The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

Kalogeropoulos, Grigoris I.; Mitrouli, Marilena; Pantelous, Athanasios A.; Triantafyllou, Dimitrios

doi:10.1007/978-3-642-00464-3_35

Cited by 5 publications

(17 citation statements)

References 5 publications

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“…As it has been extensively discussed in [6], concerning the mathematical tools used in our discretization process, only the complex Weierstrass canonical form (WCF) and some fundamental elements of matrix pencil theory are required, see also [5]. In the next lines of this section, some known results are presented.…”

Section: Problem Formulation and Preliminariesmentioning

confidence: 98%

Designing the sampling period of a discretized LTI descriptor (regular) system with inputs

Karageorgos

Pantelous

Kalogeropoulos

2011

Int. J. Control Autom. Syst.

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In this brief paper, a new sharper upper bound for the error ||x(kT) -x k || that derives from the procedure of discretization of the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is calculated and fully discussed. Practically speaking, considering numerous applications in engineering (especially in robotics and digital control) and computer science, we are very interested in determining such kind of upper bounds, since they are significant in the design process of the sampling period T.

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Section: Problem Formulation and Preliminariesmentioning

confidence: 98%

Designing the sampling period of a discretized LTI descriptor (regular) system with inputs

Karageorgos

Pantelous

Kalogeropoulos

2011

Int. J. Control Autom. Syst.

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“…Our decoupling strategy was initially used to understand the underlying structure of linear constant coefficient DAEs via the Weierstraß canonical form [8,9]. Assuming f(Ex) = 0 and that the matrix pencil (E, A) is regular, (1) can be written as a Weierstraß-Kronecker canonical form which leads to an equivalent decoupled system…”

Section: Weierstraß Canonical Formmentioning

confidence: 99%

“…are commonly known as the slow and fast parts of the solution, respectively. In [9], only complex-valued matrix pencils are considered and even for a real-valued pencil the matrix J in (3a) will be complex-valued in general, since in the Weierstraß canonical form J contains the (possibly complex) eigenvalues of the ODE part of the pencil. This can be circumvented by using the quasi-Weierstraß form introduced in [10].…”

Section: Weierstraß Canonical Formmentioning

confidence: 99%

Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

et al. 2021

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We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term $$\mathbf{f}(\mathbf{E}\mathbf{x}),$$ f ( E x ) , where $$\mathbf{E}$$ E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet accurately represent the original large-scale system behavior. However, direct reduction and simulation of nonlinear differential-algebraic equations is difficult due to hidden constraints which affect the choice of numerical integration methods and model-order reduction techniques. We propose an extension of index-aware model-order reduction methods to a special class of nonlinear differential-algebraic equations without any kind of linearization. The proposed model-order reduction approach involves automatic decoupling of nonlinear differential-algebraic equations into nonlinear ordinary differential equations and algebraic equations, based on the decoupling of the linear differential equations obtained by ignoring the nonlinear term, thanks to an additional structural condition. This allows applying standard model-order reduction techniques to both parts without worrying about the index. The same procedure can also be used to simulate nonlinear differential-algebraic equations by standard integration schemes.

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“…There are several approaches used to analyze the descriptor system and most notable of such methods is the Kronecker canonical form. Though it serves the purpose of analyzing a descriptor system, computation of the KCF for a regular pencil is still a challenge, see [16] for details. We summarize the main problem addressed in this section as follows:…”

Section: 2 and X(t) Satisfiesmentioning

confidence: 99%

“…The differential and algebraic parts of a descriptor system can be decoupled using KCF. It is known, however, that computing KCF for a descriptor system is still a challenge, see [16] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors

Reddy

Engwerda²

2010

SSRN Journal

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In this article we address the problem of finding feedback Nash equilibria for linear quadratic differential games defined on descriptor systems. First, we decouple the dynamic and algebraic parts of a descriptor system using canonical projectors. We discuss the effects of feedback on the behavior of the descriptor system. We derive necessary and sufficient conditions for the existence of the feedback Nash equilibria for index 1 descriptor systems and show that there exist many informationally non-unique equilibria corresponding to a single solution of the game. Further, for descriptor systems with index greater than 1, we give a regularization based approach and discuss the associated drawbacks.

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The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

Cited by 5 publications

References 5 publications

Designing the sampling period of a discretized LTI descriptor (regular) system with inputs

Designing the sampling period of a discretized LTI descriptor (regular) system with inputs

Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors

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