A parameterization of an adaptive infinite impulse response (IIR) filter's poles is developed, based on balanced realization theory. From this, we develop a local approximation of the actual adapted pole parameters, in which convergence speed is related to a certain eigenvalue spread. This, in turn, is shown to relate to the Hankel singular values of the system to be identified, as well as certain coefficient sensitivity functions of the adapted filter. The local approximation is not restricted to stationary points. At these points, however, it is equivalent to a Hessian approximation, with the benefit of decomposing the Hessian matrix into terms related to the aforementioned singular values and sensitivity functions. The description of the adaptation process by means of the developed approximation leads to a greater understanding of the effects on convergence speed of factors, such as the Hankel singular values of the system, its order, the distribution of its poles, and the choice of adapted parameters. In particular, the use of direct form and lattice parameters are compared in detail. Based on these properties, a new adaptive IIR algorithm with faster convergence and relatively low computational complexity has recently been proposed, which is briefly mentioned. Results also indicate a potential for variable gain algorithms