Hamiltonian Realizations of Nonlinear Adjoint Operators

This paper addresses state-space realizations for nonlinear adjoint operators. In particular the relationship among nonlinear Hilbert adjoint operators, Hamiltonian extensions and port-controlled Hamiltonian systems are clarified. The characterization of controllability, observability and Hankel operators, and controllability and observability functions will be derived based on it. Furthermore a duality between the controllability and observability functions will be proven. The state-space realizations of such operators provide new insights to nonlinear control systems theory

Nonlinear input-normal realizations based on the differential eigenstructure of hankel operators

This paper investigates the differential eigenstructure of Hankel operators for nonlinear systems. First, it is proven that the variational system and the Hamiltonian extension with extended input and output spaces can be interpreted as the Gâteaux differential and its adjoint of a dynamical input-output system, respectively. Second, the Gâteaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear Hankel operator which is closely related to the Hankel norm of the original system. Third, a new characterization of the nonlinear extension of Hankel singular values are given based on the differential eigenstructure. Finally, a balancing procedure to obtain a new input-normal/output-diagonal realization is derived. The results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems.

Alternative linear structures for classical and quantum systems

The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different non equivalent ways, "evading", so to speak, the von Neumann uniqueness theorem.Comment: 32 pages, two figures, to be published in IJMP

Duality and singular value functions of the nonlinear normalized right and left coprime factorizations

This paper considers the nonlinear left coprime factorization (NLCF) of a nonlinear system. In order to study the balanced realization of such NLCF first a dual system notion is introduced. The important energy functions for the original NLCF and their relation with the dual NLCF are studied and relations between these functions are established. These developments can be used for studying a relation between the singular value functions of the NLCF and the normalized right coprime factorization (NRCF) of a nonlinear system. The singular value functions are a useful tool for model reduction of unstable nonlinear systems.