Higher-Order Sliding Mode Consensus

• Higher-Order Sliding Mode Consensus is a distributed control framework that leverages sliding manifolds based on higher-order derivatives to achieve exact tracking and synchronization.
• EDCHO and REDCHO schemes demonstrate finite-time convergence and robustness to initialization mismatches and topology changes, ensuring dynamic average consensus and leader-following control.
• Recent developments integrate integral sliding modes and neuro-adaptive strategies to mitigate chattering and accommodate heterogeneous nonlinear network dynamics.

Searching arXiv for papers on higher-order sliding mode consensus and related dynamic average consensus. Higher-order sliding mode consensus denotes a class of distributed consensus and synchronization schemes in which sliding-mode structures of order greater than one, or sliding manifolds built from higher-order state or signal derivatives, are embedded into the inter-agent coupling. In the literature represented here, the topic covers exact dynamic average consensus of time-varying signals and their derivatives on connected undirected graphs, robust exact dynamic consensus under initialization mismatch and isolated topology changes, leader-following synchronization of high-order perturbed agents under multiple time-varying delays via integral sliding mode, and neuro-adaptive leader-follower tracking for higher-order heterogeneous nonlinear teams (Aldana-López et al., 2022, Aldana-López et al., 2022, Gholami, 2023, Chaudhari et al., 29 Jul 2025).

1. Core formulations and mathematical setting

Two problem classes dominate the cited formulations. The first is dynamic average consensus (DAC), where each agent $i$ measures a time-varying scalar reference $r_i(t)$ or $u_i(t)$, and the network seeks exact agreement on the average

$$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$

together with derivatives $s^{(\mu)}(t)$ or $\bar{u}^{(\mu)}(t)$, $\mu=0,\dots,m$. The graph is undirected and connected, with adjacency matrix $A$, degree matrix $D_g$, Laplacian $L=D_g-A$, incidence matrix $r_i(t)$0, and projection onto the disagreement subspace

$r_i(t)$1

The second class is leader-following consensus for higher-order plants, where each follower tracks a leader through distributed error coordinates built from neighbor differences and leader pinning.

A central distinction is between consensus algorithms defined on internal observer states and consensus controllers for physical plants. In EDCHO and REDCHO, the states $r_i(t)$2 are algorithmic variables, and the outputs are constructed as $r_i(t)$3 or $r_i(t)$4. In leader-following plant formulations, the states are plant coordinates and the sliding variables are built from tracking errors. For nonlinear higher-order leader-follower systems, the distributed synchronization error and composite sliding variable are

$r_i(t)$5

The principal schemes represented in the literature differ in graph assumptions, robustness mechanisms, and convergence guarantees.

Scheme	Setting	Main guarantee
EDCHO (Aldana-López et al., 2022)	Undirected connected DAC	Finite-time exact consensus on the average and its derivatives
REDCHO (Aldana-López et al., 2022)	DAC with initialization mismatch and isolated topology changes	Robust exact dynamic consensus in a large region of attraction; bounded terminal error otherwise
ISM + consensus (Gholami, 2023)	High-order perturbed LTI followers with multiple time-varying delays	Leader-following synchronization under LMI-certified delay bounds
DNN-SMC (Chaudhari et al., 29 Jul 2025)	Higher-order heterogeneous nonlinear leader-follower MAS	Asymptotic tracking with bounded DNN weights and compact-set invariance

2. Exact dynamic average consensus via distributed higher-order sliding modes

EDCHO formulates dynamic average consensus as a distributed higher-order sliding-mode differentiator. Each agent $r_i(t)$6 maintains an internal state

$r_i(t)$7

and outputs

$r_i(t)$8

Only $r_i(t)$9 is exchanged with neighbors. The coupling uses the signum-power notation

$u_i(t)$0

with exponents that descend through the cascade: $u_i(t)$1 The last exponent $u_i(t)$2 is the pure signum function, and the discontinuous dynamics are treated in the Filippov sense (Aldana-López et al., 2022).

The disagreement dynamics are expressed through $u_i(t)$3, yielding a distributed version of Levant’s arbitrary-order exact differentiator. Gains are selected recursively: $u_i(t)$4 where $u_i(t)$5 render Levant’s differentiator finite-time stable and $u_i(t)$6 is chosen sufficiently large. Under a connected undirected graph, zero-sum initialization

$u_i(t)$7

and bounded disagreement in the $u_i(t)$8-st derivatives of the references,

$u_i(t)$9

EDCHO achieves finite-time exact dynamic average consensus: $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$0

The proof structure combines invariant sums, disagreement-space dynamics, contraction arguments on tree graphs, averaging arguments for general connected graphs, and homogeneity under the transformation

$$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$1

This homogeneity implies that contraction time windows can be arranged as a geometric sequence with finite sum, establishing finite-time convergence. Relative to first-order sliding-mode DAC, EDCHO is designed to achieve zero steady-state error for time-varying references and exact agreement on derivatives up to order $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$2. The reported simulations also indicate substantially reduced chattering relative to first-order sliding modes, especially under discretization, while linear DAC retains bounded steady-state error and does not track derivatives exactly.

3. REDCHO and robust exact dynamic consensus under topology changes

REDCHO generalizes EDCHO by removing the initialization constraint that $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$3, a requirement that becomes restrictive under connection and disconnection events. The construction introduces a strictly stable internal model with lower-triangular matrix $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$4, diagonal entries $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$5, superdiagonal ones, and the observability matrix $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$6 of $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$7, together with a scaling parameter $$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$8. The output is defined as

$$s(t)=\frac{1}{N}\sum_{i=1}^N r_i(t), \qquad \bar{u}(t)=\frac{1}{n}\sum_{i=1}^n u_i(t),$$9

and the vectorized dynamics take the compact form

$s^{(\mu)}(t)$0

with the higher-order sliding-mode coupling concentrated in

$s^{(\mu)}(t)$1

The average and disagreement components decouple structurally. Using

$s^{(\mu)}(t)$2

the average error $s^{(\mu)}(t)$3 obeys

$s^{(\mu)}(t)$4

so it converges exponentially to zero for any initial condition. The disagreement analysis is performed in the scaled coordinates

$s^{(\mu)}(t)$5

for which

$s^{(\mu)}(t)$6

Here $s^{(\mu)}(t)$7 is $s^{(\mu)}(t)$8-homogeneous of degree $s^{(\mu)}(t)$9, whereas $\bar{u}^{(\mu)}(t)$0 is $\bar{u}^{(\mu)}(t)$1-homogeneous of degree $\bar{u}^{(\mu)}(t)$2. This decomposition makes the effect of the internal-model mismatch explicit and shows how $\bar{u}^{(\mu)}(t)$3 dilates the region of attraction (Aldana-López et al., 2022).

The resulting theorem states that, for a connected undirected graph and gains chosen according to the EDCHO/Levant design for the known bound $\bar{u}^{(\mu)}(t)$4, there exist neighborhoods $\bar{u}^{(\mu)}(t)$5 and $\bar{u}^{(\mu)}(t)$6 around dynamic consensus such that initial conditions in $\bar{u}^{(\mu)}(t)$7 yield robust exact dynamic consensus,

$\bar{u}^{(\mu)}(t)$8

whereas initial conditions outside $\bar{u}^{(\mu)}(t)$9 yield uniform ultimate boundedness in $\mu=0,\dots,m$0. Both $\mu=0,\dots,m$1 and $\mu=0,\dots,m$2 can be enlarged arbitrarily by increasing $\mu=0,\dots,m$3. This is the mechanism by which REDCHO becomes robust to mismatch in the initial conditions and suitable for isolated topology change events, including joining, leaving, and merging of connected subnetworks, provided the post-event graph remains connected.

A central significance of REDCHO is that it preserves the communication footprint of EDCHO: each agent exchanges only the scalar $\mu=0,\dots,m$4. The protocol therefore extends higher-order sliding-mode exact differentiator ideas to non-ideal network operation without abandoning the single-scalar edge coupling that made EDCHO communication-efficient.

4. High-order perturbed multi-agent systems with delays: integral sliding mode and the path toward HOSM

Leader-following consensus for high-order perturbed plants with multiple time-varying delays is treated in a different framework. Each follower has the controllable-canonical LTI dynamics

$\mu=0,\dots,m$5

with matched bounded perturbation $\mu=0,\dots,m$6, while the leader evolves as

$\mu=0,\dots,m$7

Consensus is full-state tracking, $\mu=0,\dots,m$8, over a directed graph with a globally reachable virtual leader and multiple bounded time-varying delays $\mu=0,\dots,m$9, $A$0 satisfying rate constraints $A$1, $A$2 (Gholami, 2023).

The distributed controller combines a linear delayed consensus term with integral sliding mode: $A$3

$A$4

$A$5

The sliding surface is

$A$6

with

$A$7

so that $A$8. This removes the reaching phase. Under $A$9, the sliding dynamics satisfy

$D_g$0

and the perturbation is removed from the equivalent consensus-error channel.

The delayed consensus error $D_g$1 admits a stacked representation with delayed matrices $D_g$2 and $D_g$3. Closed-loop stability is established through a five-term Lyapunov–Krasovskii functional

$D_g$4

with decision matrices $D_g$5. Jensen/Park-type inequalities, the Leibniz–Newton formula, and the Schur complement lead to LMIs whose feasibility guarantees

$D_g$6

The same framework yields a lower estimate of the admissible delay margin,

$D_g$7

and an optimization procedure for the edge-dependent constant gains $D_g$8.

This design is closely related to higher-order sliding mode consensus but is not itself a classical HOSM algorithm. The paper uses integral sliding mode plus a first-order filter

$D_g$9

to alleviate chattering. It explicitly notes that a super-twisting replacement of the discontinuous term is possible,

$L=D_g-A$0

but such a substitution requires a different Lyapunov analysis. In that sense, the work provides a delay-robust integral sliding-mode consensus architecture and an explicit synthesis route that can inform higher-order sliding-mode extensions.

5. Neuro-adaptive sliding mode control for higher-order heterogeneous nonlinear teams

A more recent line addresses leader-following tracking for higher-order heterogeneous nonlinear agents with unknown dynamics. Each follower has $L=D_g-A$1-th order dynamics

$L=D_g-A$2

with bounded disturbance $L=D_g-A$3, while the leader obeys

$L=D_g-A$4

The graph is directed and satisfies the condition that $L=D_g-A$5 is nonsingular, with

$L=D_g-A$6

symmetric positive definite (Chaudhari et al., 29 Jul 2025).

The higher-order sliding variable is defined on the distributed tracking errors: $L=D_g-A$7 where the polynomial

$L=D_g-A$8

is chosen Hurwitz. Unknown nonlinearities are approximated on a compact set $L=D_g-A$9 by a DNN of the form

$r_i(t)$00

with online estimate

$r_i(t)$01

Output-layer and inner-layer weights are updated online through modular adaptation laws, with indicator functions enforcing bounded parameter sets.

The distributed controller is

$r_i(t)$02

where

$r_i(t)$03

A generalized restricted potential function $r_i(t)$04 acts as a barrier Lyapunov term on

$r_i(t)$05

ensuring that the trajectories remain within the compact set where DNN approximation is valid. The nonsmooth Lyapunov function combines the restricted potential, the lower-order error stack $r_i(t)$06, and the output and inner DNN weight errors. Under explicit gain conditions on $r_i(t)$07 and $r_i(t)$08, if $r_i(t)$09, then

$r_i(t)$10

and the DNN weights remain bounded. A corollary states that, when $r_i(t)$11 and $r_i(t)$12, all higher-order tracking errors satisfy $r_i(t)$13.

Within the terminology of higher-order sliding mode consensus, the method is adjacent rather than identical to EDCHO or REDCHO. It employs a first-order sliding mode on the composite higher-order error $r_i(t)$14, not a higher-order sliding mode algorithm such as super-twisting acting on $r_i(t)$15 and $r_i(t)$16. Its significance is different: DNN feedforward is used to reduce the switching gain needed for robustness, and the restricted potential function provides compact-set invariance for learning-based approximation.

6. Conceptual distinctions, limitations, and active directions

A recurring source of ambiguity is the meaning of “higher-order.” In EDCHO and REDCHO, higher-order sliding mode refers to Levant-type exact differentiator structures with nested sign-power feedback and homogeneity-based finite-time convergence on the disagreement dynamics. In the delayed leader-following design, “high-order” refers to plant order, while the sliding mechanism is integral sliding mode rather than a classical HOSM algorithm. In the neuro-adaptive leader-follower design, the plant is higher-order and the sliding variable is composite, but the discontinuous term is first-order $r_i(t)$17, with higher-order sliding mode mentioned only as a potential extension (Aldana-López et al., 2022, Aldana-López et al., 2022, Gholami, 2023, Chaudhari et al., 29 Jul 2025).

The limitations are correspondingly specific. EDCHO requires an undirected connected graph, bounded disagreement in the $r_i(t)$18-st derivatives of the references, and the zero-sum initialization constraint; formal robustness to measurement noise, delays, quantization, and switching topologies is not addressed. REDCHO removes the initialization constraint and is robust to isolated topology changes provided connectedness holds after each event, but persistent fast switching remains outside the analysis. The delay-robust leader-following formulation assumes matched bounded disturbances, full-state measurements, symmetric inter-agent delays, known constant adjacency weights, and a directed topology with a globally reachable leader. The neuro-adaptive design requires a compact approximation domain, explicit or precomputed access to $r_i(t)$19, and yields asymptotic rather than finite-time convergence.

The research directions identified across these works are consistent. They include directed and time-varying graphs, communication delays and quantization for HOSM DAC, sampled-data and discrete-time exactness guarantees, adaptive gain tuning, packet drops and stochastic graph processes, and systematic replacement of first-order discontinuous terms by HOSM mechanisms such as super-twisting. A plausible implication is that future higher-order sliding mode consensus designs will increasingly combine three ingredients already present separately in the cited literature: homogeneity-based exact differentiator structures, LMI-certified robustness to delays and topology-dependent couplings, and adaptive approximation layers for heterogeneous nonlinear dynamics.