A Fuzzy-Neural Adaptive Iterative Learning Control for Freeway Traffic Flow Systems

Transcription

1 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong A Fuzzy-Neural Adaptve Iteratve Learnng Control for Freeway Traffc Flow Systems Yng-Chung Wang, Chang-Ju Chen, and Chun-Hung Wang Abstract In ths paper, a fuzzy-neural adaptve teratve learnng control AILC s proposed for traffc flow systems of a sngle lane freeway wth random bounded off-ramp traffc volumes. It s assumed that the system dynamc functons and nput gans are unknown for controller desgn. An adaptve fuzzy neural network FNN controller and an adaptve robust controller are appled to compensate for the unknown system nonlnearty and nput gan respectvely. On the other hand, to deal wth the dsturbance from random bounded off-ramp traffc volumes, a dead zone lke auxlary error wth the tme-varyng boundary layer s ntroduced as a boundng parameter. Ths proposed auxlary error s also utlzed for the constructon of adaptve laws wthout usng the bound of the nput gan for all the adaptaton parameters. The traffc densty trackng error s shown to converge along the axs of learnng teraton to a resdual set whose level of magntude depends on the wdth of boundary layer. Index Terms fuzzy neural network, adaptve teratve learnng control, traffc flow systems, random bounded off-ramp traffc volumes. I. INTRODUCTION IT s well-known that the traffc congestons on freeways are one of the man traffc problems n Tawan. The freeway ramp meterng 1] s one of the most typcal control approaches to adust the traffc flow of freeway. Besdes, the PID-type control n ], neural network control n 3], and optmal control n 4] are also some popular control methodologes n the research feld of freeway ramp meterng. The authors n 1], whch s a good revew of recent freeway ramp meterng, have commented that the freeway ramp meterng can be further dvded nto three classes of control strateges: 1. fxed-tme ramp meterng control,. local ramp meterng control, 3. system ramp meterng control. In the local ramp meterng control strateges, ALINEA local ramp meterng has been wdely appled n the freeway traffc flow systems for a long perod of tme. In fact, ALINEA local ramp meterng s a tradtonal PI-type controller whch s not sutable for dealng wth hghly nonlnear systems wth uncertantes. In addton, snce very few strct mathematcal analyss can be appled to desgn the controller gans of ALINEA local ramp meterng, the system stablty can not be guaranteed by ALINEA local ramp meterng. On the other hand, hgh repeatabltes often exst n the freeway traffc flow systems. For example, traffc congestons on the same freeway always repettvely appear n the same peak tme nterval from 7 to 9 AM every Monday. Unfortunately, the aforementoned freeway ramp meterng control strateges are typcal tme-doman control approaches whch Yng-Chung Wang s wth the Department of Electronc Engneerng, Huafan Unversty, New Tape Cty, Tawan e-mal: ycwang@cc.hfu.edu.tw. Chang-Ju Chen, and Chun-Hung Wang are wth the Department of Electronc Engneerng, Huafan Unversty, New Tape Cty, Tawan. do not consder the repettve characterstcs of freeway traffc flow systems for the desgn of ramp meterng controller. Ths mples that these exstng freeway ramp meterng control approaches are not sutable to perform a repeated traffc control task for traffc flow systems. Recently, tradtonal dscrete teratve learnng control ILC schemes have been successfully appled for freeway traffc flow systems 5], 6], 7] wth a repettve task over a fnte tme nterval. However, t s assumed that the system nonlneartes satsfy global Lpschtz contnuous condton. In ths paper, the repettve trackng control problem of traffc flow systems of a sngle lane freeway wth random bounded off-ramp traffc volumes s studed. We consder a more general case n the sense that the system nonlneartes and system parameters are allowed to be unknown. An adaptve fuzzy neural network FNN controller and an adaptve robust controller are appled to compensate for the unknown system nonlnearty and nput gan respectvely. On the other hand, to deal wth the dsturbance from random bounded off-ramp traffc volumes, a dead zone lke auxlary error wth the tme-varyng boundary layer s ntroduced as a boundng parameter. Ths proposed auxlary error s also utlzed for the constructon of adaptve laws wthout usng the bound of the nput gan for all the adaptaton parameters. The traffc densty trackng error s shown to converge along the axs of learnng teraton to a resdual set whose level of magntude depends on the wdth of boundary layer. Ths paper s organzed as follows. In secton II, a problem formulaton s gven. The dscrete AILC s then presented n secton III. Based on the proposed AILC and a derved traffc densty trackng error model, the analyss of closed-loop stablty and learnng performance wll be studed extensvely n Secton IV. A smulaton example s gven n Secton V to demonstrate the effectveness of the proposed learnng controller. Fnally a concluson s made n Secton VI. II. PROBLEM FORMULATION In ths paper, we consder an uncertan traffc flow system 8] for a sngle lane freeway wth n sectons whch can perform a gven task repeatedly over a fnte tme sequence t {0, 1,,, N}. The traffc flow system for a sngle lane freeway wth one on-ramp and one off-ramp n the th secton, = 1,, n s represented as follows: ρ t 1 = f q 1 t, ρ, q T L r s q = ρ ν ν t 1 = g ν 1 t, ρ, ν, ρ 1 t 1 1 where and t denote the ndex of teraton and tme, s the th secton of a sngle lane freeway, n s the total number of sectons, ρ R s the traffc densty n the ISBN: ISSN: Prnt; ISSN: Onlne IMECS 016

2 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong th secton n vehcles per lane per klometer, ν R of the th traffc flow subsystem, we apply the unversal approxmaton technque to construct the basc struc- s the space mean speed n the th secton n klometers per hour, q R s the traffc flow leavng the th secton and ture of our AILC. An MIMO FNN 9] descrbed by enterng the 1th secton n vehcles per hour, r R Θ t Φρ 1 t, ν 1 t, ρ, ν s utlzed as the s the on-ramp traffc volume n the th secton n vehcles approxmators of f ρ per hour, s 1 t, ν 1 t, ρ, ν, = R s the off-ramp traffc volume of the 1,,, n. Here Φρ 1 t, ν 1 t, ρ, ν RM 1 th secton n vehcles per hour consdered to be an unknown random bounded dsturbance, f q 1 t, ρ, q MIMO FNN wth M beng the number of rules, Θ t s the radal bass functon vector n the rule layer of the and g ν 1 t, ρ, ν, ρ 1 t 1 are unknown real R M n s the output weght matrx of the output layer wth contnuous nonlnear functons of ν 1 t, q 1 t, ρ, Θ ν RM 1 beng the th output weght vector. In other, q, ρ 1 t1. Based on 1, the th freeway traffc words, Θ Φρ 1 t, ν 1 t, ρ, ν denotes the flow subsystem can be rewrtten as follows: th output of the MIMO FNN. In ths work, we use the th output of the MIMO FNN to unformly approxmate the ρ t 1 nonlnear functon f = f ρ 1 t, ν 1 t, ρ, ν T ρ 1 t, ν 1 t, ρ, ν of the th r L t s traffc flow subsystem on a compact set A c R 4 1. An mportant aspect of the above approxmaton property s that there exst an optmal parameter vector Θ for the th output of the MIMO FNN such that the functon approxmaton Now, gven a specfed teraton-varyng desred traffc densty traectory of the th freeway traffc flow subsystem t, ν 1 t, ρ, ν between the th out- error ɛ ρ 1 ρ put of the optmal FNN Θ Φ and nonlnear functon d t R, t {0, 1,,, N 1}, the control obectve f ρ 1 s to desgn an AILC to adust the on-ramp traffc volume t, ν 1 t, ρ, ν can be bounded by prescrbed constants ɛ on the compact set A c. More precsely, f r such that the traffc densty ρ can follow ρ d we defne Φ t Φ ρ 1 as close as possble t {1,,, N 1} when teraton t, ν 1 t, ρ, ν and the ɛ approaches nfnty. In order to acheve ths control obectve, t ɛ ρ 1 t, ν 1 t, ρ, ν for smplcty, then we have f some assumptons on the freeway traffc flow system and ρ 1 t, ν 1 t, ρ, ν = Θ Φ desred traffc densty traectores are gven as follows: ɛ and ɛ ɛ, ρ 1 t, ν 1 t, ρ, ν A c. Based on the traffc densty trackng error equaton n 3 and the th output of the MIMO FNN, we propose the fuzzyneural AILC for the th freeway traffc flow subsystem as: A1 The freeway traffc flow system s a relaxed system whose on-ramp traffc volumes r, traffc denstes ρ, space mean speeds ν and traffc flows q are related by ρ = 0, ν = 0 and q = 0, t < 0. A The traffc flow rate enterng the frst secton s q 0 t and the mean speed of the traffc enterng the frst secton s assumed to be the mean speed n the frst secton,.e., ν 0 t = ν 1 t. We also assume that the mean speed and traffc densty of the traffc extng the n 1th secton are assumed to be those n nth secton,.e., ν n1 t = νnt, ρ n1 t = ρ nt. The boundary condtons can be defned as ρ 0 t q 0 t ν t, ν 0 t ν 1 t, ρ n1 t 1 ρ nt and ν n1 t ν nt, respectvely. A3 There exsts a postve unknown constant s U such that s s U for all t {0, 1,, N}, 1. A4 There exsts a postve known constant ρ U d such that ρ d ρu d for all t {0, 1,, N 1}, 1. A5 Let traffc densty trackng errors be defned as e = ρ ρ d t. The ntal traffc densty trackng errors at each teraton e 0 are bounded. r = ψ ] δ ψ Θ Φ t ρ d t 1 where δ > 0. To further see the nsght of the proposed AILC 4, we substtute 4 nto 3 and fnd that e t 1 = f ρ 1 t, ν 1 t, ρ, ν Θ Φ t T ψ L t r Θ Φ t ρ d t 1 ] ψ δ ψ Θ Φ t ρ d t 1 T s L t = Θ Θ T Φ t ψ L t r δ L where 4 5 III. THE FUZZY-NEURAL AILC In order to fnd the approach for controller desgn later, we frst derve the traffc densty trackng error equaton as: e t 1 = f ρ 1 t, ν 1 t, ρ, ν T L r T s L t ρ d t 1 3 In order to overcome the desgn problem due to unknown nonlnear functon f ρ 1 t, ν 1 t, ρ, ν δ L = ɛ T s L t δ δ ψ Θ Φ t ρ d ]6 t 1 It s clear that δ L can be shown to be bounded by Θ as follows: δ L δ Θ ] δ ψ Φ t ρ d t 1 ɛ T s L t ISBN: ISSN: Prnt; ISSN: Onlne IMECS 016

3 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong θ Θ 1 where θ, = 1,,, n are some unknown postve constants. In order to overcome the uncertanty δ L t n 7, we now defne an auxlary error e φ t 1 of the th traffc flow subsystem as e φ t 1 = e t 1 φ t 1sat e t 1 φ 8 t 1 for t {0, 1,,, N}. We don t defne e φ 0 of the th traffc flow subsystem snce t wll not be utlzed n our desgn of controller and adaptve laws. In 8, sat s the saturaton functon defned as n 10] and φ t 1 s the wdth of the tme-varyng boundary layer for the th traffc flow subsystem whch s to be desgned later. It s noted that e φ t 1 of the th traffc flow subsystem whch can be defned as n 10] and t can be easly shown that e φ t 1sat e t1 = e φ t 1, 1. φ t1 Next, the tme-varyng boundary layer for the th traffc flow subsystem wll be desgned as follows: φ t 1 = θ Θ 1 where θ s a parameter of the th boundary to be updated later. In ths AILC, Θ, ψ n 4 and θ n 9 are desgned to compensate the unknown optmal consequent parameter vectors Θ, nput gans T L and θ, respectvely. The adaptve laws for Θ, ψ and θ at next 1th teraton are gven as follows : Θ 1 = Θ β 1 e φ t 1Φ t 1 Φ t r Θ 1 ψ 1 = ψ β e φ t 1r 1 Φ t r Θ 1 θ 1 = θ β 3 e φ t 1 Θ 1 1 Φ t r Θ for t {0, 1,,, N}, where β 1, β, β 3 > 0 are the adaptaton gans. For the frst teraton, we set Θ 1t = Θ 1 and ψ 1t = ψ1 to be any constant vector and constant, respectvely. θ 1t = θ1 > 0 t {0, 1,,, N} to be a small fxed value t {0, 1,,, N}. It s noted that θ > 0, t {0, 1,,, N} and 1. Furthermore, we wll choose ψ 1 t = ψ 1 as a nonzero constant n order to prevent the controller 4 from beng a zero nput n the begnnng of the learnng process. IV. ANALYSIS OF STABILITY AND CONVERGENCE In ths secton, we wll analyze the closed loop stablty and learnng convergence. At frst, defne the parameter errors as Θ = Θ Θ, θ g t = ψ T L, θ = ISBN: ISSN: Prnt; ISSN: Onlne θ θ. Then t s easy to show, by subtractng the optmal control gans on both sdes of 10-1, that Θ 1 = Θ β 1 e φ t 1Φ t 1 Φ t r Θ 1 ψ 1 = ψ β e φ t 1r 1 Φ t r Θ 1 θ 1 = θ β 3 e φ t 1 Θ 1 1 Φ t r Θ Now we are ready to state the man results n the followng theorem. Man Theorem. Consder the traffc flow systems n 1 satsfyng the assumptons A1-A5. If the fuzzy-neural AILC s desgned as n 4, 8, 10, 11 and 1 wth adaptve laws 10, 11 and 1 for the th freeway traffc flow subsystem and the followng condton can be satsfed: β 1 β β 3 > 0, 16 then the trackng performance and system stablty wll be guaranteed as follows: t1 The adustable parameters Θ, ψ, θ and control nputs r are bounded t {0, 1,, N}, 1. t The auxlary traffc densty trackng errors e φ t1 are bounded t {0, 1,, N}, 1 and lm e φ t 1 = 0, t {0, 1,, N} t3 The traffc densty trackng error e t 1 are bounded t {0, 1,, N}, 1 and lm e t 1 θ t Θ t 1, t {0, 1,, N} Proof : t1 Defne the cost functons of performance as follows V = 1 β 1 Θ Θ 1 β ψ 1 β 3 θ Then, the dfference between V 1 t and V t can be derved as follows : V 1 V = 1 Θ 1 1 Θ β Θ Θ 1 = 1 ψ1 β ψ 1 θ1 β θ 3 e φ t 1 Θ Φ t 1 Φ t r Θ 1 β 1 e φ t 1 Φ t 1 Φ t r Θ 1 IMECS 016

4 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong e φ t 1 ψ r 1 Φ t r Θ 1 β e r φ t 1 1 Φ t r Θ 1 e φ t 1 θ Θ 1 1 Φ t r Θ 1 β 3 e φ t 1 Θ 1 1 Φ t r Θ 1 17 Snce 5 can be rewrtten as Θ Φ t ψ r = e t 1 δ L t 18 Ths mples that e φ t 1 Θ Φ t e φ t 1 ψ r = e t 1e φ t 1 e φ t 1δ L t 19 Substtutng 19 nto 17, we have V 1 V e t 1e φ t 1 e φ t 1δ L 1 Φ t r Θ 1 β 1 e φ t 1 Φ t 1 Φ t r Θ 1 β e φ t 1 r 1 Φ t r Θ 1 e φ t 1 θ Θ 1 1 Φ t r Θ 1 β 3 e φ t 1 Θ 1 1 Φ t r Θ 1 0 If we substtue 8 nto 0 and usng the fact that δ L θ Θ 1 n 7, we can derve that V 1 V e φ t 1 1 Φ t r Θ 1 e φ t 1 θ Θ 1 1 Φ t r Θ 1 e φ t 1 θ Θ 1 1 Φ t r Θ 1 e φ t 1 θ Θ 1 1 Φ t r Θ 1 β 1 e φ t 1 Φ t 1 Φ t r Θ 1 β e r φ t 1 1 Φ t r Θ 1 β 3 e φ t 1 1 Φ t r Θ 1 β 1 β β 3 e φ t 1 1 Φ t r Θ 1 1 If we choose β 1, β and β 3 such that k β 1 β β 3 > 0, then we have V 1 V ke φ t 1 1 Φ t r Θ 0 1 for 1. Snce V 1 t s bounded t {0, 1,,, N} due to Θ 1 t = Θ 1 t Θ = Θ 1 Θ, ψ1 t = ψ 1 t T L = ψ 1 T L and θ 1t = θ1 θ = θ 1 θ are bounded t {0, 1,,, N}, we conclude that from that V t, and hence Θ, ψ and θ, are bounded 1. The boundedness of r s then guaranteed by usng 4. Ths proves t1 of the man theorem. t By summng from 1 to leads to V V 1 =1 ke φ t 1 1 Φ t r Θ 1 Snce V 1 s bounded and V t must be nonnegatve, we have lm e φ t 1 1 Φ t r Θ 1 = 0 t {0, 1,,, N}. Snce 1 Φ t r Θ 1 are bounded for all 1 and t {0, 1,,, N}, ths readly mples that lm e φ t 1 = 0 t3 The boundedness of e t 1 at each teraton over {0, 1,,, N} can be concluded from 8 because φ t1 s bounded. Ths mples that the bound of e t 1 wll satsfy lm e t 1 = e t 1 φ t 1 = θ t Θ t 1, t {0, 1,,, N}. Ths proves t3 of the man theorem. Q.E.D. Remark 1 : Accordng to t3 of the man theorem, t s necessary to prevent the boundary layers to be large values n the learnng process. Hence we usually set the ntal values of θ 1 and the adaptaton gan β 3 n 1 as small constants. Ths mples that θ Θ 1, t {0, 1,, N} wll reman n a reasonable small value for all 1. Remark : In our early work 10], the desgn of adaptaton gan s dependent of the upper bound of the nput gan functon. However, n ths proposed controller, the upper bounds of nput gans T L are not necessary for our fuzzy neural AILC desgn. In other words, the convergent condton n 16 s less restrcted than that gven n our prevous work 10]. ISBN: ISSN: Prnt; ISSN: Onlne IMECS 016

5 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong V. SIMULATION EXAMPLE In ths secton, we apply the proposed AILC for an unknown long segment of a sngle lane freeway n 5], 6], 7] whch s subdvded nto 1 sectons. The dfference equaton of the th traffc flow subsystem of a sngle lane freeway wth one on-ramp and one off-ramp s gven as follows, ρ t 1 = ρ T L q 1 t q r s ] Extng flow n off ramp 4 Traffc demand n on ramp b q = ρ ν ν t 1 = ν T ] V ρ τ t ν T ] ν L t ν 1 t ν V ρ νt τl = ν free 1 ρ 1 t ρ ] ρ κ ] ρ ρ am ] l where ρ, ν, q, r, s are respectvely the traffc densty, space mean speed, traffc flow, on-ramp traffc volume, off-ramp traffc volume, = 1,, 1. Here, the teratve-varyng desred traffc densty traectory of the th traffc flow subsystem s chosen as ρ d = snπ/5veh/km. In ths smulaton, we select the length of the th secton, the samplng perod, the free speed and maxmum possble densty per lane to be L = 0.5km, T = 15/3600h, ν free = 80km/h and ρ am = 80veh/km respectvely. The freeway traffc flow system parameters τ = 0.01h, ν = 35km /h, κ = 13veh/km, l = 1.8, m = 1.7 R are respectvely the street geometry, vehcle characterstcs, drvers behavors, etc.. Besdes, we assume that the traffc flow enterng the frst secton s q 0 t = 1500veh/h. Furthermore, the ntal traffc densty and space mean speed of the th traffc flow subsystem at the begnnng of each teraton are chosen as ρ 0 = snπ/5veh/km, ν 0 = snπ/5km/h, respectvely. The offramp traffc volume of the th secton s s = 0 for = 1,, 3, 5,, 1 and the off-ramp traffc volume of the 4th secton s 4 t s shown n Fgure 1a. The control obectve s to make the traffc densty ρ of the th traffc flow subsystem to track as close as possble the desred teratve-varyng traffc densty traectory ρ d t for all t {1,, 500}. In order to acheve the control obectve, the fuzzy-neural dscrete AILC n 4, 8, 10, 11, and 1 s appled wth the desgn parameters β 1 = , β = , β 3 = so that k β 1 β β 3 = 0.1. Furthermore, we set δ = n 4 and the ntal control parameters at the frst teraton are chosen as Θ 1t = Θ 1 = 0.5, 0.5, 0.5, 0.5, 0.5], ψ 1t = ψ1 = 0.1 and θ 1t = θ1 = 1.5, = 1,, 1, respectvely. In the followng, we only nvestgate the learnng performance of the 7th traffc flow subsystem due to the lmtatons on length of the paper. In order to verfy the robustness aganst teraton-varyng ntal resettng traffc densty errors e 7 0 and the bounded off-ramp traffc volumes s 7 t of the 7th traffc flow subsystem, we show max t {1,,500} e φ7 t wth respectve to teraton n Fgure 1 b. It mples that the m asymptotcal convergence proves the techncal result gven n t of the man theorem. Because the learnng process s almost completed at the 5th teraton, the traffc densty errors of the 7th secton e 5 7t s shown n Fgure 1c to prove the result n t3 of the man theorem. It s clear that the traectory of e 5 7t satsfes θ7t 5 Θ7 5 t 1 e 5 7t θ7t 5 Θ7 5 t 1, t {1,, 500} n Fgure 1 c. In order to verfy the nce traffc densty trackng performance at the 5th teraton, we show the relaton between traffc densty ρ 5 7t and desred traffc densty traectory ρ 5 d7 t n Fgure 1 d for t {0, 1,,, 500}. To see the control behavor that ρ 5 7t s close to ρ 5 d7 t for t {0, 1,,, 500} except the ntal ffty dscrete-tme, the traectores between ρ 5 7t and ρ 5 d7 t are shown agan n Fgure 1 e only for the tme sequence t {0, 1,,, 100}. It s clear that ρ 5 7t converges to ρ 5 d7 t after t 50. Fnally, Fgure 1f shows the bounded learned control nput r7t 5 for the 7th traffc flow subsystem. c d VI. CONCLUSION A dscrete fuzzy neural AILC s proposed n ths paper for repeatable traffc flow systems wth ntal resettng traffc densty errors, teraton-varyng desred traectores and random off-ramp traffc volumes. We frst derve a trackng error model to establsh the man control structure. The MIMO FNN s appled n the man structure to compensate for the lumped uncertantes from unknown system nonlneartes. ISBN: ISSN: Prnt; ISSN: Onlne IMECS 016

6 Proceedngs of the Internatonal MultConference of Engneers and Computer Scentsts 016 Vol I, IMECS 016, March 16-18, 016, Hong Kong e f 7] Z. S. Hou, J. W. Yan, J.-X. Xu and Z. J. l, Modfed teratve-learnngcontrol-based ramp meterng strateges for freeway traffc control wth teraton-dependent factors, IEEE Transactons on Intellgent Transportaton Systems, Vol. 13, Issue:, pp , 01. 8] M. Papageorgou, J. M. Blossevlle and H. Had-Salem, Macroscopc modelng of traffc flow on the Boulevard Perpherque n Pars, Transportaton Research Part B, Vol.3 No. 1, pp. 9 47, ] Y.C. Wang and C.J. Chen, Repettve trackng control of nonlnear systems usng renforcement fuzzy-neural adaptve teratve learnng controller, Appled Mathematcs and Informaton Scences, vol. 6, no. 3, pp , ] Y.C. Wang and C.J. Chen, Desgn and analyss of fuzzy-neural dscrete adaptve teratve learnng control for nonlnear plant, Internatonal Journal of Fuzzy Systems, Vol. 15, no., pp , Fg. 1. a s 4 t versus tme t; bmax t {1,,500} e t versus control teraton ; ce 5 7 t sold lne and θ5t φ7 7 Θ 5 7 t 1, θ7 5t Θ 5 7 t 1 dotted lnes versus tme t {1,,, 500}; dρ 5 7 t sold lne and ρ d7 t dotted lne versus tme t {0, 1,, 500} at the 5th control teraton; eρ 5 7 t and ρ5 d7 t versus tme t {0, 1,, 100} at the 5th control teraton; fr7 5t versus tme t. For further compensaton of the lumped uncertantes nduced by functon approxmaton errors and random off-ramp traffc volumes of the freeway, a dead-zone lke auxlary traffc densty error functons wth tme-varyng boundares are then constructed. By the auxlary traffc densty error functons, the adaptve laws for the control parameters and tme-varyng boundary layer are desgned to guarantee the closed-loop stablty and learnng error convergence. Based on a Lyapunov lke analyss, we show that all adustable parameters and the nternal sgnals reman bounded and the traffc densty trackng errors asymptotcally converge to a resdual set whose sze depends on the wdth of boundary layer as teraton goes to nfnty. ACKNOWLEDGMENT Ths work s supported by Mnstry of Scence and Technology, Tawan, under Grants MOST104-1-E and MOST104-1-E REFERENCES 1] M. Papageorgou and A. Kotsalos, Freeway ramp meterng: an overvew, IEEE Transactons on Intellgent Transportaton Systems, Vol. 3, No. 4, pp , 00. ] M. Papageorgou, H. Had-Salem, J. M. Blossevlle, ALINEA: A local feedback control law for on-ramp meterng, Transportaton Research Record, No. 130, pp , ] H.M. Zhang, S.G. Rtche, R. Jayakrshnan, Coordnated traffcresponsve ramp control va nonlnear state feedback, Transportaton Research Part C, Vol. 9, No. 5, pp , ] A. Kotsalos, Coordnated and ntegrated control of motor-way networks va nonlnear optmal control, Transportaton Research Part C, Vol. 10, No. 1, pp , 00. 5] Z. S. Hou, J. X. Xu and H. W. Zhong, Freeway trame control usng teratve learnng control based ramp meterng and speed sgnalng, IEEE Transactons on Vehcular Technology, Vol. 56, Issue:, pp , ] Z. S. Hou, J. X. Xu, and J. W. Yan, An teratve learnng approach for densty control of freeway traffc flow va ramp meterng, Transp. Res., Part C, Vol. 16, No. 1, pp , 008. ISBN: ISSN: Prnt; ISSN: Onlne IMECS 016