About K-Positivity Properties of Time-Invariant Linear Systems Subject to Point Delays

Abstract

This paper discusses nonnegativity and positivity concepts and related properties for the state and output trajectory solutions of dynamic linear time-invariant systems described by functional differential equations subject to point time delays. The various nonnegativities and positivities are introduced hierarchically from the weakest one to the strongest one while separating the corresponding properties when applied to the state space or to the output space as well as for the zero-initial state or zero-input responses. The formulation is first developed by defining cones for the input, state and output spaces of the dynamic system, and then extended, in particular, to cones being the three first orthants each being of the corresponding dimension of the input, state, and output spaces.

On the excitability of a class of positive continuous time-delay systems

“…Positive systems are also described in [24], which is related to Leontieff models widely used in the economical production system [24]. They may be characterized in general arbitrary cones and also, in particular, for abstract delayed equations rather than in the first orthant (see for instance [24,25,33]). However, it is preferred in this manuscript to give a clear simple characterization of positivity characterized in the first orthant, which has a clear insight in related engineering and biology/ecology problems.…”

On Minimal Realizations and Minimal Partial Realizations of Linear Time‐Invariant Systems Subject to Point Incommensurate Delays

“…The oscillatory behavior under delays and possible unmodeled dynamics is investigated in 10, 21 . Different aspects and conditions of positivity of the solutions and equilibrium points have been recently described in [16][17][18]22 , either in the first orthant or in generic cones. The central purpose of this paper is concerned with the realization theory and associated properties of controllability and observability of dynamic systems under linear delayed dynamics.…”