State space construction for continuous time transfer function matrices via Nerode equivalence

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous-time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self-contained manner. Both scalar and multivariable cases in the continuous-time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2-norm bounded analytical vector function in the open left-half of the complex plane. 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 characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.