Some considerations on nonminimality, ill-conditioning, and instability of Padé approximants

Critical conditions of model simplification are pointed out and investigated. In particular, the problem of setting up an rth order model for an nth order system (with r<n) using the Padé technique is considered. The different situations of model uniqueness, non-existence, and non-minimality are classified according to the ranks of Hankel matrices formed from the original system Markov parameters (for example, the MacLaurin expansion coefficients of its transfer function). The case is then analysed where the reduced-order model parameters are given by an ill-conditioned set of equations so that small changes in the original system cause large variations of the model. Finally, the problem of reduced-order model instability is investigated and its relation to the ill-conditioned situation is pointed out.