An invariant subspace method for large-scale algebraic Riccati equation

Amodei, Luca; Buchot, Jean-Marie

doi:10.1016/j.apnum.2009.09.006

Cited by 28 publications

(44 citation statements)

References 27 publications

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“…The equivalence between both procedures is easily established using the relation P = E X E. We refer to [2] for more details about the GARE.…”

Section: The Approximating Nonlinear Systemmentioning

confidence: 99%

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

Tiago

2015

Applied Numerical Mathematics

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“…The equivalence between both procedures is easily established using the relation P = E X E. We refer to [2] for more details about the GARE.…”

Section: The Approximating Nonlinear Systemmentioning

confidence: 99%

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

Tiago

2015

Applied Numerical Mathematics

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“…In , we have defined a family of approximations

\overset{X}{false}

of the solution X by considering stable invariant subspaces of H of dimension k ⩽ n . The approximations

\overset{X}{false}

are defined by

\overset{X}{false} MathClass-rel= Z {(Z^{MathClass-bin蜧} Y)}^{MathClass-bin+} Z^{MathClass-bin蜧} MathClass-punc,

where ( Z * Y ) + denotes the Moore‐Penrose pseudo‐inverse of the matrix ( Z * Y ) and Y ,

Z MathClass-rel\in {double-struckC}^{n MathClass-bin\times k}

satisfy Equation where

{\overset{Λ}{false}}_{s} MathClass-rel\in {double-struckC}^{n MathClass-bin\times n}

is replaced by

Λ_{s} MathClass-rel\in {double-struckC}^{k MathClass-bin\times k}

with

σ (Λ_{s}) MathClass-rel\subset {double-struckC}^{MathClass-bin-}

.…”

Section: The General Solution Of the Algebraic Riccati Equationmentioning

confidence: 99%

“…The feedback matrix gain is then defined by using the stabilizing solution of the algebraic Bernoulli equation (ABE) associated with the linear differential system ( S ) (Equation ). From the general expression of the solution of the ARE, introduced in , we derive the explicit factorized form of the unique stabilizing solution X of the ABE. We show in Proposition that the matrix X involves the invariant subspace associated with the unstable eigenvalues of the system matrix and the solution of a Lyapunov equation of dimension equal to the number k of unstable eigenvalues (counting the multiplicities).…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 presents the main results concerning the general expression of the solution of ARE obtained from the stable invariant subspaces of the Hamiltonian matrix. We present the particular factorized form used in to get low‐rank approximations of the solution of ARE. The same factorized form gives a new expression of the solution of ARE.…”

Section: Introductionmentioning

confidence: 99%

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A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

Amodei

Buchot

2011

Numerical Linear Algebra App

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“…Besides the classical one [3], [20], several other methods were proposed to solve Riccati equations [21]- [24]. A class of method is based on the connection of the eigenvalue problems for an Hamiltonian matrix.…”

Section: Riccati Equation and Applicationsmentioning

confidence: 99%

Low-rank Gramian applications in dynamics and control

Freitas

Rommes

Martins

2011

2011 International Conference on Communications, Computing and Control Applications (CCCA)

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This paper presents an efficient linear system reduction method that computes approximations to the controllability and observability Gramians of large sparse power system descriptor models. The method exploits the fact that a Lyapunov equation solution can be decomposed into low-rank Choleski factors, which are determined by the Alternating Direction Implicit (ADI) method. Advantages of the method are that it operates on the sparse descriptor matrices and does not require the computation of spectral projections onto deflating subspaces of finite eigenvalues, which are needed by other techniques applied to descriptor models. The Gramians, which are never explicitly formed, are used to compute reduced-order statespace models for large-scale systems. Extensions of the method's application to algebraic Riccati equation computation are also considered. Numerical results for small-signal stability power system descriptor models show that the method is efficient for large-scale systems reduced-order model (ROM) computation.

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An invariant subspace method for large-scale algebraic Riccati equation

Cited by 28 publications

References 27 publications

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

Low-rank Gramian applications in dynamics and control

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