An algebraic framework generalizing the concept of transfer functions to nonlinear systems

Eigenvalues for a nonlinear time-delay system

“…However, analogous problem remains open also in the case of nonlinear systems without delays. Though, in such a case one can suspect that the eigenvalues of a nonlinear system can be identified with the poles of a transfer function of a nonlinear system in the sense of [16], respectively as roots of the so-called Ore determinant [13]. Since such a transfer function formalism is available also for nonlinear systems with delays [17] the problem left for the future research is to show that the roots of the respective Ore determinant are the eigenvalues of a nonlinear system with delays.…”

State elimination for nonlinear neutral state-space systems

“…, z j . Hence, G n−1 will depend only on z n and, therefore, will finally determine the univariate polynomial that can be identified with the input-output representation (10). Note that z n , as the variable with the lowest order according to (17), is the only one included in the resulting univariate polynomial.…”

“…This technique has many applications in control theory (see e. g. [27]), and can be applied here to the elimination process of the system of polynomial equations (12) and (15) in order to get the input-output representation (10). That is, to find a representation with the variables corresponding only to the inputs, outputs, their derivatives, and time delays.…”

“…Eventually, any transfer function approach to a control problem deals, in general, with input-output properties of a system. See for instance [17,25] for the applications to the systems with delays, and for instance [10,11] for the extension of the transfer function formalism to the nonlinear case.…”

Submersive rational difference systems and their accessibility

“…Let Σ stand for the rational difference system in (14), and call y = g an output equation. We assume that Σ is submersive, and that Ω Σ , H ∞ and the H i are the same as those in Section 5.…”