Lethargy results in LTI System Modelling

Abstract

Modelling of linear time-invariant (LTI) systems is studied in an approximate modelling framework motivated by modern robust control theory. As convolution-type LTI systems are represented by their unit impulse responses (i.e. through a sequence of real (complex) numbers), this paper provides also system theoretic results on modelling of sequences and their associated transfer functions. The modelling results derived in this paper are mostly of lethargy type, i.e. they establish limitations in approximate modelling of LTI systems. Non-existence of coprime factorizations is established for a number of different types of LTI systems. This leads to the somewhat surprising conclusion that many LTI systems are not even stabilizable and that an even larger class of LTI systems cannot be well-approximated by rational transfer function models. In addition, it is discussed that often the rate of rational approximation of LTI systems is not essentially better than the rate of approximation in some basis (such as finite impulse response (FIR) models). Furthermore, different definitions of the important finite power signal setup are studied and new results given.