Estimation of feedback gains and delays in connected vehicle systems

Transcription

1 26 American Conrol Conference (ACC) Boson Marrio Copley Place July 6-8, 26 Boson, MA, USA Esimaion of feedback gains and delays in conneced vehicle sysems Jin I Ge and Gábor Orosz Absrac In his paper, we presen differen parameer esimaion mehods ha may be used o idenify he dynamics of human-driven vehicles in conneced vehicle sysems By exploiing he informaion received via wireless vehicle-ovehicle communicaion from wo consecuive vehicles ahead, he esimaion algorihms idenify he driver reacion ime and feedback gains simulaneously We compare he algorihms in erms of convergence rae and esimaion accuracy, and presen a sysemaic way o improve boh performance measures The esimaed parameers can hen be used in conneced cruise conrol algorihms ha incorporae he ransmied signals in a vehicle s longiudinal moion conrol I INTRODUCTION Various advanced driver assisance sysems have been proposed over he pas decades in order o improve road ransporaion Specifically, adapive cruise conrol (ACC) uses onboard sensors such as radar/lidar o obain moion informaion faser and more accuraely han human drivers, and provides more adequae commands for he longiudinal moion conrol [] Wih he availabiliy of affordable wireless communicaion devices, conneced cruise conrol (CCC) has been proposed [2] o exploi moion informaion of vehicles over a longer range, so ha undesired acceleraion/deceleraion of he CCC vehicle can be furher suppressed This may lead o higher road efficiency, less fuel consumpion and improved safey in conneced vehicle sysems Exising research on CCC design generally assumes a priori knowledge on he dynamics of preceding vehicles whose signals are used in he CCC conroller [3], [4] Such an assumpion may no hold, when he ad-hoc wireless communicaion allows vehicles o join and leave he conneced vehicle sysem Thus, i is necessary o consider online i- denificaion of car-following dynamics of preceding vehicles [5] We choose he parameric model o be a car-following model wih driver reacion ime, as i has been found o be able o describe a wide range of driving behaviors The driver reacion ime is included, because i has been shown ha he large driver reacion ime has a significan influence on he performance of conneced vehicle sysems which include human-driven, ACC, and CCC vehicles [6] Alhough in his paper we assume ha all non-ccc vehicles are humandriven, he idenificaion algorihms are able o deal wih ACC vehicles, as he laer can also be described by carfollowing models bu wih smaller reacion ime Jin I Ge and Gábor Orosz are wih he Deparmen of Mechanical Engineering, Universiy of Michigan, Ann Arbor, Michigan 489, USA gejin@umichedu, orosz@umichedu This work was suppored by he Naional Science Foundaion (award number 35456) The mehodology for parameer idenificaion in sysems wihou ime delay has been well developed over he years Many adapive laws were designed based on Lyapunov echnique [7] While here exis some resuls concerning parameer esimaion in ime delay sysems [8], [9], esimaing he delay ime and feedback gains simulaneously is sill a challenging problem [], [] Such esimaors ypically lead o sysems wih sae-dependen delay, where i is difficul o design he convergence rae and region of aracion In his paper, we presen hree mehods for simulaneous idenificaion of he feedback gains and delay The firs (and mos sraighforward) mehod convers he problem of esimaing delay ime ino esimaing feedback gains To overcome he large compuaional need and slow convergence rae of he firs mehod, we propose he second mehod which explicily esimaes he delay ime parallel o he feedback gains The hird mehod is proposed o improve he convergence rae of he second mehod by inroducing addiional nonlineariies Numerical simulaions are used o compare he performance of he hree mehods II CONNECTED VEHICLE SYSTEM SETUP We consider he configuraion in Fig where a group of vehicles ravel on a single lane We assume ha vehicle receives he moion informaion of N vehicles ahead hrough wireless vehicle-o-vehicle (V2V) communicaion By using signals from wo consecuive vehicles ahead (vehicles i and i+), vehicle is able o idenify he dynamics of vehicle i and hus uses moion informaion of vehicle i for is longiudinal conrol In his way, vehicle can include feedback erms on he headway and velociy signals from muliple vehicles ahead, which we refer as conneced cruise conrol [6], [2] A Nonlinear car-following model The non-ccc vehicles (i = 2,, N ) are assumed o be human-driven and heir dynamics is described by he convenional car-following model ḣ i () = v i+ () v i (), v i () = α ( V (h i ( τ)) v i ( τ) ) + β ( v i+ ( τ) v i ( τ) ) Here he do sands for differeniaion wih respec o ime, h i denoes he headway, ie, he bumper-o-bumper disance beween vehicle i and is predecessor), and v i denoes he velociy of vehicle i; see Fig According o () he acceleraion of a non-ccc vehicle is deermined by wo erms: he difference beween he headway-dependen desired () /$3 26 AACC 6

2 2 i i + N v v 2 v i v i+ v N l h l l h i l l V v max h s h go h Fig : A sring of N vehicles wih a CCC vehicle a he ail receiving signals from human-driven vehicles ahead via wireless V2V communicaion The red arrows denoe informaion flow The velociy of vehicle i is denoed by v i The bumper-o-bumper disance beween vehicle i and vehicle i + is called he headway and denoed as h i The lengh of every vehicle is assumed o be l : The range policy funcion defined by (2) velociy and he acual velociy and he velociy difference beween he vehicle and is predecessor The gains α and β are used by he human drivers o correc velociy errors, and τ is he human reacion ime The desired velociy is deermined by a nonlinear range policy funcion if h h s, h h s V (h) = v max if h s < h < h go, (2) h go h s v max if h h go, which is shown in Fig Tha is, 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, he desired velociy increases wih he headway linearly Many oher range policies may be chosen, bu he qualiaive dynamics remain similar if he above characerisics are kep [3] The model (,2) is adaped from [3], [4] and can be obained as a simplificaion of he physics-based model presened in [2] Noe ha he model (,2) does no describe dynamics in emergency siuaions For he same reason, we do no consider feedback erms relaed wih he moion of vehicles behind B Linearizaion We assume he dynamics of he conneced vehicle sysem (,2) o be in he viciniy of he uniform flow equilibrium where all vehicles ravel wih he same consan velociy and mainain he same consan headway, ha is, h i () h, v i () v = V (h ), (3) for i =,, N Here he equilibrium velociy v is deermined by he head vehicle in a vehicle sring, while he equilibrium headway h can be calculaed from he range policy (2) We define he headway perurbaions δh i () = h i () h and velociy perurbaions δv i () = v i () v, for i =,, N, and linearize () abou he equilibrium (3) This yields he linear delay differenial equaion δḣi() = δv i+ () δv i (), δ v i () = (α + β)δv i ( τ) + αn δh i ( τ) + β δv i+ ( τ), for i = 2,, N Here N = V (h ) is he derivaive of he range policy (2) a he equilibrium and he corresponding ime headway is h = /N Le us inroduce he noaion x() = δv i (), u () = δh i (), u 2 () = δv i+ () (5) Based on he dynamics (4) of vehicle i, we wrie ou he parameric model (4) ẋ = ax( τ) + b u ( τ) + b 2 u 2 ( τ), (6) where he delay ime τ and he feedback gains a = α β, b = αn, b 2 = β (7) are unknown a priori and o be deermined hrough parameer idenificaion III SIMULTANEOUS ESTIMATION OF GAINS AND DELAY A Direc Lyapunov mehod of approximaed delay This mehod was originally proposed in [] I bypasses he esimaion of delay ime by inroducing muliple ficiious delays whose corresponding gains are zero, ie, he parameric model (6) is rewrien as n ( ẋ = ai x( τ i ) + b i u ( τ i ) + b 2i u 2 ( τ i ) ), i= where τ < τ 2 < < τ n, and here exiss j {,, n} such ha he real delay τ = τ j Thus, and (8) a j = α β, b j = αn, b 2j = β, (9) a i = b i = b 2i =, () for i j Then by idenifying he zero gains we are able o weed ou he fake delays and obain he real delay ime and corresponding feedback gains Based on [], he esimaion algorihm for (8) is given by n x() = (ãi x( τ i ) + b i u ( τ i ) + b 2i u 2 ( τ i ) ) i= + cˆx(), ã i () = γ iˆx()x( τ i ), b i () = γ iˆx()u ( τ i ), b 2i () = γ 2iˆx()u 2 ( τ i ), () where ildes are used o denoe esimaed sae and parameers, he sae error is ˆx = x x, and he esimaion gains are c >, γ ki >, k =,, 2, i =,, n 6

3 Define he vecors ˆx Z = Â ˆB, Â = ˆB 2 â â n, ˆBk = ˆbk ˆbkn, (2) ha conain he esimaion errors â i = ã i a i, ˆb ki = b ki b ki, k =, 2, i =,, n Using (8,), we can wrie he esimaion model as a linear ime-varying sysem: where c Xτ T Uτ T U2τ T Ż = Γ X τ Γ U τ Γ 2 U 2τ Γ k = diag [ γ k γ kn ], k =,, 2, X τ () = [ x( τ ) x( τ n ) ] T, U τ () = [ u ( τ ) u ( τ n ) ] T, U 2τ () = [ u 2 ( τ ) u 2 ( τ n ) ] T Z, (3) (4) We would like he fixed poin Z() of (3) o be (a leas) Lyapunov sable Thus, we define he Lyapunov funcion candidae V = 2 ZT = 2 ˆx2 + n i= Γ Γ Γ 2 Z ( â 2 i + ˆb2 2 γ i γ i + ) ˆb2 i γ 2i, 2i and using (3), he Lie derivaive becomes (5) V = cˆx 2 (6) Since (5) is posiive definie and (6) is negaive semidefinie, Z() is sable in he sense of Lyapunov, and he sysem (3) converges o he manifold of ˆx() asympoically On he oher hand, he convergence of he esimaed parameers (lim Â() =, lim ˆB () =, lim ˆB2 () = ) resuls from he convergence of ˆx given sufficienly rich signals x(), u (), u 2 (), ie, he persisen exciaion condiion Here his condiion requires piecewise coninuous signals wih a sufficien number of disconinuiies a non-commensurable poins [], and in each coninuous subinerval i also requires a sufficien number of Fourier componens Such jumps may be common in elecronic signals, bu hey are seldom observed in mechanical sysems such as he velociy of a car Thus, i may no be easy o implemen his mehod o esimae he parameers of a car-following model We also noe ha o obain a more accurae esimaion on he acual delay ime τ, finer meshes of ficiious delay τ i are required, which increases he dimension of he adapive law () This would no only require more compuaional power, bu also significanly slow down he convergence rae If he idenificaion becomes slower han he variaion of driver behavior his mehod would evenually fail B Mehod of sae-dependen delay To obain a more accurae esimaion of he delay ime wihou relying on a high-dimensional parameric model, we consider an adapive law where boh he delay ime and he feedback gains in (6) are idenified simulaneously In paricular, we propose x = ãx( τ) + b u ( τ) + b 2 u 2 ( τ) + cˆx, ã = γ ˆxx( τ), b = γ 2ˆxu ( τ), b 2 = γ 3ˆxu 2 ( τ), τ = γ 4ˆx ζ, (7) where we use ildes o denoe esimaed sae and parameers, he sae error is defined as ˆx = x x, and ζ = ãẋ( τ) + b u ( τ) + b 2 u 2 ( τ) (8) Due o he adapive law of τ, (7) is a nonlinear sysem wih sae-dependen delay To discuss he convergence of he algorihm, we denoe he parameer errors by â = ã a, ˆb = b b, ˆb2 = b 2 b 2, ˆτ = τ τ, (9) and formulae he dynamics of ˆx using (6,7) ˆx = cˆx() + ax( τ) ãx( τ) + b u ( τ) b u ( τ) + b 2 u 2 ( τ) b 2 u 2 ( τ) = cˆx() âx( τ) ˆb u ( τ) ˆb 2 u 2 ( τ) + ã(x( τ) x( τ)) + b (u ( τ) u ( τ)) + b 2 (u 2 ( τ) u 2 ( τ)) (2) Assuming consan feedback gains in he parameric model (6) we obain â = ã, ˆb = b, ˆb 2 = b2, ˆτ = τ (2) While he convergence of approximaed delay mehod () can be proven wih negaive semi-definieness of he Lie derivaive of a quadraic Lyapunov funcion (6), here we could no esablish he negaive semi-definieness direcly using similar quadraic Lyapunov funcions, due o he esimaed delay τ Thus we resor o he indirec Lyapunov mehod by firs isolaing he nonlineariy due o he imevarying delay τ and esablishing convergence for he linearized dynamics We assume x(), u () and u 2 () are bounded and have bounded derivaives up o he hird order, and perform Taylor expansion o exrac ˆτ from erms wih ime-varying delay x( τ) x( τ) = ˆτẋ( τ) 2 ˆτ 2 ẍ( τ) + O(3), u ( τ) u ( τ) = ˆτ u ( τ) 2 ˆτ 2 ü ( τ) + O(3), u 2 ( τ) u 2 ( τ) = ˆτ u 2 ( τ) 2 ˆτ 2 ü 2 ( τ) + O(3), (22) 62

4 where O(n) denoes n-h order erms in he sae error ˆx and he parameer errors â, ˆb, ˆb 2, ˆτ Assuming consan gains and delay ime in he parameric model (6), we have ẍ() = aẋ( τ) + b u ( τ) + b 2 u 2 ( τ), (23) and (2) is rewrien as ˆx = cˆx() x( τ)â u ( τ)ˆb u 2 ( τ)ˆb 2 + ẍ()ˆτ + g (â, ˆb, ˆb 2, ˆτ), where he higher-order erms are given by g = ẋ( τ)âˆτ + u ( τ)ˆb ˆτ + u 2 ( τ)ˆb 2ˆτ x() 2 ˆτ 2 + O(3) Similarly, we rewrie (7,2) as â = γ x( τ)ˆx() + g (ˆx, ˆτ), ˆb = γ 2 u ( τ)ˆx() + g 2 (ˆx, ˆτ), ˆb 2 = γ 3 u 2 ( τ)ˆx() + g 3 (ˆx, ˆτ), ˆτ = γ 4 ẍ()ˆx() + g 4 (ˆx, â, ˆb, ˆb 2, ˆτ), where he higher-order erms are g = γ ẋ( τ)ˆxˆτ + O(3), g 2 = γ 2 u ( τ)ˆxˆτ + O(3), (24) (25) (26) g 3 = γ 3 u 2 ( τ)ˆxˆτ + O(3), g 4 = γ 4 (ẋ( τ)â + u ( τ)ˆb + u 2 ( τ)ˆb 2 x ˆτ )ˆx + O(3) (27) We define he sae variable Y = [ˆx â ˆb ˆb2 ˆτ ] T and hen wrie (24,26) ino sae space form Ẏ = M()Y + N(Y, ), (28) where he coefficien marix is given by [ ] c v() M() = Γv T, () Γ = diag [ γ γ 2 γ 3 γ 4 ], v() = [ x( τ) u ( τ) u 2 ( τ) ẍ() ], and he higher-order erms are colleced in (29) N(Y, ) = [ g g g 2 g 3 g 4 ] T (3) When he fixed poin Y () of (28) is uniformly asympoically sable, boh he sae error ˆx and he parameer errors â, ˆb, ˆb 2, ˆτ are guaraneed o decay o zero I has been found ha given c >, he uniform asympoic sabiliy of Ẏ = M()Y is equivalen o he persisen exciaion condiion of he signal v() [5] Tha is, here exiss posiive consans T, δ, and ɛ such ha for all and a uni vecor w wih he same dimension as v(), here is a 2 [, + T ] such ha 2+δ v(θ)w T dθ ɛ (3) 2 Fig 2 : The non-coninuous sign funcion (33): The non-smooh piecewise linear funcion (34) wih he widh of boundary layer w = 2ɛ (c): The infiniely smooh hyperbolic angen funcion (35) wih he widh of boundary layer w 2 = ɛln(2 + 3) We se ɛ = 4 in (b,c) This esablishes he local convergence of his algorihm However, he size of he basin of aracion depends on he nonlinear erm N(Y, ) While we assume he inpu signals x(), u (), u 2 () are bounded, he bounds in general depend on he magniude of differen frequency componens in u 2 () These bounds, ogeher wih he choice of adapaion gains γ, γ 2, γ 3, γ 4, deermine he basin of aracion We believe ha he size of he basin of aracion can be esimaed hrough bifurcaion analysis Here, due o page limi, we only poin ou ha he algorihm is observed o perform well even for iniial guesses wih relaively large parameer errors, when considering properly chosen adapaion gains and sufficienly rich signals based on experiences from parameer esimaion in non-delayed sysems Meanwhile, having an adapive law for he delay ime insead of a grid approximaion has been found o be imporan o ensure he performance of he parameer esimaion Consequenly, his mehod is more pracical for applicaions like conneced vehicle sysems C Modified mehod of sae-dependen delay I is noed in sliding mode conrol ha he swiching beween differen conrol laws in differen regions of he sae space may increase he convergence rae [6] Moivaed by he esimaor proposed in [], we consider inroducing he sign funcion ino our adapive law and rewrie i as x = ãx( τ) + b u ( τ) + b 2 u 2 ( τ) + cˆx, ã = γ f(ˆx)x( τ), b = γ 2 f(ˆx)u ( τ), b 2 = γ 3 f(ˆx)u 2 ( τ), τ = γ 4 f(ˆx) ζ, (32) where ζ is given by (8) and we may use he swiching funcion if ˆx <, f(ˆx) = sign(ˆx) = if ˆx =, (33) if ˆx >, ha is ploed in Fig 2 However, he sign funcion canno be perfecly implemened numerically, which ypically leads o high-frequency 63

5 ã b 2 - ã -5 - b b2 (c) u (d) b (c) τ (d) Fig 3 (a,b,c): Esimaed feedback gains using () The blue curves are gains corresponding o τ = [s], and he red curves corresponds o τ 2 = 5 [s] The dashed lines mark he real value of a, b, and b 2, respecively (d): The inpu signal u 2 () = δv i+ () used by he esimaor (8) is disconinuous a 27, 52, 5 [s] In each coninuous subinerval u 2 () = 6 j= sin(ω j + ρ j ) wih frequency componens ω j {2, 6, 5, 38, 53, 7} [rad/s] oscillaion (called chaering) in he viciniy of he sliding surface To eliminae chaering, we define a boundary layer ˆx ɛ, ɛ > and define he swiching funcion as if ˆx > ɛ, f(ˆx) = p(ˆx) = ˆx/ɛ if ɛ ˆx ɛ, (34) if ˆx < ɛ, as shown in Fig 2 Here ɛ is used o adjus he gradien of he swiching funcion, and consequenly he widh w = 2ɛ of he boundary layer Noe ha inside he boundary layer, (34) is linear and he esimaor (32,34) is equivalen o (7) In order o avoid non-smoohness when enering he boundary layer, we may use a hyperbolic angen funcion f(ˆx) = anh (ˆx/ɛ ), (35) as shown in Fig 2(c) Again, ɛ deermines he gradien, and by calculaing he hird-order inflecion poins ( 3ˆx anh(ˆx/ɛ ) = ) we obain he widh of he boundary layer as w 2 = ɛln(2+ 3) Indeed, as ɛ decreases, he widh of boh boundary layers in (34) and (35) end o zero, and (34) and (35) converges o (33) We noe ha when implemening algorihms ha conain he dynamics of τ, a lower bound and an upper bound on he esimaion τ are needed In his paper, we se τ [, τ max ], where τ max is he maximum delay ime known a priori IV SIMULATIONS In his secion, we implemen he mehod of approximaed delay (), and he original and modified mehods of saedependen delay (7,32) o esimae he delay ime and feedback gains in he parameric car-following model (6), and compare heir performances Fig 4 Esimaed feedback gains and delay ime using he esimaor (7) The dashed lines mark he real parameer values The signals are generaed using (4,5) wih u 2 () = δv i+ () = 2 3 ( cos() + cos(2) + cos(7) ) The esimaor gains are γ = 5, γ 2 = 5, γ 3 = 6, γ 4 = We use v max = 3 [m/s], h s = 5 [m], h go = 35 [m] ha corresponds o realisic raffic daa [3] in he range policy (2), which resuls in he consan slope N = [/s] corresponding o he consan ime headway h = /N = [s] Moreover, we se he equilibrium a (h, v ) = (2 [m], 5 [m/s]) and use he gains α = 6 [/s], β = 9 [/s] and reacion ime τ = 5 [s] Then he real parameer values in he parameric model (6) are a = 5 [/s], b = 6 [/s], b 2 = 9 [/s]; see (7) For he firs mehod (), we assume one ficiious delay ime τ = [s] aside from he real delay ime τ 2 = 5 [s] The signals x(), u (), u 2 () are generaed by he linearized car-following model (4,5) wih non-smooh velociy perurbaion u 2 () wih frequency componens ω j {2, 6, 5, 38, 53, 7} [rad/s] The rajecory of esimaed parameers are shown in Fig 3(a,b,c) Noe ha while he parameers converge wihin 5 [s], he velociy profile of vehicle i + conains 6 frequency componens and 3 disconinuiies o mee he persisen exciaion condiion (see he capion of Fig 3) Such non-smooh velociy profile is no commonly observed in cars on road, as vehicles can be viewed as low-pass filers Thus, he convergence rae of his esimaor may be significanly slower in real-world implemenaion Also, he number of esimaed parameers (3n) increases wih he number of ficiious delays (n), leading o observed deerioraion of convergence rae in simulaions Meanwhile, as he mesh for delay ime becomes finer, he ime inerval beween mesh poins shorens As a resul, he numerical algorihm will have increasing difficuly in correcly idenifying gains corresponding o each mesh poin Thus, i is difficul o preserve he convergence rae in Fig 3 when we do no sar wih he exac guess of τ 2 = 5 [s] Fig 4 and Fig 5 show he performance of he esimaors (7) and (32,35), respecively Signals x(), u (), u 2 () are generaed by he linearized car-following model (4,5) wih 64

6 ã b (c) b τ (d) Fig 5 Esimaed feedback gains and delay ime using he esimaor (32,35) The noaions and inpu signals are he same as in Fig 4 The esimaor gains are γ = 5, γ 2 = 5, γ 3 = 4, γ 4 = 35, and we se ɛ = 4 a sinusoidal velociy perurbaion u 2 () = δv i+ () = 2 3( cos()+cos(2)+cos(7) ) In boh figures, he feedback gains and delay converge o he real values wihin reasonable amoun of ime, producing more accurae esimaions han in Fig 3 However, he convergence rae is faser in Fig 5, which illusraes he poenial benefis of he modified esimaor (32,35) V CONCLUSION We performed parameer esimaion for car-following models wih driver reacion ime using hree differen esimaors Firs, we invesigaed he mehod of muliple delays, which is a direc exension of parameer esimaion in sysems wihou delay However, i required non-coninuous signals for persisen exciaion and he convergence rae deerioraed rapidly as he number of ficiious delays were increased Then we proposed he mehod of sae-dependen delay, which is able o provide accurae esimaion of delay ime and feedback gains, wihou sacrificing he convergence rae Finally, we improved he second mehod by inroducing swiching behaviors using he sign funcion and is smooh approximaions, and demonsraed he improvemen of he convergence rae We conclude ha he laer wo mehods are suiable for applicaions in conneced vehicle design, and fuure research may include he robusness and global sabiliy analysis of he adapive laws as nonlinear sysems wih sae-dependen delays [4] W B Qin, M M Gomez, and G Orosz, Sabiliy and frequency response under sochasic communicaion delays wih applicaions o conneced cruise conrol design, IEEE Transacions on Inelligen Transporaion Sysems, p submied, 25 [5] R Pandia and D Caveney, Preceding vehicle sae predicion, in 23 IEEE Inelligen Vehicles Symposium, 23, pp 6 [6] J I Ge and G Orosz, Dynamics of conneced vehicle sysems wih delayed acceleraion feedback, Transporaion Research Par C, vol 46, pp 46 64, 24 [7] P A Ioannou and J Sun, Robus Adapive Conrol Dover Publicaions, 22 [8] S Diop, I Kolmanovsky, P Moraal, and M V Nieuwsad, Preserving sabiliy/performance when facing an unknown ime-delay, Conrol Engineering Pracice, vol 9, no 2, pp , 2 [9] O Gomez, Y Orlov, and I V Kolmanovsky, On-line idenificaion of SISO linear ime-invarian delay sysems from oupu measuremens, Auomaica, vol 43, no 2, pp , 27 [] Y Orlov, L Belkoura, J-P Richard, and M Dambrine, Adapive idenificaion of linear ime-delay sysems, Inernaional Journal of Robus and Nonlinear Conrol, vol 3, pp , 23 [] S Drakunov, W Perruquei, J-P Richard, and L Belkoura, Delay idenificaion in ime-delay sysems using variable srucure observers, Annual Reviews in Conrol, vol 3, no 2, pp 43 58, 26 [2] L Zhang and G Orosz, Moif-based analysis of conneced vehicle sysems: delay effecs and sabiliy, IEEE Transacion on Inelligen Transporaion Sysems, p acceped, 25 [3] G Orosz, R E Wilson, and G Sépán, Traffic jams: dynamics and conrol, Philosophical Transacions of he Royal Sociey A, vol 368, no 928, pp , 2 [4] G Orosz, J Moehlis, and F Bullo, Delayed car-following dynamics for human and roboic drivers, in Proceedings of he ASME IDETC/CIE Conference ASME, 2, pp , paper no DETC [5] A P Morgan and K S Narendra, On he sabiliy of nonauonomous differenial equaions ẋ = [A + B()]x wih skew symmeric marix B(), SIAM Journal of Conrol and Opimizaion, vol 5, no, pp 63 75, 977 [6] J-J E Sloine and W Li, Applied Nonlinear Conrol Prenice Hall, 99 REFERENCES [] P Seiler, A Pan, and K Hedrick, Disurbance propagaion in vehicle srings, Auomaic Conrol, IEEE Transacions on, vol 49, no, pp , 24 [2] G Orosz, Conneced cruise conrol: modeling, delay effecs, and nonlinear behavior, Vehicle Sysem Dynamics, p submied, 24 [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 ASME, 23, p V3T42A6, paper no DSCC