## 2 Nonlinear Feedback Control Analysis of Visual Information in Foggy Section of Expressway
When \(\mathfrak{t}\in[0,+\infty)\), we multiply both ends of Equation (2) with \(e^{-st}\), and the following equation can be obtained through the Laplace transform [31]:  
\[Y(s)(\tau_{\mathfrak{t}}s+1)=X(s)+\tau_{\mathfrak{t}}y_{0} \tag{3}\]  
The deformation is as follows:  
\[Y(s)=\frac{X(s)}{\tau_{\mathfrak{t}}s+1}+\frac{\tau_{\mathfrak{t}}y_{0}}{\tau_ {\mathfrak{t}}s+1}=\frac{1}{\tau_{\mathfrak{t}}s+1}[X(s)+\tau_{\mathfrak{t}}y_ {0}] \tag{4}\]  
Supposing the following:  
\[F(s)=\frac{1}{\tau_{\mathfrak{t}}s+1} \tag{5}\]  
Equation (4) can be expressed as follows:  
\[Y(s)=F(s)[X(s)+\tau_{\mathfrak{t}}y_{0}] \tag{6}\]  
\(F(s)\) is the transfer function of the input and output control system. \(\frac{X(s)}{\tau_{\mathfrak{t}}s+1}\) represents the output produced by the input, abbreviated as \(Y_{i}(s)\); and \(\frac{\tau_{\mathfrak{t}}y_{0}}{\tau_{\mathfrak{t}}s+1}\) represents the output generated by the initial condition, abbreviated as \(Y_{c}(s)\). Equation (4) can be expressed as follows:  
\[Y(s)=Y_{i}(s)+Y_{c}(s) \tag{7}\]  
\(Y_{c}(s)\) represents the visual information input to the driver driving on the road section, where agglomerate fog takes place frequently, in normal weather at a speed within the design range. \(Y_{i}(s)\) represents the visual information input resulting from changes in visibility, driving speed, and other factors in agglomerate fog environments. For the purpose of accuracy, \(Y_{c}(s)\) is substituted by \(Y_{n}(s)\), \(Y_{i}(s)\) is substituted by \(Y_{\mathfrak{f}}(s)\), Equation (7) can be transformed into the following:  
\[Y(s)=Y_{f}(s)+Y_{n}(s) \tag{8}\]  
The free flow state of vehicles running on the expressway at the design speed in normal weather is regarded as the equilibrium state of the system. According to the relative motion theorems, the relative motion of the driver's eyes and the lane in this equilibrium state can be regarded as a dynamic process where the eyes are stationary, tracking a moving object. The linear feedback system observed by the eyes during driving is expressed as follows:  
\[h(x_{t}-y_{t})=y_{t}+1 \tag{9}\]  
In the Equation (9), \(x_{t}\) represents the target position, \(y_{t}\) represents eye's position, \(h\) represents filter or system gain with time invariant. When the transfer function is \(Y(z)/X(z)\), the \(z\) transformation [31] of a linear system is equal to the following:  
\[\left\{ \begin{array}{l}H(z)(X(z)-Y(z))=Y(z)\\ H(z)X(z)=Y(z)(1+H(z))\\ \frac{H(z)}{1+H(z)}=\frac{Y(z)}{X(z)}\end{array} \right. \tag{10}\]  
From Equations (3) and (4), obtained through Laplace transform, it can be seen that the visual information received by the eyes during driving in an agglomerate fog environment and the feedback system of the eyes form a nonlinear system. In other words,the visibility-based speed control by the driver would further act on the visual information. Speed, visibility, and visual information constitute a dynamic nonlinear system in which the three interact with each other. It is necessary to perform the analysis according to the input-output relation equation of the feedback servomechanism, and nonlinear transformation should be performed for the equation. From the perspective of systematology, when the driver driving on the expressway receives information through multiple senses such as vision, audition, and touch, establishes a personal comprehensive anticipation system based on the driving environment, and eventually makes decisions on acceleration or deceleration and lane selection, which he/she believes are safe and optimal. After the vehicle enters the foggy section, the visual information received by the driver will be incomplete due to the fog, or an illusion may even come into being due to light scattering by the fog. As a result, the balance of the personal comprehensive anticipation system formed in a fog-free environment is upset, thus impairing the correctness of the driver's decision-making and leading to decision delays or erroneous judgments.