Diagonal Stability of Discrete-Time $k$-Positive Linear Systems With Applications to Nonlinear Systems

A linear dynamical system is called
$k$
-positive if its dynamics maps the set of vectors with up to
$k-1$
sign variations to itself. For
$k=1$
, this reduces to the important class of positive linear systems. Since stable positive linear time-invariant systems always admit a
diagonal
quadratic Lyapunov function, i.e., they are diagonally stable, we may expect that this holds also for stable
$k$
-positive systems. We show that, in general, this is not the case both in the continuous-time and discrete-time (DT) case. We then focus on DT
$k$
-positive linear systems and introduce the new notion of the
DT $k$-diagonal stability
. It is shown that this is a necessary condition for the standard DT diagonal stability. We demonstrate an application of this new notion to the analysis of a class of DT nonlinear systems.