Meaning of state-space representation and its relation to canonical form Part 2. Controllability and constructibility

State space is one of the key concepts of systems theory. In the previous paper (Nakano et al. 1987) a system-theoretic framework about the choice of a state space was developed and the canonical forms of linear control theory were interpreted. Four realizations of a single-input and single-output discrete basic linear system were constructed. The isomorphism between the realizations and the universal state space representation shows systemic meanings of the state space. In this paper, using the same approach, two other realizations—a controllability realization and a constructibility realization—are constructed. By the matrix representations of these realizations the meanings of the controllability and constructibility canonical forms are shown in relation to the universal state-space representation.