On the Kronecker canonical form of a full row rank matrix pencil and applications

2019 ◽  
Author(s):  
Sophia D Karathanasi ◽  
Nicholas P Karampetakis
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Transformation Matrices ◽  
Rank Matrix ◽  
The Right ◽  
Kronecker Canonical Form

Abstract The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

Control Constraint Realization for Multibody Systems

2009 ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic under-actuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that Differential-Algebraic Equations (DAE) need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

scholarly journals Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Closed Form ◽  
Lie Group ◽  
Iterative Scheme ◽  
Differential Algebraic Equations ◽  
Group Method ◽  
Computational Results ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Lie Group Method

We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.

scholarly journals On generalized inverses of singular matrix pencils

2011 ◽  
Vol 21 (1) ◽  
pp. 161-172 ◽  
Author(s):  
Klaus Röbenack ◽  
Kurt Reinschke
Keyword(s):  
Linear Time ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Algebraic Equations ◽  
Singular Matrix ◽  
Linear Time Invariant ◽  
Matrix Pencils ◽  
Linear Differential

On generalized inverses of singular matrix pencilsLinear 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.

scholarly journals On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Zaiyong Feng ◽  
Ning Chen
Keyword(s):  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Differential Algebraic Equations ◽  
Equivalent Transformation ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
Matrix Pair ◽  
Kronecker Canonical Form

The existence and uniqueness of the solution of a new kind of system—linear fractional differential-algebraic equations (LFDAE)—are investigated. Fractional derivatives involved are under the Caputo definition. By using the tool of matrix pair, the LFDAE in which coefficients matrices are both square matrices have unique solution under the condition that coefficients matrices make up a regular matrix pair. With the help of equivalent transformation and Kronecker canonical form of the coefficients matrices, the sufficient condition for existence and uniqueness of the solution of the LFDAE in which coefficients matrices are both not square matrices is proposed later. Two examples are given to justify the obtained theorems in the end.

Stability of Linear Mechanical Systems With Holonomic Constraints

1993 ◽  
Vol 46 (11S) ◽  
pp. S160-S164 ◽  
Author(s):  
Peter C. Mu¨ller
Keyword(s):  
Singular Systems ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Algebraic Equations ◽  
Holonomic Constraints ◽  
Lyapunov Matrix Equation ◽  
Lyapunov Matrix ◽  
Complete Set ◽  
And Control

Singular systems (descriptor systems, differential-algebraic equations) are a recent topic of research in numerical mathematics, mechanics and control theory as well. But compared with common methods available for investigating regular systems many problems still have to be solved making also available a complete set of tools to analyze, to design and to simulate singular systems. In this contribution the aspect of stability is considered. Some new results for linear singular systems are presented based on a generalized Lyapunov matrix equation. Particularly, for mechanical systems with holonomic constraints the well-known stability theorem of Thomson and Tait is generalized.

Control Constraint Realization for Multibody Systems

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Alessandro Fumagalli ◽  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Paolo Mantegazza
Keyword(s):  
Linear Systems ◽  
Multibody Systems ◽  
Matrix Pencil ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Control Constraint ◽  
Practical Observation ◽  
Alternative Condition ◽  
Deformable Bodies

This paper discusses the problem of control constraint realization applied to generic underactuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that differential-algebraic equations need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

scholarly journals Condensed Forms for Linear Port-Hamiltonian Descriptor Systems

2019 ◽  
Vol 35 ◽  
pp. 65-89 ◽  
Author(s):  
Lena Scholz
Keyword(s):  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Hamiltonian Formulation ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Constant Rank ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
Structure Preserving ◽  
Strangeness Index

Motivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by $\mu\leq1$.

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