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Optimized connected cruise control with time delay
Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of the 12th IFAC Workshop on Time Delay Systems June 28-30, 2015. Ann Arbor, USA Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems Proceedings of 12th IFAC Workshop on June 28-30, 2015. Ann Arbor, MI, USA Available online at www.sciencedirect.com June June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USA
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Optimized connected cruise Optimized connected cruise Optimized connected cruise Optimizedwith connected cruise time delay with time delay with time delay with time delay
control control control control
Jin I. a bor Jin I. Ge Ge and and G´ G´ a bor Orosz Orosz Jin a Jin I. I. Ge Ge and and G´ G´ abor bor Orosz Orosz Department of Mechanical Engineering, Department of Mechanical Engineering, Department of Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Mechanical Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA University of Michigan, Ann Arbor, MI 48109, (e-mail: [email protected], [email protected]). University [email protected], Michigan, Ann Arbor, MI 48109, USA USA (e-mail: [email protected]). (e-mail: (e-mail: [email protected], [email protected], [email protected]). [email protected]). Abstract: In this paper, we optimize the connectivity structure for connected cruise control Abstract: In this we the connectivity structure for cruise control Abstract: In this paper, paper, we optimize optimize the connectivity structure for connected connected cruise control (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time Abstract: In this paper, we optimize the connectivity structure for connected cruise control (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time delay. We consider that a CCC-equipped vehicle receives position and velocity signals through (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time delay. We consider that a CCC-equipped vehicle receives position and velocity signals through delay. We consider that a CCC-equipped vehicle receives position and velocity signals through wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize aa delay. We consider that a CCC-equipped vehicle receives position and velocity signals through wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize aa cost function defined by headway and velocity errors and the acceleration of the CCC vehicle wireless vehicle-to-vehicle (V2V) communication fromand multiple vehicles ahead andCCC minimize cost defined by and errors the of vehicle cost function function defined by headway headway and velocity velocity errors and contains the acceleration acceleration of the the CCC vehicle over an infinite horizon. The resulting optimal feedback distributed delay. We show cost function defined by headway and velocity errors and the acceleration of the CCC vehicle over an infinite horizon. The resulting optimal feedback contains distributed delay. We show over an horizon. The resulting optimal feedback We that the feedback gains and the distribution kernels can be becontains obtaineddistributed recursivelydelay. as signals signals from over an infinite infinite horizon. Thethe resulting optimal feedback contains distributed delay. We show show that the feedback gains and distribution kernels can obtained recursively as from that the feedback gains and the distribution kernels can be obtained recursively as signals from vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as the that the feedback gains and the distribution kernels can be obtained recursively as signals from vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as the vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as number of cars between the source and the CCC vehicle increases. The performance of the vehicles farther ahead become available. Moreover, the gains exhibit exponential decay of as the the number of cars between the source and the CCC vehicle increases. The performance number of cars between the source and the CCC vehicle increases. The performance of the developed is verified number of CCC cars controller between the source by andnumerical the CCCsimulations. vehicle increases. The performance of the developed CCC controller is verified by numerical simulations. developed CCC controller is numerical simulations. developed CCC controller Federation is verified verifiedofby by numerical simulations. © 2015, IFAC (International Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Connected cruise control, Optimization, Connectivity structure, Keywords: Connected cruise control, Optimization, Connectivity structure, Time Time delay delay Keywords: Keywords: Connected Connected cruise cruise control, control, Optimization, Optimization, Connectivity Connectivity structure, structure, Time Time delay delay 1. INTRODUCTION ity of traffic systems: information from vehicles farther 1. INTRODUCTION INTRODUCTION ity of of traffic traffic systems: systems: information information from from vehicles vehicles farther farther 1. ity downstream has less influence on the CCC vehicle and 1. INTRODUCTION ity of traffic has systems: information fromCCC vehicles farther downstream less influence on the vehicle and Connected cruise control (CCC) can be used to maintain downstream has less influence on the CCC vehicle and including motion information from vehicles farther ahead downstream has less influence on the CCC vehicle and Connected cruise cruise control control (CCC) (CCC) can can be be used used to to maintain including motion information from vehicles farther ahead Connected smooth traffic flow by exploiting motion information from vehicles ahead does not change feedback laws concerning the signals Connected cruise control (CCC)wireless can be vehicle-to-vehicle used to maintain maintain including including motionthe information vehicles farther farther ahead smooth traffic flow by exploiting wireless vehicle-to-vehicle does not not change change the feedback from laws concerning concerning the signals signals smooth traffic flow by exploiting wireless vehicle-to-vehicle (V2V) communication; see Ge and Orosz (2014a), Zhang does the feedback laws the from closer vehicles. The optimal gains are determined in smoothcommunication; traffic flow by exploiting wireless vehicle-to-vehicle does not change the The feedback lawsgains concerning the signals (V2V) see Ge and Orosz (2014a), Zhang from closer vehicles. optimal are determined in (V2V) communication; see Ge and Orosz (2014a), Zhang and Orosz (2015), Orosz (2014), Qin et al. (2014). The from closer vehicles. The optimal gains are determined in terms of the weighting factors used in the optimization (V2V) communication; see Ge and Orosz (2014a), Zhang from closer vehicles. Thefactors optimalused gains are determined in and Orosz (2015), Orosz (2014), Qin et al. (2014). The terms of the weighting in the optimization and Orosz (2015), Orosz (2014), et (2014). The CCC controller receives motion of terms of the factors used in optimization and the parameters of the non-CCC and Orosz (2015), Oroszinformation (2014), Qin Qinabout et al. al.the (2014). The terms ofdriver the weighting weighting factors used in the thevehicles. optimization CCC controller receives information about the motion of and the driver parameters of the non-CCC vehicles. CCC controller receives information about the of multiple vehicles ahead and assists the human driver or CCC controller about the motion motion of and and the the driver driver parameters parameters of of the the non-CCC non-CCC vehicles. vehicles. multiple vehiclesreceives ahead information and assists assists the the human driver or or multiple vehicles ahead and human driver actuates the vehicle based on these signals. Including every 2. CONNECTED CRUISE CONTROL multiple vehicles ahead and assistssignals. the human driver or actuates the vehicle based on these Including every 2. CONNECTED CRUISE CONTROL actuates the vehicle based on signals. Including every 2. available signal in the feedback loop requires one to tune actuates vehicle based on these these signals. Including 2. CONNECTED CONNECTED CRUISE CRUISE CONTROL CONTROL available the signal in the the feedback loop requires one to toevery tune available signal in feedback loop requires one tune aavailable large number of gains. Thus, it is necessary to optimize We consider a string of n + 1 vehicles on single signal in the feedback loop requires one to tune a large large number number of of gains. gains. Thus, Thus, it it is is necessary necessary to to optimize optimize We consider a string of n + 1 vehicles traveling traveling on on aaa single single athe consider aa string of n vehicles traveling connectivity that weigh the lane as shown in Fig. the vehicle at the tail a large number ofstructure, gains. Thus, itis, is to necessary to available optimize We We consider string of 1(a) n+ + 11where vehicles traveling on a single the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a) where the vehicle at the tail the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a) where the vehicle at the tail motion information in order to determine feedback gains implements a connected cruise control algorithm using the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a)cruise wherecontrol the vehicle at theusing tail motion information in order to determine feedback gains implements a connected algorithm motion information in order to feedback gains aavelocity connected cruise control algorithm using systematically. Ploeg al. (2014) and Wang et al. (2014) position and signals of n preceding vehicles. The motion information inet order to determine determine feedback gains implements implements connected cruise control algorithm using systematically. Ploeg et al. (2014) and Wang et al. (2014) position and velocity signals of n preceding vehicles. The systematically. Ploeg al. (2014) and etsmall al. position and velocity of n vehicles. The discussed optimization vehicle platoons with numof the CCC vehicle is given by systematically. Ploeg et etof (2014) and Wang Wang al. (2014) (2014) position and dynamics velocity signals signals n preceding preceding discussed optimization optimization ofal. vehicle platoons withetsmall small num- longitudinal longitudinal dynamics of the the of CCC vehicle is is vehicles. given by by The discussed of vehicle platoons with numlongitudinal dynamics of CCC vehicle given bers of cars. Ge and Orosz (2014b) discussed optimization ˙ discussed optimization of vehicle platoons withoptimization small num- longitudinal dynamics of=the CCC h v2 (t) − vv1vehicle (t) ,, is given by bers of of cars. cars. Ge and and Orosz Orosz (2014b) discussed h˙˙ 11 (t) (t) − bers (2014b) discussed optimization (1) 1 (t) of connectivity structure aa general scenario where aa bers of cars. Ge Ge and Orosz in (2014b) discussed optimization h (t) = = vu(t) vv22 (t) (t) − v (t) , ˙h˙ 11 (t) (1) 1 of connectivity structure in general scenario where v = , 2 (t), − v1 (t) , (1) of connectivity structure in aa general scenario where aa CCC vehicle receives V2V signals from n preceding vev ˙ (t) = u(t) 1 (1) of connectivity structure in general scenario where CCC vehicle receives V2V signals from n preceding vev ˙ (t) = u(t) , 1 v ˙ (t) = u(t) , where the dot stands for differentiation with respect to CCC vehicle receives V2V signals from n preceding ve1 hicles that are driven by human drivers. However, in these CCC vehicle receives V2V signals from n preceding vewhere the dot stands for differentiation with respect to hicles that that are are driven driven by by human human drivers. drivers. However, However, in in these these where the dot stands for differentiation with respect to is the headway (i.e., the bumper-to-bumper time t, h hicles 1 works driver reaction time delay was omitted, which may where the dot stands for differentiation with respect to hicles that arereaction driven by human drivers. However, in these is the headway (i.e., the bumper-to-bumper time t, h 1 works driver time delay was omitted, which may is the headway (i.e., the bumper-to-bumper time t, h distance between the CCC vehicle and the vehicle imme1 works driver reaction time delay was omitted, which may significantly influence the dynamics of vehicle strings; see is the headway (i.e., the bumper-to-bumper time t, h 1 works driver reaction time delay was omitted, which may between the CCC vehicle and the vehicle immesignificantly influence influence the the dynamics dynamics of of vehicle vehicle strings; strings; see see distance distance between the vehicle and the vehicle immeand is the velocity the CCC vehicle; significantly Orosz et al. (2010), Ge Orosz distanceahead), between thevv11CCC CCC vehicle andof the vehicle immesignificantly influence theand dynamics of(2014a), vehicle Zhang strings;and see diately diately ahead), and is the velocity of the CCC vehicle; Orosz et al. (2010), Ge and Orosz (2014a), Zhang and diately ahead), and vv1u(t) is the the velocity of input the CCC CCC vehicle; see Fig. 1(a). Finally, is the control that will Orosz et al. (2010), Ge and Orosz (2014a), Zhang and (2015). diately ahead), and is velocity of the vehicle; Orosz (2015). et al. (2010), Ge and Orosz (2014a), Zhang and see Fig. 1(a). Finally, 1u(t) is the control input that will be be Orosz see Finally, u(t) the control input that will designed using LQR based the position and velocity of Orosz (2015). see Fig. Fig. 1(a). 1(a). Finally, u(t) is ison the control input that will be be Orosz (2015). designed using LQR based on the position and velocity of In this paper we assume that non-CCC vehicles are human designed using LQR based on the position and velocity of other vehicles. In this paper we assume that non-CCC vehicles are human designed using LQR based on the position and velocity of other vehicles. In this paper we assume that non-CCC vehicles are human driven, and their driver reaction time is modeled as a other vehicles. In this paper we assume that non-CCC vehicles are human driven, and their driver reaction time is modeled as a other vehicles. the dynamics of non-CCC vehicles, we ignore the driven, their driver reaction time is− modeled asas-aa For discrete time delay in the range of 0.4 1 [s]. We driven, and and time the dynamics of non-CCC vehicles, we ignore the discrete timetheir delaydriver in the thereaction range of of 0.4 is− − modeled [s]. We Weasasas- For For the dynamics of non-CCC vehicles, we ignore the driveline and engine dynamics, and use the car-following discrete time delay in range 0.4 11 [s]. sume drivers with identical parameters, but the proposed For the dynamics of non-CCC vehicles, we ignore the discrete time delay in the range of 0.4 − 1 [s]. We asdriveline and engine dynamics, and use the car-following sume drivers with identical parameters, but the proposed driveline and engine dynamics, and use the car-following model sume drivers with identical parameters, but the proposed control design can also be implemented for non-identical driveline and engine dynamics, and use the car-following sume drivers with identical parameters, but the proposed model control design design can can also also be be implemented implemented for for non-identical non-identical model ˙˙ i (t) = vi+1 (t) − vi (t) , control parameters. We assume that the CCC vehicle receives poh control design can also be implemented for non-identical parameters. We assume that the CCC vehicle receives popo- model h = vvi+1 − vvi (t) ,, (t) ˙ i (t) parameters. We assume that the CCC vehicle receives h (t) = = α (t) − sition and velocity signals from n non-CCC vehicles ahead parameters. We assume that then CCC vehicle receives pohv˙˙ ii (t) vi+1 (t) − vii (t) (t) , − vi (t − τ ) (2) V (h (t − τ )) i+1 sition and velocity signals from non-CCC vehicles ahead i (2) v ˙ (t) = α V (h (t − τ )) − v (t − τ ) sition and velocity signals from n non-CCC vehicles ahead V (hii (t − τ )) − vii (t − τ ) (Fig. 1(a)), and formulate the optimization problem using sition and velocity signals from n non-CCC vehicles ahead (2) vv˙˙ ii (t) = α (Fig. 1(a)), and formulate the optimization problem using (2) (t) = α V (h (t − τ )) − v (t − τ ) + β v (t i i − τ ) − vi (ti − τ ) , i+1 (Fig. 1(a)), and formulate the optimization problem using linear quadratic regulation (LQR), where the headway + β v (t − τ ) − v (t − τ ) , (Fig. 1(a)), and formulate the optimization problem using i+1 i linear quadratic regulation (LQR), where the headway + β v (t − τ ) − v (t − τ ) , i+1 i linear quadratic regulation (LQR), where headway + β vi+1 − τv)i − vi (t −the τ ) headway , and velocity fluctuations and the acceleration of the CCC ii = 2, .. .. .. ,, n, where h and and i (t linear quadratic regulation where the the headway and velocity fluctuations and(LQR), the acceleration acceleration of the the CCC for for = 2, n, where h vvi denote denote the headway and i and and velocity fluctuations and the of CCC for i = 2, . . . , n, where h and denote the headway and vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is i i and velocity fluctuations andobtained the acceleration of the CCC for i = 2,of. . the . , n,i-th where hi andsee vi Fig. denote theThe headway and vehicle are penalized. The control law is both velocity vehicle; 1(a). gain α is vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the necessary and sufficient for optimality. We show that the for the difference between the desired velocity and gains of the optimized controller follow the spatial causalactual velocity of the vehicle, while the gain β is for the necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the gains of of the optimized optimized controller follow follow the spatial spatial causal- actual velocity of the vehicle, while the gain β is for the gains gains of the the optimized controller controller follow the the spatial causalcausal- actual actual velocity velocity of of the the vehicle, vehicle, while while the the gain gain β β is is for for the the Copyright IFAC 2015 468 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 2015 468 Copyright IFAC 2015 468 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 468Control. 10.1016/j.ifacol.2015.09.423
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˜˙ 1 (t) = v˜2 (t) − v˜1 (t) , h v˜˙ 1 (t) = u(t) , ˜˙ i (t) = v˜i+1 (t) − v˜i (t) , h ˜ i (t − τ ) − v˜i (t − τ ) v˜˙ i (t) = α f ∗ h + β v˜i+1 (t − τ ) − v˜i (t − τ ) ,
(a) v1
1
v2
2
h1
v3 h2
vn
3
n
V vmax
0 hst
vn+1 hn
469
n+1 (b)
(5)
for i = 2, . . . , n. Here f ∗ = V (h∗ ) is the derivative of the range policy (3) at the equilibrium and for h ∈ (hst , hgo ) the corresponding time headway is th = 1/f ∗ . Here f ∗ = 1 [1/s], yielding the time headway th = 1 [s] (cf. (3)). hgo
h
Fig. 1. (a): A string of n + 1 vehicles with a CCC vehicle at the tail receiving signals from multiple vehicles via V2V communication. (b): The range policy (3). velocity difference between the vehicle and its immediate predecessor. Model (2) considers identical vehicles, but the method proposed here is applicable when considering nonidentical gains αi , βi and delays τi , i = 2, . . . , n. The desired velocity is determined by the headway using the range policy 0, h ≤ hst , h − hst , hst < h < hgo , V (h) = vmax (3) hgo − hst vmax , h ≥ hgo ,
shown in Fig. 1 (b). That is, the desired velocity is zero for small headways (h ≤ hst ) and equal to the maximum speed vmax for large headways (h ≥ hgo ). Between these, the desired velocity increases with the headway linearly. Here, we consider vmax = 30 [m/s], hst = 5 [m], hgo = 35 [m] that corresponds to realistic traffic data. Many other range policies may be chosen, but the qualitative dynamics remain similar if the above characteristics are considered; see Orosz et al. (2010).
T T Let us define the vector X = [xT 1 , . . . , xn ] , where xi = ˜ i , v˜i ]T , and rewrite the system (5) as [h ˙ X(t) = AX(t) + BX(t − τ ) + Du(t) + φ(t) , (6) vn+1 (t − τ )]T denotes the where φ(t) = [0, . . . , 0, v˜n+1 (t), β˜ disturbance and the coefficient matrices are given by 0 0 D1 A0 A 1 A0 A1 B0 B1 0 , B= , D= A= .. .. .. .. ... , . . . .
A0
with block matrices 0 −1 0 1 , A1 = , A0 = 0 0 0 0 0 0 0 0 , B0 = , B1 = αf ∗ −α − β 0 β
B0
0 (7)
(8) 0 D1 = . 1
Note that A, B are upper-triangular because vehicles only react to the motion of vehicles ahead (except in emergency situations that are not considered here). We assume the non-CCC vehicles are plant stable, i.e., they are able to maintain the uniform flow (4) when the vehicles ahead travel with constant speed v ∗ ; see Orosz (2014). Then (6,7) is stabilizable, and we define cost function tf u2 (t) + X T (t)ΓX(t) dt , (9) Jtf (u, X) = 0
3. CCC DESIGN USING LQR In this section, we formulate the CCC design as a linear quadratic regulator (LQR) with delay. Since the CCC vehicle tries to maintain constant velocity and headway without using large acceleration/deceleration, we minimize a cost function containing its headway and velocity fluctuations and its acceleration. The solution gives the gains for the CCC vehicle with respect to the current and delayed headways and velocities of the vehicles ahead. The dynamics of the connected vehicle system (1,2) is assumed to be in the vicinity of equilibrium where all vehicles travel with the same constant velocity and maintain constant headway, that is, hi (t) ≡ h∗ , vi (t) ≡ v ∗ = V (h∗ ) , (4) for i = 1, . . . , n + 1. Here the equilibrium velocity v ∗ is determined by the head vehicle, while the equilibrium headway h∗ can be calculated from the range policy (3). ˜ i (t) = hi (t) − h∗ We define the headway perturbations h and velocity perturbations v˜i (t) = vi (t)−v ∗ , i = 1, . . . , n+ 1, and linearize (1,2) about the equilibrium (4), yielding the delay differential equation (DDE) 469
where the first term corresponds to the magnitude of the CCC vehicle’s acceleration with weighting factor set to 1. The second term corresponds to the fluctuation of vehicles’ headway and velocity around the uniform traffic flow X(t) ≡ 0 with weighting factor
(10) Γ = diag[γ1 , γ2 , . . . , γ2n−1 , γ2n ] ∈ R2n×2n , where γ2k−1 and γ2k are the weights for the headway and velocity errors of the k th vehicle. Since only the CCC vehicle is controlled, the choice of γ2k−1 and γ2k , k = 2, . . . , n does not influence the optimal control input, so we set these coefficients to zero. We define the augmented state Y (t) = [X T (t) 1]T to place the disturbance term φ(t) in (6) into a time-variant coefficient matrix. We also rescale the time using the driver reaction time τ , i.e., tˆ = t/τ , tˆf = tf /τ . By abuse of notation, we drop the hat immediately and obtain ˜ ˜ (t − 1) + Du(t) ˜ Y˙ (t) = A(t)Y (t) + BY , (11)
where the dot denotes the derivative with respect to the rescaled time and τ A τ φ(t) ˜ ˜ = τB 0 , D ˜ = τ D . (12) A(t) = , B 0 0 0 0 0
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The cost function (9) can be rewritten accordingly tf ˜ (t) dt , Jtf (u, Y ) = u2 (t) + Y T (t)ΓY 0 Γ 0 ˜= where Γ . 0 0
(13)
−1
cf. (18), where the matrices P1 , Q1 (θ) are given by
The optimal control input of (11,13) is given by 0 T ˜ Q(t, θ)Y (t + θ)dθ , (14) u(t) = −D P(t)Y (t) + −1
see Kolmanovskii and Myshkis (1992), where the matrices P(t) and Q(t, θ) are obtained by solving the partial differential equation (PDE) ˜ − P(t)D ˜D ˜ T P(t) + Γ ˜ ˙ ˜ T P(t) + P(t)A − P(t) =A + Q(t, 0) + QT (t, 0) , T ˜ − PD ˜D ˜ T Q(t, θ) + R(t, 0, θ) , (∂θ − ∂t )Q(t, θ) = A ˜D ˜ T Q(t, θ) , (∂ξ + ∂θ − ∂t )R(t, ξ, θ) = −QT (t, ξ)D (15) with boundary conditions P(tf ) = 0 , ˜, Q(tf , θ) = 0 , Q(t, −1) = PT B (16) R(tf , ξ, θ) = 0 ,
Thus (19) and (20) lead to the simplified controller 0 u(t) = −τ DT P1 X(t) + Q1 (θ)X(t + θ)dθ , (21)
˜ T Q(t, θ) , R(t, −1, θ) = B
where P(t) is symmetric and RT (t, ξ, θ) = R(t, θ, ξ). Given the structure of coefficient matrices (12), the matrices P(t), Q(t, θ) and R(t, ξ, θ) can be constructed as Q1 Q2 R 1 R2 P1 P2 , Q= , R= , (17) P= P 3 P4 Q3 Q4 R3 R4
where P1 , Q1 , R1 ∈ R2n×2n , P2 , Q2 , R2 ∈ R2n×1 , P3 , Q3 , R3 ∈ R1×2n , and P4 , Q4 , R4 are scalars. Recall that P(t) is symmetric, and thus, P1 (t) = PT 1 (t), T P2 (t) = PT (t). Moreover, R(t, ξ, θ) = R (t, θ, ξ) yields 3 T R1 (t, ξ, θ) = RT 1 (t, θ, ξ), R2 (t, ξ, θ) = R3 (t, θ, ξ). Thus, the optimal controller (14) becomes 0 u(t) = −τ DT P1 (t)X(t) + Q1 (t, θ)X(t + θ)dθ −1 0 Q2 (t, θ)dθ . (18) + P2 (t) + −1
In Appendix A we substitute (17) into (15,16) and discuss the solution in detail; see (A.1,A.3,A.5,A.7). While (18) gives a general solution to the finite-horizon delayed optimization problem with disturbance, it cannot be implemented in real time since the disturbance φ(t) is unknown a priori. Here we ignore the disturbance φ(t) (that is, ignore v˜n+1 (t)) by considering Q2 (t, θ) ≡ 0 . (19) P2 (t) ≡ 0 , This simplification does not influence the stability of the multi-vehicle system, and we will discuss the effects of disturbance briefly in Section 3.3. Since P1 (t), Q1 (t, θ), R1 (t, ξ, θ) are given by (A.1), which is an initial value problem backwards in time, we consider the steady-state solution in t: P1 (t) ≡ P1 , Q1 (t, θ) ≡ Q1 (θ), R1 (t, ξ, θ) ≡ R1 (ξ, θ), (20) which is equivalent to setting time horizon tf → ∞ in the cost function (9). 470
τ AT P1 + τ P1 A − τ 2 P1 DDT P1 + Q1 (0) + QT 1 (0) = −Γ , T 2 T ∂θ Q1 (θ) = τ A − τ P1 DD Q1 (θ) + R1 (0, θ) , T (∂ξ + ∂θ )R1 (ξ, θ) = −τ 2 QT 1 (ξ)DD Q1 (θ) ,
(22)
with boundary conditions
R1 (−1, θ) = τ BT Q1 (θ) ,
Q1 (−1) = τ P1 B ,
(23)
which can be attained by setting tf → ∞ in (A.1,A.2). 3.1 Decomposition of the solution The numerical solution of (22,23) with general A, B, D matrices is given in Ross and Flugge-Lotz (1969). Here we take advantage of the upper-triangular block structure of A, B in order to obtain an analytical solution. Since only the second row in D is non-zero, cf. (7,8), only the second rows of P1 , Q1 (θ) appear in the controller (21). We introduce the notation Q11 (θ) · · · Q1n (θ) P11 · · · P1n .. , .. P1 = ... . . . ... , Q1 (θ) = ... . . Pn1 · · · Pnn Qn1 (θ) · · · Qnn (θ) (24) where Pji , Qji (θ) ∈ R2×2 , i, j = 1, . . . , n, and rewrite (21) as 0 n x (t) + Q (θ)x (t + θ)dθ , P u(t) = −τ DT 1i i 1i i 1 −1
i=1
(25)
˜ i , v˜i ] . This shows that we only need to where xi = [h derive P1i , Q1i (θ), i = 1, . . . , n to construct the controller. Substituting (24) into (22,23), we obtain equations for each block Pij , Qij (θ), Rij (ξ, θ), i, j = 1, . . . , n, which can be solved recursively. Specifically, P11 and Q11 (θ) are given by ˆ T P11 + τ P11 A0 + Q11 (0) + QT (0) + diag[γ1 , γ2 ] = 0 , τA T
0
11
ˆ 0 Q11 (θ) + R11 (0, θ) , ∂θ Q11 (θ) = A T (∂ξ + ∂θ )R11 (ξ, θ) = −τ 2 QT 11 (ξ)DD Q11 (θ) ,
(26)
with boundary conditions Q11 (−1) = 0 ,
R11 (−1, θ) = 0 ,
(27)
where ˆ 0 = τ AT − τ 2 P11 D1 DT . A 0 1 The solution of (26,27) is given by √ √ − γ1 1 γ1 (γ2 + 2 γ1 ) , P11 = √ √ τ − γ1 γ2 + 2 γ1 Q11 (θ) ≡ 0,
R11 (ξ, θ) ≡ 0 .
Then, to obtain P12 , Q12 (θ), Q21 (θ) we need to solve
(28)
(29)
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ˆ 0 P12 + τ P12 A0 + τ P11 A1 + Q12 (0) + QT (0) = 0 , A 21 ˆ ∂θ Q12 (θ) = A0 Q12 (θ) + R12 (0, θ) , 2 T T T ∂θ Q21 (θ) = τ AT 0 − τ P12 D1 D1 Q11 (θ) + τ A0 Q21 (θ)
where ˆ 1i Q
+ RT 12 (θ, 0) ,
T (∂ξ + ∂θ )R12 (ξ, θ) = −τ 2 QT 11 (ξ)DD Q12 (θ) ,
τ QT 21 (θ)B0
0, = τ P12 B0 , τ P B +P 1i 0 1(i−1) B1 ,
i = 1, i = 2, i ≥ 3.
(44)
3.2 Constructing the CCC controller
(30)
with boundary conditions Q12 (−1) = τ P12 B0 , Q21 (−1) = 0 ,
471
(31)
Denote P1i
× × , =− ai bi
× × Q1i (θ) = − , fi (θ) gi (θ)
(45)
R12 (−1, θ) = 0 , R12 (θ, −1) = . Then (30,31) give the solution R12 (ξ, θ) ≡ 0 , (32) Q21 (θ) ≡ 0, while the equations for Q12 (θ) simplify to ˆ 0 Q12 (θ), Q12 (−1) = τ P12 B0 , ∂θ Q12 (θ) = A
where fi (θ) = (ai0 + ai1 (θ + 1))eλ1 (θ+1) + ai2 eλ2 (θ+1) ,
yielding the solution
With the optimized controller (25,45,46), the dynamics of the vehicle string (in rescaled time) is given by the DDE ˜˙ 1 (t) = τ v˜2 (t) − v˜1 (t) , h n ˜ i (t) + bi v˜i (t) v˜˙ 1 (t) = τ 2 ai h
ˆ
Q12 (θ) = τ eA0 (θ+1) P12 B0 . (33) Finally, the equation for P12 simplifies to ˆ0 ˆ 0 P12 + τ P12 A0 + τ P11 A1 + τ eA P12 B0 = 0 , A yielding the solution (34) vec(P12 ) = M0 vec(P11 ) . where vec(·) gives a column vector by stacking the columns of the matrix on the top of each other and (35) M0 = − L−1 (AT 1 ⊗ I) , where 1 ˆ0 ˆ 0 + AT ⊗ I + BT ⊗ e A . (36) L= I⊗A 0 0 τ For P1i , Q1i (θ), i = 3, · · · , n, (22,23,24) yield ˆ 0 P1i + τ P1i A0 + τ P1(i−1) A1 + Q1i (0) + QT (0) = 0 , A i1
ˆ 0 Q1i (θ) + R1i (0, θ) , ∂θ Q1i (θ) = A T T ∂θ Qi1 (θ) = AT 0 Qi1 (θ) + A1 Q(i−1)1 (θ) + R1i (θ, 0) T − PT 1i D1 D1 Q11 (θ) ,
T (∂ξ + ∂θ )R1i (ξ, θ) = −τ 2 QT 11 (ξ)DD Q1i (θ) ,
with boundary conditions Q1i (−1) = τ (P1i B0 + P1(i−1) B1 ) , Qi1 (−1) = 0, T R1i (θ, −1) = τ QT i1 (θ)B0 + Q(i−1)1 (θ)B1 ,
R1i (−1, θ) = 0 . Following the steps above, (37,38) yield R1i (ξ, θ) ≡ 0 , Qi1 (θ) ≡ 0, and ˆ Q1i (θ) = τ eA0 (θ+1) P1i B0 + P1(i−1) B1 , for i = 3, . . . , n, while P1i is given by vec(P1i ) = (M1 )i−2 M0 vec(P11 ) , where ˆ0 T A ), M1 = − L−1 (AT 1 ⊗ I + B1 ⊗ e and L is given in (36).
(37)
(38)
(39) (40) (41) (42)
Note that we can rewrite (29,33,40) in the compact form ˆ ˆ 1i , (43) Q1i (θ) = eA0 (θ+1) Q 471
(46) gi (θ) = (bi0 + bi1 (θ + 1))eλ1 (θ+1) + bi2 eλ2 (θ+1) , ˆ 0 . The expressions for and λ1 , λ2 are the eigenvalues of A ai0 , ai1 , ai2 , bi0 , bi1 , bi2 are given in Appendix B.
+
i=1 0
−1
˜ fi (θ)hi (t + θ) + gi (θ)˜ vi (t + θ) dθ ,
˜˙ i (t) = τ v˜i+1 (t) − v˜i (t) , h ˜ i (t − 1) − (α + β)˜ v˜˙ i (t) = τ αf ∗ h vi (t − 1) + β˜ vi+1 (t − 1) , (47) for i = 2, . . . , n, cf. (5). The optimized gains ai , bi of the CCC controller are related through (41). We found that the eigenvalues of M1 (cf. (42)) are inside the unit circle for realistic values of weighting factors γ1 , γ2 , human gains α, β, and driver reaction time τ . Thus (41) is a contracting map, and consequently, ai , bi decay with the vehicle index i = 2, . . . , n. As an example, we consider γ1 = 1 [1/s2 ], γ2 = 4 [1/s], α = 0.4 [1/s], β = 0.5 [1/s] and τ = 0.4 [s]. In this case, M1 has two zero eigenvalues and two non-zero eigenvalues 0.55 and 0.13. Fig. 2 shows the exponential decay of ai and bi in a (5 + 1) vehicle chain (red circles) and a (10+1) vehicle chain (blue crosses) using the aforementioned parameter values. The exact match between the red circles and the blue crosses for vehicles 2 to 5 demonstrate that the existing optimized gains are not changed by adding feedback terms on vehicles farther away. This corresponds to the fact that the gains a1 , b1 are not influenced by the connectivity structure (cf. (29) and (45)), and ai , bi are calculated recursively using (34) and (41). For the parameters considered above, we have the gains a1 ≈ 2.50 [1/s], b1 ≈ −6.12 [1/s].
Note that Q1i (θ) contains the same distribution matrix ˆ ˆ 0 (cf. (28)) depends eA0 (θ+1) for all i (cf. (43)), where A on the driver reaction time τ and weighting factors γ1 and γ2 through P11 (cf. (29)). The distribution functions fi (θ), gi (θ) are linear combinations of the elements of ˆ eA0 (θ+1) (cf. (B.3) in Appendix B). We plot fi (θ) and gi (θ) for i = 2, . . . , 5 in Fig. 3 using the same parameters as in Fig. 2. Note that f1 (θ) ≡ 0 and g1 (θ) ≡ 0, i.e., the
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1.6
ai
˜ h
bi
10
10
v˜
1.2 0.8
5
5 0.8
0
0
0.4 0.4
0
2
4
6
8
i
10
0
-5
2
4
6
8
i
10
0
Fig. 2. The optimized headway and velocity gains ai , bi , i = 2, . . . , n of the CCC vehicle for a string of (n + 1) vehicles. Red circles correspond to n = 5 while blue crosses are for n = 10. The human gains are set to α = 0.4 [1/s], β = 0.5 [1/s], and the driver reaction time is τ = 0.4 [s]. The weighting factors are γ1 = 1 [1/s2 ], γ2 = 4 [1/s]. 0
0.25
fi
i=2
0.2
i=3
0.15
0.05
i=3
-0.2 -0.3
i=4 i=5
0.1
i=4
-0.1
-0.4
i=2
-0.5 0 -1
-0.8
-0.6
-0.4
-0.2
θ
0
-1
-0.8
-0.6
-0.4
-0.2
0
θ
20
40
60
t
-10
0
20
40
60
t
Fig. 4. Headway and velocity fluctuations of a (5 + 1)-car vehicle string with the same parameters as in Fig. 2. The thick red curves correspond to the CCC vehicle using information from vehicles 2 − 5, while the thick black curves are when the CCC vehicle only uses information from vehicle 2. The thin curves correspond to the non-CCC vehicles. The black dashed curve is the disturbance profile (48).
i=5
gi
-5
-10
Fig. 3. The distribution functions fi (θ), gi (θ) for a (5 + 1)vehicle system with the same human parameters as in Fig. 2. The blue, red, yellow, purple, and green curves correspond to i = 1, . . . , 5, respectively. optimized controller does not use delayed information of CCC vehicle itself. While for each i = 2, . . . , n, fi (θ) and gi (θ) may increase or decrease with θ ∈ [−1, 0], in general the magnitude of fi (θ) and gi (θ) decays with i. Since the relation between LQR and pole assignment can be extended to the delayed cases, the controller derived from LQR places the eigenvalues of the system in the left half of the complex plane, i.e., guarantees stability of the equilibrium. 3.3 Simulations In this section, we test the performance of the proposed CCC controller using numerical simulations. We consider a vehicle string with 5 + 1 vehicles with initial condition X(t) ≡ 0, t ∈ [−1, 0] and a triangular velocity disturbance signal 0 −1 ≤ t ≤ 0 , 0 < t ≤ 8, −t v˜n+1 (t) = t − 16 (48) 8 < t ≤ 24 , −t + 32 24 < t ≤ 32 , 0 t > 32 , see the dashed curve in Fig. 4(b). In Fig. 4, we plot the corresponding headway and velocity perturbations for the system (47) with the same parameter as in Fig. 2. The velocity and headway of the CCC vehicle (red thick curves) converge to the equilibrium. We also show the case when the CCC vehicle loses connectivity with vehicles 3, 4, 5 (black thick curves), i.e., ai = 0, bi = 0, fi (θ) ≡ 472
0, gi (θ) ≡ 0, for i = 3, 4, 5 in (47). In this case, the CCC vehicle only relies on the link from vehicle 2. Although its headway and velocity perturbations still converge to zero, demonstrating the robustness of the proposed controller, the black curves have larger amplitude than the red curves. This shows the potential benefits of having feedback terms from multiple vehicles ahead. Notice that the amplitude of CCC vehicle’s velocity response is smaller than the disturbance input v˜n+1 (t), i.e., the velocity perturbation is attenuated as it reaches the tail vehicle. This property is called string stability, and is highly desirable to avoid amplification of congestion waves; see Seiler et al. (2004), Orosz et al. (2010). Although string stability is not considered in this paper, this property can be achieved by choosing appropriate weighting factors γ 1 , γ2 . 4. CONCLUSION In this paper, we proposed a connected cruise control design where the connectivity structure was optimized using linear quadratic regulation while considering driver reaction time delay. We found that the feedback gains on signals from nearby vehicles were not influenced by dynamics of vehicles farther downstream, and the gains decreased exponentially with the number of cars between the CCC vehicle and the signaling vehicle. The performance of the CCC controller was tested using numerical simulation. The proposed control algorithm can be extended to vehicle systems with heterogeneous driver parameters, and multiple CCC vehicles. When considering non-zero time delay in the wireless communication, the closed-loop system also contains an actuator delay; see Kristic (2009). A detailed discussion on the choice of the design parameters γ1 , γ2 to ensure plant and string stability will be considered in the future. Also, nonlinear CCC algorithms and partial knowledge about the motion of non-CCC vehicles will be investigated. ACKNOWLEDGEMENTS This work was supported by the National Science Foundation (Award Number 1351456).
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REFERENCES Ge, J.I. and Orosz, G. (2014a). Dynamics of connected vehicle systems with delayed acceleration feedback. Transportation Research Part C, 46, 46–64. Ge, J.I. and Orosz, G. (2014b). Optimal control of connected vehicle systems. In Conference on Decision and Control, IEEE, 4107–4112. Kolmanovskii, V. and Myshkis, A. (1992). Applied Theory of Functional Differential Equations. Kluwer Academic Publisher. Kristic, M. (2009). Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhauser. Orosz, G. (2014). Connected cruise control: modeling, delay effects, and nonlinear behavior. Vehicle System Dynamics, submitted. Orosz, G., Wilson, R.E., and St´ep´ an, G. (2010). Traffic jams: dynamics and control. Philosophical Transactions of the Royal Society A, 368(1928), 4455–4479. Ploeg, J., Shukla, D., van de Wouw, N., and Nijmeijer, H. (2014). Controller synthesis for string stability of vehicle platoons. IEEE Transactions on Intelligent Transportation Systems, 15(2), 845–865. Qin, W.B., Gomez, M.M., and Orosz, G. (2014). Stability analysis of connected cruise control with stochastic delays. In Proceedings of American Control Conference, 4624–4629. IEEE. Ross, D. and Flugge-Lotz, I. (1969). An optimal control problem for systems with differential-difference equation dynamics. SIAM Journal on Control, 7(4), 609–623. Seiler, P., Pant, A., and Hedrick, K. (2004). Disturbance propagation in vehicle strings. IEEE Transactions on Automatic Control, 49(10), 1835–1842. Wang, M., Daamen, W., Hoogendoorn, S.P., and van Arem, B. (2014). Rolling horizon control framework for driver assistance systems. part II: Cooperative sensing and cooperative control. Transportation Research Part C, 40, 290–311. Zhang, L. and Orosz, G. (2015). Motif-based analysis of connected vehicle systems: delay effects and stability. IEEE Transactions on Intelligent Transportation Systems, submitted. Appendix A. THE SOLUTION OF LQR PROBLEM WITH DELAY Here we present a detailed solution to the LQR problem with time delay. Since (11,12) is constructed to include disturbance φ(t) in the LQR format, and the optimal controller (18) is given using partitioned matrices (17). We write (15,16) into four groups, where P1 (t), Q1 (t, θ), R1 (t, ξ, θ) are independent from the disturbance and can be solved using only the coefficient matrices A, B, D and the weighting factor Γ. That is, for the first group we obtain the PDE ˙ 1 (t) = τ AT P1 (t) + τ P1 (t)A + Q1 (t, 0) + QT (t, 0) −P 1 + Γ − τ 2 P1 (t)DDT P1 (t) , (∂θ − ∂t )Q1 (t, θ) = τ AT − τ 2 P1 (t)DDT Q1 (t, θ) + R1 (t, 0, θ) ,
T (∂ξ + ∂θ − ∂t )R1 (t, ξ, θ) = −τ 2 QT 1 (t, ξ)DD Q1 (t, θ) , (A.1) with boundary conditions
473
473
P1 (tf ) = 0 , Q1 (tf , θ) = 0 ,
Q1 (t, −1) = τ P1 (t)B , (A.2) R1 (tf , ξ, θ) = 0 , R1 (t, −1, θ) = τ BT Q1 (t, θ) . Using P1 (t) and Q1 (t, θ) obtained from (A.1,A.2), we can calculate Q2 (t, θ) and R2 (t, ξ, θ) by solving (∂θ − ∂t )Q2 (t, θ) = τ AT − τ 2 P1 (t)DDT Q2 (t, θ) + R2 (t, 0, θ) , (A.3) T (∂t − ∂ξ − ∂θ )R2 (t, ξ, θ) = τ 2 QT 1 (t, ξ)DD Q2 (t, θ) , with boundary conditions R2 (tf , ξ, θ) = 0 , Q2 (tf , θ) = 0 , (A.4) Q2 (t, −1) = 0 , R2 (t, −1, θ) = τ BT Q2 (t, θ) . Note that the disturbance φ(t) does not appear in (A.3) either. As a matter of fact, (A.3,A.4) result in Q2 (t, θ) ≡ 0 and R2 (t, ξ, θ) ≡ 0.
The dynamics of P2 (t) and Q3 (t, θ) are driven by the disturbance φ(t): ˙ 2 (t) = τ AT − τ 2 P1 (t)DDT P2 (t) + P1 (t)φ(t) −P + Q2 (t, 0) + QT (t, 0) , T3 T (∂θ − ∂t )Q3 (t, θ) = φ (t) − τ 2 PT Q3 (t, θ) 2 (t)DD
+ RT (A.5) 2 (t, θ, 0) , with boundary conditions Q3 (tf , θ) = 0 , Q3 (t, −1) = τ PT P2 (tf ) = 0 , 2 (t)B . (A.6) Although P4 (t), Q4 (t, θ), R4 (t, ξ, θ) do not appear in the optimal control (18), they appear in the minimal cost function, and are given by the PDE ˙ 4 (t) = τ φT (t)P2 (t) + τ P3 (t)φ(t) + Q4 (t, 0) + QT (t, 0) −P 4
− τ 2 P3 (t)DDT P2 (t) , (∂θ − ∂t )Q4 (t, θ) = τ φT (t) − τ 2 P3 (t)DDT Q4 (t, θ) + R4 (t, 0, θ) ,
T (∂ξ + ∂θ − ∂t )R4 (t, ξ, θ) = −τ 2 QT 2 (t, ξ)DD Q2 (t, θ) , (A.7) with boundary conditions P4 (tf ) = 0 , Q4 (tf , θ) = 0 , Q4 (t, −1) = 0 , (A.8) R4 (t, −1, θ) = 0 . R4 (tf , ξ, θ) = 0 ,
Appendix B. THE DISTRIBUTION FUNCTIONS ˆ
Since Q1i (θ) given in (43,44) contains eA0 (θ+1) , we write ˆ
(B.1) eA0 (θ+1) = KeJ0 (θ+1) K−1 , ˆ where J0 is the Jordan form of A0 , and K is the corresponding transformation matrix. Denote the eigenvalues ˆ 0 as λ1 , λ2 , then of A ˜b11 ˜b12 ˆ 0 (θ+1) ˜12 a ˜11 a A + (θ + 1) ˜ ˜ eλ1 (θ+1) = e a ˜21 a ˜22 b21 b22 c˜11 c˜12 λ2 (θ+1) + , (B.2) e c˜21 c˜22 where ˜bij = 0 when K contains generalized eigenvectors. Substituting (B.2) into (43,44), we obtain (45,46) with ˆ 1i , ˆ 1i , a21 a ˜22 ]Q [ai1 bi1 ] = [˜b21 ˜b22 ]Q [ai0 bi0 ] = [˜ ˆ 1i . [ai2 bi2 ] = [˜ c21 c˜22 ]Q
(B.3)
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Optimized connected cruise Optimized connected cruise Optimized connected cruise Optimizedwith connected cruise time delay with time delay with time delay with time delay
control control control control
Jin I. a bor Jin I. Ge Ge and and G´ G´ a bor Orosz Orosz Jin a Jin I. I. Ge Ge and and G´ G´ abor bor Orosz Orosz Department of Mechanical Engineering, Department of Mechanical Engineering, Department of Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Mechanical Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA University of Michigan, Ann Arbor, MI 48109, (e-mail: [email protected], [email protected]). University [email protected], Michigan, Ann Arbor, MI 48109, USA USA (e-mail: [email protected]). (e-mail: (e-mail: [email protected], [email protected], [email protected]). [email protected]). Abstract: In this paper, we optimize the connectivity structure for connected cruise control Abstract: In this we the connectivity structure for cruise control Abstract: In this paper, paper, we optimize optimize the connectivity structure for connected connected cruise control (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time Abstract: In this paper, we optimize the connectivity structure for connected cruise control (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time delay. We consider that a CCC-equipped vehicle receives position and velocity signals through (CCC) using linear quadratic regulation (LQR) while taking into account driver reaction time delay. We consider that a CCC-equipped vehicle receives position and velocity signals through delay. We consider that a CCC-equipped vehicle receives position and velocity signals through wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize aa delay. We consider that a CCC-equipped vehicle receives position and velocity signals through wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize wireless vehicle-to-vehicle (V2V) communication from multiple vehicles ahead and minimize aa cost function defined by headway and velocity errors and the acceleration of the CCC vehicle wireless vehicle-to-vehicle (V2V) communication fromand multiple vehicles ahead andCCC minimize cost defined by and errors the of vehicle cost function function defined by headway headway and velocity velocity errors and contains the acceleration acceleration of the the CCC vehicle over an infinite horizon. The resulting optimal feedback distributed delay. We show cost function defined by headway and velocity errors and the acceleration of the CCC vehicle over an infinite horizon. The resulting optimal feedback contains distributed delay. We show over an horizon. The resulting optimal feedback We that the feedback gains and the distribution kernels can be becontains obtaineddistributed recursivelydelay. as signals signals from over an infinite infinite horizon. Thethe resulting optimal feedback contains distributed delay. We show show that the feedback gains and distribution kernels can obtained recursively as from that the feedback gains and the distribution kernels can be obtained recursively as signals from vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as the that the feedback gains and the distribution kernels can be obtained recursively as signals from vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as the vehicles farther ahead become available. Moreover, the gains exhibit exponential decay as number of cars between the source and the CCC vehicle increases. The performance of the vehicles farther ahead become available. Moreover, the gains exhibit exponential decay of as the the number of cars between the source and the CCC vehicle increases. The performance number of cars between the source and the CCC vehicle increases. The performance of the developed is verified number of CCC cars controller between the source by andnumerical the CCCsimulations. vehicle increases. The performance of the developed CCC controller is verified by numerical simulations. developed CCC controller is numerical simulations. developed CCC controller Federation is verified verifiedofby by numerical simulations. © 2015, IFAC (International Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Connected cruise control, Optimization, Connectivity structure, Keywords: Connected cruise control, Optimization, Connectivity structure, Time Time delay delay Keywords: Keywords: Connected Connected cruise cruise control, control, Optimization, Optimization, Connectivity Connectivity structure, structure, Time Time delay delay 1. INTRODUCTION ity of traffic systems: information from vehicles farther 1. INTRODUCTION INTRODUCTION ity of of traffic traffic systems: systems: information information from from vehicles vehicles farther farther 1. ity downstream has less influence on the CCC vehicle and 1. INTRODUCTION ity of traffic has systems: information fromCCC vehicles farther downstream less influence on the vehicle and Connected cruise control (CCC) can be used to maintain downstream has less influence on the CCC vehicle and including motion information from vehicles farther ahead downstream has less influence on the CCC vehicle and Connected cruise cruise control control (CCC) (CCC) can can be be used used to to maintain including motion information from vehicles farther ahead Connected smooth traffic flow by exploiting motion information from vehicles ahead does not change feedback laws concerning the signals Connected cruise control (CCC)wireless can be vehicle-to-vehicle used to maintain maintain including including motionthe information vehicles farther farther ahead smooth traffic flow by exploiting wireless vehicle-to-vehicle does not not change change the feedback from laws concerning concerning the signals signals smooth traffic flow by exploiting wireless vehicle-to-vehicle (V2V) communication; see Ge and Orosz (2014a), Zhang does the feedback laws the from closer vehicles. The optimal gains are determined in smoothcommunication; traffic flow by exploiting wireless vehicle-to-vehicle does not change the The feedback lawsgains concerning the signals (V2V) see Ge and Orosz (2014a), Zhang from closer vehicles. optimal are determined in (V2V) communication; see Ge and Orosz (2014a), Zhang and Orosz (2015), Orosz (2014), Qin et al. (2014). The from closer vehicles. The optimal gains are determined in terms of the weighting factors used in the optimization (V2V) communication; see Ge and Orosz (2014a), Zhang from closer vehicles. Thefactors optimalused gains are determined in and Orosz (2015), Orosz (2014), Qin et al. (2014). The terms of the weighting in the optimization and Orosz (2015), Orosz (2014), et (2014). The CCC controller receives motion of terms of the factors used in optimization and the parameters of the non-CCC and Orosz (2015), Oroszinformation (2014), Qin Qinabout et al. al.the (2014). The terms ofdriver the weighting weighting factors used in the thevehicles. optimization CCC controller receives information about the motion of and the driver parameters of the non-CCC vehicles. CCC controller receives information about the of multiple vehicles ahead and assists the human driver or CCC controller about the motion motion of and and the the driver driver parameters parameters of of the the non-CCC non-CCC vehicles. vehicles. multiple vehiclesreceives ahead information and assists assists the the human driver or or multiple vehicles ahead and human driver actuates the vehicle based on these signals. Including every 2. CONNECTED CRUISE CONTROL multiple vehicles ahead and assistssignals. the human driver or actuates the vehicle based on these Including every 2. CONNECTED CRUISE CONTROL actuates the vehicle based on signals. Including every 2. available signal in the feedback loop requires one to tune actuates vehicle based on these these signals. Including 2. CONNECTED CONNECTED CRUISE CRUISE CONTROL CONTROL available the signal in the the feedback loop requires one to toevery tune available signal in feedback loop requires one tune aavailable large number of gains. Thus, it is necessary to optimize We consider a string of n + 1 vehicles on single signal in the feedback loop requires one to tune a large large number number of of gains. gains. Thus, Thus, it it is is necessary necessary to to optimize optimize We consider a string of n + 1 vehicles traveling traveling on on aaa single single athe consider aa string of n vehicles traveling connectivity that weigh the lane as shown in Fig. the vehicle at the tail a large number ofstructure, gains. Thus, itis, is to necessary to available optimize We We consider string of 1(a) n+ + 11where vehicles traveling on a single the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a) where the vehicle at the tail the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a) where the vehicle at the tail motion information in order to determine feedback gains implements a connected cruise control algorithm using the connectivity structure, that is, to weigh the available lane as shown in Fig. 1(a)cruise wherecontrol the vehicle at theusing tail motion information in order to determine feedback gains implements a connected algorithm motion information in order to feedback gains aavelocity connected cruise control algorithm using systematically. Ploeg al. (2014) and Wang et al. (2014) position and signals of n preceding vehicles. The motion information inet order to determine determine feedback gains implements implements connected cruise control algorithm using systematically. Ploeg et al. (2014) and Wang et al. (2014) position and velocity signals of n preceding vehicles. The systematically. Ploeg al. (2014) and etsmall al. position and velocity of n vehicles. The discussed optimization vehicle platoons with numof the CCC vehicle is given by systematically. Ploeg et etof (2014) and Wang Wang al. (2014) (2014) position and dynamics velocity signals signals n preceding preceding discussed optimization optimization ofal. vehicle platoons withetsmall small num- longitudinal longitudinal dynamics of the the of CCC vehicle is is vehicles. given by by The discussed of vehicle platoons with numlongitudinal dynamics of CCC vehicle given bers of cars. Ge and Orosz (2014b) discussed optimization ˙ discussed optimization of vehicle platoons withoptimization small num- longitudinal dynamics of=the CCC h v2 (t) − vv1vehicle (t) ,, is given by bers of of cars. cars. Ge and and Orosz Orosz (2014b) discussed h˙˙ 11 (t) (t) − bers (2014b) discussed optimization (1) 1 (t) of connectivity structure aa general scenario where aa bers of cars. Ge Ge and Orosz in (2014b) discussed optimization h (t) = = vu(t) vv22 (t) (t) − v (t) , ˙h˙ 11 (t) (1) 1 of connectivity structure in general scenario where v = , 2 (t), − v1 (t) , (1) of connectivity structure in aa general scenario where aa CCC vehicle receives V2V signals from n preceding vev ˙ (t) = u(t) 1 (1) of connectivity structure in general scenario where CCC vehicle receives V2V signals from n preceding vev ˙ (t) = u(t) , 1 v ˙ (t) = u(t) , where the dot stands for differentiation with respect to CCC vehicle receives V2V signals from n preceding ve1 hicles that are driven by human drivers. However, in these CCC vehicle receives V2V signals from n preceding vewhere the dot stands for differentiation with respect to hicles that that are are driven driven by by human human drivers. drivers. However, However, in in these these where the dot stands for differentiation with respect to is the headway (i.e., the bumper-to-bumper time t, h hicles 1 works driver reaction time delay was omitted, which may where the dot stands for differentiation with respect to hicles that arereaction driven by human drivers. However, in these is the headway (i.e., the bumper-to-bumper time t, h 1 works driver time delay was omitted, which may is the headway (i.e., the bumper-to-bumper time t, h distance between the CCC vehicle and the vehicle imme1 works driver reaction time delay was omitted, which may significantly influence the dynamics of vehicle strings; see is the headway (i.e., the bumper-to-bumper time t, h 1 works driver reaction time delay was omitted, which may between the CCC vehicle and the vehicle immesignificantly influence influence the the dynamics dynamics of of vehicle vehicle strings; strings; see see distance distance between the vehicle and the vehicle immeand is the velocity the CCC vehicle; significantly Orosz et al. (2010), Ge Orosz distanceahead), between thevv11CCC CCC vehicle andof the vehicle immesignificantly influence theand dynamics of(2014a), vehicle Zhang strings;and see diately diately ahead), and is the velocity of the CCC vehicle; Orosz et al. (2010), Ge and Orosz (2014a), Zhang and diately ahead), and vv1u(t) is the the velocity of input the CCC CCC vehicle; see Fig. 1(a). Finally, is the control that will Orosz et al. (2010), Ge and Orosz (2014a), Zhang and (2015). diately ahead), and is velocity of the vehicle; Orosz (2015). et al. (2010), Ge and Orosz (2014a), Zhang and see Fig. 1(a). Finally, 1u(t) is the control input that will be be Orosz see Finally, u(t) the control input that will designed using LQR based the position and velocity of Orosz (2015). see Fig. Fig. 1(a). 1(a). Finally, u(t) is ison the control input that will be be Orosz (2015). designed using LQR based on the position and velocity of In this paper we assume that non-CCC vehicles are human designed using LQR based on the position and velocity of other vehicles. In this paper we assume that non-CCC vehicles are human designed using LQR based on the position and velocity of other vehicles. In this paper we assume that non-CCC vehicles are human driven, and their driver reaction time is modeled as a other vehicles. In this paper we assume that non-CCC vehicles are human driven, and their driver reaction time is modeled as a other vehicles. the dynamics of non-CCC vehicles, we ignore the driven, their driver reaction time is− modeled asas-aa For discrete time delay in the range of 0.4 1 [s]. We driven, and and time the dynamics of non-CCC vehicles, we ignore the discrete timetheir delaydriver in the thereaction range of of 0.4 is− − modeled [s]. We Weasasas- For For the dynamics of non-CCC vehicles, we ignore the driveline and engine dynamics, and use the car-following discrete time delay in range 0.4 11 [s]. sume drivers with identical parameters, but the proposed For the dynamics of non-CCC vehicles, we ignore the discrete time delay in the range of 0.4 − 1 [s]. We asdriveline and engine dynamics, and use the car-following sume drivers with identical parameters, but the proposed driveline and engine dynamics, and use the car-following model sume drivers with identical parameters, but the proposed control design can also be implemented for non-identical driveline and engine dynamics, and use the car-following sume drivers with identical parameters, but the proposed model control design design can can also also be be implemented implemented for for non-identical non-identical model ˙˙ i (t) = vi+1 (t) − vi (t) , control parameters. We assume that the CCC vehicle receives poh control design can also be implemented for non-identical parameters. We assume that the CCC vehicle receives popo- model h = vvi+1 − vvi (t) ,, (t) ˙ i (t) parameters. We assume that the CCC vehicle receives h (t) = = α (t) − sition and velocity signals from n non-CCC vehicles ahead parameters. We assume that then CCC vehicle receives pohv˙˙ ii (t) vi+1 (t) − vii (t) (t) , − vi (t − τ ) (2) V (h (t − τ )) i+1 sition and velocity signals from non-CCC vehicles ahead i (2) v ˙ (t) = α V (h (t − τ )) − v (t − τ ) sition and velocity signals from n non-CCC vehicles ahead V (hii (t − τ )) − vii (t − τ ) (Fig. 1(a)), and formulate the optimization problem using sition and velocity signals from n non-CCC vehicles ahead (2) vv˙˙ ii (t) = α (Fig. 1(a)), and formulate the optimization problem using (2) (t) = α V (h (t − τ )) − v (t − τ ) + β v (t i i − τ ) − vi (ti − τ ) , i+1 (Fig. 1(a)), and formulate the optimization problem using linear quadratic regulation (LQR), where the headway + β v (t − τ ) − v (t − τ ) , (Fig. 1(a)), and formulate the optimization problem using i+1 i linear quadratic regulation (LQR), where the headway + β v (t − τ ) − v (t − τ ) , i+1 i linear quadratic regulation (LQR), where headway + β vi+1 − τv)i − vi (t −the τ ) headway , and velocity fluctuations and the acceleration of the CCC ii = 2, .. .. .. ,, n, where h and and i (t linear quadratic regulation where the the headway and velocity fluctuations and(LQR), the acceleration acceleration of the the CCC for for = 2, n, where h vvi denote denote the headway and i and and velocity fluctuations and the of CCC for i = 2, . . . , n, where h and denote the headway and vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is i i and velocity fluctuations andobtained the acceleration of the CCC for i = 2,of. . the . , n,i-th where hi andsee vi Fig. denote theThe headway and vehicle are penalized. The control law is both velocity vehicle; 1(a). gain α is vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the vehicle are penalized. The obtained control law is both velocity of the i-th vehicle; see Fig. 1(a). The gain α is necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the necessary and sufficient for optimality. We show that the for the difference between the desired velocity and gains of the optimized controller follow the spatial causalactual velocity of the vehicle, while the gain β is for the necessary and sufficient for optimality. We show that the for the difference between the desired velocity and the gains of of the optimized optimized controller follow follow the spatial spatial causal- actual velocity of the vehicle, while the gain β is for the gains gains of the the optimized controller controller follow the the spatial causalcausal- actual actual velocity velocity of of the the vehicle, vehicle, while while the the gain gain β β is is for for the the Copyright IFAC 2015 468 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 2015 468 Copyright IFAC 2015 468 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 468Control. 10.1016/j.ifacol.2015.09.423
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˜˙ 1 (t) = v˜2 (t) − v˜1 (t) , h v˜˙ 1 (t) = u(t) , ˜˙ i (t) = v˜i+1 (t) − v˜i (t) , h ˜ i (t − τ ) − v˜i (t − τ ) v˜˙ i (t) = α f ∗ h + β v˜i+1 (t − τ ) − v˜i (t − τ ) ,
(a) v1
1
v2
2
h1
v3 h2
vn
3
n
V vmax
0 hst
vn+1 hn
469
n+1 (b)
(5)
for i = 2, . . . , n. Here f ∗ = V (h∗ ) is the derivative of the range policy (3) at the equilibrium and for h ∈ (hst , hgo ) the corresponding time headway is th = 1/f ∗ . Here f ∗ = 1 [1/s], yielding the time headway th = 1 [s] (cf. (3)). hgo
h
Fig. 1. (a): A string of n + 1 vehicles with a CCC vehicle at the tail receiving signals from multiple vehicles via V2V communication. (b): The range policy (3). velocity difference between the vehicle and its immediate predecessor. Model (2) considers identical vehicles, but the method proposed here is applicable when considering nonidentical gains αi , βi and delays τi , i = 2, . . . , n. The desired velocity is determined by the headway using the range policy 0, h ≤ hst , h − hst , hst < h < hgo , V (h) = vmax (3) hgo − hst vmax , h ≥ hgo ,
shown in Fig. 1 (b). That is, the desired velocity is zero for small headways (h ≤ hst ) and equal to the maximum speed vmax for large headways (h ≥ hgo ). Between these, the desired velocity increases with the headway linearly. Here, we consider vmax = 30 [m/s], hst = 5 [m], hgo = 35 [m] that corresponds to realistic traffic data. Many other range policies may be chosen, but the qualitative dynamics remain similar if the above characteristics are considered; see Orosz et al. (2010).
T T Let us define the vector X = [xT 1 , . . . , xn ] , where xi = ˜ i , v˜i ]T , and rewrite the system (5) as [h ˙ X(t) = AX(t) + BX(t − τ ) + Du(t) + φ(t) , (6) vn+1 (t − τ )]T denotes the where φ(t) = [0, . . . , 0, v˜n+1 (t), β˜ disturbance and the coefficient matrices are given by 0 0 D1 A0 A 1 A0 A1 B0 B1 0 , B= , D= A= .. .. .. .. ... , . . . .
A0
with block matrices 0 −1 0 1 , A1 = , A0 = 0 0 0 0 0 0 0 0 , B0 = , B1 = αf ∗ −α − β 0 β
B0
0 (7)
(8) 0 D1 = . 1
Note that A, B are upper-triangular because vehicles only react to the motion of vehicles ahead (except in emergency situations that are not considered here). We assume the non-CCC vehicles are plant stable, i.e., they are able to maintain the uniform flow (4) when the vehicles ahead travel with constant speed v ∗ ; see Orosz (2014). Then (6,7) is stabilizable, and we define cost function tf u2 (t) + X T (t)ΓX(t) dt , (9) Jtf (u, X) = 0
3. CCC DESIGN USING LQR In this section, we formulate the CCC design as a linear quadratic regulator (LQR) with delay. Since the CCC vehicle tries to maintain constant velocity and headway without using large acceleration/deceleration, we minimize a cost function containing its headway and velocity fluctuations and its acceleration. The solution gives the gains for the CCC vehicle with respect to the current and delayed headways and velocities of the vehicles ahead. The dynamics of the connected vehicle system (1,2) is assumed to be in the vicinity of equilibrium where all vehicles travel with the same constant velocity and maintain constant headway, that is, hi (t) ≡ h∗ , vi (t) ≡ v ∗ = V (h∗ ) , (4) for i = 1, . . . , n + 1. Here the equilibrium velocity v ∗ is determined by the head vehicle, while the equilibrium headway h∗ can be calculated from the range policy (3). ˜ i (t) = hi (t) − h∗ We define the headway perturbations h and velocity perturbations v˜i (t) = vi (t)−v ∗ , i = 1, . . . , n+ 1, and linearize (1,2) about the equilibrium (4), yielding the delay differential equation (DDE) 469
where the first term corresponds to the magnitude of the CCC vehicle’s acceleration with weighting factor set to 1. The second term corresponds to the fluctuation of vehicles’ headway and velocity around the uniform traffic flow X(t) ≡ 0 with weighting factor
(10) Γ = diag[γ1 , γ2 , . . . , γ2n−1 , γ2n ] ∈ R2n×2n , where γ2k−1 and γ2k are the weights for the headway and velocity errors of the k th vehicle. Since only the CCC vehicle is controlled, the choice of γ2k−1 and γ2k , k = 2, . . . , n does not influence the optimal control input, so we set these coefficients to zero. We define the augmented state Y (t) = [X T (t) 1]T to place the disturbance term φ(t) in (6) into a time-variant coefficient matrix. We also rescale the time using the driver reaction time τ , i.e., tˆ = t/τ , tˆf = tf /τ . By abuse of notation, we drop the hat immediately and obtain ˜ ˜ (t − 1) + Du(t) ˜ Y˙ (t) = A(t)Y (t) + BY , (11)
where the dot denotes the derivative with respect to the rescaled time and τ A τ φ(t) ˜ ˜ = τB 0 , D ˜ = τ D . (12) A(t) = , B 0 0 0 0 0
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The cost function (9) can be rewritten accordingly tf ˜ (t) dt , Jtf (u, Y ) = u2 (t) + Y T (t)ΓY 0 Γ 0 ˜= where Γ . 0 0
(13)
−1
cf. (18), where the matrices P1 , Q1 (θ) are given by
The optimal control input of (11,13) is given by 0 T ˜ Q(t, θ)Y (t + θ)dθ , (14) u(t) = −D P(t)Y (t) + −1
see Kolmanovskii and Myshkis (1992), where the matrices P(t) and Q(t, θ) are obtained by solving the partial differential equation (PDE) ˜ − P(t)D ˜D ˜ T P(t) + Γ ˜ ˙ ˜ T P(t) + P(t)A − P(t) =A + Q(t, 0) + QT (t, 0) , T ˜ − PD ˜D ˜ T Q(t, θ) + R(t, 0, θ) , (∂θ − ∂t )Q(t, θ) = A ˜D ˜ T Q(t, θ) , (∂ξ + ∂θ − ∂t )R(t, ξ, θ) = −QT (t, ξ)D (15) with boundary conditions P(tf ) = 0 , ˜, Q(tf , θ) = 0 , Q(t, −1) = PT B (16) R(tf , ξ, θ) = 0 ,
Thus (19) and (20) lead to the simplified controller 0 u(t) = −τ DT P1 X(t) + Q1 (θ)X(t + θ)dθ , (21)
˜ T Q(t, θ) , R(t, −1, θ) = B
where P(t) is symmetric and RT (t, ξ, θ) = R(t, θ, ξ). Given the structure of coefficient matrices (12), the matrices P(t), Q(t, θ) and R(t, ξ, θ) can be constructed as Q1 Q2 R 1 R2 P1 P2 , Q= , R= , (17) P= P 3 P4 Q3 Q4 R3 R4
where P1 , Q1 , R1 ∈ R2n×2n , P2 , Q2 , R2 ∈ R2n×1 , P3 , Q3 , R3 ∈ R1×2n , and P4 , Q4 , R4 are scalars. Recall that P(t) is symmetric, and thus, P1 (t) = PT 1 (t), T P2 (t) = PT (t). Moreover, R(t, ξ, θ) = R (t, θ, ξ) yields 3 T R1 (t, ξ, θ) = RT 1 (t, θ, ξ), R2 (t, ξ, θ) = R3 (t, θ, ξ). Thus, the optimal controller (14) becomes 0 u(t) = −τ DT P1 (t)X(t) + Q1 (t, θ)X(t + θ)dθ −1 0 Q2 (t, θ)dθ . (18) + P2 (t) + −1
In Appendix A we substitute (17) into (15,16) and discuss the solution in detail; see (A.1,A.3,A.5,A.7). While (18) gives a general solution to the finite-horizon delayed optimization problem with disturbance, it cannot be implemented in real time since the disturbance φ(t) is unknown a priori. Here we ignore the disturbance φ(t) (that is, ignore v˜n+1 (t)) by considering Q2 (t, θ) ≡ 0 . (19) P2 (t) ≡ 0 , This simplification does not influence the stability of the multi-vehicle system, and we will discuss the effects of disturbance briefly in Section 3.3. Since P1 (t), Q1 (t, θ), R1 (t, ξ, θ) are given by (A.1), which is an initial value problem backwards in time, we consider the steady-state solution in t: P1 (t) ≡ P1 , Q1 (t, θ) ≡ Q1 (θ), R1 (t, ξ, θ) ≡ R1 (ξ, θ), (20) which is equivalent to setting time horizon tf → ∞ in the cost function (9). 470
τ AT P1 + τ P1 A − τ 2 P1 DDT P1 + Q1 (0) + QT 1 (0) = −Γ , T 2 T ∂θ Q1 (θ) = τ A − τ P1 DD Q1 (θ) + R1 (0, θ) , T (∂ξ + ∂θ )R1 (ξ, θ) = −τ 2 QT 1 (ξ)DD Q1 (θ) ,
(22)
with boundary conditions
R1 (−1, θ) = τ BT Q1 (θ) ,
Q1 (−1) = τ P1 B ,
(23)
which can be attained by setting tf → ∞ in (A.1,A.2). 3.1 Decomposition of the solution The numerical solution of (22,23) with general A, B, D matrices is given in Ross and Flugge-Lotz (1969). Here we take advantage of the upper-triangular block structure of A, B in order to obtain an analytical solution. Since only the second row in D is non-zero, cf. (7,8), only the second rows of P1 , Q1 (θ) appear in the controller (21). We introduce the notation Q11 (θ) · · · Q1n (θ) P11 · · · P1n .. , .. P1 = ... . . . ... , Q1 (θ) = ... . . Pn1 · · · Pnn Qn1 (θ) · · · Qnn (θ) (24) where Pji , Qji (θ) ∈ R2×2 , i, j = 1, . . . , n, and rewrite (21) as 0 n x (t) + Q (θ)x (t + θ)dθ , P u(t) = −τ DT 1i i 1i i 1 −1
i=1
(25)
˜ i , v˜i ] . This shows that we only need to where xi = [h derive P1i , Q1i (θ), i = 1, . . . , n to construct the controller. Substituting (24) into (22,23), we obtain equations for each block Pij , Qij (θ), Rij (ξ, θ), i, j = 1, . . . , n, which can be solved recursively. Specifically, P11 and Q11 (θ) are given by ˆ T P11 + τ P11 A0 + Q11 (0) + QT (0) + diag[γ1 , γ2 ] = 0 , τA T
0
11
ˆ 0 Q11 (θ) + R11 (0, θ) , ∂θ Q11 (θ) = A T (∂ξ + ∂θ )R11 (ξ, θ) = −τ 2 QT 11 (ξ)DD Q11 (θ) ,
(26)
with boundary conditions Q11 (−1) = 0 ,
R11 (−1, θ) = 0 ,
(27)
where ˆ 0 = τ AT − τ 2 P11 D1 DT . A 0 1 The solution of (26,27) is given by √ √ − γ1 1 γ1 (γ2 + 2 γ1 ) , P11 = √ √ τ − γ1 γ2 + 2 γ1 Q11 (θ) ≡ 0,
R11 (ξ, θ) ≡ 0 .
Then, to obtain P12 , Q12 (θ), Q21 (θ) we need to solve
(28)
(29)
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ˆ 0 P12 + τ P12 A0 + τ P11 A1 + Q12 (0) + QT (0) = 0 , A 21 ˆ ∂θ Q12 (θ) = A0 Q12 (θ) + R12 (0, θ) , 2 T T T ∂θ Q21 (θ) = τ AT 0 − τ P12 D1 D1 Q11 (θ) + τ A0 Q21 (θ)
where ˆ 1i Q
+ RT 12 (θ, 0) ,
T (∂ξ + ∂θ )R12 (ξ, θ) = −τ 2 QT 11 (ξ)DD Q12 (θ) ,
τ QT 21 (θ)B0
0, = τ P12 B0 , τ P B +P 1i 0 1(i−1) B1 ,
i = 1, i = 2, i ≥ 3.
(44)
3.2 Constructing the CCC controller
(30)
with boundary conditions Q12 (−1) = τ P12 B0 , Q21 (−1) = 0 ,
471
(31)
Denote P1i
× × , =− ai bi
× × Q1i (θ) = − , fi (θ) gi (θ)
(45)
R12 (−1, θ) = 0 , R12 (θ, −1) = . Then (30,31) give the solution R12 (ξ, θ) ≡ 0 , (32) Q21 (θ) ≡ 0, while the equations for Q12 (θ) simplify to ˆ 0 Q12 (θ), Q12 (−1) = τ P12 B0 , ∂θ Q12 (θ) = A
where fi (θ) = (ai0 + ai1 (θ + 1))eλ1 (θ+1) + ai2 eλ2 (θ+1) ,
yielding the solution
With the optimized controller (25,45,46), the dynamics of the vehicle string (in rescaled time) is given by the DDE ˜˙ 1 (t) = τ v˜2 (t) − v˜1 (t) , h n ˜ i (t) + bi v˜i (t) v˜˙ 1 (t) = τ 2 ai h
ˆ
Q12 (θ) = τ eA0 (θ+1) P12 B0 . (33) Finally, the equation for P12 simplifies to ˆ0 ˆ 0 P12 + τ P12 A0 + τ P11 A1 + τ eA P12 B0 = 0 , A yielding the solution (34) vec(P12 ) = M0 vec(P11 ) . where vec(·) gives a column vector by stacking the columns of the matrix on the top of each other and (35) M0 = − L−1 (AT 1 ⊗ I) , where 1 ˆ0 ˆ 0 + AT ⊗ I + BT ⊗ e A . (36) L= I⊗A 0 0 τ For P1i , Q1i (θ), i = 3, · · · , n, (22,23,24) yield ˆ 0 P1i + τ P1i A0 + τ P1(i−1) A1 + Q1i (0) + QT (0) = 0 , A i1
ˆ 0 Q1i (θ) + R1i (0, θ) , ∂θ Q1i (θ) = A T T ∂θ Qi1 (θ) = AT 0 Qi1 (θ) + A1 Q(i−1)1 (θ) + R1i (θ, 0) T − PT 1i D1 D1 Q11 (θ) ,
T (∂ξ + ∂θ )R1i (ξ, θ) = −τ 2 QT 11 (ξ)DD Q1i (θ) ,
with boundary conditions Q1i (−1) = τ (P1i B0 + P1(i−1) B1 ) , Qi1 (−1) = 0, T R1i (θ, −1) = τ QT i1 (θ)B0 + Q(i−1)1 (θ)B1 ,
R1i (−1, θ) = 0 . Following the steps above, (37,38) yield R1i (ξ, θ) ≡ 0 , Qi1 (θ) ≡ 0, and ˆ Q1i (θ) = τ eA0 (θ+1) P1i B0 + P1(i−1) B1 , for i = 3, . . . , n, while P1i is given by vec(P1i ) = (M1 )i−2 M0 vec(P11 ) , where ˆ0 T A ), M1 = − L−1 (AT 1 ⊗ I + B1 ⊗ e and L is given in (36).
(37)
(38)
(39) (40) (41) (42)
Note that we can rewrite (29,33,40) in the compact form ˆ ˆ 1i , (43) Q1i (θ) = eA0 (θ+1) Q 471
(46) gi (θ) = (bi0 + bi1 (θ + 1))eλ1 (θ+1) + bi2 eλ2 (θ+1) , ˆ 0 . The expressions for and λ1 , λ2 are the eigenvalues of A ai0 , ai1 , ai2 , bi0 , bi1 , bi2 are given in Appendix B.
+
i=1 0
−1
˜ fi (θ)hi (t + θ) + gi (θ)˜ vi (t + θ) dθ ,
˜˙ i (t) = τ v˜i+1 (t) − v˜i (t) , h ˜ i (t − 1) − (α + β)˜ v˜˙ i (t) = τ αf ∗ h vi (t − 1) + β˜ vi+1 (t − 1) , (47) for i = 2, . . . , n, cf. (5). The optimized gains ai , bi of the CCC controller are related through (41). We found that the eigenvalues of M1 (cf. (42)) are inside the unit circle for realistic values of weighting factors γ1 , γ2 , human gains α, β, and driver reaction time τ . Thus (41) is a contracting map, and consequently, ai , bi decay with the vehicle index i = 2, . . . , n. As an example, we consider γ1 = 1 [1/s2 ], γ2 = 4 [1/s], α = 0.4 [1/s], β = 0.5 [1/s] and τ = 0.4 [s]. In this case, M1 has two zero eigenvalues and two non-zero eigenvalues 0.55 and 0.13. Fig. 2 shows the exponential decay of ai and bi in a (5 + 1) vehicle chain (red circles) and a (10+1) vehicle chain (blue crosses) using the aforementioned parameter values. The exact match between the red circles and the blue crosses for vehicles 2 to 5 demonstrate that the existing optimized gains are not changed by adding feedback terms on vehicles farther away. This corresponds to the fact that the gains a1 , b1 are not influenced by the connectivity structure (cf. (29) and (45)), and ai , bi are calculated recursively using (34) and (41). For the parameters considered above, we have the gains a1 ≈ 2.50 [1/s], b1 ≈ −6.12 [1/s].
Note that Q1i (θ) contains the same distribution matrix ˆ ˆ 0 (cf. (28)) depends eA0 (θ+1) for all i (cf. (43)), where A on the driver reaction time τ and weighting factors γ1 and γ2 through P11 (cf. (29)). The distribution functions fi (θ), gi (θ) are linear combinations of the elements of ˆ eA0 (θ+1) (cf. (B.3) in Appendix B). We plot fi (θ) and gi (θ) for i = 2, . . . , 5 in Fig. 3 using the same parameters as in Fig. 2. Note that f1 (θ) ≡ 0 and g1 (θ) ≡ 0, i.e., the
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1.2
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1.6
ai
˜ h
bi
10
10
v˜
1.2 0.8
5
5 0.8
0
0
0.4 0.4
0
2
4
6
8
i
10
0
-5
2
4
6
8
i
10
0
Fig. 2. The optimized headway and velocity gains ai , bi , i = 2, . . . , n of the CCC vehicle for a string of (n + 1) vehicles. Red circles correspond to n = 5 while blue crosses are for n = 10. The human gains are set to α = 0.4 [1/s], β = 0.5 [1/s], and the driver reaction time is τ = 0.4 [s]. The weighting factors are γ1 = 1 [1/s2 ], γ2 = 4 [1/s]. 0
0.25
fi
i=2
0.2
i=3
0.15
0.05
i=3
-0.2 -0.3
i=4 i=5
0.1
i=4
-0.1
-0.4
i=2
-0.5 0 -1
-0.8
-0.6
-0.4
-0.2
θ
0
-1
-0.8
-0.6
-0.4
-0.2
0
θ
20
40
60
t
-10
0
20
40
60
t
Fig. 4. Headway and velocity fluctuations of a (5 + 1)-car vehicle string with the same parameters as in Fig. 2. The thick red curves correspond to the CCC vehicle using information from vehicles 2 − 5, while the thick black curves are when the CCC vehicle only uses information from vehicle 2. The thin curves correspond to the non-CCC vehicles. The black dashed curve is the disturbance profile (48).
i=5
gi
-5
-10
Fig. 3. The distribution functions fi (θ), gi (θ) for a (5 + 1)vehicle system with the same human parameters as in Fig. 2. The blue, red, yellow, purple, and green curves correspond to i = 1, . . . , 5, respectively. optimized controller does not use delayed information of CCC vehicle itself. While for each i = 2, . . . , n, fi (θ) and gi (θ) may increase or decrease with θ ∈ [−1, 0], in general the magnitude of fi (θ) and gi (θ) decays with i. Since the relation between LQR and pole assignment can be extended to the delayed cases, the controller derived from LQR places the eigenvalues of the system in the left half of the complex plane, i.e., guarantees stability of the equilibrium. 3.3 Simulations In this section, we test the performance of the proposed CCC controller using numerical simulations. We consider a vehicle string with 5 + 1 vehicles with initial condition X(t) ≡ 0, t ∈ [−1, 0] and a triangular velocity disturbance signal 0 −1 ≤ t ≤ 0 , 0 < t ≤ 8, −t v˜n+1 (t) = t − 16 (48) 8 < t ≤ 24 , −t + 32 24 < t ≤ 32 , 0 t > 32 , see the dashed curve in Fig. 4(b). In Fig. 4, we plot the corresponding headway and velocity perturbations for the system (47) with the same parameter as in Fig. 2. The velocity and headway of the CCC vehicle (red thick curves) converge to the equilibrium. We also show the case when the CCC vehicle loses connectivity with vehicles 3, 4, 5 (black thick curves), i.e., ai = 0, bi = 0, fi (θ) ≡ 472
0, gi (θ) ≡ 0, for i = 3, 4, 5 in (47). In this case, the CCC vehicle only relies on the link from vehicle 2. Although its headway and velocity perturbations still converge to zero, demonstrating the robustness of the proposed controller, the black curves have larger amplitude than the red curves. This shows the potential benefits of having feedback terms from multiple vehicles ahead. Notice that the amplitude of CCC vehicle’s velocity response is smaller than the disturbance input v˜n+1 (t), i.e., the velocity perturbation is attenuated as it reaches the tail vehicle. This property is called string stability, and is highly desirable to avoid amplification of congestion waves; see Seiler et al. (2004), Orosz et al. (2010). Although string stability is not considered in this paper, this property can be achieved by choosing appropriate weighting factors γ 1 , γ2 . 4. CONCLUSION In this paper, we proposed a connected cruise control design where the connectivity structure was optimized using linear quadratic regulation while considering driver reaction time delay. We found that the feedback gains on signals from nearby vehicles were not influenced by dynamics of vehicles farther downstream, and the gains decreased exponentially with the number of cars between the CCC vehicle and the signaling vehicle. The performance of the CCC controller was tested using numerical simulation. The proposed control algorithm can be extended to vehicle systems with heterogeneous driver parameters, and multiple CCC vehicles. When considering non-zero time delay in the wireless communication, the closed-loop system also contains an actuator delay; see Kristic (2009). A detailed discussion on the choice of the design parameters γ1 , γ2 to ensure plant and string stability will be considered in the future. Also, nonlinear CCC algorithms and partial knowledge about the motion of non-CCC vehicles will be investigated. ACKNOWLEDGEMENTS This work was supported by the National Science Foundation (Award Number 1351456).
IFAC TDS 2015 June 28-30, 2015. Ann Arbor, MI, USA
Jin I. Ge et al. / IFAC-PapersOnLine 48-12 (2015) 468–473
REFERENCES Ge, J.I. and Orosz, G. (2014a). Dynamics of connected vehicle systems with delayed acceleration feedback. Transportation Research Part C, 46, 46–64. Ge, J.I. and Orosz, G. (2014b). Optimal control of connected vehicle systems. In Conference on Decision and Control, IEEE, 4107–4112. Kolmanovskii, V. and Myshkis, A. (1992). Applied Theory of Functional Differential Equations. Kluwer Academic Publisher. Kristic, M. (2009). Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhauser. Orosz, G. (2014). Connected cruise control: modeling, delay effects, and nonlinear behavior. Vehicle System Dynamics, submitted. Orosz, G., Wilson, R.E., and St´ep´ an, G. (2010). Traffic jams: dynamics and control. Philosophical Transactions of the Royal Society A, 368(1928), 4455–4479. Ploeg, J., Shukla, D., van de Wouw, N., and Nijmeijer, H. (2014). Controller synthesis for string stability of vehicle platoons. IEEE Transactions on Intelligent Transportation Systems, 15(2), 845–865. Qin, W.B., Gomez, M.M., and Orosz, G. (2014). Stability analysis of connected cruise control with stochastic delays. In Proceedings of American Control Conference, 4624–4629. IEEE. Ross, D. and Flugge-Lotz, I. (1969). An optimal control problem for systems with differential-difference equation dynamics. SIAM Journal on Control, 7(4), 609–623. Seiler, P., Pant, A., and Hedrick, K. (2004). Disturbance propagation in vehicle strings. IEEE Transactions on Automatic Control, 49(10), 1835–1842. Wang, M., Daamen, W., Hoogendoorn, S.P., and van Arem, B. (2014). Rolling horizon control framework for driver assistance systems. part II: Cooperative sensing and cooperative control. Transportation Research Part C, 40, 290–311. Zhang, L. and Orosz, G. (2015). Motif-based analysis of connected vehicle systems: delay effects and stability. IEEE Transactions on Intelligent Transportation Systems, submitted. Appendix A. THE SOLUTION OF LQR PROBLEM WITH DELAY Here we present a detailed solution to the LQR problem with time delay. Since (11,12) is constructed to include disturbance φ(t) in the LQR format, and the optimal controller (18) is given using partitioned matrices (17). We write (15,16) into four groups, where P1 (t), Q1 (t, θ), R1 (t, ξ, θ) are independent from the disturbance and can be solved using only the coefficient matrices A, B, D and the weighting factor Γ. That is, for the first group we obtain the PDE ˙ 1 (t) = τ AT P1 (t) + τ P1 (t)A + Q1 (t, 0) + QT (t, 0) −P 1 + Γ − τ 2 P1 (t)DDT P1 (t) , (∂θ − ∂t )Q1 (t, θ) = τ AT − τ 2 P1 (t)DDT Q1 (t, θ) + R1 (t, 0, θ) ,
T (∂ξ + ∂θ − ∂t )R1 (t, ξ, θ) = −τ 2 QT 1 (t, ξ)DD Q1 (t, θ) , (A.1) with boundary conditions
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P1 (tf ) = 0 , Q1 (tf , θ) = 0 ,
Q1 (t, −1) = τ P1 (t)B , (A.2) R1 (tf , ξ, θ) = 0 , R1 (t, −1, θ) = τ BT Q1 (t, θ) . Using P1 (t) and Q1 (t, θ) obtained from (A.1,A.2), we can calculate Q2 (t, θ) and R2 (t, ξ, θ) by solving (∂θ − ∂t )Q2 (t, θ) = τ AT − τ 2 P1 (t)DDT Q2 (t, θ) + R2 (t, 0, θ) , (A.3) T (∂t − ∂ξ − ∂θ )R2 (t, ξ, θ) = τ 2 QT 1 (t, ξ)DD Q2 (t, θ) , with boundary conditions R2 (tf , ξ, θ) = 0 , Q2 (tf , θ) = 0 , (A.4) Q2 (t, −1) = 0 , R2 (t, −1, θ) = τ BT Q2 (t, θ) . Note that the disturbance φ(t) does not appear in (A.3) either. As a matter of fact, (A.3,A.4) result in Q2 (t, θ) ≡ 0 and R2 (t, ξ, θ) ≡ 0.
The dynamics of P2 (t) and Q3 (t, θ) are driven by the disturbance φ(t): ˙ 2 (t) = τ AT − τ 2 P1 (t)DDT P2 (t) + P1 (t)φ(t) −P + Q2 (t, 0) + QT (t, 0) , T3 T (∂θ − ∂t )Q3 (t, θ) = φ (t) − τ 2 PT Q3 (t, θ) 2 (t)DD
+ RT (A.5) 2 (t, θ, 0) , with boundary conditions Q3 (tf , θ) = 0 , Q3 (t, −1) = τ PT P2 (tf ) = 0 , 2 (t)B . (A.6) Although P4 (t), Q4 (t, θ), R4 (t, ξ, θ) do not appear in the optimal control (18), they appear in the minimal cost function, and are given by the PDE ˙ 4 (t) = τ φT (t)P2 (t) + τ P3 (t)φ(t) + Q4 (t, 0) + QT (t, 0) −P 4
− τ 2 P3 (t)DDT P2 (t) , (∂θ − ∂t )Q4 (t, θ) = τ φT (t) − τ 2 P3 (t)DDT Q4 (t, θ) + R4 (t, 0, θ) ,
T (∂ξ + ∂θ − ∂t )R4 (t, ξ, θ) = −τ 2 QT 2 (t, ξ)DD Q2 (t, θ) , (A.7) with boundary conditions P4 (tf ) = 0 , Q4 (tf , θ) = 0 , Q4 (t, −1) = 0 , (A.8) R4 (t, −1, θ) = 0 . R4 (tf , ξ, θ) = 0 ,
Appendix B. THE DISTRIBUTION FUNCTIONS ˆ
Since Q1i (θ) given in (43,44) contains eA0 (θ+1) , we write ˆ
(B.1) eA0 (θ+1) = KeJ0 (θ+1) K−1 , ˆ where J0 is the Jordan form of A0 , and K is the corresponding transformation matrix. Denote the eigenvalues ˆ 0 as λ1 , λ2 , then of A ˜b11 ˜b12 ˆ 0 (θ+1) ˜12 a ˜11 a A + (θ + 1) ˜ ˜ eλ1 (θ+1) = e a ˜21 a ˜22 b21 b22 c˜11 c˜12 λ2 (θ+1) + , (B.2) e c˜21 c˜22 where ˜bij = 0 when K contains generalized eigenvectors. Substituting (B.2) into (43,44), we obtain (45,46) with ˆ 1i , ˆ 1i , a21 a ˜22 ]Q [ai1 bi1 ] = [˜b21 ˜b22 ]Q [ai0 bi0 ] = [˜ ˆ 1i . [ai2 bi2 ] = [˜ c21 c˜22 ]Q
(B.3)