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Guaranteed-cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies
Jianxiang Xi, Zhiliang Fan, Hao Liu, Tang Zheng
TL;DR
The paper addresses the tradeoff between consensus performance and energy use in high-order multi-agent networks with Lipschitz nonlinear dynamics. It decomposes the network dynamics and develops a guaranteed-cost criterion; simulations show agents converging to the consensus trajectory with finite cost below the guaranteed bound.
Problem
Consensus regulation must meet performance requirements while agents have limited energy, and prior guaranteed-cost work did not address nonlinear dynamics.
Source: §1 · pp. 2–3Method
A structured time-varying transformation separates consensus and disagreement dynamics, enabling a Riccati-based guaranteed-cost criterion and an LMI approach to cost minimization.
Source: §1 · p. 2, §4 · p. 8, §6 · p. 14Results
Simulations show all agents converging to the consensus trajectory while the cost converges to a finite value below β.
Source: §5 · p. 14Takeaways & Limitations
The consensus dynamics follows each agent's own dynamics with initial state equal to the average of the agents' initial states.
Source: §6 · p. 14The approach assumes undirected, connected switching topologies; it is not valid for directed topologies that may have asymmetric, nondiagonalizable Laplacians.
Source: §6 · p. 14
Abstract
from arXiv · showhide
Guaranteed-cost consensus for high-order nonlinear multi-agent networks with switching topologies is investigated. By constructing a time-varying nonsingular matrix with a specific structure, the whole dynamics of multi-agent networks is decomposed into the consensus and disagreement parts with nonlinear terms, which is the key challenge to be dealt with. An explicit expression of the consensus dynamics, which contains the nonlinear term, is given and its initial state is determined. Furthermore, by the structure property of the time-varying nonsingular transformation matrix and the Lipschitz condition, the impacts of the nonlinear term on the disagreement dynamics are linearized and the gain matrix of the consensus protocol is determined on the basis of the Riccati equation. Moreover, an approach to minimize the guaranteed cost is given in terms of linear matrix inequalities. Finally, the numerical simulation is shown to demonstrate the effectiveness of theoretical results.
1. INTRODUCTION
High-order multi-agent consensus is difficult to design because general agent dynamics lack the structure of integrator models, while practical applications also require balancing regulation performance against energy use. This paper addresses open problems in guaranteed-cost consensus with Lipschitz nonlinear dynamics and switching topologies by deriving the consensus dynamics, a Riccati-based criterion, and a minimum-cost approach.
1. INTRODUCTION
- High-order agents lack the structural features of first- and second-order integrators, making their consensus criteria harder to determine.Earlier high-order linear network studies cited here did not consider optimization performance.Source: §1 · p. 2
- Practical consensus design must trade off regulation performance against agents’ limited energy for sensing, communication, and movement.The cited first-order optimal criterion requires a complete interaction graph, while cited second-order work either omits energy or uses a suboptimal criterion.Source: §1 · p. 2
- The paper addresses three open problems for nonlinear networks with switching topologies: determining the consensus function, designing the protocol gain, and minimizing guaranteed cost.Earlier guaranteed-cost criteria for high-order networks either had variable dimensions tied to the number of agents or did not address nonlinear dynamics.Source: §1 · p. 2
- A structured time-varying orthonormal transformation decomposes the network into consensus and disagreement subsystems and yields an explicit consensus dynamics and initial state.The consensus dynamics is intrinsically coincident with each agent’s own dynamics, but its initial state differs.Source: §1 · p. 2
- A Riccati-based sufficient criterion gives a guaranteed-cost upper bound with a variable dimension independent of the number of agents, and an LMI approach minimizes the cost.The transformation structure and Lipschitz condition support linearizing the nonlinear effects on disagreement dynamics.Source: §1 · pp. 2–3
- The paper’s four stated advances cover nonlinear dynamics, explicit consensus dynamics and initial states, linearization of disagreement effects, and minimum guaranteed cost under mild assumptions.The cited earlier criteria - did not cover nonlinear dynamics or guarantee that their costs were minimum.Source: §1 · p. 3
2. PROBLEM DESCRIPTION
The problem is to achieve guaranteed-cost consensus for identical high-order nonlinear agents whose interaction topology switches among connected undirected graphs. The section specifies the agent and protocol setting, cost objective, assumptions, and three design questions.
2. PROBLEM DESCRIPTION
- Each of the N identical agents has state dynamics with a nonlinear term, and the nonlinear function is assumed Lipschitz with constant γ > 0.The passage also specifies that the nonlinear function is continuous and differentiable.Source: §2 · p. 3
- The stated controllability requirement is that (A, B) be controllable to ensure positive distance to unobservability for (A^T, B^T).The passage cautions that controllability alone, sufficient for linear networks, does not suffice when Lipschitz nonlinear dynamics are present.Source: §2 · p. 4
- The switching topologies are assumed connected and undirected, so each topology’s Laplacian is symmetric with a simple zero eigenvalue.The switching signal is piecewise constant with a positive minimum interval between switches.Source: §2 · p. 4
- The network uses a neighbor-state consensus protocol with gain matrix K and symmetric nonnegative interaction weights that switch over time.The switching signal selects among undirected graphs, with switching times separated by at least a positive dwell time.Source: §2 · p. 4
- Guaranteed-cost consensus requires the agents’ states to approach a common consensus function while the consensus-regulation cost remains bounded by a positive constant.The definition applies to any bounded initial state; a protocol is guaranteed-cost consensualizable if some gain achieves that property.Source: §2 · p. 5
- The section poses three design questions: characterize nonlinear effects on the consensus function, choose a gain for guaranteed-cost consensus, and determine and minimize the cost.The stated setting is nonlinear multi-agent networks with switching connected topologies.Source: §2 · p. 5
3. CONSENSUS FUNCTIONS
The section characterizes the consensus function under nonlinear dynamics and switching topologies. A structured time-varying transformation separates consensus and disagreement subsystems, although nonlinear terms remain in both.
3. CONSENSUS FUNCTIONS
- The section gives an explicit expression for the consensus dynamics and determines its initial state.The consensus dynamics contains the nonlinear term.Source: §3 · pp. 5–6
- Both transformed subsystems retain nonlinear terms, so the transformation does not completely separate consensus from disagreement dynamics.The section presents an approach for determining the consensus dynamics and its initial state despite this coupling.Source: §3 · p. 6
- When consensus is achieved, ˜x1(t)/√N is a valid candidate for the consensus function.Consensus is characterized by the disagreement component tending to zero.Source: §3 · p. 7
- Switching topologies do not impact the consensus function, while the consensus dynamics of nonlinear networks includes the nonlinear term.Earlier methods cited in the section assumed fixed topologies and linear agent dynamics.Source: §3 · p. 8
4. GUARANTEED-COST CONSENSUALIZATION CRITERIA
The section develops guaranteed-cost consensus criteria for high-order networks with Lipschitz nonlinearities and switching topologies. It uses a structured transformation to handle nonlinear effects and a Riccati equation to determine the protocol gain, then gives an LMI approach to minimize guaranteed cost.
4. GUARANTEED-COST CONSENSUALIZATION CRITERIA
- The transformation's structure and the Lipschitz condition linearize nonlinear effects in the disagreement dynamics, allowing the gain matrix to be determined.The method addresses nonlinear terms in both consensus and disagreement dynamics.Source: §4 · pp. 8–10
- Theorem 2 gives guaranteed-cost consensualization with K = B^T P when P > 0 satisfies the Riccati criterion, whose variable dimension is independent of the number of agents.The guaranteed cost is bounded under this criterion.Source: §4 · p. 8
- The Lyapunov argument establishes convergence: its derivative is nonpositive and vanishes only when all disagreement states are zero.The resulting disagreement state converges to zero, so the network achieves consensus.Source: §4 · pp. 9–10
- A positive-definite Riccati solution requires (A, B) to be stabilizable; in the nonlinear-free case, the stated bound yields a unique positive-definite solution.The Riccati equation's terms capture impacts of nonlinear dynamics and the cost function on guaranteed-cost consensualization.Source: §4 · pp. 10–11
- The minimum-cost formulation uses an LMI approach under a zero-mean initial-state-error assumption with identity second moment, yielding β* = 0.5N · trace {P}.The paper frames the minimum in expectation because guaranteed cost depends on the agents' initial states.Source: §4 · p. 11
- For fixed x(0) and P, guaranteed cost is directly proportional to the number of agents.The paper relates this scaling to the minimum guaranteed cost in Theorem 4.Source: §4 · p. 12
5. NUMERICAL SIMULATIONS
The six-agent simulation uses switching interaction topologies and reports consensus of the agents’ state trajectories alongside a bounded cost trajectory.
5. NUMERICAL SIMULATIONS
- The simulated network has six agents, with γ = 0.333 and states xi = [xi1, xi2, xi3, xi4]T.The agent dynamics are described by equation (1).Source: §5 · p. 12
- The switching set contains four undirected interaction topologies, with the switching signal shown separately.Figure 1 presents the topology set and Figure 2 the switching signal σ(t).Source: §5 · p. 12, Figure 1 · p. 12, Figure 2 · p. 13
- All six agents’ state trajectories converge to the consensus-function curve, while JT converges to a finite value below β.The authors state that the simulation demonstrates guaranteed-cost consensus for this Lipschitz nonlinear network.Source: Figure 3 · p. 13, Figure 4 · p. 13, §5 · p. 14
6. CONCLUSIONS
The paper presents guaranteed-cost consensus results for high-order networks with Lipschitz nonlinearities and switching topologies, while restricting its approach to undirected, connected topologies.
6. CONCLUSIONS
- The consensus dynamics is explicitly characterized, coincides intrinsically with each agent’s own dynamics, and starts from the average of all agents’ initial states.The paper also gives a Riccati-equation-based consensualization criterion whose variable dimension is independent of the number of agents; its guaranteed-cost upper bound depends on all agents’ initial states.Source: §6 · p. 14
- An LMI-based approach is proposed to determine the minimum guaranteed cost.The criterion’s guaranteed-cost upper bound is associated with the initial states of all agents.Source: §6 · p. 14
- The approach assumes every switching topology is undirected and connected; directed topologies may require new methods because their Laplacians can be asymmetric and non-diagonalizable.The paper identifies simplification of the quadratic cost and elimination of nonlinear effects as challenges for directed switching topologies.Source: §6 · p. 14
7. APPENDIX
The appendix defines undirected graphs and states the spectral property of the Laplacian for a connected graph.
7. APPENDIX
- An undirected graph is specified by a node set, an edge set, and a symmetric adjacency matrix.The adjacency entries are positive for edges and diagonal entries are zero.Source: §7 · p. 14
- A node’s neighbor set contains nodes connected to it by an edge, and a graph is connected when an undirected path joins every pair of nodes..Source: §7 · p. 14
- For a connected graph, the Laplacian has a simple zero eigenvalue and its other N − 1 eigenvalues are positive.The all-ones vector is associated with a zero eigenvalue.Source: §7 · p. 14