Drivers' Lane Choice Model Based on Mixed Fuzzy Multi-person and Multi-objective Non-cooperative Game
-
摘要: 建立汽车安全驾驶辅助系统（包括安全驾驶预警系统）是保证交通安全的有效手段.准确预测车辆集群态势是汽车安全辅助驾驶的前提，车道选择是车辆集群态势发生转移最为根本的原因，也是交通流理论研究的基本内容.以往研究没有综合考虑车辆集群复杂态势下各运动实体特征及其操控者类型，以及多个车道间车辆的冲突对车道选择的影响.为此，本文综合考虑各运动实体特征及其操控者类型，基于混合模糊多人多目标非合作博弈方法，建立城市快速路基本路段上的驾驶员车道选择模型.通过分析各方驾驶员在不同车道选择策略下的收益，确定换道博弈的Nash均衡，得到驾驶员最优车道选择策略.研究结果表明：基于混合模糊多人多目标非合作博弈方法建构的驾驶员车道选择模型，其预测准确率可达到85.2%.Abstract: Vehicle safety driving assistance system (including driving safety alerting system) is an effective means to ensure traffic safety while accurate prediction of vehicle cluster situation is the premise of automobile safety assistant driving system. The drivers' lane choice process is not only the root cause of the transformation of vehicle cluster situation, but also the basic topic of traffic flow research. Previous studies do not synthetically consider the characteristics of individual traffic entities and the types of manipulators under the complex vehicle cluster situation, neither the influence of vehicles conflicts under multiple lanes on lane choice is taken into account. Therefore, in this paper, considering the types of vehicle manipulators and the characteristics of each movement entity, the model of drivers' lane choice on basic segment of urban expressway is built based on mixed fuzzy multi-person and multi-objective non-cooperative game. Drivers' profits under the different combinations of lane choice behaviors are analyzed, Nash equilibrium in the game process is confirmed, and the drivers' optimal lane choice strategy in a dynamic game is obtained. The results show that the model's prediction accuracy of lane change reaches 85.2%.
-
1. Introduction
Fractional calculus and fractional differential equations have received considerable interest in the recent forty years. Fractional derivative means that the order of differentiation can be an arbitrary real number and even it can be a complex number. Fractional derivative modelling has been applied to many scientific and engineering fields, such as quantum mechanics [1], viscoelasticity and rheology [2], electrical engineering [3], electrochemistry [4], biology [5], biophysics and bioengineering [6], signal and image processing [7], mechatronics [8], and control theory [9]-[11]. Although few mathematical issues of fractional derivative remain unsolved, most of the difficulties have been overcome, and the applications of fractional calculus in above fields indicate that the fractional models can depict the property and behavior of a real-world problem more accurately. For a comprehensive review of fractional calculus, we refer readers to some monographs [12]-[14] and references therein. In contrast to integer order derivative, the way of identifying fractional derivative is not unique. There are several types of definitions, such as Riemann-Liouville derivative, Caputo derivative, Grünwald-Letnikov derivative, and so on. More details can be found in [13, Chapter 2]. In the recent years, the study of dynamical system with fractional order derivative becomes more and more popular [15]-[19]. Moreover, the dynamics in fractional dynamical system is more interesting.
Returning back to the fractional derivative, since it has several different definitions, how to develop a generalized form which can unify all the existing fractional derivatives becomes one important topic in fractional calculus [20]-[22]. Recently, a class of new generalized fractional integral and generalized fractional derivative is introduced in [22]. The new generalized fractional integral and generalized fractional derivative depend on a scale function and a weight function, which makes them more general. When the scale function and the weight function reduce to some specific cases, the generalized fractional operators will reduce to Riemann-Liouville fractional integral, Riemann-Liouville fractional derivative and Caputo fractional derivative and so on. However, the study of this new generalized fractional integral and generalized fractional derivative are in the very beginning stage now [23]-[26]. In [24], we show that in generalized fractional diffusion equation, the scale function allows the response domain to be scaled differently. It is required that the scale function should be strictly monotonically increasing or decreasing. A convex increasing scale function will compress the response domain towards to the initial time. A concave increasing scale function will stretch the response domain away from the initial time. The weight function allows the response to be assessed differently at different time, since in many applications, we may require an event to be weighed differently at different time point. For example, modeling of memory of a child may require a heavy weight at current time point, whereas the same for an older person may require more weight on the past. To be an initial attempt of application to chaotic dynamical systems, in this paper, we define a class of new generalized fractional chaotic systems by replacing the original derivatives with the new generalized fractional derivative, then apply a finite difference scheme to study the numerical solutions of two different generalized fractional chaotic systems, namely generalized fractional Lotka-Volterra system (GFLVS) and generalized fractional Lorenz system (GFLS). Their complex dynamics will be discussed, and the dynamic behavior depending on the weight and scale function will be shown graphically.
The rest of this paper is organized as follows: In Section 2, the preliminaries of fractional calculus are given. The new generalized fractional integral and generalized fractional derivative are shown. A finite difference approach for solving equations with generalized fractional derivative is carried out. In Section 3, we define the chaotic systems using the generalized fractional derivative of Caputo type, i.e., the GFLVS and GFLS. Some interesting dynamics of those two systems are shown graphically. Finally, the conclusions are drawn in Section 4.
2. Mathematical Preliminaries
In this section, we introduce the preliminaries of generalized fractional derivatives, and show a proper numerical method for differential equations with such derivatives.
2.1 Generalized Fractional Calculus
Let us begin with the common fractional operators. In calculus, the $n$ -fold integral of an integrable function $u(t)$ is defined as
$ I^{n}u(t)=\overbrace{\int^t_0\cdots\int^t_0}^{n\ {\rm times}}u(s)ds\cdots{ds}= \int^t_0\frac{(t-s)^{n-1}}{(n-1)!}u(s)ds $
where $t\geq{0}$ , and $u(0)$ is well-defined. Replacing the positive integer $n$ by a real number $\alpha>0$ , we have the following definition.
Definition 1[13]: The left Riemann-Liouville fractional integral of order $\alpha>0$ of a function $u(t)$ is defined as
$ \begin{align} \left(I^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(\alpha)}\int^t_0(t-s)^{\alpha-1}u(s)ds \end{align} $
(1) provided the integral is finite, where $\Gamma(\alpha)$ is the Gamma function.
The Riemann-Liouville fractional integral plays an important role in defining fractional derivatives. There are two basic approaches to define the fractional derivative, i.e., "first integration then differentiation" and "first differentiation then integration". The corresponding fractional derivatives are called Riemann-Liouville fractional derivative and Caputo fractional derivative, and the definitions are given as follows.
Definition 2[13]:The left Riemann-Liouville fractional derivative of order $n-1 < \alpha < n$ of a function $u(t)$ is defined as
$ \begin{align} \left(D^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n-\alpha)}\left(\frac{d^n}{dt^n}\right) \int^t_0(t-s)^{n-\alpha-1}u(s)ds \end{align} $
(2) provided the right side of the identity is finite.
Definition 3[13]: The left Caputo fractional derivative of order $n-1 < \alpha < n$ of a function $u(t)$ is defined as
$ \begin{align} \left({^cD}^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n-\alpha)}\int^t_0(t-s)^{n-\alpha-1}u^{(n)}(s)ds \end{align} $
(3) provided the right side of the identity is finite.
Besides above, there also exist right Riemann-Liouville integral and derivative, and right Caputo fractional derivative [13]. Mathematically, the Riemann-Liouville and Caputo fractional operators are used in applications frequently. In most real-world models, we always employ the left Caputo fractional derivative. One reason is that we will study generalized fractional dynamical system later, and the derivative is taken with respect to time variable. In physical models, time is always running forward. The other reason is that in the differential equations with Caputo fractional derivative, the initial conditions are taken in the same form as for integer-order differential equations which have clear physical meanings in the practical application and can be easily measured [14]. In what follows, we will introduce the generalized fractional integral and derivative proposed in [22]. They extend nearly all the existing fractional operators. Now we list the generalized fractional integral and derivative defined on positive half axis. They will be used to define the generalized fractional chaotic systems in next section.
Definition 4[22]: The left generalized fractional integral of order $\alpha>0$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(I^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{-1}}{\Gamma(\alpha)}\int^{t}_{0} \frac{w(s)\sigma'(s)u(s)}{[\sigma(t)-\sigma(s)]^{1-\alpha}}ds \end{align} $
(4) provided the integral exists, where $\sigma'(s)$ indicates the first derivative of the scale function $\sigma$ .
Definition 5[22]: The left generalized derivative of order $m$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(D^m_{[\sigma, w;L]}u\right)(t) = [w(t)]^{-1}\left[\left(\frac{1}{\sigma'(t)}D_t\right)^m(w(t)u(t))\right] \end{align} $
(5) provided the right-side of equation is finite, where $m$ is a positive integer.
Definition 6[22]: The Caputo type left generalized fractional derivative of order $\alpha>0$ of a function $u(t)$ with respect to a scale function $\sigma(t)$ and a weight function $w(t)$ is defined as
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \left(I^{m-\alpha}_{0+;[\sigma, w]}D^m_{[\sigma, w;L]}u\right)(t) \end{align} $
(6) provided the right-side of equation is finite, where $m-1\leq$ $\alpha$ $ < $ $m$ , and $m$ is a positive integer. Particularly, when $0 < $ $\alpha$ $ < $ $1$ , we have
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{-1}}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds. \end{align} $
(7) 2.2 Finite Difference Method
Now we introduce a finite difference method for solving differential equations with generalized fractional derivative. Consider the following generalized fractional differential equation:
$ \begin{align} \begin{cases} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t)=f(t, u(t)), \quad 0 < t\leq T\\ u(0)=u_0 \end{cases} \end{align} $
(8) where $0 < \alpha < 1$ and $T$ is the final time. Without loss of generality, on a uniform mesh $0=t_0 < t_1 < \cdots < $ $t_j < $ $t_{j+1} < \cdots < t_N=T$ , the Caputo type generalized fractional derivative of $u(t)$ can be approximated as
$ \begin{align} (D^{\alpha}_{0+;[\sigma, w]}& u)(t_{j+1}) \nonumber\\ &= \frac{[w(t_{j+1})]^{-1}}{\Gamma(1-\alpha)}\int^{t_{j+1}}_{0}\frac{[w(s)u(s)]'} {[\sigma(t_{{j+1}})-\sigma(s)]^{\alpha}}ds \nonumber\\ &= \frac{w_{j+1}^{-1}}{\Gamma(1-\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{[w(s)u(s)]'}{\left[\sigma(t_{{j+1}})-\sigma(s) \right]^{\alpha}}ds \nonumber\\ & \approx \frac{w_{j+1}^{-1}}{\Gamma(1-\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{\frac{w_{k+1}u_{k+1}-w_ku_k}{t_{k+1}-t_k}} {\left[\sigma_{j+1}-\sigma(s)\right]^{\alpha}}ds \nonumber\\ & \approx\sum^{j}_{k=0}\left(A^j_ku_{k+1}-B^j_ku_k\right) \end{align} $
(9) where
$ \begin{align*} A^j_k =&\ \frac{w^{-1}_{j+1}w_{k+1}}{\Gamma(2-\alpha)(\sigma_{k+1}-\sigma_k)} \\ & \times \left[(\sigma_{j+1}-\sigma_k)^{1-\alpha}- (\sigma_{j+1}-\sigma_{k+1})^{1-\alpha}\right]\\ B^j_k =&\ \frac{w^{-1}_{j+1}w_{k}}{\Gamma(2-\alpha)(\sigma_{k+1}-\sigma_k)} \\ & \times \left[(\sigma_{j+1}-\sigma_k)^{1-\alpha}- (\sigma_{j+1}-\sigma_{k+1})^{1-\alpha}\right] \end{align*} $
$k=0, 1, 2, \ldots, j$ , $u_j=u(t_j)$ , $w_j=w(t_j)$ , and $\sigma_j=\sigma(t_j)$ .
Therefore, we obtain the finite difference scheme:
$ \begin{align} \sum^{j}_{k=0}\left(A^j_ku_{k+1}-B^j_ku_k\right)=f(t_{j+1}, u_{j+1}) \end{align} $
(10) and the corresponding iteration scheme as
$ \begin{align} u_{j+1}=\begin{cases} \frac{1}{A^j_j}\left[f_j-\sum\limits^{j-1}_{k=0} \left(A^j_ku_{k+1}-B^j_ku_k\right)+B^j_ju_j \right], \\ \qquad \qquad \qquad \qquad \qquad\qquad j=1, 2, \ldots, N-1\\ \frac{1}{A^0_0}\left(f_0+B^0_0u_0\right), \qquad\qquad \ \, j=0 \end{cases} \end{align} $
(11) where $f_j=f(t_j, u_j)$ .
In what follows, we will apply this method to solve the generalized fractional chaotic systems. The numerical analysis of the above scheme can be found in [26].
3. Dynamic Behavior of Generalized Fractional Chaotic Systems
In this section, we introduce two nonlinear dynamical systems but redefine them with Caputo type generalized fractional derivative. The classical and fractional senses are special cases of the new generalized fractional system below.
3.1 Generalized Fractional Lotka-Volterra and Generalized Fractional Lorenz System
Replacing the derivative with the generalized fractional derivative defined by (7), we define the generalized fractional Lotka-Volterra system (GFLVS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = ax - bxy + mx^2 - sx^2z\\ D^{\alpha_2}_{0+;[\sigma, w]}y = -cy +dxy\\ D^{\alpha_3}_{0+;[\sigma, w]}z = -pz + sx^2z \end{cases} \end{align} $
(12) where $0 < \alpha_1, \alpha_2, \alpha_3 < 1$ ( $\alpha_1$ , $\alpha_2$ , $\alpha_3$ can be the equal or different) are the orders of the derivative and parameters $a$ , $b$ , $c$ , $d$ are positive. $a$ represents the natural growth rate of the prey in the absence of predators, $b$ represents the effect of predator on the prey, $c$ represents the natural death rate of the predator in the absence of prey, $d$ represents the efficiency and propagation rate of the predator in the presence of prey, and $m$ , $p$ , $s$ are positive constants.
By selecting the parameters $a=1$ , $b=1$ , $c=1$ , $d=1$ , $m$ $=$ $2$ , $s=2.7$ , $p=3$ and the initial condition $[x_0, y_0, z_0]$ $=$ $[1.5, 1.5, 1.5]$ , when $\alpha_1=\alpha_2=\alpha_3=0.95$ , (12) represents the generalized fractional Lotka-Volterra chaotic system and the phase portraits of the system (12) are described through Figs. 1(a) and 1(b). In Fig. 1(a), the chaotic phenomenon is shown. Moreover, the GFLVS reduces to the fractional Lotka-Volterra system as $\sigma(t)=t$ and $w(t)=1$ . In Fig. 1(b), we see that when the scale function is specified as a power function, and the weight function is taken as an exponential function, the chaotic attractor vanishes and then a stable equilibrium point appears.
图 1 Phase portraits of GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 1 Phase portraits of GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Similarly, we define the generalized fractional Lorenz system (GFLS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = r(y-x)\\ D^{\alpha_2}_{0+;[\sigma, w]}y = x(\rho-z)-y\\ D^{\alpha_3}_{0+;[\sigma, w]}z = xy-\beta{z} \end{cases} \end{align} $
(13) where $r$ is the Prandtl number, $\rho$ is the Rayleigh number and $\beta$ is the size of the region approximated by the system. The fractional order $0 < \alpha_1, \alpha_2, \alpha_3 < 1$ may take different values.
By taking the parameters $r=10$ , $\rho=28$ , $\beta= {8}/{3}$ , and the initial condition $[x_0, y_0, z_0]=[0.5, 0.5, 0.5]$ , when $\alpha_1$ $=$ $\alpha_2=\alpha_3=0.99$ , (13) represents the generalized fractional Lorenz chaotic system and the phase portraits of the system (13) are described through Figs. 1(c) and 1(d). In Fig. 1(c), the chaotic attractor of fractional Lorenz system is presented. When we take scale function as a power function, and weight function as exponential function, the GFLS remains chaotic. However, the shape of the attractor changes, which is shown in Fig. 1(d).
3.2 Analysis of the Influence of Scale and Weight Functions
Now we analyze the influence of the scale and weight functions on the responses of generalized fractional differential equation. For simplicity, we consider
$ \begin{align} D^{\alpha}_{0+;[\sigma, w]}u(t) = Au(t) + f(t) \end{align} $
(14) where $A\neq{0}$ is a constant.
Equation (14) is equivalent to
$ \begin{align} \frac{[w(t)]^{-1}}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds = Au(t) + f(t). \end{align} $
(15) Let $v(t) = w(t)u(t)$ , we have
$ \begin{align} \frac{1}{\Gamma(1-\alpha)}\int^{t}_{0} \frac{v(s)'}{[\sigma(t)-\sigma(s)]^{\alpha}}ds = Av(t) + w(t)f(t). \end{align} $
(16) According to [13], we deduce the solution of (16) as:
$ \begin{align} v(t) =&\ E_{\alpha}\left(A[\sigma(t)-\sigma(0)]^{\alpha}\right)v_0 \nonumber\\ & +\int^t_0(\sigma(t)-\sigma(s))^{\alpha-1} \nonumber\\ &\times E_{\alpha, \alpha}[A(\sigma(t)-\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $
(17) which implies that
$ \begin{align} u(t)=&\ \frac{w(0)}{w(t)}E_{\alpha}\left(A[\sigma(t)-\sigma(0)]^{\alpha}\right)u_0 \nonumber\\ & +\frac{1}{w(t)}\int^t_0(\sigma(t)-\sigma(s))^{\alpha-1} \nonumber\\ & \times E_{\alpha, \alpha}[A(\sigma(t)-\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $
(18) where $u_0$ is the initial condition, and $E$ is the Mittag-Leffler function.
In (18), we observe that how the weight and scale functions influence the behavior of (14). First of all, the weight function cannot be zero in the domain, otherwise solution $u(t)$ will go to infinity. Second, the scale function cannot be periodic, and if it is, the generalized fractional derivative will be infinity at $t=s$ . For an intuitive comprehension, we present some numerical simulations in the following.
3.3 Dynamics of GFLVS and GFLS Depend on Scale and Weight Functions
The fractional chaotic systems are sufficiently generalized by using the generalized fractional derivative, since many existing fractional derivatives, as well as integer order derivatives, are special cases of the generalized fractional derivative. In our numerical experiments, we find many interesting dynamical behaviors of generalized fractional chaotic systems which are never found in common fractional or integer order chaotic systems. Here we present some particular simulation results. However, our discussion depends on Figs. 2 and 3, and others figures are not shown here.
图 2 Influence of scale function $\sigma(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 2 Influence of scale function $\sigma(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).
图 3 Influence of weight function $w(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).Fig. 3 Influence of weight function $w(t)$ on GFLVS (top row, (a) and (b)) and GFLS (bottom row, (c) and (d)).First, we simulate the influence of scale function on dynamics of chaotic systems. In GFLVS, we take fractional order $\alpha_1=\alpha_2=\alpha_3=0.95$ , weight function $w(t)=\exp(1.2t)$ , and other parameters are the same as before. In GFLS, we select fractional order $\alpha_1=\alpha_2=\alpha_3=0.99$ , weight function $w(t)=\exp(0.1t)$ , and other parameters are the same as before. The dynamic behaviors of GFLVS and GFLS with scale function $\sigma(t)=t$ and $t^{1.14}$ are individually presented in Fig. 2.
Second, we simulate the influence of weight function on dynamics of chaotic systems. In GFLVS, we take fractional order $\alpha_1=\alpha_2=\alpha_3=0.95$ , scale function $\sigma(t)=t$ , and other parameters are the same as before. In GFLS, we select fractional order $\alpha_1=\alpha_2=\alpha_3=0.99$ , scale function $\sigma(t)=t$ , and other parameters are the same as before. The dynamic behaviors of GFLVS with weight function $w(t)$ $=$ $\exp(0.8t)$ , $\exp(1.3t)$ , and GFLS with weight function $w(t)=\exp(2+0.5t)$ and $\exp(2+0.2t)$ are presented in Fig. 3.
Finally, to end this section, we make some remarks based on the numerical experiments above. Some other figures are not listed here for shortening the length of paper.
1) The GFLVS is chaotic with scale function $\sigma(t)=t$ , weight function $w(t)$ is a nonzero constant, and fractional order $\alpha_i=0.95$ , $i=1, 2, 3$ [27]. However, From Fig. 1(a), Fig. 2(a) and Fig. 3(a), we may see that as the weight function varies, the chaotic attractor vanishes and then a limit cycle emerges or the system converges to a stable equilibrium point. Furthermore, from Fig. 2(a) and Fig. 2(b), we observe that as the scale function varies, the limit cycle tends to be a stable equilibrium point. From Fig. 3(a) and Fig. 3(b), it is shown that as the weight function varies, the limit cycle can be generated from a stable equilibrium point.
2) The GFLS is chaotic with scale function $\sigma(t)=t$ , weight function $w(t)$ is a nonzero constant, and fractional order $\sum^{3}_{i=1}\alpha_i>2.91$ [28]. In simulation, on one hand, Figs. 1(c) and 1(d), indicate that with suitable scale and weight functions, the GFLS also has a chaotic attractor. On the other hand, Fig. 1(c), Fig. 1(d), Fig. 2(c), Fig. 2(d), and Fig. 3(d) imply that the scale and weight functions can influence the shape and position of chaotic attractor. From Figs. 3(c) and 3(d), we observe that with some suitable weight function, the chaotic attractor tends to be an asymptotically stable equilibrium point.
3) Our previous work [23]-[26] verified that in generalized fractional integral and generalized fractional derivative, the basic property of scale function $\sigma(t)$ is that it changes the time axis, which means that if the time domain is specified as $[0, T]$ , then the response of the dynamical system is obtained over $[\sigma(0), \sigma(T)]$ , provided the scale function is monotone increasing. Since the chaotic dynamical systems are sensitive to the initial conditions, when we take different scale functions in generalized fractional chaotic system, many different dynamical behaviors will be drawn.
4) A similar observation to weight function can be found in [23]-[26], which shows that in generalized fractional integral and generalized fractional derivative, the basic property of weight function $w(t)$ is that it puts different weights for function in different positions of domain. The classical fractional operators have memory property which makes them excellent tools to model the diffusion process with heredity. Generally, in left Caputo type generalized fractional derivative, the monotonic increasing weight function is coincident with the inner memory property of fractional operator, while the monotonic decreasing weight function can destroy this inner property. One can also follow our numerical method and try other scale and weight functions in numerical experiments.
5) In Figs. 2 and 3, one can observe that both changing the scale and weight functions make the systems change between different dynamical behavior (e.g., limit cycle and stable equilibrium point). These phenomena can be regarded as general cases for generalized fractional chaotic systems. We shall guess that either scale function or weight function would influence the dynamics of generalized fractional chaotic systems. In Fig. 2, the weight function is fixed so that the influence of scale function on GFLVS and GFLS is presented. Similarly, in Fig. 3, the scale function is fixed so that the influence of weight function on GFLVS and GFLS is shown. From (18), we clearly see that the scale function plays an important role in scaling the long time behavior of dynamics since it is located in the generalized exponential function, and the weight function provides a different average since it lies inside the integral, and it is a variable coefficient simultaneously. Apparently, the behavior of function $u$ depends on the changing of scale and weight functions.
4. Conclusions
In this paper, we presented a class of new generalized fractional chaotic system, using the new generalized fractional derivative proposed recently. Many dynamical systems with integer or fractional order derivatives can be extended by replacing the derivative with the generalized fractional derivative. Therefore, the new generalized fractional dynamical systems considered in this paper can exhibit more complex dynamic behaviors. In simulations, we show that the dynamical behaviors of such systems not only depend on fractional order, but also depend on the scale and weight functions.
Acknowledgement: The author is grateful to Professor O. P. Agrawal (SIUC, USA) for introducing him theory of generalized fractional calculus, suggesting the basic idea of this paper, as well as his kind help and continuous encouragement in the recent years.
-
图 1 三车道场景目标车所处车辆集群态势图
Fig. 1 Vehicle cluster situation for target vehicle under the three-lane condition
图 2 三车道场景驾驶员车道选择博弈分析图
Fig. 2 Game analysis of drivers' lane choice behavior under three-lane condition
图 5 动态人车环境信息采集系统组成
Fig. 5 Dynamic human-vehicle-environment information acquisition systems
图 7 保守型、普通型、激进型驾驶员换道频率模拟结果
Fig. 7 Simulation results of lane changing frequency for conservative, common, and radical drivers
图 8 换道次数仿真值与实际测量值对比图
Fig. 8 Comparison of simulation and measured values in lane-changing times
图 9 左车道、中间车道和右车道利用率仿真值与实际测量值对比图
Fig. 9 Comparison of simulation and measured values for left, middle, and right lanes in lane occupancy rate
表 1 不同类型驾驶员感知各分区域车辆对目标车综合作用力的贡献率
Table 1 Contribution rates of integrated force from different types of drivers in sub-area to the target vehicle
目标车位置 目标车位于左车道 目标车位于中间车道 目标车位于右车道 贡献率驾驶员类型 $\xi _i^{ql}$ $\xi _i^{hl}$ $\xi _i^{lyq}$ $\xi _i^{lyq}$ $\xi _i^{gyq}$ $\xi _i^{gyh}$ $\xi _i^q $ $\xi _i^h $ $\xi _i^{zq} $ $\xi _i^q $ $\xi _i^h $ $\xi _i^{zq} $ $\xi _i^{ql} $ $\xi _i^{hl} $ $\xi _i^{lyq} $ $\xi _i^{lyq} $ $\xi _i^{gyq} $ $\xi _i^{gyh} $ 激进型 0.244 0.161 0.186 0.226 0.086 0.097 0.219 0.152 0.166 0.219 0.152 0.166 0.244 0.161 0.186 0.226 0.086 0.097 普通型 0.251 0.201 0.181 0.193 0.081 0.093 0.226 0.187 0.152 0.226 0.187 0.152 0.251 0.201 0.181 0.193 0.081 0.093 保守型 0.265 0.236 0.173 0.168 0.076 0.082 0.241 0.191 0.111 0.241 0.191 0.111 0.265 0.236 0.173 0.168 0.076 0.082 
下载: 导出CSV
表 2 冲突收益语义项与三角形模糊数之间的对应关系
Table 2 Correspondence between semantic driving conflict items and triangular fuzzy numbers
冲突收益评价语义项 三角形模糊数 冲突收益评价语义项 三角形模糊数 很差 (0, 0, 0.2) 较好 (0.6, 0.7, 0.8) 差 (0, 0.1, 0.3) 好 (0.7, 0.8, 0.9) 较差 (0, 0.2, 0.4) 很好 (0.8, 0.9, 1) 中等 (0.3, 0.5, 0.7) 非常好 (0.9, 1, 1)
下载: 导出CSV
表 3 不同类型驾驶员对各收益的模糊权重情况
Table 3 Fuzzy weight of the beneflt for difierent types of drivers
驾驶员类型 模糊权重 行车安全收益${\not{\omega }}_i^1 $ 行车时间收益${\not{\omega }}_i^2$ 冲突收益${\not{\omega }}_i^3 $ 激进型 0.3 0.5 0.2 普通型 0.35 0.4 0.25 保守型 0.4 0.3 0.3
下载: 导出CSV
表 4 实验可采集数据
Table 4 Collected experiment data types
数据 目标车与周围车辆相对距离(m) 目标车与周围车辆相对速度(m/s) 左前车 左后车 前车 后车 右前车 右后车 左前车 左后车 前车 后车 右前车 右后车 代码 $\Delta d_2 $ $\Delta d_3 $ $\Delta d_4 $ $\Delta d_5$ $\Delta d_6 $ $\Delta d_7 $ $\Delta v_2 $ $\Delta v_3 $ $\Delta v_4 $ $\Delta v_5 $ $\Delta v_6 $ $\Delta v_7 $
下载: 导出CSV
表 5 车道选择模型参数标定
Table 5 Parameter calibration of lane choice model
参数 $v_1 $ $v_2 $ $v_3 $ $v_4 $ $d_1 $ $d_2 $ $d_3 $ $d_4 $ 数值 激进型 -3.2 -1.7 1.7 3.2 8.3 22.4 36.2 50.5 普通型 -4.8 -2.5 2.5 4.8 12.6 25.8 40.1 60.5 保守型 -8.7 -4.5 4.5 8.7 22.7 35.3 50.5 70.5
下载: 导出CSV
表 6 道路实车实验结果
Table 6 Verification results of actual driving experiment
驾驶员编号 预测次数 预测结果与实测结果对比 准确率(%) 相符次数 不符次数 1 80 71 9 88.8 2 80 68 12 85.0 3 80 67 13 83.8 4 80 69 11 86.3 5 80 65 15 81.3 6 80 67 13 83.8 7 80 70 10 87.5 8 80 68 12 85.0 均值 80 68.125 11.875 85.16
下载: 导出CSV
表 7 驾驶模拟实验结果
Table 7 Verification results of virtual driving experiment
驾驶员编号 预测次数 预测结果与实测结果对比 准确率(%) 相符次数 不符次数 1 80 70 10 87.5 2 80 69 11 86.3 3 80 66 14 82.5 4 80 71 9 88.8 5 80 67 13 83.8 6 80 70 10 87.5 7 80 65 15 81.3 8 80 68 12 85.0 均值 80 68.25 11.75 85.31
下载: 导出CSV
表 8 微观仿真结果与实测数据对比分析表
Table 8 Comparative analysis of microscopic simulation and measured values
评价指标 实测值 模拟1结果 误差(%) 平均速度(m/s) 10.2 10.66 4.51 平均密度(辆/km) 19.3 18.52 4.04 平均延误(s) 7.2 6.93 3.75
下载: 导出CSV
-
[1] Gipps P G. A model for the structure of lane-changing decisions. Transportation Research Part B:Methodological, 1986, 20(5):403-414 doi: 10.1016/0191-2615(86)90012-3 [2] Hidas P. Modelling lane changing and merging in microscopic traffic simulation. Transportation Research Part C:Emerging Technologies, 2002, 10(5-6):351-371 doi: 10.1016/S0968-090X(02)00026-8 [3] Halati A, Lieu H, Walker S. CORSIM-corridor traffic simulation model. In:Proceedings of the 1997 Traffic Congestion and Traffic Safety in the 21st Century:Challenges, Innovations, and Opportunities. New York, USA:American Society of Civil Engineers, 1997. 570-576 https://www.mendeley.com/research-papers/corsim-corridor-traffic-simulation-model/ [4] Yang Q, Koutsopoulos H N. A microscopic traffic simulator for evaluation of dynamic traffic management systems. Transportation Research Part C:Emerging Technologies, 1996, 4(3):113-129 doi: 10.1016/S0968-090X(96)00006-X [5] Ahmed K I. Modeling Drivers' Acceleration and Lane Changing Behavior[Ph.D. dissertation], Massachusettes Institute of Technology, America, 1999. https://www.mendeley.com/research-papers/modeling-drivers-acceleration-lane-changing-behavior/ [6] Toledo T. Integrated Driving Behavior Modeling[Ph.D. dissertation], Massachusettes Institute of Technology, America, 2003. http://dspace.mit.edu/handle/1721.1/29285 [7] Keyvan-Ekbatani M, Knoop V L, Daamen W. Categorization of the lane change decision process on freeways. Transportation Research Part C:Emerging Technologies, 2016, 69:515-526 doi: 10.1016/j.trc.2015.11.012 [8] Balal E, Cheu R L, Sarkodie-Gyan T. A binary decision model for discretionary lane changing move based on fuzzy inference system. Transportation Research Part C:Emerging Technologies, 2016, 67:47-61 doi: 10.1016/j.trc.2016.02.009 [9] Kita H. A merging-giveway interaction model of cars in a merging section:a game theoretic analysis. Transportation Research Part A:Policy and Practice, 1999, 33(3-4):305-312 doi: 10.1016/S0965-8564(98)00039-1 [10] 杨晓芳, 张盛, 付强.基于博弈论的完全信息下的驾驶行为研究.公路交通科技, 2015, 32(7):105-111 http://d.wanfangdata.com.cn/Periodical/gljtkj201507017Yang Xiao-Fang, Zhang Sheng, Fu Qiang. Research of driving behavior under condition of complete information based on game theory. Journal of Highway and Transportation Research and Development, 2015, 32(7):105-111 http://d.wanfangdata.com.cn/Periodical/gljtkj201507017 [11] 刘小明, 郑淑晖, 蒋新春.基于动态重复博弈的车辆换道模型.公路交通科技, 2008, 25(6):120-125 http://d.wanfangdata.com.cn/Periodical/gljtkj200806024Liu Xiao-Ming, Zheng Shu-Hui, Jiang Xin-Chun. Lane changing model based on discrete dynamic game. Journal of Highway and Transportation Research and Development, 2008, 25(6):120-125 http://d.wanfangdata.com.cn/Periodical/gljtkj200806024 [12] Talebpour A, Mahmassani H S, Hamdar S H. Modeling lane-changing behavior in a connected environment:a game theory approach. Transportation Research Part C:Emerging Technologies, 2015, 59:216-232 doi: 10.1016/j.trc.2015.07.007 [13] Wang M, Hoogendoorn S P, Daamen W, van Arem B, Happee R. Game theoretic approach for predictive lane-changing and car-following control. Transportation Research Part C:Emerging Technologies, 2015, 58 (Part A):73-92 http://www.sciencedirect.com/science/article/pii/S0968090X15002491 [14] 彭金栓, 付锐, 石磊磊, 张琼.驾驶人车道变换决策分析.武汉理工大学学报, 2011, 33(12):46-50 doi: 10.3963/j.issn.1671-4431.2011.12.011Peng Jin-Shuan, Fu Rui, Shi Lei-Lei, Zhang Qiong. Research of driver's lane change decision-making mechanism. Journal of Wuhan University of Technology, 2011, 33(12):46-50 doi: 10.3963/j.issn.1671-4431.2011.12.011 [15] 彭金栓, 付锐, 郭应时, 袁伟, 王畅.基于有限零和灰色博弈的车道变换决策分析.科技导报, 2011, 29(3):52-56 http://d.wanfangdata.com.cn/Periodical/kjdb201103012Peng Jin-Shuan, Fu Rui, Guo Ying-Shi, Yuan Wei, Wang Chang. Analysis of lane change decision making based on the finite and zero-sum grey game theory. Science and Technology Review, 2011, 29(3):52-56 http://d.wanfangdata.com.cn/Periodical/kjdb201103012 [16] 吴磊, 王晓辉, 杨新月, 王晓原.交通态势识别及状态转换机制研究.交通标准化, 2007, (2-3):61-66 http://d.wanfangdata.com.cn/Periodical/jtbzh200702009Wu Lei, Wang Xiao-Hui, Yang Xin-Yue, Wang Xiao-Yuan. Study on the recognition of traffic situation and its state transition mechanism. Communications Standardization, 2007, (2-3):61-66 http://d.wanfangdata.com.cn/Periodical/jtbzh200702009 [17] Wang X Y, Zhang J L, Ban X G, Tan D R. Dynamic feature extraction method of driver's propensity under complicated vehicle group. Advances in Mechanical Engineering, 2013, 2013:1-10 doi: 10.1155/2013/287653 [18] 张青, 武东, 潘小东.有偏好信息的多目标N人非合作模糊对策及其解.山东理工大学学报(自然科学版), 2009, 23(2):24-26 http://d.wanfangdata.com.cn/Periodical/sdgcxyxb200902006Zhang Qing, Wu Dong, Pan Xiao-Dong. Multi-objective N-person noncooperative fuzzy games with preference information and its solving method. Journal of Shandong University of Technology (Natural Science Edition), 2009, 23(2):24-26 http://d.wanfangdata.com.cn/Periodical/sdgcxyxb200902006 [19] Xiao M, Shao X Y, Gao L, Luo Z. A new methodology for multi-objective multidisciplinary design optimization problems based on game theory. Expert Systems with Applications, 2015, 42(3):1602-1612 doi: 10.1016/j.eswa.2014.09.047 [20] 陈志旺, 白锌, 杨七, 黄兴旺, 李国强.区间多目标优化中决策空间约束、支配及同序解筛选策略.自动化学报, 2015, 41(12):2115-2124 http://www.aas.net.cn/CN/abstract/abstract18784.shtmlChen Zhi-Wang, Bai Xin, Yang Qi, Huang Xing-Wang, Li Guo-Qiang. Strategy of constraint, dominance and screening solutions with same sequence in decision space for interval multi-objective optimization. Acta Automatica Sinica, 2015, 41(12):2115-2124 http://www.aas.net.cn/CN/abstract/abstract18784.shtml [21] 巩敦卫, 刘益萍, 孙晓燕, 韩玉艳.基于目标分解的高维多目标并行进化优化方法.自动化学报, 2015, 41(8):1438-1451 http://www.aas.net.cn/CN/abstract/abstract18718.shtmlGong Dun-Wei, Liu Yi-Ping, Sun Xiao-Yan, Han Yu-Yan. Parallel many-objective evolutionary optimization using objectives decomposition. Acta Automatica Sinica, 2015, 41(8):1438-1451 http://www.aas.net.cn/CN/abstract/abstract18718.shtml [22] 张青, 胡子祥, 黄天民.混合模糊多目标多人非合作对策及其解.江南大学学报(自然科学版), 2005, 4(1):94-96, 108 http://d.wanfangdata.com.cn/Periodical/jiangndxxb200501022Zhang Qing, Hu Zi-Xiang, Huang Tian-Min. Mixed fuzzy multi-objective many-person noncooperative games and its solving method. Journal of Southern Yangtze University (Natural Science Edition), 2005, 4(1):94-96, 108 http://d.wanfangdata.com.cn/Periodical/jiangndxxb200501022 [23] Hranac R, Margiotta R, Alexiadis V. Next Generation Simulation (NGSIM) High-level Verification and Validation Plan. Washington, DC, USA:Federal Highway Administration, 2004. [24] Montanino M, Punzo V. Trajectory data reconstruction and simulation-based validation against macroscopic traffic patterns. Transportation Research Part B:Methodological, 2015, 80:82-106 doi: 10.1016/j.trb.2015.06.010 期刊类型引用(3)
1. 曾超，王文军，陈朝阳，张超飞，成波. 模拟驾驶实验中同步装置设计及实验教学应用. 实验技术与管理. 2020(03): 99-102 . 
百度学术2. 吴思凡，杜煜，徐世杰，杨硕，杜晨. 基于长短期记忆-异步优势动作评判的智能车汇入模型. 汽车技术. 2019(10): 42-47 . 百度学术3. 成英，高利，陈雪梅，赵亚男. 有人与无人驾驶车辆交叉口驾驶博弈模型. 北京理工大学学报. 2019(09): 938-943 . 百度学术其他类型引用(17)
-
下载:
计量
- 文章访问数: 2005
- HTML全文浏览量: 247
- PDF下载量: 601
- 被引次数: 20
