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arXiv · 2102.02144

Diagonal Stability of Discrete-time $k$-Positive linear Systems with Applications to Nonlinear Systems

Chengshuai Wu·Michael Margaliot
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Abstract

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 (LTI) 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 (CT) and discrete-time (DT) case. We then focus on DT $k$-positive linear systems and introduce the new notion of DT $k$-diagonal stability. It is shown that this is a necessary condition for standard DT diagonal stability. We demonstrate an application of this new notion to the analysis of a class of DT nonlinear systems.

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Chengshuai Wu, Michael Margaliot. 2021-02-03. Diagonal Stability of Discrete-time $k$-Positive linear Systems with Applications to Nonlinear Systems. https://arxiv.org/abs/2102.02144

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