H-infinity-norm minimization and the stabilization of systems with repetitive dynamics

Multi-pass or repetitive systems are 2-D systems evolving with respect to one discrete and one discrete or continuous variable. Physically, they are systems involving repetitive action with interaction between successive passes through the system dynamics. The paper describes the stability theory for such processes and demonstrates the relationship between stability criteria and the H-infinity-norm of an appropriate transfer-function matrix (TFM). Based upon feedback control structures, the problem of stabilization is formulated as an H-infinity-norm optimization problem with solution having a natural interpretation in terms of robustness margins. The role of high-gain feedback and control structure in stabilization is illustrated by a number of simple examples. The presentation aims to define problems to stimulate research in the area rather than to provide complete solutions.