On the solution of large Sylvester‐observer equations

Calvetti, Daniela; Lewis, Bryan W.; Reichel, Lothar

doi:10.1002/nla.254

Cited by 30 publications

(27 citation statements)

References 15 publications

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“…They include (i) the well-known Hessenberg-Schur algorithm [7] for the Sylvester equation, the Hessenberg-observer algorithm by Van Dooren [8], the Hessenberg-observer algorithm by Carvalho and Datta [9], the SVD-based algorithm by Datta and Sarkissian [10], the QR-based algorithm by Carvalho et al [11]. There also exist large-scale and parallel algorithms [12][13][14] for the Sylvesterobserver equation. The paper by Bischof et al [12] has developed a parallel algorithm, while the paper by Calvetti et al [13] has proposed several important modifications of the algorithm proposed by Datta and Saad [14] for large-scale solution of the problem.…”

Section: X(t) = Ax(t)+bu(t) Y = Cx(t)mentioning

confidence: 97%

“…There also exist large-scale and parallel algorithms [12][13][14] for the Sylvesterobserver equation. The paper by Bischof et al [12] has developed a parallel algorithm, while the paper by Calvetti et al [13] has proposed several important modifications of the algorithm proposed by Datta and Saad [14] for large-scale solution of the problem.…”

Section: X(t) = Ax(t)+bu(t) Y = Cx(t)mentioning

confidence: 99%

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A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems

Carvalho¹,

Datta

2010

Numer. Linear Algebra Appl.

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We propose a new algorithm for block-wise solution of the generalized Sylvesterobserver equation XA − F XE = GC, where the matrices A, E, and C are given, the matrices X, F , and G need to be computed, and the matrix E may be singular. The algorithm is based on an orthogonal decomposition of the triplet (A, E, C) to the observer-Hessenberg-triangular form. It is a natural generalization of the widelyknown observer-Hessenberg algorithm for the Sylvester-observer equation : XA − F X = GC, which arises in state estimation of a standard first-order state-space control system. An application of the proposed algorithm is made to state and velocity estimations of second-order control systems modeling a wide variety of vibrating structures. For dense un-structured data, the proposed algorithm is more efficient than the recently proposed SVD-based algorithm of the authors, it is heavily composed of BLAS-3 (Level 3 Basic Linear Algebra Subprograms) operations, which make the algorithm an ideal candidate for high-performance computing.

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Section: X(t) = Ax(t)+bu(t) Y = Cx(t)mentioning

confidence: 97%

Section: X(t) = Ax(t)+bu(t) Y = Cx(t)mentioning

confidence: 99%

A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems

Carvalho¹,

Datta

2010

Numer. Linear Algebra Appl.

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“…Let H m = [h i, j ] be an m × m unreduced upper Hessenberg matrix, and define the following quantities: Moreover, if the set {μ j } j=1,...,m is invariant under complex conjugation, then the matrixĤ m is real [5]. Next, we give some results that will be used later.…”

Section: Application Of the Global Arnoldi Process To Solution Of Thementioning

confidence: 99%

The global Arnoldi process for solving the Sylvester-Observer equation

Datta

Heyouni

Jbilou

2010

Comput. Appl. Math.

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Abstract.In this paper, we present a method and associated theory for solving the multi-input Sylvester-Observer equation arising in the construction of the Luenberger observer in control theory. The proposed method is a particular generalization of the algorithm described by Datta and Saad in 1991 to the multi-output. We give some theoretical results and present some numerical experiments to show the accuracy of the proposed algorithm.Mathematical subject classification: 65F10, 65F30.

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“…When E = I in (1), it arises in solutions of related problems of eigenstructure assignment, eigenvalue assignment and observer design for conventional linear systems [5,6]. For the solution of (1), there exist several numerical solutions, such as the SVD-based block algorithm [7,8] and the large-scale algorithms [9,10]. It is well known that one can obtain only a special solution by applying a numerical method.…”

Section: Introductionmentioning

confidence: 99%

Solving the generalized Sylvester matrix equation AV+BW=EVF via a Kronecker map

Zhu

Duan

et al. 2008

Applied Mathematics Letters

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This note considers the solution to the generalized Sylvester matrix equation AV + BW = E V F with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, some properties of the Sylvester sum are first proposed. By applying the Sylvester sum as tools, an explicit parametric solution to this matrix equation is established. The proposed solution is expressed by the Sylvester sum, and allows the matrix F to be undetermined.

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On the solution of large Sylvester‐observer equations

Cited by 30 publications

References 15 publications

A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems

A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems

The global Arnoldi process for solving the Sylvester-Observer equation

Solving the generalized Sylvester matrix equation AV+BW=EVF via a Kronecker map

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