(721d) Economic Model Predictive Control with Triggered Evaluations: State and Output Feedback

In recent years, there are strong calls for the development of economic model predictive control (EMPC) to maximize the economic profit from a process. Unlike the traditional two-layer real-time optimization structure, EMPC addresses the economic consideration of a process within the framework of model predictive control (MPC) with a general economic cost function which in general is not quadratic. It has been shown that EMPC typically leads to time-varying process operation instead of steady-state operation. In the literature, several results of EMPC have been developed. In [1], an EMPC design with a terminal constraint for nonlinear systems was proposed. In this type of design, the first step is to determine the economically optimal steady states; and then an EMPC scheme is designed to optimize a general economic cost function which requires that the closed-loop system state settles to a steady state at the end of the prediction horizon.An application of EMPC to cyclic processes as well as a closed-loop stability analysis was also discussed. The above results have demonstrated that EMPC can lead to significant economic profits. However, the computational complexity of an EMPC optimization problem is typically much higher than a traditional MPC optimization problem due to the generality of the cost function.

In a recent work [2], a two-layer design for optimal time-varying operation of nonlinear systems was proposed. In this two-layer design, an EMPC is used in the upper layer to compute dynamic economic optimization policies for process operation and a traditional Lyapunov-based MPC is used to ensure the closed-loop system state follows the optimal time-varying trajectories computed by the upper layer. In order to improve the computational efficiency, the upper-layer EMPC is evaluated at the beginning of each finite-time operating window. This two-layer design provides an approach to achieve improved economic performance while reducing the computational burden. However, this two-layer design requires the re-design of the control system structure and may not be applicable to other existing EMPC designs.

In this work, the objective is to develop a more general approach to reduce the computational burden of EMPC designs. Specifically, event-triggered approach will be adopted in this work to reduce the number of evaluations of an EMPC optimization problem. Event-triggered approaches have been widely used in the design of control systems that have shared communication and computation resources and have been used in state estimation in wireless sensor networks to reduce information transmission in the network while maintaining stability and performance. However, event-triggered evaluation has not been studied within the framework of EMPC. The adoption of event-triggered evaluation in EMPC is not a trivial task. The time-varying operation of EMPC needs to be considered in the design of the triggering condition; the potential open-loop operation needs to be considered in the design of the EMPC and the interplay between the threshold of the triggering condition and the stability of the closed-loop system needs to be carefully studied.

In this work, we will develop event-triggered approaches in the framework of EMPC. The developed approaches can be applied to other existing EMPC designs with appropriate re-designs. First, we consider the case that state feedback is available and revise the LEMPC design in [3] to have triggered evaluations. The triggering condition is designed based on the difference between the actual system state and its predicted value. At a sampling time, if the triggering condition is satisfied, the EMPC is re-evaluated. Subsequently, we consider the case that only output feedback is available and integrate triggered evaluation to an output feedback LEMPC. In this case, a robust moving horizon estimator is used to reconstruct the state information from output measurements and the corresponding triggering condition is based on the difference between the measured and predicted output as well as its time derivatives. In both cases, revised implementation strategies for the event-triggered EMPC are provided and sufficient conditions that ensure the closed-loop stability are derived. A chemical process is used to illustrate the effectiveness of the proposed designs.

[1] M. Diehl, R. Amrit, and J. B. Rawlings. A {L}yapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56:703--707, 2011.

[2] M. Ellis and P. D. Christofides. Optimal time-varying operation of nonlinear process systems with economic model predictive control. Industrial & Engineering Chemistry Research, in press.

[3] M. Heidarinejad, J. Liu, and P. D. Christofides. Economic model predictive control of nonlinear process systems using lyapunov techniques. AIChE Journal, 58:855--870, 2012.