A stabilizing gradient-based economic MPC for unstable processes: toward the enlargement of the domain of attraction

Abstract

This work proposes a stabilizing gradient-based economic MPC with enlargement of the domain of attraction, based on the novel combination of three ingredients: terminal equality constraints solely on open-loop non-stable states, an admissible articial steady-state, and a terminal cost. A further enlargement of the domain of attraction is achieved by including slack variables to soften the bound constraints of states, without affecting the stabilizing propertiesor capacity to drive the closed-loop system toward the economic target. Finally, a case study based on an unstable reactor is used to demonstrate the properties of the proposed strategy.

Handling disturbances with a zone tracking-oriented stabilizing MPC: enlargement of the domain of attraction

“…The slack variable, δ k , is applied to soften the bounds on the state constraints and provide additional degrees of freedom for the controller to mitigate disturbances that excite the process. In this sense, S must be orders of magnitude higher than the other tuning matrices (Santana et al, 2020). Additionally, to comply with physical constraints of the system, one can include bound constraints to limit the slack variables in Problem P0.…”

“…Constraint ( 7) is such that x(0) is the initial condition of Problem P0, taken as the measured states at time step k. k 2 is a scalar to increase the prediction horizon in (8), aiming at ensuring the feasibility of this constraint on the infinite horizon, by imposing (8) up to time N + k 2 (Santana et al, 2020). Its value can be estimated from the steps described by Rawlings and Muske (1993).…”

“…Remark 1. Problem P0 combines two major ingredients presented by Santana et al (2020) to enlarge the domain of attraction, allowing accommodation of disturbances. The first element is imposing terminal equality constraints solely on non-stable states (unstable and integrating states), which permanently enlarge the domain of attraction.…”

“…Proof. The proof of this theorem builds on the concepts of recursive feasibility and convergence (Santana et al, 2020). Consider that ∆u * ,…”

“…From this perspective, Santana et al (2020) proposed two ingredients to enlarge the domain of attraction: (i) enforce solely non-stable states to be at an artificial steady-state at the end of the control horizon, through a terminal equality constraint, and (ii) soften the states bounds with slack variables, allowing to better accommodate unmeasured disturbances. The control law can simultaneously provide both stabilizing properties and feasibility of the optimization problem, even for a small control horizon.…”