Two-time-scale distributions and singular perturbations

Abstract

vious advantages.Two-time-scale (TTS) distributions are introduced. For a class of stable systems, it is shown that every TTS distribution has a two-frequency-scale (TFS) Laplace transform. Conversely, it is shown that the impulse response of any stable TFS transfer function, and hence any stable (standard) singularly perturbed system, can be characterized in terms of a stable TTS distribution. A time domain decomposition for TTS distributions is obtained which parallels the slow and fast decompositions of singularly perturbed systems and also the frequency domain decompositions of TFS transfer functions. It is shown that every stable TTS

Realization theory for two-time-scale distributions through approximation of Markov parameters

“…The set of stable TTS distributions, that is, those with <½ ji ð0Þ > 0 for all 1 i n j and j ¼ 1, 2, is denoted by D . As was shown in Oloomi and Shafai (2004), a notable feature of a TTS distribution is that it can be decomposed in terms of two simpler distributions, namely, let hðt, Þ ¼ dðÞðtÞ þ h a ðt, Þ 2 D , where d() is analytic at ¼ 0 and h a ðt, Þ 2 L 1 ½0, 1Þ Define the slow distribution, h S (t), and the fast distribution, h F (t/), associated with h(t/) as h S ðtÞ :¼ hðt, Þj ¼0 :¼ eðtÞ þ h aS ðtÞ,…”

“…The concept of TTS distributions was introduced in Oloomi and Shafai (2004). The definition is repeated here for reference.…”

“…A notable feature of these distributions is that they can be decomposed in terms of two simpler distributions. The TTS distributions were first introduced by Oloomi and Shafai (2004) as the impulse response characterization of singularly perturbed systems. This characterization provides a framework for the study of TTS systems from their input-output behavior in the time domain -the obvious advantage being the coordinate free setting in which the problem is posed.…”

“…Takingĥðs, Þ as the Laplace transform of h(t, ), it is shown in Oloomi and Shafai (2004) thatĥ S ðsÞ andĥ F ðsÞ are indeed the Laplace transforms of h S (t) and h F (t/), respectively.…”