(46c) On the Inherent Robustness of Suboptimal Economic Model Predictive Control

One of the primary purposes of plant operation is profitability. For continuous processes, this goal has traditionally been achieved by conducting economic optimization at steady state and using tracking controllers to dynamically guide the plant to this steady state. The most popular methodology for tracking control in large plants is model predictive control (MPC). MPC uses a dynamic plant model to predict plant behavior over a finite time horizon and minimizes the deviation of the plant state from the target steady state, subject to constraints on the process inputs and outputs, but without regard for economic information.

Recently, however, increasing attention is being paid to dynamic economic optimization. This trend is being driven by several factors, among them dynamically varying economic objectives, such as time of day electricity pricing, and improving technological and modeling capabilities. One method to implement dynamic economic optimization is to consider the model predictive control methodology used for the dynamic regulation problem, but replace the MPC objective function with a function representing the plant's economics. The resulting controller is termed an economic model predictive controller (EMPC) and has been the subject of significant recent research [1].

The nominal stability of economic model predictive control has been proven for formulations with several kinds of terminal set constraints [2] or a sufficiently long horizon [3], for systems that are (economically) optimally operated at steady state. While these nominal stability results give some confidence to the application of economic model predictive control, any realistic control implementation will involve process disturbances and measurement noise. Furthermore, dynamic economic optimization problems are near universally nonlinear and nonconvex, meaning that optimization algorithms may converge to a local minimum, or may not even converge at all within a specified time limit. Algorithms allowing the model predictive controller to return a non-optimal input are referred to as suboptimal model predictive control. It is known that the nominal, suboptimal model predictive control law is inherently robust with respect to disturbances [4]. It is therefore natural to ask what the robustness properties of the nominal, suboptimal economic model predictive control law are.

To investigate these issues, we consider in this talk discrete time nonlinear systems with bounded additive process disturbances and measurement errors. First, we construct an appropriate terminal cost, terminal set and terminal control law for the economic problem. Next, we require that the optimization algorithm produces a nominally feasible input trajectory with objective value no worse than the warm start at each time step. Given any feasible input sequence for initialization, we show that it is possible to maintain closed-loop feasibility without any online optimization by constructing a warm start for the MPC optimization problem appropriately at each sample time. Furthermore, we describe the set of initial states under which economic model predictive control is robustly asymptotically stable, which is equivalent to input to state stability when considering the process disturbances and measurement errors as inputs. The conclusion of the theoretical analysis is that, given several assumptions on the model and cost functions, suboptimal EMPC has the same robustness properties as optimal MPC.

Finally, these theoretical results are confirmed with a simulation study of a continuous nonlinear polymerization reactor. We show that the suboptimal economic model predictive controller stabilizes the nominal system at the optimal steady state, and when the system is disturbed, the suboptimal controller guides the system to an invariant set centered at the optimal steady state whose size shrinks to zero as the magnitude of the disturbances shrink to zero. The economic performance of the controller increases as more optimization is allowed. These results give support to the general practice of using the nominal, suboptimal economic control law in real systems with bounded disturbances.

References

[1] J. B. Rawlings, D. Angeli, and C. Bates. Fundamentals of economic model predictive control. In IEEE Conference on Decision and Control (CDC), pages 3851-3861, Maui, HI, December 2012.

[2] R. Amrit, J. B. Rawlings, and D. Angeli. Economic optimization using model predictive control with a terminal cost. Annual Rev. Control, 35:178-186, 2011.

[3] L. Grüne. Economic receding horizon control without terminal constraints, Automatica, 49(3):725-734, March 2013.

[4] G. Pannocchia, J. B. Rawlings, S. J. Wright. Conditions under which suboptimal nonlinear MPC is inherently robust. Systems & Control Letters, 60(9):747-755, September 2011.