Realization of Proper Nonlinear Systems and Its Application to the Disturbance Observer for Van der Pol Oscillator

Control systems can be represented using either higher-order input-output differential equations or state-space representations, depending on the control problem being addressed. While a state-space realization can always be obtained for linear systems described by proper higher-order input-output differential equations, the same does not hold for nonlinear systems. This paper focuses on finding the conditions under which a state-space realization can be derived for proper nonlinear systems. Two necessary and sufficient conditions for this realization problem are presented: the first concerns the existence of a realization in observer canonical form, and the second addresses general realizability. To address these problems, the paper employs the polynomial/transfer function formalism for nonlinear systems, which is analogous to the linear case but incorporates differential forms and skew polynomial rings. The applicability of the proposed approach is demonstrated through an academic example and a practical case study. Specifically, the paper introduces a disturbance observer for the Van der Pol oscillator. Unlike traditional nonlinear disturbance observers, the proposed observer relies solely on the system output, although it requires adherence to restrictive integrability conditions.