(268a) A Safe-Parking Framework for Fault-Tolerant Control of Transport-Reaction Processes

Transport-reaction processes are characterized by significant convection and diffusion phenomena coupled with a chemical reaction. Such processes are essential in the production of various industrial products. Examples include tubular reactors and packed-bed reactors. For such processes, the distinguishing feature for the control problem is that it involves the regulation of distributed variables by using spatially-distributed control actuators and measurement sensors. The dynamic models of transport-reaction processes over finite spatial domains typically consists of highly dissipative partial differential equations (PDE), such as parabolic PDEs. These parabolic PDEs possess a highly dissipative differential operator which is characterized by an eigenspectrum which can be partitioned into a finite slow part and an infinite stable fast complement [10]. Due to the infinite-dimensional nature of the transport-reaction processes, the control designs for lumped parameter systems cannot be directly implemented on transport-reaction systems.

To develop finite dimensional approximations of the infinite dimensional system for use in controller design, some of the existing approaches include discretizing the spatial domain (e.g, [15]), which could possibly lead to high dimensional control designs, and exploiting the separation in the eigenspectrum of the parabolic operator via Galerkin's method. The reduced order model has been subsequently used to design nonlinear controllers for quasi-linear parabolic PDE systems (and other highly dissipative PDE systems, see the book [2] for details and references). These order reduction techniques have been used to design controllers for other classes of dissipative PDE systems, and address issues such as lack of full state measurement [1] and uncertainty [3]. Subsequently, the work in [7] has developed a general framework for the analysis and control of parabolic PDEs with input constraints via Lyapunov-based bounded controllers. To address the issue of state constraints satisfaction, Model Predictive Controllers (MPC) were designed using modal decomposition techniques [5, 6]. In particular, the results in [5, 6] addressed the problem of designing finite dimensional MPC formulations that ensure satisfaction of state constraints for the infinite dimensional system based on satisfaction of more stringent state constraints imposed on the finite dimensional system.

While MPC formulations with explicitly characterized stability regions are available for lumped parameter systems (see, e.g., [14, 13]), an issue which has yet to be addressed for MPC of parabolic PDE systems is that of identifying, a priori (i.e., before controller implementation), the set of initial conditions of the infinite dimensional system from where feasibility of the optimization problem and closed-loop stability are guaranteed. Preparatory to the design of the safe-parking framework, one of the contributions of the present work is to develop a Lyapunov-based MPC for the control of parabolic PDE systems modeled by parabolic PDEs that provides an explicit characterization of feasibility and therefore the stability region.

The stability guarantees of the control designs (including the proposed Lyapunov-based MPC), however, do not hold in the presence of actuator fault that prevents the implementation of the control action prescribed by the control law, and motivate the design of fault-tolerant approaches to preserve process stability and safety. Existing fault-tolerant approaches for distributed parameter systems shadow those of lumped parameter systems and follow the robust/reliable, or reconfiguration-based fault-tolerant control designs. Such approaches assume the availability of sufficient control effort or redundant control configurations to preserve operation at the nominal equilibrium point in the presence of faults. Specifically, within robust/reliable schemes, the robustness of the active control configuration is used to handle faults as disturbances (e.g., [16]). Reconfiguration-based approaches (see eg., [8, 9, 4]), on the other hand, assume the existence of a backup, redundant control configuration that can preserve nominal operation.

In contrast, handling faults which prevent the ability to operate at the nominal operating point has received limited attention. In particular, the case where a fault results in a scenario where the nominal operating point is no longer an equilibrium point for any allowable values of the functioning actuators has not been sufficiently addressed. Without a framework to handle such faults, ad-hoc approaches could result in the process being driven to a hazardous operating point, or to a state from where nominal operation cannot be resumed even upon fault-repair, thus resulting in a temporary shut down of the process which can have substantially negative economic ramifications.

Recently, in [11] a 'safe-parking' framework was developed that preserves process safety and enables smooth resumption of nominal operation on fault repair. This is accomplished by identifying appropriate 'safe-park' points where the process can be temporarily 'parked' until nominal operation can be resumed. More recently in [12], this safe-parking framework was generalized to handle the availability of limited measurements and the presence of disturbances and uncertainty. However, the safe-parking framework of [11, 12] considers lumped parameter systems described by ordinary differential equations (ODEs). In summary, the problem of designing a predictive controller which provides an explicitly characterized stability and feasibility region, along with a mechanism which handles faults that preclude the possibility of nominal operation has not been addressed for transport-reaction processes described by parabolic PDE systems. Implementing control and fault handling schemes without accounting for the distributed nature of the process can lead to the inability to control and handle faults in the process. In particular, the stability guarantees of an MPC design based on lumped parameter approximation of the infinite dimensional system may not hold for the infinite dimensional system. Furthermore, a safe parking framework implemented without accounting for the infinite dimensional nature of the process could lead to the inability to preserve process safety and resume nominal operation.

Motivated by these considerations, this work addresses the problem of designing a Lyapunov-based predictive controller and handling actuator faults in quasi-linear parabolic PDEs subject to input constraints. To this end, by exploiting the separation of the eigenspectrum of the differential operator via Galerkin's method, a finite dimensional ODE system which captures the dominant dynamics of the PDE system is constructed. This ODE system is used as the basis for the synthesis of a Lyapunov-based predictive controller that enforces closed-loop stability and provides, simultaneously, an explicit characterization of the stability region. This predictive controller is then used to develop a safe-park framework which handles faults which preclude the ability to maintain nominal operation. The key idea in the safe-park framework is to operate the plant using the depleted control at an appropriate 'safe-park' location to prevent onset of hazardous situations as well as enable smooth resumption of nominal operation upon fault-repair. Specifically, a candidate parking location is termed a safe-park distribution if 1) the process state at the time of failure resides in the stability region of the safe-park candidate (subject to depleted control action), and 2) the safe-park candidate resides within the stability region of the nominal control configuration. In determining the safe-park distribution, dynamic considerations (via stability regions) are incorporated over and above the steady state considerations (via determining existence of equilibrium distributions for acceptable values of the functioning actuators). The proposed framework is illustrated on a diffusion-reaction process.

References

[1] J. Baker and P. D. Christofides. Finite dimensional approximation and control of nonlinear parabolic PDE systems. Int. J. Contr., 73:439-456, 2000.

[2] P. D. Christofides. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes,. Birkhäuser, Boston, 2000.

[3] P. D. Christofides and P. Daoutidis. Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. J. Math. Anal. Appl., 216:398-420, 1997.

[4] M. A. Demetriou, N.H. El-Farra, and P. D. Christofides. Actuator and controller scheduling in nonlinear transport-reaction processes. Chem. Eng. Sci., 63:3537-3550, 2008.

[5] S. Dubljevic, P. Mhaskar, N. H. El-Farra, and P. D. Christofides. Predictive control of transport-reaction processes. Comp. & Chem. Eng., 29:2335-2345, 2005.

[6] S. Dubljevic, P. Mhaskar, N. H. El-Farra, and P. D. Christofides. Predictive control of parabolic PDEs with state and control constraints. Int. J. Rob. & Non. Contr., 16:749-772, 2006.

[7] N. H. El-Farra, A. Armaou, and P. D. Christofides. Analysis and control of parabolic PDE systems with input constraints. Automatica, 39:715-725, 2003.

[8] N. H. El-Farra and P. D. Christofides. Coordinated feedback and switching for control of spatially distributed processes. Comput. Chem. Eng. Res., 28:111-128, 2004.

[9] N. H. El-Farra and S. Ghantasala. Actuator fault isolation and reconfiguration in transport-reaction processes. AIChE J., 53:1518-1537, 2007.

[10] A. Friedman. Partial Differential Equations. Holt, Rinehart & Winston, New York, 1976.

[11] R. Gandhi and P. Mhaskar. Safe-parking of nonlinear process systems. Comp. & Chem. Eng., 32:2113-2122, 2008.

[12] M. Mahmood, R. Gandhi, and P. Mhaskar. Safe-parking of nonlinear process systems: Handling uncertainty and unavailability of measurements. Chem. Eng. Sci., 63:5434 - 5446, 2008.

[13] M. Mahmood and P. Mhaskar. Enhanced stability regions for model predictive control of nonlinear process systems. AIChE J., 54:1487-1498, 2008.

[14] P. Mhaskar, N. H. El-Farra, and P. D. Christofides. Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Syst. & Contr. Lett., 55:650-659, 2006.

[15] W. H. Ray. Advanced Process Control. McGraw-Hill, New York, 1981.

[16] Z. D. Wang, B. Huang, and H. Unbehauen. Robust reliable control for a class of uncertain nonlinear state-delayed systems. Automatica, 35:955-963, 1999.