Remark on the solution of linear time‐invariant descriptor systems

Müller, Peter C.

doi:10.1002/pamm.200510066

Cited by 8 publications

(6 citation statements)

References 4 publications

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“…A typical argumentation in these works is that inconsistent initial values cause distributional solutions in a way that the state trajectory is composed of a continuous function and a linear combination of Dirac's delta impulse and some of its derivatives. However, some frequency domain considerations in [109] refute this approach. This justifies that we do only consider weakly differentiable solutions as defined in the behavior…”

Section: Lemma 21 (Inclusions For Reachability Spaces)mentioning

confidence: 99%

Controllability of Linear Differential-Algebraic Systems—A Survey

Berger

Reis

2013

Surveys in Differential-Algebraic Equations I

115

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Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, strong and complete controllability are described and defined in time-domain. Equivalent criteria that generalize the Hautus test are presented and proved. Special emphasis is placed on normal forms under state space transformation and, further, under state space, input and feedback transformations. Special forms generalizing the Kalman decomposition and Brunovsky form are presented. Consequences for state feedback design and geometric interpretation of the space of reachable states in terms of invariant subspaces are proved.

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Section: Lemma 21 (Inclusions For Reachability Spaces)mentioning

confidence: 99%

Controllability of Linear Differential-Algebraic Systems—A Survey

Berger

Reis

2013

Surveys in Differential-Algebraic Equations I

115

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“…(9a) with zero initial value constraint (9b) and Eq. (9a) with no constraint at all is an evident cause for a wrong solution expression (as pointed out in [7]).…”

Section: Consistent-inconsistent Decompositionmentioning

confidence: 97%

“…Remark 1. Through somewhat complicated computation, the formula (8) was eventually arrived at very recently in [7] and [8]. In a slightly different form it appeared in [12].…”

Section: Consistent-inconsistent Decompositionmentioning

confidence: 99%

“…Descriptor systems (singular systems, differential-algebraic equations) are an interesting research topic in numerical mathematics, mechanics, and control theory recently [1][2][3][4][5][6][7][8][9]. Among many interesting phenomena, we point out the inconsistent initial value problem, which results in, for example, the need for reinitialization in numerical integration [6] and impulse elimination in control [2] for this type of systems.…”

Section: Introductionmentioning

confidence: 99%

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Consistent‐inconsistent decomposition to initial value problem of descriptor linear systems

Yan

2008

Z Angew Math Mech

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“…The inhomogeneous solution results from an integration (convolution) of the input signal u. Contrary to that, the infinite eigenvalues correspond to implicit ODEs having impulse solutions (Müller, 2005), where the input signal might be differentiated. In the time domain, each ν i × ν i Jordan block N i can be interpreted as a chain of ν i successive differentiators.…”

Section: Matrix Pencils and Indexmentioning

confidence: 99%

On generalized inverses of singular matrix pencils

Röbenack¹,

Reinschke²

2011

International Journal of Applied Mathematics and Computer Science

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Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

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Remark on the solution of linear time‐invariant descriptor systems

Abstract: For linear time‐invariant descriptor systems the correct solution in the sense of distributions is discussed for inconsistent initial conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cited by 8 publications

References 4 publications

Controllability of Linear Differential-Algebraic Systems—A Survey

Controllability of Linear Differential-Algebraic Systems—A Survey

Consistent‐inconsistent decomposition to initial value problem of descriptor linear systems

On generalized inverses of singular matrix pencils

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