Toward a more efficient implementation of distributed-delay elements

This note studies lumped-delay approximations of distributed-delay elements arising in various problems involving delay compensation or preview utilization. Several approaches to reduce implementation complexity, understood as the number of delays required to attain a required accuracy level, are put forward. First, we derive the H
²
-optimal coefficients for a given delay pattern. This can be seen as a problem-oriented alternative to traditionally used Newton-Cotes numerical integration rules. Second, we propose a computationally efficient method for finding the H
²
-optimal delay pattern, which uses dynamic programming on multistage graphs. Third, we propose a convex optimization approach, based on ℓ
¹
heuristics, of reducing the number of delays under hard bounds on the H
^∞
-norm of the approximation error. Numerical simulations demonstrate efficiency of the proposed algorithms.