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On spectra of quadratic operator pencils with rank one gyroscopic linear part

Boyko O., Martynyuk O., Pivovarchik V.

Opuscula Mathematica

2018

Vol. 38, no. 4

483--500

The spectrum of a selfadjoint quadratic operator pencil of the form [formula] is investigated where M ≥ 0, G ≥ 0 are bounded operators and A is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator G the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator G. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators A are given. Keywords: q

Interconnection and damping assignment passivity-based control of an underactuated 2-DOF gyroscope

Cordero G., Santibáñez V., Dzul A., Sandoval J.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 4

661--677

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2-DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.