Designing Network Motifs in Connected Vehicle ... - Semantic Scholar

Proceedings of the ASME 2013 Dynamic Systems and Control Conference DSCC2013 October 21-23, 2013, Palo Alto, California, USA

DSCC2013-4081

DESIGNING NETWORK MOTIFS IN CONNECTED VEHICLE SYSTEMS: DELAY EFFECTS AND STABILITY

Linjun Zhang∗ Department of Mechanical Engineering University of Michigan Ann Arbor, Michigan 48109 Email: [email protected]

´ Gabor Orosz Department of Mechanical Engineering University of Michigan Ann Arbor, Michigan 48109 Email: [email protected]

controllers must be modular and function even when connections are lost or new connections are established; (ii) the controllers must be robust against the delays arising in the communication channels. We propose a modeling framework that allows the existence of uniform traffic flow independent of the communication structure and analyze the limitations of the controllers with different delay configurations. In order to make the arising connected systems tractable we use a motif-based approach inspired by recent results in systems biology [7]. Analyzing the dynamics of simple motifs allows one to evaluate the effects of communication delays on plant and string stability. By combining these motifs, one can build complicated networks of prescribed behavior in a systematic way. The results are summarized using linear stability diagrams and numerical simulations are used to demonstrate the nonlinear behavior. The proposed approach can be used to design cooperative adaptive cruise control (CACC) systems where information about the motion of the vehicle in front (sensed by a radar) is integrated with the information about the motion of distant vehicles (received through wireless communication). Moreover, the presented framework is also applicable when human drivers react to the motion of the preceding vehicle while they are assisted by controllers that act upon the data received from distant vehicles. We show that by appropriately selecting the control gains for the long-distance communication links, plant and string stability can be achieved even in cases when the reaction time of human drivers is too large to allow stabilization for any shortdistance gain combinations.

ABSTRACT Arising technologies in vehicle-to-vehicle (V2V) communication allow vehicles to obtain information about the motion of distant vehicles. Such information can be presented to the driver or incorporated in advanced autonomous cruise control (ACC) systems. In this paper, we investigate the effects of multi-vehicle communication on the dynamics of connected vehicle platoons and propose a motif-based approach that allows systematical analysis and design of such systems. We investigate the dynamics of simple motifs in the presence of communication delays, and show that long-distance communication can stabilize the uniform flow when the flow cannot be stabilized by nearest neighbor interactions. The results can be used for designing driver assist systems and communication-based cruise control systems.

INTRODUCTION Adaptive cruise control (ACC) is receiving increasing attention these days due to the demand for safety and driver comfort in modern automobiles [1]. In traditional radar-based ACC systems, the vehicle only has information available about the motion of the preceding vehicle [2, 3]. However, emerging wireless communication technologies enable vehicles to observe the motion of distant vehicles, which can potentially increase the efficiency and safety of vehicular networks. At the same time, the modeling, analysis and design of connected vehicle platoons present additional challenges due to increased connectivity and delays caused by intermittency and packet drops in vehicle-to-vehicle (V2V) communication [4]. These delays can impact the performance of the transportation system significantly [5, 6]. In this paper we address the following two challenges: (i) the ∗ Address

all correspondence to this author.

1

Copyright © 2013 by ASME

V

V vmax

t i ,0

F1 (h)

t i ,1

t i ,i -1 vi

F2 (h)

t i ,i - 2 ĂĂ

hi

vmax

vi -1

hi -1

vi -2

0 hstop

v1

h1

hgo

v0

h

0 hstop

hgo

h

V vmax

Figure 1. STRUCTURE OF THE COMMUNICATION NETWORK WHEN A VEHICLE RECEIVES INFORMATION FROM MULTIPLE VEHICLES

F3 (h)

AHEAD. THE VELOCITIES, HEADWAYS AND COMMUNICATION DELAYS ARE DENOTED BY v j , h j AND τi, j , RESPECTIVELY.

0 hstop

Figure 2.

NETWORKED CAR-FOLLOWING MODELS In this section, we establish fundamental modeling principles of connected vehicular systems. Nonlinear models are constructed such that they guarantee the existence of the uniform flow equilibrium for arbitrary connectivity. A scenario where a vehicle communicates with multiple vehicles ahead is depicted in Fig. 1, where vi denotes the velocity of the ith vehicle, hi is the distance between the ith and the (i − 1)th vehicles (that is often called the headway), and τi, j represents the communication delay for the information sent from the jth to the ith vehicle for j = 0, 1, . . . , i − 1. We assume that the wireless communication provides the information about the positions and velocities of other vehicles so that the inter-vehicle distances and relative velocities can be calculated. Then, the dynamics of the ith vehicle is governed by

i−1

∑ γi, j βi− j

( ) v j (t − τi, j ) − vi (t − τi, j ) ,

h

RANGE POLICIES GIVEN BY (2).

summarized as   if h < hstop , 0 , V (h) = F(h) , if hstop ≤ h ≤ hgo ,   vmax , if h > hgo ,

(2)

where the function F(h) is continuous and strictly monotonically increasing such that F(hstop ) = 0 and F(hgo ) = vmax . The three possible choices h − hstop , hgo − hstop [ ( )] h − hstop vmax F2 (h) = 1 − cos π , 2 hgo − hstop [ ( ( ))] π 2h − hgo − hstop vmax 1 + tanh tan , F3 (h) = 2 2 hgo − hstop F1 (h) = vmax

h˙ i (t) = vi−1 (t) − vi (t) , ( ( ) ) i−1 i 1 v˙i (t) = ∑ γi, j αi− j V h (t − τ ) − v (t − τ ) i, j i i, j k i − j k=∑ j=0 j+1 +

hgo

(1)

(3)

are shown in Fig. 2. The simplest choice is the linear function F1 (h). However, the sharp changes in the derivatives at h = hstop and h = hgo lead to discontinuities in the jerk which can discomfort the driver. Therefore, one may use nonlinear functions F2 (h) and F3 (h) that result in smooth and infinitely smooth derivatives at h = hstop and h = hgo . In this paper, we use F2 (h) with hstop = 5 [m], hgo = 35 [m] and vmax = 30 [m/s]. A common goal in traffic systems is to achieve the so-called uniform flow equilibrium where equidistant vehicles travel with the same velocity:

j=0

where αi− j and βi− j are control gains related to the distances and the velocity differences, and the network structure is given by the adjacency matrix { 1, if vehicle i receives information from vehicle j, γi, j = 0, otherwise.

vi (t) ≡ v∗ ,

In addition, the range policy V (h) gives the desired velocity as a function of the inter-vehicle distance h. When the distance is below a threshold h < hstop , V (h) should be zero for safety reasons. For large headway h > hgo , the vehicle aims to maintain the allowable maximum velocity vmax . Between hstop and hgo , the function increases monotonically with h. This can be

hi (t) ≡ h∗ ,

(4)

for all i ∈ N. Notice that the right hand side of (1) is designed in a way that the system possesses the equilibrium v∗ = V (h∗ ) independent of the connectivity structure γi, j and control gains αi− j , βi− j .

2


In this paper, we discuss two kinds of stability of the uniform flow: plant stability and string stability. Plant stability means that, if the leading vehicle of the platoon is moving with a constant velocity v∗ , then the velocities of the following vehicles converge to v∗ . On the other hand, the system is said to be string stable if velocity fluctuations are attenuated as they propagate along the platoon [8]. In case of conventional vehicular traffic, each vehicle reacts to the motion of the vehicle in front. Then, considering identical vehicles one may define string stability using two consecutive vehicles since this implies string stability for an arbitrary pair of vehicles in the platoon. However, for connected vehicle systems one may observe string stability when comparing the velocities of a chosen vehicle pair while other pairs may still be string unstable. To avoid such ambiguity we analyze leader-to-tail string stability and compare the fluctuations in the velocities of the first and last vehicles in the platoon. At the linear level, plant and string stability can be evaluated by analyzing transfer functions that give algebraic relationships between velocities in the Laplace domain. In particular, we need to analyze the leader-to-tail transfer function. Plant stability is guaranteed when all poles of this transfer function are located in the left half complex plane. Moreover, by exploiting that the input signal can be represented by its Fourier components and that superposition holds for linear systems, attenuation of velocity fluctuations can be ensured if the magnitude of the leader-totail transfer function is smaller than one for all frequencies. One may calculate the leader-to-tail transfer function using link transfer functions which can be derived from the differential equations and act as complex gains along the communication links. In fact, the leader-to-tail transfer function is the sum of the products of link transfer functions along the paths between the first and last vehicles in the platoon as demonstrated below.

Motif 1

t No. 1

T10

No. 0

s T 20

Motif 2

No. 2

x

t

T21

T10

No. 1

d

Motif 3

No. 3

T30

x

t

t

T32

T21

No. 1

T10

d

T41

No. 2

Network

No. 4

No. 0

s

x

t

x

T43

T32

T21

No. 3

No. 2

No. 0

T20

t No. 1

T10

No. 0

Figure 3. CAR-FOLLOWING MOTIFS AND A SIMPLE NETWORK BUILT FROM THESE MOTIFS. THE COMMUNICATION DELAYS AND THE LINK TRANSFER FUNCTION ARE SHOWN ALONG THE LINKS.

( ( ) ) i 1 + αn V ∑ hk (t − τi,i−n ) − vi (t − τi,i−n ) n k=i−n+1 ( ) + βn vi−n (t − τi,i−n ) − vi (t − τi,i−n ) . (5) Defining the perturbations h˜ i = hi − h∗ and v˜i = vi − v∗ , the linearization of (5) about the uniform flow equilibrium (4) becomes h˙˜ i (t) = v˜i−1 (t) − v˜i (t) , v˙˜i (t) = φ1 h˜ i (t − τi,i−1 ) − κ1 v˜i (t − τi,i−1 ) + β1 v˜i−1 (t − τi,i−1 ) ( ) i ˜ + φn h (t − τ ) − κn v˜i (t − τi,i−n ) i,i−n ∑ k

FUNDAMENTAL NETWORK MOTIFS In this section, we define network motifs as simple networks which can be used to construct more complex connected vehicle systems, and calculate the corresponding leader-to-tail transfer functions. Motif n is defined as the network where the tail vehicle observes the motion of the vehicle immediately ahead as well as the motion of the leader n-vehicles ahead. Some of the fundamental motifs are shown in Fig. 3 together with a simple network built from motifs 1, 2 and 3. For each motif a simplified delay setup is considered where σ and δ are communication delays, τ represents the human reaction time, while ξ either represents the ACC computational time (ξ < τ ) or denotes the human reaction time (ξ = τ ) when the vehicle is not equipped with a radar. For the sake of simplicity we derive all formulae for the latter case. Using the general model (1) for motif n (where γi,i−1 = γi,0 = 1 and γi, j = 0 otherwise), one can derive

k=i−n+1

+ βn v˜i−n (t − τi,i−n ) ,

(6)

where

φn =

αnV ′ (h∗ ) , n

κn = α n + βn .

(7)

First, we calculate the leader-to-tail transfer functions for the first two motifs and then provide a general formula for the leader-to-tail transfer function of motif n. Using this we calculate the the leader-to-tail transfer function for the simple network in Fig. 3. Motif 1 represents the conventional car-following scenario where each vehicle has information about the motion of the vehicle immediately ahead. Considering that vehicle No. 1 follows vehicle No. 0 as shown in Fig. 3, we have

h˙ i (t) = vi−1 ( (t) ( − vi (t) , ) ) v˙i (t) = α1( V hi (t − τi,i−1 ) − vi (t − τi,i−1)) + β1 vi−1 (t − τi,i−1 ) − vi (t − τi,i−1 )

3


In the Laplace domain, we obtain h˙ 1 (t) =v0 (t) − v1 (t) , ( ( ) ) v˙1 (t) =α1 V (h1 (t − τ )) − v1 (t − τ ) + β1 v0 (t − τ ) − v1 (t − τ ) . (8)

V˜2 (s) = T21 (s)V˜1 (s) + T20 (s)V˜0 (s) , where the link transfer functions are (β1 s + φ1 )e−sτ , s2 + (κ1 s + φ1 )e−sτ + (κ2 s + φ2 )e−sσ (β2 s + φ2 )e−sσ . T20 (s) = 2 s + (κ1 s + φ1 )e−sτ + (κ2 s + φ2 )e−sσ

Linearizing this about the equilibrium (4) we obtain h˙˜ 1 (t) =v˜0 (t) − v˜1 (t) , v˙˜1 (t) =φ1 h˜ 1 (t − τ ) − κ1 v˜1 (t − τ ) + β1 v˜0 (t − τ ) ,

T21 (s) = (9)

(16)

Substituting (10) into (15), we obtain the leader-to-tail transfer function

cf. (6). Taking the Laplace transform of (9) with zero initial conditions yields

G20 (s) = V˜1 (s) = T10 (s)V˜0 (s) ,

(15)

V˜2 (s) N(s) , = T21 (s)T10 (s) + T20 (s) = ˜ D(s) V0 (s)

(17)

(10) where

where the link transfer function T10 (s) =

(β1 s + φ1 )e−sτ , s2 + (κ1 s + φ1 )e−sτ

( )2 N(s) = (β1 s + φ1 )e−sτ ) ( + (β2 s + φ2 ) s2 + (κ1 s + φ1 )e−sτ e−sσ , ( ) D(s) = s2 + (κ1 s + φ1 )e−sτ + (κ2 s + φ2 )e−sσ ( ) × s2 + (κ1 s + φ1 )e−sτ .

(11)

is equal to the leader-to-tail transfer function defined by G10 (s) =

V˜1 (s) = T10 (s) . V˜0 (s)

(18)

For motif n we have (12) Vñ (s) = Tn n−1 (s)Vñ−1 (s) + Tn0 (s)V˜0 (s) ,

Motif 2 consists of vehicles No. 0, No. 1, and No. 2. Thus (5) leads to

(19)

and substituting V˜k (s) = Tk k−1 (s)V˜k−1 (s) for k = 0, . . . n − 1 results in the leader-to-tails transfer function

h˙ 1 (t) = v0 (t) − v1 (t) , ( ) ( ) v˙1 (t) = α1 V (h1 (t − τ )) − v1 (t − τ ) + β1 v0 (t − τ ) − v1 (t − τ ) , h˙ 2 (t) = v1 (t) − v2 (t) , ( ) ( ) v˙2 (t) = α1 V (h2 (t − τ )) − v2 (t − τ ) + β1 v1 (t − τ ) − v2 (t − τ ) ( ( ) ) ) 1( + α2 V h1 (t − σ ) + h2 (t − σ ) − v2 (t − σ ) 2 ( ) + β2 v0 (t − σ ) − v2 (t − σ ) . (13)

Gn0 (s) =

Vñ (s) = Tn n−1 (s) · · · T21 (s)T10 (s) + Tn0 (s) . V˜0 (s)

(20)

This represents the fact that for motif n the information about the leader’s motion can reach the last vehicle in two routes: through a direct (but delayed) communication channel as well as through a series of n − 1 vehicles (that also include delays). These two signals are summed by the tail vehicle. At the linear level this is represented by the link transfer function Tn0 along the direct communication link and the product of transfer functions Tn n−1 · · · T21 T10 along the platoon. Finally, the network shown in Fig. 3 is composed of motifs 1, 2 and 3. Indeed, the leader-to-tail transfer function is the sum of the products of the link transfer functions along the paths connecting the leading vehicle to the tail vehicle. There are three paths linking vehicle No. 0 to vehicle No. 4, resulting in

According to (6) and (7), the linearized equation becomes h˙˜ 1 (t) = v˜0 (t) − v˜1 (t) , v˜˙1 (t) = φ1 h˜ 1 (t − τ ) − κ1 v˜1 (t − τ ) + β1 v˜0 (t − τ ) , h˙˜ 2 (t) = v˜1 (t) − v˜2 (t) , v˙˜2 (t) = φ1 h˜ 2 (t − τ ) + φ2 h˜ 2 (t − σ ) + β1 v˜1 (t − τ )

G40 (s) = T43 (s)T32 (s)T21 (s)T10 (s) + T43 (s)T32 (s)T20 (s) + T41 (s)T10 (s) .

+ β2 v˜0 (t − σ ) − κ1 v˜2 (t − τ ) − κ2 v˜2 (t − σ ) + φ2 h˜ 1 (t − σ ) . (14)

4

(21)


Figure 4. TOP: STABILITY DIAGRAMS FOR MOTIF 1. PLANT AND STRING STABLE DOMAINS ARE INDICATED BY LIGHT GRAY AND DARK GRAY, RESPECTIVELY. RED CURVES SHOW THE PLANT STABILITY BOUNDARY (23), GREEN CURVES SHOW THE STRING STABILITY BOUNDARIES WITH ZERO FREQUENCY

(25), WHILE THE OTHER CURVES CORRESPOND TO STRING STABILITY BOUNDARIES WITH POSITIVE FREQUEN-

CIES (26). BOTTOM: CRITICAL FREQUENCIES RELATED TO PLANT AND STRING STABILITY LOSSES.

Figure 5. ZOOM OF THE PANEL FOR τ

The corresponding curves divide the parameter plane into plant stable and plant unstable domains. When α1 and β1 are chosen from the plant stable domain, the infinitely many characteristic roots of (22) are located in the left half of complex plane. When the control gains are selected at the plant stability boundary, there exist a pair of complex conjugate characteristic roots, indicating that a Hopf bifurcation takes place in the corresponding nonlinear system. Now, we investigate the string stability of the motif 1. Using algebraic manipulations one may show that the string stability condition |G10 (jω )| < 1, ∀ω ∈ R+ is equivalent to = 0.2 [s] IN FIG. 4 WITH DIF= 0.8 [1/s].

( ( ) f1 (ω ) = ω 2 2α1 ω sin(ωτ ) +V ′ (h∗ ) cos(ωτ )

FERENT GAIN COMBINATIONS MARKED BY A–G AT β1

) (24) +2β1 ω sin(ωτ ) − ω 2 − α1 (α1 + 2β1 ) < 0 .

STABILITY ANALYSIS OF MOTIF 1 In this section, we analyze the stability of the motif 1, followed by stability charts that allow one to choose control gains which ensure both the plant stability and the string stability. To find the plant stability condition for Motif 1, we analyze the characteristic equation, which is given by the denominator of transfer function (11): s2 + (κ1 s + φ1 )e−sτ = 0 .

As f1 (0) = 0 and f1′ (0) = 0 always hold, to ensure string stability at ω = 0, we need f1′′ (0) < 0, which leads to the boundaries

α1 = 2(V ′ (h∗ ) − β1 ) , α1 = 0 .

In this case, string instability arises so that |G10 (jω )| ≥ 1 for 0 ≤ ω ≤ ω while |G10 (jω )| < 1 for ω > ω ; see already case G in Fig. 6 showing that the system is string unstable for low frequencies but string stable for sufficiently high frequencies. At the stability boundary, we have ω = ωcr = 0. If the maximum of f1 (ω ) occurs at ω > 0, the string stability boundary is given by f1 (ω ) = 0 and f1′ (ω ) = 0. Solving these two equations results in

(22)

Multiplying both sides with esτ , substituting s = jΩ, where Ω ∈ R+ , separating the real and the imaginary parts, and substituting definitions (7), we obtain the plant stability boundary in parametric form: Ω2 cos(Ωτ ) , V ′ (h∗ ) β1 =Ω sin(Ωτ ) − α1 .

α1 =

(25)

(23)

α1 =

5

1( P1 ± P2

√

) P12 + P2 P0 ,


40

1.5

40

max |G10 (jω)| = 1.38

0 −20 −40 −10

20 1.5

Im(s)

|G10 (jω)|

A.

Im(s)

20

1

−6

−4

−2

0

ω

0 0

2

0

1

0.5

−20

0.5

−8

|G10 (jω)|

2

2

4

6

ω 8

ω = 2.31

−40 −10

10

ω [rad/s]

Re(s)

−8

−6

−4

Re(s)

−2

0

2

0 0

2

4

6

8

10

ω [rad/s]

40

Figure 7. EIGENVALUES AND AMPLIFICATION RATIOS FOR MOTIF 1 WHEN α1 = 0.6 [1/s], β1 = 1.3 [1/s], τ = 0.4 [s].

2

|G10 (jω)|

B.

Im(s)

20 0 −20 −40 −10

1.5 1

where

0.5

−8

−6

−4

−2

0

ω

0 0

2

2

4

ω 6

8

( ) P0 = ω 2 sin(ωτ ) − ωτ cos(ωτ ) , ) ( P1 = V ′ (h∗ ) ωτ + sin(ωτ ) cos(ωτ ) − ω , ( ) P2 = 2V ′ (h∗ )τ − 1 sin(ωτ ) − ωτ cos(ωτ ) .

10

ω [rad/s]

Re(s) 40 2

|G10 (jω)|

C.

Im(s)

20 0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

ω

0 0

2

2

4

In this case, string instability arises so that |G10 (jω )| ≥ 1 for ω ≤ ω ≤ ω and |G10 (jω )| < 1 for 0 < ω < ω and ω > ω ; see case C in Fig. 6, which shows that the system is string unstable in the mid-frequency range but string stable for low and high frequencies. At the stability boundary, we have ω = ω = ωcr > 0. The plant and string stability domains are shown in the first row of Fig. 4 for h∗ = 20 [m], that is, v∗ = V (h∗ ) = 15 [m/s] where V ′ (h∗ ) = maxV ′ (h) = π2 [1/s]. Although multiple string stable domains show up, only the one inside the plant stable domain is of interest. As the communication delay increases, this domain shrinks and finally disappears when the two end points (denoted by *-s) meet, indicating that no control ( gains) can ensure plant and string stability for τ > τcr = 1/ 2V ′ (h∗ ) ≈ 0.32 [s]. The second row of Fig. 4 shows the critical frequencies Ω and ωcr where *-s mark ωcr = 0. Note that in case of τ = 0, loss of string stability only occurs for 0 < ω < ω . To demonstrate the behavior in different parameter domains, we select the panel for τ = 0.2 [s], fix β1 = 0.8 [1/s], and mark the points A–G in the zoomed figure Fig. 5. Point B is selected to be at the plant stability boundary while points D and F are chosen at the string stability boundary. To calculate the characteristic roots, we apply numerical continuation, namely the package DDE-biftool [9]. The corresponding eigenvalues and amplification ratios (Bode plots) are presented in Fig. 6. Among the eigenvalue diagrams shown in the left column of Fig. 6, case A has eigenvalues with positive real parts and thus the system is plant unstable. Case B has purely imaginary eigenvalues, corresponding to the Hopf bifurcation. For the cases C–G, all eigenvalues have negative real parts, i.e., the selected gains can guarantee plant stability of the system. From the right column of Fig. 6, it can be seen that max |G10 (jω )| > 1 in cases A–C and G, indicating that the system is string unstable in some frequency range. In cases A–C, string instability occurs for ω ≤ ω ≤ ω while for case G, the system is string unstable for 0 ≤ ω ≤ ω . This corresponds to the fact that in case F string instability occurs at ωcr = 0 but in case D ωcr > 0. Finally, in case

ω 6

8

10

8

10

8

10

8

10

8

10

ω [rad/s]

Re(s) 40 2

|G10 (jω)|

Im(s)

20

D.

0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

ωcr

0 0

2

2

4

6

ω [rad/s]

Re(s) 40 2

|G10 (jω)|

Im(s)

20

E.

0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

0 0

2

2

4

6

ω [rad/s]

Re(s) 40 2

|G10 (jω)|

Im(s)

20

F.

0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

0 0

2

2

4

6

ω [rad/s]

Re(s) 40 2

|G10 (jω)|

Im(s)

20

G.

0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

2

0 0

ω 2

Re(s)

4

6

ω [rad/s]

Figure 6. EIGENVALUES AND AMPLIFICATION RATIOS FOR MOTIF 1, CORRESPONDING TO THE POINTS A–G IN FIG. 5.

β1 =

α1

(27)

(( ) ) V ′ (h∗ )τ − 1 sin(ωτ ) − ωτ cos(ωτ ) + ω sin(ωτ ) + ωτ cos(ωτ )

,

(26)

6


Figure 8. STABILITY DIAGRAMS (TOP) AND RELATED FREQUENCIES (BOTTOM) FOR MOTIF 2 FOR τ = 0.4 [s], α1 = 0.6 [1/s] AND β1 = 1.3 [1/s]. THE SAME NOTATION IS USED AS IN FIG. 4 WHILE THE PLANT AND STRING STABILITY BOUNDARIES ARE GIVEN BY (29) AND (32,33), RESPECTIVELY.

form ( ( ) ) Ω2 cos(Ωσ ) − φ1 cos Ω(σ − τ ) + κ1 Ω sin Ω(σ − τ ) , V ′ (h∗ ) ( ) ( ) φ1 sin Ω(σ − τ ) β2 =Ω sin(Ωσ ) − − κ1 cos Ω(σ − τ ) − α2 . Ω (29)

α2 =2

Figure 9.

ZOOM OF THE PANEL FOR

σ = 0.2

When analyzing the string stability of motif 2, it can be shown that the condition |G20 (jω )| < 1, ∀ω ∈ R+ is equivalent to [s] IN FIG. 8 WITH

f2 (ω ) = R1 − R2 α2 (α2 + 2β2 ) + R3 α2 + R4 β2 < 0 ,

DIFFERENT GAIN COMBINATIONS MARKED BY POINTS a–l.

where

E, max |G10 (jω )| < 1 indicates that the system is string stable for all frequencies.

R1 =φ14 + 2φ12 β12 ω 2 + β14 ω 4 −C12 , R2 =ω 2C1 , R3 =C4 sin(ωσ ) +C5 cos(ωσ ) , R4 =2ω C2 sin(ωσ ) − 2ω C3 cos(ωσ ) ,

STABILITY ANALYSIS OF MOTIF 2 In this section, we derive conditions for the plant stability and string stability of motif 2. The characteristic equation of the Motif 2 (the denominator of (17)) implies that either (22) or s2 + (κ1 s + φ1 )e−sτ + (κ2 s + φ2 )e−sσ = 0 ,

(30)

(31)

and the coefficients Ci , i = 1, . . . 5 are defined in the Appendix. To guarantee string stability at ω = 0, f2′′ (0) < 0 is required, which results in the boundaries ( ) α2 = −α1 + 2 V ′ (h∗ ) − β1 − β2 , α2 = −2α1 .

(28)

hold. This means that motif 2 loses plant stability if vehicle No. 1 is plant unstable. Let us consider that vehicle No. 1 is plant stable. Then, the plant stability boundary is given by (28). Multiplying by esσ , substituting s = jΩ, where Ω ∈ R+ , separating the real and the imaginary parts, and using some algebraic manipulation, we obtain the plant stability boundary in parametric

(32)

If the maximum of f2 (ω ) occurs at ω > 0, the string stability boundary is given by f2 (ω ) = 0 and f2′ (ω ) = 0, which yields √ −2R2 S1 + R3 + R4 S2 ± ∆2 α2 = , 2R2 (1 + 2S2 )

7


40 2

|G20 (jω)|

Im(s)

a.

0 −20 −40 −10

1.5

h.

1

|G20 (jω)|

2 20

−6

−4

−2

0

ω ω

0 0

2

1 0.5

0.5

−8

1.5

2

4

6

ω [rad/s]

Re(s)

8

0 0

10

ωcr,1 ωcr,2 2

4

6

ω [rad/s]

8

10

40

0 −20 −40 −10

i.

1

|G20 (jω)|

|G20 (jω)|

Im(s)

b.

1.5

−6

−4

−2

0

ω ω

0 0

2

j.

1 0.5

0.5

−8

2

1.5

|G20 (jω)|

2

2 20

2

4

6

ω [rad/s]

Re(s)

8

0 0

10

1.5 1 0.5

ωcr,1 2

ωcr,2 4

6

ω [rad/s]

8

0 0

10

ωcr,1 2

ω2

ω2

4

6

ω [rad/s]

8

10

8

10

40

0 −20 −40 −10

k.

1

−8

−6

−4

−2

0

ω ω

0 0

2

2

4

6

ω [rad/s]

8

10

|G20 (jω)|

Im(s)

0

−40 −10

1.5

−6

−4

−2

0

2

4

6

ω [rad/s]

8

2

|G20 (jω)|

Im(s)

0 −20

1.5 1

−4

−2

0

0 0

2

2

4

6

ω [rad/s]

Re(s)

8

10

2

|G20 (jω)|

Im(s)

20 0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

−2

0

ωcr

0 0

2

2

Re(s)

4

6

8

10

4

6

8

10

ω [rad/s]

40 2

|G20 (jω)|

Im(s)

20

g.

0 −20 −40 −10

1.5 1 0.5

−8

−6

−4

10

0 0

ω1 2

ω1 ω2 ω2 4

6

ω [rad/s]

We fix α1 = 0.6 [1/s], β1 = 1.3 [1/s] and τ = 0.4 [s] where motif 1 is plant stable but string unstable as shown in Fig. 7. Fig. 8 depicts the stability diagrams for motif 2 in the (α2 , β2 )plane for different σ values. The top row shows the plant and string stability boundaries while the corresponding frequencies Ω and ωcr are displayed below. The string stable domain (dark gray) within the plant stable domain (light gray) shrinks with the increase of communication delay σ and disappears for σ > σcr ≈ 0.4. In fact, the string stable domains are quite different compared to those in Fig. 4, that is, the gains (α1 , β1 ) used when reacting the motion of the car immediately in front are different from the gains (α2 , β2 ) needed to exploit the signals received through long-distance commination. Again, to represent the behavior in different parameter domains we choose σ = 0.2 [s], fix β2 = 0.7 [1/s], and mark the points a–g in the zoomed figure Fig. 9. The corresponding eigenvalues and the amplification ratios are shown in Fig. 10. These are similar to cases A–G in Fig. 5 except that string instability always occurs in the mid-frequency range ω ≤ ω ≤ ω ; cf. panels E–G and e–g. Moreover, one may observe in Fig. 9 that the

40

f.

8

∆2 =(−2R2 S1 + R3 + R4 S2 )2 + 4R2 (1 + 2S2 )(R1 + R4 S1 ) . (34)

0.5

−6

6

S1 =

20

−8

4

ω [rad/s]

R′1 R2 − R1 R′2 , R4 R′2 − R′4 R2 R′ R2 − R3 R′2 S2 = 3 ′ , R4 R2 − R′4 R2

10

40

−40 −10

2

ω2

where

Re(s)

e.

0.5

ω1

ωcr

0 0

2

1

1 0.5

−8

1.5

Figure 11. EIGENVALUES AND AMPLIFICATION RATIOS FOR MOTIF 2, CORRESPONDING TO POINTS h–l IN FIG. 9.

2

−20

l.

1

0 0

40 20

2

1.5

0.5

0.5

Re(s)

d.

|G20 (jω)|

|G20 (jω)|

Im(s)

c.

1.5

|G20 (jω)|

2

2 20

−2

0

2

0 0

ω ω

Re(s)

2

ω [rad/s]

Figure 10. EIGENVALUES AND AMPLIFICATION RATIOS FOR MOTIF 2, CORRESPONDING TO POINTS a–g IN FIG. 9.

β2 = S1 + S2 α2 ,

(33)

8


24

20 10 0 5

10

20 18 16 0

15

5

t [s]

10

14 13 12 0

15

21 20 19

5

10

15

0

5

t [s]

16 15 14 13

22

10

15

10

15

t [s]

21 20 19

5

t [s]

10

15

0

22

17

Velocity [m/s]

Headway [m]

Velocity [m/s]

15

18

17

12 0

16

t [s]

18

b.

22

17

5

10

16 15 14 13 12 0

15

Headway [m]

−10 0

18

22

Headway [m]

30

Velocity [m/s]

Headway [m]

a.

Velocity [m/s]

40

21 20 19

5

t [s]

10

15

0

t [s]

5

t [s]

18

Headway [m]

Velocity [m/s]

e.

16 15 14 13 12 0

Figure 13. SIMULATIONS FOR THE NONLINEAR MODEL (13) SHOWING THAT THE ADDITIONAL LINK MAY ENSURE STRING STA-

22

17

21

BILITY. STATES OF VEHICLES NO. 0, NO. 1 AND NO. 2 ARE REPRESENTED BY DASHED, RED AND BLUE CURVES RESPECTIVELY. THE

20 19

5

10

15

0

t [s]

5

10

TOP ROW IS WITHOUT LONG-DISTANCE COMMUNICATION: α2 = 0 [1/s], β2 = 0 [1/s]. THE BOTTOM ROW CORRESPONDS TO POINT e

15

t [s]

IN FIG. 9:

Figure 12. SIMULATIONS FOR THE NONLINEAR MODEL (13) TESTING PLANT STABILITY FOR POINTS a, b, e IN FIG. 9. THE STATES OF

α2 = 1 [1/s], β2 = 0.7 [1/s].

VEHICLE NO. 1 AND VEHICLE NO. 2 ARE SHOWN BY RED AND BLUE CURVES, RESPECTIVELY. DASHED LINES INDICATE THE LEADER

v∗ + vamp sin(ω t) with vamp = 1 [m/s] and ω = 2.31 [rad/s]. At this frequency, the chosen α1 , β1 combination gives amplitude ratio |G10 (jω )| = 1.38 for motif 1 as shown in Fig. 7. Without wireless communication between vehicles No. 0 and No. 2, motif 2 becomes the cascade of motif 1-s and thus the systems is string unstable, as shown in the top row of Fig. 13. However, using appropriate gains for the long-distance communication allows the attenuation of oscillations between vehicles No. 0 and No. 2, and leads to leader-to-tail string stability as shown at the bottom row of Fig. 13. Notice that fluctuations are amplified as they propagate from vehicle No. 0 to vehicle No. 1 but attenuated by vehicle No. 2. This shows that the design principles derived at the linear level may be applicable at the nonlinear level.

TRAVELING WITH CONSTANT VELOCITY.

string stability curves intersect each other at points h and i. The corresponding Bode plots are shown in Fig. 11 demonstrating that string instability is lost through two independent frequencies ωcr,1 and ωcr,2 simultaneously. Consequently, string instability can occur in two separate frequency ranges ω1 ≤ ω ≤ ω1 and ω2 ≤ ω ≤ ω2 which may merge when parameters are varied as illustrated by the cases j, k and l.

NUMERICAL SIMULATIONS To verify the stability charts obtained for motifs 1 and 2, we investigate the behavior of the nonlinear system (13) in the vicinity of the uniform flow equilibrium. As in Fig. 10, we choose the delays τ = 0.4 [s] and σ = 0.2 [s] and fix the parameters α1 = 0.6 [1/s] and β1 = 1.3 [1/s]. We consider the equilibrium v∗ ≡ 15 [m/s], h∗ ≡ 20 [m] and assume the initial velocities v1 (t) ≡ 12 [m/s], v2 (t) ≡ 16 [m/s] and initial headways h1 (t) ≡ 19 [m], h2 (t) ≡ 21 [m] along the interval t ∈ [−τ , 0]. In order to test plant stability, we select α2 and β2 corresponding to cases a, b and e in Fig. 10 and use v0 (t) ≡ v∗ . The simulation results are shown in Fig. 12. Clearly, vehicle No. 1 (red curve) is plant stable as its velocity approaches v∗ . In case a, v2 and h2 diverge, implying that vehicle No. 2 (blue curve) is plant unstable. In case b, v2 and h2 keep oscillating with the same amplitude, since the control gains are chosen at the plant stability boundary. Finally, in case e, v2 and h2 converge to the equilibrium, i.e., vehicle No. 2 is plant stable. To test string stability, we use the oscillatory input v0 (t) =

CONCLUSIONS In this paper, we proposed a motif-based approach that can be used to analyze and design connected vehicle systems. By analyzing the plant stability and the leader-to-tail string stability of fundamental motifs we determined the limitations caused by the sensing and communication delays in such networks. We demonstrated that, by choosing the control gains appropriately for the long-distance communication, one can ensure string stability of systems that cannot be stabilized otherwise. The analytical results were presented using stability charts, which were confirmed by using numerical simulations. Our future work will include the study of more complicated networks with high level of heterogeneity.

9


Appendix The coefficients in (31) are defined as

REFERENCES [1] Orosz, G., Wilson, R. E., and Stépán, G., 2010. “Traffic jams: dynamics and control”. Philosophical Transactions of the Royal Society A, 368(1928), pp. 4455–4479. [2] Vahidi, A., and Eskandarian, A., 2003. “Research advances in intelligent collision avoidance and adaptive cruise control”. IEEE Transactions on Intelligent Transportation Systems, 4(3), pp. 143–153. [3] Zhou, J., and Peng, H., 2005. “Range policy of adaptive cruise control vehicles for improved flow stability and string stability”. IEEE Transactions on Intelligent Transportation Systems, 6(2), pp. 229–237. [4] Caveney, D., 2010. “Cooperative vehicular safety applications”. IEEE Control Systems Magazine, 30(4), pp. 38–53. [5] Qin, W. B., 2013. “Digital effects and delays in connected vehicles: linear stability and simulations”. In Proceedings of the ASME Dynamical Systems and Control Conference, ASME. [6] Ge, J. I., Avedisov, S. S., and Orosz, G., 2013. “Stability of connected vehicle platoons with delays acceleration feedback”. In Proceedings of the ASME Dynamical Systems and Control Conference, ASME. [7] Alon, U., 2006. An Introduction to Systems Biology: Design Principles of Biological Circuits. CRC Press, Taylor & Francis Group. [8] Swaroop, D., and Hedrick, J. K., 1996. “String stability of interconnected systems”. IEEE Transactions on Automatic Control, 41(3), pp. 349–357. [9] Roose, D., and Szalai, R., 2007. “Continuation and bifurcation analysis of delay differential equations”. In Numerical Continuation Methods for Dynamical Systems, B. Krauskopf, H. M. Osinga, and J. Galan-Vioque, eds., Understanding Complex Systems, Springer, pp. 359–399.

C1 = −2ω 3 κ1 sin(ωτ ) − 2ω 2 φ1 cos(ωτ ) + ω 4 + κ12 ω 2 + φ12 , C2 = 2φ1 ω 3 α1 sin(2ωτ ) − ω 4 (κ12 − β12 ) cos(2ωτ ) ( ) + ω − 3κ1 ω 4 + (β12 − κ12 )κ1 ω 2 − 2φ12 α1 sin(ωτ ) + ω 2 φ1 (−3ω 2 + 2κ1 β1 − β12 − κ12 ) cos(ωτ ) + ω 6 + 2κ12 ω 4 + 2ω 2 φ12 , C3 = −ω 4 (κ12 − β12 ) sin(2ωτ ) − 2ω 3 φ1 α1 cos(2ωτ ) ( ) + ω 2 φ1 − ω 2 − β12 + 2κ1 β1 − κ12 sin(ωτ ) ( ) + ω κ1 ω 4 + (κ12 − β12 )κ1 ω 2 + 2α1 φ12 cos(ωτ ) , C4 = 4φ1 ω 4 κ1 sin(2ωτ ) − 2ω 3 (κ12 ω 2 − φ12 ) cos(2ωτ ) ( ) − 2κ1 ω 2 3ω 4 + κ12 ω 2 + φ12 sin(ωτ ) ( ) − 2φ1 ω 3ω 4 + κ12 ω 2 + φ12 cos(ωτ ) + 2ω 7 + 4κ12 ω 5 + 4ω 3 φ12 +V ′ (h∗ )C3 , ( ) C5 = −2ω 3 φ12 − κ12 ω 2 sin(2ωτ ) + 4κ1 ω 4 φ1 cos(2ωτ ) ( ) + 2ωφ1 ω 4 + κ12 ω 2 + φ12 sin(ωτ ) ( ) − 2κ1 ω 2 ω 4 + κ12 ω 2 + φ12 cos(ωτ ) +V ′ (h∗ )C2 . (35)

10
