Consensus of Hybrid Multi-agent Systems

I. Introduction

• Earlier work addressed consensus for continuous-time and discrete-time agents separately, including switching topologies, delays, gossip algorithms, and first- or second-order dynamics.The introduction also cites prior switched systems combining continuous-time and discrete-time subsystems.Source: §I · pp. 2–3
• Consensus is agreement among agents on a consistent quantity of interest through control inputs based on local information.The introduction situates consensus among coordination problems such as flocking, containment, formation, coverage, and distributed estimation –.Source: §I · pp. 1–2
• The paper targets consensus when continuous-time and discrete-time agents coexist and interact, a setting not covered by the existing homogeneous or heterogeneous results described here.The authors identify differing time scales as making protocol design and analysis difficult.Source: §I · p. 3
• The paper proposes a hybrid multi-agent system and a definition of consensus, presents three protocols, and derives necessary and sufficient consensus conditions using graph theory.The stated objective is to obtain criteria under different topologies.Source: §I · p. 3

A. Graph theory

• The graph model represents agents as vertices and information links as directed edges; its Laplacian is L = D − A and satisfies L1_n = 0.The degree matrix contains the sums of incoming edge weights.Source: A. Graph theory · p. 4
• A stochastic matrix with positive diagonal entries has all eigenvalues other than 1 strictly inside the unit circle when its associated graph has a spanning tree.A spanning tree is equivalent to eigenvalue 1 having algebraic multiplicity one.Source: A. Graph theory · p. 4
• When a stochastic matrix is SIA, its powers converge to 1_nν^T, with ν a nonnegative normalized stationary vector.The stated stationary condition is P^Tν = ν and 1_n^Tν = 1.Source: A. Graph theory · pp. 4–5
• I_n − HL is SIA if and only if the associated graph has a spanning tree, provided 0 < h_i < 1/d_ii.SIA means its powers converge to a rank-one stochastic limit.Source: A. Graph theory · p. 5

B. Hybrid multi-agent system

• The system has n agents, of which m < n are continuous-time; without loss of generality, agents 1 through m are assigned that type.The agents' dynamics are introduced after this setup.Source: B. Hybrid multi-agent system · p. 5
• The sampling period is h, while x_i and u_i denote agent i's position-like state and control input, respectively.Initial conditions are specified as x_i(0) = x_i0.Source: B. Hybrid multi-agent system · p. 5
• Each agent is a graph vertex, each directed edge represents an available information link, and agents update their states from information received from neighbors.The paper assumes that at least one communication link exists.Source: B. Hybrid multi-agent system · p. 5
• Consensus is introduced as a condition on the agents' states for every initial condition.The supplied definition begins with this condition but does not include its full mathematical statement.Source: B. Hybrid multi-agent system · p. 6

III. Main results

• The section considers the hybrid system's consensus problem under three kinds of control inputs, each treated as a consensus protocol.Source: §III · p. 6

A. Case 1

• Agents communicate with neighbors and update their control inputs at sampling times t_k under protocol (4).The sampling period is h = t_{k+1} − t_k > 0.Source: A. Case 1 · p. 6
• With h < 1, protocol (4) solves consensus if and only if the communication network has a directed spanning tree.The theorem also identifies the consensus state using the vector ν.Source: A. Case 1 · p. 6
• Without a directed spanning tree, the system cannot reach consensus under protocol (4).Source: A. Case 1 · p. 7

B. Case 2

• Each continuous-time agent can observe its own state in real time, while interactions occur at sampling times t_k.This is the distinguishing observation assumption in Case 2.Source: B. Case 2 · p. 8
• Protocol (8) solves consensus if and only if the network has a directed spanning tree, with consensus state ν^T x(0).The matrix (I_n − HL) is SIA when the graph has a directed spanning tree, yielding convergence to the consensus state.Source: B. Case 2 · pp. 8–9
• If the network lacks a directed spanning tree, the hybrid system cannot achieve consensus under protocol (8).Source: B. Case 2 · p. 9
• With real-time observation, the sampling-period restriction depends only on the out-degrees of discrete-time dynamic agents.Source: B. Case 2 · p. 9

C. Case 3

• The model assumes an undirected communication network and selects an adjacent pair at each sampling time with the probability specified in (A2).The sampling period satisfies h < 1/max_i,j∈I_n{a_ij}.Source: C. Case 3 · pp. 9–10
• At each sampling time, a communicating pair is selected and interacts according to whether its agents are continuous-time or discrete-time.Continuous-time pairs communicate over (t_k, t_{k+1}], while mixed pairs have their own specified control inputs.Source: C. Case 3 · p. 10
• Under the stated assumptions and sampling bound, the system achieves consensus in mean sense if and only if the network is connected.The bound is h < 1/max_i,j∈I_n{a_ij}.Source: C. Case 3 · pp. 9–10
• The proof represents sampled states by stochastic matrices whose positive off-diagonal entries correspond to communicating edges, linking convergence to network connectivity.The matrices have positive diagonal entries, and connectivity is used with the stated lemmas to establish convergence.Source: C. Case 3 · pp. 11–12
• The hybrid model also includes pure discrete-time systems when m = 0 and pure continuous-time systems when m = n.Source: C. Case 3 · p. 13

IV. Simulations

• The section presents simulation examples to demonstrate the effectiveness of the theoretical results.Source: §IV · p. 13
• For a six-agent example with agents 1–3 continuous-time and 4–6 discrete-time, protocol (4) yields trajectories consistent with Theorem 1.The initial state is [−13, 14, 3, −9, −3, 6]^T and h = 0.2; G1 has a directed spanning tree.Source: §IV · p. 13, Figure 1 · p. 13
• For a network with a directed spanning tree, protocol (8) produces trajectories consistent with Theorem 2.The example uses G2 and reports the sampling-period condition h < 1.Source: §IV · p. 13

• Fig. 2 shows all agents’ state trajectories under protocol (4) on communication network G1.Source: Figure 2 · p. 14
• Fig. 3 depicts the directed graph G2.Source: Figure 3 · p. 14
• Fig. 4 shows all agents’ state trajectories under protocol (8) on communication network G2.Source: Figure 4 · p. 14
• Fig. 6 shows trajectories under protocols (12)–(15) on G3, and the simulation is reported consistent with Theorem 3’s sufficiency.Source: §7 · p. 15, Figure 6 · p. 15
• Fig. 5 depicts the undirected graph G3.Source: Figure 5 · p. 15

• The protocols assume all agents communicate with their neighbors and update their strategies at sampling times.Source: §V · p. 15
• Under a sampling-period restriction, consensus is achievable if and only if the communication network has a directed spanning tree.Source: §V · p. 15
• For another protocol, the conclusions assume continuous-time agents can observe their own states in real time and state that consensus can be solved under a sampling-period restriction.Source: §V · p. 15
• The paper also proposes a gossip-like consensus protocol and reports a necessary-and-sufficient condition for its solvability.Source: §V · p. 16