The hierarchical lattices of a finite machine

Abstract

The concept of dynamical consistency is introduced to define a high-level transition function for an arbitrary partition of the state space of a finite input-state machine M . The existence of a dynamically consistent transition from a partition member X i to a member X j corresponds to the existence, for every state x in X i , of some ( x dependent) control sequence which takes that state directly into X j . This paper analyses what are termed in-block controllable partition machines and between-block controllable partition machines, and establishes the existence of the associated in-block controllable hierarchical lattice IBCP( M ). It is shown that any partition machine in IBCP( M ) is between-block controllable if and only if the base machine M is controllable. This setting allows the specification of state to state control trajectories of M to be decomposed into high-level and low-level components.