-
摘要: 针对常用混合动力汽车(Hybrid electric vehicle,HEV)中锂离子电池在功率波动较大时难以满足需求,以及单个驱动周期内HEV燃油能耗大且能量不能很好回收等问题,研究采用锂离子电池和超级电容器混合储能系统(Lithium-ion battery and super-capacitor hybrid energy storage system,Li-SC HESS)与内燃机共同驱动HEV运行.结合比例积分粒子群优化算法(Particle swarm optimization-proportion integration,PSO-PI)控制器和Li-SC HESS内部功率限制管理办法,提出一种改进的基于庞特里亚金极小值原理(Pontryagin's minimum principle,PMP)算法的HEV能量优化控制策略,通过ADVISOR软件建立HEV整车仿真模型,验证该方法的有效性与可行性.仿真结果表明,该能量优化控制策略提高了HEV跟踪整车燃油能耗最小轨迹的实时性,节能减排比改进前提高了1.6%~2%,功率波动时减少了锂离子电池的出力,进而改善了混合储能系统性能,对电动汽车关键技术的后续研究意义重大.
-
关键词:
- 混合动力汽车 /
- 混合储能系统 /
- PSO-PI控制 /
- 庞特里亚金极小值原理
Abstract: Common hybrid electric vehicles (HEVs) usually suffer lithium-ion battery power fluctuations and it is difficult for them to meet the demand. This also constraints the performance of the energy storage system in which there is a the large amount of fuel consumption with less energy recovery of HEV in a single drive cycle. This paper selects a lithium-ion battery and super-capacitor hybrid energy storage system (Li-SC HESS) to drive the hybrid electric vehicle running together with an internal combustion engine. Besides, combined with particle swarm optimization-proportion integration (PSO-PI) controller and Li-SC HESS internal power limit management approach, this paper presents an improved HEV energy optimization control strategy based on the Pontryagin's minimum principle (PMP) algorithm. A simulation model of HEV is established by ADVISOR software to verify the effectiveness and feasibility of the strategy. Results show that the energy optimization control strategy can improve the real-time ability of tracking the smallest track of HEV fuel consumption function for the purpose of energy conservation. The energy-saving emission reduction ratio is improved by 1.6% to 2.0%. Moreover, when power fluctuates the output of lithium-ion battery is reduced, thus improving the performance of the Li-SC HESS. The research is significant to the follow-up study on the key technologies in electric vehicles. -
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.
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.
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.
-
-
[1] 赵秀春, 郭戈.混合动力电动汽车能量管理策略研究综述.自动化学报, 2006, 42(3):321-334 http://www.aas.net.cn/CN/abstract/abstract18823.shtmlZhao Xiu-Chun, Guo Ge. Survey on energy management strategies for hybrid electric vehicles. Acta Automatica Sinica, 2016, 42(3):321-334 http://www.aas.net.cn/CN/abstract/abstract18823.shtml [2] Kim T, Qiao W, Qu L Y. Real-time state of charge and electrical impedance estimation for lithium-ion batteries based on a hybrid battery model. In:Proceedings of the 2013 Annual IEEE Applied Power Electronics Conference and Exposition. Long Beach, USA:IEEE, 2013. 563-568 [3] 杨培刚, 周育才, 刘志强, 贾光瑞, 熊少华.基于ADVISOR的纯电动汽车复合电源建模与仿真.电力科学与技术学报, 2015, 30(3):66-71 doi: 10.3969/j.issn.1673-9140.2015.03.010Yang Pei-Gang, Zhou Yu-Cai, Liu Zhi-Qiang, Jia Guang-Rui, Xiong Shao-Hua. Modeling and simulation of pure electric vehicle with composite power source based on ADVISOR. Journal of Electric Power Science and Technology, 2015, 30(3):66-71 doi: 10.3969/j.issn.1673-9140.2015.03.010 [4] Chang W Y. Estimation of the state of charge for a LFP battery using a hybrid method that combines a RBF neural network, an OLS algorithm and AGA. International Journal of Electrical Power and Energy Systems, 2013, 53:603-611 doi: 10.1016/j.ijepes.2013.05.038 [5] Gan M, Peng H, Dong X P. A hybrid algorithm to optimize RBF network architecture and parameters for nonlinear time series prediction. Applied Mathematical Modelling, 2012, 36(7):2911-2919 doi: 10.1016/j.apm.2011.09.066 [6] Sharafi M, ELMekkawy T Y. Multi-objective optimal design of hybrid renewable energy systems using PSO-simulation based approach. Renewable Energy, 2014, 68:67-79 doi: 10.1016/j.renene.2014.01.011 [7] 许世景, 吴志新.基于PMP的HEV全局最优能量管理策略研究.中国机械工程, 2014, 25(1):138-141 doi: 10.3969/j.issn.1004-132X.2014.01.026Xu Shi-Jing, Wu Zhi-Xin. Investigation of global optimization energy management strategy of HEV based on PMP. China Mechanical Engineering, 2014, 25(1):138-141 doi: 10.3969/j.issn.1004-132X.2014.01.026 [8] 夏超英, 张聪.混合动力系统能量管理策略的实时优化控制算法.自动化学报, 2015, 41(3):508-517 http://www.aas.net.cn/CN/abstract/abstract18629.shtmlXia Chao-Ying, Zhang Cong. Real-time optimization control algorithm of energy management strategy for hybrid electric vehicles. Acta Automatica Sinica, 2015, 41(3):508-517 http://www.aas.net.cn/CN/abstract/abstract18629.shtml [9] Santucci A, Sorniotti A, Lekakou C. Power split strategies for hybrid energy storage systems for vehicular applications. Journal of Power Sources, 2014, 258:395-407 doi: 10.1016/j.jpowsour.2014.01.118 [10] Masih-Tehrani M, Ha'iri-Yazdi M R, Esfahanian V, Safaei A. Optimum sizing and optimum energy management of a hybrid energy storage system for lithium battery life improvement. Journal of Power Sources, 2013, 244:2-10 doi: 10.1016/j.jpowsour.2013.04.154 [11] Schneider M, Biel K, Pfaller S, Schaede H, Rinderknecht S, Glock C H. Optimal sizing of electrical energy storage systems using inventory models. Energy Procedia, 2015, 73:48-58 doi: 10.1016/j.egypro.2015.07.559 [12] 张培建, 吴建国.基于在线辨识的特征模型预测函数控制研究.计算机仿真, 2010, 27(11):299-302 doi: 10.3969/j.issn.1006-9348.2010.11.076Zhang Pei-Jian, Wu Jian-Guo. Research of characteristic model predictive functional control based on on-line identification. Computer Simulation, 2010, 27(11):299-302 doi: 10.3969/j.issn.1006-9348.2010.11.076 [13] Su X Y, Zhou W S, von Bally G, Vukicevic D. Automated phase-measuring profilometry using defocused projection of a Ronchi grating. Optics Communications, 1992, 94(6):561-573 doi: 10.1016/0030-4018(92)90606-R [14] 夏超英, 杜智明.丰田PRIUS混合动力汽车能量优化管理策略仿真分析.吉林大学学报(工学版), 2017, 47(2):373-383 http://d.old.wanfangdata.com.cn/Periodical/jlgydxzrkxxb201702005Xia Chao-Ying, Du Zhi-Ming. Simulation analysis on energy optimization strategy for Toyota PRIUS hybrid electric vehicle. Journal of Jilin University (Engineering and Technology Edition), 2017, 47(2):373-383 http://d.old.wanfangdata.com.cn/Periodical/jlgydxzrkxxb201702005 [15] 秦大同, 杨官龙, 刘永刚, 林毓培.插电式混合动力汽车能耗优化控制策略的研究.汽车工程, 2015, 37(12):1366-1370, 1377 doi: 10.3969/j.issn.1000-680X.2015.12.002Qin Da-Tong, Yang Guan-Long, Liu Yong-Gang, Lin Yu-Pei. A research on energy consumption optimization control strategy for plug-in hybrid electric vehicle. Automotive Engineering, 2015, 37(12):1366-1370, 1377 doi: 10.3969/j.issn.1000-680X.2015.12.002 [16] 雷嗣军, 宋小文.遗传算法与ADVISOR联合优化仿真汽车动力传动系统.机械科学与技术, 2010, 29(9):1137-1141 http://www.cnki.com.cn/Article/CJFDTOTAL-JXKX201009003.htmLei Si-Jun, Song Xiao-Wen. Optimization of an automobile transmission system with genetic algorithm and ADVISOR. Mechanical Science and Technology for Aerospace Engineering, 2010, 29(9):1137-1141 http://www.cnki.com.cn/Article/CJFDTOTAL-JXKX201009003.htm [17] 李宝磊, 施心陵, 苟常兴, 吕丹桔, 安镇宙, 张榆锋.多元优化算法及其收敛性分析.自动化学报, 2015, 41(5):949-959 http://www.aas.net.cn/CN/abstract/abstract18669.shtmlLi Bao-Lei, Shi Xin-Ling, Gou Chang-Xing, Lv Dan-Ju, An Zhen-Zhou, Zhang Yu-Feng. Multivariant optimization algorithm and its convergence analysis. Acta Automatica Sinica, 2015, 41(5):949-959 http://www.aas.net.cn/CN/abstract/abstract18669.shtml [18] 王琪, 孙玉坤, 黄永红.一种蓄电池-超级电容器复合电源型混合动力汽车制动力分配策略研究.电工技术学报, 2014, 29(S1):155-163 http://d.old.wanfangdata.com.cn/Conference/8625048Wang Qi, Sun Yu-Kun, Huang Yong-Hong. Research on a distribution strategy of braking force used in hybrid electric vehicles with battery-ultracapacitor hybrid energy storage system. Transactions of China Electrotechnical Society, 2014, 29(S1):155-163 http://d.old.wanfangdata.com.cn/Conference/8625048 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)
-