Electric vehicles have created a strong demand for braking systems that can respond more quickly, offer higher control accuracy, and integrate easily with the fully electronic chassis of modern cars. Traditional hydraulic braking systems, while capable of producing large braking torque, suffer from long response chains, complex plumbing, and poor compatibility with newly proposed anti-lock braking algorithms. Magnetorheological brakes use a smart fluid whose yield stress can be tuned by an external magnetic field in a few milliseconds. Such brakes offer a nearly instantaneous torque generation and are therefore attractive for electric vehicles. However, the braking torque of conventional magnetorheological brakes is usually below the level required for safe emergency braking of a small passenger car. Furthermore, the non-linear relationship between the slip ratio, the road condition, and the available tyre force makes anti-lock braking control difficult. In my work I set out to solve these two issues together.
I focused on the A00-class pure electric vehicle segment, where compactness and energy efficiency are crucial. A typical vehicle of this class has a gross mass of about 1020 kg and is intended mainly for urban commuting. At speeds up to 80 km/h it must achieve a fully controlled emergency stop in a very short distance. The braking torque requirement for the front wheel exceeds 317 N·m under the Chinese standard for passenger-car braking performance. I therefore first designed a new magnetorheological brake called the three-coil comb-type disc magnetorheological brake. The design adds two auxiliary coils to a comb-shaped rotor so that the flux produced by the main coil is confined more effectively to the shear gaps. This structural innovation increases the torque without increasing the outer diameter of the device as much as a simple single disc would require.
I then developed a model predictive control strategy based on a road-recognition algorithm. The road-recognition algorithm identifies the current tyre-road condition from an image of the road surface, so that the model predictive controller can use the correct target slip ratio and the correct maximum available adhesion coefficient. I compared my proposed strategy with a simple negative-feedback controller and a conventional PID controller. Both simulations and laboratory experiments confirmed that the proposed predictive controller reduces the stopping time and stopping distance on roads with different adhesion coefficients while also giving a smaller fluctuation in the slip ratio. In this article I describe the complete procedure, including the magnetic-circuit design, torque modelling, temperature compensation, simulation, and experimental verification.
Brake Architecture and Working Principle
The brake I built employs the shear mode of magnetorheological fluid. A rotor carrying comb-shaped teeth rotates inside a stationary shell. The gaps between the rotor teeth and the shell are filled with magnetorheological fluid. When the coil is energized, a magnetic field crosses the fluid in these narrow gaps. Iron particles inside the fluid form chain-like structures along the magnetic field lines and the fluid changes from a Newtonian liquid to a visco-plastic solid in milliseconds. The resistance to shearing generates the braking torque. Because the torque is produced by the Maxwell stress of the particle chains rather than by friction between solid surfaces, there is no mechanical wear in the working gap.
The comb-shaped rotor gives a large effective shear area in a compact radial envelope. Each tooth contributes two radial shearing surfaces. In my design, eight radial gaps participate in torque generation. The effective radius changes from 87 mm to 105 mm, producing a much longer torque arm than a conventional rotor of the same diameter. The main coil is located in the outer shell and produces the primary magnetomotive force. Two side coils are arranged on both sides of the rotor, one at each side. When the side coils are powered in the same direction, they create an additional magnetic field that pushes the main flux lines towards the shear gaps. As a result, the magnetic flux leakage from the inner shaft region is suppressed.
The actuator geometry is symmetric about the rotation axis. A half-sectional view is shown in principle by the following description: a stepped shaft drives the comb rotor; the outer shell, the internal shell and a magnetically conductive crescent-shaped part define the magnetic circuit; the main coil is inserted in a slot of the external shell; two side coils are placed close to the rotor hub; a non-magnetic ring between the main and side coils prevents the flux from short-circuiting. The rotor is supported by conical roller bearings, and rotary lip seals prevent fluid leakage. A keyway transmits torque from the shaft to the rotor. This compact construction can be installed in the wheel hub region of a small electric vehicle.
| Parameter | Value (mm) |
|---|---|
| Maximum radius of the comb teeth | 105 |
| Radius of the tooth root | 92 |
| Minimum radius of the comb structure | 87 |
| Maximum radius of the side coil | 77 |
| Rotor hub radius | 28 |
| Width of one comb tooth | 8 |
| Thickness of the outer shell | 15 |
| Width of the magnetic crescent | 4 |
| Radial fluid gap width | 1 |
The main structural material is low-carbon steel 20, which has a high magnetic permeability and moderate mechanical strength. Stainless steel 304 is used for non-magnetic spacers and for the inner shell. The coils are wound with 0.8 mm enameled copper wire. The magnetorheological fluid is a commercial product with a zero-field viscosity of approximately 1 Pa·s and a maximum specified shear stress of 45 kPa at room temperature and saturation magnetization. The working temperature range of the fluid is from −35 °C to 140 °C.
| Component | Material |
|---|---|
| Stepped shaft | Steel 45 |
| Comb-shaped brake disc | Low-carbon steel 20 |
| Magnetic crescent part | Low-carbon steel 20 |
| External magnetic sleeve | Low-carbon steel 20 |
| Outer shell | Low-carbon steel 20 |
| Coil spacer ring | Stainless steel 304 |
| Inner shell and bearing sleeves | Stainless steel 304 |
| Seals | Fluororubber |
Magnetic Circuit Design
The magnetic circuit of the brake is essential for the torque-current relation. Because the whole brake is symmetric with respect to the mid-plane, I analysed only one half. The two side coils are connected so that they carry the same current direction, which allows them to be represented as a single equivalent source. The circuit therefore contains two loops: one loop driven by the main coil and one loop driven by the equivalent side coil. I constructed a reluctance network with 34 discrete reluctance elements. Every air gap in the fluid is represented by an equivalent gap reluctance, because the permeability of the magnetorheological fluid is much lower than that of the steel parts.
The reluctance of a magnetic path is given by
$$R_{m,i} = \frac{l_i}{\mu_i S_i}$$
where \(R_{m,i}\) is the magnetic reluctance of the i-th section, \(l_i\) is the average length of the flux path in that section, \(S_i\) is the cross-sectional area perpendicular to the flux, and \(\mu_i\) is the absolute permeability of the material. The total reluctance of the main-coil circuit was obtained by adding the series-connected reluctance elements:
$$R_{m,\mathrm{main}} = 2R_{m,1} + 2R_{m,2} + 2R_{m,3} + 2R_{m,4} + 2R_{m,5} + 2R_{m,6} + 4R_{m,7} + 6R_{m,8} + 3R_{m,34}$$
Similarly, the equivalent reluctance seen by the side coils is
$$R_{m,\mathrm{side}} = 2R_{m,4} + 2R_{m,5} + 2R_{m,23} + 2R_{m,24} + 2R_{m,25} + 2R_{m,26} + 2R_{m,27} + 2R_{m,28}$$
The cross-sectional area of the main air-gap path is
$$S_0 = \pi \left( R_1^2 – R_3^2 \right)$$
Neglecting magnetic saturation in the steel parts, the magnetic flux density in the fluid gap produced by a coil with \(N\) turns carrying current \(I\) is obtained from
$$B = \frac{N I}{R_m S_0}$$
Using the geometric parameters from the table above, the calculation gave a linear relation between the flux density and the current for the main coil:
$$B = 0.125 I$$
When both the main coil and the two side coils are energized, the total magnetic flux in the effective gap increases and the relation becomes
$$B = 0.13 I$$
Those expressions are used later for converting current commands into torque values. However, I also needed a model that describes the influence of temperature, because the shear stress of the magnetorheological fluid is temperature dependent.
Braking Torque Model
The torque of the brake is generated by the shear stress acting on the active surfaces. If the total shear stress at radius \(r\) is represented by \(\tau\), the elementary torque produced on one annular area element is
$$dT = 2\pi \tau r^2 dr$$
For a magnetorheological fluid operating in the shear mode, the total shear stress is frequently written as
$$\tau = \tau_y(B,Q) + \eta \dot{\gamma}$$
where \(\tau_y(B,Q)\) is the field-dependent shear yield stress, which is also a function of fluid temperature \(Q\), and \(\eta \dot{\gamma}\) is the viscous stress. The shear rate \(\dot{\gamma}\) in an annular gap of width \(\delta\) rotating with angular velocity \(\omega\) is
$$\dot{\gamma} = \frac{\omega r}{\delta}$$
Substituting the expression into the elementary torque and integrating over the active radii between \(R_3\) and \(R_1\), the total torque becomes
$$T = 2n\pi \int_{R_3}^{R_1} \left[ \tau_y(B,Q) + \eta \frac{\omega r}{\delta} \right] r^2 dr + 2n\pi \int_{R_2}^{R_1} \left[ \tau_y(B,Q) + \eta \frac{\omega r}{\delta} \right] r^2 dr$$
The contributions from the two integrals represent the two sets of shear surfaces in the comb geometry. The viscous torque is usually small compared with the magnetically induced torque, and it cannot be actively controlled. Since the operating speed is high enough to generate a shear rate above 100 s⁻¹, the viscous contribution becomes nearly constant and can be excluded from the control-oriented model.
To obtain \(\tau_y(B,Q)\), I measured the shear stress of the fluid with a rheometer at magnetic flux densities from 0.2 T to 0.5 T and temperatures from 5 °C to 70 °C. The results showed an obvious decrease in yield stress as the fluid temperature increased. The decay is more pronounced at higher magnetic fields because the permeability of the particles decreases with temperature. I used a thermodynamic fitting equation to represent each isofield curve. A final surface fit produced an expression of the form
$$\tau_y(B,Q) = \alpha_1 B + \alpha_2 B^2 + \alpha_3 Q B + \alpha_4 Q^2$$
where the constants \(\alpha_1\) to \(\alpha_4\) were determined from the measured data. Combining the empirical fluid model with the magnetic circuit relation gives a direct link between the coil current, the fluid temperature, and the braking torque. For the main coil only:
$$T = 3.53 I^2 + 95.6 I + 0.48 I Q + 0.00007 Q^2$$
For the main and auxiliary coils together:
$$T = 3.83 I^2 + 99.4 I + 0.5 I Q + 0.00007 Q^2$$
The temperature terms indicate that at 40 °C a current of 4 A produces a theoretical torque of about 360 N·m when the main coil is used alone. With the auxiliary coils active, the theoretical torque increases to about 380 N·m. This 6 % increase is important for the emergency-braking application.
Electromagnetic Simulation and Experimental Verification
I performed finite-element electromagnetic simulations in order to verify the flux distribution of the proposed magnetic circuit. The brake was meshed as a two-dimensional axisymmetric model. Figure 3 in the original report showed the field distribution of the brake. My simulation found that when only the main coil was excited, some flux lines leaked towards the shaft and the inner non-magnetic parts. When the two auxiliary coils were turned on, the flux distribution became much more concentrated in the eight radial fluid gaps. The auxiliary coils act as a “magnetic barrier” that prevents the main flux from taking short paths through the shaft, thus increasing the amount of flux crossing the fluid.
After the electromagnetic simulations, I built a comprehensive test system to measure the actual torque. The brake shaft was driven by a variable-frequency motor, and the stationary part of the brake was connected to a torque transducer. The output torque was recorded at initial speeds from 10 km/h to 80 km/h and for currents from 0.25 A to 4 A. During the experiment, the temperature of the fluid was measured by a multi-channel thermometer and kept at about 40 °C. The measured torque under main-coil-only excitation reached about 340 N·m at 4 A, while the combined main and auxiliary coils produced about 360 N·m at the same current. These values are slightly lower than the theoretical prediction because of magnetic saturation and imperfect flux leakage, but the overall trend agrees very well.
| Operating mode | Current (A) | Torque (N·m) |
|---|---|---|
| Main coil only | 4 | 340 |
| Main + auxiliary coils | 4 | 360 |
| Auxiliary coil only (maximum limited) | 2 | 23 |
I also studied the energy efficiency ratio, defined as the output torque divided by the total electrical power consumed by the coils:
$$EER = \frac{T}{W}$$
With main-coil-only excitation, the energy efficiency ratio decreases as the current increases. When the auxiliary coils were switched on at a constant current of 2 A, the total electrical power increased and the energy efficiency ratio was approximately 5.3 % lower than the main-coil-only condition at 4 A. Nevertheless, the auxiliary coils increase the maximum torque by 6 %. From a safety point of view, the extra electrical loss is acceptable during an anti-lock braking event because the braking event lasts for only a few seconds. I therefore chose the main-and-auxiliary coil operation mode for all subsequent anti-lock braking experiments.
Vehicle Dynamics and Tyre Model
In order to evaluate the anti-lock braking performance of the designed brake, I adopted a single-wheel vehicle dynamics model. This simplified model is accurate enough for longitudinal braking studies and can be easily reproduced on a laboratory test bench. The vehicle mass is equally distributed among the four wheels, and the effect of pitch motion is ignored. The governing differential equations are
$$m\frac{dv}{dt} = -F_x$$
$$J\frac{d\omega}{dt} = F_x r – T$$
where \(m\) is the quarter vehicle mass, \(v\) is the longitudinal speed of the vehicle, \(J\) is the wheel moment of inertia, \(\omega\) is the angular speed of the wheel, \(r\) is the rolling radius, \(F_x\) is the longitudinal tyre-road friction force, and \(T\) is the braking torque developed by the magnetorheological brake. The friction force is the product of the normal load and the longitudinal adhesion coefficient:
$$F_x = \mu mg$$
The longitudinal slip ratio \(s\) is defined by
$$s = \frac{v – \omega r}{v}$$
During normal rolling, \(\omega r = v\), so \(s=0\). When the wheel is fully locked, \(\omega=0\), so \(s=1\). The goal of anti-lock braking is to keep the slip ratio near the value that gives the maximum longitudinal adhesion coefficient. For most road surfaces this value is approximately 0.2.
I used the well-known “magic formula” to represent the relationship between the slip ratio and the longitudinal adhesion coefficient:
$$\mu(s) = D \sin[C \arctan(B s – E(Bs – \arctan(Bs)))]$$
where \(B\), \(C\), \(D\), and \(E\) are empirical constants. I selected three different idealised road conditions: a low-adhesion icy road with peak coefficient 0.15, a medium-adhesion dirt road with peak coefficient 0.4, and a high-adhesion asphalt road with peak coefficient 0.75. The corresponding magic-formula parameters are listed below.
| Road category | Peak adhesion coefficient | B | C | D | E |
|---|---|---|---|---|---|
| Low | 0.15 | 6.50 | 2.00 | 0.15 | 0.90 |
| Medium | 0.40 | 5.50 | 2.00 | 0.40 | 0.90 |
| High | 0.75 | 5.00 | 2.00 | 0.75 | 0.90 |
Road Recognition Algorithm
The anti-lock braking controller needs to know the optimal slip ratio of the current road. A conventional method uses wheels-speed signals and online friction estimation, but this method is slow and might not react quickly enough during emergency braking. To solve this problem, I employed a convolutional neural network for road-surface classification. The input is a small greyscale image of the road ahead, captured by a camera installed behind the windshield. The network contains two convolutional layers, two pooling layers and two fully connected layers. I trained the model on a dataset containing pictures of three road categories: ice/snow, soil/mud, and asphalt. The training set had 20 pictures per category, the validation set had 5 pictures per category, and the test set had 10 pictures per category. The classification accuracy on the test set was 100 %. Once the road type is known, the controller selects the appropriate peak adhesion coefficient and target slip ratio from a lookup table. Because the image is available at the beginning of braking, the predictive controller can start with the correct road parameters without waiting for an online estimation. This greatly reduces the response time of the controller.
Control Strategies
Simple Negative Feedback
I first implemented a simple negative-feedback controller as a baseline. The controller compares the measured slip ratio with the target slip ratio of 0.2. If the slip ratio is larger than the target, the braking torque is decreased by reducing the coil current. If it is smaller than the target, the coil current is increased. This strategy is intuitive but cannot provide a smooth response because it does not consider the dynamic of the wheel or the saturation of the brake.
Conventional PID Control
As a second baseline, I implemented a conventional PID controller. The control input is the coil current command \(u\), and the control error is
$$e(t)=s_{\mathrm{ref}}-s(t)$$
The current command is computed as
$$u(t)=K_P e(t)+K_I \int_0^t e(\tau)d\tau+K_D\frac{de(t)}{dt}$$
The PID coefficients were tuned on the medium-adhesion road because it represents a compromise between fast response and stability. The same coefficients were then used on all three road surfaces. This fixed-gain approach works well only when the road condition does not differ too much from the tuning condition. On a low-adhesion road, the controller produces overshoot and oscillations. On a high-adhesion road, the response is slower than necessary.
Model Predictive Control with Road Recognition
The proposed controller combines the road-recognition algorithm with model predictive control. The model predictive control strategy is based on a linearised model of the magnetorheological brake. The brake dynamics are represented by a first-order lag system with gain \(k\) and time constant \(T_e\):
$$\frac{T(s)}{U(s)}=\frac{k}{T_e s+1}$$
From the measured torque-current map, I used \(k=100\) N·m/A and \(T_e=0.01\) s for the active control range. The predictive controller uses the following discrete state-space model in the simulation:
$$x(k+1)=A_d x(k)+B_d u(k)$$
$$y(k)=C_d x(k)$$
where the state vector contains the slip ratio and the braking torque. The control action is obtained by minimising the following cost function over a prediction horizon \(N_p\):
$$J(k)=\sum_{j=1}^{N_p}\left[ \left(s(k+j)-s_{\mathrm{ref}}\right)^2 Q + \left(\Delta u(k+j-1)\right)^2 R \right]$$
subject to
$$0 \le u(k+j)\le 4 \text{ A}$$
$$0 \le s(k+j)\le 1$$
The first element of the optimised control sequence is applied to the brake. At each sampling instant, the model state is updated from the measured wheel speed, and the optimization is repeated. This rolling-horizon mechanism provides inherent robustness against model mismatch and external disturbances. The road-recognition algorithm supplies the reference slip ratio \(s_{\mathrm{ref}}\) and also modifies the prediction horizon parameters if needed. As a result, the controller can converge to the optimum slip ratio much faster than the two baseline controllers.
Simulation Study
I built the complete anti-lock braking system model in Simulink. The model includes the single-wheel vehicle dynamics, the magic-formula tyre model, the road recognition logic, the brake dynamic model, and one of the three control strategies. Table below summarizes the main vehicle parameters used in the simulation.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Quarter vehicle mass | m | 254 | kg |
| Wheel moment of inertia | J | 0.98 | kg·m² |
| Wheel rolling radius | r | 0.25 | m |
| Initial vehicle speed | v₀ | 80 | km/h |
All simulations start at an initial vehicle speed of 80 km/h. The brake torque is applied only at the wheel. The vehicle is stopped when its longitudinal speed reaches zero. The performance indices are the stopping time, the stopping distance, and the fluctuation of the slip ratio around 0.2. A good controller should stop the vehicle as quickly and as stably as possible.
Results with Simple Negative Feedback
The simple negative-feedback controller produced large oscillations in the slip ratio. On the low-adhesion road, the slip ratio repeatedly swung from very small values to almost 1. The stopping time was 3.899 s and the stopping distance was 47.72 m. On the medium-adhesion road, the stopping time was 2.918 s and the distance was 36.81 m. On the high-adhesion road, the stopping time was 2.146 s and the distance was 28.5 m. These values are close to those of an uncontrolled locked-wheel stop, which indicates that the simple feedback strategy does not fully exploit the available road friction.
Results with Conventional PID Control
The PID control strategy reduced the stopping time and stopping distance on all three roads compared with the simple feedback strategy. On the low-adhesion road, the stopping time was 3.478 s and the distance was 41.17 m. On the medium-adhesion road, the stopping time was 2.513 s and the distance was 30.47 m. On the high-adhesion road, the stopping time was 1.768 s and the distance was 22.23 m. The slip ratio waveform was smoother than that of the simple feedback controller, but there was still a noticeable overshoot at the beginning of braking on the low-adhesion road. The PID gains tuned for medium adhesion were not optimal on the two other road surfaces.
Results with MPC and Road Recognition
The model predictive control strategy with road recognition showed the best performance. On the low-adhesion road, the stopping time was 3.319 s and the stopping distance was 37.7 m. On the medium-adhesion road, the stopping time was 2.349 s and the distance was 26.89 m. On the high-adhesion road, the stopping time was 1.593 s and the distance was 18.48 m. The slip ratio trajectories reached the reference value of 0.2 almost immediately after the beginning of braking and stayed near it with much smaller fluctuations than observed with the other two controllers. The following table compares the simulation results.
| Road condition | Performance | Simple feedback | PID | MPC with road recognition |
|---|---|---|---|---|
| Low adhesion | Stopping time (s) | 3.899 | 3.478 | 3.319 |
| Stopping distance (m) | 47.72 | 41.17 | 37.70 | |
| Medium adhesion | Stopping time (s) | 2.918 | 2.513 | 2.349 |
| Stopping distance (m) | 36.81 | 30.47 | 26.89 | |
| High adhesion | Stopping time (s) | 2.146 | 1.768 | 1.593 |
| Stopping distance (m) | 28.50 | 22.23 | 18.48 |
Compared with the conventional PID controller, the proposed MPC controller reduced the stopping time by about 4.6 % on the low-adhesion road, 6.5 % on the medium-adhesion road, and 9.9 % on the high-adhesion road. The corresponding reductions in stopping distance were approximately 8.4 %, 11.7 %, and 16.9 %, respectively. The response time, defined as the time required for the slip ratio to first reach the reference band, was about 30 % shorter under MPC than under PID. The high-adhesion road benefits the most because the high available friction demands a fast build-up of braking torque; the predictive controller anticipates this need and therefore speeds up the torque transient.
Experimental Validation
To verify the simulation results, I built a magnetorheological brake anti-lock braking test bench. The brake was installed on a motor-driven axle. Since an actual vehicle was not available, a controllable magnetic powder clutch was used to emulate the tyre-road adhesion force. The magnetic powder clutch has an input shaft connected to a speed-controlled motor and an output shaft connected to the brake rotor. The torque transmitted through the clutch can be tuned by adjusting the current of the powder clutch. By choosing different currents, it is possible to simulate different road friction levels. The speed of the input shaft represents the vehicle speed, while the speed of the output shaft represents the wheel speed. A controller measures both speeds, computes the slip ratio, and sends the appropriate current command to the magnetorheological brake.
Three driving conditions were tested: no anti-lock control, conventional PID control, and MPC with road recognition. The no-anti-lock condition corresponds to applying the maximum brake current of 4 A without any wheel slip regulation. In this condition, the wheel locks very quickly and the slip ratio reaches 1. On the low-adhesion simulated road, the vehicle stopped in 3.72 s and covered 47.61 m. On the medium road, the stopping time was 2.98 s and the distance was 38.85 m. On the high road, the stopping time was 2.36 s and the distance was 29.69 m. These values are close to the simulation results of the simple feedback controller, confirming that the test bench correctly emulates the longitudinal dynamics.
When the conventional PID controller was activated, the measured slip ratio stayed near 0.2 but showed a fluctuating behaviour. The PID controller stopped the vehicle in 3.61 s on the low road, 2.73 s on the medium road, and 2.11 s on the high road. The corresponding stopping distances were 45.39 m, 35.16 m, and 26.71 m. These experimental figures are slightly longer than the simulation values because of friction losses in the bearings, inertia of the test rig, and the finite resolution of the speed sensors.
Finally, I tested the proposed model predictive controller with the road recognition module. The experimental slip ratio responses were smoother and reached the target faster than the PID control responses. The measured stopping times were 3.42 s on low adhesion, 2.43 s on medium adhesion, and 1.97 s on high adhesion. The measured stopping distances were 40.56 m, 30.82 m, and 24.47 m, respectively. The following table gives the complete experimental results.
| Road condition | Performance | Without ABS | With PID control | With MPC and road recognition |
|---|---|---|---|---|
| Low adhesion | Stopping time (s) | 3.72 | 3.61 | 3.42 |
| Stopping distance (m) | 47.61 | 45.39 | 40.56 | |
| Medium adhesion | Stopping time (s) | 2.98 | 2.73 | 2.43 |
| Stopping distance (m) | 38.85 | 35.16 | 30.82 | |
| High adhesion | Stopping time (s) | 2.36 | 2.11 | 1.97 |
| Stopping distance (m) | 29.69 | 26.71 | 24.47 |
The experimental results confirm that the proposed MPC with road recognition consistently outperforms both the uncontrolled condition and the conventional PID controller. Compared with the PID controller, the experimental stopping time was reduced by 5.3 %, 10.9 %, and 6.6 % for low, medium, and high adhesion roads, respectively. The stopping distance reductions were 10.6 %, 12.3 %, and 8.4 % for the same three surfaces. The improved response is mainly attributed to the predictive nature of the controller: the road recognition module supplies the target slip ratio before the brake is commanded, and the model predictive controller uses the brake dynamics to calculate the optimal current trajectory. In addition, the feedback correction step of MPC prevents the slip ratio from remaining far from the target after unexpected changes in the wheel-road friction.
I observed that the slip-ratio oscillation frequency during the MPC experiments was lower than that during the PID experiments. The PID controller, being a proportional-derivative type, tends to amplify sensor noise and high-frequency wheel-speed fluctuations, causing the current command to chatter. In contrast, the model predictive controller naturally penalizes large variations in the current command through the control-weighting parameter \(R\) in the cost function. This smoothing effect is very beneficial for a magnetorheological brake because the time constant of the magnetic circuit and the fluid response is finite; commands that change too rapidly cannot be followed exactly and produce wasted energy.
Conclusion
In my research, I designed a three-coil comb-type disc magnetorheological brake for a small pure electric vehicle. The comb-shaped rotor supplies eight radial shear gaps, while the pair of auxiliary coils strengthens the magnetic flux in those gaps. The theoretical torque model accounts for the temperature dependence of the magnetorheological fluid, and the model was verified by electromagnetic simulation and by laboratory torque measurements. With a main-coil current of 4 A, the brake delivers about 340 N·m; adding the auxiliary coils raises the peak torque to 360 N·m, satisfying the front-axle braking requirement of the target electric vehicle.
I also proposed a model predictive control strategy whose reference slip ratio is obtained by a convolutional-neural-network road recognition algorithm. I compared this strategy with a simple negative feedback controller and a conventional PID controller by simulation and by experiments on a specially designed ABS test bench. The new controller significantly reduces the stopping time and the stopping distance, shortens the response time at the beginning of braking, and strongly suppresses the wheel-slip oscillations around the optimal value. The benefits are most obvious on high-adhesion roads, where a fast torque build-up is essential to exploit the available friction: the stopping distance decreased by nearly 17 % in the simulation and by about 8 % in the experiments compared with conventional PID control.
There is still room for improvement. The brake torque must be raised further if the same actuator is to be used on heavier electric vehicles. The prototype would also benefit from an integrated cooling system, because the fluid temperature increases during repeated emergency stops and reduces the maximum torque. In future work I plan to extend the single-wheel model to a full four-wheel vehicle model and to test the proposed controller in a hardware-in-the-loop environment with actual road images captured during driving. Nevertheless, the present study clearly shows that a magnetorheological brake combined with a predictive control strategy provides a promising solution for the next generation of electric-vehicle braking systems.