Abstract: Over the past three decades there has been interest in using Padé approximants K with n = deg(K) < deg(G) = N as “reduced-order models” for the transfer function G of a linear system. The attractive feature of this approach is that by matching the moments of G we can reproduce the steady-state behavior of G by the steady-state behavior of K, for certain classes of inputs. Indeed, we illustrate this by finding a first-order model matching a fixed set of moments for G, the causal inverse of a heat equation. A key feature of this example is that the heat equation is a minimum phase system, so that its inverse system has a stable transfer function G and that K can also be chosen to be stable. On the other hand, elementary examples show that both stability and instability can occur in reduced order models of a stable system obtained by matching moments using Padé approximants and, in the absence of stability, it does not make much sense to talk about steady-state responses nor does it make sense to match moments. In this paper, we review Padé approximants, and their intimate relationship to continued fractions and Riccati equations, in a historical context that underscores why Padé approximation, as useful as it is, is not an approximation in any sense that reflects stability. Our main results on stability and instability states that if N ≥ 2 and ℓ, r ≥ 0 with 0 < ℓ + r = n < N there is a non-empty open set U ℓ,r of stable transfer functions G, having infinite Lebesque measure, such that each degree n proper rational function K matching the moments of G has ℓ poles lying in ${\mathbb C}^{-}$ . and r poles lying in ${\mathbb C}^{+}$ . The proof is constructive.

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{{Explor lien
   |wiki=    Wicri/Musique
   |area=    OperaV1
   |flux=    Main
   |étape=   Exploration
   |type=    RBID
   |clé=     ISTEX:D54ADC434A7F8EBCC51E3DCD370800A9BA1A263E
   |texte=   On the Stability and Instability of Padé Approximants
}}