Connectivity and Set Tracking of Multi-agent Systems Guided by Multiple Moving Leaders

1 Introduction

• The problem addresses practical settings where multiple leaders and targets have unmeasurable variables and interaction topologies change over time.Multiple leaders are assigned to increase control effectiveness, communication or sensing range, reliability, and energy efficiency.Source: §1 · p. 2
• Unlike earlier set input-to-state stability work for fixed sets [33], the paper generalizes SISS and SiISS to moving sets with time-varying shapes and switching networks.Source: §1 · p. 3
• SISS and SiISS yield explicit estimates of convergence rate and tracking error for agents entering the convex hull spanned by multiple leaders.Source: §1 · p. 3
• The paper relates connectivity to set tracking, finding that various jointly connected conditions are necessary and/or sufficient for distributed coordination.Source: §1 · p. 3
• The analysis uses graph theory and nonsmooth methods because conventional ISS Lyapunov and Laplacian approaches may fail for moving sets, switching topologies, and uncertain nonlinear dynamics.Source: §1 · pp. 3–4

2 Problem Formulation

• Followers receive information from neighboring followers and connected leaders, while leaders evolve under their own control inputs and followers experience disturbances.The interaction weights may depend on states, time, or relative measurements.Source: §2 · pp. 6–7
• The formulation allows switching directed interaction graphs and assumes a positive lower bound between switching instants.The model also permits nonlinear, nonconstant interaction weights without requiring global information.Source: §2 · pp. 6–8
• Uniform joint L-connectivity requires the union graph over every fixed-length time window to contain a path from some leader to each follower.JLC instead requires the union over [t, ∞) to be L-connected.Source: §2 · pp. 6–7
• The tracking target is the time-varying convex polytope formed by the active leaders, and set tracking concerns followers’ distance from this moving set.The paper generalizes set input-to-state stability and its integral variant to this moving set.Source: §2 · pp. 8–9

3 Convergence Estimation

• The analysis targets each follower’s distance from the time-varying convex hull spanned by the moving leaders.The distance must be estimated as the leaders’ positions, and thus the target set’s shape, change over time.Source: §3 · p. 9
• Lemma 3.1 states a result for every pair of nonnegative times t and t0 and each index i from 1 through n.Source: §3 · p. 10

Lemma 3.2 D+p

• The upper Dini derivative of the maximal follower distance is bounded using the disturbance and the current distance from the moving set.The bound takes a maximum over agents attaining that maximal distance.Source: Lemma 3.2 D+p · pp. 11–12

4 Connectivity and SISS

• The connectivity analysis addresses nonlinear agent dynamics with disturbances and leaders whose velocities are unmeasurable.The main-results analysis treats the uniformly jointly L-connected case and uses dwell-time bounds to assemble convergence estimates across intervals.Source: §4 · p. 12, §4.1 · pp. 12–13

4.1 Main results

• Under uniformly jointly L-connected switching topologies, the section establishes a necessary-and-sufficient condition for set input-to-state stability with respect to the moving leader set.The proof builds interval-wise convergence estimates from dwell-time arcs and uses a counterexample when connectivity fails.Source: §4.1 · pp. 12–13, §4 · p. 12, §4.2 · pp. 14, 16
• For inputs satisfying condition (24), global set tracking occurs if and only if the switching topology is uniformly jointly L-connected.The result follows from the SISS theorem; the theorem’s proof notes that prolonged intervals without leader connectivity can prevent followers from entering the moving set in finite time.Source: §4.1 · p. 14, §4.2 · p. 16

5.1 Main results

• UJLC is sufficient for SiISS, JLC is necessary, and for bidirectional follower graphs JLC is necessary and sufficient.The section also treats acyclic follower graphs through a separate necessary-and-sufficient SiISS result.Source: §5.1 · pp. 17–18
• With condition (36), UJLC ensures set tracking, while JLC characterizes global set tracking in the bidirectional or acyclic cases.In general directed cases, the stated sufficient condition is UJLC.Source: §5.1 · pp. 18–19
• The section distinguishes SISS from SiISS: SISS is equivalent to UJLC, whereas UJLC is sufficient but not necessary for SiISS.JLC remains necessary for SiISS; without tail connectivity, some followers may cease to be influenced by the leader set.Source: §5.1 · pp. 17, 19

5.2 Proofs

• The proofs estimate convergence over dwell-time intervals and paste estimates across successive jointly leader-connected subintervals to establish SiISS.The argument accounts for the integral of the input while bounding convergence across intervals where arcs persist for at least the dwell time.Source: §5.2 · pp. 19, 22, 24

• The paper establishes an equivalence between uniformly jointly leader-connected (UJLC) topology and SISS for uncertain agents tracking a moving multi-leader set.It also shows UJLC is sufficient for SiISS with agent disturbances and unmeasurable leader velocities.Source: §6 · p. 25
• For bidirectional or acyclic communication topologies, JLC is necessary and sufficient for SiISS.The paper also reports set tracking in special cases using SISS and SiISS.Source: §6 · p. 25

• A differential inequality bounds the follower error during a leader-to-follower arc that persists for a dwell-time interval.The displayed estimate combines exponential decay with a term proportional to the input bound.Source: §A.1 · pp. 25–26
• After the dwell-time interval, the estimate continues to apply even when no follower-leader connection is present.The proof then combines the interval estimates to obtain a continuous bound.Source: §A.1 · pp. 26–27

• The argument treats the dwell-time interval and the subsequent interval as separate cases before combining their estimates.The combined bounds yield the stated conclusion.Source: §A.2 · pp. 27–28