J. Cent. South Univ. Technol. (2010) 17: 1271-1278

DOI: 10.1007/s11771-010-0631-0

Effect of information delay on string stability of platoon of automated vehicles under typical information frameworks

XIAO Ling-yun(肖凌云)^{1, 2}, GAO Feng(高峰)¹

1. School of Transportation Science and Engineering, Beijing University of Aeronautics and Astronautics,

Beijing 100191, China;

2. Defective Product Administrative Center, General Administration of Quality Supervision,

Inspection and Quarantine of China, Beijing 100088, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: The effect of the information delay, which was caused by the nature of the distance sensors and wireless communication systems, on the string stability of platoon of automated vehicles was studied. The longitudinal vehicle dynamics model was built by taking the information delay into consideration, and three typical information frameworks, i.e., leader-predecessor framework (LPF), multiple-predecessors framework (MPF) and predecessor-successor framework (PSF), were defined and their related spacing error dynamics models in frequency domain were proposed. The string stability of platoon of automated vehicles was analyzed for the  LPF, MPF and PSF, respectively. Meanwhile, the related sufficient string stable conditions were also obtained. The results demonstrate that the string stability can be guaranteed for the LPF and PSF with considering the information delay, but the ranges of the control gains of the control laws are smaller than those without considering the information delay. For the MPF, the “weak” string stability, which can be guaranteed without considering the information delay, cannot be obtained with considering the information delay. The comparative simulations further demonstrate that the LPF shows better string stability, but the PSF shows better string scalable performance.

Key words: string stability; information delay; information framework; automated vehicle; intelligent transportation system

1 Introduction

Nowadays, the increasing traffic accidents and traffic congestion have been extremely serious social problems, which should demand the governmental, academic and industrial entities to take strong measures to deal with together. During the few past decades, the concept of automated vehicle following control has been proposed to automatically maintain a desired spacing from the immediate preceding vehicle to improve driving comfort, traffic safety and traffic capacity [1]. String stability has been a significant topic to study automated vehicle following control since mid 1970s [2]. Intuitively, the term “string stability” indicates that spacing errors do not amplify as they propagate upstream from one vehicle to another vehicle [3]. This property ensures that any perturbation of the speed or position of the leading vehicle will not result in amplified fluctuations to speed and position of the following vehicle, and string instability is not only likely to provide poor ride quality but also can result in collision [3-4].

To achieve the purpose of the automated vehicle following control, the distance sensors and wireless communication systems are necessary to be used to access the velocity information, position information and/or acceleration information from the leading vehicle, preceding vehicles and successive vehicles [5]. At present, the distance sensors usually include radar, lidar and video camera, which measure the distance and the relative velocity of the successive two vehicles [1]. And several wireless communication systems have been proposed to implement the vehicle-to-vehicle (V2V) communication and the vehicle-to-infrastructure (V2I) communication, such as the IEEE- and ASTM-adopted dedicated short range communication (DSRC) [1, 5-6], wireless local area network (WLAN) [5] and cognitive radio (CR) [7]. Often, the wireless communication systems are applied to assisting the distance sensors so as to detect the long range information and then to provide higher traffic capacity and traffic safety.

The analysis of string stability of platoon of automated vehicles was conducted extensively and some string stable controllers were established in Refs.[8-13]. However, most of the controllers and their related string stable conditions were proposed based on the simple longitudinal vehicle dynamics models without considering the information delay. In practice, onboard sensors typically operate with a scan frequency between 3 and 10 Hz. This implies potential information delay varying between 0.10 and 0.33 s [14]. In addition, the wireless communication systems often involve the information delay due to the packet losses, transmission time and the time to choose and analyze the transmitted data [5]. It is reasonable to doubt that the string stability cannot be obtained in this more practical scenario if the string stable conditions are proposed without considering the information delay. Currently, only a few studies on string stability with considering the information delay have been conducted. For instance, MAHAL [5] analyzed the effect of the information delay on the string stability based on leader-predecessor framework (LPF) but without obtaining the detailed string stable conditions. The contribution of this work involves the way to analyze the effect of the information delay on the string stability for three typical frameworks based on Refs.[15-16].

2 Longitudinal vehicle dynamics model

The string of N+1 automated vehicles was assumed to run in only one spatial dimension and be controlled by the identical control law with considering the identical longitudinal vehicle dynamics model. Let x₀(t), v₀(t) and a₀(t) denote the position, velocity and acceleration of the leading vehicle and x_i(t), v_i(t) and a_i(t) denote the position, velocity and acceleration of the ith following vehicle in the platoon, respectively. Then the spacing error

δ_i(t)=x_i-1(t)-x_i(t)-L-D                         (1)

is obtained. Here, δ_i(t) denotes the spacing error of the ith vehicle, which is the discrepancy between the real spacing and the desired spacing; L denotes the length of the vehicle; D denotes the desired constant distance. For the convenience of analysis, v_i(t)=v₀(t), a_i(t)=a₀(t)=0 and δ_i(t) (1≤i≤N) are assumed at the initial state.

A dynamics model for the motion of a vehicle in the longitudinal direction must take into account the powertrain, longitudinal tire forces, aerodynamic drag forces, rolling resistance forces and gravitational forces. The powertrain consists of the internal combustion engine, the torque converter, the transmission and the wheels [15]. A simple functional description of such a system is shown in Fig.1. By making some appropriate assumptions [5, 15, 17] and applying the feedback linearization technique [8, 10, 12, 15], the linearized longitudinal vehicle dynamics model is obtained as follows:

                      (2)

where   

and U_i(t)=u_i(t). Let u_i(t) denote the control law established to satisfy certain performance objectives; and τ denote the “lumped” time lag of the actuators, such as engine response, throttle actuator, braking actuator and transmission line.

As mentioned previously, the automated vehicle following control system involves all-pervasive information delays during sensing or accessing the related information. Various parasitic information delays are combined into “lumped” time delay that is represented by symbol Δ. Then, the linearized longitudinal vehicle dynamics model (2) is replaced by

                   (3)

where U_i(t-Δ)=u_i(t-Δ).

3 Typical information frameworks

Any design of automatic vehicle following controller begins with the select of information frameworks and associated spacing policies [1, 8, 13]. The information framework refers to the accessible information applied to establishing control law from the related automated vehicles, including single-predecessor framework (SPF) [13], LPF [5, 12], multiple-predecessors framework (MPF) [12], and predecessor-successor framework (PSF) [12, 18-19]. The spacing policy refers to the desired steady state distance that automates vehicle attempts to maintain from the immediate preceding vehicle, including velocity dependent spacing policies (such as constant time headway [8, 15] and variable time headway [8, 10-11]) and velocity independent spacing policies (such as constant distance [12-13, 16]). In fact, the constant time headway spacing policy [1, 15] is accepted by the adaptive cruise control (ACC) system when applying the SPF, but due to the potential high traffic capability, the constant distance spacing policy is broadly accepted in the automated vehicle following control when applying the LPF, MPF and PSF.

Fig.1 Schematic diagram of longitudinal vehicle dynamics model

The linearized longitudinal vehicle dynamics model is applied to establishing control laws and obtaining the string stable conditions, so that the analysis can be performed in the frequency domain using Laplace transforms and transfer functions. As mentioned previously, δ_i(t) denotes the spacing error of the ith vehicle in time domain, then δ_i(s) denotes the spacing error of the ith vehicle in frequency domain. In the SPF, the spacing error dynamics model in frequency domain takes the form as δ_i(s)=G(s)δ_i-1(s) with a transfer function G(s).

While applying the MPF or LPF, the spacing error dynamics model in frequency domain takes the form as

δ_i(s)=with a set of transfer functions

G_r(s), where m denotes m preceding vehicles of the ith vehicle. If m preceding vehicles have the same effect on the ith vehicle, the set of transfer functions G_r(s) can take the same form that is denoted as G_m(s), then the spacing error dynamics model is replaced by

                        (4)

Similarly, if the PSF is applied, the spacing error dynamics model in frequency domain takes the form as δ_i(s)=G_p(s)δ_i-1(s)+G_s(s)δ_i+1(s) with a pair of transfer functions G_p(s) and G_s(s). If the immediate preceding vehicle and the immediate following vehicle have the same effect on the ith vehicle, the pair of transfer functions can take the same form that is denoted by G_ps(s), then the spacing error dynamics model is replaced by

                  (5)

4 Analysis of string stability

In this section, the effect of the information delay on the string stability of platoon of automated vehicles were conducted respectively for the LPF, MPF and PSF and then the stable string conditions were derived from the analysis of string stability if the string stability can still be obtained.

4.1 Leader-predecessor framework (LPF)

The LPF refers that the control law of each automated vehicle is established based on the position, velocity and/or acceleration information from itself, the leading vehicle and the single or multiple preceding vehicles of it. This framework, which is illustrated in Fig.2, is proposed as an effective way to fix the string instability when applying constant distance spacing policy and is commonly applied in the area of automated highway systems (AHS) [5, 13].

Fig.2 Schematic diagram of LPF

In Eq.(1), the spacing error between successive two vehicles with constant distance spacing policy is offered and then the spacing error between the (i-r)th preceding vehicle and the ith vehicle is defined as δ_ri(t)=x_i-r(t)- x_i(t)-rL-rD, and then the sum of m spacing errors is obtained as

     (6)

Similarly, the spacing error between the leading vehicle and the ith vehicle is defined as δ_0i(t)=x₀(t)-x_i(t)- iL-iD. Hence, the control law based on the information of the leading vehicle and m preceding vehicles is proposed as

      (7)

where  

and k_d, k_p, and denote the control gains that are positive in general.

After the combination of the longitudinal vehicle dynamics model (3) and the proposed control law (7), and some basic algebraic calculations, the related spacing error dynamics model in time domain takes the form as

(8)

Taking the Laplace transformation on both sides of the spacing error dynamics model (6), then the spacing error dynamics model in frequency domain (4) is obtained, and G_im(s) takes the form as

 (9)

The sufficient string stable condition [20] is m|G_im(jw)|＜1 forw＞0, where G_im(jw) is derived from G_im(s) by substituting s=jw. If assuming m|G_im(jw)|= then a and b can be expressed as a=m²k_d²w²+ m²k_p² and

(10)

Case 1: If ≥then taking into account the fact that -sin(Δw)≥-Δw and -cos(Δw)≥-1 for ＞0, Eq.(10) can be simplified as

b≥

                     (11)

Clearly, if ≥0 and ≥0 hold, and the right

hand of inequality (11) is more than zero for ＞0, then b＞0 and m|G_im(jw)|=＜1 for ＞0 are obtained. If k′_d=k_d=k_D and k′_p=k_p=k_P, then we obtain:

      (12)

where

Case 2: If ＜ taking into account the fact that sin(Δw)≥-Δw and -cos(Δw)≥-1 for ＞0, Eq.(10) can be simplified as

b≥

                    (13)

Assuming that k′_d=k_d=k_D and k′_p=k_p=k_P, but condition ≥0 and condition ≥0 cannot be obtained simultaneously, the right hand of inequality (13) is not always more than zero. In other words, the string stability cannot be guaranteed. Finally, the sufficient string stable condition for the LPF takes the form as inequality (12) with consideration of the information delay.

If without consideration of the information delay, that is, Δ=0, then b is replaced by

   (14)

which is derived from Eq.(10) with Δ=0. Hence, if ≥0 and 1-2(mk_d+≥0 hold, the right hand of inequality (14) is more than zero for＞0. Then the sufficient string stable condition for the LPF without consideration of the information delay is obtained as

                          (15)

For instance, assuming m=2 and τ=0.2 s, then the string stable condition (15) will take the specific form as 0＜k_P≤0.579 and 0＜k_D≤0.833. But if the information delay is considered and specified as Δ=0.2 s, string stable condition (12) will take the specific form as 0＜k_P≤0.156 and 0＜k_D≤0.432. Clearly, the ranges of the control gains, such as, k_P and k_D, of the control law (7) are smaller than those without consideration of the information delay. If the control gains are selected in the following ranges, 0.156＜k_P≤0.579 and 0432＜k_D≤0.833, the string stability cannot be guaranteed. Thus, analysis of string stability with considering the information delay provides more practical constraints of the selection of the control gains, and the result is valuable for the practical design of the controllers of the automated vehicles.

4.2 Multiple-predecessors framework (MPF)

The MPF refers that the control law of each automated vehicle is established based on the position, velocity and/or acceleration information from itself and the multiple preceding vehicles of it. This framework, which is illustrated in Fig.3, is proposed as a “weak” way to fix the string instability when applying constant distance spacing policy.

Fig.3 Schematic diagram of MPF

It is well known that the “weak” string stability can be obtained when applying MPF with constant distance spacing policy without consideration of information delay [13]. It is necessary to determine whether the “weak” string stability can be guaranteed when information delay is considered. Then the control law based on the information of m preceding vehicles is proposed as

           (16)

where control gains k_P, k_I and k_D are positive in general.

Following the procedures and calculations applied above, the spacing error dynamics model in frequency domain (4) is obtained, and G_im(s) takes the form as

       (17)

Similarly, assuming m|G_im(jw)|=then we obtain

 (18)

If k_D≥τk_P and k_P≥τk_I, taking into account the fact that sin(Δw)≥-Δw, -sin(Δw)≥-Δw, cos(Δw)≥-1 and -cos(Δw)≥-1 for ＞0, Eq.(18) can be simplified as

b≥

             (19)

Because τk_I-k_P-Δk_I＜0, even though the condition 1-2mτk_D(τ+Δ)2mk_PτΔ≥0 holds, it is impossible to obtain b＞0 and m|G_im(jw)|=＜1 for ＞0 in this case. Similarly, it is also impossible to obtain b＞0 and m|G_im(jw)|=＜1 for ＞0 in the following cases, such as, k_D＜τk_P and k_P＜τk_I, k_D≥τk_P and k_P＜τk_I, or k_D＜τk_P and k_P≥τk_I. Hence, control law (16) cannot guarantee string stability even the “weak” string stability when the information delay is considered.

Obviously, the above analysis of string stability for the MPF demonstrates that the string stability cannot be guaranteed with consideration of the information delay. In practice, if the platoon of automated vehicles uses the MPF, the good stable string performance cannot be obtained.

4.3 Predecessor-successor framework (PSF)

The PSF refers that the control law of each automated vehicle is established based on the position, velocity and/or acceleration information from itself and the immediate preceding vehicle and the immediate following vehicle of it. This framework, which is illustrated in Fig.4, is proposed as a proper way to fix the string instability for the platoon of automated vehicles that apply the constant distance spacing policy.

Fig.4 Schematic diagram of PSF

SEILER et al [19] and BAROOAH et al [21] demonstrated that the linear control law, which is only based on the position and velocity information, for PSF with constant distance policy, such as u_i(t)= k_xδ_ps(t), suffers from limitations that make it impossible to achieve good stable string performance for arbitrarily large platoon. Then, the control law that adds the acceleration information of the ith vehicle is proposed as

             (20)

wheredenotes the acceleration of the ith vehicle in the platoon; δ_ps=x_i-1(t)-2x_i(t)+x_i+1(t) andv_i-1(t)- 2v_i(t)+v_i+1(t); and k_a, k_D and k_P denote the positive control gains.

Following the procedures and calculations applied above, the spacing error dynamics model in frequency domain (5) is obtained, and G_ps(s) takes the form as

  (21)

The sufficient string stable condition [20] is 2|G_ps(jw)|＜1 for ＞0, where G_ps(jw) is derived from the spacing error propagation transform function (21) by substituting s=jw. Assuming 2|G_ps(jw)|=, then we obtain

    (22)

Case 1: If k_D≥k_Pτ, taking into account the fact that sin(Δw)≥-Δw, -sin(Δw)≥-Δw and -cos(Δw)≥-1 for ＞0, Eq.(22) can be simplified as

b≥

           (23)

Hence, if (k_a-1)²-4k_D(τ+Δ)+4k_PτΔ＞0, τ＞2k_aΔ and k_a＞1 hold simultaneously, the right hand of inequality (23) is more than zero for ＞0, then b＞0 and 2|G_im(jw)|=＜1 for ＞0 will be obtained, and the following condition will yield

                            (24)

Case 2: If k_D＜k_Pτ, taking into account the fact that sin(Δw)≥-Δw and -cos(Δw)≥-1 for ＞0, Eq.(22) can be simplified as

b≥

            (25)

Following the same analysis of case 1, the following condition is obtained

                            (26)

Clearly, ＞ is obtained because

τ＞2Δ. Thus, 0＜k_P＜ is obtained. Finally, the

sufficient string stable condition for the PSF takes the form as condition (24) with consideration of the information delay.

Without consideration of the information delay, that is, Δ=0, then b is replaced as

b=τ²w⁶+[(k_a-1)²-4k_Dτ]w⁴+4k_P(k_a-1)w²           (27)

Hence, if k_a≥1 and (k_a-1)²-4k_Dτ≥0 hold simultaneously, the right hand of Eq.(27) is more than zero. Then, the sufficient string stable condition takes the form as

                                (28)

without consideration of the information delay.

For instance, assuming τ=0.4 s, then the string stable condition (28) will take the specific form as 1≤k_a,   0＜k_D＜0.625(k_a-1)² and 0＜k_P, but if the information delay is considered and specified as Δ=0.15 s, the string stable condition (24) will take the specific form as 1＜ k_a＜0.627, 0＜k_D＜0.069 and 0＜k_P＜0.174. By comparing the two specific string stable conditions, the string stable cannot be guaranteed if k_a＞3.07, k_D＞0.069 and k_P＞0.174. Clearly, the ranges of the control gains of the control law (20) are decreased by the information delay.

5 Comparative simulations

To corroborate the above results, a serial of numerical simulations were conducted by applying a platoon of 16 automated vehicles. The leading vehicle was labeled 0 and the following automated vehicles were labeled from 1 to 15. The velocity and acceleration profiles of the leading vehicle were shown in Fig.5. In the simulations, the vehicle parameters are selected as τ=0.3 s, Δ=0.1 s, L=5 m and D=5 m. The specified values of the control gains applied by control laws (7), (16) and (20) in the simulations are listed in Table 1.

Fig.6 demonstrates excellent tracking performance of the following automated vehicles for the LPF with the information delay. It shows that the spacing errors of the 15 following automated vehicles in the platoon smoothly decrease upstream, and the string stability is guaranteed perfectly if the control gains are selected under the constraint of string stable condition (12) even though the information delay exists.

Fig.5 Velocity (a) and acceleration (b) profiles of leading vehicle

Table 1 Values of control gains in simulations

Fig.7 demonstrates bad tracking performance of the following automated vehicles for the MPF with the information delay. It shows that the spacing errors of the following automated vehicles amplify as they propagate upstream from one vehicle to the other vehicle. Actually, several different groups of control gains are selected to conduct the comparative simulations, but the string stability, even though the “weak” string stability cannot be obtained for the MPF.

Fig.8 also demonstrates good tracking performance of the following automated vehicles for the PSF with information delay. It is also meant that the string stability can be guaranteed if the control gains are selected under the constraint of the string stable condition (24), even though the information delay exists.

Fig.6 Spacing error profile for LPF

Fig.7 Spacing error profile for MPF

Fig.8 Spacing error profile for PSF

Comparing Fig.6 with Fig.8, the LPF causes the smaller spacing error of the ith vehicle than the PSF does. In other words, the LPF provides much better stable string performance. However, the PSF only depends on the local information which is measured by the on-board sensors [16]. This shows that the PSF provides much better scalable string performance.

6 Conclusions

(1) The effect of the information delay on the string stability is analyzed and simulated for three typical information frameworks. The information delay decreases the ranges of the control gains of the control laws (7) and (20) for the LPF and PSF respectively, and causes string instability for the MPF.

(2) The sufficient string stable conditions (12) and (24) demonstrate that the design of the controllers of the automated vehicles should consider the effect of the information delay on the control gains. In practice, if the inappropriate control gains are chosen, the string stability is probably not obtained with potential decrease in driving comfort and traffic safety.

(3) The results of string stability analysis demonstrate that choosing proper information framework to construct the platoon of automated vehicles is very important for string stability. For instance, if the MPF is chosen, the string stability is not obtained due to the effect of the parasitic information delay of communication systems and sensing systems.

(4) The results of the comparative simulations demonstrate that choosing proper information framework is very important for string scalability. For the small/ medium platoon, the LPF is the first choice because LPF leads to much better stable string performance than the PSF. For the large platoon, the PSF is the first choice because PSF only depends on the local information measured by the on-board distance sensors.

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(Edited by CHEN Wei-ping)

Corresponding author: XIAO Ling-yun, Assitant Research Fellow; Tel: +86-10-58811974; E-mail: xiaoly@dpac.gov.cn