Bilinear Case Research Articles

Realizability of multilinear input/output maps

Abstract This paper examines the realization theory of multilinear input/output maps. It is shown that the existence of a transfer function representation with separable denominator does not guarantee finite state realizability as in the linear and bilinear cases. Conditions for quasi-reachability of a class of multilinear realizations are demonstrated using linear methods, and a form of stability condition obtained for an associated class of multilinear transfer functions. Conditions for observability, which may also be derived from the theory of polynomial response maps, are outlined.

An algebraic structure of discrete-time biaffine systems

New results on the realization of finite-dimensional, discrete-time, internally biaffine systems are presented in this paper. The external behavior of such systems is described by multiaffine functions and the state space is constructed via Nerode equivalence relations. We prove that the state space is an affine space. An algorithm which amounts to choosing a frame for the affine space is presented. Our algorithm reduces in the linear and bilinear case to a generalization of algorithms existing in the literature. Explicit existence criteria for span-canonical realizations as well as an affine isomorphism theorem are given.

Minimal realizations and canonical forms for bilinear systems

The minimal realization theory for input–output maps that arise from finite- dimensional, continuous time, bilinear systems is discussed. It is shown that an observed bilinear system (i.e. a bilinear system together with an observation functional, but without a fixed initial state) is completely determined by its input–output correspondence, i.e. its multivalued input–output map. A precise formulation and proof of the result are given that continuous canonical forms for bilinear systems do not exist. This is done by adapting to the bilinear case an idea due to Hazewinkel and Kalman for the case of linear systems. In addition, the proof presented here has the advantage of not involving any algebraic geometry, which makes it considerably simpler than the original proof of the linear systems result.

Sensitivity analysis in. multilinear systems†

The primary objective of this study is to examine tho sensitivity properties of a multilinear plant embedded in a feedback structure. As a prelude tho existing linear theory ia summarized and extended. Tho bilinear case is then treatod in detail. Extrapolation to the multilinear ease has been carried out by Porter and DeSantis (1974 a). New results includo the following : (i) the use of causality structure as a component of sensitivity analysis in closed-loop linear and non-linear systems, (ii) the establishment of terminal equivalence conditions for multilinear plants and multilinear compensators, (iii) the development of a sensitivity operator in the multilinear setting including existence conditions, (iv) extensions of linear results to largo parameter perturbations.

Bilinear optical polarizability of silver

The linear dielectric constant and the bilinear optical polarizability of silver are calculated in the long wavelength limit, using a simple 2-band model for the metal consisting of a completely filled flatd-band and a half-filled free-electron-like conduction band. It is shown that the results thus obtained, compare well with the experimental results for the linear dielectric constantɛ(ω), forħω between 1 eV and 5 eV. In the bilinear case, the reflectivity of the second harmonic wave, when the incident laser beam is polarized perpendicular to the plane of incidence, is shown to have a broad resonance whenɛ(2ω) becomes very small. However, there is no resonance in the reflectivity of the second harmonic wave in silver at the threshold frequency for the interband absorption of this wave.