Phase conditions for stability of multi-loop control systems

Utilizing the concepts of the internal model control and the structured singular value, Grosdidier and Morari (1986) have proposed the mu-interaction measure for multiloop control system designs. Under the mu-interaction measure plot that constrains the magnitude of each closed-loop transfer function, a stable multiloop control system can be designed. However, for some processes, the plot fails to provide a stable multiloop control system with integral action and a dynamically better control system is discarded. Here we introduce a phase stability condition that reduces the above possibility when used together with the mu-interaction measure. The small gain theorem with linear fractional transformations is used to calculate the phase condition. As in the gain margin and phase margin design for single-input single-output processes, a reliable multiloop control system can be designed with the gain constraints of the mu-interaction measure and the proposed phase constraints. (C) 2000 Elsevier Science Ltd. All rights reserved.