Optimal control of connected vehicle systems

Transcription

1 53rd IEEE Conference on Decision and Conrol December 15-17, 214. Los Angeles, California, USA Opimal conrol of conneced vehicle sysems Jin I. Ge and Gábor Orosz Absrac In his paper, linear quadraic racking LQ) is used o opimize he conrol gains for conneced cruise conrol CCC). We consider a vehicle sring where he CCC vehicle a he ail receives posiion and velociy signals hrough wireless vehicle-o-vehicle V2V) communicaion from oher vehicles ahead ha are no equipped wih CCC). An opimal feedback law is obained by minimizing a cos funcion defined by headway and velociy errors and he acceleraion of he CCC vehicle on an infinie horizon. We show ha he feedback gains can be obained recursively as signals from vehicles farher ahead become available, and ha he gains decay exponenially wih he number of cars beween he source of he signal and he CCC vehicle. he effecs of he cos funcion on he head-oail sring sabiliy are invesigaed and he robusness agains variaions in human parameers is esed. he analyical resuls are verified by numerical simulaions. I. INRODUCION Conneced cruise conrol CCC) has been proposed o mainain smooh raffic flow in heerogeneous conneced vehicle sysems by exploiing vehicle-o-vehicle V2V) communicaion [1]. he CCC conroller receives informaion abou he moion of muliple vehicles ahead, and acuaes he vehicle or assiss he driver based on hese signals. he influence of conneciviy srucures, signal ypes, packe drops, and communicaion delays on he longiudinal moion of vehicular chains ha include CCC vehicles has been invesigaed [2] [5]. Our goal here is o opimize he feedback gains in order o maximize he benefi of conneciviy and reduce he complexiy of uning gains individually in large sysems; see [6], [7] for iniial aemps using simple configuraions. Moreover, he design parameers should be chosen so ha addiional performance requiremenssuch as sring sabiliy) are saisfied. In his paper we opimize he gains of a CCC vehicle ha receives posiion and velociy informaion from muliple human-driven vehicles ahead. he goal of opimizaion is o obain a CCC conroller ha ensures he sabiliy of uniform raffic flow i.e. he aenuaion of perurbaions along he vehicular chain), while minimizing velociy and headway error and acceleraion of he CCC vehicle. his problem is solved by using linear quadraic racking LQ) wih design parameers being he weighs on he error erms and he acceleraion erm in he cos funcion. We show ha he gains of he opimized conroller follow he spaial causaliy of raffic sysems: informaion from vehicles farher downsream have less influence on he CCC vehicle and does *his work was suppored by he Naional Science Foundaion Award Number ) Jin I. Ge and Gábor Orosz are wih he Deparmen of Mechanical Engineering, Universiy of Michigan, Ann Arbor, Michigan 4819, USA. Corresponding gejin@umich.edu, orosz@umich.edu no change he feedback laws on signals from closer vehicles. he opimal gains are deermined by he weighs used in he opimizaion design parameers) and he driver parameers of oher vehicles. he range of design parameers ensuring head-o-ail sring sabiliy, and heir robusness agains variaions of driver parameers are also demonsraed. Finally, simulaions are performed o demonsrae he effeciveness of he opimal design. II. CONNECED CAR-FOLLOWING MODELS We consider a chain of n+1 vehicles raveling on a single lane as shown in Fig. 1a). he ail vehicle he las vehicle of he chain) implemens a CCC algorihm using posiion and velociy signals received hrough V2V communicaion from n preceding vehicles, while oher vehicles are human driven and only ransmi informaion abou heir moion. he dynamics of he CCC vehicle is modeled by ḣ 1 ) = v 2 ) v 1 ), v 1 ) = u), 1) where he do sands for differeniaion wih respec o ime, h 1 is he headway i.e., he bumper-o-bumper disance beween he CCC vehicle and he vehicle immediaely ahead), and v 1 is he velociy of he CCC vehicle; see Fig. 1a). Finally, u) is he conrol inpu ha will be designed using LQ based on he velociy and headway of oher vehicles he laer obained from posiion informaion). For simpliciy, we consider ha vehicles i = 2,..., n are idenical and are described by he car-following model ḣ i ) = v i+1 ) v i ), v i ) = α V h i )) v i ) ) + β v i+1 ) v i ) ), 2) ha can be obained as a simplificaion of he physics-based model presened in [1]. Here h i and v i denoe he headway and velociy of vehicle i; see Fig. 1a). he firs erm in he second equaion represens he driver s inenion o drive a a disance-dependen velociy given by V h)), while he second erm represens he driver s aim o mach he velociy o ha of he vehicle immediaely ahead. he corresponding gains are denoed α and β. We remark ha he proposed algorihm can also be applied in case of non-idenical drivers as well. he desired velociy in 2) is deermined by he range policy if h h s, v V h) = max 2 1 cos ) ) π h hs h go h s if h s < h < h go, v max if h h go, 3) /14/$ IEEE 417

2 i = 2,..., n, and linearize 2) abou he equilibrium 4): h i ) = ṽ i+1 ) ṽ i ), ṽ i ) = α f hi ) ṽ i ) ) + β hi ), 5) Fig. 1. a): A chain of n + 1 vehicles wih a CCC vehicle a he ail receiving signals from oher vehicles via V2V communicaion. b): he nonlinear range policy 3) used in his paper. which is shown in Fig. 1b). he desired velociy is zero for small headways h h s ) and equal o he maximum speed v max for large headways h h go ). Beween hese, i increases wih he headway monoonically. o ensure smooh longiudinal dynamics, he funcion 3) and is derivaive are chosen o be coninuous a h s and h go. Here we consider v max = 3 [m/s], h s = 5 [m], h go = 35 [m] ha corresponds o realisic raffic daa [8]. Many oher range policies may be chosen, bu he qualiaive dynamics remain similar if he above characerisics are kep [2], [9], [1]. We remark ha model 1,2,3) may no adequaely describe longiudinal dynamics when vehicles are driven near physical limis, e.g., ire force sauraion due o emergency braking or severe maneuvering. In his paper we focus on he poenials of wireless conneciviy, while undersanding such limiaions is lef for fuure research. III. LINEAR QUADRAIC RACKING OF UNIFORM RAFFIC FLOW In his secion, he opimal conrol problem under disurbance is formulaed, where he CCC vehicle is racking he uniform flow sae. he cos funcion is consruced in order o minimize he headway and velociy errors and he acceleraion of he CCC vehicle. he soluion gives he gains for he CCC vehicle wih respec o he headways and velociies of oher vehicles. he dynamics of he conneced vehicle sysem 1,2) is invesigaed in he viciniy of an equilibrium where all vehicles ravel wih he same consan velociy and mainain consan headways. While he equilibrium velociy v is deermined by he head vehicle he firs vehicle in he chain), he equilibrium headway h i is obained for each non-ccc vehicle using a range policy v = V i h i ), i = 2,..., n. When considering idenical range policies cf. 3)), he vehicles are equidisan and we obain he uniform flow equilibrium h i ) h, v i ) v = V h ), 4) for i = 2,..., n. We define headway perurbaions h i ) = h i ) h and velociy perurbaions ṽ i ) = v i ) v, for i = 2,..., n. Here f = V h ) is he derivaive of he range policy a he equilibrium and he corresponding ime headway is h = 1/f. In his paper, we use h, v ) = 2 [m], 15 [m/s]), which resuls in he maximum slope f = π/2 [1/s] corresponding o he minimum ime headway h = 2/π.64 [s]; cf. 3) wih v max = 3 [m/s], h s = 5 [m], and h go = 35 [m]. For he CCC vehicle, we define headway perurbaion h 1 ) = h 1 ) h 1 and velociy perurbaion ṽ 1 ) = v 1 ) v. hen 1) yields he linearized dynamics h 1 ) = ṽ 2 ) ṽ 1 ), ṽ 1 ) = u). 6) Le s define he sae x = [ h 1, ṽ 1,..., h n, ṽ n ] R 2n, and wrie dynamics 5,6) in he form ẋ) = Ax) + Bu) + Dṽ n+1 ), 7) where u) is he inpu, ṽ n+1 ) is he disurbance, and he coefficien marices ake he form A 1 A 2 B 1 A 3 A 4 A =......, B =., D =. A 3 A 4 A 3 D 1 8) where he block marices are given by [ ] [ ] [ ] A 1 =, A 2 =, A 4 =, β [ ] [ 1 A 3 = αf, B α β 1 =, D 1] 1 = [ 1 β]. 9) Since our goal is o rack he uniform flow equilibrium x cf. 4)) under velociy disurbance ṽ n+1 ) from he head vehicle, we minimize he cos funcion J τ u, x) = τ x )Qx) + r u 2 ) ) d. 1) he firs erm corresponds o he variaion of he headways and velociies which we call racking errors, he second erm corresponds o he magniude of he CCC vehicle s acceleraion, and τ is he ime horizon we use τ laer). he weigh marix Q is chosen o be diagonal, ha is, Q = diag [q 1, q 2,..., q 2n 1, q 2n ]), 11) where q 2i 1 and q 2i are he weighs on he headway and velociy errors for vehicle i, respecively. Since only vehicle 1 has he CCC conroller, hi, ṽ i, i = 2,..., n are no conrollable and he choice of q 2i 1, q 2i, i = 2,..., n does no influence he opimal conrol inpu. hus we se q 2i 1 = q 2i =, i = 2,..., n. 418

3 3 P a) w b) Fig. 2. a): he soluion P) of 13) wih A = 1, B = 1, Q = 1, r = 1, τ = 3 [s]. b): he corresponding soluion w) of 14) wih perurbaion ṽ n+1 ) = sin2), using P) shown in panel a) red dashed curve) and using he approximaion P = P) blue solid curve). Based on he linear quadraic racking heory [11], he soluion of he opimal conrol problem 7,8,1,11) is given by u) = 1 r B P)x) + w) ), 12) where P) R 2n 2n is a symmeric, posiive definie marix ha saisfies he Riccai differenial equaion Ṗ) = 1 r P)BB P) A P) P)A Q, 13) wih end boundary condiion Pτ) =, while w) R 2n is he soluion of ẇ) = A 1 r BB P) ) w) P)Dṽn+1 ), 14) wih end boundary condiion wτ) =. Noe ha wihou disurbance, i.e., ṽ n+1 ), he LQ problem 7,1) simplifies o an LQR problem wih inpu u) = 1 r B P)x), where P) is he soluion of 13). According o [11], when he sysem 7,8) is sabilizable, hen 13) has uniformly bounded soluion. Moreover, in he infinie-ime horizon τ, he soluion P) can be approximaed by a consan marix given by he algebraic Riccai equaion A P + PA + Q 1 r PBB P =. 15) From now on we use P wihou o denoe his imeindependen marix. Subsiuing P ino 14), and considering ha A 1 r BB P) is Hurwiz, he boundary value problem can be viewed as an iniial value problem wih a special iniial condiion ha eliminaes he ransiens. A simple example is given in Fig. 2 for he simplified problem ẋ) = x) + u) + ṽ n+1 ), x, u R using he weighs Q = 1, r = 1. Panel a) shows ha P) is approximaely consan when τ. Panel b) shows ha using consan P = P) insead of P) in 14) only influences w) near ime τ. hus for large τ, P) can be approximaed by P = P) and he laer can be used o calculae w). Le us define he noaion P = Π Π 1n..... Π n1... Π nn, 16) where Π ij = Π ji R 2 2, i, j = 1,..., n. Considering infinie ime horizon τ, he feedback law 12) gives he acceleraion of he CCC vehicle in 6) as u) = n αi hi ) + β i ṽ i ) ) 1 r w 2), 17) i=1 where w 2 ) denoes he second elemen of w) and he gains α i and β i are given by α i = 1 r Π 1i[2, 1], β i = 1 r Π 1i[2, 2], 18) for i = 1,..., n, where [k, l] represens he elemen in he k h row and l h column. Due o he paricular form of he coefficien marices 8), for i = j = 1, 15) yields Π 11 B 1 Π 11 Π 11 A 1 A 1 Π 11 diag[q 1, q 2 ]) =, 19) where B 1 = 1 r B 1B 1 ; c.f. 9). Moreover, using 15) i can be shown ha he firs row of block marices in P cf. 16)) saisfy he recursive equaions A 1 Π 11 B 1 ) Π12 + Π 12 A 3 = Π 11 A 2, A 1 Π 11 B 1 ) Π1j + Π 1j A 3 = Π 1j 1) A 4, 2) where j = 3,..., n. Also, he second row of block marices saisfy he recursive equaions A 3 Π 22 + Π 22 A 3 = Π 21 B 1 Π 12 Π 12A 2 A 2 Π 12, A 3 Π 2j + Π 2j A 3 = Π 21 B 1 Π 1j Π 2j 1) A 4 A 2 Π 1j, 21) where j = 3,..., n and Π 21 = Π 12. For he remaining n 2 rows of block marices, we obain he recursive equaions A 3 Π ij +Π ij A 3 = Π i1 B 1 Π 1j Π ij 1) A 4 A 4 Π i 1)j, 22) where i = 3,..., n, j = i,..., n and Π i1 = Π 1i. hus, he soluion of he Riccai equaion 15) can be obained by solving 19,2,21,22) consecuively. In he physically realisic case q 1, q 2, r >, he only feasible soluion of 19) is given by q Π 11 = 1 q q1 3r q 1 r q 1 r q 2 r + 2, 23) q 1 r 3 and hus 18) yields α 1 = q 1 /r, β 1 = q 2 /r + 2 q 1 /r. 24) Moreover, according o 18), he gains α i, β i obained from 2) can be rewrien as vecπ 12 ) = M vecπ 11 ), vecπ 1i ) = MvecΠ 1i 1) ), 25) 419

4 .8 α i i.8 β i Fig. 3. he opimized headway and velociy gains α i, β i, i = 2,..., n of he CCC vehicle in a n + 1)-car plaoon for n = 5 red circles) and for n = 1 blue crosses). where vecπ 1i ), i = 3,..., n, is a vecor obained by wriing columns of Π 1i ino a vecor and we have M = I A ) 1 Π 11 B 1 + A 1A 3 I) 4 I), M = I A ) 1 Π 11 B 1 + A 1A 3 I) 2 I). herefore, i 26) vecπ 1i ) = M i 2 M vecπ 11 ) 27) is a map beween Π 11 and Π 1i, for i = 2,..., n, and hus α i, β i, i = 2,..., n can be obained as funcions of α 1, β 1, α, β, f ; cf. 9,18,26). Moreover, 27) indicaes ha he feedback gains of vehicle i are deermined by he carfollowing dynamics of vehicles 1, 2,..., i 1 and are no influenced by he dynamics of vehicles i+1,..., n+1. his propery means ha our CCC design is scalable, since he values of gains can be kep consan regardless how many vehicles ahead are moniored. Finally, we noe ha 21,22) are only needed o obain w); cf. 14). One may show ha he eigenvalues of M in 27) are inside he uni circle for human gains α >, β >. herefore, 27) is a conracing map in realisic scenarios and α i and β i converge o zero following geomeric series as i increases. his indicaes ha he CCC vehicle relies more on signals obained from closer vehicles, and his characerisic behavior is no influenced by he choice of weighs q 1 and q 2 in he cos funcion 1,11). As an example, we consider q 1 = 2 [1/s 2 ], q 2 = 4, r = 1 [s 2 ] and assume non-ccc vehicles wih gains α =.6 [1/s] and β =.9 [1/s]. In his case 24) gives he gains α [1/s 2 ], β [1/s]. he exponenial decay of he gains α i, β i wih he vehicle index i = 2,..., n is demonsraed graphically in Fig. 3 for a 5 + 1) vehicle chain red circles) and for a 1 + 1) vehicle chain blue crosses). his is suppored by he fac ha he eigenvalues of M in 27) are λ 1 =.61, λ 2 =.37 and λ 3,4 =, which are locaed inside he uni circle. We remark ha he gains on signals for vehicles 7 1 are small, indicaing ha close-o-opimal design can be obained when only observing approximaely 5 6 vehicles ahead. In his sense, he benefis of increasing he number of cars ahead saurae, and having very long connecions may no be favorable as hey only make he nework srucure more complicaed. IV. HEAD-O-AIL SRING SABILIY In his secion, we discuss how he opimal CCC design influences he longiudinal sabiliy of he conneced vehicle sysem. In paricular, we analyze plan sabiliy and sring sabiliy. he plan sabiliy of a CCC vehicle is given as follows: suppose ha he vehicles whose signals are used by a CCC vehicle are driven a he same consan velociy, hen he velociy of he CCC vehicle approaches his consan velociy. Sring sabiliy characerizes he aenuaion of velociy flucuaions as hey propagae upsream [12]. However, sring sabiliy can only be ensured for he CCC vehicle as we have no conrol over he non-ccc vehicles). herefore, we evaluae he head-o-ail sring sabiliy, i.e., compare he velociy flucuaions of he head vehicle and he CCC vehicle a he ail. Noice ha his definiion allows ha vehicles in he middle may amplify he velociy flucuaions of vehicles ahead. Despie he presence of such inra-plaoon sring insabiliy, a CCC vehicle can be used o ensure heado-ail sring sabiliy. We consider he velociy perurbaion ṽ n+1 of he head vehicle as he inpu and he velociy perurbaion ṽ 1 of he ail vehicle as he oupu. Since perurbaion signals can be represened using Fourier componens and superposiion holds for linear sysems, he head-o-ail sring sabiliy is ensured when sinusoidal signals are aenuaed beween he head and he ail vehicles for all exciaion frequencies. hus, we consider he periodic exciaion ṽ n+1 ) = v amp sinω) and seady-sae soluion of 7,12,14,15), see Fig. 2b) for illusraion. hen aking he Laplace ransform of 14) wih zero iniial condiion and reversed ime while using P) P leads o W s) = si A + 1 r BB P) PDṼn+1s), 28) where W s) is he Laplace ransform of w), Ṽ n+1 s) is he Laplace ransform of ṽ n+1 ). aking he Laplace ransform of he sysem 7,12) wih zero iniial condiions and eliminaing he velociies of he oher vehicles and he headways, we obain he head-o-ail ransfer funcion Γs) = Ṽ1s) Ṽ n+1 s) = 1 α 1 Γ n 1 s) + sf n+1s) G 1 s) G 1 s) n + αi + β i s α i )Γ s) ) ) Γ n i s). 29) i=2 Here Ṽ1s) and Ṽn+1s) denoe he Laplace ransform of ṽ 1 ) and ṽ n+1 ), respecively, and Γ s) = F s) G s), G s) = s 2 + α + β)s + αf F s) = βs + αf, G 1 s) = s 2 β 1 s + α 1, F n+1 s) = α n + ββ n )s + Π 1n [1, 1] + βπ 1n [1, 2]. 3) Plan sabiliy a he linear level is deermined by he denominaor of he ransfer funcion in 29). he sysem 411

5 α =.6 [1/s] q 2 a) q 2 α =.9 [1/s] b) β =.6 [1/s] q 1 q 1 β =.9 [1/s] q 2 c) q 2 A B C d) q 1 q 1 Fig. 4. Sring sabiliy chars of a 5+1)-car plaoon in he q 1, q 2 )-plane for differen human parameers α, β as indicaed. he sring sable domains are shaded. is linearly plan sable, if and only if all soluions of he characerisic equaion G n s)g 1 s) = are locaed in he lef half complex plane. Noice ha plan sabiliy is only influenced by he human parameers α, β and he CCC gains α 1, β 1. Using Rouh-Hurwiz crieria, we obain he condiions for plan sabiliy Fig. 5. Magniude of ransfer funcion as a funcion of he exciaion frequency. Panels a c) correspond o poins marked A C in Fig. 4 c). β a) q 2 = 1 β b) q 2 = 5 β c) q 2 = 1 q1 = 1 α α α β d) q 1 = 2 β e) q 1 = 5 β f) q 1 = 1 q2 = 1 α >, α + β >, α 1 >, β 1 <. 31) In he following analysis, we only consider plan sable human parameers α, β. Soluion 23) provides he gains α 1, β 1 ha have o saisfy 31). hese can be used o obain all oher gains α i and β i, i = 2,..., n; see 18,27). A he linear level he necessary and sufficien condiion of head-o-ail sring sabiliy is given by Γiω) 2 1 = ω 2 fω) <, ω >, 32) where Γiω) is defined by 29,3); see [4], [13]. he order of fω) increases wih he number of vehicles n. Sring sabiliy is violaed when he maximum of fω) is larger han, and hus, he sring sabiliy boundary is given by he equaions fω cr ) =, fω cr ) ω =, 33) subjec o 2 fω cr ) ω 2 <, where ω cr indicaes he locaion of he maximum of fω). o obain sring sabiliy chars, we solve 33) numerically and plo he sring sabiliy boundary in he q 1, q 2 )-plane and in he α, β)-plane. Noe ha in pracical ranges of human parameers α and β he sring insabiliy only occurs a zero frequency i.e., ω cr = ). When he human parameers α and β change, he range of weighs q 1, q 2, r ha resuls in a sring sabiliy changes. Wihou loss of generaliy, we fix r = 1 [s 2 ] and only consider he change of weighs q 1, q 2. As observed in he previous secion, gains on vehicles i, i > 6, are small, and herefore we consider n = 5. α α α Fig. 6. Sring sabiliy chars of a 5 + 1)-car plaoon in he α, β)-plane. he noaion is he same as in Fig. 4. In Fig. 4 we fix he human parameers α =.6,.9 [1/s] and β =.6,.9 [1/s] and shade he sring sable domains in he q 1, q 2 )-plane. I has been shown [3] ha wihou CCC, he sysem is sring unsable when α + 2β 2f >, 34) and hus, we have a sring unsable sysem for q 1 = q 2 =. Fig. 4a) shows ha increasing he weighs q 1, q 2 on racking errors is beneficial for sring sabiliy, and ha no sring sabiliy exiss for q 1 1, q 2 4. However, q 1 and q 2 canno be chosen independenly. Similar resuls are observed in panels b d). Comparing he four panels, one can noice ha increasing eiher α or β increases he size of he sring sable domain, while he minimum q 2 ensuring sring sabiliy decreases o zero. However, for q 2 = he range of sring sable q 1 is very small. We remark ha sring sabiliy loss in he conneced vehicle sysem analyzed here only happens a zero frequency. Fig. 5 demonsraes such sabiliy loss for he poins A C marked in Fig. 4c). As q 2 decreases, he magniude of ransfer funcion 29) increases and i exceeds 1 in he lowfrequency domain in case C. he robusness of opimized CCC designs is invesigaed 4111

6 by varying he human parameers α and β and he resuls are summarized in Fig. 6. We fix q 1 = 1 [1/s 2 ] in panels a c) and q 2 = 1 in panels d f), and shade he sring sable domains. Increasing α and β improves sring sabiliy in each case. For fixed q 1, increasing q 2 enlarges he sring sabile area cf. panels a) and b)), bu oo large q 2 resuls in smaller sring sabiliy region cf. panels b) and c)). For fixed q 2, sring sabiliy increases wih larger q 1 ; see d f). We remark ha sring sabiliy may be los when increasing q 1 even furher, alhough q 1 < 1 is considered o be physically realisic. hus weighing heavily on eiher h 1 or v 1 is derimenal for sring sabiliy. Finally, o evaluae he performance of our CCC algorihm, we consider a 5+1)-car sysem wih sring unsable human parameers and invesigae he evoluion of headway and velociy errors by numerical simulaions. Fig. 7 shows he simulaion resuls of he 5+1)-car sysem wih gains generaed using differen design parameers q 1 and q 2. he simulaion resuls are presened for he parameers corresponding o poins A and C in Fig. 4c), while using he disurbance signal ṽ n+1 ) = v amp sinω) wih ampliude v amp = 1 [m/s] and frequency ω =.3 [rad/s]; see Fig. 5a,c) for he amplificaion plos. he simulaion resuls demonsrae ha case A is head-o-ail sring sable, as he CCC vehicle s velociy flucuaion hick blue curve) has smaller ampliude han he velociy inpu dashed curve) in Fig. 7a). On he oher hand, he CCC vehicle s velociy flucuaion in Fig. 7c) has larger ampliude han he velociy inpu, indicaing sring insabiliy. Noe ha in boh cases he ampliude of velociy flucuaions are amplified by non-ccc vehicles, because he human parameers α =.6 [1/s], β =.9 [1/s] are sring unsable; see 34). Sill, he CCC vehicle is able o mainain sring sabiliy when q 1, q 2 are chosen appropriaely. On he oher hand, he comparison of panels b) and d) shows a rade-off. While an increased weigh on velociy error q 2 ensures sring sabiliy, he relaive weigh on q 1 decreases, and he headway error increases hough sill aenuaed compared o head vehicle). V. CONCLUSION In his paper, we proposed a conneced cruise conrol design based on linear quadraic racking and analyzed he head-o-ail sring sabiliy. I was shown ha he gains depend on he human parameers and he design parameers in he cos funcion. We found ha he opimal gains on preceding vehicles are no influenced by dynamics of vehicles farher downsream, and ha he gains decrease wih he number of cars beween he CCC vehicle and he signaling vehicle. he opimized CCC is shown o be able o sabilize an oherwise sring unsable sysems when he weighs on he headway and velociy errors are chosen appropriaely. he design was robus agains variaions of human parameers, and he resuls were verified using numerical simulaions. Fuure research includes opimizing nonlinear CCC algorihms while considering more complicaed conneciviy srucures and imperfec communicaion v a) v c) 21 h b) h d) Fig. 7. Velociy and headway responses of a 5 + 1)-car vehicle sring wih human parameers α =.6 [1/s], β =.9 [1/s]. a,b): using design parameers q 1 = 2 [1/s 2 ], q 2 = 4. c,d): using design parameers q 1 = 2 [1/s 2 ], q 2 = 1. hick curves denoe he CCC vehicle, hin curves are used for non-ccc vehicles, while dashed curves indicae he velociy disurbance of he head vehicle. REFERENCES [1] G. Orosz, Conneced cruise conrol: modeling, delay effecs, and nonlinear behavior, Vehicle Sysem Dynamics, p. submied, 214. [2] G. Orosz, R. E. Wilson, and G. Sépán, raffic jams: dynamics and conrol, Philosophical ransacions of he Royal Sociey A, vol. 368, no. 1928, pp , 21. [3] L. Zhang and G. Orosz, Designing nework moifs in conneced vehicle sysems: delay effecs and sabiliy, in Proceedings of he ASME Dynamical Sysems and Conrol Conference. 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