PaperlayerView exact revision on arXiv

Source-linked AI summary

Abstract homogeneous chains: a Lyapunov framework for high-order sliding modes in multi-agent systems

Rodrigo Aldana-López

arXiv:2609.12250v1eess.SYcs.SY

TL;DR

Multi-agent high-order sliding-mode algorithms lack a Lyapunov framework that covers graph-coupled arbitrary-order dynamics and their gain design. This paper models such dynamics as abstract homogeneous chains and constructs a homogeneous Lyapunov function with recursive optimization-based gain proposals. It establishes global finite-time stability and applies the framework to distributed algorithms and formation tracking.

  • Problem

    Graph-induced nonlinear feedback couplings are not directly covered by existing generalized multivariable super-twisting frameworks.

    Source: §I · p. 1

  • Method

    The paper defines abstract homogeneous chains and combines homogeneity with convex analysis to construct a Lyapunov function and recursive finite-dimensional gain-proposal procedure.

    Source: §I · p. 2, A. Auxiliary results · p. 12

  • Results

    The AHC class is globally finite-time stable at arbitrary order, and the framework supplies gain proposals for distributed HOSM algorithms and a new formation observer.

    Source: §III · p. 3, §VIII · p. 11

  • Takeaways & Limitations

    The framework extends arbitrary-order gain design across several multi-agent applications, including leader–follower differentiation and multi-leader affine formation tracking.

    Source: §I · p. 2, §VIII · p. 11

    The construction is not common across different interaction graphs and does not directly cover switching networks; certified gain thresholds generally require application-specific representations and optimization tools.

    Source: §VIII · p. 11

Abstract

from arXiv · show

This work develops a Lyapunov framework for a broad class of arbitrary-order sliding-mode algorithms in multi-agent systems. We introduce abstract homogeneous chains, a class of nonlinear error systems characterized by common convexity and homogeneity properties. For this class, we establish global finite-time stability for arbitrary order, construct a homogeneous Lyapunov function, and derive a recursive optimization-based gain-proposal procedure. The framework addresses several gaps in existing dynamic average consensus and distributed differentiation results: it provides a recursive numerical optimization-based gain-proposal procedure for EDCHO at arbitrary order, extends REDCHO convergence from local to global, and provides arbitrary-order numerical gain-proposal rules for leader-follower distributed differentiation, previously available only at first order. It also provides a new arbitrary-order observer for multi-leader affine formation tracking with global finite-time convergence.

I. INTRODUCTION

Multi-agent high-order sliding-mode methods have advanced faster than their supporting Lyapunov theory, in part because graph-coupled nonlinearities fall outside existing frameworks. The paper introduces abstract homogeneous chains to extend Lyapunov analysis and gain proposal to this setting.

I. INTRODUCTION

  • EDCHO lacks an arbitrary-order explicit threshold for one gain, REDCHO has only a local convergence guarantee, and leader–follower gain proposal is available only at first order.Graph-induced coupling through incidence matrices, Laplacians, and related operators also prevents existing generalized super-twisting frameworks from directly covering these systems.Source: §I · p. 1
  • Abstract homogeneous chains generalize the super-twisting abstraction to arbitrary-order multi-agent error dynamics and support global finite-time stability and recursive numerical gain proposals.The framework addresses the EDCHO, REDCHO, and leader–follower gaps and also yields a new arbitrary-order multi-leader affine-formation protocol.Source: §I · p. 2

A. Notation

  • The notation section defines vector and matrix conventions and componentwise operators used to express the systems and their nonlinearities.It also specifies stacking, block-diagonal notation, and the sign convention for differential inclusions.Source: A. Notation · p. 2

B. Convex analysis

  • The Fenchel–Young gap is nonnegative, vanishes when y = ∇U(x), and measures mismatch through ∇U without requiring comparable coordinates.This makes it a natural building block for the paper’s Lyapunov function.Source: B. Convex analysis · p. 2

C. Homogeneity

  • Weighted homogeneity scales a function according to a dilation that assigns distinct weights to the components of the chain state.The paper defines the weighted dilation and a homogeneous gauge adapted to those weights.Source: C. Homogeneity · p. 2

III. ABSTRACT HOMOGENEOUS CHAINS

Abstract homogeneous chains provide a shared representation for arbitrary-order error dynamics, with structural conditions that support a global finite-time stability result. The section then connects that representation to distributed observer problems, including leader–follower differentiation.

III. ABSTRACT HOMOGENEOUS CHAINS

  • An AHC represents a chain state q = col(q0, …, qm) in X^(m+1) using graph-compatible dynamics and specified homogeneity and convexity structure.The conditions are chosen to enable the Lyapunov construction while encompassing classical and networked high-order sliding-mode dynamics.Source: §III · p. 3
  • The scalar robust exact differentiator is an AHC when its parameters are chosen according to the first row of Table I, with D = [−1, 1] and γ = 0.Its dynamics are those of the arbitrary-order robust exact differentiator in.Source: §III · p. 3
  • For systems satisfying the AHC conditions, there exist positive gains that make the origin globally finite-time stable.The conditions include convex, positive-definite homogeneous potentials, a suitably regular homogeneous set-valued map, and a compact convex disturbance set.Source: §III · p. 3

A. Leader-follower distributed differentiation

  • The applications derive AHC error systems for distributed observer problems over a fixed connected undirected graph.For leader–follower differentiation, at least one agent must access the leader signal; under the stated assumption, the protocol achieves its objective from every initial condition, and the framework supplies recursive numerical gain proposals for general order.Source: A. Leader-follower distributed… · pp. 3–4, §IV · p. 3

B. Leaderless distributed differentiation

The section casts leaderless dynamic average consensus as an AHC error system and applies the framework to arbitrary-order distributed differentiation. It gives global finite-time convergence results and an optimization-based gain proposal for EDCHO, while extending REDCHO's convergence result from local to global.

B. Leaderless distributed differentiation

  • Each agent estimates derivatives of its local signal using auxiliary variables, and only ẑ_i,0(t) = z_i(t) − η_i,0(t) is exchanged.The objective is for every agent to generate derivative estimates for the dynamic average consensus problem.Source: B. Leaderless distributed… · p. 4
  • The error transformation maps stacked estimates through (G^-1 ⊗ P), after which the resulting dynamics are identified as an AHC.Convergence of the transformed variables to zero implies the consensus objective.Source: B. Leaderless distributed… · pp. 4–5
  • Under Assumption 2 and γ ≥ 0, positive gains k0, . . . , km make the origin of (16) globally finite-time stable; for γ > 0, (14) achieves (13) for every initial condition.Source: B. Leaderless distributed… · p. 5
  • EDCHO's prior gain design left one gain merely “sufficiently large”; the paper proposes numerical optimization-based gains at arbitrary order and extends REDCHO from local to global convergence.EDCHO and REDCHO are identified as protocols in and, respectively.Source: B. Leaderless distributed… · p. 5

C. Multi-leader affine formation tracking

The section treats multi-leader affine formation tracking as a distributed estimation problem: followers cannot evaluate their desired reference directly from their local neighbor interactions. A new observer reconstructs the reference and its derivatives, with global finite-time convergence under the stated leader-motion assumption.

C. Multi-leader affine formation tracking

  • The nominal configuration's stress matrix characterizes admissible formations, and the leaders determine followers uniquely when the nominal leader positions are affinely independent.Under that condition, Ω_FF is symmetric positive definite.Source: C. Multi-leader affine formation… · p. 5
  • Followers cannot evaluate the closed-form desired reference because they lack leader positions and the global stress matrix, so the section formulates its recovery as distributed estimation.Each follower interacts only with its neighbors.Source: C. Multi-leader affine formation… · p. 5
  • The observer reconstructs each follower's reference position and its first m derivatives, providing estimates that can be used by a local trajectory-tracking controller.The observer is described as new; it can also be adapted as a controller for first-order agents.Source: C. Multi-leader affine formation… · pp. 5–6
  • Under Assumption 3, the observer error is globally finite-time stable, so every follower achieves the reference objective from every initial condition.Assumption 3 bounds how fast the leaders may maneuver and limits the (m+1)-th derivative of the reference.Source: C. Multi-leader affine formation… · p. 6
  • The observer can use relative position measurements and local coordinate frames without requiring agents to share a global frame.The passage attributes this to rotation equivariance of the operator used in the observer.Source: C. Multi-leader affine formation… · p. 6

V. PROOF OF THE MAIN RESULT

The proof normalizes the chain, constructs a homogeneous Lyapunov function from Fenchel–Young gaps, and uses its derivative to select gains recursively; the case γ ≥ 0 follows from γ = 0 by time reparameterization.

V. PROOF OF THE MAIN RESULT

  • The proof uses the constructed function's properties to support recursive gain selection for the finite-time Lyapunov inequality.The proof overview specifies gain selection first for γ = 0 and then obtains γ ≥ 0 by time reparameterization.Source: §V · p. 6, B. Lyapunov construction · p. 7

A. Normalization

  • An invertible coordinate change removes the scale L and replaces the original gains with normalized gains, preserving global finite-time stability.The proof uses homogeneity to transform the dynamics and establishes stability equivalence between the original and normalized systems.Source: A. Normalization · p. 6

B. Lyapunov construction

  • The Lyapunov function combines Fenchel–Young gaps of the potentials with positive weights, yielding a continuously differentiable, positive-definite, radially unbounded function.Its positive homogeneity supplies the radial growth bound used to establish radial unboundedness.Source: B. Lyapunov construction · pp. 6–7

C. Derivative along solutions

The derivative analysis rewrites the Lyapunov function in coordinates adapted to the potentials and separates nonnegative terms from cross terms with no definite sign. This yields a minimum-based derivative bound along solutions.

C. Derivative along solutions

  • The adapted coordinates compare each level through potential gradients, and the quantities h_μ collect differences between ∇U_μ(ξ_0) and ∇U_μ(ξ_μ+1).These quantities organize the terms that arise in differentiating the Lyapunov function.Source: C. Derivative along solutions · p. 7
  • Convexity and positive definiteness make the functions H_μ and terms B_μ nonnegative, while the remaining cross terms have no definite sign.The nonnegative terms vanish when adjacent coordinates coincide; ψ_μ vanishes when ξ_0 = ξ_μ.Source: C. Derivative along solutions · pp. 7–8
  • For almost every time along each solution, the Lyapunov derivative equals −Φ, where Φ is the minimum defined in (39).The lemma establishes this derivative relation for γ = 0 and positive gains.Source: C. Derivative along solutions · p. 8

D. Recursive domination

Recursive domination characterizes where each nonnegative term vanishes, then selects gains backward so the Lyapunov derivative remains positive away from the origin. Homogeneity and the resulting derivative bound support the finite-time stability proof.

D. Recursive domination

  • The resulting derivative inequality has exponent α = 1 − 1/((m + 1)dV), yielding the bound required for the finite-time stability argument.The proof relates the homogeneous state bound to V and then applies Lemma 3.Source: D. Recursive domination · pp. 8–9
  • Φ is homogeneous of degree q, and the recursive domination argument uses the nonnegative Bν terms to enforce its positivity.The sets Cν are closed, contain the origin, and are invariant under the stated dynamics.Source: D. Recursive domination · pp. 8–9
  • Bν(ξ(x)) = 0 on Cν−1 exactly when ξν = ξν+1, placing such states in Cν and enabling the recursive argument.The base case uses positive definiteness of Q + Q⊤ and injectivity of ∇U0; subsequent cases use the stated properties of Bν.Source: D. Recursive domination · p. 9
  • Backward induction first establishes Φ > 0 on Cm−1 outside zero, then selects each ekν to extend positivity from Cν to Cν−1.On Cν−1, the dependence on ekν is isolated in the nonnegative term βνekνBν, while the remaining terms depend on later gains.Source: D. Recursive domination · p. 9
  • The recursion applies a finite threshold at each step, using the induction hypothesis and the zero-set characterization to ensure Φ stays positive on the preceding set.The thresholds follow from a domination argument applied on Cν−1; repeating through ν = m−1,…,0 gives a positive bound on the full state space.Source: D. Recursive domination · p. 9

E. Proof of Theorem 1

The proof removes the time-varying parameter γ by rescaling the states and changing time, reducing the dynamics to the γ = 0 case. A Lyapunov decay inequality then establishes global finite-time stability for every γ ≥ 0.

E. Proof of Theorem 1

  • The γ = 0 gain construction supplies the normalized gains, which are converted to gains for the original dynamics before applying the rescaling argument.The proof chooses km > 1/cS and ekm > 0, then obtains the remaining normalized gains from Lemma 4.Source: E. Proof of Theorem 1 · p. 9
  • Rescaling each state by exp(γrμt) cancels the γ-dependent term, and a time change converts the remaining dynamics to the γ = 0 system.The common factor in the transformed dynamics is dτ/dt = exp(γt/(m + 1)) > 0.Source: E. Proof of Theorem 1 · pp. 9–10
  • Every solution reaches the origin in finite time and remains there, with Lyapunov stability, so the origin is globally finite-time stable.The derivative inequality has α ∈ (0, 1), and the settling-time bound is T ≤ V(x(0))^(1−α)/(c(1−α)).Source: E. Proof of Theorem 1 · p. 10

VI. GAIN PROPOSAL

The gain-proposal algorithm turns the proof’s recursive thresholds into finite-dimensional optimization problems. Under the theorem’s conditions, the resulting gains satisfy its global finite-time stability conclusion.

VI. GAIN PROPOSAL

  • At each recursive step, a finite-dimensional optimization problem characterizes the gain threshold using the AHC data and selected Lyapunov coefficients.The thresholds generally require numerical approximation, and the resulting gains are proposals rather than closed-form values.Source: §VI · p. 10
  • Algorithm 1 selects β0,…,βm, km, and ekm, then proceeds backward through ν = m−1,…,0 to determine thresholds and gains.It converts the normalized gains into k0,…,km after the recursive steps.Source: §VI · p. 10
  • The threshold optimization can be restricted to normalized states satisfying ξ0 = ··· = ξν, and gains selected by Algorithm 1 satisfy Theorem 1.The normalization represents each nonzero homogeneous ray; the proof establishes equivalence with the threshold definition.Source: §VI · p. 10

VII. NUMERICAL EXAMPLE

The order-2 affine formation observer reaches follower references in finite time in the numerical example, while its Lyapunov function decreases to zero. The conclusion presents the broader arbitrary-order framework and identifies graph dependence and certified gain computation as limitations.

VII. NUMERICAL EXAMPLE

  • For the planar example, cS is normalized to 1, the terminal condition becomes k2 > 1, and the proposed gains are (k0, k1, k2) = (5.77, 8.32, 2).The threshold estimates are ω1 = 5.81 and ω0 = 23.09; the nonconvex optimization searches propose gains that are then evaluated in simulation.Source: §VII · p. 11

VIII. CONCLUSIONS

  • The framework establishes arbitrary-order global finite-time stability and recursive sufficient gain conditions, with applications to EDCHO, REDCHO, distributed differentiation, and affine formation tracking.It provides an arbitrary-order numerical gain-proposal procedure for EDCHO, strengthens REDCHO convergence from local to global, and gives a new arbitrary-order affine formation observer.Source: §VIII · p. 11

VII. NUMERICAL EXAMPLE

  • Follower observer states reach their references in finite time, while V decreases to zero and numerically evaluated −V̇ remains nonnegative up to numerical precision.Figure 2 plots horizontal state components and references for μ = 0, 1, 2, followed by V and −V̇ on a logarithmic scale.Source: Figure 2 · p. 11, §VII · p. 11

VIII. CONCLUSIONS

  • Graph-dependent potentials and Q do not directly cover switching networks, and certified threshold computation may require application-specific representations and optimization tools.The authors identify both issues as directions for future work.Source: §VIII · p. 11

APPENDIX

The auxiliary results establish structural properties of homogeneous convex functions and their conjugates, then use these properties to support homogeneous domination and smoothness arguments. In particular, the conjugate is twice continuously differentiable, and a threshold gain yields a positive homogeneous bound.

A. Auxiliary results

  • For p in (1, 2), U is strictly convex with bijective gradient, while U* is positive definite and homogeneous of degree p/(p−1).The gradients are inverses, and the Fenchel–Young gap is nonnegative, vanishing exactly when y = ∇U(x).Source: A. Auxiliary results · p. 12
  • The Hessian of U* scales as ∇²U*(λ^(p−1)y) = λ^(2−p)∇²U*(y), and the Fenchel–Young gap scales with degree p.The gap vanishes exactly when y = ∇U(x), equivalently x = ∇U*(y).Source: A. Auxiliary results · p. 12
  • For any k above the finite threshold k*, kb(x) − a(x) is bounded below by ckϱ(x)^q on C.The assumptions include a(x) < 0 wherever b(x) = 0 outside the origin, with a and b homogeneous of the same positive degree.Source: A. Auxiliary results · p. 12
  • The constructed U is continuously differentiable, convex, positive definite, and homogeneous of degree 1 + r; its conjugate U* is twice continuously differentiable.Positive definiteness follows from M being injective on X; the proof handles points where some Mℓx = 0 by defining a linear map Tx.Source: A. Auxiliary results · pp. 12–13
  • The proof establishes local Lipschitz continuity of ∇U* and continuity of Tx, yielding continuous second derivatives of U*.The Jacobian of ∇U* at y = ∇U(x) is Tx.Source: A. Auxiliary results · pp. 12–14

B. Proof of the application corollaries

The corollary proofs verify that the leader-follower, leaderless, and affine formation error systems meet the abstract homogeneous-chain conditions. The resulting gain choices give global finite-time stability and the stated application conclusions.

B. Proof of the application corollaries

  • The three application error systems satisfy the required properties of Uμ, verified using the respective matrices L + B, D⊤, and ΩFF.The stated graph and matrix conditions make the corresponding Q matrices symmetric positive definite; the set-valued maps also satisfy the conditions used for cS.Source: B. Proof of the application corollaries · p. 14
  • For leader-follower tracking, the selected gains make the abstract homogeneous-chain origin globally finite-time stable, yielding the stated application result.The proof uses γ = 0.Source: B. Proof of the application corollaries · p. 14
  • For leaderless consensus, positive gains ensure global finite-time stability; the application conclusion follows for γ > 0 or, under (17), for γ = 0.The proof uses the disturbance set D = {d ∈ 1⊥ : ∥d∥ ≤ 1}.Source: B. Proof of the application corollaries · p. 14
  • For affine formation tracking, the selected gains make the abstract homogeneous-chain origin globally finite-time stable, yielding the stated application result.The proof identifies the error system as an abstract homogeneous chain with γ = 0.Source: B. Proof of the application corollaries · p. 14
Loading 2609.12250v1…