On the span reachability of polynomial state-affine systems

The purpose of this report is to derive an explicit condition for the span reachability of a discrete polynomial state-affine system described by
x(k+1)=(A_{0} +\Sigma\min{i=1}\max{r}u^{i}(k)A_{i})x(k)+ \sum\min{i=1}\max{r} u^{i}(k)B_{i}, (k=0,1,...)
(1) where
r
is a positive integer,
x \in R^{n}, u \in R^{1},u^{i}
denotes the ith power of
u
, and A
i
and B
i
are matrices of appropriate dimensions. In order to define input sequences which can construct reachable state vectors from the origin to span the whole state space, a generalized type of the Vandermonde's matrix is newly defined and utilized fully. Although the algebraic structure of (1) is more complicated than discrete bilinear systems, the result turns out to be quite analogous to each other.