27 12 2010 12 : 1000 8152(2010)12 1669 05 Control Theory & Applications Vol. 27 No. 12 Dec. 2010 Lyapunov,, (, 400044) : (intelligent transportation system, ITS),.,, (full velocity difference, FVD), Lyapunov. FVD,. : ; ; Lyapunov; : TP391.9 : A Lyapunov stability analysis for the full velocity difference car-following model LI Yong-fu, SUN Di-hua, CUI Ming-yue (College of Automation, Chongqing University, Chongqing 400044, China) Abstract: Car-following model is an important part for microscopic traffic flow in the intelligent transportation system(its), and the stability analysis is one of the key issues in car-following theory. To make an effective stability analysis, the present paper, from the control theory viewpoint, conducts the stability analysis based on the Lyapunov function for the full velocity difference(fvd) model. In the final, the simulation is carried out, and results validate the effectiveness of this method. Key words: intelligent transportation system; car-following model; Lyapunov function; stability analysis 1 (Introduction),,,,.,., [1 12]. 1995, Bando [4] (optimal velocity, OV),, OV [4 12]. 1998, HelbingTilchOV, OV. OV,, HelbingTilch, GF(general force) [5]. GF OV,, Treiber,,, OVGF [6]., 2002,, FVD(full velocity difference) [7].,,, [8, 9].,,.,,, FVDLyapunov,. : 2009 11 16; : 2010 04 19. : (CDJXS10170002); (20090191110022); 863 (2006AA04A124).
1670 27 2 (Car-following model) 1995, Bando [4],. (optimal velocity),, OV(optimal velocity): k[v ( x i (t)) v i (t)], i1, 2,, N, (1) x i (t) > 0, v i (t) > 0 dv i(t) i, m, m/sm/s 2. t, s, N > 0N N, k > 0k R, s 1, x i (t) x i+1 (t) x i (t)t, i(i i + 1). V ( ), x i (t),. [4] V ( x i (t)) v max 2 [tanh( x i(t) x c ) + tanh(x c )], (2) : v max, x c, tanh( ). 1998, HelbingTich [5] OV, OV. OV,, (general force, GF) [5], k [V ( x i (t)) v i (t)] + λh( v i (t)) v i (t), (3) : H( )Heaviside, λ 0λ R, s 1, v i (t) v i+1 (t) v i (t)., OVGF, Treiber [6],,,,., 2002, [7], FVD(full velocity difference): k [V ( x i (t)) v i (t)] + λ v i (t), (4) k > 0k R, s 1, λ 0λ R, s 1, v i (t) v i+1 (t) v i (t). FVDOVGF [7]. FVD,, [10 12]. 3 (Stability analysis),,,,,. FVD,, Lyapunov : 1 : [ ] [ ] [ ] [ ] ẋ 1 a b x 1 m + u, (5) ẋ 2 c d x 2 n : a, b, c, d, m, n R, x 1, x 2. a 0, d 0, bc < 0, Lyapunov: E(x) px 2 1 + qx 2 2, p, q Rp > 0, q > 0., : E(x) ξ(x 1 + x 2 ) 2 + ψ(x 1 x 2 ) 2 + px 2 1 + qx 2 2, (6) ξ, ψ, p, q R, ξ 0, ψ 0, p > 0, q > 0. Ė(x) 2[(a + c)ξ + (a c)ψ + ap]x 2 1 + 2[(a + b + c + d)ξ + (b + c a d)ψ + bp + cq]x 1 x 2 + 2[(b + d)ξ + (d b)ψ + dq]x 2 2. (7) Ė(x) 0 (a + b + c + d)ξ + (b + c a d)ψ+ bp + cq 0, (a + c)ξ + (a c)ψ + ap 0, (b + d)ξ + (d b)ψ + dq 0, (8), ξ ψ 0, bp + cq 0, ap 0, (9) dq 0,, a 0, d 0, p > 0, q > 0, bc < 0, p c, q b, p c, q b. Lyapunov
12 : Lyapunov 1671 ( c)x 2 1 + bx 2 2, p c, q b, [ ] c < 0, b > 0, E(x) (10) cx 2 1 + ( b)x 2 2, p c, q b, P [δv i (t) δy i (t)] T k λ kλ, A, 1 0 c > 0, b < 0, [ ] [ ] T λ 1 b, c, p > 0, q > 0, B, C, 1 0 Lyapunov: E(x) px 2 1 + qx 2 2. (11) 1 FVD, k > 0, λ 0, FVDLyapunov., Konishi [13,14], FVD: d 2 x i (t) k[v (y 2 i (t)) dx i(t) ] + λ v i (t), y i (t) x i (t) x i+1 (t) x i (t), (12) v i (t) v i+1 (t) v i (t), i 1,, N, (12): k[v (y i (t)) v i (t)]+ λ(v i+1 (t) v i (t)), (13) dy i (t) v i+1 (t) v i (t). v 0, [14] [v i (t) y i (t)] T [v 0 V 1 (v 0 )] T, (14) (13)(14) : dδv i (t) k[λδy i (t) δv i (t)]+ λ(δv i+1 (t) δv i (t)), (15) dδy i (t) δv i+1 (t) δv i (t), δv i (t) v i (t) v 0, δy i (t) y i (t) V 1 (v 0 ). ΛOVy i (t) V 1 (v 0 ), : Λ dv (y i(t)) yi(t)v dy i (t) 1 (v 0). δv i (t)δy i (t), (15) : dδv i (t) [ ] [ ] k λ kλ δv i (t) dδy i (t) + 1 0 δy i (t) [ ] λ (16) δv i+1 (t), 1 δv i (t) [1 0] [ δv i (t) δy i (t) ]. (16) { P AP + Bδv i+1 (t), δv i (t) CP. 1, Lyapunov: (17) E(x) δv i (t) 2 + kλδy i (t) 2, (18) Ė(x) 2δv i (t)[( k λ)δv i (t) + kλδy i (t)] + 2kΛδy i (t)( δv i (t)) 2(k + λ)δv i (t) 2. (19) k > 0, λ 0, k + λ > 0, Ė(x) < 0. FVDLyapunov. 4 (Numerical simulation),,.,., i [15] []i []i. (20) (i + 1)i, 1. 1 Fig. 1 The relation between two successive vehicles 1,, : : V i (s) G(s)V i+1 (s), (21) V n (s) L(δv n (t)), V n+1 (s) L(δv n+1 (t)). L( )Laplace. [14], (17)G(s) G(s) C(sI A) 1 B
1672 27 [ ] 1 [ ] s + k + λ kλ λ [1 0] 1 s 1 λs + kλ d(s) λs + kλ s 2 + (k + λ)s + kλ. (22) Konishi[13,14], : 1 [13,14] (22)d(s), G(s)H 1, Ḡ(s) sup Ḡ(iω) > 1, (23) ω [0, ) FVD., k 1 s 1, λ 0.2 s 1 ; k 1 s 1, λ 1 s 1 ; k 2 s 1, λ 0.2 s 1. 2(a)G(s) G(jw), : k 1 s 1, λ 0.2 s 1,, 0; k 1 s 1, λ 1 s 1,,, k, λ ; k 2 s 1, λ 0.2 s 1,,, λ, k, k. 1,,. 2(b), 3, FVD,, k 2 s 1, λ0.2 s 1., 21: 3, FVD,, k 2 s 1, λ 0.2 s 1. (b) 2 G(s) Fig. 2 The characteristic of the transfer function G(s) 34t 100 s t 500 s, : k 2 s 1, λ 0.2 s 1, x c 2 m, v 0 0.964 m/s, v max 2 m/s, N 100. 3(a)4(a)t 100 s,, t 500 s, 2 m, 0.964 m/s,, 3(b)4(b).,. (a) t 100 s (a) (b) t 500 s 3 Fig. 3 The headway distribution of each vehicle
12 : Lyapunov 1673 (a) t 100 s (b) t 500 s 4 Fig. 4 The velocity distribution of each vehicle 5 (Conclusion). FVD,,,, Lyapunov,,. 3, k 2 s 1, λ 0.2 s 1. (References): [1] LIGHTHILL M H, WHITHAM G B. On kinematics wave II: a theory of traffic flow on long crowed roads[j]. Proceeding of the Royal Society, 1955, A229(1178): 281 345. [2] RICHARDS P I. Shock waves on the highway[j]. Operations Research, 1956, 4(1): 42 51. [3] PAYNE H J. Models of freeway traffic and control[j]. Mathematical Methods of Public Systems, 1971, 1(1): 51 61. [4] BANDO M, HASED K, NAKYAAMA A, et al. Dynamics model of traffic congestion and numerical simulation[j]. Physics Review E, 1995, 51(2): 1035 1042. [5] HELBING D, TILCH B. Generalized force model of traffic dynamics[j]. Physics Review E, 1998, 58(1): 133 138. [6] TREIBE M R, HENNECKE A, HELBING D. Derivation, properties and simulation of a gas-kinetic-based nonlocal traffic model[j]. Physics Review E, 1999, 59(1): 239 253. [7] JIANG R, WU Q S, ZHU Z J. A new continuum model for traffic flow and numerical tests[j]. Transportation Research, Part B, 2002, 36(5): 405 419. [8]. [J]., 2003, 52(11): 2750 2756. (XUE Yu. A car-following model with stochastically considering the relative velocity in traffic flow[j]. Acta Physica Sinica, 2003, 52(11): 2750 2756.) [9],,,. [J]., 2002, 51(3): 492 496. (XUE Yu, DONG Liyun, YUAN Yiwu, et al. Numerical simulation on traffic flow with the consideration of relative velocity[j]. Acta Physica Sinica, 2002, 51(3): 492 496.) [10] GE H X. Two velocity difference model for a car following theory[j]. Physica A, 2008, 387(21): 5239 5245. [11] ZHAO X M, GAO Z Y. A new car-following model: full velocity and acceleration difference model[j]. The European Physical Journal B, 2005, 47(1): 145 150. [12],,. [J]., 2010, 30(7): 1326 1332. (SUN Dihua, LI Yongfu, TIAN Chuan. Car-following model based on the information of multiple ahead & velocity difference[j]. System Engineering Theory & Practice, 2010, 30(7): 1326 1332.) [13] KONISHI K, KOKAME H, HIRATA K. Coupled map car-following model and its delayed-feedback control[j]. Physic Review E, 1999, 70(4): 4000 4007. [14] KONISHI K, KOKAME H, HIRATA K. Decentralized delayedfeedback control of an optimal velocity traffic model[j]. The European Physical Journal B, 2000, 15(4): 715 722. [15] WENG Y L, WU T J. Car-following model for vehicular traffic[j]. Journal of Zhejiang University-Science A, 2002, 3(4): 412 417. : (1983 ),,, E-mail: laf1212@163.com, ; (1962 ),,,,, E-mail: d3sun@163.com; (1974 ),,, E-mail: cuimingyue@sina.com.