On Scalability of Platoon of Automated Vehicles for Leader-predecessor Information Framework Lingyun Xiao, Feng Gao, Jiangfeng Wang School of Transportation Science and Engineering Beihang University, Beijing, 100191, China Email:
[email protected][email protected] Abstract-It is well known and broadly accepted that the leader-predecessor framework has been proposed to fix the string instability which is caused by applying the single-predecessor information framework. However, a reasonable but easily neglected fact is that the communication delay between the leading vehicle and the following vehicle is larger than the communication delay between the immediate preceding vehicle and the following vehicle when the large platoon is under consideration. The scalability issue arisen immediately is that whether or not the proposed control laws guaranteeing string stability for the small/medium platoon are suitable for the large platoon. This paper firstly proposes PD control law and sliding model control law for constant distance spacing policy and constant time headway spacing policy respectively, and then proves that both of the two control laws can guarantee string stability for the small/medium platoon. Secondly, it demonstrates that the PD control law for constant distance spacing policy can not guarantee string stability for the large platoon but the sliding model control law for constant time headway spacing policy can. In other words, the sliding model control law for constant time headway spacing policy can make the platoon applying leader-predecessor information framework scalable. I. INTRODUCTION
The purpose of the automatic vehicle following control is to automatically maintain a desired spacing from its preceding vehicle to improve driving comfort, traffic safety and traffic capacity. String stability has been a significant topic of the automatic vehicle following control since mid 1970s [1], [2]. A precise definition of string stability was provided by Chu [1]. Intuitively, the term "string stability" indicates that spacing errors do not amplify as they propagate upstream from one vehicle to another vehicle [3], [4]. This property ensures that any perturbation of the velocity or position of the leading vehicle will not result in amplified fluctuations to the following vehicles' velocity and position [5]. If spacing and velocity errors amplify as they propagate upstream, it is not only likely to provide poor ride quality but also could result in collision [6]. It is well known that the string stability cannot be guaranteed when the platoon consisting of automated vehicles applies the single-predecessor information framework with the constant distance spacing policy [7], [8], [9], but this string instability can be fixed by applying leader-predecessor information framework [7], [8], [10] which adds the velocity and position information of the leading vehicle to construct control laws. Most physical systems often involve time delays and lags,
978-1-4244-3504-3/09/$25.00 ©2009 IEEE
which may significantly affect the stability of systems in some circumstances. The automatic vehicle following control system involves parasitic time delays and lags in engine, actuators, sensors, drive-line and communication system [11]. For the sake of practical design and deployment, it is inevitable to prove whether or not the proposed control laws still guarantee string stability under consideration of the parasitic time delays and lags and, if satisfied, to obtain the string stable conditions which are constrained by the values of the parasitic time delays and lags. Huang and Ren [12] have demonstrated that the "weak" string stability can be guaranteed by applying leaderpredecessor information framework under consideration of the parasitic time delays and lags. In fact, the main parasitic time delay is the communication delay between the leading vehicle and each following vehicle, which amplifies with the size of the platoon increasing. For the small/medium platoon, the difference between the leading communication delay and the preceding communication delay is so small that it can be neglected during the string stability analysis. However, this difference should be taken into consideration for the large platoon. Hence, the property in which the proposed control law can guarantee string stability for both the small/medium platoon and the large platoon is called as scalability [13]. The remainder of this paper is structured as follows: In section II, we formulate the longitudinal vehicle dynamics model under consideration of "lumped" parasitic delay and lag. In section III, we propose PD control law and sliding model control law for constant distance spacing policy and constant time headway spacing policy respectively and demonstrate both of the two control laws can guarantee string stability for small/medium platoon. In section IV, we analyze the string stable issue of the two control laws for the large platoon. In section V, we provide the comparative numerical simulations and conclude thereafter in section VI. II. LONGITUDINAL VEHICLE DYNAMICS MODEL
The string of N + 1 automated vehicles assume to operate in only one spatial dimension and be controlled by the identical control law with identical longitudinal vehicle dynamics model. Let Xo (t ), Vo (t) and ao (t) denote the position, velocity and acceleration of the leading vehicle respectively and Xi (t), Vi (t) and ai (t) (1 ::; i ::; N) denote the position, velocity and acceleration of the i th following vehicle in the string
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with a pair of transfer function Go (s) and G 1 ( s) for the leaderpredecessor information framework.
respectively. Then we obtain the spacing error as:
ei(t) - D, constant distance 8i () t ={ ei(t) - hVi(t), constant time headway
(1)
where 8i (t) denotes the spacing error of the i th vehicle which is the deviation between the real spacing and desired spacing, ei(t) = Xi-l (t) - Xi(t) - L denotes the real spacing between the i th vehicle and the (i - 1) th vehicle, L denotes the length of the vehicle, D denotes the desired constant distance and h denotes the desired constant time headway. For convenience of analysis, we assume that Vi(t) = vo(t), ai(t) = ao(t) = 0 and 8i (t) = 0 (1 ::; i ::; N) hold 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 [14]. In order to arrive at a simple longitudinal vehicle model, one should make some appropriate assumptions [13], [15], [16] and use the feedback linearization technique to obtain the linearized model as:
Xi(t) = AiXi(t) + BiUi(t) where Xi
= [~:l
Ai
= [~ ~], B i =[~]
(2)
As mentioned previously, automated vehicle following control system involves all-pervasive parasitic delays and lags. In this note, the various parasitic delays and lags are combined into two "lumped" time delay and lag which represent by symbol ~ and T respectively. For convenience, the delay-lag factor has been introduced as: DL * ( s) = e- ~s j (T s+ 1). Then, the longitudinal vehicle model (2) is replaced by introducing the delay-lag factor as [11]:
where Xi =
Ui(t -
+ BiUi(t -~)
(3)
[~:], Ai = [~ ~ ~1] ,Bi = [~] , Ui(t-~) = ai 0
0
--::;:
In this section, we propose two control laws based on the constant distance spacing policy and the constant time headway spacing policy respectively for small/medium platoon, and then analyze the string stable issue of the two control laws, and finally obtain the related string stable conditions.
A. Constant Distance Spacing Policy In equation (1), the spacing error between successive two vehicles with constant distance spacing policy has been offered and then the spacing error 80i ( t) between the leading vehicle and the i th vehicle is defined as:
80i (t) = xo(t) - Xi(t) - iL - iD.
Ui(t) = kD(8i (t)
+ 80i (t)) + kp(8i (t) + 80i (t))
(7)
where 8i (t) = Vi-l (t) - Vi(t), 80i (t) = vo(t) - Vi(t), kD and k p denote the control gains which are positive in general. Under consideration of the parasitic time delays and lags within vehicle dynamics, the control law (7) is replaced as:
TUi(t)
+ Ui(t) = kD( 8i (t +k p (8 i (t
~)
+ 80i (t - ~)) + 80i (t - ~)).
-~)
(8)
Combination with the longitudinal vehicle model (3) and after some algebraic calculations (the detailed calculations can be referred from [11]), we obtain the spacing error dynamics model in time domain of successive two vehicles in a platoon, such as,
T 8i (t) + 8i (t) + 2k D8i (t -~) + 2k p 8i (t -~) k D 8i - 1 (t -~) + k p 8i - 1 (t - ~). (9)
-::;:
~).
The linearized longitudinal vehicle dynamics model is applied to establish control laws and obtain the string stable conditions, so that the analysis can be performed in the frequency domain, using Laplace transforms and transfer functions. As mentioned previously, 8i (t) denotes the spacing error of the i th vehicle in time domain, then 8i (s) denotes the spacing error of i th vehicle in frequency domain. In leader-predecessor information framework, the spacing error dynamics model in frequency domain takes the form as:
(6)
Huang and Ren [12] have demonstrated that when the acceleration of the leading vehicle is applied to establish control law, only the "weak" string stability can be guaranteed. In other words, the string stability can't be guaranteed for V w > O. Hence, we propose the PD control law only based on the position and velocity information, such as,
and Ui(t) = Ui(t),
Ui (t) denotes the control law applied to the i th vehicle.
Xi(t) = AiXi(t)
III. STRING STABILITY OF SMALL/MEDIUM PLATOON
Taking the Laplace transformation on both sides of equation (9), we obtain the spacing error dynamics model (4) in frequency domain and G(s) takes the form as:
G(s) - TS 3
(kDS + kp)e-L:!..s + s2 + 2kDse-~s + 2kpe-~s
(10)
with a transfer function G(s ). If the above spacing error dynamics model is not available after the basic algebraic calculations, the more general model takes the form as [17]:
which is called as spacing error propagation transfer function between successive two vehicles in the presence of parasitic delays and lags. The sufficient string stable condition [18] is IG(jw) I < 1, for V w > 0, where G(jw) is derived from the spacing error propagation transform function (10) by substituting s = jw. If we assume IG(jw)1 = y'aj(a + b), then we obtain
(5)
(11)
(4)
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ei(t) - hVi(t) = VO(t) - 2Vi(t) + Vi-l (t) - (i + l)hvi(t), the
and
b
= (3k'b - 4kp cos(~w))w2 - 4(kD - kp7) sin(~w)w3 +7 2W 6 + 3k~ + (1 - 4kD7COS(~w))w4. (12)
Taking into account the fact that - sin(~w) 2: -~w and -cos(~w) 2: -1 for V w > 0 and holding the condition kD 2: kP7, then we obtain
desired control law is obtained as: 1 . .
Ui(t) = H [eOi(t) + ei(t) + -X8zp (t)]
(17)
where H = (i + l)h, eOi(t) = vo(t) - Vi(t) and ei(t) = Vi-l(t) - Vi(t). Under consideration of the parasitic time delays and lags within vehicle dynamics, the control law (17)
b 2: 72w6+3k~+(3k'b-4kp )w2+(1+4kp7~-4kD(7+~))w4. is replaced as: (13). 1 . . Hence, if 3k'b :?: 4k p and 1 + 4k p rD.. :?: 4k D (r + D..) hold rUi(t)+Ui(t) = H[eOi(t-D..)+ei(t-D..)+,X8/ p (t-D..)]. (18)
and because 7 W + 3k~ > 0 holds for V w > 0, the right hand of the inequality (13) is more than zero. Clearly, we obtain b > 0 and then obtain IG(jw)1 = y'a/(a + b) < 1 for V w > O. Finally, we obtain the sufficient string stable condition as: 2
6
k < 8(7 + ~)2 - 37~ - 4(7 + ~)F(7, ~) < p 1272~2 ' 2(r + D..) - y'4(r + D..)2 _ 3rD.. (14) o < kD ::; 37~
o
where F(7,~) = y'4(7 + ~)2 - 37~. Remark 1: If without consideration of the parasitic time delays and lags, the string stable condition takes the form as 3k'b 2: 4kp. The control gains kD and kp are not constrained with the upper bounds like the condition (14). Intuitively, if the control gains which are selected under the constraint of the condition 3k'b 2: 4kp during the design and implementation process may potentially lead to string unstable.
B. Constant Time Headway Spacing Policy In our earlier work [11], we have conducted string stability analysis and obtained the associated string stable condition, h > 2 (7 + ~), for single-predecessor information framework with constant time headway spacing policy. However, if the sum of the parasitic delays and lags of some specific vehicles is more than and equal to the constant time headway, such as 2 (7 + ~) 2: h, especially for the commercial heavy vehicle (CHV) [19], the earlier result in [11] is not available to guarantee the string stability. Hence, adding the information of the leading vehicle is an effect option to fix this kind of string instability. In equation (1), the spacing error between successive two vehicles with constant time headway spacing policy has been offered and then the spacing error 80i ( t) between the leading vehicle and the i th vehicle is defined as: (15) where the real distance eOi (t) between the leading vehicle and the i th vehicle takes the form as: (16)
Defining a sliding surface SI = 8zp = 80i + 8i , an asymptotically converging algorithm can be obtained by imposing the requirement, 81 = --XS1 , where -X > 0 is the convergence rate of the sliding surface. Since 8zp (t) = eOi(t) - ihvi(t) +
Combination with the longitudinal vehicle model (3) and after some algebraic calculations and taking the Laplace transformation, we obtain the spacing error dynamics model (5) with Go(s) = G l (s) = G(s) which takes the form as:
G(s)
= Hrs 3
(s + -X)e-~S + H s2 + (H'x + 2)se-.6.s
+ 2'xe-.6.s·
(19)
The sufficient string stable condition [18] is IGo(jw) I + IG 1 (jw)1 = 2IG(jw)1 < 1 for V w > 0, where G(jw) is derived from G(s) by substituting s = jw. If we assume 2IG(jw)1 = y'a/(a + b), then we obtain (20) a=4(w 2 +-X 2) and
b
H 27 2W 6 + H(H - 27(2 + H-X) cos(~w))w4 -2H(2 + H-X - 27-X) sin(~w)w3 (21) +H(4-X(1 - cos(~w)) + H-X 2)w 2.
Taking into account the fact that 1 - cos(~w) 2: 0, - sin(~w) 2: -~w and - cos(~w) 2: -1 for V w > 0 and holding the condition (i + l)h 2: 27, then we obtain
b >
H(H - 4(7 +~) - 2H-X(7 +~) + 4-X7~)w4 +H 2-X 2w 2 + H 27 2W 6. (22)
Hence, if H - 4(7 +~) - 2H-X(7 +~) + 4-X7~ 2: 0 holds and because H 2-X 2w 2 + H 27 2W 6 > 0 holds for V w > 0, the right hand of the inequality (22) is more than zero. Clearly, we obtain b > 0 and then obtain 2IG(jw)1 = y'a/(a + b) < 1 for V w > O. Finally, we obtain the sufficient string stable condition as: 4 h>-·-1(~+7),
t+
O 0 and holding the condition kD 2: kP7, then we obtain
b 2:
IV. STRING STABILITY OF LARGE PLATOON In this section, we analyze the string stable issue of the proposed two control laws for large platoon by applying the different values for the leading communication delay and the preceding communication delay.
A. Constant Distance Spacing Policy As mentioned above, the leading communication delay is larger than the preceding communication delay for the large platoon and it is not reasonable to ignore the difference between of them. In this paper, the leading communication delay is denoted by ~l and the preceding communication delay is denoted by ~p. Hence, the above proposed control law (7) is replaced by:
7Ui(t)
+ Ui(t) =
k D(8 i (t - ~p) + 80i (t - ~l)) +kp(8i (t - ~p) + 80i (t - ~l)).(24)
Assuming that the i th following vehicle's leading communication delay is equal to the (i - 1) th following vehicle's, then we obtain
+ +
7Ui-l(t)
Ui-l(t) = kD(8i - 1(t - ~p) + 80i - 1 (t - ~l)) k p (8 i - 1 (t - ~p) + 80i - 1 (t - ~l)). (25)
Taking equation (25) -equation (24) and after some basic algebraic calculations, we obtain the spacing error dynamics model in time domain of successive two vehicles in a platoon, such as, 7
·8·i (t) + 8i (t) + kD(8i (t -
~p)
+8i (t - ~l)) + kp(8i (t - ~p) + 8i (t - ~l)) kD 8Pi _ 1 (t - ~p) + k p 8i - 1 (t - ~p). (26) Taking the Laplace transformation on both sides of equation (26), we obtain the spacing error dynamics model (4) in frequency domain and G(s) takes the form as:
G
S
-
( )-
TS 3
(kDS + kp )e-.6. p s + s2 + (kDS + kp )(e-.6. p s + e-.6. 1 s)·
(27)
The sufficient string stable condition [18] is IG(jw) I < 1, for V w > 0, where G(jw) is derived from the spacing error propagation transform function (27) by substituting s = jw. If we assume IG(jw)1 = y'aj(a + b), then we obtain
a = kbw2
+ k~
(28)
= 7 2W6 + (kb(l + 2COS(~l -
+ ~p + ~l))w4.
(30)
B. Constant Time Headway Spacing Policy For the large platoon, the leading communication delay, ~l, and the preceding communication delay, ~P' are applied to the related control law (17) which is replaced by:
7Ui(t)
+ Ui(t) =
1 .
.
+ ei(t - ~p) ~l) + 8i (t - ~p))].(31)
H [eOi(t - ~l) +A(8oi (t -
Combination with the longitudinal vehicle model (3) and after some algebraic calculations and taking the Laplace transformation, we obtain the spacing error dynamics model (5) with Go(s) taking the form as:
Go(s) =
(s + A)e-~IS H7S 3 + H s2 + F(s)
(32)
and G1(s) taking the form as:
G1(s) =
(s + A)e-~pS H7S 3 + H s2 + F(s)
(33)
where F(s) = (H' AS + s + A)e-~IS + (hAS + s + A)e-~pS and H' = (i - l)h. The sufficient string stable condition [18] is IGo(jw) I + IG1(jw)1 < 1 for V w > 0, where Go(jw) and G1(jw) are derived from Go(s) and G1(s) by substituting s = jw respectively. If we assume IGo(jw)I+IG1(jw)1 = y'aj(a + b), then we obtain (34) Taking into account the fact that - sin ~pw 2: -~pw, 2: -~lW, -cos~pw 2: -1, -COS~lW 2: -1, sin(~l - ~p)w 2: -(~l - ~p)w and COS(~l - ~p)w 2: -1 for V w > 0 and holding the conditions h 2: 7 and H' 2: h, then we obtain -sin~lw
b >
H 27 2W6 + H(H - 2(27 + ~l + ~p) - 2(H ' (7 - ~l) +h(7 - ~p) + 7(~l + ~p)))w4 + (H '2 A2 2
2
2
+h A - 2H hA - 4HA - 2H A +2hA2(~l - ~p) - 4)w 2 - 4A 2.
+COS~lW))w2
(29)
k~ - (kb + 4k p )w 2 + (1 + 2kp7~p
Because -k~ < 0 and -(kb + 4kp) < 0 hold, even though the condition 1+2kp7~p+2kp7~l-2kD(27+~p+~l)2: 0 holds, it is impossible to obtain b > 0 for V w > 0 and then it is impossible to obtain IG(jw)1 < 1, for V w > O. Hence, the proposed control law (7) can't guarantee string stability for the large platoon by applying the leader-predecessor information framework with constant distance spacing policy.
~p)w) - 2kp(cos~pw
+ k~(l + 2COS(~l - ~p)w) -2(kD - kp7)(sin~pw + sin~lw)w3 +(1- 2kD7(COS~pw+ COS~lW))w4.
-
+2kp7~l - 2kD(27
and
b
7 2W6
I
I
(~l
-
~p)
(36)
In practice, the control gain A is selected as small value during the design and implementation process and the large platoon
1106
b
H 27 2W6
27(H' -X + 1) COS~lW - 27(h-X + 1) cos~pw)w4 -2H((1 + H' -X - 7-X) sin~lw + (1 + h-X - 7-X) sin~pw)w3 + (H'2-X 2 + h 2-X 2 + 2H' -X + 2h-X - 2 -
2-X 2
+ H(H -
+2(H' h-X + H' -X + h-X + 1) COS(~l - ~p)w - 2H-XcoS~lW - 2H-Xcos~pw)w2 +2w-X 2(H' - h) sin(~l - ~p)w + 2-X 2 COS(~l - ~p)w.
TABLE I SPECIFIC PARAMETERS FOR CD POLICY
Parameters
~l
Small/Medium Platoon 0.2s Large Platoon
O.4s
~p
T
Kp
KD
(35)
TABLE II SPECIFIC PARAMETERS FOR CTH POLICY
D
L
Parameters
~l
~p
T
A
h
L
0.2s
0.2s 0.35 0.65 5m 5m
Small/Medium Platoon
0.2s
0.2s
0.2s
0.5
Is
5m
0.2s
0.2s 0.35 0.65 5m 5m
Large Platoon
O.4s
0.2s
0.2s
0.5
Is
5m
leads to the large value of Hand H'. Hence, the terms, such as, 4-X 2 , h 2-X 2 and 2h-X2(~l - ~p) can be ignored comparing to the other terms. Then the inequality (36) is replaced as:
meant that the PD control law can guarantee string stability for small/medium platoon but not for large platoon with constant distance spacing policy. Fig. 3 and Fig. 4 illustrate that the spacing errors of both the first 10 following vehicles and the b > H 27 2W6 + H(H - 2(27 + ~l + ~p) - 2(H' (7 - ~l) last 10 following vehicles smoothly decrease upstream. It is +h(7 - ~p) + 7(~l + ~p)))w4 + (H'2-X 2 meant that the sliding model control law can guarantee string , , 2 2 -2H h-X - 4H-X - 2H -X (~l - ~p) - 4)w . (37) stability for both small/medium platoon and large platoon with constant time headway spacing policy. Hence, if the conditions H - 2(27 + ~l + ~p) - 2(H' (7 VI. CONCLUSION ~l) + h(7 - ~p) + 7(~l + ~p)) 2: 0 and H'2 -X 2 - 2H' h-X4H-X-2H' -X2(~l-~p)-4 2: 0 hold, and because H 27 2W6 > In this paper, the scalability issue of platoon of automated o holds for V w > 0, the right hand of the inequality (37) is vehicles for the leader-predecessor information framework has more than zero. Intuitively, we obtain b > 0 and then obtain been discussed with the consideration of different values of the IGo(jw)1 + IG 1 (jw)1 = y'aj(a + b) < 1 for V w > O. leading communication delay and preceding communication Hence, the string stability can be guaranteed by applying the delay due to the inevitable natural properties of wireless leader-predecessor information framework with constant time communication system. The proposed PD control law which headway spacing policy for large platoon. guarantees string stability for small/medium platoon is not available for the large platoon with constant distance spacing V. COMPARATIVE SIMULATIONS policy, but the sliding model control law can guarantee string To corroborate the above results, a serial of numerical stability for both small/medium platoon and large platoon simulations have been conducted by applying a platoon of 21 with constant time headway spacing policy. In other words, automated vehicles. The leading vehicle is labeled 0 and the the latter control law can make the platoon applying leaderfollowing automated vehicles are labeled from 1 to 20. The predecessor information framework scalable. leading vehicle has specific velocity profile which displays ACKNOWLEDGMENT that the leading vehicle runs with the steady initial velocity of 20m j s and then accelerates after an interval of 20s and finally The authors would like to acknowledge the support of China reaches a steady state with constant velocity of 40m j s. In the Scholarship Council and the comments of Professor Swaroop simulations, the leading communication delays take the same Darbha, Department of Mechanical Engineering, Texas A&M values as the preceding communication delays of the first 10 University, College Station. following vehicles (1-10), but the leading communication delays take the larger values than the preceding communication REFERENCES delays of the last 10 following vehicles (11-20). The values of parameters applied in the simulations are listed in Table I and [1] K. Chu, "Decentralized Control ofHigh-speed Vehicular Strings", Transportation Sicence, Vol. 8, No.3, pp. 361-384, Nov. 1974. Table II for the constant distance spacing policy and constant [2] L. Peppard, "String Stability of Relative-motion PID Vehicle Control time headway spacing policy respectively. Systems", IEEE Transaction on Automatic Control, pp. 579-581, Oct. 1974. Fig. 1 and Fig. 2 illustrate that the spacing errors of the [3] S. Darbha and J. Hedrick, "String Stability of Interconnected Systems," first 10 following vehicles decrease upstream but the spacing IEEE Transactions on Automatic Control, Vol. 41, No.3, pp. 349-357, errors of the last 10 following vehicles amplify upstream. It is Mar. 1996.
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[4] S. Darbha and K. Rajagopal, "A Review of Constant Time Headway Policy for Automatic Vehicle Following", 2001 IEEE ITS Conference Proceedings, Oakland, CA, USA, Aug. 2001, pp. 65-69. [5] J. Zhou and H. Peng, "Range Policy of Adaptive Cruise Control Vehicle for Improved Flow Stability and String Stability", IEEE Transactions on ITS, Vol. 6, No.2, pp. 229-237, Jun. 2005. [6] R. Rajamani and S. Shladover, "An Experimental Comparative Study of Autonomous and Co-operative Vehicle-follower Control Systems", Transportation Research Part C 9, pp. 15-31, 2001. [7] J. Eyre, D. Yanakiev, and I. Kanellakopoulos, "A Simplified Framework for String Stability Analysis of Automated Vehicles", Vehicle System Dynamics, Vol. 30, No.5, pp. 375-405, Nov. 1998. [8] S. Darbha and J. Hedrick, "Constant Spacing Strategies for Platooning in Automated Highway Systems", Journal of Dynamics Systems, Measurement, and Control, Vol. 121, pp. 462-470, Sep. 1999. [9] P. Seiler, A. Pant and K. Hedrick, "Disturbance Propagation in Vehicle Strings", IEEE Transaction on Automatic Control, Vol. 49, No. 10, pp. 1835-1841, Oct. 2004. [10] J. Hedrick, D. McMahon, V. Narendran, and S. Darbha, "Longitudinal Vehicle Controller Design for NHS Systems", Proceedings of the American Control Conference, Boston, USA, 1991, pp, 3107-3112. [11] L. Xiao, S. Darbha and F. Gao, "Stability of String of Adaptive Cruise Control Vehicles with Parasitic Delays and Lags", Proc. of the 11th International IEEE Conf. on ITS, Beijing, China, Oct. 2008, pp. 11011106.
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Fig. 4.
Spacing Error of the Last 10 vehicles in Platoon (CTH Policy)
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