Control Theory & Applications. Lyapunov. : (intelligent transportation system, ITS),

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1 : (2010) Control Theory & Applications Vol. 27 No. 12 Dec Lyapunov,, (, ) : (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 , 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,. : ; : : (CDJXS ); ( ); 863 (2006AA04A124).

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 qx 2 2, p, q Rp > 0, q > 0., : E(x) ξ(x 1 + x 2 ) 2 + ψ(x 1 x 2 ) 2 + px 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 [(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,

3 12 : Lyapunov 1671 ( c)x bx 2 2, p c, q b, [ ] c < 0, b > 0, E(x) (10) cx ( 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 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) 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

4 [ ] 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 m/s, v max 2 m/s, N (a)4(a)t 100 s,, t 500 s, 2 m, m/s,, 3(b)4(b).,. (a) t 100 s (a) (b) t 500 s 3 Fig. 3 The headway distribution of each vehicle

5 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): [2] RICHARDS P I. Shock waves on the highway[j]. Operations Research, 1956, 4(1): [3] PAYNE H J. Models of freeway traffic and control[j]. Mathematical Methods of Public Systems, 1971, 1(1): [4] BANDO M, HASED K, NAKYAAMA A, et al. Dynamics model of traffic congestion and numerical simulation[j]. Physics Review E, 1995, 51(2): [5] HELBING D, TILCH B. Generalized force model of traffic dynamics[j]. Physics Review E, 1998, 58(1): [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): [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): [8]. [J]., 2003, 52(11): (XUE Yu. A car-following model with stochastically considering the relative velocity in traffic flow[j]. Acta Physica Sinica, 2003, 52(11): ) [9],,,. [J]., 2002, 51(3): (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): ) [10] GE H X. Two velocity difference model for a car following theory[j]. Physica A, 2008, 387(21): [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): [12],,. [J]., 2010, 30(7): (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): ) [13] KONISHI K, KOKAME H, HIRATA K. Coupled map car-following model and its delayed-feedback control[j]. Physic Review E, 1999, 70(4): [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): [15] WENG Y L, WU T J. Car-following model for vehicular traffic[j]. Journal of Zhejiang University-Science A, 2002, 3(4): : (1983 ),,, laf1212@163.com, ; (1962 ),,,,, d3sun@163.com; (1974 ),,, cuimingyue@sina.com.

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