Distributed Robust Control of Linear Multi-Agent Systems with Parameter Uncertainties

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

💡 Research Summary

This paper addresses the distributed robust control problem for linear multi‑agent systems whose agents share identical nominal dynamics but are subject to distinct norm‑bounded parametric uncertainties, a situation the authors term “weakly heterogeneous.” The communication topology is represented by an undirected connected graph, and only a subset of agents have access to their absolute states; all agents can measure relative states with respect to their neighbors.

The authors propose a distributed controller of the form
(u_i = c K\big(\sum_{j} a_{ij}(x_i - x_j) + d_i x_i\big)),
where (c>0) is a coupling gain, (K) is a common feedback matrix, (a_{ij}) are adjacency entries, and (d_i>0) for agents that can use absolute information (otherwise (d_i=0)). The uncertain dynamics are written as (\dot{x}_i = (A + \Delta A_i)x_i + B u_i) with (\Delta A_i = D F_i E) and (|F_i|_2 \le \delta).

By stacking all agent states and exploiting the Kronecker product, the closed‑loop network dynamics become
(\dot{x} = \big(I_N\otimes A + c,\tilde{L}\otimes B K + (I_N\otimes D)\Delta (I_N\otimes E)\big)x),
where (\tilde{L}=L + \mathrm{diag}(d_1,\dots,d_N)) and (L) is the Laplacian.

Key theoretical contribution (Theorem 1). The network is quadratically stable for all admissible uncertainties if and only if, for each eigenvalue (\lambda_i) of (\tilde{L}), the matrix (A + c\lambda_i B K) is Hurwitz and the transfer function
(T_i(s)=E(sI - A - c\lambda_i B K)^{-1}D)
has an (H_\infty) norm smaller than (1/\delta). This result decouples the original high‑dimensional robust stabilization problem into (N) independent scalar‑parameter robust (H_\infty) problems, each of the same dimension as a single agent. Consequently, the design of the feedback gain (K) can be performed independently of the graph topology.

Design algorithm for continuous‑time systems.

1. Solve the LMI
   (\begin{bmatrix} AP+PA^\top - \tau BB^\top & \delta DP^\top & EP\ \delta DP & -I & 0\ E^\top P & 0 & -I \end{bmatrix}<0)
   to obtain a positive matrix (P) and a scalar (\tau>0).
2. Set (K = -\frac{1}{2}B^\top P^{-1}).
3. Choose the coupling gain (c) such that (c \ge \tau / \lambda_{\min}(\tilde{L})).

Theorem 2 proves that this two‑step procedure yields a controller guaranteeing quadratic stability for any admissible uncertainties. Moreover, the LMI feasibility condition is both necessary and sufficient; thus the maximal allowable uncertainty bound (\delta_{\max}) can be obtained by maximizing (\delta) subject to the same LMI.

For discrete‑time agents, a similar formulation is derived, leading to an LMI‑based existence condition that ensures Schur stability of the closed‑loop system. Because eigenvalues of the stochastic matrix associated with the graph must lie inside the unit disk, the discrete‑time condition is slightly more conservative than its continuous counterpart.

Extension to (H_\infty) performance with external disturbances. The agents are further subjected to disturbances (\omega_i) and performance outputs (z_i = Cx_i). The same distributed controller (with the same (K) and (c)) is employed. By stacking the disturbance and output vectors, the closed‑loop system can again be decomposed into (N) independent scaled (H_\infty) problems. Hence, the design guarantees both robust quadratic stability and a prescribed (H_\infty) attenuation level for the overall network.

Illustrative examples. The authors validate their theory on three representative weakly heterogeneous systems: (i) a mass‑spring network with uncertain spring constants, (ii) a set of Lorenz‑type chaotic oscillators with differing parameters, and (iii) discrete‑time double integrators with unknown model parameters. Simulations demonstrate that the proposed distributed controller stabilizes the networks across the entire admissible uncertainty range and achieves the desired disturbance attenuation.

Overall significance. By converting a high‑dimensional distributed robust control problem into a set of low‑dimensional, decoupled (H_\infty) problems, the paper provides a clear, non‑conservative synthesis method that separates feedback gain design from graph topology. The LMI‑based procedure is computationally tractable and yields explicit bounds on allowable uncertainties. The extension to disturbance rejection further broadens the applicability to realistic networked control scenarios where both parametric uncertainties and external perturbations coexist. This work thus advances the state‑of‑the‑art in distributed robust control of weakly heterogeneous multi‑agent systems, offering both rigorous theoretical guarantees and practical design tools.