Equivalence of Different Realization Methods for Higher Order Nonlinear Input-Output Differential Equations

Several state space realizability conditions and realization algorithm for nonlinear single-input single-output higher order input-output differential equation are compared. Three different necessary and sufficient state space realizability conditions, respectively, given in terms of integrability of the subspaces of one-forms, involutivity of the conditionally invariant distributions, and commutativity of iterative Lie brackets of vector fields, are proved to be equivalent. Moreover, the algorithm-based solutions are demonstrated for computing the integrable basis of the subspaces of oneforms. Finally, explicit formulas are provided for calculation of the differentials of the state coordinates which, in case the necessary and sufficient realizability conditions are satisfied, can be integrated to obtain the state coordinates.