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Consensus of Hybrid Multi-agent Systems
Yuanshi Zheng, Jingying Ma, Long Wang
TL;DR
The paper addresses consensus in hybrid multi-agent systems combining continuous-time and discrete-time agents. It proposes and analyzes consensus protocols, establishing sampling-period conditions under which consensus is equivalent to a directed spanning tree in the communication network.
Problem
The paper addresses how to achieve consensus in systems combining continuous-time and discrete-time agents.
Method
It studies neighbor communication and sampling-time updates, including a gossip-like protocol, under sampling-period conditions.
Results
Under 0 < h < 1/max_i∈I_n{d_ii}, consensus occurs if and only if the communication network has a directed spanning tree.
Takeaways & Limitations
The paper characterizes when hybrid agents can reach consensus under the stated sampling-period condition and communication-network structure.
Takeaways & Limitations
The analysis assumes neighbor communication with sampling-time updates and real-time observation of each continuous-time agent’s own state.
Abstract
from arXiv · showhide
In this paper, we consider the consensus problem of hybrid multi-agent system. First, the hybrid multi-agent system is proposed which is composed of continuous-time and discrete-time dynamic agents. Then, three kinds of consensus protocols are presented for hybrid multi-agent system. The analysis tool developed in this paper is based on the matrix theory and graph theory. With different restrictions of the sampling period, some necessary and sufficient conditions are established for solving the consensus of hybrid multi-agent system. The consensus states are also obtained under different protocols. Finally, simulation examples are provided to demonstrate the effectiveness of our theoretical results.
I. Introduction · II. Preliminaries · A. Graph theory
The supplied preliminaries define weighted directed graphs, graph connectivity, Laplacians, stochastic matrices, and their spectral links to spanning trees and consensus-related convergence. They establish that powers of I_n − HL converge to a rank-one limit exactly when the graph has a spanning tree.
- A. Graph theory: A. Graph theory: A weighted directed graph consists of vertices, edges, and a nonnegative adjacency matrix, with neighbors defined by positive edge weights.The neighbor set is N_i = {j : a_ij > 0}.
- A. Graph theory: A. Graph theory: Directed paths, directed trees, and directed spanning trees formalize reachability from a root to all other vertices.An undirected graph is connected when a path exists between every pair of distinct vertices.
- A. Graph theory: A. Graph theory: The degree matrix is diagonal, the Laplacian is L = D − A, and the Laplacian satisfies L1_n = 0.The degree entries sum the weights associated with each agent’s neighbors.
- A. Graph theory: A. Graph theory: A stochastic matrix has row sums equal to 1, and an SIA matrix satisfies lim_k→∞ P^k = 1y^T.The graph associated with P contains (s_i, s_j) exactly when p_ji > 0.
- A. Graph theory: A. Graph theory: A stochastic matrix has eigenvalue 1 with algebraic multiplicity one if and only if its associated graph has a spanning tree.Positive diagonal entries imply every other eigenvalue satisfies |λ| < 1.
- A. Graph theory: A. Graph theory: If eigenvalue 1 is simple and all other eigenvalues satisfy |λ| < 1, then the stochastic matrix is SIA and its powers converge to 1_nν^T.The limiting vector ν is nonnegative and satisfies A^Tν = ν.
- A. Graph theory: A. Graph theory: For H = diag{h_1, h_2, . . . , h_n} with 0 < h_i < 1, lim_k→∞[I_n − HL]^k = 1_nν^T if and only if G has a spanning tree.The proof constructs P = I_n − HL as a stochastic matrix with positive diagonal entries and the same associated graph as G.
- A. Graph theory: A. Graph theory: The constructed matrix P has positive off-diagonal entries exactly where a_ij > 0, so spanning-tree convergence follows from the stochastic-matrix lemmas.Without a spanning tree, eigenvalue 1 has algebraic multiplicity m > 1, preventing the required rank-one limit.
B. Hybrid multi-agent system
The hybrid multi-agent system combines continuous-time and discrete-time agents, whose state interactions are modeled through a communication graph. Consensus is defined for the system under arbitrary initial conditions.
- System composition: The system contains n agents, including m continuous-time agents with m < n and n − m discrete-time agents.Agents 1 through m are assumed to be continuous-time dynamic agents.
- Agent dynamics: Each agent has a position-like state x_i, control input u_i, sampling period h, and initial condition x_i(0) = x_i0.The collective initial state is x(0) = [x10, x20, · · · , xn0]^T.
- Communication graph: Agents are represented as graph vertices, while directed edges encode available information links between agents.An edge (s_i, s_j) ∈ E corresponds to an information link from agent i to agent j.
- Communication graph: Each agent updates its current state using information received from neighboring agents.The model assumes communication behavior exists, with at least one pair satisfying a_ij > 0.
- Consensus definition: The paper defines consensus as a property of the hybrid multi-agent system that must hold for any initial conditions.This definition is introduced as Definition 1, although the supplied passage is truncated after this condition.
III. Main results
Section III examines consensus in the hybrid multi-agent system under three distinct control inputs, or consensus protocols.
- Consensus is analyzed under three kinds of control inputs, referred to as consensus protocols.
A. Case 1
Case 1 considers agents that communicate and update control inputs at sampling times. Under protocol (4), consensus is characterized by the sampling-period condition h < 1 and the existence of a directed spanning tree.
- Case 1: Agents communicate with neighbours and update their control inputs at sampling times t_k, with sampling period h = t_{k+1} − t_k > 0.The protocol is formulated for the hybrid multi-agent system under this sampling-time update scheme.
- Case 1: For h < 1, protocol (4) achieves consensus if and only if the directed communication network G has a directed spanning tree.The theorem establishes both sufficiency and necessity of the graph condition under the sampling-period restriction.
- Case 1: Under the sufficient conditions, the proof shows that the sampled-state relation and the limiting inter-sampling behavior together imply the consensus result.The argument uses the graph condition and h < 1, then establishes lim t→∞ ∥x_i(t) − x_i(t_k)∥ = 0 for i ∈ I_m.
- Case 1: When G lacks a directed spanning tree, equation (2) does not hold, and the hybrid multi-agent system cannot reach consensus.The necessity argument directly links failure of the spanning-tree condition to failure of consensus.
B. Case 2
Case 2 assumes continuous-time agents observe their own states in real time while interactions occur at sampling times. Under protocol (8), consensus is characterized by the network topology and sampling-period restriction, with consensus state νT x(0).
- Case 2: Continuous-time agents observe their own states in real time, while agent interactions occur at sampling times tk.This distinguishes Case 2 from Case 1.
- Theorem 2: Under protocol (8), consensus is achieved if and only if the directed communication network G has a directed spanning tree, provided h < 1.Here h = tk+1−tk > 0 is the sampling period.
- Theorem 2: The consensus state is νT x(0), where L^T Hν = 0.The state convergence follows from the SIA property of In − HL when G has a directed spanning tree.
- Theorem 2: If G lacks a directed spanning tree, the hybrid multi-agent system cannot achieve consensus.Thus, the spanning-tree condition is necessary as well as sufficient.
- Remark 1: With real-time observation by continuous-time agents, the sampling-period restriction h is related only to the out-degrees of discrete-time dynamic agents.This is the comparison stated in Remark 1 for Theorems 1 and 2.
C. Case 3
Case 3 studies gossip-like interactions on an undirected hybrid communication network, with protocols covering continuous–continuous, continuous–discrete, and discrete–discrete agent pairs. Under these assumptions, mean consensus is achieved if and only if the communication network is connected, while the model includes purely discrete- and continuous-time systems as special cases.
- Case 3: Case 3 assumes gossip-like interactions and an undirected communication network.At each sampling time, an interacting edge is selected probabilistically.
- Case 3: The protocol specifies interaction rules for continuous–continuous, continuous–discrete, and discrete–discrete agent pairs.Continuous agents communicate over each sampling interval, while the corresponding control inputs depend on the selected agent types.
- Case 3: Mean consensus occurs if and only if the undirected communication network G is connected.The sufficiency proof models the sampled states as a stochastic linear system, while necessity follows from disconnected components preventing consensus.
- Case 3: For 0 < h < 1, each interaction matrix Φij is stochastic with positive diagonal entries and positive symmetric entries only for communicating pairs.These properties identify G with the graph associated with the interaction matrices and support convergence when G is connected.
- Case 3: The hybrid model reduces to a discrete-time multi-agent system when m = 0 and to a continuous-time multi-agent system when m = n.Thus, the formulation provides a unified framework for both system types.
IV. Simulations
The simulations illustrate the theoretical results for hybrid multi-agent systems using six agents, two communication networks, and multiple consensus protocols. Examples 1 and 2 show trajectories consistent with the sufficiency of Theorem 1 and the results of Theorem 2, respectively.
- Simulation setup: The simulation study evaluates the theoretical results for hybrid multi-agent systems.The section states that simulations were provided to demonstrate the effectiveness of the paper’s theoretical results.
- Simulation setup: Six agents are used, with agents 1–3 continuous-time and agents 4–6 discrete-time.The initial state is x(0) = [−13, 14, 3, −9, −3, 6]T and the sampling period is h = 0.2.
- Example 1: Example 1 uses network G1, which has a directed spanning tree, and its trajectories are consistent with the sufficiency of Theorem 1 under consensus protocol (4).The stated sampling-period condition is h < 1.
- Example 2: Example 2 uses network G2, which has a directed spanning tree, and its trajectories are consistent with Theorem 2 under consensus protocol (8).The stated sampling-period condition is h < 1.
- Example 3: Example 3 considers the hybrid system under control inputs (12)–(15) with the communication network depicted in Fig. 5.The supplied passage begins describing a sampling-period condition and probabilistic edge selection but is incomplete.
7. The state trajectories
The section presents state-trajectory simulations for three communication networks under protocols (4), (8), and (12)–(15). The reported simulation results are consistent with the sufficiency of Theorem 3.
- Fig. 2 shows the state trajectories of all agents under consensus protocol (4) with communication network G1.
- Fig. 3 presents the directed graph G2 used with consensus protocol (8).
- Fig. 4 shows the state trajectories of all agents under consensus protocol (8) and communication network G2.
- Fig. 5 presents the undirected graph G3 used for the final simulation.
- Fig. 6 shows the state trajectories of all agents under consensus protocols (12)–(15) and network G3.The simulation results are stated to be consistent with the sufficiency of Theorem 3.
V. Conclusions
The paper establishes consensus conditions for hybrid multi-agent systems combining continuous-time and discrete-time agents under sampling-based communication. It identifies directed spanning trees and necessary-and-sufficient sampling-period conditions for consensus protocols, including a gossip-like protocol.
- System and communication: The hybrid multi-agent system combines continuous-time and discrete-time dynamic agents whose strategies are updated when sampling occurs.Agents communicate with their neighbours and update their strategies in the sampling time.
- Consensus condition: Under 0 < h < 1/max_i∈I n{d_ii}, consensus is achieved if and only if the communication network has a directed spanning tree.This condition is stated for the sampling-period restriction in the first protocol setting.
- Gossip-like protocol: With real-time self-state observation by continuous-time agents, a gossip-like consensus protocol is proposed with necessary and sufficient conditions for consensus when 0 < h < 1.The conclusions separately consider continuous-time agents observing their own states in real time.