Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function

Yi, Sun; Ulsoy, A. Galip

doi:10.1109/acc.2007.4282376

Cited by 19 publications

(25 citation statements)

References 30 publications

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“…Therefore, any requirement on the controllability of the time-delay system (23) is considered in the matrix condition (19). The topics of controllability and observability for linear time-delay systems are thoroughly treated in [28,29], and the references therein.…”

Section: Remarkmentioning

confidence: 99%

Robust output regulation of linear time-delay systems: A state predictor approach

Yoon

Lin

2015

Int. J. Robust. Nonlinear Control

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Summary This paper considers the robust output regulation problem for linear systems in the presence of state, input, and output delays. First, a state feedback control law is constructed from a state predictor, recently developed for systems with state and input delays. Necessary and sufficient conditions for the existence of a solution to the regulator equations are presented. Next, an error feedback control law for the output regulation problem is constructed by means of an error‐based state predictor. Finally, the proposed error feedback solution is extended to solve the output regulation problem in the presence of model uncertainty. Numerical examples demonstrate the effectiveness of the proposed predictor‐based solutions. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: Remarkmentioning

confidence: 99%

Robust output regulation of linear time-delay systems: A state predictor approach

Yoon

Lin

2015

Int. J. Robust. Nonlinear Control

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“…Definition 1 (Controllability of a TDS) [36]: The system (1) is point-wise controllable if, for any given initial conditions g(t) and x 0 , there exists 0 < t 1 < 1, and an admissible (i.e. measurable and bounded on a finite time interval) input u(t) for t 2 [0, t 1 ] such that x(t 1 , 0, g(t), u(t)) = 0.…”

Section: Controllability and Observability Gramians Of Tdssmentioning

confidence: 99%

“…check the point-wise controllability of linear TDSs [33]. In [36], the Lambert W function is used to define the Gramian matrices analytically. Algebraic Gramians are derived from the time-domain solution to system (1) using the matrix Lambert W function [36] x t ð Þ ¼…”

Section: Controllability and Observability Gramians Of Tdssmentioning

confidence: 99%

Gramian‐based model order reduction of parameterized time‐delay systems

Wang

Zhang

Wang

et al. 2012

Circuit Theory & Apps

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Time-delay systems (TDSs) frequently arise in circuit simulation especially in high-frequency applications. Model order reduction (MOR) techniques can be used to facilitate the simulation of TDSs. On the other hand, many kinds of variations, such as temperature and geometric uncertainties, can have significant impact on the transient responses of TDSs. Therefore, it is important to preserve parametric dependence during the MOR procedure. This paper presents a new parameterized MOR scheme for TDSs with parameter variations. We derive parameterized reduced-order models (ROMs) for TDSs using balanced truncation by approximating the Gramians in the multi-dimensional space of parameters. The resulting ROMs can preserve the parametric dependence, making it efficient for repeated simulations under different parameter settings. Numerical examples are presented to verify the accuracy and efficiency of our proposed algorithm.

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“…The matrix Lambert W function arises in the numerical solution and stability analysis of delay differential (systems of) equations [5], [9], [25], [32], [33], where the principal Lambert W function of a matrix, W 0 (A), is used to deduce properties of the stability of the system. Cepeda-Gomez and Michiels [9] show with examples that the principal branch is not sufficient to determine the stability of all systems, but rather W −1 (A) is needed as well.…”

Section: ) (O) the Colors Of The Curves That Separate Two Adjacent Rmentioning

confidence: 99%

An Algorithm for the Matrix Lambert $W$ Function

Fasi

Higham

Iannazzo³

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. An algorithm is proposed for computing primary matrix Lambert W functions of a square matrix A, which are solutions of the matrix equation W e W = A. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert W function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert W function in a numerically reliable way.

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Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function

Cited by 19 publications

References 30 publications

Robust output regulation of linear time-delay systems: A state predictor approach

Robust output regulation of linear time-delay systems: A state predictor approach

Gramian‐based model order reduction of parameterized time‐delay systems

An Algorithm for the Matrix Lambert $W$ Function

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