Realisation and Approximation of Linear Infinite-Dimensional Systems with Error Bounds

Abstract

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.