Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

Abstract

We consider a class of continuous-time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces, and which include the fluid-level dynamics of the Max-Weight scheduling policy as a special case. We study the effect of an external disturbance/perturbation on the state trajectory, and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation. † A. Sharifnassab and S. J. Golestani are with the

When do Trajectories have Bounded Sensitivity to Cumulative Perturbations?

“…In this paper, we investigate sensitivity of some generalizations of this class of systems, as well as other classes of practical interest. In particular, we show that the result in [1] does not generalize to the (negative) gradient field of a convex function.More concretely, we provide examples of gradient field of a twice continuously differentiable convex function, and subgradient field of a piecewise constant convex function with infinitely many pieces, that have unbounded sensitivity. Moreover, we give a necessary and sufficient condition for a linear dynamical system to have bounded sensitivity, in terms of the spectrum of the underlying matrix.The proofs and the development of our examples involve some intermediate results concerning transformations of dynamical systems, which are also of independent interest.…”

“…In this paper, we investigate sensitivity of some generalizations of this class of systems, as well as other classes of practical interest. In particular, we show that the result in [1] does not generalize to the (negative) gradient field of a convex function.…”

Jumping Fluid Models and Delay Stability of Max-Weight Dynamics under Heavy-Tailed Traffic

“…In view of (4.4), we have ξ λ (x) ∈ D λ (x). Moreover, it is shown in Lemma 2(a) of [17] that ξ λ (x) has the minimum norm among all vectors in D λ x , i.e., (9.2)…”

Fluctuation Bounds for the Max-Weight Policy with Applications to State Space Collapse

“…In this subsection, we review some definitions and results from [29]. A dynamical system is identified with a setvalued function F : R n → 2 R n and the associated differential inclusioṅ x(t) ∈ F (x(t)).…”

“…On the technical side, the proof of the sensitivity bound (2) exploits a similar bound from our earlier work [29] on the sensitivity of a class of hybrid subgradient dynamical systems to fluctuations of external inputs or disturbances. The main challenges here concern the transition from discrete to continuous time, as well as the presence of boundary conditions, as queue sizes are naturally constrained to be non-negative.…”