(235b) Online Construction of Achievable and Feasible Funnels for Transient Constraints of Model Predictive Controllers

Model predictive control (MPC) is widely accepted in the chemical process industry due to its multivariable constrained control capability. Designing the output constraints of an MPC is a crucial task since the improper selection of such constraints can render the controller infeasible when a disturbance moves the process far away from its intended operating region [1]. In the existing literature, soft output constraints [2] and dynamic flexibility analysis [3] have been proposed to address this challenge. However, the past approaches either have challenges to achieve the tightest possible feasible constraints or require high computational cost, which may hinder their applications for cases when frequent updates to the MPC implementation are needed. In this work, an approach to construct the dynamic achievable output set and feasible output constraints is introduced using a novel dynamic operability mapping [4].

For a linear system, the achievable output set (AOS) at a fixed predicted time is the smallest convex hull that contains all the images of the extreme points of the available input set (AIS) when propagated through the dynamic model. Given a collection of AOS’s at all predicted times, referred to as the achievable funnel, a set of output constraints is infeasible if its intersection with the achievable funnel is empty. Under the influence of a stochastic disturbance, the achievable funnel is shifted according to the definition of the expected disturbance set (EDS). If the EDS is bounded, the intersection of all achievable funnels at each disturbance realization is the tightest set of transient output constraints. Additionally, given a fixed setpoint, an AOS is referred to as a feasible AOS if there always exists a series of inputs from the AIS that bring any output to the setpoint regardless of the realization of the disturbance within the EDS. The collection of all feasible AOS’s is referred to as the feasible funnel that provides the tightest feasible output constraints. It can be shown that the construction of the achievable and feasible funnels are two special cases of dynamic operability mapping. Thus, a novel developed theory and an algorithm to update the dynamic operability mapping according to the current state variables and the disturbance propagations are proposed to reduce the online computational time of the constraint calculation task.

The proposed framework is applied to a linearized high-dimensional reaction system, such as a water gas shift membrane reactor [5], to demonstrate the developed theory as well as the online calculation of the output constraints, and the linear transformation of the AOS’s due to the effect of random disturbances.

References

[1] P.O. Scokaert, J.B. Rawlings, Feasibility Issues in Linear Model Predictive Control, AIChE Journal. 45 (1999) 1649–1659.

[2] J.B. Rawlings, Tutorial Overview of Model Predictive Control, IEEE Control Systems Magazine. 20 (2000) 38–52.

[3] V.D. Dimitriadis, E.N. Pistikopoulos, Flexibility Analysis of Dynamic Systems, Ind. Eng. Chem. Res. 34 (1995) 4451–4462.

[4] S. Dinh, F.V. Lima, Dynamic Operability Analysis for the Calculation of Transient Output Constraints of Linear Time-Invariant Systems, In Proceedings of The 14th International Symposium on Process Systems Engineering, Kyoto, Japan. (2022), Accepted for publication.

[5] B.A. Bishop, F.V. Lima, Novel Module-Based Membrane Reactor Design Approach for Improved Operability Performance, Membranes. 11 (2021) 157.