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Distributed Robust Consensus Control of Multi-agent Systems with Heterogeneous Matching Uncertainties
Zhongkui Li, Zhisheng Duan, Frank Lewis
TL;DR
The paper studies distributed consensus for linear multi-agent systems with heterogeneous matching uncertainties, including leaderless and leader-follower cases. It develops static and adaptive protocols that bound consensus errors and extends them to bounded nonmatching disturbances, with stabilizability as a sufficient existence condition.
Problem
Different matching uncertainties make multi-agent systems heterogeneous and prevent consensus algorithms developed for homogeneous systems from applying directly.
Method
The paper develops distributed continuous static and adaptive consensus protocols for undirected leaderless systems and leader-follower systems with unknown bounded leader input.
Results
The protocols produce ultimately bounded consensus errors that exponentially converge to small adjustable residual sets, while adaptive designs require neither global graph information nor uncertainty and leader-input upper bounds.
Takeaways & Limitations
The proposed protocols also remain applicable after redesign for bounded external disturbances that do not satisfy the matching condition.
Takeaways & Limitations
For bounded nonmatching disturbances, the stated sufficient existence condition is controllability, and larger ε generally trades faster convergence for a higher-gain K.
Abstract
from arXiv · showhide
This paper considers the distributed consensus problem of linear multi-agent systems subject to different matching uncertainties for both the cases without and with a leader of bounded unknown control input. Due to the existence of nonidentical uncertainties, the multi-agent systems discussed in this paper are essentially heterogeneous. For the case where the communication graph is undirected and connected, a distributed continuous static consensus protocol based on the relative state information is first designed, under which the consensus error is uniformly ultimately bounded and exponentially converges to a small adjustable residual set. A fully distributed adaptive consensus protocol is then designed, which, contrary to the static protocol, relies on neither the eigenvalues of the Laplacian matrix nor the upper bounds of the uncertainties. For the case where there exists a leader whose control input is unknown and bounded, distributed static and adaptive consensus protocols are proposed to ensure the boundedness of the consensus error. It is also shown that the proposed protocols can be redesigned so as to ensure the boundedness of the consensus error in the presence of bounded external disturbances which do not satisfy the matching condition. A sufficient condition for the existence of the proposed protocols is that each agent is stabilizable.
1 Introduction
The paper frames consensus as distributed agreement under local information, then addresses heterogeneous multi-agent systems with different matching uncertainties through static and adaptive robust protocols. It covers leaderless and leader-follower settings, nonmatching disturbances, and existence conditions for the proposed protocols.
- Motivation: Consensus uses distributed control policies based only on local information to make autonomous agents agree on quantities of interest.The paper highlights applications including spacecraft formation flying, sensor networks, and cooperative surveillance.
- Problem setting: Different matching uncertainties make the agents essentially heterogeneous and render existing consensus algorithms for homogeneous systems inapplicable.The uncertainties may be time-varying, nonlinear, and unknown.
- Main approach: A continuous static protocol for undirected connected graphs yields uniformly ultimately bounded consensus error with exponential convergence to a small residual set.Its design requires Laplacian eigenvalues and upper bounds of the matching uncertainties.
- Main approach: A fully distributed adaptive protocol removes the need for Laplacian eigenvalues and uncertainty upper bounds, while providing adjustable residual sets and explicit convergence rates.The protocol addresses the global-information limitation identified in earlier consensus designs.
- Extensions: Static and adaptive protocols also address leader-follower consensus when the leader has an unavailable, possibly nonzero control input.The paper additionally redesigns the protocols for bounded nonmatching disturbances, and identifies stabilizability of each agent as a sufficient existence condition.
2 Notation and Graph Theory
This section introduces matrix and vector notation and defines directed and undirected communication graphs, adjacency matrices, Laplacian matrices, and their connectivity-related eigenvalue property.
- Notation: The notation includes identity matrices, block-diagonal matrices, Kronecker products, vector 2-norms, and minimum and maximum eigenvalues of symmetric matrices.IN is the identity matrix of dimension N, while A⊗B denotes the Kronecker product.
- Graph theory: A directed graph consists of nodes and ordered edges, with parent, child, neighbor, path, subgraph, and undirected-graph concepts defined from those edges.An undirected graph contains both edge directions whenever either direction is present.
- Graph matrices: The adjacency matrix records graph edges, and the Laplacian uses node indegrees on its diagonal and negative adjacency entries off-diagonal.For undirected graphs, both adjacency and Laplacian matrices are symmetric.
- Graph properties: Zero is a Laplacian eigenvalue with the all-ones right eigenvector, while all nonzero eigenvalues have positive real parts.Zero is simple exactly when the directed graph has a directed spanning tree.
3 Distributed Robust Leaderless Consensus
The paper models leaderless consensus for identical nominal linear agents with heterogeneous, possibly time-varying, nonlinear, and unknown matching uncertainties. For connected undirected graphs, it develops continuous static and adaptive protocols that bound consensus error, with adaptive control avoiding key global information requirements.
- Problem: Heterogeneous uncertainties make otherwise identical nominal linear agents substantially harder to coordinate through consensus.The uncertainties may be time-varying, nonlinear, unknown, and nonidentical across agents.
- Distributed Static Consensus Protocol: A continuous static protocol based on neighboring relative states makes the consensus error uniformly ultimately bounded and exponentially convergent to a small residual set.The protocol includes linear and nonlinear terms, with the nonlinear component suppressing uncertainty effects.
- Distributed Static Consensus Protocol: Under the static protocol, reducing the boundary-layer parameter κ makes the residual set arbitrarily small, but does not provide asymptotic stability.The residual set also depends on graph and system quantities, including the smallest nonzero Laplacian eigenvalue.
- Distributed Adaptive Consensus Protocol: The adaptive protocol uses local neighboring-state information and avoids requiring the smallest nonzero Laplacian eigenvalue or uncertainty upper bounds.Its gains are adapted online, while σ-modification terms help ensure ultimate boundedness of the consensus error and adaptive gains.
- Distributed Adaptive Consensus Protocol: With suitable feedback gains, the adaptive protocol ensures uniform ultimate boundedness and exponential convergence of both the consensus error and adaptive gains to residual sets.The convergence rates and residual sets depend on design parameters and uncertainty-growth constants.
4 Distributed Robust Leader-Follower Consensus with a Leader of Nonzero Control Input
The paper formulates leader-follower consensus when the leader has bounded, unknown, possibly time-varying control input and followers face nonidentical matching uncertainties. It proposes distributed static and adaptive protocols that keep the consensus error uniformly ultimately bounded, with exponential convergence to residual sets under stated conditions.
- Problem: The leader’s control input is possibly nonzero, time varying, bounded, and inaccessible to followers.The leader-follower setting is therefore harder than the case with zero leader input.
- Assumptions: The communication graph contains a leader-rooted directed spanning tree, while the follower subgraph is undirected.The leader receives no follower information, and only a subset of followers can access its state.
- 4.1 Distributed Static Consensus Protocol: The distributed static protocol uses neighboring relative states and parameters constrained by the smallest eigenvalue of L1, the leader-input bound γ, and an LMI solution P.The protocol includes a modified nonlinear function to handle the leader’s nonzero input, with c2 related only to the upper bound of u0.
- Problem formulation: The leader-follower consensus error is the difference between each follower state and the leader state, and consensus is achieved when this error converges to zero.The stacked error is denoted ζ.
- 4.1 Distributed Static Consensus Protocol: The static protocol makes the leader-follower consensus error uniformly ultimately bounded and exponentially convergent to a residual set.The convergence rate is faster than exp(−αt).
- 4.2 Distributed Adaptive Consensus Protocol: The adaptive protocol removes the need for the global eigenvalue, leader-input bound, and matching-uncertainty bounds required by the static design.Its adaptive gains and consensus error are uniformly ultimately bounded, with convergence faster than exp(−δt) under the theorem’s conditions.
5 Robustness With Respect To Bounded Non-matching Disturbances
The paper extends its consensus protocols to bounded external disturbances that do not satisfy the matching condition, redesigning static and adaptive protocols to retain bounded consensus error. Under stated connectivity, disturbance, and LMI conditions, the error converges exponentially to explicit residual sets.
- Problem: Bounded non-matching disturbances are introduced as a robustness case beyond the earlier matching-condition assumption.The section focuses on the leaderless problem and assumes each disturbance is bounded.
- Static protocol: The distributed static protocol is redesigned to guarantee boundedness of the consensus error under non-matching disturbances.The result applies when the communication graph is connected and the stated assumptions hold, with K and Q selected through the given LMI.
- Scope and limitation: Existence of the redesigned protocol requires a positive-definite Q satisfying the LMI, which is equivalent to controllability and introduces a convergence-rate versus gain trade-off.Larger ε gives faster convergence but generally requires a higher-gain K.
- Adaptive protocol: The adaptive protocol is likewise redesigned, and both the consensus error and adaptive gains are uniformly ultimately bounded.The adaptive design uses K = −B^T Q^-1 and Γ = Q^-1BB^TQ^-1, where Q > 0 solves the stated LMI.
- Adaptive protocol: The adaptive design yields exponential convergence to residual sets, with a rate faster than exp(−σt) under the stated parameter conditions.For smaller adaptation parameters satisfying the additional inequality, the error converges to a further residual set at a rate faster than exp(−̺t).
- Scope and limitation: Non-matching disturbances enlarge the residual sets because their bounds and the largest eigenvalue of the communication graph Laplacian enter the resulting bounds.This contrasts with the residual sets for the earlier matching-uncertainty cases.
6 Simulation Examples
The simulations examine leaderless and leader-follower consensus protocols on mass-spring systems and Chua’s circuits with heterogeneous uncertainties. The figures present communication graphs, state trajectories, and adaptive gains.
- The section introduces numerical examples to illustrate the theoretical results.
- Figures: Figures report leaderless and leader-follower communication graphs, state trajectories, and adaptive gains for the two simulation settings.
- Mass-spring systems: The mass-spring example models agents with bounded unknown spring constants and rewrites their dynamics as linear systems with matching uncertainty.The uncertainty term is kiExi, with ||kiExi|| ≤ ki||xi||.
- Mass-spring systems: Because the spring constants are unknown, the example uses the adaptive protocol (23) for consensus.The simulation sets m = 2.5 kg and computes feedback gains by solving LMI (8).
- Chua’s circuits: The Chua’s-circuit example uses a leader indexed by 0 and followers with nonidentical nonlinear components.The circuits are represented in compact form as linear nominal dynamics plus uncertainty, with the leader’s nonlinear term treated as a virtual control input.
7 Conclusion
The paper develops distributed static and adaptive protocols for robust consensus with heterogeneous matching uncertainties, with and without a leader having bounded unknown control input. The protocols yield bounded consensus errors, including under bounded external disturbances that need not satisfy the matching condition.
- Distributed continuous static and adaptive protocols are designed for multi-agent systems with heterogeneous matching uncertainties, covering cases with and without a leader.
- The consensus error is ultimately bounded and exponentially converges to small adjustable residual sets.
- The adaptive protocols require neither global communication-graph information nor upper bounds on uncertainties and the leader’s control input.
- The protocols can be redesigned to ensure bounded consensus error under bounded external disturbances that do not necessarily satisfy the matching condition.