Necessary and sufficient conditions for stabilizability of positive LTI systems

Small-gain stability theorems for positive Lur’e inclusions

“…We note that statement (i) of Corollary 3.3 holds, but that statements (ii) and (iii) do not. More generally, for arbitrary g : R + → R + , whenever y * > 0 is such that 21) in particular meaning that the estimates in both (ii) and (iii) do not hold, then x * := (I − A) −1 Bg(y * ) is a non-zero equilibrium of (3.18) and the zero equilibrium of (3.18) cannot then be globally asymptotically or exponentially stable. The papers [49,64] consider attractivity and stability properties of the non-zero equilibrium x * when m = p = 1 and under conditions on A, B, C and g which ensure that y * > 0 in (3.21) is unique.…”

“…In addition to numerous important examples, control of positive systems is motivated by the challenges which nonnegativity constraints place, since subtraction is not always well-defined in positive cones, for instance. As such, the following fundamental facets of linear control theory all require different treatments for positive systems: reachability and controllability [15,16], observability [17], realisability [18,19] and stabilisability [20,21]. However, the additional structure afforded by positivity is often intuitive, mathematically helpful and thus simplifies matters.…”

“…Consequently, the problem is one of static output feedback control design for linear positive systems via the choice of k, which has been considered in, for example [21]. …”

Differential positivity on compact sets

“…k 2 + λk 1 ε where λ := max x∈C,|ϑ|x=1,K(x,ϑ)≥0 λ(x, ϑ) and λ(x, ϑ) refers to (15). By continuity, for ε sufficiently small there exists k 3 > 0 such that…”

“…Under mild conditions, the ray λv ∈ K given by the Perron-Frobenius eigenvector v ∈ K is an attractor for the system dynamics, [2], [3]. This fundamental property is exploited in a number of applications [5], [12], [15], [14] Differential positivity extends linear positivity to the nonlinear setting. A nonlinear system is differentially positive if its linearization along trajectories makes a cone (field) invariant [7].…”