On the approximation of delay elements by feedback
The functional equation for the output \(y(t)\) of a closed-loop system with a time-delay \(T\) in the direct path, a gain \(a\) in the feedback loop, and input \(u(t)\) is \[ y(t)= u(t- T)+ ay(t- T).\tag{1} \] Problem: Assuming that the input is zero for \(t< 0\) and a power of \(t\) for \(t\geq 0\), find a solution of (1) of the form \(y(t)= q(t)+ p(t)\), for \(t\geq 0\), such that \(q(t)\) is a polynomial and \(p(t)\) is periodic. If \(p(t)\) is periodic with period \(\tau\): \[ p(t)={1\over 2} a_0+ \sum_{k\geq 1} \Biggl(a_k\cos{2\pi kt\over \tau}+ b_k\sin{2\pi kt\over \tau}\Biggr) \] then \(\tau= T\) and \(a=1\), or \(\tau= 2T\), \(a= -1\). If we set \(u(t)= t^m\), and look for polynomial solutions of the form \[ q(t)= \sum^{m+1}_{k=1} a_kt^k, \] we find \(a_k= {m!\over k!(m- k+ 1)!} T^{m-k} B_{m- k+1}\), \(k= 1,\dots, m+ 1\).
scientific article; zbMATH DE number 1441453