Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices

Abstract In the investigation performed we give, on half-infinity discrete intervals, formulas for solution of initial problem of linear discrete systems x ( k + 1 ) = A x ( k ) + B x ( k − m ) + f ( k ) with constant square matrices A, B such that A B = B A , det A ≠ 0 and with a vector function f ( k ) . Corresponding representations are obtained with the aid of so-called discrete matrix delayed exponential, which permits to represent solutions in a matrix form similarly as for ordinary differential systems with constant matrices, or as well as for differential systems with constant matrices and constant delay.