Resilience to Non-Compliance in Coupled Cooperating Systems

Abstract

This letter explores the implementation of a safe control law for systems of dynamically coupled cooperating agents. Under a CBF-based collaborative safety framework, we examine how the maximum safety capability for a given agent, which is computed using a collaborative safety condition, influences safety requests made to neighbors. We provide conditions under which neighbors may be resilient to non-compliance of neighbors to safety requests, and compute an upper bound for the total amount of non-compliance an agent is resilient to, given its 1-hop neighborhood state and knowledge of the network dynamics. We then illustrate our results via simulation on a networked susceptible-infected-susceptible (SIS) epidemic model.

Figures (4)

• Figure 1: Simulated networked SIS model from \ref{['eq:sis_networked']}, with each node implementing the collaborative safety framework from butler2024collaborativesafety, where $n=3$ and $\eta_i=10, \kappa_i=1$ for all $i \in [3]$. Each node's safety constraint $\bar{x}_i$ is represented by the dotted line of corresponding color and the input constraint $\mathcal{U}_i(t)$ for all nodes $i \in [3]$ is shown by the black dotted line, where $\mathcal{U}_i(t) = [0,0.75]$ if $t < 10$ and $\mathcal{U}_i(t) = [0,0.6]$ if $t \geq 10$. Note that the system failure at $t=10$ causes node $1$ to violate its safety constraint.
• Figure 2: The same simulation from Figure \ref{['fig:eta10_kappa1']} except $\eta_i=\kappa_i=0.3$ for all $i \in [3]$. Note that, unlike in Figure \ref{['fig:eta10_kappa1']}, node $1$ is able to satisfy its safety constraint for all $t$.
• Figure 3: A numerical approximation of $\nu_1^*$ for $x_2 = x_3 = 0.1$ with $x_1 \in [0, \bar{x}_1]$, where $\nu_1$ is incremented by $\delta_{\nu} = 0.01$ until $u_1^* = u_1^c$. Notice as $x_1$ approaches $\bar{x}_1$, the resilience boundary increases to include larger values of $\nu_1$ that would enable resilience for node $1$ to non-compliant neighbors.
• Figure 4: The error tolerance in neighbor request fulfillment $\epsilon_i(x, \nu_i)$, defined by \ref{['eq:neighbor_error_tol']}, computed for $x_2 = x_3 = 0.1$ with $x_1 \in [0, \bar{x}_1]$ and $\nu_1 \in [0, 0.8]$. Notice as $x_1$ approaches the barrier of its safety constraints it requires a larger amount of error tolerance from its neighbors.

Theorems & Definitions (8)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 2
• Theorem 1
• proof