SOLUTION TO THE USER EQUILIBRIUM DYNAMIC TRAFFIC ROUTING PROBLEM USING FEEDBACK LINEARIZATION

Pergamon Transpn Res.-B, Vol. 3, No. 5, pp. 343±360, 998 # 998 Elsever Scence Ltd. All rgts reserved Prnted n Great Brtan 09-65/98 $9.00+0.00 PII: S09-65(97)0003-3 SOLUTION TO THE USER EQUILIBRIUM DYNAMIC TRAFFIC ROUTING PROBLEM USING FEEDBACK LINEARIZATION PUSHKIN KACHROO* Bradley Department o Electrcal and Computer Engneerng, Vrgna Polytecnc Insttute and State Unversty, Blacksburg, VA 406-0, U.S.A. and KAAN OÈ ZBAY Department o Cvl and Envronmental Engneerng, Rutgers, Te State Unversty o New Jersey, Pscataway, NJ 08855-0909, U.S.A. AbstractÐIn ts paper, te Dynamc Tra c Routng problem s de ned as te real-tme pont dverson o tra c durng non-recurrent congeston. Ts dynamc tra c routng problem s ten ormulated as a eedback control problem tat determnes te tme-dependent splt parameters at te dverson pont or routng te ncomng tra c ow onto te alternate routes n order to aceve a user-equlbrum tra c pattern. Feedback lnearzaton tecnque s used to solve ts spec c user-equlbrum ormulaton o te Dynamc Tra c Routng problem. Te control nput s te tra c splt actor at te dverson pont. By transormng te dynamcs o te system nto canoncal orm, a control law s obtaned wc cancels te nonlneartes o te system. Smulaton results sow tat te perormance o ts controller on a test network s qute promsng. # 998 Elsever Scence Ltd. All rgts reserved Keywords: control, closed loop, tra c dverson Tra c varables. NOTATION q q, j ;j v r s U x y u e J (.,.) tra c volume enterng lnk tra c volume enterng lnk j o route tra c densty n lnk tra c densty n lnk j o route average tra c speed n lnk ramp tra c ow enterng lnk ramp tra c ow extng lnk tra c splt actor at a node nput ow at a node state vector measurement vector nput vector error vector objectve uncton travel tme uncton *Autor or correspondence. Tel: 00 540 3 7740; Fax: 00 540 3 54; e-mal: puskn@vt.edu 343

344 Puskn Kacroo and Kaan OÈ zbay Secton parameters max v k k jp v j tra c jam densty or lnk ree ow tra c speed or lnk travel tme parameter or lnk travel tme parameter or lnk j route ree ow tra c speed or lnk j route Oters t t L y j A B F G tme nal tme or nte orzon optmal control problems Le dervatve o scalar wt respect to vector eld dervatve o te order j o y state varable or nternal dynamcs transton matrx decouplng matrx transton scalar decouplng scalar. INTRODUCTION Real-tme control o tra c dverson durng non-recurrent congeston contnues to be a callengng topc. Wt te advent o ntellgent transportaton systems (ITS), te need or real-tme models and algortms tat wll control te dverson becomes especally evdent. Several researcers tred to solve ts on-lne control problem by adoptng d erent approaces. Among te models tat ave been developed or determnng dverson routes and dvertng te tra c onto tese routes n te context o advanced tra c management systems (ATMS)/advanced traveller normaton systems (ATIS) are expert systems, eedback control, and matematcal programmng models. Expert system-based strateges ave been used or dverson (Gupta et al., 99; Ketseldou, 993). Gartner and Ress (987) developed a multlevel control system ncorporatng local-level and corrdor-level controls. Ts ramework recognzes and empaszes te completon o te eedback loop between te system outputs and te control nputs. Peraps te most popular way o solvng te dynamc tra c assgnment (DTA)/dynamc tra c routng (DTR) problem s troug te use o tradtonal optmzaton-based approaces tat attempt to optmze te objectve unctons or te nomnal model over te `plannng orzon' (Peeta and Mamassan, 995). Papageorgou (990) and Papageorgou and Messmer (99) ave desgned some bang-bang controllers, controllers based on te lnear quadratc regulator (LQR) prncple, and controllers based on te rollng orzon tecnque. Kacroo and OÈ zbay (996) desgned a uzzy eedback control law or pont dverson. Te DTR eedback controller dscussed n ts paper wll be used as part o an overall ncdent management ramework n order to allevate congeston caused by ncdents (Hobeka et al., 993; Kacroo et al., 997b). Real-tme route gudance d ers rom te dverson concept used n te plannng concept n tat t s not a long-term polcy decson concernng route coce, but rater t s a reacton to an mmedate stuaton acng te motorst (Hall, 974). Te statc and quasdynamc tecnques or tra c assgnment (Ress et al., 99) are useul or acevng pre-set system optmal or user-equlbrum condtons but cannot be used or te eedback control problem. For real-tme tra c ow control, were on-lne sensor normaton and actuaton metods are avalable, te optmzaton-based tecnques used n some o te models descrbed n te prevous secton are not very well suted. Most o tese models are not spec cally desgned or on-lne control problems, suc as te DTR problem dscussed n ts paper. Tey do ave several drawbacks due to te act tat tey were orgnally desgned as extensons o tradtonal optmzatontype models or o -lne applcatons. Most o tese models are not desgned to take ull advantage o te on-lne sensor normaton, but tey do assume a perect knowledge o te transportaton system or a spec c tme perod. Te computatonal requrements o tese models are also very

Dynamc tra c routng problem 345 g due to te complexty o te models and soluton algortms tey use. Tereore, altoug tey are perect tools or o -lne plannng/evaluaton problems, tey are not well suted or on-lne control problems. On te oter and, tere exst real-tme eedback control approaces tat are spec cally desgned or suc on-lne systems. Te drawback o te exstng lnear eedback control tecnques tat ave prevously been tred s tat te system sould reman n te lnear regon around te equlbrum or te trajectory at all tmes or te controller to be vald. Snce te system s nonlnear, tme varyng, and contans uncertantes, eedback control laws tat andle suc systems sould be used. Te major motvaton o ts paper s to rst provde a realstc DTR model spec cally developed or eedback control and to use te eedback lnearzaton tecnque to develop e cent and robust eedback control laws or ts nonlnear tra c ow model. Note tat te tecnque developed n ts paper s vald or sngle orgn, sngle destnaton problems. For mult-orgn, multdestnaton problems, we can use oter tecnques, suc as nonlnear H control, wc s te topc o researc beng conducted by te autors... Prelmnary consderatons or usng eedback control or tra c dverson Feedback control or DTR can be an e ectve soluton or allevatng tra c congeston durng major ncdents. However, te success o suc a system depends on te e ectve modelng o te system as well as te desgn o te approprate control law. Te desgner o te controller needs to address ssues suc as controllablty and observablty o te tra c system, actuaton and sensng, robustness, and stablty o te closed-loop system sown n Fg..... Actuaton and sensng. Te actuaton o ts system can be aceved n many ways, ncludng varable message sgns (VMS), n-vecle gudance, and gway advsory rado (HAR). State varables suc as tra c densty and average tra c speed can be sensed usng varous types o tra c sensors suc as nductve loops, tra c cameras, and transponders, wc can be used as sensors.... Controllablty and observablty. Te desgner sould analyze te system beore desgnng te controller to determne te system s controllable and observable. Controllablty mples tat a sutable control law can be devsed n order to obtan a desred response rom te system. Observablty mples tat te system state varables can be observed rom te sensed output. For nstance, te system s not controllable, ten we mgt decde to add more actuaton nrastructure, suc as VMS or HAR, and te system s not observable, we mgt add more sensors to te system...3. Robustness and stablty. Te e ectveness o te control desgn can be measured n terms o ts robustness, stablty, and transent caracterstcs. A robust controller wll perorm well even n te presence o uncertantes n te nomnal model o te system. Models representng tra c systems cannot represent te system ully, and tereore tere are uncertantes n te system tat must be dealt wt. A control law sould provde stablty to te system and desrable transent response. For nstance, a good DTR control law would mnmze te tme or te system to cange rom a congested state to a normal ow state. 3. SYSTEM DYNAMICS Many researcers ave studed and desgned optmal open loop controllers utlzng space and tme dscretzed models o tra c ow (Fresz et al., 989; Tan et al., 993). Some researcers ave Fg.. Block dagram or DTR eedback control.

346 Puskn Kacroo and Kaan OÈ zbay also desgned eedback control laws usng smlar models (Papageorgou, 983, 990; Papageorgou and Messmer, 99; Messmer and Papageorgou, 995). Te reason or te popularty o tese models s tat tere are many tecnques avalable to deal wt dscretzed systems. Te same s also true or eedback control, and ence, n order to utlze te varous lnear and nonlnear (Kuo, 987; Isdor, 989; Mosca, 995) control tecnques avalable or lumped parameter systems, te dstrbuted parameter model s space dscretzed (Papageorgou, 983). For ts, te gway s subdvded nto several sectons, as sown n Fg.. Te ollowng ordnary d erental equatons (ODE) wt te gven varable relatonsps can be used to model a space dscretzed reeway: d dt ˆ q t q t r t s t Š; ˆ ; ;...n q t ˆ t v t v ˆ v max 3 Here, r t and s t terms ndcate te on-ramp and o -ramp ows, s te densty o te tra c as a uncton o x and tme t, q(t) s te ow at gven x, (t) v s te ree ow speed, and max s te jam densty. Equaton () and te output eqns (4) gve te matematcal model or a gway, wc can be represented n a standard nonlnear state space orm or control desgn purposes. Te standard state space orm s y j ˆ g j ; ;... n ; j ˆ ; ;...; p 4 d x t ˆ x t ; u t Š dt y t ˆg x t ; u t Š 5 x 0 ˆx 0 were, x ˆ ; ;...; n Š T and u t ˆq 0 t : Tere are varous oter proposed models, wc are more detaled n te descrpton o te system dynamcs. Te penomenon o sock waves, wc s very well represented n te PDE representaton o te system, s modeled by expressng te tra c ow between two contguous sectons o te gway as te wegted sum o te tra c ows n tose two sectons wc correspond to te denstes n tose two sectons. A dynamc relatonsp nstead o a statc one lke eqn (3) as also been proposed by Papageorgou (983) and used successully. In te uture, we wll look nto desgnng eedback controllers usng te models wc can represent spllback also. In general, owever, te plosopy bend eedback control desgn s to use a relatvely smple nomnal model or control desgn, and te robustness o te controller sould take care o te uncertantes n te system model. 4. FEEDBACK CONTROL FOR TRAFFIC In te dscretzed tra c ow model, te reeway s dvded nto sectons wt aggregate tra c denstes. Sensors are used to measure varables suc as denstes, tra c ow, and tra c average Fg.. Hgway dvded nto sectons.

Dynamc tra c routng problem 347 speeds n tese sectons, wc can be used by te eedback controller to gve approprate commands to actuators lke VMS and HAR. Tere are essentally two ways to desgn controllers or suc nonlnear tra c systems. One way s to desgn te controller by lnearzng te nonlnear dynamcs o te system about ts equlbrum, or a trajectory; te oter way s to desgn drectly or te nonlnear system. Te rst way s easer, snce an mmense amount o lterature s avalable descrbng te varous desgn tecnques, especally or lnear tme nvarant (LTI) systems, but te results are vald only were te lnearzaton s applcable. On te oter and, desgn o controllers drectly or te nonlnear system s muc more d cult; owever, te results are usually global. Some o te lnear control tecnques are LQR, lnear quadratc gaussan (LQG), proportonal ntegral dervatve (PID), H, and prevew control; some o te nonlnear control tecnques nclude descrbng uncton desgn, eedback lnearzaton, sldng mode control, and nonlnear H. Qualtatve metods suc as uzzy control and expert systems can also be utlzed. Qualtatve metods are usually easy to desgn, but are d cult to tune and analyze. Some o tese ssues are dscussed by Papageorgou and Messmer (99). In te next secton, we present a user-equlbrum ormulaton o te DTR problem or te dscretzed tra c ow model gven n Secton. Ten, n te ollowng sectons, tree d erent controllers or tree d erent versons o te same DTR problem ormulaton are developed usng te eedback lnearzaton tecnque. 4.. DTR ormulaton We rst present a DTR ormulaton or te two alternate routes problem wc s ten generalzed or n routes case. Te two routes are dvded nto n and n sectons, respectvely. For smplcty, we are consderng te statc velocty relatonsp, and gnorng te e ect o downstream ow. Hence, te model used s d dt ;j ˆ q ;j t q ;j t Š; ; j ˆ ; ; ; ;...; ; n ; ; ; ; ;...; ; n 6 wt relatonsps () and (3). Te control nput s gven by t U t ˆq ;0 t ; 044; t U t ˆq ;0 t 7 Te ow U t s measured as a uncton o tme, and te splttng rate t s te control nput. Te output measurement could be te ull state vector,.e. vector o ows o all te sectons, or a subset o tat. Te control problem can be stated as: nd 0 t, te optmal t, wc mnmzes J ˆ t " # X m Xm p j dt 8 0 ˆ were :;: s te travel tme uncton and t s te nal tme. Note tat a eedback soluton s needed or te problem, not an open loop optmal control. Hence, we can eter decde te structure o te eedback control, suc as a PID control wt constant gans, and solve numercally or te optmal values o te gans, or we can state te control objectve or a standard eedback control problem, suc as steady state asymptotc stablty gven by Lt t! ˆ m " # X m Xm p j! 0 9 m and some transent beavor caracterstcs suc as a spec ed settlng tme or percent oversoot. Now we ormulate te same DTR ormulaton or a generalzed case, or an n alternate route problem, as ollows: Problem: Fnd 0, ˆ ; ;...; n, wc mnmze

348 Puskn Kacroo and Kaan OÈ zbay t ( ) X X j 0 ; ˆ ; ;...n ˆ n k ;k Xn p 4 j;p dt 0 0 k;p k ˆ ; ;...; n, p ˆ ; ;...; n, and te summatons are taken over te total number o combnatons o n and p, and not permutatons, so tat k; p ˆ ; s consdered te same as k; p ˆ ;, and ence only one o tese two wll be n te summaton), or wc guarantee ˆ jˆ Lt t! e! 0 were "( ) ( ) # X n e ˆ ; Xn X n k j; ;...; ;k Xn p j;p ;... ˆ jˆ ˆ jˆ wt some transent beavor caracterstcs lke a spec ed settlng tme or percent oversoot or te system d dt ;j ˆ q ;j t q ;j t Š; ; j ˆ ; ;...; ; n ; ; ;...; ; n ;...; n; ;...; n; n n wt gven ull and partal state observaton, and nput constrants X n ˆ q ;0 t ˆU t and X n ˆ ˆ 3 5. FEEDBACK LINEARIZATION TECHNIQUE Feedback lnearzaton s an approprate tecnque or developng eedback controllers or nonlnear systems smlar to te DTR model descrbed above. Te eedback lnearzaton tecnque s applcable to an nput a ne square multple nput multple output (MIMO), system. Te detals on exact nonlnear decouplng tecnque (eedback lnearzaton) can be ound n Isdor (989), Slotne and L (99), Lgtll and Wtam (955), and Godbole and Sastry (995), and are bre y summarzed ere or te DTR applcaton. Let us consder te ollowng square MIMO system: _x t ˆ x Xp g x u 4 Ts can be wrtten n a compact orm as ˆ y j ˆ j x j ˆ ; ;...; p X _x t ˆ x g x u : y ˆ x 5 were xr n, x : R n! R n, g x : R p! R n, ur p, and yr p. Te vector elds o x and g x are analytc unctons. It s assumed tat or te system, eac output y j as a de ned relatve degree j. Te concept o relatve degree mples tat te output s d erentated wt respect to tme j tmes, ten te control nput appears n te equaton. Ts can be succnctly represented usng Le dervatves. A de nton o a Le dervatve s gven below, ater wc te de nton o relatve degree n terms o Le dervatves s stated. De nton (Le dervatve): Le dervatve o a smoot scalar uncton : R n! R wt respect to a smoot vector eld : R n! R n s gven by L ˆ @ @x. Here, L denotes te Le dervatve o order zero. Hger-order Le dervatves are gven by L ˆ L L ).

Dynamc tra c routng problem 349 De nton (relatve degree): Te output y j o te system as a relatve degree j, 9 an nteger, s.t. L g L` x 0 8` < j, 844p, 8xU, andl g L j x 6ˆ0. U R n wc s n a gven negborood o te equlbrum pont o te system. Te total relatve degree o te system r s de ned to be te sum o te relatve degrees o all te output varables,.e., r ˆ Pp j. By successvely takng te Le dervatves o eac o te output varables up to ter respectve relatve degrees, we obtan 6 4 y y y p 3 ˆ 7 6 5 4 L L L p 3 3 x L g L x... L gp L x x L gp L x... L gp L x u 7 5 6 7 4 5 p x L g L p x... L gp L p x jˆ 6 Ts can be wrtten as y ˆ A x B x u 7 were y ˆ y y... y p T 8 A x ˆ L x L x... L p T p x 9 3 L g L x... L gp L x L gp L x... L gp L x B x ˆ 6 7 4 5 L g L p x... L gp L p x 0 I te decouplng matrx B x s nvertble, ten we can use te eedback control law eqn () to obtan te decoupled dynamcs eqn (). u ˆ B x A x vš y ˆ v were T v ˆ v v... v p Te vector v can be cosen to render te decoupled system eqn (7) stable wt desred transent beavor. Now, te relatve degree o te system r s less tan te order o te system n, ten te closed loop system sould also ave stable nternal dynamcs. In order to study tat, one can de ne state varables x ; ˆ ; ;...; n r, wc are ndependent o te state varables r related to te output o te system, and are also ndependent o eac oter. Te nternal dynamcs o te system can ten be wrtten as: _ ˆ w ; P ; u 3 wt k ˆ ; ;...; n r and ˆ ; ;...; p

350 Puskn Kacroo and Kaan OÈ zbay w k ; ˆL k x P k ; ˆL g k x 4 5 Te eedback controller desgned or eqn (7) usng te eedback lnearzaton tecnque sould also guarantee te stablty o te nternal dynamcs descrbed n eqn (3). Note tat or a sngle nput case, we could use te act tat L g k x ˆ0 to coose te ndependent nternal dynamcs state varables, but or te multple nput case, ts condton s not vald, unless te vectors o g are nvolutve. Note tat te control law eqn () as to satsy te constrants eqn (3). In case te constrants are not sats ed, te control varables take extreme values, and te desred perormance o te egenvalues s not aceved. Te moment te tra c condton canges, suc tat te control varables belong to te easble set, te perormance o te system comes back to te desred state. As an example, to aceve a ast response tme our controller mgt try to overcompensate, but n realty te constrants wll produce a slower rate tan desred by te desgn; neverteless, te system wll move towards equal travel tmes. 6. SAMPLE PROBLEM (TWO ALTERNATE ROUTES WITH ONE SECTION) In order to llustrate te deas dscussed above, we ave desgned a eedback control law or te two alternate routes problem wt a sngle secton eac. Te control s based on eedback lnearzaton tecnque or nonlnear systems. Te tecnque s based on de nng a d eomorpsm and perormng te transormaton on te state varables n order to convert tem nto te canoncal orm. I te relatve degree o te system s less tan te system order, ten te nternal dynamcs are studed to ensure tat t s stable. Te detals o ts tecnque are gven n Godbole and Sastry (995). In ts problem, te system order s two and te relatve degree s one. Te space dscretzed ow equatons used or te two alternate routes are: _ ˆ v U 6 m _ ˆ v U U m We ave consdered a smple rst order travel tme uncton, wc s obtaned by dvdng te lengt o a secton by te average velocty o te vecles on t. Accordng to tat, te travel tme can be calculated as k ˆd = v 8 m k ˆd = v m were d and d are secton lengts, v and v are te ree ow speeds o eac secton, and m and m are te maxmum ( jam) denstes o eac secton. Snce we need to equate te travel tmes accordng to te UE DTR ormulaton dscussed n te prevous secton, we take te new transormed state varable y as te d erence n travel tmes. D erentatng te equaton representng y n terms o te state varables ntroduces te nput splt actor nto te dynamc equaton. Tereore, tat transormed equaton can be used to desgn te nput tat cancels te nonlneartes o te system and ntroduces a desgn nput, wc can be used to place te poles o te error equaton or asymptotc stablty. Tese steps are sown below. Te varable y s equal to te d erence n te travel tme on te two sectons. 7 9 y ˆ k k k 3 k 4 30

Dynamc tra c routng problem 35 were k ˆ d m v ; k ˆ m ; k 3 ˆ d m ; k 4 ˆ m v Ts equaton can be d erentated wt respect to tme to gve te travel tme d erence dynamcs. _y ˆ k _ k k 3 _ k 4 By substtutng eqns (6) and (7) n (3), we obtan _y ˆ k v m U k k 3 v m U k 4 3 3 Ts equaton can be rewrtten n te ollowng orm: were F ˆ k v k m G ˆ _y ˆ F G k 3 k 4 v U m k k k 3 k 4 U 33 34 35 Hence, a eedback lnearzaton control law can be desgned to cancel te nonlneartes and provde te desred error dynamcs. Te eedback control law gven n eqn () s used, ˆ G F 36 wc gves te closed loop dynamcs as y ˆ 37 As was mentoned earler, snce te relatve degree o te system s one, and te system order s two, we need to test te stablty or boundedness o te second transormed state varable gven by ˆ 38 Te state varable s bounded snce te denstes on te sectons cannot exceed te correspondng jam denstes. Ts assumes tat no tra c rom ramps enters a secton wt jam densty, and also tat at te node, te tra c ow nto te sectons s zero jam densty s reaced. Ts s a reasonable assumpton snce measured tra c densty wll never become ger tan te jam densty. 4 m m 39 and ence te overall system s exponentally stable y! 0 we coose ˆ Ky, K > 0, and y asymptotcally goes to zero as y t ˆy 0 e Kt. Ts mples tat wen a splttng value based on (36) s utlzed, te d erence n travel tme o two alternate routes wll go to zero at an exponental rate. Hence, te closed loop tra c system controlled by te proposed eedback lnearzaton law s exponentally stable and as desred transent beavor. Note tat nput saturaton occurs, te rate o convergence cannot be guaranteed.

35 Puskn Kacroo and Kaan OÈ zbay 7. SAMPLE PROBLEM (TWO ALTERNATE ROUTES WITH TWO SECTIONS) Now, we extend te above problem to a case wt two sectons and ollow te same steps or desgnng a new controller or ts extended system. Te space dscretzed ow equatons used or te two alternate routes are: _ ˆ v U 40 m _ ˆ v m _ ˆ v m v m U U _ ˆ v v m m 4 4 43 We ave consdered a smple rst order travel tme uncton, wc s obtaned by dvdng te lengt o a secton by te average velocty o vecles on t. Accordng to tat, we approxmate travel tme as d t ˆ v m d v m 44 d t ˆ v m d v m 45 were d and d are secton lengts, v and v are te ree ow speeds o eac secton, and m and m are te maxmum (jam) denstes o eac secton. Te system can be wrtten n te standard nonlnear nput a ne orm _x t ˆ x; t g x; t u t y t ˆ x; t 46 were ˆ 6 4 x ˆ Š 0 ; u t ˆ v m v m v m v m v m U v m 3 7 5 47 48 g ˆ 6 4 U O U O 3 7 5 49

Dynamc tra c routng problem 353 Snce we need to equate te travel tmes on alternate routes accordng to te UE DTR problem ormulaton presented n te prevous secton, we take te new transormed state varable y as te d erence n travel tmes. D erentatng te equaton representng y n terms o te state varables ntroduces te nput splt actor nto te dynamc equaton. Tereore, tat transormed equaton can be used to desgn te nput tat cancels te nonlneartes o te system and ntroduces a desgn nput, wc can be used to place te poles o te error equaton or asymptotc stablty. Tese steps are sown below. Te varable y s equal to te d erence n te travel tme on te two sectons. " # " # d y t ˆ v m d d v m v m d v m 50 Ts equaton can be d erentated wt respect to tme to gve te travel tme d erence dynamcs: _y ˆ k _ k k 3 _ k 4 k 5 _ k 6 k 7 _ k 8 By substtutng eqns (40)±(43) n (5), we obtan _y ˆ k v m U k k 5 v m U U k 6 k 3 v m k 4 k 7 v m v m k 8 v m 5 5 Ts equaton can be rewrtten n te ollowng orm: were F ˆ k v m k k 5 v m U k 6 _y ˆ F G k 3 v m k 4 k 7 v m v m k 8 v m 53 54 k G ˆ k k 5 k 6 ŠU 55 Hence, a eedback lnearzaton control law smlar to te one gven by eqn () can be desgned to cancel te nonlneartes and provde te desred error dynamcs. Te law used s wc gves te closed loop dynamcs as ˆ G F 56 y ˆ 57

354 Puskn Kacroo and Kaan OÈ zbay Te relatve degree o te system s one, and te system order s our, and we need to test te stablty or boundedness o te tree nternal states. Now our task s to obtan te oter tree ndependent state varables. Snce ts s a sngle nput system, to obtan tese state varables, we sould satsy L g k x ˆ 0 as ollows: @ ˆ @ ; ˆ ; ; 3 @ @ ˆ ˆ 58 3 ˆ Te state varable s bounded snce te denstes on te sectons cannot exceed te correspondng jam denstes. We use te same argument as or te prevous example or ts clam. m m 59 3 4 and ence te overall system s exponentally stable y! 0 we coose ˆ Ky, K > 0, and y asymptotcally goes to zero as y t ˆy 0 e Kt. Ts mples tat wen a splttng value based on eqn (56) s utlzed, te d erence n travel tme o two alternate routes wll go to zero at an exponental rate. Hence, te closed loop tra c system controlled by te proposed eedback lnearzaton law s exponentally stable and as desred transent beavor. 8. SOLUTION FOR THE ONE-ORIGIN, ONE-DESTINATION CASE WITH MULTIPLE ROUTES WITH MULTIPLE SECTIONS. In ts secton, we gve a generalzed soluton or te n alternate route DTR problem descrbed n Secton 3.. Te space dscretzed ow equatons used or te n alternate routes and n sectons are gven by eqns () and (3). Te number o sectons or eac alternate route I s denoted by n. We are consderng ull state observaton, wc s used or estmatng (sensng) te travel tmes on te varous alternate routes. Te dynamcs can be wrtten as _ j v j j j v j j j 60 j mj mj wen ; j ˆ ; ;...; ; n ; ; ;...; ; n ;...; n; ;...; n; n n _ j U v j j j j mj 6 wen ; j ˆ ; ; ; ;...; n; We ave consdered a smple rst order travel tme uncton, wc s obtaned by dvdng te lengt o a secton by te average velocty o vecles on t. Accordng to tat, we approxmate travel tme or a route as t ˆXn d j 6 jˆ v j j mj Te system can be wrtten n te standard nonlnear nput a ne orm: _x t ˆ x; t g x; t u t y t ˆ x; t 63

Dynamc tra c routng problem 355 were x ˆ... n... n... nnn Š 0 ; u t ˆ... n Š 64 Te output vector s denoted by y, and s gven by: were y ˆ y y...y...y n Š y ˆ t t 65 66 Ts equaton can be d erentated wt respect to tme to gve te travel tme d erence dynamcs. _y ˆ Xn jˆ k j k j _ j j m j Xn jˆ k j k j _ j 67 j mj were k jp denotes a constant p ˆ or tat belongs to secton j o route, smlar to te constant k descrbed or eqn (30). Te system (67) s n te orm o eqn (6) and can be represented n te orm o eqn (7) by determnng te values o A x and B x. Te control law () provdes us wt user equlbrum or te DTR problem. Te nput appears n all te output equatons ater d erentatng tem one tme. Hence, te relatve degree o te system s n. Te order o te system s Pn n. Snce te denstes on te ˆ P sectons are bounded by jam densty values, te ndependent state varables ; ˆ ; ;...; n ˆ n n are also bounded. 9. SIMULATION Several smulaton studes are perormed to demonstrate te utlzaton o te eedback lnearzaton tecnque presented n ts paper. Te test network, wc conssts o two alternate routes, s sown n Fg. 3. Tree d erent smulaton scenaros tat were cosen are: (a) model wtout any parametrc uncertantes and ull user complance, (b) model wt parametrc uncertantes and ull user complance, (c) model wt parametrc uncertantes and partal user complance. Te nput uncton s assumed to be a snusodal uncton wc reaces a pre-de ned peak value and ten settles at a constant value or te rest o te smulaton perod. Ts uncton emulates te peak our demand tat reaces ts maxmum value at a certan tme, and ten settles at a constant value wen te peak perod s over. In ts spec c smulaton study, te peak perod s assumed to be. 9.. Scenaro : model wt ull user complance and wtout any uncertantes In ts scenaro, te controller as perect knowledge o te parameters o te tra c model. In addton, ull complance o te users to te dverson commands s assumed by te controller. Te system dynamcs model also smulates ull complance o te users to te controller s dverson Fg. 3. Sample network.

356 Puskn Kacroo and Kaan OÈ zbay Fg. 4. (a) D erences n travel tmes (s) or scenaro ; (b) splt actors or scenaro. Fg. 5. (a) D erences n travel tmes (s) or scenaro ; (b) splt actors or scenaro.

Dynamc tra c routng problem 357 Fg. 6. (a) D erences n travel tmes (s) or scenaro 3; (b) splt actors or scenaro 3. commands. In ts case, snce te controller as te complete knowledge o te system dynamcs, t s able to perorm exact cancellaton o te system nonlneartes and attan exponental error convergence. Ts result s sown n Fg. 4(a). Fgure 4(b) sows splt actors. 9.. Scenaro : model wt ull user complance and uncertantes In ts scenaro, te controller does not ave perect knowledge o te parameters o te tra c model. In order to smulate te e ects o uncertantes, 30% errors ave been assumed. However, ull complance o te users to te dverson commands s assumed by te controller and s also smulated by te system model. In ts case, snce te controller does not ave complete knowledge o te system dynamcs, t s not able to perorm exact cancellaton o te system nonlneartes and attan exponental error convergence. However, as can be seen n Fg. 5(a) and (b), te results obtaned by usng ts controller, even wt suc relatvely large parametrc uncertanty, are gly encouragng. 9.3. Scenaro 3: model wt partal user complance and uncertantes In ts scenaro, we assume bot partal user complance (80%), and te exstence o parametrc uncertanty n te model. As can be seen n Fg. 6(a) and (b), te uctuatons o d erences n travel are muc ger tan te prevous scenaros, and t takes te controller a longer tme to attan error convergence. However, even wt partal user complance and arly large parametrc uncertantes, te system stablzes and te d erences n travel tmes asymptotcally converge at a desrable rate. 9.4. Scenaro 4: model wt partal user complance and uncertantes, and a lnear PI controller In ts scenaro, we assume bot partal user complance (80%), and te exstence o parametrc uncertanty n te model, and we use a lnear PI (proportonal, ntegral) controller. As s evdent ere, te perormance o ts controller s neror to te eedback lnearzaton controller [Fg. 7(a) and (b)].

358 Puskn Kacroo and Kaan OÈ zbay Fg. 7. (a) D erences n travel tmes (s) or scenaro 4; (b) splt actors or scenaro 4. Fg. 8. (a) D erences n travel tmes (s) or scenaro 5; (b) splt actors or scenaro 5.

Dynamc tra c routng problem 359 Fg. 9. Planned deployment ste or dverson. 9.5. Scenaro 5: model wt partal user complance, uncertantes, and dynamc velocty relatonsp In ts scenaro, we assume bot partal user complance (80%), and te exstence o parametrc uncertanty n te model. We also consder a dynamc relatonsp or velocty to represent te sock wave dynamcs n te model. Te results o usng te eedback lnearzaton to uncertantes obtaned by under-modelng o ts knd are also satsactory [Fg. 8(a) and (b)]. In all te scenaros, tere s no congeston created n te two routes. In general, te control algortm cannot prevent congeston tere s a very large n ow o tra c. Consder te stuaton were te n ow tra c s so g tat t produces tra c wc s greater tan te overall capacty o te two routes; ten te splt actor control cannot avod congeston. In tat case, some tra c rom te n ow tsel would ave to be dverted. 9.6. Smulaton envronment Te smulaton envronment we used s SIMNON (Elmqvst, 975), wc s a specal programmng language developed n te Lund Insttute o Tecnology, Sweden, or smulatng dynamc systems descrbed as ordnary d erental equatons, as d erence equatons, or as combnatons o bot. Ts program s avalable n DOS and Wndows envronments. 0. DEPLOYMENT ISSUES We are makng plans or deployng ts strategy at te Su olk area n Vrgna (Fg. 9). Te mornng rus our ow s rom pont A to B, and te evenng rus our n te oter drecton. Altoug ts problem s really a network level problem wt multple nodes and routes, te majorty o te tra c ows only n two alternate routes. Te eedback lnearzaton tecnque can be appled to ts problem or calculatng te desred splt rates, and ten anoter algortm could be desgned to come up wt te means to control te varous ways to aceve tat. For nstance, we would use varable message sgns and gway advsory rado. Wat messages to put on tese systems s part o a separate researc topc, but te results o tose could be combned wt te eedback control desgn to aceve an overall control system. Te eedback control law s desgned wt te nomnal model, and te robustness propertes o te controller are used to andle te uncertantes n te nomnal model. We are currently desgnng controllers or network level problems usng eedback control. Kacroo et al. (997a,998) and Kacroo and OÈ zbay (997) deal wt ormulatons and control solutons or tose problems.. CONCLUSIONS In ts paper, we ave addressed te real-tme tra c control problem or pont dverson. A eedback model s developed or control purposes, and eedback lnearzaton tecnque s used to desgn ts eedback controller. Frst, te smplest case, wt two alternate routes consstng o a sngle secton eac, s studed and a eedback controller usng eedback lnearzaton tecnque s

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