State space theory of nonlinear two-terminal higher-order elements
An analysis of the state-space properties of two terminal higher-order elements (h.o.e.) is given. These h.o.e.'s, introduced to give a logically complete formulation for nonlinear circuit theory, in the two terminal case are described explicitly by a relationship \(v^{(\alpha)}=f(i^{(\beta)}),\) where v and i stand for port voltage and current respectively and the (\(\alpha)\)-index denotes the \(\alpha\)-th derivative with respect to time. Both \(\alpha\) and \(\beta\) may take any integral value, negative values symbolizing integration. Theorem 2.1 states conditions under which the state-space axioms are satisfied by these elements. A definition of input-observability is given and Theorem 3.1 proves that it is satisfied by all state representations considered by the authors. Additional conditions for local controllability are described in Theorem 3.2. Passivity is defined and sufficient conditions for activity of an h.o.e. obtained (Theorems 4.1, 4.2). For the linear case, necessary and sufficient conditions for passivity are given for a certain subclass which includes the linear elements of classical circuit theory (Theorem 4.3). Losslessness is also discussed and a necessary and sufficient condition for losslessness for a subclass of linear h.o.e.'s is derived (Theorems 5.2, 5.3). Also, a characterization of a class of h.o.e.'s that are not lossless is obtained (Theorem 5.1).
scientific article; zbMATH DE number 3860987