938 research outputs found
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
J-spectral factorization and equalizing vectors
For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a -spectral factorization. One of these conditions is in terms of equalizing vectors. A second one states that the existence of a -spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory. Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard-Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given. For Riesz-spectral systems another necessary and sufficient conditions for the existence of a -spectral factorization in terms of the Hamiltonian is added
A Survey of Riccati Equation Results in Negative Imaginary Systems Theory and Quantum Control Theory
This paper presents a survey of some new applications of algebraic Riccati equations. In particular, the paper surveys some recent results on the use of algebraic Riccati equations in testing whether a system is negative imaginary and in synthesizing state feedback controllers which make the closed loop system negative imaginary. The paper also surveys the use of Riccati equation methods in the control of quantum linear systems including coherent H∞ control.This work was supported by the Australian Research Council (ARC) under grants FL110100020 and DP160101121 and the Air Force Office of Scientific Research (AFOSR), under agreement number FA2386-16-1-4065
A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations
The worst situation in computing the minimal nonnegative solution of a
nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when
the corresponding linearizing matrix has two very small eigenvalues, one with
positive and one with negative real part. When both these eigenvalues are
exactly zero, the problem is called critical or null recurrent. While in this
case the problem is ill-conditioned and the convergence of the algorithms based
on matrix iterations is slow, there exist some techniques to remove the
singularity and transform the problem to a well-behaved one. Ill-conditioning
and slow convergence appear also in close-to-critical problems, but when none
of the eigenvalues is exactly zero the techniques used for the critical case
cannot be applied.
In this paper, we introduce a new method to accelerate the convergence
properties of the iterations also in close-to-critical cases, by working on the
invariant subspace associated with the problematic eigenvalues as a whole. We
present a theoretical analysis and several numerical experiments which confirm
the efficiency of the new method
Decay rate estimations for linear quadratic optimal regulators
Let be the optimal control of the open-loop system
in a linear quadratic optimization problem. By using
different complex variable arguments, we give several lower and upper estimates
of the exponential decay rate of the closed-loop system .
Main attention is given to the case of a skew-Hermitian matrix .
Given an operator , for a class of cases, we find a matrix that
provides an almost optimal decay rate.
We show how our results can be applied to the problem of optimizing the decay
rate for a large finite collection of control systems , , and illustrate this on an example of a concrete mechanical system. At the
end of the article, we pose several questions concerning the decay rates in the
context of linear quadratic optimization and in a more general context of the
pole placement problem.Comment: 25 pages, 1 figur
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