Systematic discretization of input-output maps of linear infinite dimensional systems (revised)

Author(s) : Michael Schmidt

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 29-2008

MSC 2000

93C20 Systems governed by partial differential equations

35B37 PDE in connection with control problems

Abstract :
Many model reduction techniques take a semi-discretization of a PDE model as starting point and aim then at an accurate approximation of its input/output map. In this contribution, we discuss the direct discretization of the i/o map of the infinite-dimensional system for a general class of linear time-invariant systems with distributed inputs and outputs. First, the input and output signals are discretized in space and time, resulting in the matrix representation of an approximated i/o-map. In a generalized sense, the matrix contains the Markov parameters of a corresponding time-discrete multi-input-multi-output system. Second, the system dynamics is approximated in form of the underlying evolution operator, in order to calculate the matrix representation numerically. The discretization framework, corresponding error estimates, a SVD-based system reduction method and a numerical application in an optimization problem are presented, and illustrated for a heat control system.

Keywords : input-output map, discretization, infinite-dimensional control system, time-discrete MIMO system, model reduction, optimization, feedback control