Control Design Problems for Multidimensional Behaviours

In this paper we study a generalized tracking and disturbance rejection problem for multidimensional linear behaviours. Given a multidimensional plant, our first goal is to design a compensator to be connected to the plant through regular partial interconnection, in such a way that the overall controlled system is autonomous and stable, when no exogenous signal acts on the system. On the other hand, when exogenous signals affect the controlled system evolution, we want to impose that a suitable linear combination of the overall system trajectories is “negligibile” in a sense we will clarify within the paper. This problem set-up formalizes a number of classical control problems, first of all tracking of some (reference) signal together with rejection of another (disturbance) signal. The adopted approach is extremely general and it is based on the idea of describing all behaviour trajectories as the sum of a “transient signal” and a “steady state” component, a decomposition that relies on Gabriel’s localization theory. Necessary and sufficient conditions for the problem solvability are provided, and the compensators that satisfy the control goal are characterized in terms of an internal model condition.