898 research outputs found
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback
We develop a complete state-space solution to H_2-optimal decentralized
control of poset-causal systems with state-feedback. Our solution is based on
the exploitation of a key separability property of the problem, that enables an
efficient computation of the optimal controller by solving a small number of
uncoupled standard Riccati equations. Our approach gives important insight into
the structure of optimal controllers, such as controller degree bounds that
depend on the structure of the poset. A novel element in our state-space
characterization of the controller is a remarkable pair of transfer functions,
that belong to the incidence algebra of the poset, are inverses of each other,
and are intimately related to prediction of the state along the different paths
on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic
Contro
Linear Quadratic Games: An Overview
In this paper we review some basic results on linear quadratic differential games.We consider both the cooperative and non-cooperative case.For the non-cooperative game we consider the open-loop and (linear) feedback information structure.Furthermore the effect of adding uncertainty is considered.The overview is based on [9].Readers interested in detailed proofs and additional results are referred to this book.linear-quadratic games;Nash equilibrium;affine systems;solvability conditions;Riccati equations
Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction
We review a family of algorithms for Lyapunov- and Riccati-type equations
which are all related to each other by the idea of \emph{doubling}: they
construct the iterate of another naturally-arising fixed-point
iteration via a sort of repeated squaring.
The equations we consider are Stein equations , Lyapunov
equations , discrete-time algebraic Riccati equations
, continuous-time algebraic Riccati equations
, palindromic quadratic matrix equations , and
nonlinear matrix equations . We draw comparisons among these
algorithms, highlight the connections between them and to other algorithms such
as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge
Solving rank structured Sylvester and Lyapunov equations
We consider the problem of efficiently solving Sylvester and Lyapunov
equations of medium and large scale, in case of rank-structured data, i.e.,
when the coefficient matrices and the right-hand side have low-rank
off-diagonal blocks. This comprises problems with banded data, recently studied
by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for
large-scale interconnected systems", Automatica, 2016, and by Palitta and
Simoncini in "Numerical methods for large-scale Lyapunov equations with
symmetric banded data", SISC, 2018, which often arise in the discretization of
elliptic PDEs.
We show that, under suitable assumptions, the quasiseparable structure is
guaranteed to be numerically present in the solution, and explicit novel
estimates of the numerical rank of the off-diagonal blocks are provided.
Efficient solution schemes that rely on the technology of hierarchical
matrices are described, and several numerical experiments confirm the
applicability and efficiency of the approaches. We develop a MATLAB toolbox
that allows easy replication of the experiments and a ready-to-use interface
for the solvers. The performances of the different approaches are compared, and
we show that the new methods described are efficient on several classes of
relevant problems
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