A note on the degenerate discrete-delay system

The discrete-delay system
x_{k+1}=Ax_{k} + Bx_{k-l}, l \geq 1, k= 0,1,2,3,... \infty
(
A
is invertible,
l
is the delay) is said to be degenerate with respect to the vector
d
if
y_{k}=d^{T}x_{k}\equiv 0, k \geq l_{0}
, for all choices of initial points
(x_{0},x_{-1},x_{-2},... ,x_{-l})
. In this note we shall show that if the above system is degenerate, then the minmum value of l
0
is (
l+ 1
). Given
A,d,l
, necessary and sufficient conditions are obtained such that the above system is degenerate at
l_{0}=l+ 1
, and the construction of
B
is given in this case. These constructions are used to find delayed state-feedback controls for the linear discrete system to steer the output to zero in a minimum number of steps.