A note on controlled invariance for behavioral nD systems

Abstract

In this chapter we extend the notion of invariance of nD behaviors introduced in Pereira and Rocha (European Control Conference 2013, ECC’13. ETH Zurich, Switzerland, pp. 301–305, 2013) [4], Rocha and Wood (Int. J. Appl. Math. Comput. Sci. 7(4):869–879, 1997) [7] to the controlsetting. More concretely, we introduce a notion which is the behavioral counterpart of classical controlled invariance, using the framework of partial interconnections. In such interconnections, the variables are divided into two sets: the variables to-be-controlled and the variables on which it is allowed to enforce restrictions (called control variables). In particular we focus on regular partial interconnection, i.e., interconnections in which the restrictions of the controller do not overlap with the ones already implied by the laws of the original behavior. For some particular cases, complete characterizations of controlled invariance and controller construction procedures are derived for both 1D and nD behaviors.