A normal form for pure differential algebraic systems
type: Item
instance of: scholarly article
title: A normal form for pure differential algebraic systems
zbMATH Open document ID: 1165.34002
author: Stephan Trenn
published in: Linear Algebra and its Applications
publication date: 2009-02-10T00:00:00Z
review text: The paper deals with differential algebraic equations (DAEs) of the form \[ E \dot x = Ax + bu, \quad y=cx, \eqno(1) \] where \(x \in {\mathbb R}^n\) and \(y \in {\mathbb R}^1\) are variables depending on the real time \(t\), \(\dot x = \frac{dx}{dt}\), \(E\) and \(A\) are \((n \times n)\)-matrices with constant coefficients, \(b\) is a constant column vector, \(c\) is a constant row vector, \(u\) is a scalar input (control). The equations \(E \dot x = Ax + bu\) are also called descriptor system, quasilinear systems or generalized vector field of a constant rang, the studying of such DAEs goes back to Weierstrass and Kronecker. In the case \(\det E \neq 0\) the DAE (1) turns into a standard ODE. In this paper the author mainly consider the antipodal case, so-called pure DAEs: \(\det E = 0\), \(\det A \neq 0\), and the matrix \(A^{-1}E\) is nilpotent. The rational function \(g(s) = c(Es-A)^{-1}b\) is called the transfer function of (1). In particular, the transfer functions of pure DAEs (1) are polynomials. The author proves that for a given rational function \(g(s)\) there exists DAE (1) with the transfer function \(g(s)\), which is a composition of some pure DAE and some ODE. The main result of the paper is the normal form (Theorem 20) for pure DAEs (1), which allows to simplify the matrices \(E, A\) and vectors \(b ,c\). Here, the equivalence of two DASs (1) with the same dimension \(n\) and corresponding \((c_1, E_1, A_1, b_1)\) and \((c_1, E_1, A_1, b_1)\) means that there exist invertible \((n \times n)\)-matrices \(S\) and \(T\) such that \(c_2=c_1 T\), \(E_2=S E_1 T\), \(A_2=S A_1 T\), \(b_2=S b_1\). Using this normal form, the author gives characterizations of impulsive controllability and impulsive observability for pure DAEs (1).
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 15A99
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B07
zbMATH DE Number: 5505614
zbMATH Keywords: differential algebraic equations
zbMATH Keywords: descriptor systems
zbMATH Keywords: normal form
zbMATH Keywords: impulse controllability
zbMATH Keywords: impulse observability
reviewed by: Alexey O. Remizov
MaRDI profile type: Publication
full work available at URL: j.laa.2008.10.004
OpenAlex ID: W2004695528
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DOI: 10.1016/J.LAA.2008.10.004
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description: scientific article; zbMATH DE number 5505614