Stability tests and performance bounds for a class of 2D linear systems

Repetitive, or multipass, processes are a class of 2D systems characterized by a recursive action with interaction between successive outputs or pass profiles. This interpass interaction is the source of the unique control problem for these processes in that it can cause the output sequence to exhibit oscillations which increase in amplitude from pass to pass. Previous work has developed an abstract stability theory and applied it to subclasses, such as discrete nonunit memory linear processes which are considered here, to produce basic stability tests. This article begins by reviewing the known stability tests and concludes that, at best, they only produce highly qualitative indicators of relative stability and performance. Hence, unlike (say) Bode and Nyquist tests for standard linear systems, they are of limited appeal as a basis for computer-aided control systems design. To remove this difficulty, step response data is used to develop new simulation-based tests which yield, at no extra cost, unique computable performance measures. Further, the undoubted advantages of having such measures available is clearly shown by developing a (virtually) complete solution to controller design for one subclass.