(150c) Supervisory Control Of Autonomous Hybrid Systems: Enforcing Synchronous Process--Controller Transitions Under Uncertainty

Process systems wherein the overall dynamic behavior is shaped by strong interactions between continuous dynamics and discrete components are referred to as hybrid systems, and constitute an important class of systems in the study of process control. The distinguishing feature of a hybrid system is its multi-modal structure characterized by a finite collection of continuous dynamical subsystems coupled with discrete events that trigger the transitions between them. The continuous dynamics often 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 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], [2], [3]). More recently, the fusion of hybrid systems tools with advances in nonlinear process control has aided the formulation and solution of several practical control problems for nonlinear hybrid systems [4], [6], [7]. Research on this front has focused mainly on hybrid systems where a plant supervisor tries to verify or enforce a prescribed switching schedule in order to satisfy a higher operational objective. For this purpose, some form of a priori (though not necessarily exact) knowledge of the mode transitions is assumed in the problem formulation. For hybrid systems where mode switchings are not necessarily triggered by the control system, however, a priori knowledge of either the timing or the sequence of transitions between the constituent modes is in general not feasible or cannot be easily obtained. 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. A key consideration in the control of this class of hybrid systems is the degree of synchronicity between the process transitions on the one hand and the controller transitions, on the other. To ensure controllability of the overall system, the designer must ensure that as the process transitions into a new dynamical mode, the control system is also able to transition to the appropriate controller in a timely fashion so that the process and controller transitions coincide. Failure to do so may result in asynchronous process/controller transitions and lead to instabilities and overall performance deterioration.

In an effort to address this problem, we developed in [5] a supervisory control architecture that integrates within the hybrid control structure a model-based monitoring scheme that detects mode transitions and identifies the active mode at any given time. The scheme involves the use a set of dynamic filters that simulate the expected behavior of each mode/controller pair and are run in parallel with the process. The expected behavior of each pair is then matched against the actual behavior of the closed-loop process and a residual function that captures the discrepancy between the two behaviors is computed for each filter. Analyzing the pattern of residuals then allows the identification of the timing and sequence of mode transitions. In this scheme, however, even though the timing and sequence of transitions are unknown, it is assumed that the dynamics of the constituent modes are exactly known. In practice, the continuous dynamics of a hybrid system are typically uncertain 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 schemes. Specifically, in the presence of significant mismatch between a process mode and the model used to describe it, the residual will be non-zero even in the absence of mode transitions, thus making it difficult to decipher if and when a transition took place. Unless the filter is re-designed to be robust with respect to the model uncertainty, the supervisory control system cannot ensure synchronous process/controller transitions. Beyond corrupting the mode identification task, failure to account for the presence of parametric uncertainty and disturbances can induce closed-loop instabilities or lead to deterioration in the performance of the controllers responsible for stabilizing the continuous modes.

Motivated by these considerations, we focus in this contribution on the problem of controlling hybrid process systems with both uncertain continuous dynamics and unknown mode transitions. A robust supervisory control scheme that brings together tools from robust fault detection and robust control theory is developed to enforce synchronous process/controller transitions in the presence of uncertainty. Initially, a family of robust feedback controllers that enforce practical stability in the constituent subsystems are designed. To handle the uncertainty in the timing and sequence of mode transitions, a set of robust state observers that recreate the expected dynamic behavior of each mode/controller pair are then constructed. The observers are designed using the unknown input observer principle to ensure that the observation errors, which are used as residual signals, are decoupled from the model uncertainty and/or disturbances, and are thus sensitive only to the mode transitions. 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 will return a zero residual, while the rest will return nonzero values. A mode transition is marked by a change of the zero residual to a non-zero value. Once a mode transition is detected and the active mode is identified, the supervisor switches to the corresponding controller to enforce robust closed-loop stability. For hybrid systems where complete uncertainty decoupling is not possible, a bound that captures the size of the residual in the absence of mode transitions is derived and used as a threshold for detection purposes. A key feature of this approach is that the residual bound is a function of the achievable degree of asymptotic uncertainty attenuation and can be made arbitrarily small by properly tuning the robust controllers. Essentially, by controlling the degree of uncertainty attenuation, the asymptotic closeness between the process and observer outputs (i.e., the residual size) can be made as small as desired in the absence of mode transitions, and thus practically insensitive to the uncertainty. The supervisory monitoring and control schemes 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] El-Farra, N. H., ``Integrating Feedback and Supervisory Control of Hybrid Nonlinear Processes with Uncertain Mode Transitions," AIChE Annual Meeting, paper 654b, San Francisco, CA, 2006.

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

[7] 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.