Outer transfer functions of differential-algebraic systems

¹ Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany.
achim.ilchmann@tu-ilmenau.de
² Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany.
timo.reis@math.uni-hamburg.de

Received: 4 October 2014
Revised: 21 June 2015
Accepted: 26 November 2015

Abstract

We consider differential-algebraic systems (DAEs) whose transfer function is outer: i.e., it has full row rank and all transmission zeros lie in the closed left half complex plane. We characterize outer, with the aid of the Kronecker structure of the system pencil and the Smith–McMillan structure of the transfer function, as the following property of a behavioural stabilizable and detectable realization: each consistent initial value can be asymptotically controlled to zero while the output can be made arbitrarily small in the ℒ²-norm. The zero dynamics of systems with outer transfer functions are analyzed. We further show that our characterizations of outer provide a simple and very structured analysis of the linear-quadratic optimal control problem.