(712h) Economic Performance Improvement for Lyapunov-Based Economic Model Predictive Control Using Disturbance Probability Distributions

Lyapunov-based economic model predictive control (LEMPC) [1] is an economic model predictive control (EMPC) [2] scheme that can dictate two modes of operation. In the first mode, LEMPC can impose time-varying operation through a state constraint while maintaining the state in state-space within the stability region of a nonlinear process operated under a Lyapunov-based controller. In the second mode of operation, LEMPC utilizes a contractive constraint to drive the closed-loop state closer to the origin (in the sense that it is driven into Lyapunov level sets defined by a smaller upper bound on the Lyapunov function) even in the presence of uncertainty. Due to the guaranteed closed-loop stability properties of this LEMPC paradigm, it has been used over the past years to promote various control purposes such as process safety [3] and production management [4]. However, the constraints that define the two modes of operation of the LEMPC to ensure that the stability region is a forward invariant set for the closed-loop process under LEMPC in the presence of disturbances are activated based on the location of the process state in state-space (i.e., the contractive constraint is activated whenever the measurement of the closed-loop state is outside a subset of the stability region at a sampling time). This subset of the stability region is chosen such that even if the worst-case upper bound on the disturbance was reached throughout the entire sampling period and the process is operating in a time-varying fashion (the first mode of operation), the state would still be within the stability region at the end of the sampling period. This is a highly conservative approach for designing the region in which time-varying operation is allowable, given that the worst-case upper bound on the disturbance is unlikely to be obtained throughout an entire sampling period under normal operation of a chemical process. Therefore, the economic performance of the LEMPC design may be significantly improved if the subset of the stability region that dictates the activation of the first or second mode of operation is chosen in a less conservative way.

Motivated by the above considerations, we propose an LEMPC design in which the region in state-space within which time-varying operation is allowable is expanded compared to the traditional LEMPC design by accounting for the probability distribution of the disturbance magnitude (rather than assuming that the upper bound occurs throughout the entire sampling period). Specifically, the region in which time-varying operation is allowed is chosen such that if the disturbance at any given time throughout a sampling period takes a value within its bounds with a probability P according to a disturbance probability distribution obtained from routine process operating data, then the closed-loop state is maintained within the stability region throughout the sampling period. The role that the probability distribution shape plays in the size of the region of time-varying operation is evaluated, and the closed-loop stability and feasibility properties of this less conservative LEMPC design are investigated. Practical suggestions are made for obtaining a disturbance magnitude probability distribution and for designing the LEMPC when the probability distribution itself has uncertainty since it is obtained from limited process data. A chemical process example is used to evaluate the economic benefits of designing the LEMPC with a less conservative state constraint activating the first or second mode of operation compared to the traditional LEMPC design methodology.

[1] Heidarinejad M, Liu J, Christofides PD. Economic model predictive control of nonlinear process systems using Lyapunov techniques. AIChE Journal. 2012;58:855-870.

[2] Ellis M, Durand H, Christofides PD. A tutorial review of economic model predictive control methods. Journal of Process Control. 2014;24:1156-1178.

[3] Alanqar A, Durand H, Albalawi F, Christofides PD. An economic model predictive control approach to integrated production management and process operation. AIChE Journal; 2017; in press.

[4] Albalawi F, Alanqar A, Durand H, Christofides, PD. A feedback control framework for safe and economically-optimal operation of nonlinear processes. AIChE Journal. 2016;62:2391-2409.