2.1 Detailed longitudinal dynamics modeling

The longitudinal motion of an EV is governed by the balance between the traction force and the cumulative resistance forces [44], as shown in Fig 1. According to Newton’s second law, the fundamental dynamics can be expressed as [45]:

(1)

where m is the total vehicle mass. v is the longitudinal velocity. F_traction is the net traction force at the drive wheels (positive for propulsion, negative for regenerative or friction braking). F_resistive is the total resistance force.

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Fig 1. The schematic diagram of longitudinal dynamics model of electric vehicle.

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The total resistive force F_resistive typically comprises the following components:

1. a) Aerodynamic Drag Force (F_aero)(2)where is the air density. C_d is the aerodynamic drag coefficient (dimensionless). A_f is the vehicle frontal area (m²). v_wind is the effective headwind velocity. For controller design, the wind effect is often treated as part of a bounded disturbance or ignored for nominal modeling. The approximation is used, recognizing that k_a may be uncertain in practice.
2. b) Rolling Resistance Force (F_roll)(3)where g is the gravitational acceleration. C_r is the rolling resistance coefficient (dimensionless). is the road inclination angle or grade (rad). The coefficient C_r varies with tire pressure, road surface, and vehicle speed, introducing parametric uncertainty.
3. c) Grading Force (F_grade):(4)
   The road grade is a time-varying exogenous input that is often not directly measured.
4. d) Lumped Disturbance Force (d_f(t)):

This term encapsulates unmodeled dynamics (e.g., internal drivetrain losses, bearing friction) and external disturbances (e.g., varying wind gusts, minor road irregularities). It is assumed to be bounded.

Remark 1 (Parameter Variability): The vehicle mass m can change significantly with passenger and cargo load. Coefficients , and the effective grade are also subject to uncertainty and variation during operation. This motivates the use of an adaptive control strategy that does not rely on precise prior knowledge of these parameters.

The traction force F_traction is generated by the electric motor(s) and transmitted through the powertrain. For control design, the motor torque is typically the primary command. Assuming a single dominant motor or a properly torque-vectored multi-motor setup, the relationship can be simplified to:

(5)

where T_m is the motor torque command (Nm). This serves as the control input u. is the combined efficiency of the transmission and final drive (dimensionless, for driving). i_g is the total gear ratio. r_w is the effective wheel radius.

Assumption 1 (Actuator): The motor torque T_m can be commanded and realized by the motor controller without significant dynamics or delay relative to the vehicle’s longitudinal dynamics. The torque is bounded by physical constraints: . For the stability analysis, we initially consider an unbounded control input, but saturation must be considered in practical implementation.

2.2 Control-oriented state-space formulation

To cast the problem into a canonical form suitable for nonlinear control synthesis, define the state variables as [46]:

(6)

(7)

The output to be controlled is typically the velocity x₂ or position x₁. For this study, we focus on velocity tracking, which is fundamental for cruise control and a prerequisite for accurate position tracking. Therefore, define y = x₂.

The complete dynamics can now be written as:

(8)

where, is a lumped, normalized disturbance term.

This system is of the form:

(9)

where:

• State vector: .
• Control input: u = T_m.
• Unknown nonlinear functions:
• is the bounded lumped disturbance.

Assumption 2 (System Properties):

1. a) The sign of the control gain function g(x) is known, positive, and nonzero over the operational domain . This is physically justified as the powertrain efficiency and gear ratio are positive, i.e., .
2. b) The function g(x) is bounded such that , where g_min and g_max are known positive constants. This follows from the physical bounds on mass, efficiency, and gear ratio.
3. c) The functions f(x), g(x), and the disturbance are smooth enough, and is bounded such that , where is a finite positive constant, but its exact value is unknown.
4. d) The desired velocity trajectory y_d(t) and its first two time derivatives, and , are continuous, bounded, and known.

Remark 2 (Model Uncertainty): The functions f(x) and g(x) contain the uncertain parameters . Therefore, they are considered unknown for the purpose of controller design. The control strategy must be robust to these uncertainties and the disturbance .

2.3 Control objective

The primary control objective is to synthesize a direct adaptive control law for the electric vehicle longitudinal system that ensures robust and precise velocity tracking of a smooth desired trajectory y_d(t), despite the presence of unknown nonlinearities f(x), g(x), and bounded disturbances . Specifically, the controller must guarantee that the tracking error and all closedloop signals are Uniformly Ultimately Bounded (UUB). Furthermore, the sliding variable s, which encapsulates the tracking error dynamics, should converge to a small residual set around zero, with its ultimate bound adjustable via controller design parameters. This objective is to be achieved without explicit knowledge of the system’s nonlinear functions or the disturbance bounds, instead leveraging an RBF neural network to directly approximate the ideal stabilizing control law online. The overall structure of the proposed controller is illustrated in Fig 2.

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Fig 2. The proposed architecture diagram for direct robust adaptive tracking control for longitudinal motion of electric vehicles.

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3.1 Preliminaries and sliding surface design

To facilitate controller design and achieve robust tracking, we first define a sliding surface that encapsulates the tracking error dynamics, as shown in Fig 3. Define the tracking error vector where:

(10)

(11)

where (desired velocity) and (implied desired position).

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Fig 3. Robust adaptive tracking control for EV longitudinal motion.

https://doi.org/10.1371/journal.pone.0346228.g003

The sliding variable s is defined as:

(12)

where c > 0 is a design parameter that determines the convergence rate of the tracking error. The choice of c ensures that the polynomial p + c = 0 is Hurwitz, implying that if , then and .

Differentiating s with respect to time yields:

(13)

where is the desired acceleration. For notational convenience, we define:

(14)

which can be interpreted as a lumped nonlinear term combining system dynamics and reference signals. Thus:

(15)

Remark 3: The function is unknown due to the unknown f(x) and potentially unknown reference accelerations in practical scenarios. However, it is assumed to be smooth and bounded for bounded arguments.

Assumption 3 (Compact Operating Domain): The system states, tracking errors, and reference signals evolve within a known compact set defined as:

(16)

where (with a small design parameter) and . The bounds are known constants determined by physical constraints and performance requirements.

3.2 Ideal feedback control law synthesis

Under ideal conditions of perfect model knowledge and absence of disturbances , a stabilizing control law can be derived using Lyapunov direct method. Consider the candidate Lyapunov function:

(17)

The time derivative of V_s along the system trajectories is:

(18)

where represents the time variation of the control gain.

To achieve exponential convergence, we desire for some , which translates to:

(19)

Solving for the control input yields the ideal feedback control law:

(20)

Theorem 1 (Ideal Controller Stability): Consider system under Assumptions 1–3 with . If the control law (Eq.20) is applied with , then:

1. a) The closed-loop system is globally asymptotically stable.
2. b) All tracking errors converge to zero: and .
3. c) The sliding variable converges exponentially:

Proof: Substituting (20) into (18) with gives:

(21)

Using the Lyapunov function (Eq.17):

(22)

since g(x) > 0. This establishes exponential stability of s, which implies exponential convergence of tracking errors.

3.3 Neural network approximation of ideal controller

The ideal controller (Eq.20) is not implementable due to unknown functions f(x), g(x), and . However, we can represent it as a continuous nonlinear mapping:

(23)

where the augmented input vector is defined as:

(24)

(25)

(26)

Remark 4: The inclusion of both s and is strategic. When is small, these variables operate at different scales, enhancing the approximation capability of the neural network by providing multi-resolution information about the tracking error dynamics.

A RBFNN is a type of feedforward neural network that employs radially symmetric basis functions (typically Gaussian) as activation units in its hidden layer to perform local approximation [41,49–51], as shown in Fig 4. Its primary role is to serve as a universal approximator capable of online learning, enabling it to directly approximate unknown continuous nonlinear functions—such as the ideal feedback control law—without requiring an explicit mathematical model of the system dynamics. The choice of RBFNN as the function approximator in this study is justified by a concise comparison with alternative techniques reported in the literature. Compared to Multi-Layer Perceptrons (MLPs), RBFNNs offer faster learning convergence due to their local approximation property and avoid the risk of local minima inherent in back-propagation training. Unlike fuzzy logic systems, which often require expert knowledge for rule base initialization, RBFNNs can be initialized using simple data-driven methods. Furthermore, the output of an RBFNN is linear in the adjustable weights, a property that greatly simplifies the derivation of adaptive laws and enables rigorous Lyapunov-based stability proofs.

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Fig 4. Mechanism diagram of radial basis function neural network.

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We employ a RBFNN to approximate this unknown mapping over the compact set . The RBFNN architecture is characterized by:

(27)

where:

• is the weight matrix (N is the number of neurons)
• is the vector of Gaussian activation functions:

(28)

with being the center and the width of the i-th neuron.

Universal Approximation Property: For any continuous function on the compact set and arbitrary approximation accuracy , there exist optimal weights and sufficiently large N such that:

(29)

where is the bounded reconstruction error.

Assumption 4 (NN Approximation): The ideal weights are bounded by an unknown constant W_max > 0 such that . The basis functions H(z) are bounded such that for all .

3.4 Direct adaptive robust controller design

Let be the estimate of the ideal weights . The proposed direct adaptive controller is:

(30)

The adaptive law for updating the weight estimates is designed with -modification for robustness:

(31)

where is a positive definite adaptation gain matrix. is the -modification parameter that prevents parameter drift and ensures boundedness of weights.

Remark 5: The -modification term introduces a leakage effect that bounds the weight estimates even in the presence of non-vanishing disturbances and approximation errors. This is crucial for practical implementation where persistent excitation cannot be guaranteed.

3.5 Closed-loop stability analysis

Define the weight estimation error as . Substituting the control law (Eq.30) into the sliding dynamics (Eq.21):

(32)

Adding and subtracting and using (Eq.29):

(33)

Substituting the expression for from (Eq.20)-(Eq.23):

(34)

Simplifying:

(35)

Consider the composite Lyapunov function candidate:

(36)

The time derivative along the system trajectories is:

(37)

Substituting (Eq.35) and noting that (since is constant):

(38)

Using Young’s inequality for the cross terms:

(39)

(40)

Combining these bounds:

(41)

Using the bounds on g(x) from Assumption :

(42)

where .

From the Lyapunov function (Eq.36), we have:

(43)

where denotes the minimum eigenvalue of .

Define:

(44)

(45)

where is the maximum eigenvalue of .

Then we can establish:

(46)

Theorem 2 (Main Stability Result): Consider the EV longitudinal dynamics (Eq.9) under Assumptions 2–4, with the direct adaptive controller (Eq.30) and adaptive law (Eq.31). Then:

1. 1) All signals in the closed-loop system are UUB [52].
2. 2) The sliding variable s converges exponentially to the compact set:(47)
3. 3) The tracking errors satisfy:(48)(49)
   Proof: Solving the differential inequality (Eq.46) yields:(50)

This proves UUB of V(t), which implies UUB of s and . The bounds on |s| follow from . The tracking error bounds follow from the definition and the triangle inequality.

3.6 Stability under parameter variations and tuning inaccuracies

While the Lyapunov-based analysis in the Closed-Loop Stability Analysis subsection establishes the UUB of all closed-loop signals under ideal conditions, it is essential to examine the system’s behavior when physical parameters vary or controller parameters are inaccurately tuned [53]. This subsection discusses the potential risks of instability and clarifies how the proposed control architecture mitigates these effects.

4.1 Simulation environment and controller configuration

The parameters for the proposed and comparative controllers are detailed below.

1. 1) Proposed adaptive RBF neural network Controller The key parameters of the designed RBF neural network-based adaptive controller are listed in Table 1.
2. 2) Benchmark Controller Configuration

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Table 1. Configuration of the proposed RBF neural network controller.

https://doi.org/10.1371/journal.pone.0346228.t001

To ensure a fair and transparent comparison, the benchmark controllers were tuned as follows:

• PID Controller: The proportional, integral, and derivative gains were optimized using the Ziegler-Nichols tuning method followed by fine-tuning to minimize the Integral Time Absolute Error (ITAE) criterion under the nominal sinusoidal reference trajectory. The final gains were set to , and K_d = 5.
• Sliding Mode Controller (SMC): The sliding surface was designed as with . The switching gain was selected based on the estimated upper bound of lumped disturbances, and a boundary layer of thickness was introduced to mitigate chattering. These parameters were tuned to achieve a balance between tracking accuracy and control smoothness under the same nominal conditions as the PID controller. Furthermore, to test robustness, an external disturbance was injected into the system dynamics during simulation.

Both benchmark controllers were evaluated under identical simulation conditions and performance metrics as the proposed controller to ensure a consistent basis for comparison.

4.2 Analysis of key design parameter influence

To provide a deeper understanding of the proposed controller, we systematically investigate the influence of its key design parameters: the sliding surface coefficient c, the convergence rate gain , the adaptation gain matrix , the -modification coefficient , and the RBFNN width b. The analysis covers robustness, sensitivity, and adaptability.

Sliding surface coefficient c: A larger c forces faster error convergence along the sliding surface but may amplify noise and require larger control effort. It also affects the bandwidth of the error dynamics.

Convergence gain : According to the stability analysis (Eq. 46), directly influences the exponential convergence rate of the sliding variable s. A larger yields faster transient response but may increase the ultimate bound C (as seen in Eq. 42), potentially leading to larger steady-state tracking errors and higher control chattering.

Adaptation gain : Higher adaptation gains accelerate the online weight update, leading to faster learning and better tracking during transients. However, excessively large can introduce high-frequency oscillations in the control signal and may cause instability if the system is noisy or if there are unmodeled dynamics.

-modification coefficient : This term prevents weight drift in the absence of persistent excitation. A small provides robustness against bounded disturbances without significantly affecting ideal adaptation, while a larger enhances robustness but may introduce bias in weight estimates, increasing the ultimate tracking error bound (see Eq. 42).

RBFNN width b: The width of the Gaussian basis functions determines the locality of the neural network’s response. A smaller b leads to more localized neurons, improving approximation accuracy in densely sampled regions but requiring more neurons to cover the input space. A larger b results in smoother function approximation but may reduce the network’s ability to capture fine details.

4.3 Simulation scenarios

To comprehensively evaluate the proposed controller, three distinct test scenarios are designed:

• Scenario A (Step Response): A step change in desired velocity from 0 to 20 m / s at t = 2 s is applied to assess transient performance, including rise time, and overshoot.
• Scenario B (Sinusoidal Tracking): A time-varying reference with varying frequency components is used to evaluate adaptability and tracking accuracy under dynamic conditions.
• Scenario C (Standard Driving Cycle): The Urban Dynamometer Driving Schedule (UDDS) is employed to validate performance under realistic stop-and-go driving conditions representative of urban environments.

Additionally, robustness tests are conducted under parametric variations: vehicle mass increased by 20% and road grade varying sinusoidally between and .

4.4 simulation results

The expanded simulation study yields several key insights as shown in Table 2. First, in Scenario A (step response), the proposed controller achieves the lowest overshoot (2.1%) compared to SMC (5.3%) and PID (8.7%), confirming its superior transient performance. Second, in Scenario B (multi-frequency tracking), the consistently low RMSE (0.08 m / s) demonstrates the RBFNN’s ability to adapt online to changing dynamics without retuning-a key advantage over fixed-gain PID and SMC. Third, under UDDS cycle conditions, the proposed controller maintains accurate tracking despite frequent acceleration/deceleration, while SMC exhibits chattering and PID shows noticeable lag. The robustness tests reveal that mass variation primarily affects PID performance (RMSE increases by 32%), whereas the proposed controller degrades gracefully (RMSE increase <10%), confirming the effectiveness of the adaptive mechanism. Finally, the convergence of the sliding variable s to a small residual set (observed in all scenarios) validates the UUB stability guarantee established in Theorem 2.

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Table 2. Quantitative performance comparison across scenarios.

https://doi.org/10.1371/journal.pone.0346228.t002

This simulation diagram is shown in Figs 5−11. This figure presents a direct comparison of the position tracking capabilities of three distinct controllers—SMC, PID control, and the proposed RBF neural network-based adaptive control—against a desired trajectory. The main plot illustrates the system’s positional response over a 10-second duration. The proposed controller demonstrates superior tracking accuracy, indicating excellent reference following. The PID controller shows noticeable deviation and oscillation, while the SMC exhibits a consistent but offset tracking error. The inset graph provides a magnified view between 1.5 and 2 seconds, crucial for evaluating transient performance and steady-state precision. Here, the proposed controller’s minimal error becomes even more apparent, confirming its enhanced ability to handle dynamic changes and settle accurately compared to the oscillatory and lagging responses of the SMC and PID controllers, respectively. Fig 6 provides a quantitative breakdown of tracking errors for three controllers at specific time points. The data lists the position errors of SMC, PID, and the proposed control method. A clear trend can be observed: the proposed controller consistently maintains the minimum absolute error in almost all sampling time instances. For example, at t = 2.0s, the error of the proposed controller is −0.070 rad, while SMC is −0.090 rad and PID is −0.100 rad. This consistent numerical advantage highlights the effectiveness bias of adaptive RBF neural networks in minimizing continuous tracking. Fig 7 depicts the control input signals required by each controller over time. The control signal generated by the proposed controller is smoother and typically lower in amplitude than the jitter signal characteristics of SMC. The input of the PID controller is between the two. The absence of high-frequency jitter in the input of the proposed controller is particularly important. Although SMC is known for its robustness, it is often affected by jitter, which can lead to excessive actuator wear and energy loss. The proposed method effectively alleviates this problem by using a smooth neural network approximation while maintaining robust performance.

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Fig 5. Comparative analysis of position tracking performance over time.

https://doi.org/10.1371/journal.pone.0346228.g005

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Fig 6. Tabulated tracking error metrics at discrete time intervals.

https://doi.org/10.1371/journal.pone.0346228.g006

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Fig 7. Control input effort comparison across controllers.

https://doi.org/10.1371/journal.pone.0346228.g007

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Fig 8. Velocity tracking performance evaluation.

https://doi.org/10.1371/journal.pone.0346228.g008

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Fig 9. Three-dimensional visualization of controller performance layers.

https://doi.org/10.1371/journal.pone.0346228.g009

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Fig 10. Quantitative performance metrics summary.

https://doi.org/10.1371/journal.pone.0346228.g010

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Fig 11. Phase portrait analysis of system dynamics.

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Fig 8 shows the speed tracking performance, comparing the actual speed achieved by each controller with the required speed curve. The proposed controller exhibits excellent accuracy in tracking the expected speed of changes and closely matches the reference value throughout the simulation process. In contrast, both SMC and PID controllers exhibit significant deviations, especially during dynamic changes in required speed. Fig 9 provides a comprehensive, multi perspective 3D view of tracking performance. This visualization allows for real-time and comprehensive comparison of how the trajectory of each controller deviates from the expected path over time. The required trajectory is displayed as a clear and smooth surface. The trajectory layer of the proposed controller is almost aligned with the required surface, thus confirming its accuracy. The trajectories of SMC and PID are clearly separated, indicating consistent tracking errors. Fig 10 lists the key performance indicators for rigorous comparison. The proposed controller performs well in all major metrics: it achieves the lowest RMSE and lowest MAE of 0.0098 rad, which means the highest tracking accuracy. The RMSE values of 0.0594 and 0.0504 rad objectively confirm that the proposed adaptive controller provides the optimal balance between accuracy and controlled actuation force. Fig 11 shows the relationship between the system position and velocity of different controllers. The trajectory illustrates the dynamic path of the system under each control law. The phase trajectory of the proposed controller seems to be closest to the ideal or expected dynamic path, and the SMC and PID trajectories show deviations in this state space, indicating different dynamic responses.

The comprehensive simulation results of the provided charts clearly demonstrate the superior performance of the proposed direct RBF neural network adaptive controller in electric vehicle tracking control. Visually and quantitatively, it outperforms traditional PID and robust sliding mode control (SMC) strategies. The key advantage of the proposed method lies in its excellent tracking accuracy, which can be demonstrated by the almost perfect overlap with the desired trajectory in the position map and the lowest RMSE/MAE value. In addition, the controller designed in this article exhibits excellent dynamic response, accurately tracking the change speed curve of other controllers when shaking, and maintaining stable and ideal state space dynamics according to the phase diagram. Although the performance indicator table shows the relevant control work, the main achievement is a significant improvement in accuracy. In summary, the proposed adaptive control scheme successfully utilizes the approximation ability of RBF neural networks to achieve robust, accurate, and smooth tracking control in the absence of clear system nonlinear dynamics, making it a highly promising solution for advanced electric vehicle motion control applications.