Nonlinear Dynhttps://doi.org/10.1007/s11071-019-04889-8

ORIGINAL PAPER

On the global dynamics of connected vehicle systems

Adam K. Kiss · Sergei S. Avedisov ·Daniel Bachrathy · Gábor Orosz

Received: 23 November 2018 / Accepted: 13 March 2019© The Author(s) 2019

Abstract In this contribution, heterogeneous con-nected vehicle systems, that include human-driven aswell as connected automated vehicles, are investigated.The reaction time delay of human drivers as well asthe communication and actuation delays of connectedautomated vehicles is incorporated in the model. Satu-rations due to the speed limit and limited accelerationcapabilities of the vehicles are also taken into account.The arising nonlinear delayed system is studied usinganalytical and numerical bifurcation analysis. Stabilityanalysis is used to identify regions in parameter spacewhere oscillations arise due to loss of linear stability ofthe equilibrium. Moreover, with the help of numericalcontinuation, bistability between the equilibrium andoscillations is shown to exist due to the presence of anisola. It is demonstrated that utilizing long-range wire-less vehicle-to-vehicle communication the connectedautomated vehicle is able to eliminate the oscillationsand keep the traffic flow smooth.

Keywords Time delays · Bistability · Vehicle-to-vehicle (V2V) communication · Connectedautomated vehicle ·Beyond-line-of-sight information ·Acceleration saturation

A. K. Kiss (B) · D. BachrathyDepartment of Applied Mechanics, Budapest University ofTechnology and Economics, Budapest 1111, Hungarye-mail: kiss_a@mm.bme.hu

S. S. Avedisov · G. OroszDepartment ofMechanical Engineering, University ofMichigan,Ann Arbor, MI 48109, USA

1 Introduction

Road transportation is expected to go through a hugetransformation in the upcoming decades with the intro-duction of connected and automated vehicles. Whilesuch technologies mainly aim at improving passengercomfort and safety, they may also be used to improvethe overall traffic flow. For example, by controllingthe motion of automated vehicles may lead to highertraffic throughput and improved fuel economy [1–7].Most prior research on the topic assumed 100% pene-tration rate of automation as well as vehicle-to-vehicle(V2V) connectivity. While the latter one is expected torise rapidly in the next few years, the former one willlikely follow in a slower pace due to the high cost ofthese vehicles. Thus, there is a need to understand thedynamics ofmixed scenarioswhere automated vehiclesare mixed into the flow of human-driven traffic. Initialresearch on mixed scenarios typically considered lin-ear dynamics [8–10], while recent experimental resultsshow that nonlinearities may play an important role inthese systems [11–13].

In this paper, we start with modeling the longitudi-nal dynamics of human-driven vehicles. Then, a con-nected cruise control (CCC) algorithm is proposed toregulate the longitudinal dynamics of connected auto-mated vehicles (CAVs) which utilize motion infor-mation from multiple human-driven vehicles ahead(within and beyond the line of sight). In particular, ourgoal is to design CCC controllers for CAVs so thatthey can mitigate traffic waves at the linear and nonlin-

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Fig. 1 Schematic representation of vehicles on a single lane. Cascade of CCCs consisting of human-driven and connected automatedvehicles. Arrows show the direction of information propagation via V2V communication

ear levels. We adopt the strategy of “do not look aheadof another CAV” leading to the formation of a cas-cade of CCCs, as depicted in Fig. 1. This allows for amodular and scalable design of connected vehicle sys-tems [14].Whenmodeling the vehicles, we incorporatetime delays to represent the human reaction time andthe communication and actuation delays of connectedautomated vehicles. Moreover, we take into accountthe speed limits and limited acceleration capabilities ofvehicles, which introduce nonlinearities into the mod-eling equations.

In order to select the control gains of the connectedautomated vehicles, linear and nonlinear analyses arecarried out while the vehicles placed on a circular road.First, linear stability analysis is presented to understandthe effects of the connected automated vehicles on thestability of the uniform flow.With the help of the insta-bility gradient method [15], we construct robust stabil-ity diagrams in the plane of the control parameters. Wedemonstrate that using beyond-line-of-sight informa-tion (obtained via V2V communication) can improvelinear stability. Then, nonlinear analysis is carried outto have an insight into the influence of limited accel-eration capabilities of vehicles. We use numerical con-tinuation techniques [16,17], to draw the skeleton ofthe traffic dynamics as a three-dimensional wire framerepresentation. For the first time, we demonstrate thatacceleration saturation leads to new dynamic behavior,where bistable regions appear through the presence ofisolas. In this scenario, no linear stability loss occurs,but larger perturbations may still trigger traffic waves.Then, the effects of connected automated vehicles areinvestigated at the nonlinear level.We demonstrate thatwhen using information only from the vehicle immedi-ately ahead, an automated vehicle cannot eliminate theisolas. However, including additional information frombeyond-line-of-sight, a connected automated vehiclecan make the uniform flow globally stable.

The paper is organized as follows: in Sect. 2, weprovide the governing equations of heterogeneous con-nected vehicle systems (including human-driven andautomated vehicles) and explain the sources of the non-linearities. Section 3 gives the linear stability analysisfor the proposed model. A case study is conducted inSect. 4 with a connected automated and two human-driven vehicles with detailed linear stability analysis.Then, in Sect. 5, we analyze the nonlinear effects ofthe acceleration saturation using numerical continua-tion. Finally, in Sect. 6, we conclude our results anddiscuss future directions.

2 Governing equations

In this section, we model the car-following behaviorand the corresponding governing equations of hetero-geneous connected vehicle systems. We consider thathuman drivers respond to the motion of the vehicleimmediately ahead, while the connected automatedvehicle’s connected cruise control algorithm can uti-lize motion information from up to k cars ahead. Weplace N vehicles on a ring of length L such that wehave M automated and N −M human-driven vehicles.The distribution of the human-driven vehicles and theconnected automated vehicles are described with thehelp of the indices ih ∈ H and ia ∈ A, such thatA + H = {1, . . . , N } and | A | + | H |= N .

Following the models in [18], the governing equa-tions can be given as:

vi = f(α

(V (hσ

i ) − vσi

) +ki∑j=1

β j

(vσi+ j − vσ

i

) )

if i = ia, (1)

vi = f(αh

(V (hτ

i ) − vτi

) + βh(vτi+1 − vτ

i

) )

if i = ih,

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On the global dynamics of connected vehicle systems

hi = vi+1 − vi ,

where the dot stands for differentiation with respect totime t and we use the abbreviated notation vϑ

i = vi (t−ϑ) to indicate delays. Also, vi denotes the velocity ofvehicle i and hi denotes the distance headway, that is,the bumper-to-bumper distance between the vehicle iand its predecessor. This can be calculated as

hi = xi+1 − xi − li , (2)

where xi denotes the position of the rear bumper ofvehicle i and li denotes the length of vehicle i such thatL = ∑N

i=1(hi +li ) is the length of the ring. It should benoted that for i = N , we have hN = L − ∑N−1

i=1 (hi +li ) − lN and vN+ j = v j due to periodicity imposed bythe ring. In case of a human-driven vehicle (i = ih),the sum of the reaction time delay and the actuationdelay is denoted by τ , while for a connected automatedvehicle (i = ia), the sum of the communication andactuation delay is denoted by σ .

The parameters α and β j are the control parametersof the connected automated vehicle, and αh and βh arethe control parameters of the human-driven vehicles.For the sake of simplicity, all human-driven vehicles areconsidered to be identical and so the automated ones.The β parameters are directly related to the velocitydifferences, while the α parameters are related to theoptimal velocity function V (h) (also called range pol-icy), that satisfies the following properties:

1. V (h) is continuous, smooth and monotonicallyincreasing (the more sparse the traffic is, the fasterthe vehicles want to travel).

2. V (h) ≡ 0 for h ≤ hst (in dense traffic vehiclesintend to stop).

3. V (h) ≡ vmax for h ≥ hgo (in sparse traffic vehi-cles intend to travel with the maximum speed, alsocalled free-flow speed)

These properties can be summarized as:

V (h) =⎧⎨⎩0 if h ≤ hst,F(h) if hst < h < hgo,vmax if h ≥ hgo,

(3)

where F(h) is strictly monotonically increasing andsatisfies F(hst) = 0 and F(hgo) = vmax. In this paper,

(a) (b)

Fig. 2 Source of the nonlinearities in the model: a saturationsdue to the speed limit in the range policy function (4); b accelera-tion saturation function (6). The parameters are amin = − 6m/s2,amax = 3m/s2, c = 0.05m/s2

Fig. 3 A simple CCC placed on a ring where the first car isautomated and the second and third cars are human driven (N =3, M = 1)

we present the results using the range policy function

F(h) = vmax

2

(1 − cos

(π

h − hsthgo − hst

)), (4)

that is smooth at hst and hgo; see Fig. 2a. For simplicity,we use the same V (h) function for human-driven andautomated vehicles.

Saturations due to limited acceleration capabilitiesof vehicles are also taken into account by the function

f (a) =⎧⎨⎩amin if a < amin,

a if amin ≤ a ≤ amax,

amax if a > amax,

(5)

where amin and amax are the lower and upper saturationbounds, respectively.

However, the nonsmoothness at amin and amax maylead to computational challenges. In order to avoidthese, we smooth the saturation function by introduc-ing a smoothness range, where the different sectionsdefined in Eq. (5) are connected to each others while

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Table 1 Parameters in thetest case Total veh. numb. N 3 max. speed vmax 30m/s

CCC veh. numb. M 1 Stop dist. hst 5m

Human delay τ 1 s Driving dist. hgo 55m

CCC delay σ 0.5 s Lower satur. limit amin − 6m/s2

Human gain αh 0.2 1/s Upper satur. limit amax 3m/s2

Human gain βh 0.4 1/s Smoothness range c 0.05m/s2

maintaining Cn continuity. In particular, tomaintain C1

continuity, we use second-order polynomials, yielding

f (a)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

amin if a ≤ amin − c,

a + (amin−a+c)2

4c if amin − c < a < amin + c,a if amin + c ≤ a ≤ amax − c,

a − (amax−a−c)2

4c if amax − c < a < amax + c,amax if a ≥ amax + c.

(6)

Additional parameters can be found in Table 1.

3 Linear stability analysis

In this section, we present the mathematical back-ground of the linear stability analysis of the connectedvehicle system. Setting the left-hand side of Eq. (1) tobe zero, we obtain the unique equilibrium:

vi (t) ≡ v∗ = V (h∗),hi (t) ≡ h∗ = L/N ,

(7)

where vehicles travel with constant speed while keep-ing constant distance. Using the vector

x(t) = [v1(t) · · · vN (t) h1(t) · · · hN (t)

]T, (8)

the equilibrium can be given as

x∗ = [v∗ · · · v∗ h∗ · · · h∗]T . (9)

Substituting the perturbation y(t) = x(t) − x∗ intoEq. (1), using a Taylor series expansion and droppingthe higher-order terms,weobtain the variational system

y(t) = Ly(t) + Py(t − τ) + Ry(t − σ). (10)

The 2N × 2N coefficient matrix in front of the non-delayed term is given by

L =[

0 0S − I 0

], (11)

where I stands for the N × N identity matrix and S isan N × N circulant matrix, defined as

S =

⎡⎢⎢⎢⎣

0 1 0 . . . 0...

. . .. . . 0

0 11 0 . . . 0

⎤⎥⎥⎥⎦ . (12)

The 2N × 2N coefficient matrices P and R in frontof the delayed terms correspond to the human-drivenvehicles and the automated vehicles, respectively. Inparticular, we have

P =[−H

((αh + βh)I + βhS

)αhκH

0 0

], (13)

where κ = V ′(h∗) is the gradient of the range policyfunction at the equilibrium. Also note that at x∗, the sat-uration function gives f ′(0) = 1. The N ×N indicatormatrix H contains zero elements except those diagonalterms which correspond to human-driven vehicles, thatis,

Hi,i ={1 if i = ih,0 if i = ia.

(14)

Moreover, we have

R =[−(I − H)

((α + ∑k

j=1 β j )I + ∑kj=1 β jS j ) ακ(I − H)

0 0

],

(15)

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On the global dynamics of connected vehicle systems

where (I − H) corresponds to the automated vehicles.In order to determine the stability of the equilibrium,

we utilize the D-subdivision method [19–21]. Substi-tuting the trial function y(t) = c eλt into Eq. (10), thecorresponding characteristic equation becomes

D(λ) = det(λI − L − P e−λτ − R e−λσ

) = 0. (16)

The real and the imaginary parts of the characteristicroots are computed using the multi-dimensional bisec-tion method (MDBM), which can find all the rootswithin a selected domain in the complex plane whenusing a sufficiently dense initial mesh [22].

The system is stable if all characteristic roots havenegative real parts, i.e., Reλi < 0 for all i = 1, 2, . . ..By substituting λ = iωcr, ωcr ≥ 0 into Eq. (16), thestability boundaries can be determined [19–21].

In case of large number of vehicles, solving Eq. (16)can be time consuming and sensitive for roundingerrors. However, in case of special patterns of distri-bution of the automated vehicles, where every (K =N/M)th vehicle is automated, Eq. (16) takes the ana-lytical form

D =(a0b

K−10

)M

−((−1)K−1a2b0b

K−21 + (−1)K a1b

K−11

)M.(17)

Here, M is the number of connected automated cars,each monitoring the motion of K − 1 human-drivenvehicles ahead, and the coefficients read as

a0 = λ(λ + (α + β1 + β2)e

−λσ) + ακe−λσ ,

b0 = λ(λ + (αh + βh)e

−λτ) + αhκe

−λτ ,

a1 = −(λβ1 + ακ

)e−λσ ,

b1 = −(λβh + αhκ

)e−λτ ,

a2 = −λβ2e−λσ .

(18)

4 Stability analysis of a simple connected vehiclesystem

In this section, the stability analysis is applied to thesimplest case of CCCs with 3 cars on a ring, where thefirst vehicle is automated, while the second and thirdcars are human driven (N = 3, M = 1); see Fig. 3.

We analyze the effects of the average headway h∗and the design parameters of the autonomous vehicle

α, β1 and β2, while the other parameters, such as theparameters of the human drivers and the time delays,are kept fixed. The applied parameters together withreaction time delays are presented in Table 1, see therelated parameter identification techniques in [10,13].

The corresponding coefficient matrices in the lin-earized system (10) read as follows:

L =

⎡⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

−1 1 0 0 0 00 −1 1 0 0 01 0 −1 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

, H =⎡⎣0 0 00 1 00 0 1

⎤⎦ . (19)

cf. (11) and (14), while the coefficient matrices of thedelayed terms are

P =

⎡⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 00 −(αh + βh) βh 0 αhκ 0βh 0 −(αh + βh) 0 0 αhκ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

,

(20)

and

R =

⎡⎢⎢⎢⎢⎢⎢⎣

−(α + β1 + β2) β1 β2 ακ 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

, (21)

cf. (13) and (15).As a first step, we analyze the influence of an auto-

mated car that does not utilize beyond-line-of-sightinformation (β2 = 0) on the traffic dynamics by draw-ing the stability chart in the plane of h∗ and α for differ-ent β1 parameters. In the stability charts in Fig. 4, thegray areas identify regions in parameter space whereoscillations arise due to loss of linear stability of theequilibrium. The coloring of the boundaries shows thefrequency of the Hopf bifurcation in the correspond-ing nonlinear system. The darkest blue corresponds toωcr = 0, that is related to a fold bifurcation. All thecharts show stable range only between h∗ ∈ [5, 55]m. Outside of this domain, the saturation in the range

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Fig. 4 Linear stability diagrams for different β1 values with-out beyond-line-of-sight information (β2 = 0). Gray areas cor-respond to the linearly unstable parameter combinations, whilewhite areas are linearly stable. The color of the stability boundary

corresponds to the frequency of arising oscillations when stabil-ity is lost. Note that the parameters α, β1 and β2 correspond tothe automated vehicle, while parameters of the human driversare given in Table 1. (Color figure online)

Fig. 5 The effect of the β1 parameter for h∗ = 30m. The whiteareas are robustly stable for any average headway h∗. In the grayarea, a stable or an unstable equilibrium may exist depending onthe average headway h∗

policy function eliminates the headway control so thatequilibrium in (7) does not exist. Furthermore, negativeα values always lead to unstable motion due to positivefeedback.

For the leftmost panel in Fig. 4 (at β1 = 0), largeunstable range in the shape of a “droplet” can be found.This is centered around h∗ = 30 m, which correspondsto the largest derivative of the range policy function(κ = 0.943 1/s). Comparing the different panels inFig. 4, the unstable droplet is shrinking when increas-ing parameter β1. However, the droplet is also shift-ing downward and merging with the unstable regionα < 0. For β1 > 1.3 1/s, the droplet is completely van-ished and a large stable area forms.However, such largecontrol parameter may require large acceleration thatmay not be possible under limited acceleration capabil-

ities. Notice that the upper unstable area is also shiftingdownward as β1 increases, but only causes instabilitiesfor unrealistically large α values.

We prefer those robust control parameters for whichthe system is stable for any available headway h∗ ∈[5, 55] m. The corresponding stability boundaries canbe found by following the edges of the droplets wherethe tangent is horizontal (at h∗ = 30m in our case) [23].That is, if we change parameter h∗, the real parts ofthe critical characteristic roots remain zero (Reλ = 0).This condition can be described as follows:

Re

(∂λ

∂h∗

)= 0, (22)

while setting the critical value λ = iωcr. This can berewritten as

Re

(−

∂D(λ)∂h∗

∂D(λ)∂λ

)= 0, (23)

using implicit differentiation of the characteristic equa-tion Eq. (16). Thus, the robust parameter boundariescan be calculated by solving the real and imaginaryparts of Eqs. (16) and (23) together while settingλ = iωcr.

In the 4-dimensional parameter space (h∗, β1, α,ωcr), Eqs. (16) and (23) give us a 1-dimensional object(curve). Still multi-dimensional bisection method canbe used to find the corresponding solutions. Figure 5shows the corresponding robust stability boundariesagainst the variation in the average headway h∗ in theplane of β1 and α control parameters, while color indi-cates the critical frequency ωcr. Two robustly stable

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On the global dynamics of connected vehicle systems

Fig. 6 Linear stability charts for different values of β1 and β2. The same notation is used as in Fig. 4

Fig. 7 Robust stability diagram in the plane of parameters β1and β2. The same notation is used as in Fig. 5 except that colorrepresents the critical values of α

domains can be observed. However, the lower one isnot practical as such small α values may not be safe.

Fig. 8 Bifurcation diagram showing peak-to-peak amplitude ofvelocity oscillations for the automated vehicle as a function of theaverage headway. Continuous and dashed curves correspond tothe model with and without acceleration saturation, respectively.The control parameters are β1 = 0.3 1/s, β2 = 0.15 1/s andα = 0.6 1/s. (Color figure online)

When the connected automated vehicle can uti-lize beyond-line-of-sight information (with nonzero

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(a) (b)

(c)

(d)

Fig. 9 Periodic orbits with acceleration saturation (continuouscurves) and without saturation (dashed curves) for parametersβ1 = 0.3 1/s, β2 = 0.15 1/s, α = 0.6 1/s and h∗ = 30m.Panels a, b phase-plane representation; panels c, d time profilesof velocity and acceleration. Here, we have Tper = 6.965s

parameter β2), the stability charts are shown in Fig. 6.It can be seen that the increase of the β2 parameterimproves stability. The unstable droplets are shrinkingbut not moving downward. The best option to improvelinear stability is to increase both β1 and β2 together.Even for relatively small parameters, the droplet van-ishes leading to robustness against changes in α andh∗. Robust stability can be obtained by solving Eq. (16)together with

Re

(−

∂D(λ)∂h∗

∂D(λ)∂λ

)= 0, Re

(−

∂D(λ)∂α

∂D(λ)∂λ

)= 0, (24)

while considering λ = iωcr. Note that Eqs. (16) and(24) consist of four equations, and in the 5-dimensionalparameter space (h, α, β1, β2, ωcr) this still results in acurve.

Figure 7 shows the stable parameter domain in the(β1, β2)-plane, where the system is linearly stable forall relevant h∗ andα parameters values. In the gray area,the system may be stable or unstable for different h∗and α parameter combinations. At the robust stabilityboundary, the unstable droplet completely disappears.

It should be noted that in this test case, the unstabledroplet disappears always at fix h∗ = 30 m but for dif-ferent α values, and the stability boundary is coloredaccording to these α values. Note that the upper unsta-ble area still exists, but only leads to instabilities forunrealistically large α values.

5 Nonlinear analysis

Using the above-introduced linear stability charts, it ispossible to select the control parameters for the auto-mated vehicle. However, these charts are valid onlywhen small perturbations are introduced around theequilibrium traffic flow. In order to ensure stability forlarge perturbations, nonlinearities have to be taken intoaccount when analyzing the global stability properties.To map out the global dynamics, we use numericalcontinuation, in particular, the package DDE-Biftool[16,17]. This allows us to follow branches of equilib-ria and periodic solutionswhile varying parameters andcan provide information about the stability and bifur-cations of these solutions. In our system, the source ofthe nonlinearities is the saturations due to range policyand limited acceleration capabilities (see Fig. 2).

First, the criticality of the Hopf bifurcation is ana-lyzed. In case of supercritical/subcritical bifurcation, abranch of stable/unstable periodic orbits emanates fromthe Hopf points. Figure 8 presents the bifurcation dia-gram along a horizontal cross section at α = 0.6 1/s inthe panel of Fig. 6 for β1 = 0.3 1/s, β2 = 0.15 1/s (seedotted line in Fig. 6).

The peak-to-peak amplitude of the velocity oscil-lations of the automated vehicle v

amp1 is plotted as a

function of the bifurcation parameter h∗. Note that forthe equilibrium (7), this amplitude is zero correspond-ing to the horizontal axis. In this case, the equilibriumis unstable for h∗ ∈ [24.44, 35.56] m and the Hopfbifurcations, denoted by H1 and H2, are supercritical.Continuous and dashed green curves correspond to themodelwith andwithout acceleration saturation, respec-tively. It can be seen that taking into account the satura-tion of the acceleration creates no difference close to thebifurcation points where the amplitude is small. How-ever, after reaching the acceleration limit, the amplitudeof the oscillation is smaller for the case with accelera-tion saturation.

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On the global dynamics of connected vehicle systems

(a) (b)

(c)(d)(e)

Fig. 10 Three-dimensional wire frame representation of theperiodic orbits where the amplitude v

amp1 is plotted as a func-

tion of h∗ and α for parameters β1 = 0.3 1/s and β2 = 01/s. a three-dimensional isometric view. b–d different projec-tions into 2-dimensional parameter planes. Green and red curves

present stable and unstable periodic orbits, respectively, whileblack curves visualize their fold bifurcation points. e the linearstability chart is extended with bistable parameter domain (darkgray area), where different types of solutions coexist. (Color fig-ure online)

For h∗ = 30 m, the oscillations of the automatedvehicle are depicted in Fig. 9 as phase-plane plots andas timeprofiles.Again the caseswith andwithout accel-eration saturation are distinguished as solid and dashedcurves. Note that the amplitude of the oscillations maybe different for different cars. Which vehicle reachesthe acceleration saturation limit depends on selectedparameters, but the amplitude of oscillations decreasesfor all vehicles due to the saturation. Furthermore, thereexist scenarios, where the acceleration is saturated atboth upper and lower boundaries.

In order to get a clear view of the global dynam-ics, the bifurcation diagram is extended along the con-trol parameter α resulting in a three-dimensional wireframe representation of the periodic orbits shown inFig. 10, where the peak-to-peak amplitude v

amp1 is plot-

ted as a function of parameters h∗ and α. The three-

dimensional isometric view (Fig. 10a) and its differ-ent projections (front, top, right) into parameter planes(Fig. 10b–d) are presented to help the visualization ofthe structure.

We generate these figures as follows. First, we per-form similar continuation of the periodic orbits as inFig. 8 for numerous α values (corresponding to hori-zontal lines in Fig. 10c, d). Second,wefix the parameterh∗ andmake similar continuation alongα (vertical linesin Fig. 10b, c). Finally, in order to explore the full set ofperiodic branches, we continue from existing periodicorbits while varying both bifurcation parameters. Notethat these steps are necessary to discover global dynam-ics leading to the bended tube-like shape in Fig. 10a,where each cross section along direction h∗ shows anisola for larger α values. Stable and unstable periodicorbits are colored with green and red curves, respec-

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(a) (b)

(c)(d)(e)

Fig. 11 Three-dimensional bifurcation diagram of the periodicorbits in case of separated isola, where the amplitude v

amp1 is

plotted as a function of h∗ and α for β1 = 0.3 1/s and β2 = 0.151/s. a Three-dimensional isometric view. b–d Different projec-tions into 2-dimensional parameter planes. Green and red curves

present stable and unstable periodic orbits, respectively, whileblack curves visualize their fold bifurcation points. e The linearstability chart extended with bistable parameter domain (darkgray area), where different types of solutions coexist. (Color fig-ure online)

(a) (b)

Fig. 12 Phase portrait representation of periodic orbits at abistable scenario for parameter point A in Fig. 10, where anunstable equilibrium (red cross) and an unstable periodic orbit(red curve) coexist with two stable periodic orbits (green curves).The parameters are h∗ = 32m, α = 1.5 1/s, β1 = 0.3 1/s andβ2 = 0 1/s. (Color figure online)

tively. Fold bifurcation points, where branches of sta-ble and unstable oscillations meet, are marked by blackcurves. These are generated by 2-parameter continua-

(a) (b)

Fig. 13 Phase portrait representation of periodic orbits at abistable scenario for parameter point B in Fig. 11, where a sta-ble equilibrium (green cross) and a stable periodic orbit (greencurve) are “separated” by an unstable periodic orbit (red curve).The parameters are h∗ = 32 m, α = 1.5 1/s, β1 = 0.3 1/s andβ2 = 0.15 1/s. (Color figure online)

tion in DDE-Biftool, and they bound the bistable areathat is highlighted as a dark gray domain in Fig. 10e,as an extension of the linear stability diagram.

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On the global dynamics of connected vehicle systems

Fig. 14 Stability charts with linearly unstable (light gray) and bistable (dark gray) domains for different values of β1 and β2 as indicated

In Fig. 11a, similar 3D-bifurcation diagram is visu-alized, but for a positive β2 value (when the automatedvehicle can have access for information beyond-line-of-sight). In this case, the bended tube-like shape isseparated from the periodic orbits arising from equi-libria. In this way, a set of isolas appear that are notconnected to periodic branches arising from equilibria(see Fig. 11a, b and d).

The appearance of isola results in a bistable solu-tion in two different scenarios: (i) two stable periodicorbits coexist or (ii) a stable equilibrium solution anda periodic orbit coexist. Scenario (i) can be found onlyabove the linearly unstable equilibrium solution, whichis already a practically unfavorable regime. In order toillustrate this case, point A is selected in Fig. 10c and eand oscillations are plotted for stable (green curves) andunstable (red curve) periodic orbits in the phase planein Fig. 12 where the unstable equilibrium is denoted

by red cross. In scenario (ii), the system may approachthe equilibrium or the a periodic orbit depending on theinitial conditions. This case is illustrated for parameterpoint B, noted in Fig. 11c and e. The correspondingphase portraits are shown in Fig. 13a and b where thestable equilibrium is denoted by green cross.

The role of the unstable oscillation is that it “sepa-rates” the basins of attractions of the two stable states(in the infinite dimensional state space). In other words,it determines the threshold for the strength of pertur-bations, which is necessary to “transfer” the systemsfrom uniform flow to stop-and-go motion. Over-the-threshold perturbations may trigger stop-and-go trafficjams even when the uniform flow is linearly stable.Again, the bistable region is denoted by a dark grayarea in Fig. 11e.

The final goal of parameter analysis is to ensureglobally stable behavior of the traffic flow, that is, to

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extend the linear stability charts in Fig. 6 with thebistable regions as in Figs. 10e and 11e. This way, wecan represent the influence of all the parameters andthe global dynamics. This is summarized in Fig. 14,where only parameters within the white globally sta-ble domains are favorable. Similar to linearly unstabledroplet, the bistable domain shrinks when increasingβ1 and β2. However, notice that the β2 parameter has astronger effect on the reduction in the unsafe bistabledomain.

It should be noted that without utilizing informationfrom beyond-line-of-sight (β2 = 0), neither the dropletnor the bistability can be eliminated within reasonablecontrol parameters. For this case, we need very largeβ1 to push the bistable domain up to the upper linearlyunstable area. On the other hand, utilizing beyond-line-of-sight information (β2 > 0), moderate parameterscan be selected. During the design of the control, withthe knowledge of the parameter uncertainties, we canselect robust control parameters from the globally sta-ble parameter domain (e.g., α = 0.5 1/s, β1 = 0.3 1/s,β2 = 0.3 1/s). In other words, these results allow usto identify those parameters where heterogeneous con-nected vehicle system is efficient and still robustly sta-ble. These parameters were used in experimental vali-dation of connected automated vehicle among human-driven vehicles in [12,13].

6 Conclusion

In this paper, heterogeneous connected vehicle sys-tems were investigated, which include human-drivenas well as connected automated vehicles. The reactiontime delay of human drivers as well as the commu-nication and actuation delays of connected automatedvehicles was taken into consideration. Saturations dueto the speed limit and limited acceleration capabilitiesof the vehicles were also incorporated. The arising non-linear delayed system was studied using analytical andnumerical bifurcation analyses. Linear stability anal-ysis was used to identify regions in parameter space,where oscillations arose due to loss of stability of theequilibrium. Robust stability charts were drawn thatcan be used to ensure linear stability independent ofthe average headway (that changes as traffic becomesmore dense) and the gain used for distance control(which is often chosen according to safety considera-tions). Moreover, with the help of numerical continua-

tion, we determined the bistable regions, where smoothtraffic flow coexists with congested states and suffi-ciently large perturbations may trigger traffic waves.We demonstrated that utilizing beyond-line-of-sightinformation (that is obtained viaV2V communication),a connected automated vehicle is able to completelyeliminate traffic waves at both the linear and nonlinearlevels.

Acknowledgements Open access funding provided byBudapest University of Technology and Economics (BME). Thiswork was partially funded by the Hungarian EMMI ÚNKP-18-3-I New National Excellence Program of the Ministry ofHuman Capacities and the National Science Foundation (AwardNo. 1351456). The research reported in this paper was sup-ported by the Higher Education Excellence Program of the Min-istry of Human Capacities in the frame of Artificial intelligenceresearch area of Budapest University of Technology and Eco-nomics (BME FIKP-MI).

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestr-icted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

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