Realization theory for two-time-scale distributions
H. Oloomi, Bahram Shafai · 2004
In this paper we investigate the problem of approximating the Markov parameters of a two-time-scale distribution. It is shown that the Markov parameters of a two-time-scale distribution can be approximated in terms of the Markov parameters of its fast distribution only. This is an approximation, which deteriorates by a factor of 1//spl epsi/ for every higher order Markov parameter. In order to use the Markov parameters of the slow distribution in the approximation scheme, an inversion map is introduced by which a two-time-scale distribution and its slow distribution are mapped into new distributions. It is then shown that every Markov parameter of the inverted distribution to within an O(/spl epsi/) quantity. An approximate expression for the Hankel matrix of the Markov parameters of a two-time-scale distribution is also obtained. This expression is in form of a telescopic series and involves the Markov parameters of the fast distribution only.