On control of nD plant with 1D controller in a behavioral framework

In this paper, we address the control problems of an n-dimensional (the dynamics evolves with n independent variables, and which is abbreviated to as n-D) plant with a 1-D controller (the dynamics evolves with only the ordinary time) for a desired 1-D or n-D controlled behavior. From the theoretical points of view, such issues are concerned with the problem under what condition it is possible to interconnect two systems in the case in which the time axes (or independent variables) of them are different. From the practical points of view, there are many cases in which the dynamics of a plant evolving with not only the time but also the spacial coordinates, e.g., the heat transfer, the vibration of the elastic membrane, and so on, should be controlled by a controller described by the set of ordinary differential equations. As main results of this paper, we first present a necessary and sufficient condition for the desired 1-D controlled behavior to be implemented by a 1-D controller in the case in which the dynamics of a plant is n-D. We then also present a necessary and sufficient condition for the desired n-D controlled behavior to be implemented in the same situation. Finally, we also consider these problems in the linear time-invariant case.