This paper presents a near-optimal algorithm for H-infinity identification of fixed order rational discrete time transfer functions that are not affinely parameterized. As an experimental condition we assume that the measured output signal is corrupted by component-wise additive disturbance whose amplitude is bounded by a known value sigma (nu). It is also assumed that the rxed order rational model and true plant do not necessarily belong to the same sets and the distance in H-infinity norm between the true plant and the set of all rxed order (stable) rational models, does not exceed a known value gamma. Provided that the input signal consists of all sequences of -1 and 1, the near-optimal algorithm identifies a rxed order model such that the additive H-infinity distance between the model and plant is asymptotically bounded by 2 sigma (nu) + gamma. A modified near-optimal algorithm is also presented which does not use the knowledge of gamma. An iterative algorithm is presented to overcome the non-convexity of the resulting optimization problems. In each iteration of this algorithm a mixed LMI-LP problem is solved. Numerical examples illustrate the results.