A signed graph G=(V,E,𝒜) represents a vehicle platooning. The set of nodes is given as V={1,2,3,…,M} representing individual vehicles in this model. The set of edges E⊆V×V represents the communication links that connect pairs of vehicles. The communication network structure is encoded in the weighted adjacency matrix 𝒜=[amn]∈RM×M, which describes the interactions between these vehicles.

A nonzero weight amn≠0 value shows that there exists an edge connecting nodes m and n such that (m,n)∈E. The neighborhood of vehicle m is defined as Nm={n∣amn≠0}. The signed graph G is undirected if the edge set meets the condition (m,n)∈E⇔(n,m)∈E.

The weights amn may be either positive or negative; a positive weight amn>0 means that the vehicles are cooperative with each other, while a negative weight amn<0 means that the vehicles are adversarial with each other. A sequence of edges such as (m1,m2),(m2,mℓ),…,(mk−1,mk). Where all intermediate nodes mℓ for ℓ=1,2,…,k are distinct, is referred to as a path from node m1 to node mk.

A graph is considered connected if, for any pair of distinct nodes m and n where m≠n, there exists a path connecting them. The Laplacian matrix ℒ=[ℓmn]∈RM×M for a signed graph G is defined such that the diagonal elements are given as ℓmm=∑n=1n≠mM|amn|.

The off-diagonal entries are ℓmn=−amn, for m≠n. The network becomes an augmented graph G^=(V^,E^) by adding a leader node, indexed as node 0. The node set transforms into V^=V∪{0} while the edge set becomes E^⊆V^×V^. A communication link from the leader to the vehicle m∈V is established if am0>0, the leader shares information with the vehicle otherwise am0=0. The diagonal matrix 𝒮0=diag(a10,a20,…,aM0) represents the leader’s influence power within the network structure. The modified Laplacian matrix takes into account the leader’s input through this expression ℒ0=ℒ+𝒮0.

Lemma 1. [46] A signed graph G is structurally balanced if it contains a spanning tree. The Laplacian matrix ℒ0, which includes the leader’s influence, becomes positive definite when the leader node acts as the root of this spanning tree in a structurally balanced network signed graph G^. This ensures both structural coherence and stability in the network.

Definition 1. [46] A signed graph G is considered structurally balanced if its node set V can be divided into two disjoint subsets V1 and V2 such that V1∩V2=∅,V1∪V2=V.

The edge weights satisfy the condition amn≥0 when both nodes m and n are in the same subset Vq while amn≤0 when m∈Vq and n∈V3−q, in which q∈{1,2}. If such a partition does not exist, the graph is termed structurally unbalanced. To formalize this concept, the set 𝒟 defined as 𝒟={D=diag(d1,d2,d3,…,dM)|dm∈{1,−1}}.

Each matrix D∈𝒟 corresponds to a particular partitioning of the graph, where nodes are assigned according to the sign of dm. V1={m∣dm=1},V2={m∣dm=−1}form∈V

The following vehicle platooning provides conditions that are equivalent to the structural balance of the graph.

Lemma 2. [46] A signed graph G is said to be connected and structurally balanced if it satisfies the following conditions

A diagonal matrix D∈𝒟 exists which transforms the matrix D𝒜D into a form where all entries are nonnegative.

Lemma 3. The bipartite consensus formulation uses the gauge transformation D=diag(d1,d2,…,dM) with di∈{1,−1}. The transformation DTD=I results in an orthogonal matrix that maintains the original norm values of the input vectors ‖Dx‖=‖x‖, ‖Dv‖=‖v‖.

The mapping x↦Dx, v↦Dv leaves all vehicle state constraints of the form |xi(t)|≤xmax and |vi(t)|≤vmax unchanged. The transformed Laplacian ℒ′=DℒD retains the spectral characteristics of ℒ and maintains a bounded Lyapunov function V(t)=12HTH. The gauge transformation maintains constraint satisfaction because it prevents the transformed consensus dynamics from increasing system state values.

Remark 1. Lemma 3 demonstrates that the gauge transformation maintains bounded vehicle states because it protects the absolute values of the states. The transformation between cooperative and competitive clusters only changes the sign of the states but keeps their magnitudes unchanged, which results in physically valid state trajectories for bipartite consensus.

Remark 2. In this work, we assume that external disturbances (such as aerodynamic drag, tire friction variation, road slope, and wind) are bounded or sufficiently small relative to the primary destabilizing effects considered. According to recent studies such as [47], network disconnections and communication delays rank among the top contributors to platoon instability, whereas external disturbances, while non-negligible, have a smaller relative impact. Under this assumption, we focus primarily on mitigating communication-based uncertainties and deception attacks.

Vehicle platooning consists of M vehicles that demonstrate nonlinear impulsive behavior. The dynamics of each vehicle are characterized by the following equations (1){x˙m(t)=vm(t),xm(t0)=xm0,for t≠tp,v˙m(t)=ze(vm(t)),vm(t0)=vm0,for t≠tp,Υxm(t)=zd(xm(t)),at t=tp,Υvm(t)=zd(vm(t)),at t=tp.

The state vector of the mth vehicle is represented by xm(t)∈Rl and vm(t)∈Rl composed of its position and velocity in Eq. (1). The system evolves continuously through ze:Rl×R+→Rl, which governs the vehicle’s dynamics between the impulsive moments and is assumed to be a continuous mapping. The discrete (impulsive) behavior is described by the function zd:Rl×R+→Rl, which is also a continuous function. Each impulsive event at time tp results in a state change of the vehicle according to Υxm(tp)=xm(tp+)−xm(tp) Υvm(tp)=vm(tp+)−vm(tp).

The state vector xm(t), vm(t) remains left-continuous which means xm(tp−)=xm(tp), vm(tp−)=vm(tp) while the post-impulse state can be equivalently expressed as xm(tp+)=xm(tp−)+Υxm(t), vm(tp+)=vm(tp−)+Υvm(t). This framework provides a rigorous basis for the analysis of how impulsive effects contribute to the finite-time or fixed-time stabilization of the overall system.

Lemma 4. [48] Suppose ψ1,ψ2,…,ψℓ≥0 with parameters 0<k≤1 and l>1. The following inequalities are satisfied (2)∑m=1ℓψmk≥(∑m=1ℓψk)k,∑m=1ℓψml≥ℓ1−l(∑m=1ℓψm)l

These results show that under the given conditions, the kth and lth powers of non-negative sequences are bounded by the sum of the original elements.

Lemma 5. [49] Let’s analyze the nonlinear impulsive system defined by Eq. (1) and suppose there be a function 𝒱(xm,vm) such that for all times t≠tp the inequality 𝒱˙(xm,vm)ze(xm,vm)≤−e(𝒱(xm,vm))w holds, and at the impulse times t=tp it must fulfills 𝒱(xm,vm(tp+))≤𝒱(xm,vm(tp−)). In this context, where e>0 and 0<w<1 the impulsive system is guaranteed to be finite-time stable. Moreover, the finite settling time 𝒯, taken by the state to reach the origin, is given as 𝒯≤𝒱(xm(0),vm(0))1−we(1−w)

Lemma 6. [50] Consider a system characterized by a bounded impulsive interval χp=tp+1−tp defined over an impulsive sequence ε={tp}(p∈N) with χ1 and χ2 as the maximum and minimum bounds of χp. Assume that there is a scalar function 𝒱(xm,vm) that satisfies the following conditions

1. For all t≠tp the function is bounded as u1‖xm(t),vm(t)‖ξ≤𝒱(xm(t),vm(t))≤u2‖xm(t),vm(t)‖ξ

2. The jump condition 𝒱(xm(tp+),vm(tp+))≤λ𝒱(xm(tp),vm(tp)) applies for every impulsive instant t=tp. Under these assumptions, the system Eq. (1) is guaranteed to reach a fixed-time stable state. The total convergence time 𝒯 is given as

For λ=1 𝒯=1ψ(1−b)+1γ(c−1)

u1>0,u2>0,p>0,ψ>0,γ>0,0<b<1,c>1,and0<λ≤1

3. The derivative of 𝒱 when t≠tp can be expressed as 𝒱˙(xm(t),vm(t))≤{−ψ𝒱b(xm(t),vm(t)),0<𝒱(xm(t),vm(t))≤1,−γ𝒱c(xm(t),vm(t)),𝒱(xm(t),vm(t))>1,0,𝒱(xm(t),vm(t))=0.

Examine a second-order vehicle platooning consisting of M vehicles that communicate through an undirected signed graph G. The following describes the dynamics of the leader’s behavior. (3){x˙0(t)=v0(t)v˙0(t)=u0(t)where (x0(t),v0(t))∈Rl denote the position and velocity of the leader while u0(t)∈Rl represents its control input vector, which determines the state evolution over time.

The dynamic behavior of each vehicle is modeled as (4){x˙m(t)=vm(t)v˙m(t)=um(t)m∈V

The control input is given as um(t)∈Rl, while, (x0(t),v0(t))∈Rl denote the position and velocity of the mth vehicle. The network contains a specific leader in certain cases, and other vehicles must follow or synchronize their states with this leader.

Assumption 1. The leaderless system Eq. (3) operates under the assumption that the interaction graph remains structurally balanced and connected, which enables reliable information exchange and vehicle state alignment. The leader-following framework uses systems Eqs. (3) and (4) under the assumption that the communication topology G contains a directed spanning tree with the leader positioned at the root. The network structure ensures that the leader’s information reaches all followers either directly or indirectly.

The set of nodes in the signed graph G can be divided into two disjoint subsets V1 and V2, according to assumption (1) reflecting the structurally balanced nature of the network. The main objective is to design a class of impulsive control strategies that can steer the vehicles to bipartite consensus within a finite or fixed time horizon, as formally stated in the following section.

Definition 2. Considering a leaderless vehicle platooning governed by system (4) is said to achieve finite-time bipartite consensus if, for some appropriate control input um and a finite time 𝒯>0 which may depend on the initial conditions of vehicles, for any (x0,v0) the following condition must satisfied, (5)limt→𝒯(xm(t),vm(t))=dmρ,m∈V

where ρ is a constant and dm∈{±1}. This means that the state (xm(t),vm(t)) converges to either ρ or −ρ within time 𝒯, based on the value of dm. Here, 𝒯 is called the settling time. The leader-following vehicle platooning described by systems (3) and (4) requires the following condition for finite time bipartite consensus (FNTBC). (6)limt→𝒯||(xm(t),vm(t))−dm(x0(t),v0(t))||=0,m∈V

If the dm=1 then the mth vehicle belongs to V1; otherwise, dm=−1. Furthermore, a system reaches fixed-time bipartite consensus when the settling time 𝒯 remains independent of initial states.

The concepts of Finite-Time Non-Singular Terminal Bipartite Consensus (FNTBC) and Fixed-Time Bipartite Consensus (FXTBC) play crucial roles in ensuring rapid coordination among networked vehicles. Although both aim to achieve bipartite consensus in a finite duration, their convergence characteristics differ significantly.

Specifically, FNTBC guarantees that the consensus among agents is reached within a finite time that depends on the initial conditions of the system. The convergence rate in this case varies with the magnitude of initial errors, leading to fast stabilization when initial discrepancies are small. In contrast, FXTBC achieves convergence within a fixed upper bound of time, completely independent of the initial conditions. This is realized by introducing additional nonlinear corrective terms with exponents greater than one, which strengthen the control action as the system approaches equilibrium.

From a practical perspective, FNTBC is efficient when the initial state differences among vehicles are moderate and known, while FXTBC is more suitable for scenarios where initial states are highly uncertain, such as large-scale vehicle platoons or dynamically changing environments.