Self-Delayed Feedback Based Car Following Control With Velocity Uncertainty of Preceding Vehicle On Gradient Roads

: Uphill and downhill roads are prevalent in mountainous areas and freeways. Despite the advancement of vehicle - to - vehicle (V2V) communication technology, the driving field of vision is still largely limited under such a complex road environment, which hinders the sensors accurately perceiving the speed of the front vehicle. As such, a fundamental question for autonomous traffic management is how to control traffic flow associated with the velocity uncertainty of preceding vehicles? This paper seeks to develop a controlling framework for corporative car following control under such complex road environment. To this end, we first propose a traffic flow model accounting for the uncertainty effect of preceding vehicles velocity on gradient road. We further design a new self - delayed feedback controller based on the velocity and headway difference between the current time step and historical time step, in an aim to enhance the robustness of traffic flow. The sufficient condition where traffic jam does not occur is derived from the perspective of the frequency domain via Hurwitz criteria and H  norm of transfer functions. The bode diagram reveals that the robustness of closed - loop traffic flow model has been significantly enhanced. We also conduct simulations to verify the theoretical analysis.


Introduction
Over the past decades, the rapid development of urban economy has spurred the growth of car ownership and traffic congestion. Along with this, a variety of approaches and countermeasures have been proposed to alleviate traffic congestion by making use of emerging technology. On one hand, numerous traffic management and planning approaches have emerged, such as bus route optimization tactics [1][2] and online electric vehicle charge scheduling [3]. On the other hand, another stream of research direction mainly concentrated on the development of traffic flow models to reproduce traffic congestion and explore the underlying mechanism. Motivated by the advancement in V2V communication technology, this paper devises a self-delayed feedback control for autonomous traffic considering uncertain velocity of preceding vehicle on gradient road. The autonomous traffic is modeled as a car-following model with a designed controller.
According to the research subjects, current traffic flow theoretical models can generally be divided into following two branches: macroscopic and microscopic traffic flow models. The former mainly includes the continuous models [4][5][6][7][8] and lattice hydrodynamic models [9][10][11][12][13][14][15], while the latter includes the car-following models [16][17][18] and the cellular automata models [19][20]. In the former category, a number of en-route vehicles are modeled as compressible continuous fluid with the time-and location-dependent vehicle density and average speed. By constructing the partial differential equations of local density and local velocity, the dynamic behavior of traffic flow can be revealed by solving the equations. Although such type of models is less computationally expensive, it cannot analyze the microscopic driving behavior of the vehicle. In contrast, the latter takes individual vehicles as the research subject, and establishes differential equations based on the speed, headway and acceleration information of vehicles to study the vehicle motion behavior. In 1995, a seminal work of car-following model were put forward by Bando, termed optimal velocity (OV) model [21] . The optimal velocity function has an inflection point and elegant mathematical and physical properties, which facilitates linear and nonlinear theoretical derivation. Since then, the OV model has attracted wide attentions, yielding a number of extended and more realistic traffic flow models. The representative works can be referred to Table 1.  [67][68] In practice, the gradient roads are not uncommon in the economically underdeveloped areas or mountainous freeways, as shown in Fig.1. The road slope may exert great influence on road traffic jam [69][70]. Despite the development of Internet of Vehicles, the driving field of vision is still limited in a traffic environment with gradient roads, which hinders the sensors perceiving the speed of the front vehicle accurately. Therefore, the velocity uncertainty of preceding vehicles should not be neglected in such complex road environment. However, to date, the existing studies on traffic flow assume that the current vehicle can accurately receive the headway or velocity information of preceding vehicles. Motivated by this fact, there is an increasing need to devise systematic traffic flow control algorithm for such traffic environment. To this end, we first propose an extended traffic flow model accounting for the effect of velocity uncertainty of preceding vehicle on the gradient road. It proceeds to design a new self-delayed feedback controller without relying on any extra traffic information. The stability criterion of closed-loop system is also derived.
The structure of this paper is as follows: In Section 2, a modified car following model taking into account the effect of velocity uncertainty of preceding vehicle on gradient road is presented. In Section 3, the stability condition of closed-loop car following model is obtained via Hurwitz criterion and H  norm of transfer functions. In Section 4, some numerical examples are conducted out to verify the conclusions of the above theoretical analysis, finally, the conclusions and follow-up research works of this paper are given in Section 5.

Model
In 1995, Bando [21] proposed an OV model to study the car following characteristic of vehicles on a single lane where the overtaking is forbidden, and the dynamic equation is expressed as: where ()  represents the instantaneous headway of the target vehicle,   V g represents the optimal velocity function, which is uniquely determined by the parameter where max v and c y respectively represent the maximum speed and safe distance. In order to eliminate the phenomenon of the excessive acceleration and unreasonable deceleration in the traditional optimal velocity model, Helbing and Tilch [71] presented an improved generalized force (GF) model expressed as follows: where () n vt  represents the velocity difference between current vehicle n and preceding vehicle n+1, and .  is the corresponding weight coefficient,

 
H g represents the Heaviside function, and Later, in order to overcome the low starting wave speed in above-mentioned generalized force model, Jiang et al. [72] presented a full velocity difference (FVD) model, which is written as follows: Since the FVD model, extensive derivative models are developed. considering that uphill and downhill roads are prevalent in mountainous areas and freeways, the literature [69] proposed a traffic flow model considering road slope information to study the impact of road slope information on traffic jam, and the kinetic equation is as follows where   op V g represents the modified optimal velocity function constrained by the slope information, and the function is taken as follows: () c y  respectively represent the speed adjustment and safety distance item affected by the acceleration of gravity. ,max f v is the maximum velocity without any slope. Fig. 2 shows a schematic diagram of the force of the vehicle on the uphill and downhill scenarios. The specific value of max () v  of can be determined by where m and g respectively represent the mass of the vehicle and the acceleration of gravity, i.e.
where c h is the safety distance of vehicles on the general road without any slope information,  is constant, which is taken as 1   to simplify the analysis. As a result, the corresponding optimal speed function can be rewritten as To date, the existing studies on traffic flow assume that the current vehicle can accurately receive the traffic information of preceding vehicle. However, as discussed previously, the driving field of vision will be still limited in a traffic environment with gradient roads even though the V2V technology is in use, such that the sensors are difficult to accurately perceive the speed of front vehicle. In other words, certain errors of perceived speed may exist between the vehicle ahead and the real data. As a result, there is an imminent need to incorporate the velocity uncertainty of preceding vehicles in the design of car following control under such complex road environment. In view of this, we introduce the perception error term of preceding vehicle on Eq. (7), yielding a new traffic flow model as follows: where () n ut is the designed control input.  is the error coefficient representing the difference between the perceived data and real data; if 0   , then the error is positive; otherwise, the error is negative; here we assume that 1   . When 0   , the proposed model is consistent with the literature [69]. Therefore, proposed new traffic flow model can be regarded as a generalized form of previous research.

Control scheme for the car-following model
In order to improve road traffic efficiency, we propose a new self-delayed feedback control for the car-following model. This control scheme only relies on the information of velocity and headway difference between the current time and historical time, specifically where 1  is delay time. k1 and k2 are the weight coefficients corresponding to the headway and the speed difference, respectively.
Applying the Laplace transform to the above formula, we have that Lemma 1 [52] : According to the classical control theory, the traffic flow is stable and the traffic jam phenomenon will disappear if the following conditions are established simultaneously: Substituting m1 and m2 into the formula (21), we have that In summary, if condition (22) or (23) Therefore, if the inequality   1 G iw  is hold for any   0, w  , then the following condition is met, specifically,   (14), where Fig. 5 and Fig. 6 correspond to downhill and uphill scenario, respectively. As shown in Fig. 5(a), the amplitude of the transfer function G is effectively reduced as the parameters k1, k2 increase; when k1=2.5 and k2=2.5, the theorem 1 is satisfied, the amplitude vertex of the transition function is less than 1. Therefore, the designed controller (14) is conducive to enhance the robustness of traffic flow. Fig. 5(b) compares the amplitude of the transfer function G under different delay time 1  . When 1 =0.5  , the amplitude of the bode curve is greater than 0, and the traffic disturbance will exacerbate over time and induce traffic jam phenomenon. As the parameter 1  continues to increase, the amplitude of bode curve keep decreases. When 1 =2.5  , the amplitude of the bode curve is approximately less than 1, which means that the traffic flow is stable. Therefore, it can be concluded that a higher value of parameter 1  is conducive to improving the robustness of traffic flow.

Simulation example
In this section, we will perform numerical simulations under periodic boundary conditions to verify the effect of designed self-delay feedback controller on suppressing traffic congestion and reducing energy consumption. The initial position information is selected as follows: where the number of vehicles N=100,  is the disturbance term, i.e., 0.1   , the other default parameters are selected as follows: Fig. 7 shows the spatiotemporal evolution of the headways with different controller gains coefficient k1, k2 under the downhill scenario. Fig. 7(a) corresponds to the traffic flow without designed controller (k1, k2=0). It shows that the headways on the road oscillate frequently with stop-and-go waves, which indicates that the traffic jam prone to occur. With the increasing of control gain coefficients k1, k2, the fluctuating frequency and amplitude of density waves have been effectively reduced. Typically, when k1=2, k2=2, the initial disturbance has been completely suppressed, and the stop-and-go phenomenon disappears. Therefore, higher values of parameter k1, k2 contributes to suppressing traffic congestion. Fig. 8 shows the instantaneous headway distribution of all vehicles corresponding to Fig.7 at t=10 300s. As we can see, the fluctuating amplitude of headways without control in Fig.8 (a) is the largest, while the fluctuating amplitude is approximately equal to 0 in Fig. 14(d) indicating that the traffic flow is stable. Fig. 9 describes the evolution of the headway between vehicles on the road under different values of parameter 1  . One can see that the headway fluctuation gradually decreases over time with the increase of parameter 1  , which suggests that a higher value of parameter 1  is conducive to improving the anti-interference ability of traffic flow. Fig. 10 shows the instantaneous headway distribution of all vehicles corresponding to Fig.9 at t=10 300s. We can see that the fluctuating amplitude of headways has been effectively reduced with the increase of parameter 1  , and the fluctuating amplitude between vehicles can be neglected in Fig.10(d). In other words, the initial disturbance has been completely diluted when the designed controller (14) is applied.

Conclusion
This paper proposes a new self-delayed feedback controller for the car-following model with velocity uncertainty of preceding vehicle on gradient road. The stability condition of closed-loop traffic flow model is obtained via Hurwitz criteria and H  norm of transfer functions. We investigate the influence of parameters k1, k2 and 1  on traffic jam from the uphill and downhill scenarios. Results show that as parameters of k1, k2 increases, the traffic congestion can be effectively suppressed. A higher value of delay time 1  will decrease the fluctuation amplitude of headway, which indicates that the robustness of the traffic flow model is enhanced. Despite the newly developed autonomous traffic flow model, there are a few strong assumptions in the current research. For example, the delay time term in designed feedback controller is constant, whereas the parameter uncertainties may be more realistic in the complex traffic environment [73]. In addition, we only verify the feasibility of designed control scheme without using real traffic data. These issues will be addressed in the follow-up research.
the National Science Foundation of China (Project No. 72071079).