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Distributed Robust Control of Linear Multi-Agent Systems with Parameter Uncertainties
Zhongkui Li, Zhisheng Duan, Lihua Xie, Xiangdong Liu
TL;DR
The paper addresses robust distributed control for linear multi-agent systems whose agents share nominal dynamics but have different norm-bounded uncertainties. It designs continuous- and discrete-time controllers using neighboring relative states and a subset of absolute states, and relates quadratic stabilization to decoupled, single-agent-sized H∞ problems.
Problem
The paper studies robust distributed control when agents have identical nominal dynamics but different norm-bounded parameter uncertainties.
Source: §1 · p. 2Method
It designs continuous- and discrete-time distributed controllers using neighboring relative states and absolute states from a subset of agents.
Source: §1 · p. 2Results
For both time domains, distributed quadratic stabilization is equivalent to H∞ control of decoupled systems with the dimensions of a single agent.
Source: §1 · p. 2Takeaways & Limitations
The continuous-time two-step design decouples feedback-gain design from communication topology without conservatism.
Source: §1 · p. 2The discrete-time sufficient existence condition is somewhat conservative because communication-matrix eigenvalues are treated as uncertainties.
Source: §4 · p. 12
Abstract
from arXiv · showhide
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.
1 Introduction
Distributed control is motivated by the practical limits of centralized controllers and by the restrictive assumption that all agents have identical dynamics. The paper studies weakly heterogeneous agents and shows how robust control can be designed through single-agent-sized H∞ problems.
1 Introduction
- Centralized controllers may be too expensive or infeasible because actuators are spatially distributed and communication and sensing are limited.Distributed strategies instead use local information.Source: §1 · p. 1
- Earlier work commonly assumed identical agent dynamics, while heterogeneous-agent studies addressed nonidentical systems using approaches including S-hull, IQC, and internal-model methods.The cited examples include robust stability, output regulation, and neural adaptive tracking.Source: §1 · p. 2
- The paper studies agents with identical nominal dynamics but different norm-bounded uncertainties, a weakly heterogeneous setting between homogeneous and heterogeneous systems.Examples include mass-spring systems with differing or uncertain spring constants and Lorenz-type systems with parameter changes.Source: §1 · p. 2
- For both continuous- and discrete-time systems, distributed quadratic stabilization is equivalent to H∞ control of decoupled systems with the dimensions of one agent.The controllers use neighboring relative states and absolute states from a subset of agents.Source: §1 · p. 2
- The continuous-time two-step design decouples feedback-gain design from topology without conservatism, while the discrete-time existence condition is sufficient and somewhat conservative.The discrete-time conservatism comes from treating communication-matrix eigenvalues as uncertainty.Source: §1 · p. 2
- The paper also considers distributed quadratic stabilization with H∞ disturbance attenuation, reducible to scaled H∞ problems for independent single-agent-sized systems.Design procedures are given for both continuous- and discrete-time cases.Source: §1 · pp. 2–3
2 Graph Theory and Notation
This section establishes matrix and graph notation used to describe communication structure and stability. In the undirected graph, edges are unordered, and connectedness means every pair of distinct nodes is linked by a path.
2 Graph Theory and Notation
- Hurwitz means continuous-time eigenvalues have negative real parts, whereas Schur stable means discrete-time eigenvalues have magnitude below one.The section also defines compatible matrix notation, positive (semi-)definite inequalities, and the Kronecker product.Source: §2 · p. 3
- An undirected graph consists of nodes and unordered edges; it is connected when every pair of distinct nodes has a path between them.The adjacency matrix is symmetric, and the Laplacian has zero as an eigenvalue with the all-ones vector as a corresponding eigenvector.Source: §2 · p. 3
- The graph-associated double-stochastic matrix has positive diagonal entries and symmetric positive weights on edges; its non-unit eigenvalues lie inside the open unit disk.The eigenvalue 1 corresponds to the all-ones vector.Source: §2 · p. 3
3 Distributed Robust Control of Uncertain Continuous-Time MultiAgent Systems
This section develops distributed robust control for uncertain continuous-time multi-agent systems over connected undirected networks, reducing design to decoupled single-agent-dimensional conditions for quadratic stabilization and stabilization with a prescribed disturbance-attenuation level. It also completes the proof by rescaling variables and establishing equivalence to (10).
3.1 Distributed Quadratic Stabilization
- Quadratic stability holds exactly when every mode A + cλ_iBK is Hurwitz and its uncertainty channel satisfies the stated H∞ bound.The modes are indexed by the eigenvalues λ_i of the augmented Laplacian; the condition is necessary and sufficient for all admissible uncertainties.Source: §3.1 · p. 5
- Theorem 1’s modal decomposition shows that the network is quadratically stable exactly when its N single-agent-dimensional modes are simultaneously quadratically stable.This equivalence relies on identical nominal agent dynamics, despite the agents having different parameter uncertainties.Source: §3.1 · p. 5
- Algorithm 1 constructs a controller that guarantees quadratic stability, and its LMI feasibility is necessary and sufficient for controller existence.The largest allowable uncertainty bound δ_max can be found by optimization subject to that LMI.Source: §3.1 · pp. 6–7
3.2 Distributed Quadratic H∞Control
- The robust H∞ network condition follows from simultaneous stability and unitary disturbance attenuation of a set of decoupled systems.The systems have the dimensions of a single agent, with augmented disturbance and performance variables.Source: §3.2 · pp. 7–8
- Algorithm 2’s controller guarantees quadratic stability with disturbance attenuation γ for all admissible uncertainties.Its feedback-gain design is decoupled from the communication topology; only the smallest augmented-Laplacian eigenvalue is used to select the coupling strength, allowing reuse across connected graphs.Source: §3.2 · p. 9
- As γ tends to infinity, Algorithm 2’s LMI condition reduces to Algorithm 1’s stabilization LMI.The reduction is stated as Corollary 1.Source: §3.2 · p. 9
- Setting Q = γP, ε = γ^-1, and τ̃ = γ^-1τ yields εQ E^T E Q + Q C^T C Q < 0, which is equivalent to (10).Source: §3.2 · p. 9
4 Distributed Robust Control of Uncertain Discrete-Time MultiAgent Systems
For uncertain discrete-time agents, the paper designs distributed controllers using neighbor-relative states and a subset of agents’ absolute states. It gives quadratic-stability conditions, a sufficient LMI-based controller design, and an extension to disturbance attenuation.
4 Distributed Robust Control of Uncertain Discrete-Time MultiAgent Systems
- The distributed controller uses neighbor-relative states and absolute states available to a subset of agents; the agent uncertainties have the form ΔA_i = D F_i E.The network is modeled as N discrete-time agents, and the controller is defined using a doubly stochastic communication matrix.Source: §4 · pp. 9–10
- Quadratic stability is characterized by conditions on the matrices A + (1 − λ̃_i)BK and their uncertainty transfer functions for every eigenvalue λ̃_i of the modified communication matrix.Theorem 5 gives the characterization under Assumption 1.Source: §4 · pp. 10–11
- Feasibility of the LMI provides a sufficient condition for a quadratically stabilizing controller, with gain K = WQ^-1; maximizing δ in the LMI gives the largest allowable uncertainty bound.Theorem 5’s condition is conservative because it bounds communication-matrix eigenvalues by κ, but a larger κ can provide robustness to graph changes.Source: §4 · pp. 11–12
- With external disturbances, quadratic stability and attenuation follow if the associated systems are simultaneously asymptotically stable with unitary disturbance attenuation.The theorem applies to all admissible agent uncertainties under Assumption 1.Source: §4 · p. 12
5 Simulation Examples
The simulations illustrate the proposed controllers on continuous-time mass-spring agents and uncertain discrete-time agents. They report a continuous-time stabilization condition and allowable uncertainty bounds for both examples.
Example 1
- The continuous-time example uses agents with common mass m = 2.5 kg and nominal spring constant k_0 = 7 N/m, while spring-constant deviations satisfy |Δk_i| ≤ δ.The example obtains a feasible solution to the LMI with δ = 10.Source: §5 · p. 13
- c ≥ 270.2295 is reported to solve the continuous-time stabilization problem for |Δk_i| ≤ 10; the simulation uses c = 275 and randomly chosen Δk_i in (7, 22].The agents are mass-spring systems with a common nominal spring constant and different uncertain spring constants.Source: §5 · pp. 13–14, Figure 2 · p. 14
- δ tends to infinity as the maximum allowable uncertainty in the continuous-time optimization, while the paper notes that very large δ generally implies a high-gain controller.The cited example uses δ = 1.5 × 10^5 to illustrate the high-gain qualification.Source: §5 · p. 14
Example 2
- δ_max = 2.5 is the largest allowable uncertainty bound reported for the discrete-time example when κ = 0.9.The resulting controller’s state trajectories are shown for network (26).Source: §5 · pp. 14–15, Figure 3 · p. 15
6 Conclusion
The paper studies robust distributed control for linear multi-agent systems with identical nominal dynamics and different norm-bounded uncertainties. For continuous- and discrete-time cases, quadratic stabilization under the proposed controllers is equivalent to H∞ control of decoupled single-agent-sized systems.
6 Conclusion
- The agents share nominal dynamics but have different norm-bounded parameter uncertainties.The controllers use neighboring agents’ relative states and a subset of agents’ absolute states.Source: §6 · p. 15
- For both continuous- and discrete-time systems, distributed quadratic stabilization is equivalent to H∞ control of decoupled systems sized like a single agent.Source: §6 · p. 15
- The paper presents algorithms for constructing the distributed controllers and identifies extending the results to consensus and formation control as future work.Source: §6 · p. 15