---
title: Higher-Order Sliding Mode Consensus
url: https://www.emergentmind.com/topics/higher-order-sliding-mode-consensus
type: topic
---

# Higher-Order Sliding Mode Consensus

Searching arXiv for recent 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 [2202.03012], [2204.12344], [2302.14033], [2507.21667].

## 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 \(D\), and projection onto the disagreement subspace
\[
P=I-\frac{1}{N}\mathbf{1}\mathbf{1}^\top.
\]
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 \(x_{i,\mu}\) are algorithmic variables, and the outputs are constructed as \(y_{i,\mu}=r_i^{(\mu)}-x_{i,\mu}\) or \(y_{i,\mu}=u_i^{(\mu)}-\sum_{\nu}G_{\mu+1,\nu+1}x_{i,\nu}\). 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
\[
e_i^m=\sum_{j\in\mathcal{N}_i} a_{ij}(x_j^m-x_i^m)+b_i(x_0^m-x_i^m),
\qquad
r_i=\sum_{m=1}^{M-1}\lambda_m e_i^m + e_i^M.
\]

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

| Scheme | Setting | Main guarantee |
|---|---|---|
| EDCHO [2202.03012] | Undirected connected DAC | Finite-time exact consensus on the average and its derivatives |
| REDCHO [2204.12344] | DAC with initialization mismatch and isolated topology changes | Robust exact dynamic consensus in a large region of attraction; bounded terminal error otherwise |
| ISM + consensus [2302.14033] | High-order perturbed LTI followers with multiple time-varying delays | Leader-following synchronization under LMI-certified delay bounds |
| DNN-SMC [2507.21667] | 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 \(i\) maintains an internal state
\[
x_i(t)=[x_{i,0}(t),\dots,x_{i,m}(t)]^\top
\]
and outputs
\[
y_{i,\mu}(t)=r_i^{(\mu)}(t)-x_{i,\mu}(t), \qquad \mu=0,\dots,m.
\]
Only \(y_{i,0}(t)\) is exchanged with neighbors. The coupling uses the signum-power notation
\[
\lceil x\rceil^\alpha = |x|^\alpha \operatorname{sign}(x),
\]
with exponents that descend through the cascade:
\[
\begin{aligned}
\dot{x}_{i,0}(t) &= x_{i,1}(t)+k_0\sum_{j=1}^N a_{ij}\Big\lceil y_{i,0}(t)-y_{j,0}(t)\Big\rceil^{\frac{m}{m+1}},\\
\dot{x}_{i,\mu}(t) &= x_{i,\mu+1}(t)+k_\mu\sum_{j=1}^N a_{ij}\Big\lceil y_{i,0}(t)-y_{j,0}(t)\Big\rceil^{\frac{m-\mu}{m+1}}, \quad \mu=1,\dots,m-1,\\
\dot{x}_{i,m}(t) &= k_m\sum_{j=1}^N a_{ij}\Big\lceil y_{i,0}(t)-y_{j,0}(t)\Big\rceil^{0}.
\end{aligned}
\]
The last exponent \(0\) is the pure signum function, and the discontinuous dynamics are treated in the Filippov sense [2202.03012].

The disagreement dynamics are expressed through \(\tilde{Y}_\mu=P Y_\mu\), yielding a distributed version of Levant’s arbitrary-order exact differentiator. Gains are selected recursively:
\[
k_\mu=\lambda_\mu\,k_{\mu-1}^{\frac{m-\mu}{m-(\mu-1)}}, \qquad \mu=1,\dots,m,
\]
where \(\lambda_\mu\) render Levant’s differentiator finite-time stable and \(k_0\) is chosen sufficiently large. Under a connected undirected graph, zero-sum initialization
\[
\sum_{i=1}^N x_{i,\mu}(t_0)=0,\qquad \mu=0,\dots,m,
\]
and bounded disagreement in the \((m+1)\)-st derivatives of the references,
\[
\big|s^{(m+1)}(t)-r_i^{(m+1)}(t)\big|\le L,
\]
EDCHO achieves finite-time exact dynamic average consensus:
\[
y_{i,\mu}(t)\equiv s^{(\mu)}(t), \qquad \forall\, t\ge t_0+T,\ \forall\, i,\ \mu=0,\dots,m.
\]

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
\[
(t,\tilde{Y}_\mu)\mapsto (\eta t,\eta^{m-(\mu-1)}\tilde{Y}_\mu).
\]
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 \(m\). 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 \(\sum_i x_{i,\mu}(t_0)=0\), a requirement that becomes restrictive under connection and disconnection events. The construction introduces a strictly stable internal model with lower-triangular matrix \(\Gamma\), diagonal entries \(-\gamma_\mu\), superdiagonal ones, and the observability matrix \(G\) of \((\Gamma,C)\), together with a scaling parameter \(\theta\ge 1\). The output is defined as
\[
y_{i,\mu}(t)=u_i^{(\mu)}(t)-\sum_{\nu=0}^{m}G_{\mu+1,\nu+1}x_{i,\nu}(t),
\]
and the vectorized dynamics take the compact form
\[
\dot{X}=(\Gamma\otimes I_n)X+F(Y_0;\theta), \qquad
Y=U-(G\otimes I_n)X,
\]
with the higher-order sliding-mode coupling concentrated in
\[
F(Y_0;\theta)=
\begin{bmatrix}
k_0\theta\,D\left\lceil D^\top Y_0\right\rceil^{\frac{m}{m+1}}\\
\vdots\\
k_m\theta^{m+1}\,D\left\lceil D^\top Y_0\right\rceil^{0}
\end{bmatrix}.
\]

The average and disagreement components decouple structurally. Using
\[
Y(t)=(I_{m+1}\otimes \mathbf{1}_n)\bar{y}(t)+\tilde{Y}(t),
\]
the average error \(e(t)=\bar{y}(t)-\bar{u}(t)\) obeys
\[
\dot{e}(t)=\tilde{\Gamma}\,e(t),
\]
so it converges exponentially to zero for any initial condition. The disagreement analysis is performed in the scaled coordinates
\[
Z(t)=\big(\Theta G^{-1}\otimes I_n\big)\tilde{Y}(t), \qquad
\Theta=\operatorname{diag}\{1,\theta^{-1},\ldots,\theta^{-m}\},
\]
for which
\[
\dot{Z}(t)\in \theta\big(H(Z(t))+Q(Z(t))\big).
\]
Here \(H\) is \(r\)-homogeneous of degree \(-1\), whereas \(Q\) is \(r\)-homogeneous of degree \(0\). This decomposition makes the effect of the internal-model mismatch explicit and shows how \(\theta\) dilates the region of attraction [2204.12344].

The resulting theorem states that, for a connected undirected graph and gains chosen according to the EDCHO/Levant design for the known bound \(L\), there exist neighborhoods \(R\) and \(R'\) around dynamic consensus such that initial conditions in \(R\) yield robust exact dynamic consensus,
\[
Y_\mu(t)\to \bar{u}^{(\mu)}(t)\,\mathbf{1}_n,\qquad \mu=0,\dots,m,
\]
whereas initial conditions outside \(R\) yield uniform ultimate boundedness in \(R'\). Both \(R\) and \(R'\) can be enlarged arbitrarily by increasing \(\theta\). 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 \(y_{i,0}(t)\). 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
\[
\dot{x}_i(t)=A x_i(t)+B u_i(t)+B\,\omega_i(t),
\]
with matched bounded perturbation \(|\omega_i(t)|\le \Gamma_i\le \Gamma\), while the leader evolves as
\[
\dot{x}_0(t)=A x_0(t).
\]
Consensus is full-state tracking, \(y_i(t)=x_i(t)\), over a directed graph with a globally reachable virtual leader and multiple bounded time-varying delays \(\tau_{ij}(t)\), \(\tau_{i0}(t)\) satisfying rate constraints \(|\dot{\tau}_{ij}(t)|\le \bar d_p<1\), \(|\dot{\tau}_{i0}(t)|\le d_l<1\) [2302.14033].

The distributed controller combines a linear delayed consensus term with integral sliding mode:
\[
u_i(t)=u_{i,\mathrm{cons}}(t)+u_{i,\mathrm{ISM}}(t),
\]
\[
u_{i,\mathrm{cons}}(t)= -\sum_{j=0}^{N}\alpha_{ij}\,k_{ij}^{\top}\big(x_i(t-\tau_{ij}(t))-x_j(t-\tau_{ij}(t))\big),
\]
\[
u_{i,\mathrm{ISM}}(t)= -\rho\,\operatorname{sign}\big(s_i(t)\big), \qquad \rho>\Gamma.
\]
The sliding surface is
\[
s_i(t)=\mathbf{1}_n^\top x_i(t)+z_i(t),
\]
with
\[
z_i(0)=-\mathbf{1}_n^\top x_i(0),
\]
so that \(s_i(0)=0\). This removes the reaching phase. Under \(\rho>\Gamma\), the sliding dynamics satisfy
\[
\dot{s}_i(t)=b\big(\omega_i(t)-\rho\,\operatorname{sign}(s_i(t))\big),
\]
and the perturbation is removed from the equivalent consensus-error channel.

The delayed consensus error \(e_i(t)=x_i(t)-x_0(t)\) admits a stacked representation with delayed matrices \(\hat A_l\) and \(\tilde A_p\). Closed-loop stability is established through a five-term Lyapunov–Krasovskii functional
\[
V(e(t))=\sum_{i=1}^{5}V_i(e(t)),
\]
with decision matrices \(P,Q_l,\bar Q_p,R_l,\bar R_p\succ 0\). Jensen/Park-type inequalities, the Leibniz–Newton formula, and the Schur complement lead to LMIs whose feasibility guarantees
\[
\|x_i(t)-x_0(t)\|\to 0,\qquad \forall i,\quad t\to\infty.
\]
The same framework yields a lower estimate of the admissible delay margin,
\[
\tau^\star=
\min\left\{
\frac{\|M_1\|}{\|M_2\|},
\frac{\|\bar M_1\|}{\|\bar M_2\|},
\frac{\|\tilde M_1\|}{\|\tilde M_2\|}
\right\},
\]
and an optimization procedure for the edge-dependent constant gains \(k_{ij}\).

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
\[
\widetilde{T}\,\dot{\upsilon}_i(t)+\upsilon_i(t)=\rho\,\operatorname{sign}\big(s_i(t)\big)
\]
to alleviate chattering. It explicitly notes that a super-twisting replacement of the discontinuous term is possible,
\[
u_{i,\mathrm{HOSM}}=-k_1|s_i|^{1/2}\operatorname{sign}(s_i)+v_i,\qquad
\dot v_i=-k_2\operatorname{sign}(s_i),
\]
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 \(M\)-th order dynamics
\[
\dot{x}_i^m=x_i^{m+1},\quad m=1,\ldots,M-1,\qquad
\dot{x}_i^M=f_i(x_i,t)+u_i(t)+\omega_i(t),
\]
with bounded disturbance \(\|\omega(t)\|\le \omega_m\), while the leader obeys
\[
\dot{x}_0^m=x_0^{m+1},\quad m=1,\ldots,M-1,\qquad
\dot{x}_0^M=f_0(x_0,t).
\]
The graph is directed and satisfies the condition that \(L+B\) is nonsingular, with
\[
q=(L+B)^{-1}\mathbf{1},\qquad
P=\operatorname{diag}\{1/q_i\},\qquad
Q=P(L+B)+(L+B)^\top P
\]
symmetric positive definite [2507.21667].

The higher-order sliding variable is defined on the distributed tracking errors:
\[
r_i=\sum_{m=1}^{M-1}\lambda_m e_i^m+e_i^M,
\]
where the polynomial
\[
s^{M-1}+\lambda_{M-1}s^{M-2}+\cdots+\lambda_1
\]
is chosen Hurwitz. Unknown nonlinearities are approximated on a compact set \(\Omega\) by a DNN of the form
\[
f_i(x_i,t)=W_i^\top \rho_i(\Phi_i(x_i))+\epsilon_i(x_i),
\]
with online estimate
\[
\hat f_i(x_i,t)=\hat W_i^\top \hat\rho_i(\hat\Phi_i(x_i)).
\]
Output-layer and inner-layer weights are updated online through modular adaptation laws, with indicator functions enforcing bounded parameter sets.

The distributed controller is
\[
u_i=
\frac{1}{d_i+b_i}\sum_{m=1}^{M-1}\lambda_m e_i^{m+1}
+\gamma_1 r_i
+\gamma_2\,\operatorname{sgn}(r_i)
-\hat f_i(x_i,t)
+c_i,
\]
where
\[
c=-(L+B)^{-1}A\,\hat W^\top \hat\rho(\hat\Phi(x)).
\]
A generalized restricted potential function \(\Upsilon(\|r\|_P)\) acts as a barrier Lyapunov term on
\[
\Omega_\mu=\{r\in\mathbb{R}^N:\|r\|_P<\mu\},
\]
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 \(\mathcal E_1\), and the output and inner DNN weight errors. Under explicit gain conditions on \(\gamma_1\) and \(\gamma_2\), if \(\|r(0)\|_P<\mu\), then
\[
\|r(t)\|_P<\mu \quad \forall t\ge 0,\qquad \|r(t)\|_P\to 0,
\]
and the DNN weights remain bounded. A corollary states that, when \(\sum_{m=1}^{M-1}\lambda_m>1\) and \(\lambda_m>0\), all higher-order tracking errors satisfy \(\|e^m(t)\|_P\to 0\).

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 \(r\), not a higher-order sliding mode algorithm such as super-twisting acting on \(r\) and \(\dot r\). 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 \(\operatorname{sgn}(r_i)\), with higher-order sliding mode mentioned only as a potential extension [2202.03012], [2204.12344], [2302.14033], [2507.21667].

The limitations are correspondingly specific. EDCHO requires an undirected connected graph, bounded disagreement in the \((m+1)\)-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 \((L+B)^{-1}\), 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.

Source: https://www.emergentmind.com/topics/higher-order-sliding-mode-consensus