A Leader–Follower Sequential Game Approach to Optimizing Parameters for Intelligent Vehicle Formation Control

Abstract

A vehicle formation control algorithm is presented to deal with the ineluctable uncertainty and guarantee the vehicle platoon Lyapunov stability (uniform boundedness, uniform ultimate boundedness, and string stability). The uncertainty here is nonlinear, possibly fast time-varying, and is assumed to be within the confines of a known set. Most important of all, the control parameters are optimized to achieve better performances. A Pareto optimal approach is employed to solve such an optimal problem. In the optimal procedure, the control parameters are considered as players and their corresponding value ranges are regarded as the decision sets. The cost function of any player includes three parts: the state cost portion, the time cost portion, and the control cost portion. The optimal parameters minimize the cost functions. The resulting vehicle formation control with optimal parameters impels the vehicle platoon both Lyapunov stability and low cost.

Appendix

Membership function: A function that the values assigned to the elements of the universal set fall within a specified range and indicate the membership grade of these elements in the set in question is called membership function.

$\alpha$-cut and strong $\alpha$-cut: For a given fuzzy set $\Omega$ defined on X and any number $\alpha \in [0,1]$, the $\alpha$-cut of fuzzy set $\Omega$ is defined as $^{\alpha }\Omega =\{x \mid \mu _{\Omega }(x)\ge \alpha \}$, the strong $\alpha$-cut of fuzzy set $\Omega$ is defined as $^{\alpha +}{\Omega } =\{x \mid \mu _{\Omega }(x)> \alpha \}$.

Fuzzy number: To qualify as a fuzzy number, a fuzzy set $\Omega$ must possess the following properties: (i) $\Omega$ is a normal fuzzy set; (ii) $\Omega$ is convex; (iii) the support of $\Omega$ must be bounded; (iv) for each $\alpha \in (0, 1]$, $^{\alpha }{\Omega }$ is a closed interval in $\mathbf{R}$.

Arithmetic Operations: Let $\Omega _1$, $\Omega _2$ denote two fuzzy numbers, and $*^{\alpha }{\Omega }{_1}=[a_1, b_1]$, $^{\alpha }{\Omega }{_2}=[a_2, b_2]$, $a_1<b_1$, $a_2<b_2$. Then, we have the following operations:

$$\begin{aligned}& ^{\alpha }{(\Omega _1+\Omega _2)} =[a_1+a_2, b_1+b_2], \end{aligned}$$

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$$\begin{aligned}& ^{\alpha }{(\Omega _1-\Omega _2)} =[a_1-b_2, b_1-a_2], \end{aligned}$$

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$$\begin{aligned}& ^{\alpha }{(\Omega _1\cdot \Omega _2)} =[\min (a_1a_2, a_1b_2, b_1a_2, b_1b_2), \nonumber \\&~~~~~~~~~~~~~~~~~\max (a_1a_2, a_1b_2, b_1a_2, b_1b_2) ], \end{aligned}$$

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$$\begin{aligned}& ^{\alpha }{(\Omega _1/ \Omega _2)} =[\min (a_1/a_2, a_1/b_2, b_1/a_2, b_1/b_2), \nonumber \\&~~~~~~~~~~~~~~~~~\max (a_1/a_2, a_1/b_2, b_1/a_2, b_1/b_2)]. \end{aligned}$$

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D-operation For a fuzzy set ${\mathcal {A}}=\{\upsilon , \mu _{N}(\upsilon ) \mid \upsilon \in {A}\}$ and any function $f(\upsilon ): A\rightarrow {\mathbf{R}}$, the D-operation of $f(\upsilon )$ is defined as

$$\begin{aligned} {D[f(\upsilon )]}=\frac{\int _{A} f(\upsilon ) \mu _{A}(\upsilon ) d \upsilon }{\int _{A} \mu _{A} (\upsilon ) d \upsilon }. \end{aligned}$$

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