On Control of n-D Plant with 1-D Controller in a Behavioral Framework

In this paper, we address the problems of controlling an n-dimensional (n-D) plant (i.e., its dynamics evolves along n independent variables) using a 1-dimensional (1-D) controller (i.e., its dynamics evolves only along the conventional time axis) for the behavior of a desired specification. Theoretically, the problem is one of finding the condition under which it is possible to interconnect two systems with different time axes (or independent variables); practically, there are many cases in which an n-D plant, whose dynamics evolves along spatial coordinates (e.g., heat transfer of a stick, vibration of an elastic membrane, and so on) in addition to a conventional time axis, must be controlled using a 1-D controller. Here, we provide a necessary and sufficient condition for implementing the behavior of a desired 1-D specification in an n-D plant using a 1-D controller. We also provide a necessary and sufficient condition for implementing the behavior of a desired n-D specification in the same situation. Finally, we consider these problems for the linear timeinvariant case.