the problem of determining which backup configuration can be activated to ensure stability. The fault-tolerant capabilities of such an approach depends on whether or not the state, at the time of the failure, resides in the stability region of the backup control configuration. The problem of ensuring that it is possible to switch to such a backup configuration at the time of occurrence of a fault is not addressed, which can be done via guiding the system trajectory towards the stability region of the backup control configurations.

Guiding the system towards a pre-computed target set calls for invoking the model predictive control approach that allows for incorporating both state and input constraints in the control design. A prerequisite to implementing the model predictive control approach for this purpose is to design a robust model predictive controller that guarantees closed-loop stability from an explicitly characterized set of initial conditions. Predictive control formulations (including those that do not consider uncertainty) typically owe their stabilizing properties to some form of `stability' constraints that essentially require some appropriate measure of the state to decrease or reach a target set by the end of the horizon, and to ensure stability, the initial feasibility of the stability constraint is assumed. In [5], a predictive controller is designed that does not assume, but provides guaranteed stabilization from an explicitly characterized set of initial conditions under input constraints, in the absence of uncertainty. The stability guarantees of existing predictive control approaches for nonlinear systems with uncertainty, however, remain contingent upon assumption of initial feasibility of the optimization problem (or the assumption of the knowledge of a set of initial conditions starting from where the optimization problem is guaranteed to be feasible) and the set of initial conditions starting from where feasibility of the optimization problem (and therefore, stability of the closed-loop system) is guaranteed, is not explicitly characterized. In a recent work [6] (the robust hybrid predictive control design), embedding the operation of predictive controllers within the stability region of Lyapunov-based bounded robust nonlinear controllers is utilized to achieve stability and explicit characterization of the stability region for the switched closed-loop system. The robust hybrid predictive controllers, however, provide the closed-loop system comprising of the fall-back controller, any predictive controller (linear or nonlinear, robust or otherwise) and a higher level supervisor with an explicit characterization of the robust stability region via switching to a fall-back controller; and the problem of designing a robust model predictive controller that guarantees stability from an explicitly characterized set of initial conditions is not considered.

In summary, the discussion above reveals that designing a robust model predictive controller that can guarantee stability from an explicitly characterized set of initial conditions is important from the perspective of guaranteeing stability for nonlinear uncertain systems in the absence of faults, as well as from the point of view of tackling the important problem of achieving tolerance with respect to faults. Motivated by these considerations, in this work we first present a robust model predictive controller that guarantees stability from an explicitly characterized set of initial conditions. The main idea in the robust model predictive controller design is to employ Lyapunov-based techniques to formulate constraints that a) explicitly account for uncertainty in the predictive control law, without making the optimization problem computationally intractable and b) allow for explicitly characterizing the set of initial conditions starting from where closed-loop stability is guaranteed. The application of the robust model predictive controller will be demonstrated via a benchmark chemical reactor. The explicit characterization of the stability region, together with the constraint handling capabilities and optimality properties of the predictive control approach are then utilized to drive the system trajectory into the stability region of candidate backup control configurations and achieve fault-tolerant control subject to failures in the primary control configuration. Finally, implementation of the proposed method to fault-tolerant control of the chemical reactor example will be demonstrated.

References:

[1] W. B. Bequette. Nonlinear control of chemical processes: A review, Ind. Eng. Chem. Res., 30:1391-1413, 1991.

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

[3] E. B. Ydstie, New vistas for process control: Integrating physics and communication networks, AIChE J., 48:422-426, 2002.

[4] P. Mhaskar, A. Gani, N. H. El-Farra, C. McFall, P. D. Christofides, and J. F. Davis, Integrated fault-detection and fault-tolerant control for process systems, AIChE J., in press, 2006.

[5] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, Predictive control of switched nonlinear systems with scheduled mode transitions, IEEE Trans. Automat. Contr., 50:1670-1680, 2005.

[6] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, Robust hybrid predictive control of nonlinear systems, Automatica, 41:209-217, 2005.