Quasi-finite-rank approximation of compression operators in sampled-data systems and time-delay systems

This article studies an approximation problem of compression operators that play important roles in the operator-theoretic approach to sampled-data systems and time-delay systems. Taking account of the ease in the use of approximated operators in such an approach, we confine ourselves to a special form of approximation that we call quasi-finite-rank approximation. We first introduce a suboptimal method for such approximation by means of what we call the fast-lifting approach, with which the problem can be reduced to a finite-dimensional one in an approximate sense. We further introduce another method that somehow applies the first method in a nested fashion for better overall performance. We also discuss the asymptotic exactness of the suboptimal optimisation methods as the integer parameter M in the fast-lifting approach tends to infinity, as well as the associated monotonic properties as to the reduction of the deviation from the optimal approximation. Numerical examples are given to demonstrate the effectiveness of these approximation techniques.