Cooperative-Game-Oriented Optimal Design in Vehicle Lateral Stability Control with Fuzzy Uncertainties

Abstract

Robustness is a major concern when engineers design lateral stability controllers for intelligent vehicles. The system uncertainty derives from the internal parametric perturbation and the external disturbance. The fuzzy set theory is adopted to describe the bound of the uncertainty. Hence the vehicle lateral dynamical system becomes a fuzzy dynamical system. An explicit robust optimal control law is then proposed to ensure the uniform boundedness and uniform ultimate boundedness of the system. The control law is deterministic and is not IF-THEN fuzzy rule based. A two-player cooperative game is established to determine the optimal value of tunable parameters. The cost functions of the players are developed based on the system performance and the control cost. The Pareto optimal is then obtained using the cooperative game theory. The effectiveness of the proposed method is proven by numerical simulations and Hardware-in-the-loop (HIL) experiments. This paper makes a breakthrough by casting the cooperative game theory and the fuzzy set theory into vehicle lateral stability control.

Appendix: Derivative Cost Functions Using D Operator

As mentioned in Sect. 4, the cost function should be deterministic to formulate the cooperative game. Therefore, we introduce the D operator to form the mapping from fuzzy set to real number.

Definition 2

Consider a fuzzy set

$$\begin{aligned} \begin{aligned} N=\left\{ (\xi ,\mu _{\Xi }(\xi )|\xi \in \Xi \right\} . \end{aligned} \end{aligned}$$

(77)

For any function $f:\Xi \rightarrow {\mathbb {R}}$, the D operation $D[f(\xi )]$ is given by

$$\begin{aligned} \begin{aligned} D[f(\xi )]=\dfrac{\int _\Xi f(\xi )\mu _\Xi (\xi )\text {d}\xi }{\int _\Xi \mu _\Xi (\xi )\text {d}\xi }. \end{aligned} \end{aligned}$$

(78)

According to definition 2, the D operator can be viewed as a defuzzification method and has the follow properties

$$\begin{aligned} \begin{array}{lcl} D[cf(\xi )]=cD[f(\xi )] &{} &{} \text {for any crisp constant } c\in {\mathbb {R}},\\ D[f(\xi )]=f(\xi ) &{} &{} \text {if } \mu _\Xi (\xi )=1, \text { for\,all } \xi \in \Xi . \end{array} \end{aligned}$$

(79)

The transient state and steady state system performance can be derived as follows

$$\begin{aligned} \begin{aligned} \eta _t&=\int _{t_0}^{+\infty } \eta _1^2(\gamma ,\epsilon ,t)\text {d}t\\&=\big ( V_0^2-2\kappa h(\gamma ,\epsilon )V_0\rho ^{-\frac{1}{\epsilon -1}}\delta ^{\frac{\epsilon }{\epsilon -1}}\\&\quad +\kappa ^2h^2(\gamma ,\epsilon )(\rho ^2)^{-\frac{1}{\epsilon -1}}(\delta ^2)^{\frac{\epsilon }{\epsilon -1}}\big ) \frac{\kappa }{2},\\ \eta _s&=\eta _2^2(\gamma ,\epsilon )=\kappa ^2h^2(\gamma ,\epsilon )(\rho ^2)^{-\frac{1}{\epsilon -1}}(\delta ^2)^{\frac{\epsilon }{\epsilon -1}}. \end{aligned} \end{aligned}$$

(80)

Based on Eqs. (77)–(80), we now propose the explicit cost functions for the two-player cooperative game

$$\begin{aligned} \begin{aligned} J_\gamma (\gamma ,\epsilon )&=\frac{\kappa }{2}D[V_0^2]-\kappa ^2 h(\gamma ,\epsilon )D[V_0]D[\rho ]^{-\frac{1}{\epsilon -1}}D[\delta ]^{\frac{\epsilon }{\epsilon -1}}\\&+\frac{\kappa ^3}{2}h^2(\gamma ,\epsilon )D[\rho ^2]^{-\frac{1}{\epsilon -1}}D[\delta ^2]^{\frac{\epsilon }{\epsilon -1}}+\gamma ^2,\\ J_\epsilon (\gamma ,\epsilon )&=\kappa ^2h^2(\gamma ,\epsilon )D[\rho ^2]^{-\frac{1}{\epsilon -1}}D[\delta ^2]^{\frac{\epsilon }{\epsilon -1}}+\epsilon ^2.\\ \end{aligned} \end{aligned}$$

(81)

We denote

$$\begin{aligned} \begin{matrix} \alpha _1=\dfrac{\kappa }{2}D[V(t_0)^2]&{}&{} \alpha _2=\kappa ^2D[V(t_0)]D[\delta ] &{}&{} \alpha _3=\dfrac{D[\delta ]}{D[\rho ]} \\ \alpha _4=\dfrac{\kappa ^3}{2}D[\delta ^2]&{}&{} \alpha _5=\dfrac{D[\delta ^2]}{D[\rho ^2]}&{}&{} \alpha _6=\kappa ^2D[\delta ^2]. \end{matrix} \end{aligned}$$

(82)

Then we have

$$\begin{aligned} \begin{aligned} J_\gamma (\gamma ,\epsilon )&=\alpha _1-\alpha _2h(\gamma ,\epsilon )\alpha _3^{\frac{1}{\epsilon -1}}+\alpha _4h^2(\gamma ,\epsilon )\alpha _5^{\frac{1}{\epsilon -1}}+\gamma ^2,\\ J_\epsilon (\gamma ,\epsilon )&=\alpha _5h^2(\gamma ,\epsilon )\alpha _6^{\frac{1}{\epsilon -1}}+\epsilon ^2. \end{aligned} \end{aligned}$$

(83)