Small-gain stability theorems for positive Lur’e inclusions

Abstract

Stability results are presented for a class of differential and difference inclusions, so-called positive Lur'e inclusions which arise, for example, as the feedback interconnection of a linear positive system with a positive set-valued static nonlinearity. We formulate sufficient conditions in terms of weighted one-norms, reminiscent of the small-gain condition, which ensure that the zero equilibrium enjoys various global stability properties, including asymptotic and exponential stability. We also consider input-to-state stability, familiar from nonlinear control theory, in the context of forced positive Lur'e inclusions. Typical for the study of positive systems, our analysis benefits from comparison arguments and linear Lyapunov functions. The theory is illustrated with examples.

Stabilization by Adaptive Feedback Control for Positive Difference Equations with Applications in Pest Management

“…In control theory jargon, the matrix H(w) in (2.7) is the steady state gain of the linear control system specified by the triple (A(w), B(w), E(w)). The second inequality in (2.6) is a so-called weighted smallgain condition, and is used extensively in [19] in the stability analysis of systems of positive Lur'e difference equations. The matrix H(σ)Σ is order q × q, and indeed is scalar if q = 1, in which case the second condition in (2.6) reduces to the scalar inequality H(σ)Σ < 1.…”