Finite input/output representative of a class of Volterra polynomial systems
In this paper, discrete infinite Volterra systems of the form \[ y(t)= \sum_{n= 1}^\infty \sum^\infty_{i_1=0} \cdots \sum^\infty_{i_n=0} a_{i_1i_2 \dots i_n} u(t-i_1) \dots u(t-i_n),\quad t =0,1, \dots, \] are studied with the goal of establishing an input/output representation of the form \[ y(t)+ a_1y(t-1)+ \cdots +a_ky(t-k)= \sum^m_{n=1} \sum_{i=(i_1,i_2, \dots, i_n)\in I_n} b_iu(t-i_1) \dots u(t-i_n) \] where \(I_n\) is finite, for each \(n=1, \dots, m\). The notions of a \(\delta\)-operator, \(\delta\)-series, and linear factorization of \(\delta\)-series are employed to attack the problem. The idea is to find \(\delta\)-polynomials \(A(\delta)\) and \(B(\delta)\) with \(A (\delta) \) being linear so that the system \(A(\delta)y(t)= B(\delta)u(t)\) yields the same output as the original Volterra system. The main result of this paper guarantees that the original system \(C\) can be factored in the form \(C=L*B\) with \(L\) a linear series and \(B\) a finite nonlinear polynomial, and that the inverse of \(L\) is a finite linear polynomial. Therefore, the original infinite system \(y(t)=Cu(t)\) can be transformed to the finite \(L^{-1} y(t)= Bu(t)\) input/output form.
scientific article; zbMATH DE number 1008767
scientific article