In this study, I investigate a cooperative braking control method for an electric vehicle that combines an anti-lock braking system with an air reverse thrust system. The central problem is that ground braking in an electric vehicle is nonlinear, time-varying, and strongly coupled, and the achievable deceleration is ultimately limited by tire-road adhesion. I therefore construct a control architecture in which fuzzy proportional-integral-derivative logic provides rapid coarse adjustment of wheel slip, while a particle-swarm-optimized back-propagation neural network supplies fine gain compensation. The electric vehicle is represented by a five-degree-of-freedom longitudinal model, and the air reverse thrust system is represented by a one-dimensional steady isentropic flow model. The simulation results show that the proposed electric vehicle braking strategy reduces braking time and braking distance while holding the slip ratio near the desired value of 0.2.
The motivation for this work is straightforward. A conventional anti-lock braking system can exploit the available tire-road friction, but it cannot create friction beyond the road adhesion limit. For an electric vehicle operating at high speed or on a low-adhesion surface, the braking distance may remain excessively long even when the anti-lock logic is well tuned. The air reverse thrust system introduces an additional longitudinal force that does not depend on road adhesion. By ejecting high-speed air forward during braking, the electric vehicle obtains a reaction force that can supplement the ground braking force. The challenge is not merely to add this force, but to coordinate it with the anti-lock braking system so that wheel slip remains stable and the additional thrust does not excite oscillation.
I organize the remainder of the article as follows. First, I define the electric vehicle braking model and the five-degree-of-freedom longitudinal dynamics. Second, I formulate the air reverse thrust model. Third, I develop the fuzzy proportional-integral-derivative controller and the particle-swarm-optimized back-propagation compensation network. Fourth, I describe the simulation setup and compare the proposed electric vehicle braking controller with conventional strategies. Finally, I discuss the physical meaning of the results, the limitations of the present simulation, and the direction of future work.
1. Electric Vehicle Braking Problem and Modeling Assumptions
For an electric vehicle in emergency braking, the longitudinal motion is governed by the balance among front and rear ground braking forces, aerodynamic drag, and the additional air reverse thrust. The wheel rotation is governed by the difference between the tire longitudinal force moment and the brake torque. Because the normal loads shift from the rear axle to the front axle during deceleration, the tire forces are coupled with the longitudinal acceleration. This coupling is the main reason why a fixed-gain controller cannot maintain optimal slip under all conditions. I therefore retain the load transfer, the wheel rotational dynamics, and the thrust force in the same model, while neglecting lateral motion and roll motion to keep the electric vehicle model tractable.
The assumptions used in the electric vehicle model are as follows. I neglect lateral motion and yaw motion. I treat the left and right wheels on each axle as an equivalent wheel. I neglect pitch inertia in the control-oriented model, but I retain the pitch-related load transfer through the longitudinal acceleration. I assume that the road is flat and that the tire longitudinal force is a function of slip ratio and normal load. I also assume that the air reverse thrust acts along the longitudinal axis of the electric vehicle and that its direction is opposite to the vehicle velocity during braking.
The main symbols used in the electric vehicle braking model are summarized in Table 1.
| Symbol | Meaning | Unit |
|---|---|---|
| m | Electric vehicle mass | kg |
| Vx | Longitudinal velocity | m/s |
| Fxf | Front axle longitudinal ground force | N |
| Fxr | Rear axle longitudinal ground force | N |
| Fw | Aerodynamic drag | N |
| Fj | Air reverse thrust | N |
| Fzf | Front axle normal load | N |
| Fzr | Rear axle normal load | N |
| Lf | Distance from center of gravity to front axle | m |
| Lr | Distance from center of gravity to rear axle | m |
| L | Wheelbase, L = Lf + Lr | m |
| hcg | Center-of-gravity height | m |
| R | Effective wheel radius | m |
| Ji | Wheel equivalent rotational inertia | kg m2 |
| Ti | Brake torque on wheel i | N m |
| λi | Slip ratio of wheel i | dimensionless |
| λ0 | Desired slip ratio | dimensionless |
The longitudinal motion equation of the electric vehicle is written as
$$
m\dot V_x=-F_{xf}-F_{xr}-F_w-F_j .
$$
The aerodynamic drag is
$$
F_w=\frac{1}{2}\rho C_D A V_x^2 ,
$$
where ρ is air density, CD is the drag coefficient, and A is the frontal area. The normal loads under braking are
$$
F_{zf}=\frac{mgL_r-m\dot V_x h_{cg}}{L},
$$
$$
F_{zr}=\frac{mgL_f+m\dot V_x h_{cg}}{L}.
$$
During braking, the longitudinal acceleration is negative, so the front normal load increases and the rear normal load decreases. This load transfer changes the available tire force and therefore changes the slip dynamics. For the electric vehicle braking controller, this effect cannot be ignored because the front and rear wheels do not reach the same slip at the same time.
The wheel rotational dynamics are
$$
J_i\dot\omega_i=F_{si}R-T_i,\qquad i=1,2,3,4,
$$
where ωi is the angular speed of wheel i and Fsi is the longitudinal tire force. The slip ratio is defined as
$$
\lambda_i=1-\frac{\omega_i R}{V_x}.
$$
The desired slip ratio is denoted by λ0. The slip error used by the controller is
$$
e_i=\lambda_i-\lambda_0 .
$$
For the electric vehicle anti-lock braking problem, the target is to keep λi near λ0 = 0.2 on a high-adhesion road. If the slip ratio is too low, the tire does not use the available adhesion fully. If the slip ratio is too high, the tire enters the unstable region and the electric vehicle may lose directional stability. The controller must therefore regulate the brake torque and the air reverse thrust together.
The instantaneous braking deceleration of the electric vehicle can be expressed as
$$
a_x=\frac{F_{xf}+F_{xr}+F_w+F_j}{m}.
$$
Integrating the deceleration gives the velocity history:
$$
V_x(t)=V_x(0)+\int_0^t a_x(\tau)\,d\tau .
$$
Under a pure rolling assumption, the reference angular speed is
$$
\omega_0=\frac{V_x}{R}.
$$
The wheel angular acceleration can be integrated from
$$
\dot\omega_i=\frac{F_{si}R-T_i}{J_i}.
$$
These equations form the core of the electric vehicle braking model. The controller acts on Ti and Fj, while the slip ratio and the wheel speed provide the feedback signals.
The electric vehicle parameters used in the simulation are listed in Table 2.
| Parameter | Value | Unit |
|---|---|---|
| Electric vehicle mass m | 1620 | kg |
| Distance from center of gravity to front axle Lf | 1.500 | m |
| Distance from center of gravity to rear axle Lr | 2.718 | m |
| Wheelbase L | 4.218 | m |
| Effective wheel radius R | 0.300 | m |
| Center-of-gravity height hcg | 0.550 | m |
| Wheel rotational inertia Ji | 0.75 | kg m2 |
| Air density ρ | 1.225 | kg/m3 |
| Drag coefficient CD | 0.30 | dimensionless |
| Frontal area A | 2.2 | m2 |
I use the above electric vehicle parameters consistently in all simulations. The initial braking speed is 70 km/h, and the road is a high-adhesion surface. The desired slip ratio is 0.2. The controller sampling period is set small enough to resolve the wheel dynamics and the thrust response.
2. Air Reverse Thrust Model for the Electric Vehicle
The air reverse thrust system produces a longitudinal force by ejecting compressed air forward. The thrust is computed from the momentum change and the pressure difference at the nozzle exit:
$$
F_j=\dot m v_e+A_e(p_e-p_a),
$$
where \dot m is the mass flow, ve is the exit velocity, pe is the exit static pressure, pa is the ambient pressure, and Ae is the exit area. For an ideal fully expanded nozzle, pe = pa, and the thrust reduces to
$$
F_j=\dot m v_e .
$$
Using the continuity equation,
$$
\dot m=\rho_e A_e v_e .
$$
For one-dimensional steady isentropic flow, the exit velocity is
$$
v_e=\sqrt{\frac{2\gamma R_g T_0}{\gamma-1}\left[1-\left(\frac{p_e}{p_0}\right)^{\frac{\gamma-1}{\gamma}}\right]},
$$
where γ is the specific heat ratio, Rg is the gas constant, T0 is the stagnation temperature, and p0 is the stagnation pressure. The mass flow can be written as
$$
\dot m=A_e p_0\sqrt{\frac{2\gamma}{(\gamma-1)R_g T_0}\left[\left(\frac{p_e}{p_0}\right)^{\frac{2}{\gamma}}-\left(\frac{p_e}{p_0}\right)^{\frac{\gamma+1}{\gamma}}\right]}.
$$
For a choked nozzle, the mass flow reaches a maximum and can be approximated by
$$
\dot m_{\max}=A_e p_0\sqrt{\frac{\gamma}{R_g T_0}}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}.
$$
In the electric vehicle braking simulation, the air reverse thrust is commanded as a bounded force. I use
$$
F_j^{cmd}=\operatorname{sat}\left(F_{j,\max},\ K_j u_j\right),
$$
where uj is the thrust control command and Kj is the thrust gain. The saturation prevents the electric vehicle model from receiving an unrealistically large thrust during the transient. The thrust is activated when the brake pedal is urgently applied and the slip ratio exceeds a threshold. It remains active for a finite period and then decays to zero.
The air reverse thrust parameters used in the simulation are summarized in Table 3.
| Parameter | Value | Unit |
|---|---|---|
| Initial thrust magnitude | 5.5 | kN |
| Thrust duration | 2.2 | s |
| Nozzle exit area Ae | 0.018 | m2 |
| Stagnation pressure p0 | 0.65 | MPa |
| Stagnation temperature T0 | 320 | K |
| Specific heat ratio γ | 1.40 | dimensionless |
| Gas constant Rg | 287 | J/(kg K) |
The air drag and the air reverse thrust differ greatly in magnitude. At the beginning of braking, the aerodynamic drag is approximately 145 N, while the air reverse thrust is approximately 5.5 kN. The thrust is therefore about 30 to 40 times the peak aerodynamic drag. The drag decays with the square of the electric vehicle speed and approaches zero near the end of braking. The thrust remains high for about 2.2 s and then falls rapidly. This behavior means that the air reverse thrust dominates the additional longitudinal force during the early and middle stages of electric vehicle braking, while the aerodynamic drag provides a small but continuous passive retardation.
The ratio between the thrust and the drag can be expressed as
$$
R_{jw}=\frac{F_j}{F_w}=\frac{2F_j}{\rho C_D A V_x^2}.
$$
Because Vx decreases during braking, the ratio increases with time even if the thrust is constant. In the electric vehicle simulation, the thrust is not constant; it is reduced near the end of the injection period. The controller therefore must coordinate the thrust reduction with the ground braking force so that the slip ratio does not overshoot when the thrust disappears.
3. Fuzzy PID Control for the Electric Vehicle Braking System
I use a fuzzy proportional-integral-derivative controller as the main controller. The continuous-time control law is
$$
u(t)=K_p(t)e(t)+K_i(t)\int_0^t e(\tau)\,d\tau+K_d(t)\frac{de(t)}{dt}.
$$
The gains are updated around baseline values:
$$
\begin{aligned}
K_p(t)&=K_{p0}+\alpha_1\Delta K_p^{Fuzzy}+\Delta K_p^{BP},\\
K_i(t)&=K_{i0}+\alpha_2\Delta K_i^{Fuzzy}+\Delta K_i^{BP},\\
K_d(t)&=K_{d0}+\alpha_3\Delta K_d^{Fuzzy}+\Delta K_d^{BP}.
\end{aligned}
$$
Here, Kp0, Ki0, and Kd0 are the baseline gains. The terms ΔKpFuzzy, ΔKiFuzzy, and ΔKdFuzzy are the fuzzy gain corrections. The terms ΔKpBP, ΔKiBP, and ΔKdBP are the neural-network compensation terms. The weights α1, α2, and α3 set the relative authority of the fuzzy correction and the neural compensation. This structure allows the fuzzy controller to respond quickly to large slip errors, while the neural network removes steady-state error and suppresses oscillation.
The fuzzy inputs are the slip error e and the error change rate ec. The input universe is [-6, 6]. The output universe for ΔKp, ΔKi, and ΔKd is [0, 6]. The fuzzy subsets are NB, NM, NS, ZE, PS, PM, and PB, which denote negative big, negative medium, negative small, zero, positive small, positive medium, and positive big. The output labels for the gain corrections are Z, S, M, and B, which denote zero, small, medium, and big. The membership functions are triangular and trapezoidal, and they overlap to provide smooth transitions. The rule base is summarized in Table 4.
| e \ ec | NB | NM | NS | ZE | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | B/Z/S | B/Z/M | B/Z/B | B/Z/B | Z/Z/M | Z/S/M | Z/Z/S |
| NM | B/S/M | M/S/M | M/Z/B | S/Z/B | Z/S/M | Z/S/M | Z/S/M |
| NS | M/M/Z | M/B/S | S/M/S | Z/B/S | S/M/S | S/M/S | Z/S/M |
| ZE | M/B/Z | S/B/Z | Z/B/S | Z/B/Z | S/B/S | S/B/S | M/B/Z |
| PS | S/M/Z | S/S/M | Z/Z/S | S/B/S | Z/S/M | Z/Z/M | S/M/S |
| PM | S/M/S | S/Z/B | B/Z/B | Z/B/S | S/Z/B | B/Z/B | M/M/S |
| PB | M/S/M | B/Z/M | M/M/S | M/S/M | B/Z/M | M/M/Z | B/S/M |
Each table entry contains three labels in the order Kp, Ki, and Kd. For example, B/Z/S means that the proportional correction is big, the integral correction is zero, and the derivative correction is small. The rule base is designed so that a large slip error produces a strong proportional action and a moderate derivative action. When the error is near zero, the integral action is allowed to remove the residual slip error without exciting oscillation. When the error change rate is large, the derivative action is increased to damp the transient.
The fuzzy inference uses the minimum operator for the antecedent and the product operator for implication. The defuzzification uses the centroid method:
$$
\Delta K_m^{Fuzzy}=\frac{\sum_{j=1}^{N}\mu_j\Delta K_{m,j}}{\sum_{j=1}^{N}\mu_j},\qquad m=p,i,d,
$$
where μj is the firing strength of rule j and ΔKm,j is the corresponding output center. The resulting gain corrections are passed to the proportional-integral-derivative law. This fuzzy controller is fast and interpretable, but its rule base is fixed. For an electric vehicle braking system with large variations in road adhesion and thrust level, the fuzzy controller alone may leave a small steady-state slip error or produce a low-amplitude oscillation. This is why I add the particle-swarm-optimized back-propagation network as a compensation controller.
4. PSO-BP Gain Compensation for the Electric Vehicle Controller
The compensation network receives the slip error, the error change rate, the electric vehicle speed, and the current thrust level. Its output is a three-dimensional gain correction vector. The network is trained offline with particle swarm optimization and then fine-tuned by back propagation. The particle swarm optimization searches for initial weights and biases that minimize a composite fitness function. The back-propagation step then adapts the network during the simulation.
The particle swarm update equations are
$$
v_{id}^{k+1}=w v_{id}^{k}+c_1 r_1\left(p_{id}-x_{id}^{k}\right)+c_2 r_2\left(p_{gd}-x_{id}^{k}\right),
$$
$$
x_{id}^{k+1}=x_{id}^{k}+v_{id}^{k+1},
$$
where xid is the position of particle i in dimension d, vid is the velocity, pid is the personal best position, pgd is the global best position, w is the inertia weight, c1 and c2 are acceleration coefficients, and r1 and r2 are random numbers in [0, 1]. The inertia weight is reduced linearly:
$$
w(k)=w_{\max}-\frac{k}{k_{\max}}\left(w_{\max}-w_{\min}\right).
$$
The fitness function used for the electric vehicle braking controller is
$$
J=w_1\int_0^{t_b}\left|e(t)\right|t\,dt+w_2\max\left|\lambda-\lambda_0\right|+w_3 s_b+w_4\left|t_b-t_{ref}\right|,
$$
where tb is the braking time, sb is the braking distance, and w1 to w4 are positive weights. The first term penalizes the integral of time-weighted absolute slip error. The second term penalizes the peak slip deviation. The third term penalizes the braking distance. The fourth term penalizes excessive deviation from the reference braking time. This fitness function balances tracking accuracy, stability, and braking efficiency for the electric vehicle.
The back-propagation network has an input layer, one hidden layer, and an output layer. The forward pass is
$$
h_j=f\left(\sum_{i=1}^{n}w_{ij}x_i+b_j\right),
$$
$$
y_m=g\left(\sum_{j=1}^{q}w_{jm}h_j+b_m\right),
$$
where xi is the input, hj is the hidden activation, ym is the output, wij and wjm are weights, bj and bm are biases, f is the hidden activation function, and g is the output activation function. The output vector is
$$
\mathbf{y}=\left[\Delta K_p^{BP},\ \Delta K_i^{BP},\ \Delta K_d^{BP}\right]^{T}.
$$
The network is trained by minimizing
$$
E=\frac{1}{2}\sum_{m=1}^{3}\left(K_m^{ref}-K_m\right)^2,
$$
where Kmref is the reference gain from the fuzzy controller or from the desired response. The weight update is
$$
\Delta w_{ij}(k)=-\eta\frac{\partial E}{\partial w_{ij}}+\mu\Delta w_{ij}(k-1),
$$
$$
w_{ij}(k+1)=w_{ij}(k)+\Delta w_{ij}(k),
$$
where η is the learning rate and μ is the momentum coefficient. The same update rule is applied to the output-layer weights. The back-propagation compensation is bounded to prevent excessive gain variation:
$$
\left|\Delta K_m^{BP}\right|\le \Delta K_{m,\max}^{BP},\qquad m=p,i,d.
$$
The final control law combines the fuzzy and neural corrections. The brake torque command for each wheel is
$$
T_i^{cmd}=\operatorname{sat}\left(T_{i,\max},\ T_{i,0}+K_T u(t)\right),
$$
where Ti0 is the baseline brake torque and KT is the torque gain. The air reverse thrust command is
$$
F_j^{cmd}=\operatorname{sat}\left(F_{j,\max},\ F_{j,0}+K_F u(t)\right).
$$
The saturation limits are set so that the electric vehicle model remains within the physical capability of the brake actuators and the air supply system. The key is that the thrust and the brake torque share the same slip error signal but have different authority. The brake torque has fast response and directly affects the wheel slip, while the air reverse thrust has a slower response and mainly affects the electric vehicle deceleration. This separation makes the cooperative control stable.
The particle swarm and back-propagation parameters used in the simulation are listed in Table 5.
| Parameter | Value |
|---|---|
| Number of particles | 40 |
| Maximum iterations | 120 |
| Inertia weight range | 0.9 to 0.4 |
| Acceleration coefficients c1, c2 | 1.5, 1.8 |
| Hidden layer neurons | 9 |
| Learning rate η | 0.035 |
| Momentum coefficient μ | 0.85 |
| BP correction limit | 0.25 |
| Fuzzy correction weights α1, α2, α3 | 0.55, 0.35, 0.30 |
5. Simulation Setup and Evaluation Metrics
I evaluate the proposed electric vehicle braking controller under high-adhesion road conditions. The initial speed is 70 km/h. The target slip ratio is 0.2. I compare four cases. The first case is the conventional anti-lock braking system. The second case is the anti-lock braking system with the air reverse thrust system. The third case is the fuzzy proportional-integral-derivative controller without neural compensation. The fourth case is the proposed particle-swarm-optimized back-propagation fuzzy proportional-integral-derivative controller. The comparison metrics are braking time, braking distance, slip ratio peak, slip ratio steady-state error, and oscillation amplitude.
| Simulation item | Setting |
|---|---|
| Initial electric vehicle speed | 70 km/h |
| Road condition | High adhesion |
| Desired slip ratio λ0 | 0.2 |
| Controller type | Fuzzy PID with PSO-BP compensation |
| Baseline controller | Conventional ABS and fuzzy PID |
| Thrust system | Air reverse thrust system with bounded force |
| Measured outputs | Vehicle speed, wheel speed, slip ratio, braking distance |
The braking time is the time from the brake command to zero electric vehicle speed. The braking distance is
$$
s_b=\int_0^{t_b}V_x(t)\,dt.
$$
The time reduction ratio and distance reduction ratio are
$$
R_t=\frac{t_{base}-t_{pro}}{t_{base}}\times 100\%,
$$
$$
R_s=\frac{s_{base}-s_{pro}}{s_{base}}\times 100\%.
$$
The slip ratio error is
$$
e_\lambda(t)=\lambda(t)-\lambda_0.
$$
I use the peak absolute slip error and the root-mean-square slip error as stability indicators:
$$
e_{\lambda,\max}=\max_{0\le t\le t_b}\left|e_\lambda(t)\right|,
$$
$$
e_{\lambda,rms}=\sqrt{\frac{1}{t_b}\int_0^{t_b}e_\lambda^2(t)\,dt}.
$$
These metrics capture both the transient overshoot and the steady-state fluctuation of the electric vehicle braking system.
6. Simulation Results and Analysis
6.1 Comparison of Fuzzy PID and PSO-BP Fuzzy PID
I first compare the conventional fuzzy proportional-integral-derivative controller with the proposed particle-swarm-optimized back-propagation fuzzy proportional-integral-derivative controller. The results are listed in Table 6.
| Controller | Braking time (s) | Braking distance (m) | Time reduction | Distance reduction |
|---|---|---|---|---|
| Fuzzy PID | 5.251 | 52.18 | 0 | 0 |
| PSO-BP Fuzzy PID | 4.555 | 45.62 | 13.3% | 12.6% |
The proposed electric vehicle controller reduces the braking time from 5.251 s to 4.555 s, which is a reduction of about 13.3%. The braking distance decreases from 52.18 m to 45.62 m, which is a reduction of about 12.6%. The improvement is not only numerical. The slip ratio response is also smoother. With the fuzzy controller alone, the slip ratio fluctuates more strongly after the initial transient. With the particle-swarm-optimized back-propagation compensation, the gain correction adapts to the slip error and the error change rate, so the slip ratio settles more quickly near the target value.
The improvement can be explained by the gain adaptation. The fuzzy controller provides a fast response when the slip error is large, but its fixed rule base cannot perfectly balance the proportional, integral, and derivative actions near the target. The neural compensation adds a small gain correction that depends on the operating point. When the electric vehicle speed is high and the thrust is active, the network increases the damping action to suppress overshoot. When the electric vehicle speed is low and the thrust is fading, the network increases the integral action to remove the residual slip error. This operating-point-dependent correction is the main reason for the shorter braking time and the smaller braking distance.
6.2 Comparison of Conventional ABS and ABS with Air Reverse Thrust
I then compare the conventional anti-lock braking system with the composite braking system that includes the air reverse thrust. The results are listed in Table 7.
| Braking system | Braking time (s) | Time reduction | Braking distance (m) | Distance reduction |
|---|---|---|---|---|
| Conventional ABS | 4.555 | 0 | 45.62 | 0 |
| ABS with air reverse thrust | 3.385 | 25.7% | 29.81 | 34.7% |
The composite electric vehicle braking system reduces the braking time from 4.555 s to 3.385 s, a reduction of 25.7%. The braking distance decreases from 45.62 m to 29.81 m, a reduction of 34.7%. The distance reduction is larger than the time reduction because the thrust is most effective in the high-speed stage. During this stage, the electric vehicle covers a large distance per unit time, so an additional deceleration force removes a large portion of the stopping distance. As the electric vehicle speed decreases, the aerodynamic drag becomes small, but the thrust has already done most of its work.
The physical interpretation is that the conventional anti-lock braking system relies on the tire-road friction. The friction force is limited by the normal load and the road adhesion coefficient. The air reverse thrust adds a force that is independent of the road adhesion. Therefore, the total braking force changes from a single-source friction braking force to a composite force that includes friction braking and active aerodynamic braking. This is the main advantage of the proposed electric vehicle braking concept.
6.3 Slip Ratio Regulation and Stability
The slip ratio is the key indicator of braking stability. I compare the slip ratio response of the conventional anti-lock braking system and the composite electric vehicle braking system. The quantitative results are listed in Table 8.
| Metric | Conventional ABS | ABS with air reverse thrust |
|---|---|---|
| Target slip ratio | 0.20 | 0.20 |
| Time to reach target region | about 0.5 s | about 0.5 s |
| Steady slip ratio | around 0.20 | around 0.20 |
| Peak slip ratio | about 0.30 | about 0.24 |
| Peak reduction | 0 | about 20% |
| Oscillation amplitude | larger in the later stage | smaller and more regular |
Both strategies bring the slip ratio into the effective control region within about 0.5 s. This shows that the composite controller does not weaken the fast pressure-establishing capability of the anti-lock braking system. In the steady stage, the conventional anti-lock braking system shows a growing fluctuation, and the slip ratio peak approaches 0.30. In contrast, the composite electric vehicle braking system keeps the slip ratio around 0.20, with a peak of about 0.24. The upper fluctuation limit is reduced by about 20%. This means that the air reverse thrust helps the electric vehicle maintain a more balanced use of tire adhesion. The thrust reduces the demand on the friction brake, so the brake torque does not need to be modulated as aggressively. As a result, the slip ratio oscillation is suppressed.
The slip ratio dynamics can be interpreted through the wheel equation. When the brake torque is large, the wheel angular speed decreases quickly and the slip ratio increases. When the thrust is added, the electric vehicle decelerates faster, which increases the denominator effect in the slip definition and changes the slip error. The controller then reduces the brake torque slightly and uses the thrust to provide part of the deceleration. This coordinated action prevents the wheel from entering the deep-slip region. The result is a more stable electric vehicle braking process.
6.4 Force Contribution and Time Response
The air reverse thrust and the aerodynamic drag have different magnitudes and different time histories. I summarize their contributions in Table 9.
| Force | Initial magnitude | Time behavior | Role in electric vehicle braking |
|---|---|---|---|
| Aerodynamic drag Fw | about 145 N | decays with Vx squared; near zero at about 3 s | passive retardation; small but continuous |
| Air reverse thrust Fj | about 5.5 kN | high for about 2.2 s; then drops rapidly | dominant additional braking force |
The thrust is about 30 to 40 times the peak aerodynamic drag. This ratio shows why the air reverse thrust can significantly shorten the braking distance of the electric vehicle. The aerodynamic drag is still included in the model because it is always present and因为它 affects the high-speed stage. However, the main braking enhancement comes from the thrust. The thrust duration is also important. If the thrust is cut off too early, the electric vehicle loses the additional deceleration in the middle stage. If the thrust is cut off too late, the slip ratio may be disturbed near the end of braking. The proposed controller manages this transition by reducing the brake torque command when the thrust decays, so the slip ratio remains near the target.
The total braking force can be written as
$$
F_{total}=F_{xf}+F_{xr}+F_w+F_j .
$$
The friction contribution and the thrust contribution are not simply additive in their effect on slip, because the thrust changes the electric vehicle deceleration and therefore the load transfer. Nevertheless, the total force equation explains the average deceleration trend. The average deceleration is
$$
a_{avg}=\frac{V_x(0)}{t_b}.
$$
For the conventional anti-lock braking case, the average deceleration is lower because the friction force is limited by adhesion. For the composite case, the average deceleration is higher because the thrust adds a road-independent force. This is consistent with the observed reduction in braking time.
6.5 Sensitivity and Robustness Discussion
I also examine the sensitivity of the proposed electric vehicle braking controller to the thrust level and the slip target. The controller retains a stable response when the thrust is reduced by 10% to 20%, because the neural compensation adjusts the gain correction and the fuzzy controller continues to regulate the slip error. When the target slip is shifted slightly around 0.2, the controller tracks the new target with a small transient. This robustness is important for an electric vehicle because the actual road adhesion and the actual thrust may differ from the nominal values.
| Variation | Expected effect | Controller response |
|---|---|---|
| Thrust reduced by 10% | Braking distance increases slightly | Fuzzy PID increases brake torque; PSO-BP compensates gain |
| Thrust reduced by 20% | Braking time increases moderately | Slip ratio remains near target; oscillation remains small |
| Target slip shifted to 0.18 | Small change in tire force | Slip error drives gain correction; stable tracking |
| Target slip shifted to 0.22 | Higher tire force but lower stability margin | Derivative gain increases to damp oscillation |
The sensitivity results are qualitative in this study, but they reveal the intended behavior of the electric vehicle braking controller. The particle-swarm-optimized back-propagation network does not replace the fuzzy controller. It acts as a trim controller. This division of labor is important because a pure neural controller may be difficult to interpret and may require extensive training data. A pure fuzzy controller may be too rigid. The combined structure retains the interpretability of fuzzy logic and the adaptability of neural compensation.
7. Discussion
The proposed electric vehicle braking strategy has three main advantages. First, the air reverse thrust system provides a road-independent braking force. This is particularly useful for an electric vehicle at high speed, where the braking distance is sensitive to the available adhesion. Second, the fuzzy proportional-integral-derivative controller provides fast slip regulation and keeps the wheel out of the unstable deep-slip region. Third, the particle-swarm-optimized back-propagation network supplies fine gain compensation, which reduces steady-state error and suppresses oscillation. The combination of these three elements is the reason for the improved braking time and braking distance.
The results also show a clear trade-off. The air reverse thrust increases the deceleration, but it also changes the slip dynamics. When the thrust is active, the electric vehicle slows down faster, and the wheel speed must follow the vehicle speed. If the brake torque is not adjusted, the slip ratio may overshoot. The cooperative controller solves this problem by using the same slip error to coordinate the brake torque and the thrust. The fuzzy controller responds quickly to the slip error, while the neural network adjusts the gain according to the operating point. This is a practical way to handle the nonlinear and time-varying nature of electric vehicle braking.
The present study has limitations. The tire model is simplified and linearized around the operating region. Road roughness and lateral wind are not included. The neural network is trained from offline simulation data, so it does not capture the full response delay of a real actuator. The air reverse thrust model is one-dimensional and steady, and it does not resolve the detailed flow field near the nozzle. These limitations mean that the numerical results should be interpreted as a control-oriented assessment rather than a final vehicle-level validation.
| Limitation | Effect on the electric vehicle braking result | Possible improvement |
|---|---|---|
| Simplified tire model | May overestimate the available friction and the stability margin | Use a nonlinear tire model with experimental identification |
| No road roughness | May underestimate wheel-speed fluctuation | Add stochastic road input and suspension coupling |
| No lateral wind | May neglect thrust misalignment and yaw disturbance | Add crosswind and nozzle angle dynamics |
| Offline neural training | May not cover all actuator delays | Use online learning or hardware-in-the-loop data |
| One-dimensional thrust model | May not capture nozzle losses and flow separation | Use computational fluid dynamics and experimental calibration |
Future work should proceed in two directions. The first direction is high-fidelity fluid simulation of the air reverse thrust system, including nozzle geometry, pressure ratio, and flow separation. The second direction is hardware-in-the-loop testing with actuator delay and energy consumption constraints. For an electric vehicle, the air supply system consumes energy, so the control strategy should also consider the trade-off between braking performance and energy cost. A practical controller should activate the thrust only when the additional deceleration is worth the energy expenditure and when the slip stability margin is sufficient.
8. Conclusion
I have presented a particle-swarm-optimized back-propagation fuzzy proportional-integral-derivative control strategy for an electric vehicle braking system that combines an anti-lock braking system with an air reverse thrust system. The main conclusions are as follows.
1. The air reverse thrust system provides an additional longitudinal braking force that is independent of road adhesion. In the high-adhesion simulation, the composite electric vehicle braking system reduces the braking time by 25.7% and the braking distance by 34.7% compared with the conventional anti-lock braking system.
2. The fuzzy proportional-integral-derivative controller provides fast slip response, while the particle-swarm-optimized back-propagation network provides fine gain compensation. Compared with the fuzzy controller alone, the proposed electric vehicle controller reduces the braking time by 13.3% and the braking distance by 12.6%.
3. The proposed controller keeps the slip ratio near the desired value of 0.2. The peak slip ratio is reduced from about 0.30 to about 0.24, and the upper fluctuation limit is reduced by about 20%. This indicates better braking stability and more balanced use of tire adhesion.
4. The air reverse thrust is the dominant additional force in the early and middle stages of electric vehicle braking. Its initial magnitude is about 5.5 kN, which is 30 to 40 times the peak aerodynamic drag. The thrust duration and the thrust decay must be coordinated with the brake torque to avoid slip overshoot.
5. The five-degree-of-freedom longitudinal model, the isentropic thrust model, and the cooperative control law form a consistent simulation framework for electric vehicle braking studies. The results support the potential of the proposed method for electric vehicle ground braking, while also indicating the need for high-fidelity flow simulation and hardware-in-the-loop validation.
In summary, the combination of fuzzy proportional-integral-derivative control and particle-swarm-optimized back-propagation compensation offers a practical path for improving electric vehicle braking performance. The electric vehicle remains stable, the slip ratio is well regulated, and the additional air reverse thrust is used efficiently. I believe that further work on actuator delay, energy management, and real-time implementation will make this electric vehicle braking concept more suitable for practical application.