Note on the Control of Processes Exhibiting Input Multiplicity

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 value. 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 2 × 2 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 model predictive control (MPC) is used for the example system, all three closed loop stationary points can be stable. Depending on the initial values 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 systematically so that only specific equilibrium points are stable.