I study the braking dynamics of an electric car because the ground braking problem is nonlinear, time-varying, and strongly coupled. In my work, I combine an anti-lock braking system, which I denote ABS, with an air reverse thrust system, which I denote ARTS, and I control the resulting composite braking power system by a cooperative strategy. The strategy fuses a fuzzy proportional-integral-derivative controller with a particle-swarm-optimized back-propagation neural network. My goal is not merely to tune one controller for one operating point, but to create a control structure in which the fuzzy PID loop gives fast response and coarse adjustment of the wheel slip ratio, while the PSO-BP network gives fine gain compensation and suppresses steady-state error and oscillation. I treat the electric car as a five-degree-of-freedom longitudinal system, and I derive the reverse thrust from one-dimensional steady isentropic flow theory. In the sections below I present the model, the control architecture, the optimization procedure, the simulation matrix, and the comparative results. I also discuss the limits of my current study and the direction of my future work.
The motivation for my study is straightforward. A conventional ABS can exploit tire-road adhesion effectively, but it remains bounded by the friction available at the tire contact patch. On a high-adhesion road, the electric car can decelerate strongly, yet on a low-adhesion road or at high speed the achievable braking distance cannot be shortened beyond a physical limit without an additional force source. I therefore introduce ARTS as an active aerodynamic brake. The system draws compressed air and discharges it forward through a nozzle, so the momentum flux produces a reaction force opposite to the motion of the electric car. Because this force does not depend on tire-road friction, it can supplement the ground braking force and alter the force balance of the electric car in a way that a friction-only ABS cannot. I keep the ARTS force adjustable by controlling the mass flow, the nozzle exit velocity, and the discharge area. The composite brake is activated when the brake pedal is triggered urgently and the slip ratio exceeds a threshold, which allows the electric car to preserve normal ABS behavior in mild braking and to call on the air reverse thrust only when the additional deceleration is needed.
In my model I make several simplifying assumptions so that the electric car braking problem remains transparent and reproducible. I neglect lateral motion and roll motion, I treat the left and right wheels on each axle as an equivalent wheel, and I neglect pitch inertia. I retain the longitudinal motion, the vertical motion, the pitch motion in a reduced form, and the rotational motion of the front and rear equivalent wheels. These assumptions are acceptable for a first comparative study of ABS and ABS/ARTS because the dominant braking variables are the longitudinal speed, the wheel angular speed, the normal load transfer, and the slip ratio. I define the electric car mass as \(m\), the longitudinal speed as \(V_x\), the front and rear longitudinal ground braking forces as \(F_{xf}\) and \(F_{xr}\), the front and rear vertical loads as \(F_{zf}\) and \(F_{zr}\), the distance from the center of gravity to the front axle as \(L_f\), the distance from the center of gravity to the rear axle as \(L_r\), the wheelbase as \(L=L_f+L_r\), the gravity acceleration as \(g\), and the center-of-gravity height as \(h_{cg}\). The aerodynamic drag is \(F_w\), and the ARTS reverse thrust is \(F_j\).
The longitudinal motion of the electric car is written as
$$
m\dot{V}_x=-F_{xf}-F_{xr}-F_w-F_j .
$$
I use a positive braking force convention in which \(F_{xf}\), \(F_{xr}\), and \(F_j\) oppose the forward motion of the electric car. The aerodynamic drag is
$$
F_w=\frac{1}{2}\rho C_D A V_x^2 ,
$$
where \(\rho\) is the air density, \(C_D\) is the drag coefficient, and \(A\) is the frontal area. I include \(F_w\) because it completes the force balance and because it becomes non-negligible at the initial speed of my braking test. In my simulation the electric car starts at \(70\ \mathrm{km/h}\), so the passive drag contributes a measurable retardation even though it is much smaller than the ARTS thrust.
Because the electric car pitches under braking, the normal loads are not constant. I express the load transfer as
$$
F_{zf}=\frac{mgL_r+m a_x h_{cg}}{L},
$$
$$
F_{zr}=\frac{mgL_f-m a_x h_{cg}}{L},
$$
where \(a_x\) is the positive deceleration magnitude. This form is convenient because it shows directly that the front axle gains load and the rear axle loses load as the electric car decelerates. The instantaneous deceleration is obtained from the force balance as
$$
a_x=\frac{F_{xf}+F_{xr}+F_j+F_w}{m}.
$$
I integrate this deceleration to obtain the speed history of the electric car:
$$
V_x(t)=V_x(0)-\int_0^t a_x(\tau)\,d\tau .
$$
The wheel motion is modeled with an equivalent wheel inertia for each axle. For the \(i\)-th equivalent wheel I write
$$
J_i\dot{\omega}_i=F_{si}R-T_i ,
$$
where \(J_i\) is the wheel inertia, \(\omega_i\) is the wheel angular speed, \(R\) is the effective rolling radius, \(F_{si}\) is the longitudinal tire force, and \(T_i\) is the braking torque. I define the braking slip ratio as
$$
\lambda_i=\frac{V_x-\omega_i R}{V_x},
$$
with the understanding that the slip ratio is evaluated only when \(V_x\) is above a small numerical threshold. The control error for the electric car brake controller is
$$
e=\lambda-\lambda_0 ,
$$
where \(\lambda_0\) is the desired slip ratio. In my study I set \(\lambda_0=0.20\) because this value lies in the high-adhesion region for the tire model I use and it allows the electric car to preserve lateral stability while still producing strong longitudinal braking. The error rate is
$$
e_c=\frac{de}{dt}.
$$
Table 1 summarizes the main parameters of the electric car used in my simulation. I keep these parameters fixed in all comparative runs so that the differences between controllers come from the control law and from the presence or absence of ARTS, not from a change in the electric car platform.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Vehicle mass | \(m\) | 1620 | kg |
| Center of gravity to front axle | \(L_f\) | 1500 | mm |
| Wheelbase-related rear distance | \(L_r\) | 2718 | mm |
| Wheel effective radius | \(R\) | 0.300 | m |
| Center-of-gravity height | \(h_{cg}\) | 550 | mm |
| Wheel inertia | \(J_i\) | 0.75 | kg·m² |
| Air density | \(\rho\) | 1.225 | kg/m³ |
| Drag coefficient | \(C_D\) | 0.30 | – |
| Frontal area | \(A\) | 2.2 | m² |
| Initial braking speed | \(V_x(0)\) | 70 | km/h |
| Desired slip ratio | \(\lambda_0\) | 0.20 | – |
For the ARTS, I derive the reverse thrust from the momentum change across the nozzle. The thrust is
$$
F_j=\dot{m}v_e+A_e(p_e-p_a),
$$
where \(\dot{m}\) is the mass flow, \(v_e\) is the nozzle exit velocity, \(A_e\) is the nozzle exit area, \(p_e\) is the nozzle exit static pressure, and \(p_a\) is the ambient pressure. The continuity relation is
$$
\dot{m}=\rho_e A_e v_e,
$$
where \(\rho_e\) is the exit gas density. In the ideally expanded case \(p_e=p_a\), the pressure term vanishes and the thrust reduces to
$$
F_j=\dot{m}v_e .
$$
For a one-dimensional steady isentropic flow, I relate the exit state to the stagnation state. The stagnation pressure is \(p_0\), the stagnation temperature is \(T_0\), and the exit Mach number is \(M_e\). I use the standard isentropic relations
$$
\frac{p_0}{p_e}=\left(1+\frac{\gamma-1}{2}M_e^2\right)^{\frac{\gamma}{\gamma-1}},
$$
$$
\frac{T_0}{T_e}=1+\frac{\gamma-1}{2}M_e^2,
$$
where \(\gamma\) is the specific heat ratio. The exit velocity follows as
$$
v_e=M_e\sqrt{\gamma R_g T_e},
$$
with \(R_g\) denoting the specific gas constant. These equations let me compute the ARTS thrust from pressure, temperature, and nozzle geometry, and they make the thrust physically interpretable rather than an arbitrary additive force. In my simulation the ARTS thrust reaches about \(5.5\ \mathrm{kN}\) at the beginning of the emergency braking event and then decreases as the electric car slows and the喷射 process approaches its end. The passive aerodynamic drag at the same instant is on the order of \(145\ \mathrm{N}\), so the ARTS thrust is the dominant additional force in the early braking phase. This is exactly why the composite brake can shorten the stopping distance of the electric car without requiring a higher tire-road friction coefficient.
I now describe the control architecture. I use a parallel cooperative structure with two modules. The first module is the fuzzy PID controller. It receives the slip error \(e\) and the error rate \(e_c\), applies a seven-level fuzzy rule base, and outputs gain corrections \(\Delta K_p^{Fuzzy}\), \(\Delta K_i^{Fuzzy}\), and \(\Delta K_d^{Fuzzy}\). The second module is the PSO-BP compensator. It receives the same error signals and, after offline training and online compensation, outputs \(\Delta K_p^{BP}\), \(\Delta K_i^{BP}\), and \(\Delta K_d^{BP}\). The final PID gains are
$$
K_p(t)=K_{p0}+\alpha_1\Delta K_p^{Fuzzy}+\Delta K_p^{BP},
$$
$$
K_i(t)=K_{i0}+\alpha_2\Delta K_i^{Fuzzy}+\Delta K_i^{BP},
$$
$$
K_d(t)=K_{d0}+\alpha_3\Delta K_d^{Fuzzy}+\Delta K_d^{BP},
$$
where \(K_{p0}\), \(K_{i0}\), and \(K_{d0}\) are the baseline gains and \(\alpha_1\), \(\alpha_2\), and \(\alpha_3\) are weighting factors that balance the fuzzy and neural contributions. I keep the weights in the interval \([0,1]\) and I choose them so that the fuzzy loop dominates the transient response while the neural loop dominates the fine correction near the target slip ratio. The control signal applied to the brake actuator and to the ARTS valve is
$$
u(t)=K_p(t)e(t)+K_i(t)\int_0^t e(\tau)\,d\tau+K_d(t)\frac{de(t)}{dt}.
$$
I implement the controller in discrete time with a sampling period \(T_s\). The integral and derivative terms are discretized as
$$
\int_0^{kT_s}e(\tau)\,d\tau\approx T_s\sum_{n=0}^{k}e(nT_s),
$$
$$
\left.\frac{de(t)}{dt}\right|_{t=kT_s}\approx\frac{e(kT_s)-e((k-1)T_s)}{T_s}.
$$
The discrete control law is therefore
$$
u(k)=K_p(k)e(k)+K_i(k)T_s\sum_{n=0}^{k}e(n)+K_d(k)\frac{e(k)-e(k-1)}{T_s}.
$$
I limit the control signal and its rate so that the brake actuator and the ARTS valve do not saturate abruptly. The anti-windup term is
$$
I(k)=I(k-1)+T_s e(k)+K_{aw}\left(u(k-1)-u_{sat}(k-1)\right),
$$
where \(K_{aw}\) is a small anti-windup gain and \(u_{sat}\) is the saturated control signal. This detail matters for the electric car because a large slip error during the first few tenths of a second can otherwise produce an excessive integral action and a visible slip overshoot after the emergency braking command.
For the fuzzy module I use \(e\) and \(e_c\) as inputs and \(\Delta K_p\), \(\Delta K_i\), and \(\Delta K_d\) as outputs. The linguistic variables are negative big, negative medium, negative small, zero, positive small, positive medium, and positive big. I denote them by NB, NM, NS, ZE, PS, PM, and PB. The input universe is \([-6,6]\), and the output universe is \([0,6]\). I use triangular membership functions of the form
$$
\mu_A(x)=\max\left(0,\min\left(\frac{x-a}{b-a},\frac{c-x}{c-b}\right)\right),
$$
where \(a\), \(b\), and \(c\) are the left foot, peak, and right foot of the triangle. I also use shoulder memberships at the two ends of the input range so that large errors remain fully represented. The fuzzy inference is Mamdani type, and the defuzzification is the centroid method:
$$
\Delta K=\frac{\sum_{j=1}^{N}\mu_j\Delta K_j}{\sum_{j=1}^{N}\mu_j}.
$$
Table 2 gives the representative structure of my rule base. I write each cell as \(\Delta K_p/\Delta K_i/\Delta K_d\) using the linguistic labels Z, S, M, and B. The table is not a set of independent scalar rules; it is a compact way to show how the three gains move together as the slip error and its rate change. The general principle is that a large positive error requires a large proportional action and a moderate derivative action, while a large negative error requires a more conservative integral action to avoid overshoot. Near zero error, the controller reduces the proportional gain and relies on the integral and neural compensation to remove the residual error.
| \(e\) | \(e_c=\) 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/Z | S/B/Z | M/B/Z |
| PS | S/M/Z | S/B/S | Z/S/M | Z/B/S | S/M/S | S/M/S | Z/S/M |
| PM | S/S/M | S/B/S | Z/S/M | S/Z/B | B/Z/B | Z/B/S | B/Z/B |
| PB | Z/Z/S | Z/Z/M | M/M/S | M/S/M | B/Z/M | M/M/S | M/M/Z |
I now explain the PSO-BP compensation. A BP neural network alone can approximate a nonlinear gain map, but its training is sensitive to initial weights and it can fall into a local minimum. I therefore use particle swarm optimization to search for a good initial weight and bias vector before I apply gradient-based fine tuning. Each particle represents a complete set of network weights and biases. For a particle \(i\) in dimension \(d\), the velocity and position updates are
$$
v_{i,d}^{k+1}=w v_{i,d}^{k}+c_1 r_1\left(p_{i,d}^{k}-x_{i,d}^{k}\right)+c_2 r_2\left(g_{d}^{k}-x_{i,d}^{k}\right),
$$
$$
x_{i,d}^{k+1}=x_{i,d}^{k}+v_{i,d}^{k+1},
$$
where \(w\) is the inertia weight, \(c_1\) and \(c_2\) are the cognitive and social acceleration coefficients, \(r_1\) and \(r_2\) are random numbers in \([0,1]\), \(p_{i,d}\) is the personal best position, and \(g_d\) is the global best position. I constrain the velocity to a maximum \(v_{\max}\) so that the swarm does not oscillate around a poor region. The fitness function is the mean squared control error over the training horizon:
$$
J_{PSO}=\frac{1}{N}\sum_{k=1}^{N}\left(\lambda_k-\lambda_0\right)^2+\beta_1\left(t_b-t_{b,ref}\right)^2+\beta_2\left(s_b-s_{b,ref}\right)^2,
$$
where \(t_b\) is the braking time, \(s_b\) is the braking distance of the electric car, and \(\beta_1\) and \(\beta_2\) are small weighting factors that discourage solutions that improve slip tracking at the cost of an excessively long stop. I include both time and distance in the fitness because the electric car braking problem is not purely a set-point tracking problem. A controller that holds the slip ratio perfectly but delays the brake torque build-up is not acceptable for an emergency stop.
After PSO provides the initial weights, I train the BP network with the standard error back-propagation rule. The network output is a three-dimensional gain correction vector. The cost function is
$$
E=\frac{1}{2}\sum_{k=1}^{N}\left(y_k^{ref}-y_k\right)^2,
$$
and the weight update is
$$
w_{ij}^{new}=w_{ij}^{old}-\eta\frac{\partial E}{\partial w_{ij}}+\mu\Delta w_{ij}^{old},
$$
where \(\eta\) is the learning rate, \(\mu\) is the momentum coefficient, and \(\Delta w_{ij}^{old}\) is the previous weight change. I use a single hidden layer because the input dimension is small and because a deeper network would increase the online computational burden for the electric car controller. The input vector is \([e,e_c,\lambda,V_x]\), and the output vector is \([\Delta K_p^{BP},\Delta K_i^{BP},\Delta K_d^{BP}]\). The network architecture and the optimization parameters are listed in Table 3 and Table 4.
| Parameter | Symbol | Value |
|---|---|---|
| Swarm size | \(N_p\) | 40 |
| Maximum iterations | \(k_{\max}\) | 250 |
| Inertia weight | \(w\) | 0.72 |
| Cognitive coefficient | \(c_1\) | 1.50 |
| Social coefficient | \(c_2\) | 1.50 |
| Maximum velocity | \(v_{\max}\) | 0.20 |
| Search interval | \([x_{\min},x_{\max}]\) | [-1, 1] |
| Item | Setting |
|---|---|
| Input nodes | 4 |
| Hidden layer nodes | 9 |
| Output nodes | 3 |
| Hidden activation | Hyperbolic tangent |
| Output activation | Linear |
| Learning rate | 0.035 |
| Momentum coefficient | 0.80 |
| Training epochs | 600 |
The control allocation between the friction brake and the ARTS is another important part of my design. I do not let the ARTS operate as an uncontrolled on-off thruster. Instead, I define a desired total braking force \(F_{des}\) from the PID output and split it into a ground braking command \(F_{g,cmd}\) and an ARTS command \(F_{j,cmd}\). I use the allocation rule
$$
F_{des}=F_{g,cmd}+F_{j,cmd},
$$
$$
F_{j,cmd}=\operatorname{sat}\left(\kappa_j F_{des},0,F_{j,\max}\right),
$$
$$
F_{g,cmd}=F_{des}-F_{j,cmd},
$$
where \(\kappa_j\) is the ARTS share and \(F_{j,\max}\) is the maximum thrust available from the nozzle at the current pressure and mass flow. I set \(\kappa_j\) higher at high speed and lower at low speed, because the ARTS is most useful when the electric car still has a large forward momentum and the ground brake is approaching its adhesion limit. This allocation keeps the tire slip ratio near its target and lets the air thrust remove part of the deceleration demand that would otherwise require a larger brake torque and a larger slip excursion.
I define the total braking distance of the electric car as
$$
s_b=\int_0^{t_b}V_x(t)\,dt,
$$
and the braking time \(t_b\) as the time at which \(V_x\) falls below a small threshold. I evaluate the slip tracking quality with the root-mean-square error
$$
e_{RMS}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left(\lambda_k-\lambda_0\right)^2},
$$
and the peak slip deviation
$$
e_{\max}=\max_k\left|\lambda_k-\lambda_0\right|.
$$
For the controller comparison I also use the time reduction ratio and the distance reduction ratio:
$$
\eta_t=\frac{t_{b,base}-t_{b,new}}{t_{b,base}}\times100\%,
$$
$$
\eta_s=\frac{s_{b,base}-s_{b,new}}{s_{b,base}}\times100\%.
$$
In my simulation matrix I first compare the fuzzy PID controller with the PSO-BP fuzzy PID controller under the same ABS-only configuration. This comparison isolates the effect of the neural compensation. I then compare the ABS-only electric car with the ABS/ARTS electric car under the same optimized controller. This second comparison isolates the effect of the air reverse thrust. Table 5 lists the baseline and optimized controller results. The optimized controller reduces the braking time from \(5.251\ \mathrm{s}\) to \(4.555\ \mathrm{s}\), which is about \(13.3\%\), and it reduces the braking distance from \(52.18\ \mathrm{m}\) to \(45.62\ \mathrm{m}\), which is about \(12.6\%\). These values show that the PSO-BP compensation improves the electric car braking response even before ARTS is introduced.
| 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% |
I then activate the ARTS on the electric car and repeat the high-adhesion braking test. The initial speed is \(70\ \mathrm{km/h}\) in both runs. The ABS-only electric car with the optimized controller stops in \(4.555\ \mathrm{s}\) over \(45.62\ \mathrm{m}\). The ABS/ARTS electric car stops in \(3.385\ \mathrm{s}\) over \(29.81\ \mathrm{m}\). The time reduction is \(25.7\%\), and the distance reduction is \(34.7\%\). I present these values in Table 6. The result is important because it shows that the additional thrust does not merely shift the operating point of the controller. It changes the force balance of the electric car so that the ground brake can remain closer to its adhesion-optimal slip while the air thrust supplies a large part of the deceleration demand.
| Configuration | Braking time (s) | Braking distance (m) | Time reduction | Distance reduction |
|---|---|---|---|---|
| ABS only | 4.555 | 45.62 | – | – |
| ABS/ARTS | 3.385 | 29.81 | 25.7% | 34.7% |
The force history explains the mechanism. At the beginning of braking, the ARTS thrust is about \(5.5\ \mathrm{kN}\), while the passive aerodynamic drag is about \(145\ \mathrm{N}\). The ARTS thrust is therefore roughly thirty to forty times larger than the drag peak. The ARTS thrust remains high for the first part of the stop and then decreases as the喷射 process ends. The drag decreases with the square of the electric car speed and becomes small near the end of the stop. Table 7 summarizes the force contribution in three phases of the braking event. I use these phases only for interpretation; the controller itself does not switch between them explicitly.
| Braking phase | Dominant force source | Role of ARTS | Control focus |
|---|---|---|---|
| Initial phase | Ground brake plus ARTS | Large additional deceleration | Fast slip establishment without overshoot |
| Middle phase | Ground brake plus ARTS | Maintains high deceleration | Slip regulation near the target |
| Final phase | Ground brake and drag | Thrust tapers to zero | Prevent wheel lock and residual oscillation |
The slip ratio behavior is equally important. In the initial phase, both the ABS-only electric car and the ABS/ARTS electric car can raise the slip ratio from zero to the neighborhood of \(0.20\) within about half a second. This means the composite controller does not weaken the brake pressure build-up. In the steady phase, however, the ABS-only electric car shows a larger oscillation amplitude, and its slip peaks approach \(0.30\). The ABS/ARTS electric car keeps the slip ratio around \(0.20\) with a smaller oscillation, and its peak is about \(0.24\). The upper fluctuation bound is therefore reduced by roughly \(20\%\). I summarize the slip metrics in Table 8. The lower oscillation is valuable for the electric car because it reduces the risk of tire lock and it keeps the longitudinal force closer to the adhesion peak instead of cycling between over-slip and under-slip regions.
| Metric | ABS only | ABS/ARTS | Observation |
|---|---|---|---|
| Target slip ratio | 0.20 | 0.20 | Same set point |
| Initial slip establishment time | about 0.5 s | about 0.5 s | Comparable speed of response |
| Steady slip band | 0.10 to 0.30 | about 0.20 ± 0.04 | Narrower band with ARTS |
| Peak slip | near 0.30 | near 0.24 | About 20% lower upper bound |
| Qualitative stability | Visible oscillation | Small oscillation | Improved electric car braking stability |
I also examine the control effort. The fuzzy PID loop provides a fast initial correction because its rule base reacts strongly to a large slip error. The PSO-BP loop then adjusts the gains around the target so that the integral action does not accumulate excessive error and the derivative action does not amplify measurement noise. In my runs, the neural compensation is most active between \(0.4\ \mathrm{s}\) and \(2.5\ \mathrm{s}\), which is exactly the period in which the electric car transitions from slip build-up to steady slip regulation. After that period, the gains remain close to their fuzzy values and the network output becomes small. This behavior is desirable because it means the PSO-BP module corrects the fuzzy controller rather than replacing it. The resulting control law remains interpretable, and the electric car controller can revert to a safe fuzzy PID mode if the neural module is disabled.
I use Table 9 to summarize the qualitative effect of each control module. The table is deliberately organized by function rather than by signal name, because I want to show how the modules divide the braking task. The fuzzy module is responsible for the coarse nonlinear mapping from slip error to gain correction. The PSO-BP module is responsible for the fine gain compensation that removes steady-state error and reduces oscillation. The force allocator is responsible for sending the correct share of the demand to the friction brake and to the ARTS. The anti-windup and saturation logic protect the electric car actuator from unrealistic commands.
| Module | Main input | Main output | Primary function |
|---|---|---|---|
| Fuzzy PID | \(e\), \(e_c\) | \(\Delta K_p^{Fuzzy}\), \(\Delta K_i^{Fuzzy}\), \(\Delta K_d^{Fuzzy}\) | Fast coarse gain adjustment |
| PSO-BP compensator | \(e\), \(e_c\), \(\lambda\), \(V_x\) | \(\Delta K_p^{BP}\), \(\Delta K_i^{BP}\), \(\Delta K_d^{BP}\) | Fine compensation and oscillation suppression |
| Force allocator | \(F_{des}\), \(V_x\) | \(F_{g,cmd}\), \(F_{j,cmd}\) | Coordinate friction brake and ARTS |
| Safety logic | Actuator limits, slip threshold | Saturated commands | Prevent lock and actuator overload |
I pay attention to the robustness of the electric car controller because the road adhesion can change during braking. In my current simulation I focus on a high-adhesion road, but I also perform a qualitative sensitivity study in which I vary the friction coefficient and the ARTS thrust scale. The purpose is not to claim a complete robustness proof, but to identify the variables that matter most. I find that the slip target remains trackable when the friction coefficient is moderately reduced, provided that the force allocator increases the ARTS share and the fuzzy PID gain scheduler does not demand an excessive ground brake torque. When the ARTS thrust is reduced, the electric car still stops safely under ABS, but the braking distance advantage shrinks. This is consistent with the physical interpretation: ARTS adds a force that is independent of road friction, so its benefit is largest when the friction brake is close to its limit or when the electric car is heavy and fast.
I express the sensitivity of the stopping distance to the ARTS thrust scale as a first-order approximation:
$$
\frac{\partial s_b}{\partial F_j}\approx-\frac{mV_x(0)}{2\left(F_{g,\max}+F_j\right)^2},
$$
where \(F_{g,\max}\) is the maximum ground braking force available at the current adhesion. The negative sign means that an increase in ARTS thrust reduces the stopping distance. The magnitude of the sensitivity decreases as the total braking force becomes large, which explains why the incremental benefit of ARTS is more visible in the early high-speed phase than in the final low-speed phase. I use this relation only as a design guide; the actual simulation integrates the full nonlinear model.
I also consider the energy and implementation implications for the electric car. The ARTS consumes compressed air or compressor work, so it is not a free braking force. In my control allocation I therefore avoid using ARTS when the demanded deceleration can be met by the friction brake alone. The ARTS is engaged only when the brake command is urgent and the slip ratio indicates that the electric car is approaching the adhesion limit. This event-based use keeps the auxiliary energy consumption bounded. For a practical electric car, the compressor and nozzle would need to be sized for a short emergency pulse rather than for continuous braking. I note that my current model does not include the compressor dynamics or the energy cost in the control objective, and this is one of the limitations I intend to address in future work.
I further examine the effect of the initial speed. A higher initial speed increases the kinetic energy that must be dissipated and also increases the passive drag, but the drag grows only with the square of speed while the ARTS thrust is controlled by pressure and mass flow. For the electric car, the benefit of ARTS is therefore expected to be more pronounced at high speed. I do not claim a full speed sweep in this study, but the force balance shows the trend clearly. The braking distance without ARTS scales approximately with the square of the initial speed when the deceleration is roughly constant. With a controlled ARTS thrust, the effective deceleration is higher in the early phase, so the distance reduction is largest when the initial speed is high. This is one reason I choose \(70\ \mathrm{km/h}\) as a representative emergency braking condition: it is high enough for the ARTS effect to be visible, yet it remains within a realistic urban and suburban operating range for an electric car.
I summarize the simulation workflow in Table 10. The workflow is reproducible and separates the plant model, the controller, the optimizer, and the evaluation metrics. I use the same electric car plant for all runs. The only changes are the controller type and the presence of the ARTS force. This separation is important because it prevents the comparison from being confounded by different vehicle parameters or different road inputs.
| Step | Action | Output |
|---|---|---|
| 1 | Build the five-degree-of-freedom electric car model | Longitudinal speed, wheel speed, normal loads, slip ratio |
| 2 | Build the ARTS thrust model | Thrust force from pressure, temperature, and nozzle geometry |
| 3 | Design the fuzzy PID controller | Baseline gain scheduling |
| 4 | Train the PSO-BP compensator | Optimized gain corrections |
| 5 | Run ABS-only and ABS/ARTS cases | Speed, slip, time, and distance histories |
| 6 | Evaluate time, distance, slip RMS, and peak slip | Comparative performance metrics |
The results support three conclusions that I consider central. First, the PSO-BP fuzzy PID controller improves the ABS-only electric car braking performance relative to a conventional fuzzy PID controller. The braking time and distance both decrease, and the slip oscillation becomes smaller. Second, the ARTS adds a large, controllable force that is independent of tire-road friction. When I combine ARTS with the optimized ABS controller, the electric car stops in \(3.385\ \mathrm{s}\) over \(29.81\ \mathrm{m}\), which is a \(25.7\%\) time reduction and a \(34.7\%\) distance reduction relative to the ABS-only case. Third, the slip ratio remains close to the ideal region around \(0.20\), and the upper fluctuation bound is reduced. This means the improvement in stopping performance does not come at the cost of braking stability. On the contrary, the electric car remains more stable because the ground brake does not need to be driven as deeply into the over-slip region.
I also note the limitations of my current study. The tire model is simplified and linearized around the operating region, so it does not capture all road roughness effects or all nonlinear tire transient behavior. The ARTS model is one-dimensional and steady, so it does not resolve the full nozzle flow field or the interaction between the jet and the external air flow around the electric car. The neural network is trained mainly on offline simulation data, so it does not yet include actuator delay, sensor noise, or compressor response delay. These limitations do not invalidate the comparative trends, but they define the boundary of my current claims. I therefore present the results as a simulation-based proof of concept for the electric car composite brake, not as a final vehicle calibration.
For future work I plan to add a higher-fidelity flow model and a hardware-in-the-loop platform. The flow model will let me examine the nozzle pressure ratio, the jet angle, and the external flow interaction more accurately. The hardware-in-the-loop platform will let me include actuator delay, sensor sampling, and energy constraints in the controller design. I also intend to extend the slip controller to a mixed-surface case and to a split-\(\mu\) case so that the electric car can coordinate the left and right brake forces more explicitly. Finally, I plan to include the compressor energy in the cost function so that the ARTS is used only when its braking benefit justifies its energy consumption. These extensions will make the electric car braking controller more realistic and more suitable for experimental validation.
In summary, I have presented a PSO-BP optimized fuzzy PID method for the composite braking power system of an electric car. I derived a five-degree-of-freedom longitudinal model, a one-dimensional ARTS thrust model, and a cooperative controller that combines fuzzy PID coarse adjustment with PSO-BP fine compensation. I used tables and equations to make the model and the results reproducible. The simulation shows that the proposed method shortens the braking time and braking distance of the electric car, keeps the slip ratio near the ideal value, and reduces oscillation. The method therefore has clear potential for improving the braking safety and stability of an electric car, especially when the friction brake alone is close to its adhesion limit. I will continue this work with higher-fidelity flow simulation, hardware-in-the-loop testing, and energy-aware control allocation for the electric car.