Properties of eventually positive linear input–output systems

Abstract

In this paper, we consider the systems with trajectories originating in the nonnegative orthant becoming nonnegative after some finite time transient. First we consider dynamical systems (i.e., fully observable systems with no inputs), which we call eventually positive. We compute forward-invariant cones and Lyapunov functions for these systems. We then extend the notion of eventually positive systems to the input-output system case. Our extension is performed in such a manner, that some valuable properties of classical internally positive input-output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have nonnegative derivatives.

Stability and performance analysis of linear positive systems with delays using input–output methods

“…Finally, since eventually positive systems can be used to efficiently represent externally positive systems, some of the results for internally positive systems are expected to remain true for these systems as well; see e.g. [58,59].…”

“…is Schur stable. Proof : The equivalence between the three first statements follows from the application of Theorem 5 on the input-output formulation of the system (58). The equivalence with the statements (iv) and (vii) comes from Theorem 6 and the fact that, for two matrices A, B of appropriate dimensions, we have that ρ(AB) = ρ(BA).…”

Periodicity and stability analysis of periodic switched positive systems

“…is exponentially stable, i.e. there exist α > 0, r > 1 satisfying ∥ z(t) ∥ ≤ αr −t ∥ z(0) ∥ , t ∈ ℕ 0 (17) Setting z(0) = y(0) for system (8) and (16), it holds that…”

“…This property was then extended to the case where the delays are unbounded [9][10][11]. Similar topics are investigated in [12][13][14][15][16][17][18].…”

Block-diagonal solutions to Lyapunov inequalities and generalisations of diagonal dominance

“…Therefore, generalisations of positivity attracted some attention as well, e.g. eventual positivity [3], [4], which inherits some properties of positivity. However, in the context of Lyapunov inequalities, perhaps, a more relevant generalisation is based on (scaled) diagonally dominant matrices.…”