On some algorithms for non-parametric identification of linear systems (Q1344338)

The authors consider an identification problem where a function \(f(t)\), \(t\geq 0\) has to be found from the integral equation \(x(t)= \int^ t_ 0 f(t- \tau) u(\tau) d\tau\) which is of the form \(x= Wu\), where \(W\) is the linear operator, asymptotically stable. A real polynomial is chosen \[ \varphi(\tau)= a_ 0 \tau^ p+ a_ 1\tau^{p-1}+\cdots+ a_ p, \] \(a_ 0> 0\), \(p\geq 2,\) and the following functions are built: \(v(\tau)= {1\over 2\pi} \int^ \infty_{-\infty} e^{-t\tau+ i\varphi(\tau)} d\tau\), \(-\infty< t<\infty\), \(w(\tau)= {1\over 2\pi} \int^ \infty_{- \infty} e^{-t\tau- i\varphi(\tau)} d\tau\), \(-\infty< t< \infty\). The following input \(u(t, \lambda)\) is supplied to \(W\): \(u(t, \lambda)= v(t- \lambda)\) and the corresponding output \(z(t, \lambda)\) is measured. Afterwards the function \(f(t, \lambda)= \int^{2\lambda}_ 0 z(t, \lambda) w(t- \tau+ \lambda) d\tau\), \(t\leq 0\), is computed. This function is considered as an approximation of the weight function \(f(t)\) of the system \(W\).