A family of virtual contraction based controllers for tracking of flexible-joints port-Hamiltonian robots:Theory and experiments

In this work, we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots in the port-Hamiltonian framework. The proposed design method, called virtual contraction based control, combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity, and the existence of (incrementally) passive maps. The method is later applied to a planar RR robot, and two nonlinear tracking control schemes in the developed controllers family are designed using different contraction analysis approaches. Experiments confirm the theoretical results for each controller

Nonlinear Eigenvalue Approach to Differential Riccati Equations for Contraction Analysis

In this paper, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian matrix. Moreover, under an assumption for the differential Hamiltonian matrix, real symmetricity, regularity, and positive semidefiniteness of solutions are characterized by nonlinear eigenvalues and eigenvectors

A geometric approach to differential Hamiltonian systems and differential Riccati equations

Motivated by research on contraction analysis and incremental stability/stabilizability the study of 'differential properties' has attracted increasing attention lately. Previously lifts of functions and vector fields to the tangent bundle of the state space manifold have been employed for a geometric approach to differential passivity and dissipativity. In the same vein, the present paper aims at a geometric underpinning and elucidation of recent work on 'control contraction metrics' and 'generalized differential Riccati equations'

Non-viscous Regularization of the Davey-Stewartson Equations: Analysis and Modulation Theory

In the present study we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in variety of parameter regimes under which the finite blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the $\alpha$-models of turbulence and therefore are called the $\alpha$-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the $\alpha$-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work of Cao, Musslimani and Titi on the non-viscous $\alpha$-regularization of the nonlinear Schr\"odinger equation

The Topography of \W_\infty-Type Algebras

We chart out the landscape of \Winfty-type algebras using \Wkpq---a recently discovered one-parameter deformation of \W_{\rm KP}. We relate all hitherto known \Winfty-type algebras to \Wkpq and its reductions, contractions, and/or truncations at special values of the parameter.Comment: 15 pages, Plain TeX, BONN-HE-92-24, US-FT-8/92, KUL-TF-92/32. [This new version contains some minor revisions due to the revision of hep-th/9207092.

Gardner's deformations of the N=2 supersymmetric a=4-KdV equation

We prove that P.Mathieu's Open problem on constructing Gardner's deformation for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry invariant solutions, whenever it is assumed that they retract to Gardner's deformation of the scalar KdV equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's deformation of the Kaup-Boussinesq equation, which is contained in the bosonic limit of the super-hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full N=2, a=4-SKdV hierarchy. Our method is applicable towards the solution of Gardner's deformation problems for other supersymmetric KdV-type systems.Comment: Extended version of the talks given by A.V.K. at 8th International conference `Symmetry in Nonlinear Mathematical Physics' (June 20-27, 2009, Kiev, Ukraine) and 9th International workshop `Supersymmetry and Quantum Symmetries' (July 29 - August 3, 2009, JINR, Dubna, Russia); 22 page

Effect of spin on electron motion in a random magnetic field

We consider properties of a two-dimensional electron system in a random magnetic field. It is assumed that the magnetic field not only influences orbital electron motion but also acts on the electron spin. For calculations, we suggest a new trick replacing the initial Hamiltonian by a Dirac Hamiltonian. This allows us to do easily a perturbation theory and derive a supermatrix sigma model, which takes a form of the conventional sigma model with the unitary symmetry. Using this sigma model we calculate several correlation functions including a spin-spin correlation function. As compared to the model without spin, we get different expressions for the single-particle lifetime and the transport time. The diffusion constant turns out to be 2 times smaller than the one for spinless particles.Comment: 7 pages, revtex, result of the spin correlation function corrected, Appendix adde