A note on the control of processes exhibiting input multiplicity

TL;DR

This paper investigates the control challenges posed by input multiplicity in nonlinear systems, demonstrating how to design controllers to stabilize only desired steady states and analyzing the stability of model predictive control in such scenarios.

Contribution

It introduces a control design method to eliminate input multiplicity by stabilizing only one equilibrium point and explores the stability properties of MPC in systems with multiple equilibria.

Findings

Designed controllers can stabilize a single desired steady state.

MPC can stabilize multiple equilibria depending on initial conditions.

Basin boundaries of the closed-loop system are computationally characterized.

Abstract

Steady state multiplicity can occur in nonlinear systems, and this presents challenges to feedback control. Input multiplicity arises when the same steady state output values can be reached with system inputs at different values. Dynamic systems with input multiplicities equipped with controllers with integral action have multiple stationary points, which may be locally stable or not. This is undesirable for operation. For a 2x2 example system with three stationary points we demonstrate how to design a set of two single loop controllers such that only one of the stationary points is locally stable, thus effectively eliminating the "input multiplicity problem" for control. We also show that when MPC is used for the example system, all three closed-loop stationary points are stable. Depending on the initial value of the input variables, the closed loop system under MPC may converge to different steady state input instances (but the same output steady state). Therefore we computationally explore the basin boundaries of this closed loop system. It is not clear how MPC or other modern nonlinear controllers could be designed so that only specific equilibrium points are stable.