Coalitional Zero-Sum Games for ${H_{\infty}}$ Leader-Following Consensus Control

Dingguo Liang; Yuezu Lv; Yunxiao Ren; Zhisheng Duan

arxiv: 2604.06089 · v1 · submitted 2026-04-07 · 📡 eess.SY · cs.SY· math.OC

Coalitional Zero-Sum Games for {H_(infty)} Leader-Following Consensus Control

Yunxiao Ren , Dingguo Liang , Yuezu Lv , Zhisheng Duan This is my paper

Pith reviewed 2026-05-10 19:32 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords multi-agent systemsleader-following consensusH-infinity controlzero-sum gamesdistributed controlalgebraic Riccati equationadversarial attacksdynamic average consensus

0 comments

The pith

Formulating multi-agent leader-following consensus under adversarial attacks as a coalitional zero-sum game yields a distributed H∞ control law.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper casts the leader-following consensus problem for multi-agent systems facing external adversarial inputs as a global coalitional min-max zero-sum game in which the agents jointly minimize a shared cost while the attacks maximize it. The optimal game policy produces an H∞ robust controller that attenuates the disturbances. To implement this without a central processor, the authors decompose the high-dimensional generalized algebraic Riccati equation into several identical lower-dimensional equations and apply a dynamic average consensus algorithm to remove the remaining coupling in the control law. A sympathetic reader would care because the method lets a network of vehicles or robots maintain formation or agreement even when some members are under attack, using only local computations.

Core claim

The paper claims that the robust leader-following control problem under adversarial attacks can be solved by formulating it as a coalitional zero-sum differential game whose solution is an H∞ controller, and that the associated high-dimensional GARE can be decomposed into uniform lower-dimensional GAREs whose solutions are combined via a dynamic average consensus algorithm to obtain a fully distributed control law.

What carries the argument

The coalitional min-max zero-sum game whose value function satisfies the high-dimensional GARE; this GARE is decomposed into lower-dimensional copies whose coupling is resolved by dynamic average consensus.

If this is right

The game-derived policy guarantees H∞ performance against disturbances modeled as attacks.
Each agent computes its control input using only local state information after the decomposition and consensus steps.
The method applies directly to formation control of multi-vehicle systems whose dynamics have been feedback-linearized.
Distributed implementation removes the need for a central node to solve or store the full high-dimensional Riccati equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the exact decomposition holds, the same game-theoretic reduction could be applied to other network disturbance-rejection tasks that currently require centralized Riccati solutions.
The dynamic average consensus step may produce transient mismatch during finite-time convergence, so performance bounds would need explicit finite-time analysis.
The linear-system assumption leaves open whether the decomposition and consensus technique extend to nonlinear agent dynamics without losing the H∞ guarantee.

Load-bearing premise

The high-dimensional GARE can be split into multiple uniform lower-dimensional GAREs while exactly preserving the optimality and robustness properties of the original centralized solution.

What would settle it

A side-by-side simulation on the same multi-vehicle formation where the distributed control inputs and closed-loop trajectories from the decomposed GAREs plus consensus differ measurably from those produced by solving the full centralized GARE.

Figures

Figures reproduced from arXiv: 2604.06089 by Dingguo Liang, Yuezu Lv, Yunxiao Ren, Zhisheng Duan.

**Figure 1.** Figure 1: Conceptual diagram of the global coalitional min-max zero-sum game framework for multi-agent systems, illustrating the structural game relationship [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 3.** Figure 3: Schematic of the ith vehicle. of vehicle i. Similar to [29], the dynamics of the ith vehicle can be reformulated as follows by feedback linearization at a fixed reference point qi = [qx,i, qy,i] T off the center of the vehicle. d dt qi q˙i = 02 I2 02 02 qi q˙i + 02 I2 ui + 02 I2 wi , (58) where ui = [ux,i uy,i] T is the control input, wi = [wx,i wy,i] T is the energy-bounded attack inpu… view at source ↗

**Figure 4.** Figure 4: The convergence of Algorithm 1. -20 -15 -10 -5 0 5 -5 0 5 10 vehicle1 vehicle2 vehicle3 vehicle4 vehicle5 vehicle6 vehicle7 vehicle8 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: The trajectories of 8 vehicles. Fig.4 shows the consensys error of Algorithm 1, the result demonstrates that Algorithm can converge very fast meet the decoupling requirements. Fig.5 illustrates the trajectories of 8 vehicles, it can be obatained that the 8 vehicles can effectively form a formatiion and transform the formation by reset their relative positions with the leader. The convergence result of the … view at source ↗

**Figure 8.** Figure 8: The worst-cast energy-bounded attack inputs of 8 vehicles. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

read the original abstract

This paper investigates the leader-following consensus problem for a class of multi-agent systems subject to adversarial attack-like external inputs. To address this, we formulate the robust leader-following control problem as a global coalitional min-max zero-sum game using differential game theory. Specifically, the agents' control inputs form a coalition to minimize a global cost function, while the attacks form an opposing coalition to maximize it. Notably, when these external adversarial attacks manifest as disturbances, the designed game-theoretic control policy systematically yields a robust $H_\infty$ control law. Addressing this problem inherently requires solving a high-dimensional generalized algebraic Riccati equation (GARE), which poses significant challenges for distributed computation and controller implementation. To overcome these challenges, we propose a two-fold approach. First, a decentralized computational strategy is devised to decompose the high-dimensional GARE into multiple uniform, lower-dimensional GAREs. Second, a dynamic average consensus-based decoupling algorithm is developed to resolve the inherent coupling structure of the robust control law, thereby facilitating its distributed implementation. Finally, numerical simulations on the formation control of multi-vehicle systems with feedback-linearized dynamics are conducted to validate the effectiveness of the proposed algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The coalitional zero-sum game framing for H∞ leader-following consensus is a reasonable extension, but the GARE decomposition into uniform lower-dimensional pieces is likely to require strong symmetry on agents and graphs.

read the letter

The main point to know is that the authors frame the H∞ leader-following consensus problem under adversarial disturbances as a coalitional min-max zero-sum game and then provide a two-fold method to compute and implement the solution in a distributed way. The decomposition of the global GARE is the part that stands out as potentially fragile. What the paper does well is to connect differential game theory directly to the robust consensus task. By letting the agents form a coalition to minimize the cost while the attacks maximize it, the resulting policy becomes an H∞ controller when disturbances are the adversary. The decentralized strategy that breaks the high-dimensional GARE into uniform lower-dimensional ones, combined with the dynamic average consensus algorithm to handle coupling, is a practical step toward distributed implementation. The simulations on multi-vehicle formation control with feedback-linearized dynamics provide concrete evidence that the closed-loop system behaves as expected under the designed law. The soft spots center on the exactness of that decomposition. The global GARE arises from the full system including the Laplacian matrix, so splitting it into identical smaller GAREs and reassembling without losing the saddle-point property or the H∞ bound typically demands identical agents and a graph structure that supports uniformity, such as regular undirected graphs. The stress-test concern about this is reasonable; if the paper only establishes the result under those conditions, the distributed guarantee is narrower than the general statement in the abstract. The consensus-based decoupling can also introduce small estimation errors that move the control away from the precise optimal policy, though these may be manageable in practice. This paper targets readers in multi-agent systems and robust control who are comfortable with Riccati-based methods and consensus protocols. It offers value to someone designing controllers for autonomous vehicle fleets or robot teams looking for a game-theoretic robust approach. A reader will get a clear new application and an implementable algorithm, provided they verify the assumptions match their setup. The paper shows coherent engagement with the literature and internally consistent reasoning, so it deserves a serious referee. The derivations around the decomposition will be the key part to examine. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper formulates the H∞ leader-following consensus problem for multi-agent systems under adversarial disturbances as a coalitional min-max zero-sum differential game. The resulting high-dimensional generalized algebraic Riccati equation (GARE) is decomposed into multiple uniform lower-dimensional GAREs, and a dynamic average consensus algorithm is used to decouple the control law for distributed implementation. Effectiveness is illustrated via numerical simulations on formation control of feedback-linearized multi-vehicle systems.

Significance. If the decomposition exactly preserves the saddle-point solution and the H∞ performance bound, the approach would offer a practical route to distributed robust control for MAS, mitigating the computational burden of global GAREs while retaining theoretical guarantees. The simulation results on multi-vehicle formation provide concrete evidence of applicability, but the strength hinges on rigorous verification that the distributed law matches the centralized H∞ optimum.

major comments (2)

[the proposed decentralized computational strategy] The central claim that the high-dimensional GARE decomposes into uniform lower-dimensional GAREs while preserving optimality and the H∞ bound (two-fold approach) requires explicit proof. For leader-following MAS the closed-loop dynamics involve the graph Laplacian; uniformity of the decomposed Riccati solutions holds only under restrictive assumptions (identical agents, regular undirected graphs). Without a general proof that the reassembled solution satisfies the original saddle-point condition, the distributed policy may lose the guaranteed disturbance attenuation level.
[the dynamic average consensus-based decoupling algorithm] The dynamic average consensus decoupling step must be shown not to perturb the control input away from the exact H∞ saddle point. Any finite-time estimation error or communication delay introduces a mismatch that can violate the min-max optimality and degrade the closed-loop H∞ norm; the manuscript should quantify the resulting performance loss or provide a stability margin.

minor comments (2)

Notation for the coalitional cost function and the disturbance attenuation level γ should be introduced with explicit definitions before the GARE is stated.
The simulation section would benefit from a direct comparison of the achieved H∞ norm against the centralized solution and against a non-decomposed baseline to quantify any performance gap.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help strengthen the theoretical foundations of our work. We address each major comment point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim that the high-dimensional GARE decomposes into uniform lower-dimensional GAREs while preserving optimality and the H∞ bound (two-fold approach) requires explicit proof. For leader-following MAS the closed-loop dynamics involve the graph Laplacian; uniformity of the decomposed Riccati solutions holds only under restrictive assumptions (identical agents, regular undirected graphs). Without a general proof that the reassembled solution satisfies the original saddle-point condition, the distributed policy may lose the guaranteed disturbance attenuation level.

Authors: We agree that an explicit proof is required to rigorously establish preservation of the saddle-point solution and the H∞ performance bound. The manuscript develops the decomposition specifically for identical agents over undirected regular graphs, where the Laplacian structure enables uniform lower-dimensional GAREs. In the revised version we will insert a new theorem together with a complete proof demonstrating that the reassembled solution satisfies the original min-max saddle-point condition under these assumptions, thereby retaining the guaranteed disturbance attenuation level. We will also clarify the assumptions and note that uniformity need not hold for non-identical agents or directed graphs. revision: yes
Referee: The dynamic average consensus decoupling step must be shown not to perturb the control input away from the exact H∞ saddle point. Any finite-time estimation error or communication delay introduces a mismatch that can violate the min-max optimality and degrade the closed-loop H∞ norm; the manuscript should quantify the resulting performance loss or provide a stability margin.

Authors: The dynamic average consensus protocol converges asymptotically to the exact average, so the distributed control law converges to the centralized H∞ saddle-point solution. To address finite-time estimation errors and bounded communication delays, the revised manuscript will include a Lyapunov-based analysis that quantifies the resulting deviation from the optimal H∞ norm and supplies an explicit stability margin together with a bound on performance degradation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard differential game theory applied to H∞ formulation; decomposition and consensus steps are constructive proposals, not reductions to inputs or self-citations

full rationale

The derivation begins with a standard formulation of the leader-following consensus problem as a coalitional min-max zero-sum game drawn from differential game theory, which directly yields the H∞ control law via the global GARE as a known consequence of the LQ setup. The subsequent two-fold approach (decomposition of the GARE into uniform lower-dimensional instances and dynamic-average-consensus decoupling) consists of explicitly constructed algorithms whose correctness is asserted via derivation rather than by redefining the target quantity in terms of itself or by load-bearing self-citation. No equation is shown to equal its own fitted parameter or prior result by construction, and the paper does not invoke uniqueness theorems or ansatzes from the authors' own prior work to force the outcome. The chain therefore remains self-contained against external benchmarks of game-theoretic H∞ control.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from differential game theory and linear-quadratic H∞ control; no explicit free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The multi-agent system dynamics are linear and the performance index is quadratic.
Required for the GARE to arise from the min-max game.

pith-pipeline@v0.9.0 · 5522 in / 1247 out tokens · 62190 ms · 2026-05-10T19:32:09.029892+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

formulate the robust leader-following control problem as a global coalitional min-max zero-sum game... high-dimensional generalized algebraic Riccati equation (GARE)... decompose the high-dimensional GARE into multiple uniform, lower-dimensional GAREs
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

quadratic performance function J(δ,u,w)=1/2∫(δᵀQδ+uᵀRu−wᵀΓw)dt

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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