32,879 research outputs found
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
A novel iterative method to approximate structured singular values
A novel method for approximating structured singular values (also known as
mu-values) is proposed and investigated. These quantities constitute an
important tool in the stability analysis of uncertain linear control systems as
well as in structured eigenvalue perturbation theory. Our approach consists of
an inner-outer iteration. In the outer iteration, a Newton method is used to
adjust the perturbation level. The inner iteration solves a gradient system
associated with an optimization problem on the manifold induced by the
structure. Numerical results and comparison with the well-known Matlab function
mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of
the method
Approximated structured pseudospectra
Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra
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