Full Interconnection Issues

As we explained in chapter 2, the behavioral approach considers systems that interact with their environment through their terminals. After a controller is interconnected to the plant, the variables living on the terminals have to obey both the laws of the plant and the controller. The plant and the controller may or may not have an input/output structure when viewed from the interfacing variables. From a general point of view, the behavioral approach is not restricted to an input/output scheme for modelling and control. An important special case when interconnecting systems (e.g. plant and controller) is the so called full interconnection case, i.e., the situation that all the variables are available for interconnection. In this case, all terminals are used to interconnect the systems.

In this chapter we will study a number of synthesis problems that occur in the behavioral approach to control. In particular, we will establish algorithms for the numerical computation of controllers that achieve pole placement and stabilization by behavioral fullinterconnection. These synthesis problems were studied before in [9]. We stress that in the present chapter we only deal with the full interconnection case. In chapter 4 we will then deal with the partial interconnection case.

However, before discussing these synthesis problems, in this chapter we will first recall the notions of implementability and regular implementability by full interconnection. Given a full plant behavior, a given behavior is called implementableif it can be achieved as controlled behavior by interconnecting the plant with a controller. It is called regularly implementable if the controller that achieves the desired behavior does not re-impose restrictions that are already present in the plant behavior. We will present a new condition for regular implementability of a given behavior. Also, numerical algorithms will be presented to check regular implementability.