The rapid adoption of electric cars, driven by the global push for sustainable and low-carbon mobility, presents unique safety challenges. Among these, a tire blowout remains one of the most critical and sudden emergencies. The high-speed capability and often higher center of gravity in electric cars can exacerbate the vehicle’s instability following a blowout. The immediate aftermath involves nonlinear dynamic changes: a reduction in the rolling radius of the affected wheel, a sharp increase in rolling resistance, and a significant drop in lateral and longitudinal stiffness. These changes disrupt the vehicle’s equilibrium, inducing severe yaw moments and body sway, which can swiftly lead to loss of control, rollover, or secondary collisions. Traditional safety systems like the Tire Pressure Monitoring System (TPMS) offer only pre-failure warnings and are ineffective for active stabilization within the critical first 0.1 seconds post-blowout. Therefore, developing fast, robust, and effective active control strategies to restore stability in electric cars immediately after a tire blowout is a paramount research focus in automotive safety and intelligent control.
Existing control strategies for blowout scenarios primarily involve active front steering or differential braking to compensate for the sudden yaw moment. However, these methods often rely on the full measurement of vehicle states or are validated under idealized conditions, struggling with real-world sensor errors, external disturbances, and unmodeled dynamics. Fixed-time control theory presents a compelling solution. Unlike finite-time control, whose convergence time depends on initial conditions, fixed-time control guarantees stability within a predetermined, bounded time regardless of the initial state, offering superior predictability and safety for time-critical events like a blowout in an electric car.
This paper addresses the stability control problem for electric cars experiencing a tire blowout. The core contributions are: (1) proposing an output feedback fixed-time control framework using an input-driven filter, eliminating the restrictive requirement for full state measurement; (2) incorporating a robust adaptive mechanism to enhance the system’s resilience against observer errors and external disturbances; and (3) providing rigorous theoretical proof of the closed-loop system’s boundedness and fixed-time convergence, supported by comprehensive simulations.
Technical Background and Vehicle Modeling
To design an effective controller, we first establish a control-oriented model of the electric car’s dynamics under a blowout event. We begin with a linear two-degree-of-freedom (2-DOF) bicycle model, focusing on lateral and yaw motions, which are most critically affected. The key parameters for the electric car model are summarized in Table 1.
| Parameter | Symbol | Parameter | Symbol |
|---|---|---|---|
| Vehicle Mass | $$m$$ | Yaw Moment of Inertia | $$I$$ |
| Distance from CoG to Front Axle | $$a_f$$ | Distance from CoG to Rear Axle | $$a_r$$ |
| Longitudinal Velocity | $$v_x$$ | Lateral Velocity | $$v_y$$ |
| Yaw Rate | $$r$$ | Sideslip Angle | $$\beta$$ |
| Front Wheel Steering Angle | $$\delta_f$$ | Front Tire Cornering Stiffness | $$C_f$$ |
| Rear Tire Cornering Stiffness | $$C_r$$ | Lateral Force | $$F_y$$ |
| Yaw Moment | $$M_z$$ | Road Friction Coefficient | $$\mu$$ |
The fundamental equations of motion for the 2-DOF electric car model are:
$$ m(\dot{v}_y + v_x r) = F_{yf} + F_{yr} $$
$$ I\dot{r} = a_f F_{yf} – a_r F_{yr} + \Delta M $$
where $$F_{yf}$$ and $$F_{yr}$$ are the lateral forces on the front and rear axles, typically modeled as linear functions of tire slip angles for small angles: $$F_{yf} = C_f \alpha_f$$, $$F_{yr} = C_r \alpha_r$$. The tire slip angles are $$\alpha_f = \delta_f – \beta – \frac{a_f r}{v_x}$$ and $$\alpha_r = -\beta + \frac{a_r r}{v_x}$$. The term $$\Delta M$$ represents an external yaw moment, which in our context is the control input intended to stabilize the electric car after a blowout.
Under steady-state cornering conditions (constant speed, constant yaw rate), we can derive the desired yaw rate $$r_d$$ that respects physical limits imposed by tire-road friction:
$$ r_d = \min\left( \left| \frac{v_x / L}{1 + K v_x^2} \delta_f \right|, \left| \frac{\mu g}{v_x} \right| \right) \cdot \text{sign}(\delta_f) $$
where $$L = a_f + a_r$$ is the wheelbase, $$g$$ is gravity, and $$K = \frac{m}{L^2}\left(\frac{a_f}{C_r} – \frac{a_r}{C_f}\right)$$ is the stability factor.
When a tire blowout occurs, it introduces an abrupt disturbance. This includes an additional yaw moment $$\Delta M_1$$ and an effective additional steering angle $$\delta_1$$ due to the force imbalance and changes in rolling radius. Accounting for these blowout effects and general external disturbances $$\mathbf{d} = [d_1, d_2]^T$$, the actual error dynamics for the electric car, representing the deviation from the desired stable motion, can be formulated as:
$$ \begin{bmatrix} \dot{e}_\beta \\ \dot{e}_r \end{bmatrix} = \mathbf{A} \begin{bmatrix} e_\beta \\ e_r \end{bmatrix} + \mathbf{B}_1 \delta_1 + \mathbf{B}_2 (\Delta M_1 + \mathbf{d}) $$
where $$e_\beta = \beta – \beta_d$$ and $$e_r = r – r_d$$ are the tracking errors for sideslip angle and yaw rate, respectively. The matrices $$\mathbf{A}, \mathbf{B}_1, \mathbf{B}_2$$ contain vehicle parameters. The blowout-induced moment $$\Delta M_1$$ is dominant but unmeasurable. For control design, we consolidate the unknown constant effects ($$\Delta M_1$$ and parts of $$\mathbf{d}$$) into unknown parameters $$\theta_1$$ and $$\theta_2$$. This allows us to express the blowout stabilization problem for the electric car in a canonical form suitable for output feedback design:
$$ \begin{align*}
\dot{x}_1(t) &= x_2(t) + a_{11} x_1(t) + \theta_1 \\
\dot{x}_2(t) &= u(t) + a_{21} x_1(t) + a_{22} x_2(t) + \theta_2
\end{align*} $$
where $$x_1 = e_\beta$$, $$x_2$$ is related to $$e_r$$, $$u$$ is the control input (synthesized from differential braking or active steering), and $$a_{11}, a_{21}, a_{22}$$ are known constants derived from the vehicle parameters. The measurable output is $$y(t) = x_1(t)$$. The control objective is to drive $$x_1$$ and $$x_2$$ to zero in fixed time.
Control System Design
The proposed control strategy has two main components: an Input-Driven Filter to estimate the unmeasured state, and a Fixed-Time Output Feedback Controller with robust adaptation.
1. Input-Driven Filter Design
Since only the output $$y = x_1$$ is measurable, we design a filter to estimate the full state vector $$\mathbf{x} = [x_1, x_2]^T$$. A linear input-driven filter is constructed as:
$$ \begin{align*}
\dot{\hat{x}}_1 &= \hat{x}_2 – l_1 (\hat{x}_1 – x_1) \\
\dot{\hat{x}}_2 &= u – l_2 (\hat{x}_1 – x_1)
\end{align*} $$
where $$\hat{\mathbf{x}} = [\hat{x}_1, \hat{x}_2]^T$$ is the state estimate, and $$l_1, l_2 > 0$$ are filter gains. Defining the estimation error as $$\mathbf{r} = \mathbf{x} – \hat{\mathbf{x}} = [r_1, r_2]^T$$, its dynamics are:
$$ \dot{\mathbf{r}} = \mathbf{A}_c \mathbf{r} + \mathbf{F}(\mathbf{x}) + \mathbf{\Theta} $$
where $$\mathbf{A}_c = \begin{bmatrix} -l_1 & 1 \\ -l_2 & 0 \end{bmatrix}$$ is made Hurwitz by proper choice of $$l_1, l_2$$, $$\mathbf{F}(\mathbf{x})$$ contains known nonlinearities, and $$\mathbf{\Theta} = [\theta_1, \theta_2]^T$$ represents the unknown constants. For any symmetric positive definite matrix $$\mathbf{Q}$$, there exists a symmetric positive definite matrix $$\mathbf{P}$$ satisfying the Lyapunov equation:
$$ \mathbf{A}_c^T \mathbf{P} + \mathbf{P} \mathbf{A}_c = -\mathbf{Q} $$
2. Fixed-Time Output Feedback Controller Design
The controller is designed using the backstepping technique based on the estimated states $$\hat{x}_1$$ and $$\hat{x}_2$$. We define the error coordinates:
$$ z_1 = \hat{x}_1, \quad z_2 = \hat{x}_2 – \alpha_1 $$
where $$\alpha_1$$ is a virtual control law for the first step.
Step 1: Consider the Lyapunov function candidate $$V_1 = \frac{1}{2} z_1^2 + \frac{1}{2\lambda} \tilde{\mu}^2$$, where $$\tilde{\mu} = \mu – \hat{\mu}$$ is the estimation error for an adaptive parameter $$\mu$$ which bounds the combined effects of $$\theta_1$$ and filter errors. Its derivative is:
$$ \dot{V}_1 \leq z_1 (z_2 + \alpha_1 + a_{11} z_1 + \hat{\mu} \Phi(z_1) + \ldots) – \frac{1}{\lambda} \tilde{\mu} \dot{\hat{\mu}} $$
where $$\Phi(\cdot)$$ is a smooth function. We design the virtual control $$\alpha_1$$ and the adaptive law for $$\hat{\mu}$$ with fixed-time convergence properties:
$$ \begin{align*}
\alpha_1 &= -k_1 z_1 – l_1 z_1^{[2p-1]} – \hat{\mu} \tanh\left(\frac{z_1}{\omega_1}\right) – a_{11} z_1 \\
\dot{\hat{\mu}} &= \lambda \left( z_1 \tanh\left(\frac{z_1}{\omega_1}\right) – b \hat{\mu}^{[2q-1]} – \rho \hat{\mu} \right)
\end{align*} $$
Here, $$z^{[a]} \triangleq |z|^a \text{sign}(z)$$, $$k_1, l_1, b, \rho, \lambda, \omega_1 > 0$$ are design constants, and $$p, q$$ are ratios satisfying $$p = p_1/p_2$$, $$q = q_1/q_2$$ with $$p_1 > p_2$$, $$q_1 < q_2$$ being positive odd integers, ensuring $$p>1$$ and $$0<q<1$$. achieving="" fixed-time="" is="" key="" p="" stability.
Step 2: Consider the Lyapunov function $$V_2 = V_1 + \frac{1}{2} z_2^2$$. Its derivative involves the actual control input $$u$$. We design the final control law as:
$$ u = -k_2 z_2 – l_2 z_2^{[2p-1]} + \dot{\alpha}_1 – \frac{\partial \alpha_1}{\partial \hat{\mu}} \dot{\hat{\mu}} – a_{21} \hat{x}_1 – a_{22} \hat{x}_2 – \frac{1}{2} z_2 \left\| \frac{\partial \alpha_1}{\partial \hat{x}_1} \right\|^2 $$
where $$k_2, l_2 > 0$$, and $$\dot{\alpha}_1$$ is computed using the filter dynamics and the adaptive law.
The composite Lyapunov function for the entire closed-loop system (estimation errors and tracking errors) is $$V = \frac{1}{2} \mathbf{r}^T \mathbf{P} \mathbf{r} + \frac{1}{2} z_1^2 + \frac{1}{2\lambda} \tilde{\mu}^2 + \frac{1}{2} z_2^2$$. Through rigorous analysis using Lyapunov theory and inequalities for fixed-time systems, it can be proven that its derivative satisfies:
$$ \dot{V} \leq -c V^p – \psi V^q + A $$
where $$c, \psi, A$$ are positive constants. This inequality is the hallmark of fixed-time stable systems. It guarantees that the system states converge to a small residual set around the origin within a fixed time $$T_{\max}$$, independent of the initial conditions. The settling time is bounded by:
$$ T \leq T_{\max} = \frac{1}{c(1-p)} + \frac{1}{\psi(1-q)} $$
Simulation Analysis and Results
To validate the proposed fixed-time output feedback controller for the electric car under a tire blowout, numerical simulations were conducted. The parameters for the electric car model used in the simulation are listed in Table 2.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Mass ($$m$$) | 2130 kg | Yaw Inertia ($$I$$) | 1110 kg·m² |
| $$a_f$$ | 1.103 m | $$a_r$$ | 1.244 m |
| $$C_f$$ | 20000 N/rad | $$C_r$$ | 22000 N/rad |
The controller parameters were set as $$l_1=1, l_2=2, k_1=12, k_2=3.8$$, with $$p=1.2$$ and $$q=0.8$$. The scenario involves a front-left tire blowout while the electric car is traveling in a straight line at different speeds. The performance is compared between the uncontrolled case (driver takes no action) and the controlled case using the proposed method.
Case 1: Initial Speed = 60 km/h
In the uncontrolled scenario, the electric car experiences significant deviation. The yaw rate peaks at about 7.7 deg/s and settles to a steady-state error of 4.7 deg/s. The sideslip angle reaches -1.62 deg. With the proposed fixed-time control applied, both the yaw rate and sideslip angle converge rapidly to near zero within approximately 4.26 seconds, demonstrating effective stabilization of the electric car.
Case 2: Initial Speed = 90 km/h
The higher speed amplifies the instability. Without control, the yaw rate and sideslip angle errors are larger. When the controller is engaged, the states of the electric car converge to zero even faster, in about 3.83 seconds, showcasing the controller’s effectiveness across different operating conditions for the electric car.
Case 3: Initial Speed = 120 km/h
This represents a high-speed emergency. The uncontrolled dynamics are more severe. The proposed controller successfully brings the yaw rate and sideslip angle of the electric car back to stability within 4.06 seconds, confirming its robustness and fixed-time convergence property irrespective of the initial speed of the electric car.
The estimation errors $$r_1$$ and $$r_2$$ from the input-driven filter for all three speeds converge to a small neighborhood of zero quickly, confirming the accuracy of the state estimation and validating the output feedback approach for the electric car system. The consistent performance across speeds highlights the practicality of this method for real-world application in electric cars.
Conclusion
This paper has presented a novel fixed-time output feedback control strategy to enhance the safety of electric cars during a tire blowout event. The key challenge of requiring full state measurement was overcome by employing an input-driven filter. A fixed-time adaptive law was designed to handle unknown disturbances arising from the blowout, and a robust adaptive term was incorporated to compensate for observer errors. The theoretical analysis proves that all signals in the closed-loop system are bounded and that the tracking errors converge to a small region near zero within a fixed time, which is independent of the electric car’s initial state. Simulation results under various driving speeds confirm the controller’s effectiveness, rapid response, and robustness. This work provides a viable and theoretically sound solution for improving the active safety systems of modern electric cars against sudden tire failures.
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