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Distributed Model Predictive Control for Optimal Consensus of Constrained Heterogeneous Multi-agent Systems
Nan Bai, Tao Liu, Qishao Wang, Zhisheng Duan
TL;DR
The paper tackles optimal consensus for constrained heterogeneous multi-agent systems when both the consensus equilibrium and control sequence must be selected. It formulates this joint choice within MPC, solves it using a distributed primal-dual method, and establishes conditions for solver convergence, recursive feasibility, and asymptotic consensus.
Problem
Existing methods either prescribe the consensus equilibrium or do not ensure that jointly optimized targets are feasible equilibria for heterogeneous agents under local constraints.
Method
The paper develops an MPC formulation that jointly optimizes control inputs and dynamically feasible consensus equilibria, solved by a distributed projected primal-dual algorithm.
Results
The proposed framework has conditions guaranteeing distributed-solver convergence, recursive feasibility, and asymptotic consensus while satisfying constraints.
Takeaways & Limitations
Optimizing the consensus equilibrium within constrained MPC integrates performance optimization with feasibility and stability analysis for heterogeneous multi-agent systems.
Abstract
from arXiv · showhide
This paper investigates the distributed optimal consensus control problem of constrained heterogeneous multi-agent systems within a model predictive control (MPC) scheme. Both the control input sequence and the dynamically feasible consensus equilibrium are optimized simultaneously within the proposed MPC framework to improve consensus performance, yielding a coupled constrained optimization problem at each prediction time. A distributed primal--dual algorithm is developed to solve the resulting optimization problem, and locally verifiable conditions are derived to guarantee its convergence. Furthermore, sufficient terminal conditions are established for the proposed MPC framework to guarantee the recursive feasibility and asymptotic consensus of the closed-loop heterogeneous multi-agent systems. Finally, numerical simulations verify the effectiveness of the proposed approach.
I. INTRODUCTION
The paper addresses optimal consensus for constrained heterogeneous multi-agent systems, where conventional methods do not select the consensus state or explicitly handle constraints. It proposes an MPC formulation and distributed solution with feasibility and consensus guarantees.
- Motivation: Conventional consensus control drives agreement but does not specify how agreement should be achieved or which consensus state is preferable.
- Motivation: Prescribed consensus equilibria, typically the origin, limit performance optimization by restricting the available equilibrium choices.
- Motivation: Existing simultaneous optimization methods do not explicitly account for state and input constraints, although such constraints arise in bounded workspaces and actuator limits.
- Contributions: The proposed MPC framework simultaneously optimizes control inputs and consensus equilibrium while restricting the equilibrium to each agent’s admissible equilibrium set.
- Contributions: The equilibrium restriction ensures the optimized common equilibrium is dynamically feasible and maintainable under heterogeneous dynamics and local constraints.
- Contributions: The paper develops a distributed projected primal-dual algorithm and derives conditions ensuring convergence, recursive feasibility, and asymptotic consensus.
B. MPC-based Problem Formulation
The MPC formulation treats both control sequences and consensus equilibria as decision variables while enforcing dynamics, local constraints, consensus, and terminal requirements. Its equilibrium restrictions connect transient optimization with steady-state feasibility.
- MPC formulation: At each prediction time, the MPC scheme optimizes predicted inputs, states, and equilibrium state-control pairs over a finite horizon.
- MPC formulation: The individual cost uses positive definite state, input, and equilibrium-related weighting matrices to evaluate predicted behavior and steady-state deviations.
- MPC formulation: The admissible equilibrium state set is nonempty, closed, convex, and contained in the dynamically feasible equilibrium conditions imposed by local state and input constraints.
- Relation to prior work: Unlike local artificial-target formulations, hard consensus constraints alone do not guarantee that the common target is a feasible equilibrium for every heterogeneous agent.
- MPC formulation: The equilibrium-input deviation term links transient performance optimization to steady-state feasibility and supports recursive-feasibility and asymptotic-consensus analysis.
III. DISTRIBUTED SOLVER FOR THE MPC OPTIMIZATION PROBLEM
The paper solves the coupled MPC problem with a distributed parallel primal-dual algorithm that exploits graph sparsity and local projections. Control and equilibrium variables are updated in parallel using local and neighboring information.
- Distributed algorithm design: The distributed solver is designed for the repeatedly solved MPC optimization problem with jointly optimized control inputs and consensus targets.
- Distributed algorithm design: The augmented Lagrangian introduces multipliers for the global consensus constraint while local constraints are enforced through projection operators.
- Distributed algorithm design: The all-agent stopping condition is verified through distributed termination detection rather than allowing agents to stop independently.
- Distributed algorithm design: Each agent updates its control and equilibrium variables in parallel using historical iterates, then exchanges multiplier information with neighbors.
B. Convergence Analysis
The distributed parallel algorithm is interpreted as a projected gradient primal-dual method whose iterates converge to the optimal solution under locally verifiable conditions. Its convergence rate can also be estimated from the algorithm parameters and graph properties.
- Algorithm 2 is a distributed parallel method whose solution sequence converges to the optimal solution when the stated conditions are satisfied.The result applies when Assumption 2 holds and Problem 1 is feasible at the prediction time.
- The algorithm can be interpreted as a projected gradient primal-dual method applied to the augmented Lagrangian.A modified penalty preserves the same consensus manifold and supports distributed decomposition.
- The modified augmented-Lagrangian penalty is equivalent to the usual consensus penalty in terms of its zero set.Both penalty forms characterize the same consensus manifold.
- The convergence conditions allow each agent to choose auxiliary parameters using only local information, enabling fully distributed implementation.The distributed execution procedure and locally selectable parameters jointly support this implementation.
- The convergence rate is estimated using the optimal solution, initialization, algorithm parameters, and the maximum eigenvalue of the communication graph Laplacian.The rate expression includes Γ0, ρ, and λmax{L}.
IV. CLOSED-LOOP ANALYSIS OF THE PROPOSED MPC FRAMEWORK
The MPC framework optimizes a dynamically feasible consensus equilibrium online together with control inputs. Under terminal conditions and exact online optimization, it guarantees recursive feasibility and asymptotic consensus while satisfying constraints.
- Closed-loop guarantees: Theorem 2 guarantees recursive feasibility and asymptotic consensus under positive weighting matrices and the stated terminal conditions.The guarantee assumes feasibility at the initial time and optimal solution of Problem 1 at every prediction time.
- Terminal conditions: The terminal set and admissible equilibrium set must satisfy an invariant-set inclusion involving the closed-loop dynamics.The condition is (Ai + BiKi)XiT ⊕ (I − (Ai + BiKi)) ˜Zi ⊆ XiT.
- Terminal conditions: Terminal conditions are imposed uniformly over the admissible equilibrium set because the consensus equilibrium is optimized online for heterogeneous agents.This distinguishes the framework from MPC with a fixed equilibrium or prespecified reference.
- Scope: The closed-loop guarantees are established for exact optimal solutions, whereas practical finite termination produces generally approximate solutions.A rigorous characterization under finite optimization errors is beyond the paper’s scope.
- Terminal design: A possible terminal-design procedure uses an algebraic Riccati equation for Pi and Ki, a Lyapunov equation for Si, and a terminal set contained in Xi.The admissible equilibrium set and scalar βi are adjusted to satisfy the terminal conditions.
- Scope: The terminal construction is broad for heterogeneous dynamics, making it difficult to prescribe a universal design method.An extension to nonlinear systems would require analogous nonlinear terminal ingredients and is left for future research.
- Computational considerations: Online re-optimization adds decision variables and inner iterations, creating a trade-off between control performance and computational complexity.A two-stage strategy can fix the optimized consensus state after consensus errors become sufficiently small and then track it locally.
V. SIMULATION EXAMPLES
Simulations evaluate the proposed approach on heterogeneous multi-agent systems using a five-agent linear-system example. The reported trajectories show rapid constrained consensus, while heterogeneous dynamics lead to distinct nonzero steady-state inputs.
- A. Heterogeneous Linear Systems: The simulation considers five heterogeneous agents connected by the communication topology shown in Fig. 1.The agents are modeled as linear systems with state and input constraints.
- A. Heterogeneous Linear Systems: The example uses prediction horizon T = 8 and control period ∆t = 2, with Ri = 0.1 and Qi = 0.1I3.Terminal weights and gains are constructed using the algebraic Riccati and Lyapunov equations.
- A. Heterogeneous Linear Systems: The proposed algorithms are applied with randomized initial states and distributed-solver parameters gui = gzi = 1 200 and ρ = 1.The initial state components are sampled from rand(−6, 6).
- A. Heterogeneous Linear Systems: Fig. 2 reports system state trajectories and control inputs, including an inset showing the three state components of Agent 1.The main panel is used to show convergence toward an optimized consensus equilibrium.
- A. Heterogeneous Linear Systems: Agents quickly reach consensus while state variables and control inputs remain within their prescribed constraints.This is the principal reported simulation outcome.
- A. Heterogeneous Linear Systems: Distinct nonzero control inputs converge because heterogeneous dynamics require different steady inputs to maintain the consensus equilibrium.The nonzero limiting inputs are attributed to the agents’ heterogeneous dynamics.
B. Integrator Systems
The integrator-system simulations evaluate constrained robot formation and larger rendezvous problems, comparing the proposed method with several consensus controllers. The proposed approach completes formation and offers broader applicability to constrained heterogeneous agents.
- Formation experiment: Five robots use double-integrator dynamics with bounded position, velocity, and control inputs to form a prescribed square around a center.The relative positions define the square formation, while the state and input constraints bound each robot's operation.
- Formation experiment: The simulation uses sampling period δ = 0.5, prediction horizon T = 10, control period Δt = 1, and masses varying randomly around nominal mass m0 = 1.The weighting matrices are Ri = 0.1I2 and Qi = I4, with terminal ingredients obtained from an algebraic Riccati equation.
- Formation experiment: The formation task is completed under the proposed MPC algorithm and distributed solver.The initial positions and velocities are randomized within the stated constraints, and Fig. 3 displays the resulting formation process.
- Rendezvous comparison: For rendezvous experiments with N = 50 and N = 100 robots, consensus is evaluated against STBC, AFGC, STMPC, and OCMPC using consensus steps and performance cost.The consensus step is defined by the first time instant satisfying the stated disagreement threshold.
- Rendezvous comparison: The proposed method jointly optimizes a feasible common equilibrium and control inputs, while competing methods either restrict equilibrium choice, use soft target agreement, or assume homogeneous agents.The comparison discussion links these design differences to additional consensus steps or higher costs for some baselines and identifies broader applicability for the proposed method.
APPENDIX I PROPOSITION 1
Proposition 1 states that each agent can implement the specified iteration process in equations (6a)–(6b). Its proof follows directly by substituting the dynamic model.
- Proof: The proof derives the conclusion by substituting dynamic model (1) into the update processes, and omits further details.The passage explicitly attributes the result to this substitution.
APPENDIX II PROOF OF THEOREM 1
Theorem 1's proof establishes convergence of the distributed primal–dual iterations by combining projected updates, graph-related matrix properties, and strong convexity of the Lagrangian.
- Proof setup: The proof begins by invoking projection and smooth-convexity lemmas before rewriting the optimization problem in an equivalent form.The local update rules are then represented compactly using block-diagonal step-size matrices.
- Distributed updates: The compact updates use GU and GZ block-diagonal matrices, with positive step-size blocks and Lipschitz constants for local cost gradients.The passage identifies Lδi as the Lipschitz constant of the gradient of Ji.
- Optimality relation: A saddle point of the Lagrangian exists under the relative interior constraint qualification, and the proof uses it to compare iterates with the optimum.The qualification follows from Assumption 2 and feasibility of Problem 1.
- Descent argument: Graph Laplacian properties make the relevant matrices positive semidefinite with positive definite leading blocks, yielding a non-increasing weighted distance and bounded primal sequences.The argument uses 2DL − L ≥ 0 for the undirected communication graph.
- Convergence conclusion: Strong convexity of the Lagrangian converts the descent inequality into convergence of (Uq, Zq) to the optimal solution (U⋆, Z⋆).The proof introduces µ > 0 to bound the Lagrangian gap by squared distances in U and Z.
APPENDIX III PROOF OF COROLLARY 1
The corollary proof bounds the approximate primal–dual solution using a bounded multiplier set and a lemma for approximate convex optimization. Choosing γ > ∥Λ⋆∥ yields the stated conclusion.
- Approximate solution bound: The proof introduces the multiplier set Bγ = {Λ | ∥Λ∥ ≤ γ} and evaluates the candidate solution against arbitrary feasible variables and multipliers.The candidate (Û, Ẑ, Λ̂) is supplied by Lemma 3.
- Lemma application: Lemma 4 applies to the convex optimization problem with an equality constraint and an optimal Lagrangian multiplier.Its variables include an approximate solution, a multiplier bound γ, and an error tolerance ϵ.
- Approximate solution bound: The argument combines convexity of J, Lemma 3, and the preceding inequality to establish the required bound for the candidate solution.The passage explicitly identifies convexity and Lemma 3 as sources of the first inequality.
- Conclusion: Because γ > ∥Λ⋆∥, applying Lemma 4 gives the desired conclusion.The supplied proof then ends after this application.
APPENDIX IV PROOF OF THEOREM 2
The proof establishes recursive feasibility by constructing shifted candidate sequences that remain within input, state, and terminal constraints. A non-increasing, lower-bounded optimal cost then supports asymptotic consensus while preserving constraints for all time.
- Recursive feasibility: Shifted optimal trajectories generate candidate input and state sequences at the next prediction time.The candidates are formed from the previous optimal trajectory and terminal feedback dynamics.
- Recursive feasibility: Terminal conditions ensure the shifted inputs satisfy input constraints and the corresponding states satisfy state and terminal constraints.These candidates therefore provide feasible solutions to the MPC problem at the next prediction time.
- Conclusion: Recursive feasibility together with asymptotic consensus ensures that all state and input constraints remain satisfied for all time.This combines the feasibility result with the closed-loop convergence conclusion.
- Cost decrease: The optimal cost sequence is non-increasing and lower-bounded by zero because the weighting matrices Qi and Ri are positive definite.The proof uses the cost decrease relation and nonnegativity of the objective.
- Asymptotic consensus: The resulting error terms converge to zero for every agent and prediction step, implying that the closed-loop multi-agent system asymptotically achieves consensus.The conclusion applies across all agents and the specified prediction horizon.