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Leader-Based Optimal Coordination Control for the Consensus Problem of Multiagent Differential Games via Fuzzy Adaptive Dynamic Programming
Huaguang Zhang, Jilie Zhang, Guang-Hong Yang, Yanhong Luo
TL;DR
The paper addresses optimal leader-following consensus coordination for nonlinear multi-agent differential games, where coupled HJ equations must be solved. It proposes online FADP with GFHM value-function approximation, policy iteration, and a single network per agent. The analysis proves UUB of local consensus and weight-estimation errors and CUUB of the control-node trajectory.
Problem
Optimal coordination must stabilize multi-agent systems, minimize agents’ performance indexes, and achieve consensus despite coupled neighborhood decisions.
Method
FADP uses GFHMs to approximate coupled HJ-equation value functions under policy iteration, with gradient-based online weight updates and a single network per agent.
Results
The proposed scheme proves that weight-estimation errors and local consensus errors are uniformly ultimately bounded, while the control-node trajectory is cooperatively uniformly ultimately bounded.
Takeaways & Limitations
Single-GFHM FADP eliminates the action network and reduces updated weights for optimal coordination control in multi-agent systems.
Abstract
from arXiv · showhide
In this paper, a new on-line scheme is presented to design the optimal coordination control for the consensus problem of multi-agent differential games by fuzzy adaptive dynamic programming (FADP), which brings together game theory, generalized fuzzy hyperbolic model (GFHM) and adaptive dynamic programming. In general, the optimal coordination control for multi-agent differential games is the solution of the coupled Hamilton-Jacobi (HJ) equations. Here, for the first time, GFHMs are used to approximate the solution (value functions) of the coupled HJ equations, based on policy iteration (PI) algorithm. Namely, for each agent, GFHM is used to capture the mapping between the local consensus error and local value function. Since our scheme uses the single-network rchitecture for each agent (which eliminates the action network model compared with dual-network architecture), it is a more reasonable architecture for multi-agent systems. Furthermore, the approximation solution is utilized to obtain the optimal coordination controls. Finally, we give the stability analysis for our scheme, and prove the weight estimation error and the local consensus error are uniformly ultimately bounded. Further, the control node trajectory is proven to be cooperative uniformly ultimately bounded.
I. INTRODUCTION
The paper frames optimal consensus coordination as a cooperative multi-agent differential game and proposes an online FADP scheme using GFHMs and a single-network architecture.
- Motivation: Optimal coordination must stabilize the multi-agent system, minimize each agent’s performance index, and drive agents to consensus.Each agent selects its control using its own and neighboring agents’ actions, creating a cooperative multi-player game.
- Motivation: Coupled Hamilton-Jacobi equations characterize the optimal coordination controls for multi-agent differential games.The equations are coupled because each agent’s action depends on its own and neighboring agents’ outcomes.
- Related work: Existing LQR methods are limited to linear systems and offline design, while MPC requires continuous sampling and repeated finite-horizon prediction.The paper instead addresses continuous nonlinear leader-following consensus online.
- Proposed approach: FADP uses GFHMs to approximate value functions while policy iteration and gradient descent update their weights online.The approach combines game theory, adaptive dynamic programming, and generalized fuzzy hyperbolic models.
- Contributions: The proposed single-network architecture removes the action network and reduces the number of updated weights compared with dual-network designs.The paper presents this architecture for solving the optimal consensus problem in multi-agent systems.
- Foundations: The paper establishes graph-theoretic foundations using weighted communication graphs, Laplacian matrices, neighbors, and fixed strongly connected topology.GFHM rule bases are also introduced as the function-approximation framework.
III. CONSENSUS ERROR DYNAMIC SYSTEM
This section models nonlinear leader-following multi-agent dynamics and derives local and global consensus-error systems from the communication graph and pinning structure.
- Node dynamics: Each agent has nonlinear dynamics with state x_i and coordination input u_i, while the leader follows its own trajectory x_0(t).The node dynamics use f(x_i) and g_i(x_i)u_i, with f(0)=0.
- Consensus errors: The local consensus error e_i measures agreement between node v_i, the leader, and its neighboring nodes.Its dynamics depend on the states and control inputs of node v_i and all its neighbors.
- Global error system: The global error vector stacks all local errors and is formed using the pinned Laplacian L+B and a Kronecker product with the identity matrix.The matrix B records leader pinning gains.
- Boundedness notions: UUB bounds local consensus errors after a finite settling time, while CUUB bounds every node’s distance from the leader trajectory.Both definitions use bounds and settling times independent of the initial time.
IV. OPTIMAL COORDINATION CONTROL
The paper formulates optimal coordination as an N-person cooperative game so agents reach consensus while minimizing their local performance indexes.
- Cooperative-game formulation: N-person cooperative games are used to design optimal coordination controls that achieve consensus while minimizing each agent’s local performance index.The formulation applies the game-theoretic machinery to the local consensus-error systems.
A. The Coupled HJ Equation
The section defines local cost functionals and coupled Hamiltonians for consensus-error dynamics, then links their optimal value functions to Nash-equilibrium coordination controls.
- Cost functionals: Each local performance index combines state-error penalties with control penalties involving the agent and relevant neighboring inputs.Weighting matrices satisfy Q_ii > 0, R_ii > 0, and R_ij ≥ 0, with neighbor dependence encoded by R_ij.
- Problem formulation: The design problem is to stabilize the local error systems, keep local cost functionals finite, and make all agents reach consensus with the leader.Admissible policies must satisfy both stability and finite-cost requirements.
- Coupled Hamiltonians: The coupled local Hamiltonians incorporate the value-function gradient, local error dynamics, and control costs.Their coupling reflects the dependence of each local problem on neighboring agents’ controls.
- Coupled HJ equations: The local optimal value functions satisfy coupled HJ equations obtained from the necessary condition of optimality.The value functions are defined under admissible coordination policies.
- Optimal controls: The optimal coordination controls are obtained from the optimal value functions solving the coupled HJ equations.The solution of these equations is identified with a Nash equilibrium.
B. Nash Equilibrium
The coupled HJ equations characterize local optimal value functions whose associated coordination policies form a Nash equilibrium. Under the stated conditions, these value functions also establish asymptotic stability of local neighborhood consensus errors, while the paper notes that ADP generally obtains only a local optimum.
- A solution of the coupled HJ equations, combined with the corresponding control policies, yields a Nash equilibrium for the multiagent game.
- The local performance values equal the corresponding optimal value functions at the Nash equilibrium.
- The local optimal value functions serve as Lyapunov functions for the local neighborhood consensus-error systems, which are therefore asymptotically stable.
- The solution of the coupled HJ equations is not unique, and ADP generally obtains a local optimum rather than the globally optimal solution.The paper states that global optimality would require exploring the entire state space, which is generally not possible.
C. Policy Iteration (PI) Algorithm for the Coupled HJ Equations
The PI algorithm solves the coupled HJ equations through repeated policy evaluation and policy improvement. Under a small maximum-singular-value condition, the iterations converge to optimal value functions and the cooperative Nash equilibrium.
- Policy iteration procedure: PI alternates policy evaluation, which solves for value functions, and policy improvement, which updates all agents’ control policies.
- Policy iteration procedure: The iterations continue until the policy improvement result no longer changes and the controls converge.
- Convergence: Under the theorem’s small ¯σ(R−1 condition, the policy iteration algorithm converges to the Nash equilibrium and optimal value functions.
- Convergence: During the convergence proof, each value function is shown to be nonincreasing and bounded below by zero.
- Design conditions: Proper weighting matrices and a small upper bound on ∥gi(xi)∥ are identified as practical requirements for the local performance design.
- GFHM integration: The GFHM-based scheme provides a single-network adaptive architecture for approximating coupled-HJ-equation solutions within PI.
V. GFHM-BASED APPROXIMATE SOLUTIONS OF THE COUPLED HJ EQUATIONS
The paper approximates each agent’s local value function from its local consensus error using a GFHM critic estimator and adapts its weights by gradient descent. The design requires fixed GFHM parameters for linearity in the estimated weights and persistent excitation for convergence toward ideal weights.
- Because each agent’s control depends on its own and neighboring information, the coupled HJ equations are difficult to approximate directly.
- GFHMs are used for the first time in the paper to estimate each agent’s local value function from its local consensus error.
- The GFHCE represents the value function using generalized inputs, estimated weights, and an estimation error, with the constant offset set to zero because the value function satisfies Vi(0)=0.
- The GFHM parameters Φi are fixed to the identity matrix so the model is linear in the estimated weights θi during stability analysis.
- Gradient descent minimizes the squared residual error of the GFHM approximation and produces adaptive updating laws for the estimated weights.
- Persistent excitation is required for estimated weights to converge toward their ideal values, so probing noise is mixed into the coordination policies.
VI. STABILITY ANALYSIS
The stability analysis uses Lyapunov-like arguments under boundedness and persistent-excitation assumptions to establish ultimate boundedness of estimation and consensus errors.
- Stability analysis: The weight-error dynamics include the coupled HJ approximation residual ε_HJi.The residual arises from function-approximation error in the coupled HJ equation.
- Stability analysis: The analysis assumes persistent excitation and upper bounds on coupled HJ residual and GFHCE approximation errors.These assumptions bound σ_i, θ_i, ε_HJi, ε_i, and Δε_i.
- Stability analysis: Under the adaptive policies and weight-update laws, local consensus errors and weight estimation errors are uniformly ultimately bounded.The theorem also states that the control-node trajectory is cooperatively uniformly ultimately bounded.
- Stability analysis: The approximation coordination control approaches the ideal coordination control, with ∥û_i − u_i∥ ≤ ε_ui as t →∞.ε_ui is specified as a small positive constant.
- Stability analysis: A Lyapunov function combines the weight estimation error and local consensus error terms.Its components include L1i = tr(θ̃_i^Tθ̃_i)/2a_i and a consensus-error term involving V_i(e_i).
VII. SIMULATION
A five-node leader-following digraph is used to evaluate the proposed optimal coordination controller, with simulations tracking weights, agent states, and local consensus errors.
- Simulation setup: The numerical example considers a five-node digraph whose leader is connected to node 3.Edge weights and the pinning gain are chosen as one.
- Simulation setup: The simulation designs optimal coordination control to make agent states reach leader consensus while minimizing the cost functional.The generalized input variable is set to e_i for simplicity.
- Simulation results: After 10s, the estimated weights θ̂_i converge to their ideal values.This behavior is reported for the weight evolution shown in Fig. 2.
- Simulation results: After 15s, every agent state reaches consensus on the leader node under the optimal coordination controls.The state evolution is shown in Fig. 3.
- Simulation results: After 15s, the local consensus error e_i goes to zero.Fig. 4 reports the evolution of the local consensus errors e_i1 and e_i2.
- Architecture: Only one GFHM is used for each agent’s critic network, eliminating the action network architecture.The paper presents this as an advantage over the dual-network methods cited in the remark.
VIII. CONCLUSION
The conclusion presents FADP with a single GFHM as an online approach for optimal coordination control and reports an example demonstrating its effectiveness.
- Conclusion: The scheme approximates coupled Hamilton-Jacobi solutions with a single GFHM under policy iteration, then uses the approximation to obtain coordination control.The architecture avoids the dual-network action model.
- Conclusion: FADP reduces the number of weights updated by eliminating the action network model.The conclusion identifies this reduced architecture as a better scheme for multi-agent optimal coordination control.
- Conclusion: A numerical example is presented to show the effectiveness of the proposed scheme.The simulation tracks weight convergence, agent-state evolution, and local consensus-error evolution.
APPENDIX A
The appendix verifies equivalence between formulations by expanding network-wide dynamics and removing terms associated with non-neighboring nodes.
- Appendix A: The derivation expands the network dynamics using stacked states, inputs, dynamics, graph matrices, and the diagonal pinning matrix.The resulting expression separates contributions from each node.
- Appendix A: Terms for non-neighboring nodes are removed because their graph weights and corresponding pinning terms are zero.The reduced expression retains the neighbor contributions relevant to node i.
- Appendix A: After this reduction, the appendix concludes that expression (14) holds.The equivalence follows from the graph structure and diagonal form of B.