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.
Highlights
Positive systems have an important relevance since the input, state, and output signals in many physical or biological systems are necessarily positive [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
An hydrological system composed of a set of lakes in which the input is the inflow into the upstream lake and the output is the outflow from the downstream lake is externally positive system since the output is always positive under a positive input [8]
Hyperstable single-input single-output systems are externally positive since the impulse response kernel is everywhere positive
Summary
Positive systems have an important relevance since the input, state, and output signals in many physical or biological systems are necessarily positive [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Hyperstable single-input single-output systems are externally positive since the impulse response kernel is everywhere positive. This implies that the associated transfer functions (provided they are time invariant) are positive real and their input/output instantaneous power and timeintegral energy are positive. Hyperstable systems of second and higher orders are not guaranteed to be externally positive since the impulse response kernel matrix is. The main objective of this paper is to study the nonnegativity/positivity properties of time-invariant continuous-time dynamic systems under constant point delays. The main new contribution of the paper is the study of a hierarchically established set of positivity concepts formulated in generic cones for a class of systems subject to point delays.
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