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 kpositive 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.