(654b) Integrating Feedback and Supervisory Control of Hybrid Nonlinear Processes with Uncertain Mode Transitions

The study of hybrid systems in process control is motivated by the hybrid nature of many process systems whose overall dynamic tendencies are shaped by an intricate interaction between lower-level continuous dynamics and upper-level discrete or logical components. In many of these systems, the continuous dynamics arise from the underlying physical laws such as mass, momentum, and energy conservation, and are typically modeled by continuous-time differential equations. Discrete events, on the other hand, can be the result of inherent physico-chemical discontinuities in the continuous dynamics, controlled 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. The theoretical challenges posed by the combined discrete-continuous interactions, together with the abundance of practical application areas where such interactions arise, have motivated significant research work on the design of supervisory and control schemes for hybrid systems (e.g., [1-5]). While the focus of much of the earlier studies has been on linear hybrid systems, more recent research efforts are increasingly directed towards control of nonlinear hybrid systems due, in part, to the realization that the underlying dynamics of many chemical processes are inherently nonlinear and must therefore be accounted for in the controller design.

The fusion of hybrid systems tools with recent advances in nonlinear process control has enabled the formulation and solution of several control problems for nonlinear hybrid systems [6]. Examples include the development of integrated feedback and supervisory control structures for hybrid nonlinear systems with input constraints under state [7] and output feedback [8] conditions, and the design of predictive controllers for switched systems [9] to enforce pre-determined switching schedules without compromising closed-loop stability. In these studies, the hybrid control problem involves mode transitions that are controlled by the plant supervisor and thus assumed to be known a priori. In many situations, however, a priori knowledge of the timing and sequence of transitions between the constituent modes is either not feasible or difficult to obtain. For example, in hybrid processes undergoing autonomous transitions -- due to intrinsic physico-chemical discontinuities (e.g., phase changes, flow reversals and transitions, saturation of control valves, irregularities in the geometry of vessels, etc.) -- a priori knowledge of the timing and sequence of switches is difficult to obtain. Even for processes with controlled transitions, unexpected disruptions in raw material supplies and energy sources may force plant operation to deviate from the nominal schedule in order to minimize production losses and maintain the overall plant objectives. The uncertainty, or lack of precise knowledge, about when and how mode transitions take place can have a detrimental effect on the ability of the supervisor to activate the appropriate controller at the right time, thus possibly leading to instability or performance deterioration.

In this paper, we present a supervisory control architecture for nonlinear hybrid systems subject to uncertain mode transitions. Both the state and output feedback control problems are considered. The design is based on the idea of integrating a model-based monitoring mechanism within the hybrid control structure to detect the mode transitions in a timely fashion and ensure proper controller switching by the supervisor. Under full state feedback conditions, we initially design a family of Lyapunov-based nonlinear feedback controllers that asymptotically stabilize the constituent subsystems. To handle the uncertainty in the timing and sequence of mode transitions, a set of nonlinear dynamic filters that each simulate the expected dynamic behavior of a given mode is then constructed and run in parallel with the process. The expected behavior of each mode is then matched against the actual behavior of the process and a residual function that captures the discrepancy between the two behaviors is computed for each filter. At any given time, only the filter replicating the active mode's behavior will yield a zero residual, while the filters replicating the inactive modes will return nonzero residuals. This unique pattern of residuals allows the detection of the timing and sequence of mode transitions by monitoring the evolution of the residuals over time. A change in the residual value of a given filter from a non-zero value to zero indicates that the corresponding mode has been switched in, while a change from zero to a non-zero value is indicative that the mode has been switched out. The filters act essentially as transition detectors and play a conceptually similar role to fault detection filters. Once a mode transition is detected and the active mode is identified, the supervisor switches to the corresponding stabilizing controller. In the output feedback setting where full state measurements are not available for the implementation of the state feedback control laws or for the detection of the mode transitions, a set of nonlinear state estimators are incorporated into the control structure to generate appropriate state estimates from output measurements. The estimators are designed to ensure a sufficiently fast convergence of the estimation error which is shown to be necessary not only for closed-loop stability but also for the ability of the transition detection filters to distinguish between estimation errors and the errors caused by mode transitions. The proposed control method is illustrated 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] Xu, X.P. and P.J. Antsaklis, ``Results and Perspectives on Computational Methods for Optimal Control of Switched Systems," Lecture Notes in Computer Science, 2623:540?555, 2003.

[5] Hespanha, J. and A. S. Morse, ``Switching between stabilizing controllers," Automatica, 38:1905?1917, 2004.

[6] Christofides, P. D. and N. H. El-Farra. Control of Nonlinear and Hybrid Process Systems: Designs for Uncertainty, Constraints and Time-Delays, 446 pages, Springer-Verlag, Berlin, Germany, 2005.

[7] El-Farra, N. H. and P. D. Christofides, ``Coordinating Feedback and Switching for Control of Hybrid Nonlinear Processes,'' AIChE J., 49:2079-2098, 2003.

[8] El-Farra, N. H. and P. D. Christofides, ``Output Feedback Control of Switched Nonlinear Systems Using Multiple Lyapunov Functions,'' Syst. Contr. Lett., 54:1163-1182, 2005.

[9] Mhaskar, P., N. H. El-Farra and P. D. Christofides, ``Predictive Control of Switched Nonlinear Systems With Scheduled Mode Transitions,'' IEEE Trans. Autom. Contr., 50:1670-1680, 2005.