The class of linear infinite-dimensional systems with finite-dimensional inputs and outputs whose impulse response h satisfies $h \in L_1 \cap L_2 (0,\infty ;\mathbb{C}^{p \times m} )$ and induces a nuclear Hankel operator is said to be of nuclear type. For this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies. Furthermore, explicit $L_\infty$ bounds on the transfer function errors, $L_1$ and $L_2$ bounds on the impulse response errors, and Hilbert-Schmidt and nuclear bounds on the Hankel operator errors are obtained. These truncations also generate an approximating sequence to the optimal Hankel-norm approximations to the original system, and various error bounds of these approximants are deduced.

We consider system identification in H_∞ in the framework proposed by Helmicki, Jacobson and Nett. An algorithm using the Jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems. It is shown that this, and similar identification algorithms, can be successfully combined with a model reduction procedure to produce low-order models. Connections with the Nevanlinna-Pick interpolation problem are explored, and an algorithm is given in which the identified model interpolates the given noisy data. Some numerical results are provided for illustration. Finally, the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions.