A Novel Fixed-Time Zeroing Neural Network for Real-Time Inter-Vehicle Distance Estimation in Autonomous Driving

1. Introduction

The rapid advancement of autonomous driving has raised rigorous demands for vehicles’ environmental perception and decision-making capabilities. As a core perception task, real-time inter-vehicle distance measurement directly determines the reliability of path planning and driving safety for intelligent vehicles. Although LiDAR, visual sensors and other traditional techniques can acquire distance information, their real-time responsiveness and anti-interference ability are greatly restricted in dynamic traffic environments. In recent years, zeroing neural networks (ZNNs) have exhibited unique advantages in real-time optimization and dynamic system control due to their inherent fixed-time convergence property.

Different from conventional neural networks optimized by cost functions, ZNNs are constructed based on error dynamics. Their fundamental principle is to drive the system state to converge to an equilibrium point by designing specific error functions. By utilizing error derivative information to dynamically adjust network parameters, ZNNs realize real-time state tracking without relying on iterative operations, thus guaranteeing superior real-time performance. In general, ZNNs have three prominent merits: strong adaptability to real-time systems, low sensitivity to initial values with high robustness, and powerful capability to handle nonlinear and time-varying problems. Since first proposed by Zhang Yunong in [1], ZNNs have become a popular research direction and been widely applied in robot trajectory tracking [2,3,4], matrix inversion [5,6,7], time-varying quadratic programming [8,9,10] and other fields. Nevertheless, research on applying ZNNs to autonomous driving ranging tasks remains scarce. In this paper, a new fixed-time ZNN will be given to solve the problem of autonomous driving.

Fixed-time stability is a special form of finite-time stability which refers to the ability of a system to converge to an equilibrium point in finite time and regardless of the initial value chosen, the convergence time is not greater than a fixed upper bound. Therefore, fixed-time stability has stronger robustness and is widely used. The concept of fixed-time stability was first proposed by Polyakovin [11], who also established the corresponding stability criteria and derived the convergence time upper bound. Exploring new stability criteria and obtaining tighter convergence time limits have long been the key challenges in this research field. Several effective criteria and time upper bounds have been reported in previous studies [12,13], including our earlier works [7,14], which also discussed parameter selection rules and parameter sensitivity analysis in [7] and [14].

Inspired by the existing theoretical frameworks, this paper develops a new fixed-time ZNN model for the real-time distance estimation problem in autonomous driving. The fixed-time stability of the proposed model is strictly proven, and the corresponding convergence time upper bound is calculated. Combined with the motion characteristics of vehicles, a time-varying optimization model for inter-vehicle distance is established and solved using the designed ZNN.

The rest of this paper is organized as follows. Section 2 presents the preliminaries, including the basic ZNN framework, the definition of fixed-time stability and related lemmas. Section 3 constructs the time-varying optimization model for inter-vehicle distance and elaborates the design of the novel fixed-time ZNN, along with the stability proof and convergence time analysis. Section 4 conducts numerical simulations to validate the effectiveness of the proposed method. Section 5 concludes this work and prospects future research directions.

2. Preliminaries

In this part, the ZNN model is introduced and the definition of fixed-time stability and some criteria are reviewed.

2.1. Fixed-Time Stability

Considering The following system is considered,

$$\dot{x}\left(t\right)=f\left(x\left(t\right)\right),f\left(0\right)=0,x\in {\mathbb{R}}^n,x\left(0\right)=x_0.$$

(1)

where $f:\mathbb{D}\to {\mathbb{R}}^n$ denotes a continuous function, $\mathbb{D}$ represents an open neighborhood of the origin $x^{*}=0$ and $x^{*}$ is an equilibrium point of (1). Next, we will present some fixed-time stability theories.

Definition 1

([11]). For (1), its solution $x\left(t\right)$ exists and is unique for $t\in [0,T\left(x_0\right))$ starting from any initial condition $x_0\in \mathbb{U}$. Then, its equilibrium point is finite-time stable if it is Lyapunov stable. $T\left(x_0\right)$ is called as convergence time. In addition, its equilibrium point is fixed-time stable if $T\left(x_0\right)\le T_{max}$ for all $x_0$. $T_{max}$ is called as upper bound on the convergence time.

Lemma 1

([12]). For (1), if there is a Lyapunov function, $V\left(x\right)$, it satisfies

$$\dot{V}\left(x\left(t\right)\right)\le -{[a{V}^{\delta }\left(x\left(t\right)\right)+b]}^k,$$

(2)

where $a>0$, $b>0$, $\delta >0$, $k>0$ and $\delta k>1$. Then, its equilibrium is fixed-time stable and

$$T_{max}^1={\displaystyle \frac{1}{b^k}}{\left({\displaystyle \frac{b}{a}}\right)}^{{\displaystyle \frac{1}{\delta }}}\left(1+{\displaystyle \frac{1}{\delta k-1}}\right).$$

(3)

Lemma 2

([12]). For (1), if there is a Lyapunov function, $V\left(x\right)$, it satisfies

$$\dot{V}\left(x\left(t\right)\right)\le -{[a{V}^{\delta }\left(x\left(t\right)\right)+b{V}^{\theta }\left(x\left(t\right)\right)]}^k,$$

(4)

where $a>0$, $b>0$, $\delta >0$, $\theta >0$, $k>0$, $\delta k>1$ and $\theta k<1$. Then, its equilibrium is fixed-time stable and

$$T_{max}^2={\displaystyle \frac{1}{b^k}}{\left({\displaystyle \frac{b}{a}}\right)}^{{\displaystyle \frac{1-\theta k}{\theta -\delta }}}\left({\displaystyle \frac{1}{1-\theta k}}+{\displaystyle \frac{1}{\delta k-1}}\right).$$

(5)

Lemma 3

([7]). For (1), if there is a Lyapunov function, $V\left(x\right)$, it satisfies,

$$\dot{V}\left(x\left(t\right)\right)\le -z{[a{V}^{\delta }\left(x\left(t\right)\right)+b{V}^{\theta }\left(x\left(t\right)\right)]}^k,$$

(6)

where $a>0$, $b>0$, $\delta >0$, $\theta >0$, $k>0$, $\delta k>1$ and $\theta k<1$, $z=min\{e^t,e^{t_0}\}$,

$$e^{t_0}={\displaystyle \frac{\frac{1}{a^k(1-\theta k)}+\frac{1}{b^k(\delta k-1)}}{ln\left\{1+\frac{1}{a^k(1-\theta k)}\right\}+ln\left\{1+\frac{1}{b^k(\delta k-1)}\right\}}}.$$

(7)

Then, its equilibrium is fixed-time stable and

$$T_{max}^3=ln\left\{1+{\displaystyle \frac{1}{a^k(1-\theta k)}}\right\}+ln\left\{1+{\displaystyle \frac{1}{b^k(\delta k-1)}}\right\}.$$

(8)

2.2. Zeroing Neural Network Model

In [1], a special recurrent neural network model, $\dot{E}\left(t\right)=-\epsilon E\left(t\right)$, was designed to solve error equation. It was called as Zhang neural network or the zeroing neural network. The ZNN model with an activation function is shown:

$$\begin{array}{c}\hfill \dot{E}\left(t\right)=-\epsilon \mathcal{F}\left(E\left(t\right)\right),\end{array}$$

(9)

where $\epsilon >0$, $\mathcal{F}\left(x\right)$ is an activation function. By selecting different activation functions, the performance of the ZNN model will undergo significant changes. There exist some forms of $\mathcal{F}\left(x\right)$, $\mathcal{F}\left(x\right)=x$ in [1], $\mathcal{F}\left(x\right)={\displaystyle \frac{1+e^{-r}}{1-e^{-r}}}{\displaystyle \frac{1-e^{-rx}}{1+e^{-rx}}}$ for $r>2$ in [15], $\mathcal{F}\left(x\right)=x^p$ for $p>3$ in [16], ${\displaystyle \frac{1}{2}}{\left|x\right|}^r\mathrm{sign}\left(x\right)+{\displaystyle \frac{1}{2}}{\left|x\right|}^{1/r}\mathrm{sign}\left(x\right)$ for $1>r>0$ in [17], ${\displaystyle \frac{1}{2}}{\left|x\right|}^r\mathrm{sign}\left(x\right)+{\displaystyle \frac{1}{2}}x+{\displaystyle \frac{1}{2}}{\left|x\right|}^{1/r}\mathrm{sign}\left(x\right)$ for $1>r>0$ in [18], $\mathcal{F}\left(x\right)=k_1{\left|x\right|}^r\mathrm{sign}\left(x\right)+k_{2}x$ for $k_1>0$, $k_2>0$ and $1>r>0$ in [19], $\mathcal{F}\left(x\right)=e^t{\mathrm{sign}\left(x\right)\left(a\right|x|}^{\delta }+{b\left|x\right|}^{\theta }{)}^k$ for $a>0$, $b>0$, $\delta >0$, $\theta >0$, $k>0$, $\delta k>1$ and $\theta k<1$ in [7], and so on.

3. Model Building and Fixed-Time ZNN Design

In this part, a time-varying optimization model is used to obtain the distance between vehicles in autonomous driving. Then, we design a fixed-time ZNN model to solve the time-varying optimization model.

Figure 1 gives the autonomous driving schematic diagram and mathematical model diagram. In order to obtain real-time distance between vehicles represented by the red dashed line in Figure 1, the following time-varying optimization model is builded:

$$\begin{array}{ccc}\hfill \mathrm{minimize}\text{:}& & d={\displaystyle \frac{1}{2}}({(x_a\left(t\right)-x_b\left(t\right))}^2+{(y_a\left(t\right)-y_b\left(t\right))}^2)\hfill \\ \hfill \mathrm{subject}\mathrm{to}\text{:}& & F(x_b\left(t\right),y_b\left(t\right),x_1\left(t\right),y_1\left(t\right))=0\hfill \\ & & G(x_a\left(t\right),y_a\left(t\right),x_2\left(t\right),y_2\left(t\right))=0\hfill \\ & & f(x_1\left(t\right),y_1\left(t\right))=0\hfill \\ & & g(x_2\left(t\right),y_2\left(t\right))=0\hfill \end{array}$$

(10)

where $F(x_b\left(t\right),y_b\left(t\right),x_1\left(t\right),y_1\left(t\right))=0$ and $G(x_a\left(t\right),y_a\left(t\right),x_2\left(t\right),y_2\left(t\right))=0$ are used to represent the point set of the corresponding structure of the car B and A, $f(x_1\left(t\right),y_1\left(t\right))=0$ and $g(x_2\left(t\right),y_2\left(t\right))=0$ are are used to represent the central motion trajectory of the car B and A, respectively.

In order to solve (Section 3) by using the ZNN model, define the following Lagrangian function,

$$\begin{array}{ccc}\hfill L=& & {\displaystyle \frac{1}{2}}({(x_a\left(t\right)-x_b\left(t\right))}^2+{(y_a\left(t\right)-y_b\left(t\right))}^2)+\hfill \\ & & {\lambda }_{1}F(x_b\left(t\right),y_b\left(t\right),x_1\left(t\right),y_1\left(t\right))+\hfill \\ & & {\lambda }_{2}G(x_a\left(t\right),y_a\left(t\right),x_2\left(t\right),y_2\left(t\right))+\hfill \\ & & {\lambda }_{3}f(x_1\left(t\right),y_1\left(t\right))+{\lambda }_{4}g(x_1\left(t\right),y_1\left(t\right)),\hfill \end{array}$$

where ${\lambda }_i$ is called the Lagrange multiplier. Then, we have

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial L}{\partial x_a}}=(x_a\left(t\right)-x_b\left(t\right))+{\lambda }_2{\displaystyle \frac{\partial G}{\partial x_a}}=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial x_b}}=-(x_a\left(t\right)-x_b\left(t\right))+{\lambda }_1{\displaystyle \frac{\partial F}{\partial x_b}}=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial y_a}}=(y_a\left(t\right)-y_b\left(t\right))+{\lambda }_2{\displaystyle \frac{\partial G}{\partial y_a}}=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial y_b}}=-(y_a\left(t\right)-y_b\left(t\right))+{\lambda }_1{\displaystyle \frac{\partial F}{\partial y_b}}=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial {\lambda }_1}}=F(x_b\left(t\right),y_b\left(t\right),x_1\left(t\right),y_1\left(t\right))=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial {\lambda }_2}}=G(x_a\left(t\right),y_a\left(t\right),x_2\left(t\right),y_2\left(t\right))=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial {\lambda }_3}}=f(x_1\left(t\right),y_1\left(t\right))=0,\hfill \\ {\displaystyle \frac{\partial L}{\partial {\lambda }_4}}=g(x_2\left(t\right),y_2\left(t\right))=0.\hfill \end{array}\right.$$

(11)

Let

$$E=[{\displaystyle \frac{\partial L}{\partial x_a}},{\displaystyle \frac{\partial L}{\partial x_b}},{\displaystyle \frac{\partial L}{\partial y_a}},{\displaystyle \frac{\partial L}{\partial y_b}},{\displaystyle \frac{\partial L}{\partial {\lambda }_1}},{\displaystyle \frac{\partial L}{\partial {\lambda }_2}},{\displaystyle \frac{\partial L}{\partial {\lambda }_3}},{\displaystyle \frac{\partial L}{\partial {\lambda }_4}}].$$

Then,

$$\dot{E}\left(t\right)=[{\displaystyle \frac{\partial L}{\partial x_a}}{\dot{x}}_a,{\displaystyle \frac{\partial L}{\partial x_b}}{\dot{x}}_b,{\displaystyle \frac{\partial L}{\partial y_a}}{\dot{y}}_a,{\displaystyle \frac{\partial L}{\partial y_b}}{\dot{y}}_b,$$

$${\displaystyle \frac{\partial L}{\partial {\lambda }_1}}{\dot{\lambda }}_1,{\displaystyle \frac{\partial L}{\partial {\lambda }_2}}{\dot{\lambda }}_2,{\displaystyle \frac{\partial L}{\partial {\lambda }_3}}{\dot{\lambda }}_3,{\displaystyle \frac{\partial L}{\partial {\lambda }_4}}{\dot{\lambda }}_4].$$

It notes that the solution to the equation $E=0$ is the optimal solution of (Section 3). In order to solve the equation $E=0$, the following fixed-time ZNN model is designed,

$$\dot{E}\left(t\right)=-\epsilon \mathcal{F}\left(E\left(t\right)\right),\dot{e_i}\left(t\right)=-\epsilon \mathcal{F}\left(e_i\left(t\right)\right)$$

(12)

where $e_i\left(t\right)$ is the ith of $E\left(t\right)$, $\mathcal{F}\left(x\right)=e^t\mathrm{sign}\left(x\right){\left|x\right|}^{\varpi \left(x\right)}$ is an activation function and $\mathrm{sign}\left(x\right)=\left\{\begin{array}{c}1,x>0\\ 0,x=0\\ -1,x<0\end{array}\right.$, $\varpi \left(x\right)=\left\{\begin{array}{c}\tau >1,1<\left|x\right|<+\infty \\ 1/\tau <1,0\le \left|x\right|\le 1\end{array}\right.$ After the solution of (Section 3) is obtained, we have the distance between the front and rear vehicles $d=\sqrt{{(x_a^{*}\left(t\right)-x_b^{*}\left(t\right))}^2+{(y_a^{*}\left(t\right)-y_b^{*}\left(t\right))}^2}$. If $d<d_{min}$, the minimum distance allowed during driving, a deceleration command is added to the following vehicle to make $d=d+\Delta$ or an acceleration command is added to the vehicle ahead so that $d=d+\Delta$.

Next, we will give the fixed-time theorems for the proposed ZNN model (12).

Theorem 1.

For (Section 3), its solution can be obtained within a fixed-time using the proposed ZNN model (12) with arbitrary initial values, and the fixed-time is

$$T_{max}=ln\left(1+{\displaystyle \frac{2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1)\epsilon }}\right)+ln\left(1+{\displaystyle \frac{2^{\frac{\tau +3}{2}}}{(\tau -1)\epsilon }}\right).$$

(13)

Proof: Let $e^{+}\left(t\right)=max\left\{\right|e_i\left(t\right)\left|\right\}$. Notes that $-|e^{+}\left(t\right)|\le e_i\left(t\right)\le \left|e^{+}\left(t\right)\right|$. Thus, $e_i\left(t\right)$ must converge to zero as $e^{+}\left(t\right)=0$. Define the following Lyapunov function $V\left(t\right)={\left(e^{+}\left(t\right)\right)}^2/2$. Then, $\dot{V}\left(t\right)=e^{+}\left(t\right){\dot{e}}^{+}\left(t\right)$. According to (12), we have

$$\begin{array}{c}\hfill \dot{V}\left(t\right)=-\epsilon e^{+}\left(t\right)\mathcal{F}\left(e^{+}\left(t\right)\right)=-\epsilon e^t{\left(e^{+}\left(t\right)\right)}^{\varpi \left(e^{+}\left(t\right)\right)+1}.\end{array}$$

(14)

Case I:$V\left(0\right)={\left(e^{+}\left(0\right)\right)}^2/2\le 1$

By (14),

$$\begin{array}{c}\hfill \dot{V}=-\epsilon e^t{\left(e^{+}\left(t\right)\right)}^{{\displaystyle \frac{1+\tau }{\tau }}}=-\epsilon e^t{\left(2V\right)}^{{\displaystyle \frac{1+\tau }{2\tau }}}=-2^{{\displaystyle \frac{1+\tau }{2\tau }}}\epsilon e^t{V}^{{\displaystyle \frac{1+\tau }{2\tau }}}<0.\end{array}$$

(15)

By integrating, it can obtain that

$$\begin{array}{ccc}\hfill & & {\int }_0^t{V}^{-{\displaystyle \frac{1+\tau }{2\tau }}}dV=-2^{{\displaystyle \frac{1+\tau }{2\tau }}}\epsilon {\int }_0^t{e}^{t}dt.\hfill \end{array}$$

(16)

Then

$$\begin{array}{c}\hfill {\displaystyle \frac{2\tau }{\tau -1}}{V}^{{\displaystyle \frac{\tau -1}{2\tau }}}{|}_0^t=-2^{{\displaystyle \frac{1+\tau }{2\tau }}}\epsilon e^t{|}_0^t.\end{array}$$

Let $V\left(t\right)=0$, we have

$$\begin{array}{ccc}\hfill t=ln\left(1+{\displaystyle \frac{\frac{2\tau }{\tau -1}{V}^{\frac{\tau -1}{2\tau }}\left(0\right)}{2^{-\frac{1+\tau }{2\tau }}\epsilon }}\right)& =& ln\left(1+{\displaystyle \frac{2^{\frac{1+3\tau }{2\tau }}\tau V^{\frac{\tau -1}{2\tau }}\left(0\right)}{(\tau -1)\epsilon }}\right)\hfill \\ & \le & ln\left(1+{\displaystyle \frac{2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1)\epsilon }}\right).\hfill \end{array}$$

(17)

Cass II:$V\left(0\right)={\left(e^{+}\left(0\right)\right)}^2/2>1$

By (14), we have

$$\begin{array}{ccc}\hfill \dot{V}& =& -\epsilon e^t{\left(e^{+}\left(t\right)\right)}^{\tau +1}=-\epsilon e^t{\left(2V\right)}^{{\displaystyle \frac{\tau +1}{2}}}\hfill \\ & =& -2^{{\displaystyle \frac{\tau +1}{2}}}\epsilon e^t{V}^{{\displaystyle \frac{\tau +1}{2}}}<0.\hfill \end{array}$$

(18)

By integrating, it can obtain that

$$\begin{array}{ccc}\hfill & & {\int }_0^t{V}^{-{\displaystyle \frac{\tau +1}{2}}}dV=-2^{{\displaystyle \frac{\tau +1}{2}}}\epsilon {\int }_0^t{e}^{t}dt.\hfill \end{array}$$

(19)

Then,

$$\begin{array}{c}\hfill -{\displaystyle \frac{2}{\tau -1}}{V}^{{\displaystyle \frac{1-\tau }{2}}}{|}_0^t=-2^{{\displaystyle \frac{\tau +1}{2}}}\epsilon e^t{|}_0^t.\end{array}$$

Let $V\left(t\right)=1$, we have

$$\begin{array}{c}\hfill t=ln\left(1+{\displaystyle \frac{2}{\tau -1}}{\displaystyle \frac{1-V^{\frac{1-\tau }{2}}\left(0\right)}{2^{-\frac{\tau +1}{2}}\epsilon }}\right)\le ln\left(1+{\displaystyle \frac{2^{\frac{\tau +3}{2}}}{(\tau -1)\epsilon }}\right).\end{array}$$

(20)

According to (17)and (20), it notes that $V\left(t\right)$ must reach zero as

$$\begin{array}{ccc}\hfill t& \le & ln\left(1+{\displaystyle \frac{2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1)\epsilon }}\right)+ln\left(1+{\displaystyle \frac{2^{\frac{\tau +3}{2}}}{(\tau -1)\epsilon }}\right)=T_{max}.\hfill \end{array}$$

From the above, the solution of (Section 3) can be obtained for $t\le T_{max}$ starting with arbitrary initial values. The proof is completed.

Remark 1.

Now, we proceed to analyze the influence of the parameter ε and τ on $T_{max}$. By computing the derivative of $T_{max}$ to ε,

$${\displaystyle \frac{d{T}_{max}}{d\epsilon }}={\displaystyle \frac{1}{1+\frac{2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1)\epsilon }}}{\displaystyle \frac{-2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1){\epsilon }^2}}+{\displaystyle \frac{1}{1+\frac{2^{\frac{\tau +3}{2}}}{(\tau -1)\epsilon }}}{\displaystyle \frac{-2^{\frac{\tau +3}{2}}}{(\tau -1){\epsilon }^2}}<0,$$

it notes that the larger ε is, the smaller $T_{max}$ is. But

$${\displaystyle \frac{d{T}_{max}}{d\tau }}={\displaystyle \frac{1}{1+\frac{2^{\frac{1+3\tau }{2\tau }}\tau }{(\tau -1)\epsilon }}}{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{-2^{\frac{1+3\tau }{2\tau }}\frac{\tau -1}{2\tau }ln2-2^{\frac{1+3\tau }{2\tau }}}{{(\tau -1)}^2}}+$$

$${\displaystyle \frac{1}{1+\frac{2^{\frac{\tau +3}{2}}}{(\tau -1)\epsilon }}}{\displaystyle \frac{1}{\epsilon }}{\displaystyle \frac{2^{\frac{\tau +3}{2}}((\tau -1)\frac{ln2}{2}-1)}{{(\tau -1)}^2}},$$

it may be greater than or less than zero. In order to determine the selection strategy of τ, let $\epsilon =1$ and draw the image of ${\displaystyle \frac{d{T}_{max}}{d\tau }}$.

From Figure 2, it notes that ${\displaystyle \frac{d{T}_{max}}{d\tau }}<0$ for $1<\tau <4.67$ and ${\displaystyle \frac{d{T}_{max}}{d\tau }}>0$ for $\tau >4.67$. Then, we have the selection strategy of τ: (1)For ${\displaystyle \frac{d{T}_{max}}{d\tau }}<0$, the larger τ is, the smaller $T_{max}$ is. (2)For $\tau >4.67$, the larger τ is, the larger $T_{max}$ is.

Remark 2.

According to the proof of lemma 1, 2 and 3 in [7,12], it is not difficult to find that, they all ignored a certain term when they estimate the upper bound of convergence time. However, our method takes the form of piecewise function, which makes the upper bound of our estimation closer to the least upper bound.

4. Numerical Simulation and Application

In the part, an application are used to show the validity of fixed-time ZNN model (12).

For (Section 3), chooses $x_1\left(t\right)=t$, $y_1\left(t\right)=(3t^2+2t+5)sint+5rand(1,1)$, $x_2\left(t\right)=t-0.01$, $y_2\left(t\right)=(3{(t-0.01)}^2+2(t-0.01))sin(t-0.01)$, $F(x_b\left(t\right),y_b\left(t\right),x_1\left(t\right),y_1\left(t\right))={(x_b\left(t\right)-x_1\left(t\right))}^2+{(y_b\left(t\right)-y_1\left(t\right))}^2-r_1^2=0$ for $r_1=1$, $x_1\left(t\right)-0.5\le x_b\left(t\right)\le x_1\left(t\right)-0.5$ and $y_b\left(t\right)>y_1\left(t\right)$ , $G(x_a\left(t\right),y_a\left(t\right),x_2\left(t\right),y_2\left(t\right))={(x_a\left(t\right)-x_2\left(t\right))}^2+{(y_a\left(t\right)-y_2\left(t\right))}^2-r_2^2=0$ for $r_2=1.1$, $x_2\left(t\right)-0.6\le x_a\left(t\right)\le x_2\left(t\right)-0.6$ and $y_a\left(t\right)>y_2\left(t\right)$.

In simulation, we set $\epsilon =4$ and $\tau =100$. By computing (13), $T_{max}=0.0773s$. Figure 3 shows the result. The solution of (Section 3) can be obtained before $T_{max}=0.0773s$.

Remark 3.

Compared with traditional LiDAR and visual sensing methods, the proposed fixed-time ZNN model achieves a much shorter response time. Meanwhile, different from conventional finite-time ZNNs whose convergence time is affected by initial states, this model maintains a fixed convergence upper bound regardless of initial error values, which greatly enhances the stability and reliability in dynamic and complex driving scenarios.

Remark 4.

The selected parameters $\epsilon =4$ and $\tau =100$ follow the parameter selection strategies discussed in Remark 1 of Section 3. Under this parameter combination, the model balances convergence speed and numerical stability well. In practical engineering deployment, users can adjust ε and τ according to the real-time requirements of autonomous driving systems: increasing ε can further shorten the convergence time, while a reasonable selection of τ can avoid numerical oscillation in extreme error conditions.

5. Conclusions

This paper proposes a novel fixed-time zeroing neural network and applies it to the real-time inter-vehicle distance estimation task for autonomous driving. A specially designed piecewise activation function is embedded into the ZNN framework, which effectively accelerates the network convergence. Theoretical analysis strictly proves the fixed-time stability of the proposed model, and a tight upper bound of the convergence time is derived analytically. In addition, we systematically analyze the influence of key parameters on the convergence performance and present targeted parameter selection strategies for practical engineering deployment. Numerical simulation results fully validate the superior performance of the proposed method in terms of convergence speed and fixed-time convergence capability. For future research, we will extend the proposed ZNN model to complex traffic scenarios involving multiple vehicles and obstacles. Meanwhile, we will further optimize the activation function to enhance the model’s anti-noise ability and adaptability to extreme driving environments, so as to promote the practical engineering application of ZNN in autonomous driving perception systems.