Aggregates of positive impulse response systems: A decomposition approach for complex networks

Abstract

Abstract-To simplify the analysis of complex dynamical networks, we have recently proposed an approach that decomposes the overall system into the sign-definite interconnection of subsystems with a Positive Impulse Response (PIR). PIR systems include and significantly generalise input-output monotone systems, and the PIR property (or equivalently, for linear systems, the Monotonic Step Response property) can be evinced from experimental data, without an explicit model of the system. An aggregate of PIR subsystems can be associated with a signed matrix of interaction weights, hence with a signed graph where the nodes represent the subsystems and the arcs represent the interactions among them. In this paper, we prove that stability is structurally ensured (for any choice of the PIR subsystems) if a Metzler matrix depending on the interaction weights is Hurwitz; this condition is non-conservative. We also show how to compute an influence matrix that represents the steady-state effects of the interactions among PIR subsystems.

On biological networks capable of robust adaptation in the presence of uncertainties: A systems-theoretic approach

“…Although, the question of systems with positive impulse response (PIR) has been fairly well-discussed in the control theory literature [35–37] the relation between imperfect adaptation (regulation) and PIR has not been well-explored. Drenstig et al (2008) first argued that a three protein network must contain at least one zero for adaptation i. e. to avoid, the system being PIR [20].…”

“…Hence, it can be concluded that the network structures capable of adaptation should contain a non-PIR (NIR) impulse response. Further, Blanchini et al (2017) showed that if a linear, time-invariant LTI dynamical system h(t) contains a positive impulse response then it should satisfy the following where, H(jω) is the Fourier transform of h(t) [35].…”

On biological networks capable of robust adaptation in the presence of uncertainties: A linear systems-theoretic approach

“…well-discussed in the control theory literature [35][36][37] the relation between imperfect adaptation (regulation) and PIR has not been well-explored. Drenstig et al (2008) first argued that a three protein network must contain at least one zero for adaptation i. e. to avoid, the system being PIR [20].…”

“…1. The set of coupled, first-order differential equations in (35) constitute a well-posed dynamical system.…”