---
title: "Composite-learning finite-time nonsingular terminal sliding mode control for intelligent vehicle p"
id: "plos-one-22-composite-learning-based-finite-time-nonsingular-terminal-sliding-mode-path"
canonical_url: "https://medichelpline.com/clinical-feed/plos-one-22-composite-learning-based-finite-time-nonsingular-terminal-sliding-mode-path"
content_type: "clinical_feed_article"
specialty: "General"
source_name: "PLOS ONE (Medicine)"
source_url: "https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0357264"
published_at: "2026-09-01T14:00:00.000Z"
evidence_level: "Journal Feed"
license: "CC-BY-NC-4.0 / Informational Use"
---
# Composite-learning finite-time nonsingular terminal sliding mode control for intelligent vehicle p
## Provenance & Clinical Metadata
- **Canonical URL:** https://medichelpline.com/clinical-feed/plos-one-22-composite-learning-based-finite-time-nonsingular-terminal-sliding-mode-path
- **Specialty:** [General](https://medichelpline.com/clinical-feed/general.md)
- **Primary Source:** PLOS ONE (Medicine)
- **Source URL:** [Original Journal Publication](https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0357264)
- **Published At:** 2026-09-01T14:00:00.000Z
- **Evidence Rating:** Journal Feed
## Executive GIST (TL;DR)
- The paper proposes a control framework — referred to as **CL-FNTSMC** — to address path tracking for intelligent vehicles under parametric uncertainties, unmodeled dynamics, and external disturbances. - A unified path tracking error model is established that lumps parametric variations, unmodeled dynamics, and external disturbances into a single uncertainty term; a nonlinear disturbance observer is used to estimate and compensate these lumped disturbances in real time. - A **composite learning** law is developed that combines tracking errors and prediction errors from a serial–parallel estimation model; this enables online parameter estimation under the weaker **interval excitation** condition rather than the restrictive persistent excitation (PE) requirement. - The controller employs a nonsingular terminal sliding surface built from a continuously differentiable nonlinear function to guarantee finite-time convergence while avoiding singularity issues present in some terminal sliding schemes. - Lyapunov-based analysis in the paper establishes practical finite-time stability of the closed-loop system with the proposed controller and learning law. - Comparative simulations are reported showing the proposed CL-FNTSMC delivers superior tracking accuracy, faster convergence, and improved disturbance rejection versus conventional NTSMC, adaptive fast NTSMC, and PID controllers under aggressive disturbances and significant parameter uncertainties. - The work emphasizes improved transient response, high steady-state precision, and robustness achieved by integrating disturbance observation, composite learning, and finite-time nonsingular terminal sliding mode control.
## Clinical Analysis & Structured Key Points
Composite learning based finite time nonsingular terminal sliding mode path tracking control for intelligent vehicles | PLOS One Browse Subject Areas ? Click through the PLOS taxonomy to find articles in your field. For more information about PLOS Subject Areas, click here . Article Authors Metrics Comments Media Coverage Peer Review Reader Comments Figures Figures Abstract This paper addresses the path tracking control problem for intelligent vehicles subject to parametric uncertainties, unmodeled dynamics, and external disturbances. A composite learning-based finite-time nonsingular terminal sliding mode control (CL-FNTSMC) strategy is proposed. Unlike conventional adaptive sliding mode controllers that rely solely on tracking errors for parameter update—and thus require the restrictive persistent excitation condition-the proposed scheme incorporates a serial–parallel estimation model to construct prediction errors, which together with tracking errors drive a composite learning law. This mechanism ensures accurate online estimation of unknown parameters under the significantly weaker interval excitation condition. A nonsingular terminal sliding surface, constructed with a continuously differentiable nonlinear function, guarantees finite time convergence while inherently avoiding singularity. Furthermore, a nonlinear disturbance observer is integrated to estimate and compensate for lumped disturbances in real time, substantially enhancing robustness. Rigorous Lyapunov-based analysis establishes the practical finite time stability of the closed loop system. Comprehensive comparative simulations under aggressive disturbances and significant parametric uncertainties demonstrate that the proposed CL-FNTSMC achieves superior tracking accuracy, faster convergence, and markedly improved disturbance rejection compared with conventional NTSMC, adaptive fast NTSMC, and PID controllers. The results confirm that the proposed framework offers an excellent balance of fast transient response, high steady-state precision, and strong robustness. Citation: Wu C (2026) Composite learning based finite time nonsingular terminal sliding mode path tracking control for intelligent vehicles. PLoS One 21(9): e0357264. https://doi.org/10.1371/journal.pone.0357264 Editor: Xingyu Wang, Jiangsu University of Science and Technology, CHINA Received: June 30, 2026; Accepted: August 13, 2026; Published: September 1, 2026 Copyright: © 2026 Changgui Wu. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: The author declares that all the settings of simulation parameters supporting the discovery of this work are included in the article. Funding: This work was supported by the Jiangsu Provincial Vocational Education Teaching Reform Research Project under Grant No. ZDZC584. Competing interests: The authors have declared that no competing interests exist. 1. Introduction Intelligent vehicles have emerged as a transformative technology within modern transportation systems, offering substantial potential to enhance traffic safety, operational efficiency, and environmental sustainability [ 1 – 3 ]. As a cornerstone enabling technology for autonomous driving, path tracking control (PTC) is instrumental in governing a vehicle’s ability to precisely adhere to a prescribed reference trajectory while maintaining dynamic stability [ 4 ]. The accuracy of PTC critically influences driving safety and ride comfort, especially in complex and rapidly changing road environments. Despite considerable research efforts, achieving high-performance path tracking for intelligent vehicles remains a persistent challenge, primarily owing to the intrinsic complexity of vehicle dynamics—which are strongly coupled, inherently nonlinear, and subject to a broad spectrum of uncertainties [ 5 ]. The challenges in PTC primarily arise from multiple sources of uncertainties, including parametric perturbations, unmodeled dynamics, and external disturbances [ 6 – 8 ]. These uncertainties can significantly degrade tracking accuracy and even compromise vehicle stability, necessitating the development of robust control strategies capable of handling such complexities while maintaining satisfactory transient and steady-state performance [ 9 ]. Various control methodologies have been proposed to address the path tracking problem of intelligent vehicles [ 10 – 12 ]. Model predictive control (MPC) has gained substantial popularity due to its ability to handle constraints and predict future system behavior [ 13 ]. In [ 14 ], Sun et al. tackle the issue of insufficient adaptability of conventional MPC to varying driving scenarios in autonomous vehicle path tracking. In [ 15 ], Nascimento et al. address the challenges of MPC design for nonholonomic mobile robot trajectory tracking by providing a comprehensive survey that covers modeling, constraint handling, stability analysis, and computational considerations. However, MPC typically relies on accurate system models and may suffer from substantial computational burden, limiting its applicability in real-time implementations [ 16 ]. Linear quadratic regulator (LQR) and PID based methods are simple [ 17 – 19 ]. In [ 20 ], Sun et al. address the challenge of coordinating lateral and longitudinal control for intelligent vehicles in the presence of varying road curvatures and future path changes by proposing a LQR based control method that incorporates preview information to optimize tracking performance. Jiang and Palaoag address the trajectory tracking problem for autonomous systems by proposing a hybrid algorithm that integrates MPC with LQR, leveraging MPC’s constraint handling capability and LQR’s optimality for enhanced tracking performance [ 21 ]. However, they often lack robustness to significant uncertainties and disturbances. Recent years have witnessed a surge of interest in sliding mode control (SMC) for vehicle path tracking, as evidenced by several comprehensive survey papers [ 22 , 23 ]. These reviews systematically categorize various SMC methodologies—including terminal sliding mode, adaptive sliding mode, and disturbance-observer-based sliding mode—and identify their respective merits and limitations when applied to ground vehicles. Notably, [ 22 ] highlights that while SMC offers inherent robustness against matched uncertainties, its application to vehicle path tracking is often hindered by chattering, singularity, and the stringent persistent excitation (PE) condition required for adaptive parameter convergence. These challenges motivate the development of integrated solutions that combine nonlinear control techniques with advanced learning mechanisms. SMC has emerged as a powerful robust control technique for nonlinear systems with uncertainties, owing to its insensitivity to matched disturbances and its capability to enforce desired system behavior through discontinuous control actions [ 24 – 26 ]. In [ 27 ], Ansari et al. address the load frequency control problem in power systems subject to parameter uncertainties and external disturbances by proposing a robust backstepping SMC that integrates the systematic design of backstepping with the robustness of sliding mode control to ensure frequency stability. In [ 5 ], Zhang et al. address the trajectory tracking control problem of permanent magnet synchronous motors subject to parameter uncertainties, load disturbances, and sliding mode chattering by proposing a non-singular fast terminal SMC scheme with disturbance compensation. In [ 28 ], Zhang et al. tackle the challenges of finite-time convergence and disturbance sensitivity in robotic manipulator control. They propose a fixed-time SMC integrated with a disturbance observer, which guarantees global fast tracking convergence with a convergent time upper bound independent of initial states, while effectively estimating and compensating for lumped disturbances. In [ 29 ], Tan et al. address the spacecraft reorientation control problem under multiple attitude constraints and actuator resource limitations by proposing an event-triggered sliding mode control scheme, which achieves robust attitude tracking while significantly reducing actuator updates and ensuring the avoidance of Zeno behavior. More recently, several studies have specifically explored SMC for autonomous vehicle applications. In [ 30 ], Wang et al. proposes an adaptive sliding mode approach for vehicle lateral control under uncertain tire-road friction conditions. In [ 31 ], Li et al. develops a finite-time terminal sliding mode controller for path tracking of autonomous vehicles, demonstrating improved transient performance. However, these methods either require stringent PE conditions for parameter convergence or lack a systematic mechanism for disturbance estimation and compensation, leaving room for further improvement. Conventional adaptive controllers rely on the persistent excitation (PE) condition for parameter convergence, which is seldom satisfied in real driving scenarios. Consequently, tracking-error-driven controllers such as AFNTSMC fail to ensure parameter convergence [ 32 ], degrading model fidelity and tracking accuracy. The composite learning mechanism overcomes this by achieving reliable estimation under interval excitation—a weaker and more practical condition than PE. Motivated by the aforementioned discussions, this paper proposes a composite learning-based finite-time nonsingular terminal sliding mode control (CL-FNTSMC) strategy for intelligent vehicle path tracking. The main contributions are threefold: A unified path tracking error model is established that explicitly accounts for lumped uncertainties encompassing parametric variations, unmodeled dynamics, and external disturbances. A nonlinear disturbance observer is incorporated to provide real-time estimation and feedforward compensation, effectively reducing the conservativeness of robust control design. A composite learning law is developed that simultaneously exploits tracking errors and prediction errors from a serial-parallel estimation model, enabling accurate online parameter estimation without the restrictive PE condition. Theoretical analysis establishes parameter convergence under interval excitation, which is strictly weaker than PE and practically more relevant. Non-singular terminal sliding surface is designed to have a nonlinear function of continuously differentiable, which ensures finite time convergence while inherently avoiding singularity. The proposed framework cooperatively integrates compound learning, finite time sliding mode and interference observation, and is supported by strict Lyapunov-based stability analysis, which ensures that tracking error, parameter estimation and interference estimation converge simultaneously. A large number of comparative simulations in different scenarios confirm that the proposed method is obviously superior to PID, traditional NTSMC and the most advanced AFNTSMC. The remainder of this paper is organized as follows. Section 2 presents the problem formulation and the vehicle path tracking model. Section 3 details the design of the proposed CL-FNTSMC strategy, including the composite learning laws, the nonsingular terminal sliding surface, the disturbance observer, and the stability analysis. Section 4 presents simulation results and comparative studies. Finally, Section 5 concludes the paper with discussions and future research directions. 2. Problem description and vehicle modeling This section establishes the mathematical foundation for path tracking control of intelligent vehicles. Both kinematic and dynamic characteristics are incorporated to capture the essential lateral behavior. The resulting model is cast into a compact state-space representation that explicitly accounts for parametric uncertainties, unmodeled dynamics, and external disturbances, thereby facilitating the subsequent control design. 2.1. Vehicle kinematic model Consider an intelligent vehicle moving in the horizontal plane. The vehicle pose is described by the global position coordinates ( X , Y ) and the yaw angle , measured with respect to a fixed inertial frame . The generalized coordinate vector is defined as . Under the nonholonomic constraint of no lateral slip at the rear axle, the kinematic equations follow the standard bicycle model: (1a) (1b) (1c) where v and denote the longitudinal velocity at the vehicle center of gravity (CG) and the yaw rate, respectively. At the kinematic level, both v and serve as control inputs. Let the desired reference path be parameterized by arc length s , with the desired pose denoted as . The path curvature relates to the desired yaw rate via . Assuming the path is tracked at the desired longitudinal speed , we have . To facilitate error-based control design, the tracking errors are defined in the vehicle body-fixed frame. The rotation matrix from the inertial frame to the body frame is given by (2) The error vector in the body frame is then defined as (3) where , , and denote the longitudinal error, lateral (cross-track) error, and yaw angle error, respectively. Differentiating (3) with respect to time and substituting (1) yields the kinematic error dynamics: (4a) (4b) (4c) In typical driving scenarios, the longitudinal error is regulated by throttle/brake control, whereas lateral control is predominantly achieved through steering. In this study, we assume that a lower-level longitudinal controller maintains a constant forward speed, i.e., , and we focus exclusively on the lateral path tracking problem. Under this assumption, and invoking the small-angle approximation and , which is valid for normal driving conditions—the lateral error dynamics simplify to (5) It is important to note, however, that the kinematic model neglects inertial effects and tire–road interaction forces, which become significant at higher speeds. Consequently, a dynamic model is required for robust controller design in practical driving scenarios. Remark 1: The kinematic model (5) provides a geometric description of the vehicle–path relationship under the assumptions of small slip angles and constant longitudinal speed. While it serves as a useful baseline for understanding the fundamental tracking geometry, it is inadequate for high-speed maneuvers where lateral acceleration and yaw dynamics dominate. The subsequent subsection addresses these limitations by introducing a dynamic vehicle model. 2.2. Vehicle dynamic model To incorporate the essential inertial and tire-force characteristics, we adopt the widely used single-track (bicycle) dynamic model. The vehicle is assumed to have front-wheel steering only, and the state variables are defined as the longitudinal velocity , lateral velocity , and yaw rate in the body frame. The equations of motion are: (6a) (6b) (6c) where m is the vehicle mass, is the yaw moment of inertia, and and are the distances from the centre of gravity (CG) to the front and rear axles, respectively; is the front steering angle, which serves as the control input; and are the longitudinal tire forces at the front and rear; and and are the lateral tire forces. For moderate lateral accelerations, the lateral tire forces can be approximated by a linear relationship with the corresponding slip angles: (7) where and are the cornering stiffnesses (positive constants), and and are the front and rear slip angles, respectively, given by: (8) To facilitate control design, we assume that the longitudinal speed is maintained at a nearly constant value by a separate controller. Under this assumption, the longitudinal dynamics can be neglected. Furthermore, we ignore the longitudinal tire forces ( ), as the steering control primarily affects the lateral motion. Substituting the linear tire force expressions into Eq. (6) and linearising about a straight-line motion with small steering angles, we obtain the following lateral-yaw dynamics: (9a) (9b) In the above derivation, we have used the small-angle approximations and , together with the slip-angle expressions and . The term v appearing in the coefficient of in Eq. (9a) arises from the centripetal acceleration term after linearisation. Remark 2: The linear tire model is valid only for small slip angles. Under higher lateral accelerations, nonlinear tire characteristics become significant and should be considered. Nevertheless, for the purpose of control design, the linear model provides a reasonable approximation. The unmodeled nonlinearities and parameter variations are treated as lumped disturbances, which will be compensated by the proposed adaptive neural network controller in the following section. 2.3. Path tracking error dynamics To facilitate the controller design for path tracking, we express the vehicle dynamics in terms of the path tracking errors defined in Eq. (3) . Under the assumptions of small yaw error and constant longitudinal speed v , the lateral error and yaw error are related to the vehicle states and by the following kinematic relations: (10) where denotes the desired yaw rate corresponding to the reference path curvature. Taking the time derivatives of the above equations and substituting the dynamic equations from Eq. (9), we obtain the second-order error dynamics: (11a) (11b) where the lumped disturbances and collectively account for the following: parametric uncertainties in vehicle parameters ( m , , , , , ), unmodeled dynamics (e.g., nonlinear tire characteristics and suspension effects), external disturbances (e.g., wind gusts, road bank angle, and measurement noise), and residual coupling errors arising from the linearisation process. All lumped disturbance terms are assumed to be bounded and slowly time-varying relative to the system dynamics. Assumption 1: The lumped disturbances and , along with their time derivatives, are bounded. That is, there exist positive constants and such that and for all , where . Assumption 2: The reference path is smooth, and its derivatives up to second order are bounded. In particular, and are bounded. 2.4. State-space representation Define the state vector as: (12) Then, the error dynamics in Eqs. (11a)–(11b) can be written in the following compact state-space form: (13) where is the control input (front steering angle), and the system matrix and input matrix are given by: (14) (15) The lumped disturbance vector is defined as: (16) For control design, we use a nominal model based on the nominal parameter values (denoted with the subscript 0), while the actual plant parameters may deviate from these nominal values. We define the parametric uncertainties as and , and incorporate these into the lumped disturbance term. Consequently, the state equation can be rewritten as: (17) where contains all uncertainties. Remark 3: The model in Eq. (13) is linear time-invariant except for the disturbance term. The controller design in the following section uses only the nominal matrices and . The effects of uncertainties and disturbances are handled by the composite learning law and the nonlinear disturbance observer. 2.5. Problem statement Given the state-space model in Eq. (13) with unknown but bounded lumped disturbances , and given a smooth reference path with bounded curvature, the control objective is to design a steering control law u such that the following requirements are satisfied: The tracking error vector converges to a small neighbourhood of the origin in finite time. That is, for a prescribed small constant , there exists a finite settling time such that for all ; The closed-loop system is robust against parametric uncertainties and external disturbances; The control input u is smooth and free of chattering, while the singularity problem inherent in conventional terminal sliding mode control is effectively avoided. Remark 4: The
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