Designing Fully Distributed Consensus Protocols for Linear Multi-agent Systems with Directed Graphs

Abstract: This paper addresses the distributed consensus protocol design problem for multi-agent systems with general linear dynamics and directed communication graphs. Existing works usually design consensus protocols using the smallest real part of the nonzero eigenvalues of the Laplacian matrix associated with the communication graph, which however is global information. In this paper, based on only the agent dynamics and the relative states of neighboring agents, a distributed adaptive consensus protocol is designed to achieve leader-follower consensus for any communication graph containing a directed spanning tree with the leader as the root node. The proposed adaptive protocol is independent of any global information of the communication graph and thereby is fully distributed. Extensions to the case with multiple leaders are further studied.

• The paper presents a fully distributed adaptive consensus protocol that uses only local information to achieve leader-follower state convergence.
• It employs adaptive coupling weights and feedback gains derived from a Linear Matrix Inequality to ensure stability without global graph knowledge.
• The approach extends to multiple leaders for containment control, offering robust solutions for large-scale, directed network applications.

This paper tackles the problem of designing control protocols for multi-agent systems (MAS) where multiple agents with linear dynamics need to reach a consensus, specifically following the state of a designated leader agent. The communication between agents is represented by a directed graph, meaning information flow can be one-way.

A key challenge addressed is that many existing consensus protocols require knowledge of global properties of the communication graph, such as the smallest real part of the non-zero eigenvalues of the graph's Laplacian matrix ($\mathcal{L}_1$). This information is typically not available to individual agents locally, making such protocols difficult to implement in a truly distributed manner, especially in large or dynamic networks.

To overcome this limitation, the paper proposes a fully distributed adaptive consensus protocol. This protocol allows each follower agent to adjust its control input based only on its own state, the relative states of its immediate neighbors (including the leader if it's a neighbor), and its own internal adaptive parameter.

Proposed Adaptive Protocol:

For each follower agent $i$ (indexed $1, \dots, N$), the control input $u_i$ and an adaptive coupling weight $c_i(t)$ are defined as:

u_i = c_i * rho_i(xi_i^T * P^(-1) * xi_i) * K * xi_i d(c_i)/dt = xi_i^T * Gamma * xi_i

where:

• $x_i$: State vector of agent $i$.
• $x_j$: State vector of neighbor $j$.
• $a_{ij}$: Adjacency matrix entry (positive if agent $j$ sends info to $i$, 0 otherwise). The leader is index 0.
• $\xi_i = \sum_{j=0}^N a_{ij}(x_i - x_j)$: The local consensus error for agent $i$, based on weighted relative states of neighbors.
• $c_i(t)$: Time-varying adaptive coupling weight for agent $i$, initialized as $c_i(0) \ge 1$.
• $P$: A positive definite matrix solution to the Linear Matrix Inequality (LMI) $AP + PA^T - 2BB^T < 0$, where $A$ and $B$ define the agent dynamics $\dot{x} = Ax + Bu$. The existence of such a $P$ is guaranteed if the pair $(A,B)$ is stabilizable.
• $K = -B^T P^{-1}$: Feedback gain matrix.
• $\Gamma = P^{-1} B B^T P^{-1}$: Gain matrix for the adaptation law.
• $\rho_i(s)$: A smooth, monotonically increasing function with $\rho_i(s) \ge 1$ for $s > 0$. The paper specifically uses $\rho_i(s) = (1+s)^3$ where $s = \xi_i^T P^{-1} \xi_i$. This function provides extra design freedom compared to previous adaptive protocols.

Key Contributions and Implementation Aspects:

1. Fully Distributed: The protocol relies only on local information ($\xi_i$) and the agent's own dynamics matrices ($A, B$ via $P$, $K$, $\Gamma$). Agents do not need to know the overall network topology or Laplacian eigenvalues.
2. Directed Graphs: It works for general directed communication graphs, provided the graph contains a directed spanning tree with the leader as the root (Assumption 1). This is a significant improvement over previous adaptive methods often restricted to undirected graphs.
3. Adaptive Weights: The coupling strength $c_i$ adapts automatically based on the local consensus error $\xi_i$. The paper proves using a novel Lyapunov function (incorporating $c_i$, $\xi_i$, and properties of the Laplacian) that these weights converge to finite positive values.
4. Convergence Guarantee: Theorem 1 proves that under Assumption 1 and stabilizability of $(A,B)$, the protocol guarantees leader-follower consensus, i.e., $\lim_{t\rightarrow \infty}\|x_i(t)- x_0(t)\|=0$ for all followers $i$.
5. Implementation Steps:
   • For the given agent dynamics $(A, B)$, solve the LMI $AP + PA^T - 2BB^T < 0$ offline to find $P$. This requires stabilizability of $(A,B)$. Alternatively, solve the corresponding Algebraic Riccati Equation.
   • Compute $K = -B^T P^{-1}$ and $\Gamma = P^{-1} B B^T P^{-1}$. These are constant matrices.
   • Initialize each follower agent $i$ with $c_i(0) \ge 1$.
   • At runtime, each agent $i$:
     • Receives states $x_j$ from its neighbors $j$.
     • Calculates its local error $\xi_i = \sum_{j=0}^N a_{ij}(x_i - x_j)$.
     • Calculates the scalar $s = \xi_i^T P^{-1} \xi_i$.
     • Calculates $\rho_i = (1+s)^3$.
     • Computes its control input $u_i = c_i \rho_i K \xi_i$.
     • Updates its adaptive weight using the differential equation $\dot{c}_i = \xi_i^T \Gamma \xi_i$ (requiring numerical integration).
6. Multiple Leaders (Containment Control): The paper extends the result (Theorem 2) to the case of multiple leaders (indexed $0, \dots, M$). If each follower has a directed path from at least one leader (Assumption 2), the same adaptive protocol drives the followers' states into the convex hull spanned by the leaders' states (containment control). The proof follows similar steps.

Practical Significance:

This work provides a practical and robust method for achieving consensus in MAS with linear dynamics over directed communication links without needing centralized information or coordination regarding the network structure. This is valuable for applications like formation control, sensor network synchronization, and distributed estimation where network topology might be complex, large-scale, or time-varying (though the paper focuses on fixed topology). The adaptive nature potentially adds robustness compared to fixed-gain protocols requiring careful tuning based on global graph properties.