Robust Control and Stabilization of Autonomous Vehicular Systems under Deception Attacks and Switching Signed Networks

1  Introduction

Vehicular networking has become essential in modern intelligent transportation systems, enabling efficient communication for safety and coordination among vehicles. With 5G technology offering ultra-reliable and low-latency links, such networks support advanced applications requiring real-time responsiveness [1]. Recent research in vehicular ad hoc networks (VANETs) has emphasized enhancing routing stability, communication reliability, and control robustness through predictive and trajectory-based approaches. Predictive modeling techniques forecast future vehicular states, enabling routing algorithms to sustain reliable communication links even under rapidly changing mobility conditions [2]. Similarly, bus-trajectory-based street-centric routing leverages predefined bus routes to optimize message delivery and reduce latency in dense urban environments [3]. Furthermore, recent advances in prescribed-time optimal control for switched nonlinear systems provide a powerful framework for achieving optimal-precision coordination within vehicular platoons, ensuring that stability and convergence are attained within a user-defined time despite switching network and system nonlinearities [4]. Decision-making and motion planning frameworks ensure smooth, oscillation-free movement by coordinating perception and control modules [5]. Focuses on the importance of protecting vehicle location privacy to ensure secure communication and prevent tracking in intelligent transportation systems. Emphasizes privacy preservation as a key factor for safety and trust in vehicular networks [6]. Furthermore, semi-global stabilization of parabolic systems with input saturation ensures stability under actuator constraints using Lyapunov and back-stepping-based control, enhancing robustness in distributed vehicular dynamics [7]. Recent advancements in intelligent transportation and autonomous control systems focus on adaptive and resilient designs to ensure stability, coordination, and safety in dynamic environments. Knowledge-guided self-learning control strategies enhance mixed vehicle platoon performance under communication delays, while cognitive risk prediction models utilize spatiotemporal features to assess human driving behavior and prevent potential hazards [8,9]. Coordinated tracking control of integrated wheel-end systems supports precise motion management, and advanced identification algorithms improve network stability and fault detection in complex control structures for the surveillance task schedule [10,11]. Enhanced localization techniques using Low Earth Orbit (LEO) satellite signals strengthen real-time positioning accuracy in uncertain conditions [12]. Furthermore, recurrent neural network–based sliding mode control provides robust stabilization for uncertain tilting quad rotor UAVs by handling external disturbances and modeling errors, while AI-driven perturbation defense techniques ensure accurate recognition of ultra high-speed weak targets, thereby reinforcing the reliability and intelligence of autonomous aerial and vehicular systems [13,14]. The intelligent vehicular systems highlights the integration of advanced testing, sensing, modeling, and resilient control strategies to ensure safety, autonomy, and robustness in dynamic environments. Adversarial scenario generation using hierarchical reinforcement architectures provides rigorous validation for intelligent vehicles, while high-fidelity ultrasonic radar in-the-loop testing enables realistic assessment of automatic parking systems [15,16]. Lane-changing models that incorporate human driving behavior and spatiotemporal decision features enhance motion prediction and cooperative driving accuracy [17,18].

Adaptive PI event-triggered control strategies for nonlinear systems with input delays enhance system responsiveness and robustness against time-varying uncertainties, contributing to improved stability in networked control environments [19]. The trade-off between code estimation error rate and terminal gain has been analyzed to strengthen communication reliability and resilience against adversarial disturbances in vehicle networks [20]. Furthermore, an improved social force model based on pedestrian collision avoidance behavior in counterflow conditions provides deeper insight into dynamic interactions within mixed traffic environments, improving safety and flow efficiency [21]. A hybrid framework combining neural networks with physics-based estimators enables accurate vehicle longitudinal dynamics modeling using limited driving data, bridging the gap between data-driven learning and physical interpretability [22]. Distributed safety-critical formation control using contracting bearing estimators and control barrier functions ensures reliable coordination among connected vehicles under sensing and communication limitations [23]. Finally, intelligent event-triggered lane-keeping security control mechanisms for autonomous vehicles under Denial-of-Service (DoS) attacks significantly enhance communication efficiency and cyber-resilience in connected vehicular networks [24]. Finally, stability and stabilization principles for sampled-data Lurie systems provide theoretical foundations supporting the implementation of these intelligent control strategies in discrete-time vehicular applications [25].

The intelligent autonomous systems and robust control highlights advances in perception, learning, and cooperative navigation to ensure safe and efficient decision-making in complex environments. Human interaction recognition using extended filtering improves behavior estimation and interaction awareness, while real-time contactless eye-blink detection enhances driver safety by monitoring fatigue without physical sensors [26,27]. Interaction-aware trajectory prediction based on transformer-driven transfer learning enables proactive motion planning by forecasting surrounding agents’ intentions, thus improving collision avoidance and cooperative driving efficiency [28,29]. Robust localization frameworks integrating multi-sensor fusion and synchronization algorithms ensure precise navigation performance despite urban signal interference and noise [30,31]. Additionally, noise-tolerant and predefined-time zeroing neural networks offer resilient synchronization and rapid dynamic computations for nonlinear systems, supporting stable control performance under disturbances [32,33]. Furthermore, vehicle-assisted service caching and task offloading in vehicular edge computing, when combined with Nash-equilibrium-based robust control under servo constraints, significantly enhance distributed coordination, reduce latency, and enable adaptive cooperative operation in autonomous and humanoid robotic systems [34,35].

Research on vehicular perception and control emphasizes adaptive learning and deception-resilient recognition to ensure reliability and safety in dynamic driving environments. A neural network incorporating plasticity effectively detects driver fatigue on real roads by adapting to changing behavioral and physiological patterns while resisting deceptive signal variations that may mask true driver conditions [36]. Likewise, a self-supervised evaluation method using a hyper graph neural network with dynamic feature learning accurately assesses insulation conditions in vehicle cable terminals, remaining robust against deceptive or corrupted sensor inputs that could threaten electrical stability and safety [37]. Together, these studies highlight how adaptive and self-learning neural models strengthen resilience and situational awareness in modern vehicular systems. Research on cyber–physical resilience and deception-resistant control in vehicular and transportation networks integrates detection, coordination, and adaptive learning to counter malicious interference and data manipulation. A DDoS detection framework using interquartile range and deep feature fusion convolutional networks identifies abnormal traffic behaviors in software-defined systems, serving as the first layer of defense against large-scale deception-based disruptions [38]. Complementing this, AST-based static fuzz mutation enhances software reliability by identifying concurrency vulnerabilities that attackers might exploit to inject deceptive commands or data anomalies [39]. Presents distributed robust finite-time consensus protocols designed for nonlinear multi-agent systems under uncertainties and disturbances. Emphasizes rapid convergence and resilience, ensuring system stability within a finite time through decentralized control strategies [40]. Perception task-oriented information sharing across autonomous vehicles further strengthens system awareness, enabling robust response to occlusion-based or deceptive sensory manipulation [41]. To ensure secure dynamic control, fixed-time safe tracking for uncertain nonlinear systems achieves rapid recovery and resilience against feedback distortion or deceptive control signals [42]. Finally, machine learning-driven analysis of transport infrastructure connectivity and conflict resolution enhances system resilience by detecting hidden anomalies and strategic deception within large-scale transportation data [43]. Moreover, recent robust adaptive control strategies further strengthen platoon coordination by incorporating disturbance estimation, compensation, and resilience against network disconnections and finite-time networks [44,45], ensuring reliable and secure operation across varying traffic and communication conditions in complex fractional-order networks.

Motivated by the challenges of maintaining coordination and stability in nonlinear vehicle platoons affected by time-varying delays, switching topologies, and deception attacks, this study presents a unified model-based control framework integrating both finite-time and fixed-time bipartite consensus schemes. The main contributions are as follows:

•   A distributed impulsive control protocol is designed to achieve Finite-Time Non-Singular Terminal Bipartite Consensus (FNTBC) and Fixed-Time Bipartite Consensus (FXTBC) under switching signed networks, ensuring rapid convergence and resilience to communication disturbances.

•   A deception-attack detection and suppression mechanism is incorporated within impulsive control events, enhancing network security and maintaining synchronization accuracy under malicious data manipulation.

•   The vehicle dynamics are modeled as second-order nonlinear systems capturing both position and velocity behaviors, accurately reflecting realistic platoon interactions.

•   A delay-compensated control strategy is introduced to handle time-varying communication delays, preserving coordinated motion despite asynchronous updates.

•   The stability and boundedness of the proposed framework are rigorously established using Lyapunov theory, impulsive differential systems, and Markovian switching models.

•   The overall framework operates using only local information, making it scalable and implementable in large-scale vehicular communication networks.

The following sections make up the rest of this paper:

1.   Section 1 provides background information and research motivation, and the primary contributions of the study.

2.   Section 2 provides a brief overview of graph theory basics before moving on to analyze impulsive systems under finite- and fixed-time stability conditions with their required assumptions and definitions, and problem formulation.

3.   Section 3 contains the main analytical results, which include leaderless and leader-following control strategies for finite-time and fixed-time bounded consensus in networked vehicle systems, supported by four principal theorems.

4.   Section 4 presents simulation results which confirm the effectiveness of the proposed control methods.

5.   Section 5 of the paper presents essential findings together with recommendations for additional research.

2  Problem Statement

2.1 Graph Theory

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.

•   The associated Laplacian matrix ℒ is positive semidefinite, having exactly one eigenvalue equal to zero.

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.

2.2 Analysis of Impulsive Systems under Finite-Time and Fixed-Time Stability

Vehicle platooning consists of M vehicles that demonstrate nonlinear impulsive behavior. The dynamics of each vehicle are characterized by the following equations

{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.(1)

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

∑m=1ℓψmk≥(∑m=1ℓψk)k,∑m=1ℓψml≥ℓ1−l(∑m=1ℓψm)l(2)

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)

•   For 0<λ<1

𝒯=χ1(1−c)ln⁡λln⁡(1−λ1−cln⁡λγχ1)+χ1(1−b)ln⁡λln⁡(1+ln⁡λψχ2λ2(1−c)−ln⁡λ)

Such that all the constants fulfill

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.

2.3 Problem Formulation

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.

{x˙0(t)=v0(t)v˙0(t)=u0(t)(3)

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

{x˙m(t)=vm(t)v˙m(t)=um(t)m∈V(4)

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,

limt→𝒯(xm(t),vm(t))=dmρ,m∈V(5)

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).

limt→𝒯||(xm(t),vm(t))−dm(x0(t),v0(t))||=0,m∈V(6)

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.

3  Main Results

This section examines control mechanisms for both leaderless and leader-driven vehicle platooning under finite-time and fixed-time boundary consensus protocols. The mechanisms operate within an impulsive control framework to achieve coordination under specified convergence criteria.

3.1 Leaderless Finite-Time Bipartite Consensus (FNTBC) and Fixed-Time Bipartite Consensus (FXTBC) in Networked Vehicles

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.

3.1.1 Leaderless Finite-Time Bipartite Consensus (FNTBC)

The finite time bipartite consensus (FNTBC) criterion from Eq. (5) for the vehicle platooning dynamics modeled in Eq. (4) requires a strategically designed impulsive control scheme that uses discontinuous impulses and nonlinear consensus dynamics to achieve finite-time boundary convergence. The impulsive control protocol is extended to handle both communication delays and malicious deception attacks for more realistic vehicle platooning dynamics. The control input for the mth vehicle is modified,

um(t−τ)={e0∑n=1Mamnsignθ⁡(xn−sign⁡(amn)xm)+r0∑n=1Mamnsignθ⁡(vn−sign⁡(amn)vm)−[∑p=1+∞α(t−tp)Q(t)∑n=1M|amn|(xm−sign⁡(amn)xn)+∑p=1+∞β(t−tp)R(t)∑n=1M|amn|(vm−sign⁡(amn)vn)]+Γ(tp)Φ(xi(tp))+Ξ(tp)ϕ(vi(tp))(7)

where,

•   xn, vn denote the position and velocity of agent n, respectively.

•   amn is the (m,n)-th entry of the adjacency matrix representing the communication graph.

•   signθ⁡(z) denotes the continuous sign function with fractional exponent θ∈(0,1), defined as signθ⁡(z)=|z|θ⋅sign⁡(z).

•   e0≥0 and r0≥0 are protocol gains for the position and velocity components.

•   Q(t) and R(t) are time-varying impulsive gain functions applied at discrete times.

•   α(t) and β(t) are Dirac delta functions with the property:

α(t)=β(t)={+∞,t=00,t≠0,∫−∞+∞α(t)dt=∫−∞+∞β(t)dt=1,

tp denotes the impulsive update times.

•   Γ(tp)Φ(xi(tp))+Ξ(tp)ϕ(vi(tp)) represents a deception attack acting on the position and velocity of agent i at time tp. The terms Γ(tp) and Ξ(tp) model the attack intensities, while Φ(⋅) and ϕ(⋅) describe the injected malicious signal components.

The uniformly spaced time sequence {tp}0∞ represents the potential occurrence of deception attacks. The system operates under 0=t0<t1<⋯<tp<⋯ conditions, and the time intervals between impulses follow tp−tp−1=q, rules with q>0 being a predetermined constant.

The sequences Γ(tp),Ξ(tp) follows a Bernoulli distribution because the probability of Γ(tp),Ξ(tp) taking the value 1 equals prob{Γ(tp),Ξ(tp)=1}=ϑ, which leads to prob{Γ(tp),Ξ(tp)=0}=1−ϑ. The function Φ(xi(tp)),ϕ(vi(tp))∈Rlx,v represents the deception attack input.

The impulsive structure enables vehicles to interact minimally yet efficiently. The proposed control protocol (7) modifies both continuous and instantaneous behavior of each vehicle to achieve finite-time convergence even with sporadic corrections.

The dynamic model for vehicle m under this scheme evolves as,

(x˙m(t),v˙m(t))=e0∑n=1Mamnsignθ⁡(xn−sign⁡(amn)xm)+r0∑n=1Mamnsignθ⁡(vn−sign⁡(amn)vm)−∑p=1+∞α(t−tp)Q(t)∑n=1M|amn|(xm−sign⁡(amn)xn)−∑p=1+∞β(t−tp)R(t)∑n=1M|amn|(vm−sign⁡(amn)vn)+Γ(tp)Φ(xi(tp))+Ξ(tp)ϕ(vi(tp))(8)

Lemma 7. The system described by Eq. (4) with the control protocol defined in (7) operates as a vehicle platooning. The weighted average state of the system, denoted by (x^(t),v^(t))=1M∑m=1Mdm(xm(t),vm(t)), In other words, (x^˙(t),v^˙(t))=0 remains constant over time. Implying that this aggregated state is invariant throughout the system’s evolution.

Remark 3. The control mechanism operates conventionally at times other than impulses. The vehicles experience major state changes during impulse moments. Dirac delta function activation causes this phenomenon, which results in an immediate and powerful state adjustment. The impulsive effects play a major role in speeding up the system’s convergence process.

{H(t)=[(x1,v1)𝒯(t),(x2,v2)𝒯(t),…,(xM,vM)𝒯(t)]𝒯,ze(H(t))=[ze1𝒯(x(t),v(t)),ze2𝒯(x(t),v(t)),…,zeM𝒯(x(t),v(t))]𝒯,zem(x(t),v(t))=e0∑n=1Mamnsignθ⁡(xn−sign⁡(amn)xm)+r0∑n=1Mamnsignθ⁡(vn−sign⁡(amn)vm),∑n=1M|amn|(xm−sign⁡(amn)xn)+∑n=1M|amn|(vm−sign⁡(amn)vn)=∑n=1Mℓmn(xn,vn).

The original system (4) can be compactly expressed as

H˙(t)={ℱ(H(t)),t≠tpH(t−)+Υ(H(t−),tp),t=tp

H˙(t)={[0IM−e0ℒθx−r0ℒθv]⊗Il⋅H(t),t≠tp[I2M−𝒟(tp)]⊗Il⋅H(tp−)+A(tp),t=tp(9)

where

{H(t)=[x(t)⊤v(t)⊤]⊤∈R2Ml,is the augmented state vector,ℒθx and ℒθvarenonlinearLaplacian-likematricesforpositionandvelocityderivedusingthesignθ⁡rule,𝒟(tp)=[00Q(tp)ℒR(tp)ℒ],representstheimpulsivefeedbackappliedateventtimetp,A(tp)=[0Γ(tp)Φ(x(tp))+Ξ(tp)ϕ(v(tp))],capturesthedeceptionattacksintroducedduringimpulseinstants.

Theorem 1. Consider a system of M vehicles described by second-order dynamics as in Eq. (4) following the impulsive control strategy, to reach finite-time bipartite consensus under deception attacks, which is defined by Eq. (7). The parameters should be chosen to meet the following conditions: e0>0, 0<θ<1, and (xm(t),vm(t))∈Rl. The impulses gain functions Q(t) and R(t) must satisfies

Q(t)≤2μmin(ℒ)μmax2(ℒ)R(t)≤2νmin(ℒ)νmax2(ℒ)

The system reaches leaderless finite time bipartite consensus (FNTBC) in finite time when the network’s interaction topology, represented as an undirected signed graph G, is structurally balanced and connected.

The settling time 𝒯 has an upper bound that can be written as

𝒯≤𝒱(x(0),v(0))1−we(1−w)r(1−w)

while w=θ+12,e=e0⋅2θ−12⋅μmin(ℒ′)θ+12,r=r0⋅2η−12⋅νmin(ℒ′)η+12.

The Laplacian matrix is denoted by (ℒ′) and is obtained from the absolute weight matrix 𝒮′=[|amn|θ+12].

Proof: Define the transformed state as xm(t)=dmxm(t)−x(t), vm(t)=dmvm(t)−v(t), where xm(t) and vm(t) represent the average position and velocity terms, then according to lemma (7), it follows that x˙m(t)=dmx˙m(t),v˙m(t)=dmv˙m(t). Let us define the Lyapunov function candidate:

𝒱(t)=12H𝒯(t)H(t),

where H(t)=[x(t)v(t)]∈R2Ml is the stacked state vector.

The impulsive dynamics are given by Eq. (9)

H˙(t)={[0IM−e0ℒθx−r0ℒθv]⊗Il⋅H(t),t≠tp[I2M−𝒟(tp)]⊗Il⋅H(tp−)+A(tp),t=tp

We compute the derivative of 𝒱(t) for t≠tp:

𝒱˙(t)=H𝒯(t)H˙(t).

Using the continuous dynamics, we get,

𝒱˙(t)=H𝒯(t)([0IM−e0ℒθx−r0ℒθv]⊗Il)H(t)=x𝒯(t)v(t)−e0v𝒯(t)ℒθxx(t)−r0v𝒯(t)ℒθvv(t).

Now, we use the identity for antisymmetric nonlinear functions κ(a,b)=−κ(b,a) on an undirected graph with symmetric weights amn=anm:

∑(m,n)∈E|amn|υmκ(υn,υm)=−12∑(m,n)∈E|amn|(υn−υm)κ(υn,υm),

where we choose υ=x+v and κ(a,b)=signθ⁡(a−b). Then:

𝒱˙(t)=−e02∑(m,n)∈E|amn| |(xn+vn)−(xm+vm)|θ+12=−e02∑(m,n)∈E(|amn|2θ+1((xn+vn)−(xm+vm))2)θ+12≤−e02(∑(m,n)∈E|amn|2θ+1((xn+vn)−(xm+vm))2)θ+12=−e02[2H𝒯(t)ℒ′H(t)]θ+12,

where ℒ′ is the Laplacian matrix corresponding to the weighted adjacency matrix 𝒮′=[|amn|2θ+1].

Since ℒ′ is positive semi-definite and H(t)𝒯ℒ′H(t)≥μmin(ℒ′)‖H(t)‖2=2μmin(ℒ′)𝒱(t).

The transformation of ℒ′=DℒD represents the gauge-transformed Laplacian, which we defined in Lemma 3. The transformation described in Lemma 3 keeps vehicle states bounded, which ensures all position and velocity constraints remain valid throughout the consensus process.

We get:

𝒱˙(t)≤−e02θμmin(ℒ′)θ+12𝒱(t)θ+12.

At impulse times t=tp, the state is updated as:

H(tp+)=[I2M−𝒟(tp)]⊗Il⋅H(tp−)+A(tp),

where

A(tp)=[0Γ(tp)Φ(x(tp))+Ξ(tp)ϕ(v(tp))]

represents the deception attack introduced during impulsive instants.

The Lyapunov function at tp+ becomes,

𝒱(tp+)=12H𝒯(tp+)H(tp+)=12H𝒯(tp−)[(I2M−𝒟(tp))2⊗Il]H(tp−)+attack-related terms.

Provided the gain matrices satisfy.

(IM−Q(tp)R(tp)ℒ)2≤IM,

and the bounds,

Q(t)≤2μmin(ℒ)μmax2(ℒ),R(t)≤2νmin(ℒ)νmax2(ℒ),

we ensure that,

𝒱(tp+)≤𝒱(tp−),

even in the presence of bounded deception attacks.

Combining the continuous-time decay and impulsive updates, we obtain,

𝒱˙(t)≤−e𝒱(t)w,where w=θ+12,e=e02θ−12μmin(ℒ′)θ+12.

This guarantees fixed-time convergence. The settling time 𝒯 satisfies,

𝒯≤𝒱(x(0),v(0))1−we(1−w),

thus, the impulsive protocol ensures fixed-time non-singular terminal bipartite consensus despite the presence of bounded deception attacks. ◻

Remark 4. The Lyapunov framework used in this study combines continuous stability analysis between impulsive events with discrete jump analysis at tp. The described structure correctly demonstrates the operational behavior of practical systems that use sampled data.

Remark 5. The main distinction between FNTBC and FXTBC lies in their dependence on initial conditions. In FNTBC, the convergence time varies according to the initial state differences, whereas in FXTBC, the convergence time is fixed and predefined, regardless of the initial conditions. This fixed-time property offers stronger robustness and predictability for practical systems such as autonomous vehicle coordination, where the control objective must be achieved within a guaranteed time bound. Consequently, FXTBC enhances reliability under uncertain or rapidly changing initial states.

3.1.2 Fixed-Time Control for Leaderless Vehicle Platooning

The fixed time bipartite consensus (FXTBC) objective described in Eq. (5) for the vehicle platooning defined in Eq. (4) requires a new impulsive control protocol, which is presented below to achieve its goal.

um(t−τ)=e1∑n=1Mamnsignθ(xn−sign(amn)xm)+r1∑n=1Mamnsignθ(vn−sign(amn)vm)+e2∑n=1Mamnsignε(xn−sign(amn)xm)+r2∑n=1Mamnsignε(vn−sign(amn)vm)−[∑p=1+∞α(t−tp)Q(t)∑n=1M|amn|(xm−sign(amn)xn)+∑p=1+∞β(t−tp)R(t)∑n=1M|amn|(vm−sign(amn)vn)]+Γ(tp)Φ(xi(tp))+Ξ(tp)ϕ(vi(tp))(10)

The control parameters are defined as e1,e2≥0, 0<θ<1, and ε>1, which will be properly chosen in the following steps. All other involved terms maintain their original definitions.

Lemma 8. The control strategy defined by protocol (10) applied to the vehicle platooning modeled by Eq. (4) results in a constant average state vector throughout time (x^(t),v^(t)) in the form of position and velocity.

Proof: The argument validating this result aligns closely with the reasoning presented in Lemma 5 of [51]. The analytical steps are nearly identical to those in the previous result, so the detailed proof is not reproduced here for brevity. ◻

Remark 6. The main difference between the control protocol in (7) and the one proposed in (10) is the presence of the term e2∑n=1Mamnsignε(xn−sign(amn)xm), r2∑n=1Mamnsignε(vn−sign(amn)vm). This extra term is important for the fixed-time bounded consensus. It is worth noting that the finite-time control approach presented in (7) is a particular case of (10) when e2=0,r2=0. The system Eq. (4) is rewritten into control protocol (10) similar to Eq. (9).

{H˙(t)=ze′(H(t)),t≠tpH(tp+)=[(IM−Q(tp)R(tp)ℒ)⊗Il]H(tp−)+A(tp),t=tp(11)

where,

•   H(t)∈R2Ml is the stacked state vector of all agents, defined as:

H(t)=[x1𝒯(t), v1𝒯(t), …, xM𝒯(t), vM𝒯(t)]𝒯

ze′(H(t))∈R2Mlis the vector field derived from the fixed-time consensus protocol (see Eq. (10)), applied component-wise to each agent’s position and velocity.

•   Q(tp)∈RM×M is the event-triggered selection matrix, indicating which agents trigger updates at the impulsive instant tp.

•   R(tp)∈RM×M is the diagonal impulse strength matrix, determining the impact of neighbors’ information at tp.

•   ℒ∈RM×M is the Laplacian matrix representing the communication topology.

•   Il∈Rl×l is the identity matrix corresponding to the local state dimension l (typically l=2 or 3 depending on whether velocity and/or acceleration are included).

The impulsive deception attack input injected at the impulsive instants tp is given by,

A(tp)=[0Γ(tp)Φ(x(tp))+Ξ(tp)ϕ(v(tp))]∈R2Ml

•   Γ(tp), Ξ(tp)∈RMl×Ml are time-varying gain matrices of the deception attack.

•   The nonlinear functions are defined elementwise as:

Φ(x(tp))=sign(x(tp))⊙|x(tp)|q,ϕ(v(tp))=sign(v(tp))⊙|v(tp)|r

where q,r∈(0,1] are the attack shaping exponents, and ⊙ denotes the Hadamard (element-wise) product.

Theorem 2. Consider the dynamical system described by Eq. (4) receives impulsive control through Eq. (10) while maintaining the conditions of Assumption (1) The system contains e1,e2>0, 0<θ<1, ε>1 constants and a time-dependent function R(t) which fulfills the following condition

Q(t)≤2μmin(ℒ)μmax2(ℒ)R(t)≤2νmin(ℒ)νmax2(ℒ)

The impulsive interval is assumed to satisfy further conditions χ1≤Υ=tp+1−tp≤χ2. Under these conditions, the leaderless finite-time bounded consensus is guaranteed. The upper limit of convergence time 𝒯 depends on the value of a parameter λ which is an upper estimate of λp defined by λp=1+R2(tp)Q2(tp)μmax(ℒ)2−R(tp)Q(tp)μmin(ℒ).

Particularly

•   When 0<λ<1

𝒯≤χ1(1−c)ln⁡λln⁡(1−λ1−cln⁡λγχ1)+χ1(1−b)ln⁡λln⁡(1+ln⁡λψχ2λ2(1−c)−ln⁡λ)

•   When λ=1

𝒯≤1ψ(1−b)+1γ(c−1)

The auxiliary parameters are defined through the following structure b=θ+12,c=ε+12,ψ=e1⋅2θ⋅μmin(ℒ1′)θ+12,γ=e2⋅2ε⋅M1−ε2⋅μmin(ℒ2′)ε+12. The matrices ℒ1′ and ℒ2′ represent Laplacian matrices which are derived from the weighted adjacency matrices 𝒮1′, 𝒮2′, where 𝒮1′=[|amn|θ+12],𝒮2′=[|amn|ε+12].

Proof: Define the transformed state as

xm(t)=dmxm(t)−x(t),vm(t)=dmvm(t)−v(t),

where xm(t) and vm(t) represent the average position and velocity terms. Then, according to Lemma 7, it follows that

x˙m(t)=dmx˙m(t),v˙m(t)=dmv˙m(t).

Let us define the Lyapunov function candidate:

𝒱(t)=12H𝒯(t)H(t),

where

H(t)=[x(t)v(t)]∈R2Ml

is the stacked state vector.

The impulsive dynamics are given by Eq. (11),

H˙(t)={[0IM−e0ℒθx−r0ℒθv]⊗Il⋅H(t),t≠tp[I2M−𝒟(tp)]⊗Il⋅H(tp−)+A(tp),t=tp

We compute the derivative of 𝒱(t) for t≠tp:

𝒱˙(t)=H𝒯(t)H˙(t).

Using the continuous dynamics, we get,

𝒱˙(t)=H𝒯(t)([0IM−e0ℒθx−r0ℒθv]⊗Il)H(t)=x𝒯(t)v(t)−e0v𝒯(t)ℒθxx(t)−r0v𝒯(t)ℒθvv(t).