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

Abstract

It is known that input-output approaches based on scaled small-gain theorems with constant Dscalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input-output approaches provide, in general, sufficient conditions only. Using these results we provide simple alternative proofs for many of the existing results on the stability of linear positive systems with discrete/distributed/neutral time-invariant/-varying delays and linear difference equations. In particular, we give a simple proof for the characterization of diagonal Riccati stability for systems with discrete-delays and generalize this equation to other types of delay systems. The fact that all those results can be reproved in a very simple way demonstrates the importance and the efficiency of the input-output framework for the analysis of linear positive systems. The approach is also used to derive performance results evaluated in terms of the L1-, L2-and L∞-gains. It is also flexible enough to be used for design purposes.

Ergodicity analysis and antithetic integral control of a class of stochastic reaction networks with delays

“…This result parallels existing ones on the stability of linear positive systems with delays for which it is well-known that the stability is equivalent to that of the delay-free system and is independent of the value of the delay; see e.g. [14,35].Similar results are obtained for bimolecular networks. When there is no delayed bimolecular reactions, it is shown that the ergodicity conditions reduces to those of the delay-free network.…”

Stability andL1× ℓ1-to-L1× ℓ1performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

“…It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called worst-case system, which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties, similarly to as in [1]. As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system -a result that is consistent with all the existing ones in the literature on linear positive systems with delays.…”

“…The objective of this section is to provide stability and performance criteria for systems of the form (1). Those criteria are novel and extend previously obtained ones on the stability analysis [39] and the L 1 / 1performance [18] to the case of uncertain systems in LFT-form.…”

“…Constant delay operators are known to have unit L 1 -gain; see e.g. [1,24]. We have the following result regarding the state positivity of the impulsive system (1).…”

“…They have been shown to be often exact in the context of linear positive systems; see e.g. [1,24,35,57].The first part of the paper is devoted to the development of stability and L 1 / 1 performance analysis conditions for linear uncertain positive impulsive systems in linear fractional form. In this context, we consider uncertainties both in the continuous-time and the discrete-time dynamics of the system.…”