On realization of nonlinear systems described by higher-order differential equations

We consider systems of smooth nonlinear differential and algebraic equations in which some of the variables are distinguished as “external variables.” The realization problem is to replace the higher-order implicit differential equations by first-order explicit differential equations and the algebraic equations by mappings to the external variables. This involves the introduction of “state variables.” We show that under general conditions there exist realizations containing a set of auxiliary variables, called “driving variables.” We give sufficient conditions for the existence of realizations involving only state variables and external variables, which can then be split into input and output variables. It is shown that in general there are structural obstructions for the existence of such realizations. We give a constructive procedure to obtain realizations with or without driving variables. The realization procedure is also applied to systems defined by interconnections of state space systems. Finally, a theory of equivalence transformations of systems of higher-order differential equations is developed.