State-space construction for scalar continuous-time transfer function via Nerode equivalence

In this paper we show that a minimal state space realisation in Jordan canonical form for linear continuous-time systems described by rational transfer function could be obtained in a natural and basic way by using the concept of Nerode equivalence. whilst the state space realisation is known, the contribution of this note is that the proposed realisation procedure is directly introduced and not like other alternatives which are all introduced in a somewhat ad hoc way. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time-invariant continuous-time system, the zero state is identified with the kernel of certain Hankel operator. This characterisation via Nerode equivalence class sheds light on the construction of state for general nonlinear input-output systems.