Robust Stability of positive continuous time systems

Abstract

In this note a s i n p l e forrr?u!a for the real stability radius of uncertain positive linear continuous time systems is established and it is shown that the real stability radius coincides with the complex one. Arbitrary disturbance norms induced by monotonic vector norms (e.g. p n o r m s , 1 < p < co) are considered. The distance of intervals of positive systems from instability is also determined.

Stability radii for positive linear time-invariant systems on time scales

“…For positive systems the situation is simpler, since the complex and the positive stability radius coincide. The discrete-time case is investigated in [6], whereas the continuous-time case is established in [5]. In the setting of Banach lattices a similar result is obtained in [7].…”

“…For positive systems we provided an easily computable formula for the stability radii. We note that although we have restricted ourselves to the case of the Euclidean norm, all results apply to monotone norms using the techniques provided in [5].…”

Robust stability of positive linear time delay systems under time-varying perturbations

“…Theorem 2.1. [36] Suppose that M ∈ R n×n is a Metzler matrix. Then (i) (Perron-Frobenius) µ(M ) is an eigenvalue of M and there exists a non-negative eigenvector…”

“…In particular, problems of robust stability of linear differential systems without delayẋ(t) = Ax(t), t ≥ 0, under the structured perturbations have been studied extensively for a long time, see e.g. [15][16][17][34][35][36]. Moreover, robust stability of the linear differential systems with delay (1) under time-invariant structured perturbations has been dealt with in [19,32,37].…”

Small-gain stability theorems for positive Lur’e inclusions

“…However, the additional structure afforded by positivity is often intuitive, mathematically helpful and thus simplifies matters. For example, in classical robust control, the complex and real stability radii, introduced in [22,23] are, in general, different for linear systems, but are known to be equal for linear positive systems, see [24].…”