Dynamic Processor Allocation for Multiple RHC Systems in Multi-Core Computing Environments

Transcription

1 9 Amercan Control Conference Hyatt Regency Rverfront, St. Lous, MO, USA June -, 9 FrB. Dynamc Processor Allocaton for Multple RHC Systems n Mult-Core Coutng Envronments Al Azm and Brandon W. Gordon Abstract Ths paper develops a new dynamc processor allocaton algorthm for multple recedng horzon controllers (RHC) executng on a mult-core parallel couter. The proposed formulaton accounts for bounded model uncertanty, sensor nose, and coutaton delay. A cost functon approprate for control of multple coupled vehcle systems on multple processors s used and an upper bound on the cost as a functon of the executon horzon s eloyed. A parallel processng adaptaton of the SOPT optmzaton package s used and the effcency factor of the parallel optmzaton routne s estmated through smulaton benchmarks. Mnmzaton of the cost functon upper bound combned wth the effcency factor nformaton results n a combnatoral optmzaton problem for dynamcally allocatng the optmal number of logcal processors for each RHC subsystem. The new approach s llustrated through smulaton of a leaderfollower control system for two 3DOF helcopters runnng on a couter wth two quad-core processors. R I. ITRODUCTIO ECEDIG horzon control (RHC) s a repeated onlne soluton of a fnte horzon open-loop optmal control problem []. Applcaton of RHC to control problems wth multple subsystems s addressed by applyng RHC to the ndvdual subsystems whle the nformaton regardng the state varables, or trajectores, are exchanged between them, whch leads to a decentralzed formulaton. In most of the decentralzed archtectures, each subsystem s optmzed on a sngle couter. Besdes, some decentralzed RHC approaches optmze a group of subsystems on a sngle couter (see [] for exale). In ths case, multple RHC processes must be scheduled n an approprate manner on a sngle processor and t was dscussed n [9], [], and []. In those approaches, the executon horzons of all subsystems were selected by solvng an optmzaton problem. However, most new couter desgns are adoptng a mult-core archtecture where multple logcal processors runnng n parallel are contaned on each physcal processor package on the couter. Ths allows an ncrease n processng speed, wth sgnfcantly roved performance than networked coutng, provded approprate algorthms are avalable to take advantage of the addtonal logcal processors (cores). It s antcpated by coutng processor A. Azm and B. W. Gordon are wth Department of Mechancal and Industral Engneerng, Concorda Unversty, 455 de Masonneuve Blvd. West, Montreal, Quebec, H3G M8, CAADA (phone: (54) ext. 758; fax: (54) ; (e-mal: a_az@encs.concorda.ca, bwgordon@encs.concorda.ca). manufactures that n the near future couters coosed of hundreds of cores wll be avalable. However, n order to beneft from such archtectures, a systematc approach for performng the coutatons n parallel and dynamcally allocatng the processors becomes necessary. For the case of RHC whch s very coutatonally expensve, the performance of RHC can be sgnfcantly roved usng more processors through reducton of the executon horzon and the coutaton delay. In addton, more colex and accurate models wll be possble as well as examnng multple scenaros for more globally optmal results. In addton, mult-core archtectures are potentally superor to dstrbuted/network coutng, snce there are much smaller communcaton delays/latences and hgher bandwdth than even ggabt class networks such as ggabt Ethernet. Ths enables the applcaton of mult-core couters to RHC problems that would not be possble on a coutatonal network, especally when usng seral algorthms such as sequental quadratc programmng (SQP), whch are currently used to solve most types of RHC optmzaton problems. Some attets have been performed to apply optmal control problems on parallel couters, ncludng the approach presented n [3] usng dynamc programmng and a space decooston scheme n whch the global optmal control problem s reduced to the optmzaton of subproblems. In [4] and [5], parallel algorthms are presented for optmal control problems wth long predcton horzon usng tme decooston technques. A two-phase parallel coutng method s presented n [6] to obtan the soluton of recedng horzon controller for constraned nonlnear systems. The approach n ths paper s dstngushed from these exstng approaches n that t consders dynamc allocaton of the processors n response to changng uncertanty and dsturbances of the RHC subsystems. In ths paper multple RHC subsystems are consdered on a sngle couter wth multple processors. However, the number of processors assgned to each subsystem s changng and s determned by a proposed algorthm. The new technque determnes the executon horzons and allocates the approprate number of processors for all subsystems. The executon horzon determnaton of each subsystem and processor allocaton whle optmzng the performance s cast nto a constraned optmzaton problem. Onlne soluton of the foresad optmzaton problem usng the updated optmzaton parameters, results n dynamcally determnng the processor allocaton /9/$5. 9 AACC 49

2 II. PROBLEM STATEMET A. Decentralzed RHC formulaton Consder dynamcally decoupled subsystems usng the RHC approach. The subsystems are supposed to have connecton wth each other by exchangng ther nformaton. The term system used n ths paper, refers to all subsystems. Furthermore, consder the followng nomnal equaton for the th subsystem: x = f( x( t), u( t), t), =,..., () whch serves as a model for the actual subsystem descrbed by: x ˆ = f xˆ (), t u (), t t + g ( xˆ, u,), t =,, () ( ) p where x () t R and ˆ x () t R are the nomnal and actual states of the th subsystem, respectvely. The nput m vector u () t R satsfes the constrants u ( t) U ( t ), where U s the allowable set of nputs for subsystem. Furthermore, t s assumed that (A-A3) n [] are also satsfed; that s, f s twce dfferentable, U s coact and convex, and subsystem () has a unque soluton for a gven ntal condton. Defnton. The set A s called the neghbourng set of subsystem, and conssts of any subsystem that ts nformaton s used n the control of subsystem. The fnte horzon cost assocated to th subsystem s defned as follows []: J( t, T, x, u, x ) = V( x( T + t; t) ) + (3) t+ T q( x( τ ; t), u( τ) ) + gj ( ( τ; t), j ( τ; t) ) dτ t x x j A where ~ x s a vector contanng the states of all neghbours of the th subsystem. g j s a functon whch defnes the nteracton between two nodes of the system x and x j. T s the optmzaton horzon of the RHC controller assocated to th subsystem. V (.) defnes the fnal penalty and x ( τ;t) s the state vector of th subsystem at tme τ whch has been saled at tme t. The optmal cost s then gven by [8]: (,, x, u, x ) nf (,, x, u, x ) * * * = () u J t T J t T (4) The optmzed trajectory resultng from (4) s defned as * * ( xt, (;), τ t u T, (;)), τ t τ (, t t+ T]. In the closed loop RHC the calculated nput u * T, (;) τ t s appled to the actual subsystem (), whle τ ( t, t + δ ] and δ s called the Executon Horzon of subsystem ( δ < T). B. Real-tme processor allocaton Assumng the control of subsystems s desred on a couter wth m p dentcal physcal processors, such that each physcal processor j has m j logcal processors. The problem under study s processor allocaton, whch s p defnng the approprate number of logcal processors and ts coutaton topology to each subsystem. The term coutaton topology s defned later. Defnton. A logcal processor s referred to as the smallest ndependent processng unt. Multple logcal processors are used to form physcal processors. Multple physcal processors can be put together n a couter. For RHC control of a sngle apparatus usng a sngle couter, the calculated nputs are appled for a perod equal to the executon horzon of the recedng horzon controller. Therefore, n real-tme lementaton, a common way s to defne a real-tme perodc task [7], wth ts perod equal to the executon horzon of that RHC. Assume the subsystems descrbed n () and () are connected to a set of multple couters for feedback control, usng RHC method, as explaned earler. From a couter control pont of vew, each control system can be handled as a perodc task n the real-tme programmng. The perod of each perodc task s equal to the executon horzon of ts related subsystem. Determnng these perods (or executon horzons) s not trval. In ths paper, a systematc approach s presented to calculate these perods. Based on that, the approprate processng unts,.e. logcal processors, are assgned to each subsystem. III. IMPLEMETATIO OF RHC O MULTI-CORE PROCESSORS Assume a sngle task, whch s lementaton of a sngle RHC system, needs to be couted on a couter wth more than one dentcal mult-core processor. In addton, assume the coutaton tme c s known f ths task s couted on one of the logcal processors only. However, f n dentcal logcal processors are used, the coutaton tme should be decreased and the mnmum coutaton tme s c n n the deal case. However, dependng on the type of the task that needs to be parallelzed and the method used n the parallelzaton, the actual coutaton tme,.e. c, s more than the deal case n practce. An effcency factor, η, s ntroduced as follows for dentcal logcal processors: η c / ( nc ) where η (,] = (5). It should be noted that f all of the n logcal processors, used n the calculaton of the foresad task, are not on a sngle processor, the effcency factor may vary dependng on the dstrbuton of cores on dfferent processors, whch shall be dscussed later. In the followng, the method used n calculaton of RHC on parallel processors s explaned. Ilementng RHC on parallel processng unts deals wth dvdng the assocated nonlnear optmzaton problem to sub-problems and calculatng them on dfferent unts. A. Parallel SQP lementaton for a sngle RHC Sequental quadratc programmng (SQP) methods are among the most effectve nonlnear programmng algorthms for solvng dfferentable nonlnear optmzaton problems 49

3 [3]. A Fortran lementaton of a SQP algorthm s presented n [3] for parallel processors. Part of ths dea correspondng to the Jacoban and cost functon calculaton, s eloyed n ths paper to apply RHC on parallel processors. In order to solve the optmzaton problem assocated wth RHC and to fnd the approprate nputs correspondng to ts predcton horzon, the SOPT optmzaton package [4] s used. To formulate the problem, every nput s estmated by a cubc splne. Thereby, the am of optmzaton s fndng the splne coeffcents of all nputs. In order to calculate the cost functon based on the optmzaton varables, the followng steps should be done: Step - Run the splne routne to fnd the nputs based on the splne coeffcents Step - Fnd the state trajectores by smulatng the subsystem from the ntal condtons and usng the calculated nputs n Step. Step 3- Calculate the cost by usng the nput and state trajectores obtaned n the prevous steps. It should be noted that, ths cost calculaton ncludes estmatng an ntegral usng trapezodal ntegraton. Therefore, the RHC optmzaton s tme consumng due to colexty of ts cost functon calculaton. Besdes, the gradent of the cost cannot normally be calculated analytcally, and t s typcally couted usng center fnte dfference approxmatons. A sle yet effectve approach for lementng ths optmzaton on parallel processors, s to calculate the gradent (or Jacoban) and the cost functon n parallel. However, n ths paper only the Jacoban s calculated n parallel. Usng ths approach, a relatvely hgh effcency factor of.9 s obtaned n the exale secton when two logcal processors were used. Remark. It should be ehaszed that the processor allocaton approach presented n ths paper, works wth any effcent parallel processng optmzaton method. However, the method should be flexble enough to work when the number of processors used n each subsystem vares dynamcally. B. Effcency factor calculaton As mentoned earler, the number of logcal processors used n calculatng a partcular task affects the effcency factor. However, there are also some other factors that affect t and are dscussed presently. The way the logcal processors access memory, can dramatcally affect the performance of the parallel algorthms, and consequently affect the effcency factor. Partcularly, by usng on-unform Memory Archtecture (UMA), separate memory s assgned to each logcal processor, whch avods the performance ht when some processors try to access the same memory. UMA can be consdered as a tghtly form of cluster coutng, and s used n the couters performng the smulatons n ths paper. Another ortant factor s the avalable cache memory (L and L) for each processor. Remark. The couter used n the exale secton has two quad-core Intel Xeon 53 seres processors. Each processor has two des, each ncludes two cores. Its L cache has 3KB for nstructons and 3KB for data. In addton, t has 4MB L cache per de that means 8MB L cache for each physcal processor [5]. Based on the foresad factors, n addton to the number of logcal processors, the dstrbuton of them on the physcal processors, also affects the effcency factor. Ths s referred to as coutaton topology n ths paper. As an exale for possblty of havng deferent coutaton topologes, assume 4 tasks are operated on a couter wth two quad-core processors, whle based on the coutatonal need of them, tasks to 4 requre 3,,, and core, respectvely. Two dfferent archtectures are presented n Fg. (-a) and Fg. (-b). Remark 3. It should be noted that the effcency factor may stll vary even wth fxed logcal processors and coutaton topology. For exale, n calculaton of an optmzaton problem, the number of teratons may vary that mght cause slghtly dfferent effcency factor. However, n ths paper, t s assumed that the effcency factor s only depends on the number of logcal processors and the coutaton topology, as long as the smulatons are done on a same couter wth the same parallel algorthm. Task Task 3 Task Task Task Task 4 Physcal Physcal Processor Processor (a) Physcal Processor Task 4 Task 3 (b) Physcal Processor Fg.. Dfferent archtectures assgned to the set of 4 tasks n two quad-core processors. In (b) each processor has tasks and there s no need to have data exchange between processors. Based on the foresad dscussons and the relaton presented n (5), the effcency factor, regardng the calculaton of a sngle RHC, can be obtaned usng the followng procedure: Step - Perform the smulaton usng only a sngle logcal processor and obtan the coutaton tme c. Step - Perform the same smulaton usng more than one processor and obtan c for dfferent number of logcal processors and every possble coutaton topologes. Step 3- Fnd the effcency factor for all cases usng (5) and store the results n a lookup table. Remark 4. The foresad procedure should be lemented several tmes usng dfferent ntal condtons and dfferent possble problem parameters, and the fnal effcency factor s the average of them. Ths task s done offlne and the values of effcency factors are used later n the dynamc processor allocaton algorthm. 493

4 IV. DYAMIC PROCESSOR ALLOCATIO ALGORITHM For the proposed approach, the executon horzons of all subsystems should be defned such that the overall performance of the system s maxmzed. However, the executon horzon s dependent on the number of logcal processors assgned to t, the effcency factor, and the coutaton tme of the task that wll be dscussed later n ths secton. Smlar to the approach presented n [], n order to evaluate the performance of the system, the followng cost functon s proposed as the cost of the closed loop system from tme t to t+t a, where T a s the perod that calculated executon horzons would be appled to: * q ( ˆ ( τ), T, ( τ; t x u k )) + t+ T a Jˆ a = dτ t * j = (6) gj ( ˆ ( τ), T, j ( τ; tk )) x x j A * where tk = t + ( k ) δ and u, ( ; T τ t k) s the optmal nput appled to subsystem. In addton, the followng s presented as the estmaton of J ˆa []: * q ( xˆ ( τ), ut, ( τ; ts, )) + T t + δ a Ja = t * dτ j ( ˆ g ( τ), T ( )), j τ; t = δ (7) s, j x x j A where t t and t > t s,. t represents the tme that new nputs are appled to subsystem based on the saled data at ts, and t ndcates the start tme of dynamc allocator. In the followng, an upper bound on (7) s presented and t shall be used to fnd the optmal number of logcal processors for each subsystem. In the Lemma 4 of [], by consderng some common assutons, the followng upper bound was presented on the J a : J G ( δ ) = a = t + δ T T s T s a t = δ + P B + B * * ( x, ( ;, ), u, ( ;, )) q τ t τ t dτ (8) () () ( ( δ ) ( δ )) t + δ * * T gj ( T ( ) ( )), τ; t s,, T, j τ; t s, j dτ a x x t + = δ j A g () () + Lj ( B ( δ ) + B ( δ )) where: Lx, δ Lx, δ b () s, Lx, δ Lx, δ b e e B ( δ) = ( e e ) + δ Lx, L x, L x, (9) ε p () Lu, Lx, δ δ Lx, δ B ( δ) = e ( e ) L ( x, ) In whch, b s, s the state measurement error bound, b s the bound on g (.,.,.) n (), L x, and L u, are the Lpschtz constants of f( x, u, t) wth respect to x and u, respectvely. ε s a constant scalar defned n Corollary of [], whch s used to bound the msmatch between the nputs n two subsequent calculatons. δ p s the last predcton horzon of subsystem, whle δ s ts current value. In the case of parallel processng, the choce of executon horzon s dependent on the number of logcal processors, the effcency factor and the coutaton tme of the task on a sngle processor, as mentoned earler. Assumng the coutaton tme (or worst case coutaton tme) s known, snce the effcency factor s dependent on the number of logcal processors and the coutaton topology, the processor allocaton problem cast nto fndng the optmal number of processors and the coutaton topology smultaneously. Ths results n a colex optmzaton problem. However, a suboptmal soluton can be obtaned, by decouplng the problem of calculatng the number of processors (n ) from coutaton topology assgnment. Ths way, by consderng effcency factor, only as a functon of n, whch referred to as η, n can be defned. Knowng n, the coutaton topology, and η as a result, can be assgned. Fnally, δ can be calculated based on the calculated η and n values. Ths procedure can be explaned n the followng: Algorthm : Step - Fnd the effcency factor as a functon of number of cores only ( η ). Ths task s done offlne and the results are stored n a lookup table. Step - Fnd δ and assocated number of logcal processors ( n ) by solvng the processor allocaton optmzaton problem presented n the followng lemma (Lemma ) Step 3- Fnd the optmum coutaton topology based on the n values. Ths task s done n Lemma. Then calculate the actual effcency factors, η, based on n and assgned coutaton topology. Step 4- Modfy δ (ncrease some of them) based on the corrected values of η. Lemma : Suppose the assutons lead to the upper bound on the J a presented n (8) are all vald. Then the followng optmzaton problem can be used to sub-optmally determne n as presented n the Step of Algorthm. mn Gc / ( ) nη n = () Subject to: n = m, n = = where G (.) s defned n (8) and δ s replaced by ts 494

5 equvalent c / ( ) nη. In addton, c s the maxmum coutaton tme of th subsystem on a sngle core and η n a functon of n. Proof: G( δ ) presents the upper bound on the performance ndex of the overall system and mnmzaton of that upper bound should result n mnmzaton of overall system cost, as dscussed n []. Based on the defnton of η, the worst case coutaton tme on n logcal processors s c = c / nη. Moreover, the salng perod of th ( ) subsystem ( δ ) should be more than or equal to c n order to prevent coutatonal overload condtons. In addton, the frst constrant ensures that all of the avalable logcal processors are used, whle t s assumed that at least one processor s assgned to each subsystem by the use of second constrant. Remark 5. The presented problem n () s a combnatoral optmzaton problem because n s a postve nteger. In the exale secton, a sle grd search method s used to solve ths optmzaton problem. A. Coutaton Topology Assgnment Assume the number of logcal processors requred for each RHC subsystem s determned. Based on that, the optmal coutaton topology should be found, whch s dscussed n the followng lemma. Lemma : Assume m p ndcates the total number of physcal processors, such that each physcal processor j has m j logcal processors; n s the number of logcal processors for subsystem, and the total subsystem number s. x j s defned as the number of logcal processors assgned to subsystem from physcal processor j. In addton, assume that the communcaton between every two logcal processors takes less tme f they are from a sngle physcal processor coared to the case that they are from dfferent physcal processors. The followng maxmzaton problem can optmally defne x j. max x xj () Subject to: = j= j j = j j j= = () C: x n, C: x m C3: xj,, j Proof: Snce logcal processors need to have communcaton wth each other for every subsystem calculaton, t s desred to select them from a sngle physcal processor, f possble. In other way, the communcaton between every two logcal processors placed on dfferent physcal processors should be mnmzed, whch can be presented as mnmzng the followng relaton: xj xk = j= k= j+ (3) Usng the constrant C, the followng equalty s vald: xj = ( n ) xj xk (4) = j= = = j= k= j+ Mnmzaton of (3) results n maxmzaton of the left hand sde of (4). Moreover, constrant C ensures that the number of logcal processors s assgned correctly whle C ndcates that each physcal processor s not assgned more than ts avalable logcal processors. V. SIMULATIO RESULTS The proposed approach s appled n smulaton to the processor allocaton of two mnature 3DOF helcopters usng RHC scheme, on a couter wth quad-core processors. The vehcle dynamcs are selected accordng to [6]. Subsystem s followng the trajectory (leader) whle subsystem, follower, tres to mantan a certan relatve poston wth respect to the leader. Based on the method n Secton III.A, one of the logcal processors s assgned for managng the optmzaton process and the remanng seven processors are used n the parallel processng of the RHC calculatons for the two helcopters. The proposed dynamc processor allocaton algorthm, whch s presented n Algorthm, s coared to the result of the case that 3 and 4 logcal processors are assgned to the leader and the follower, respectvely. The smulatons are performed usng Mcrosoft Vsual C++ 8 and Wndows XP. Besdes, the uncertanty s added to both subsystems usng random varables, such that elements of uncertanty vector g (.,.,.), presented n (), are selected randomly, whle g b, and b s the uncertanty upper bound of subsystem. b s changng based on the pattern shown n Fg.. In addton, T a s selected as. seconds. Maxmum uncertanty subystem subystem tme (s) Fg.. Changng n the uncertanty upper bound of dfferent subsystems The paths followed by two subsystems n both cases are presented n Fg. 3 and Fg. 4. Dynamc PA refers to dynamc processor allocaton, whle Statc PA s the other case of havng fxed processors for each subsystem. As llustrated n these two fgures, n Dynamc RA both subsystems have better performance than Statc RA. It should be noted that the coutaton topology 495

6 assgnment s trval for ths exale, snce there are only two subsystems and two physcal processors. y c subsystem subsystem ref. trajectory x c Fg. 3. Paths followed by both subsystems n Dynamc RA case usng the proposed algorthm y c subsystem subsystem ref. trajectory x c Fg. 4. Paths followed by both subsystems n Statc RA case In addton, the change of processor assgnment n the Dynamc RA case s shown n Fg. 5. It should be noted that at the begnnng of the Dynamc RA case, the processors are assgned smlar to the Statc RA case and after seconds, dynamc allocaton of processors starts n ths exale. umber of Processors Subsystem Subsystem tme (s) Fg. 5. umber of processors allocated to each subsystem vs. tme n the Dynamc RA case VI. COCLUSIOS In ths paper, a new algorthm for the dynamc processor allocaton of multple recedng horzon controllers runnng on a mult-core couter s developed. In the proposed approach, an executon horzon dependent cost functon was used whle t accounts for bounded model uncertanty, sensor nose, coutaton delay, and couplng n the controller cost ndex. A parallel adaptaton of SOPT optmzaton package s used by calculatng the Jacoban n parallel. By defnng the effcency factor dependng on the number of logcal processors used to lement each RHC, two combnatoral optmzaton problems are presented. The frst optmzaton problem, defnes the optmal number of logcal processors to each RHC, and the second problem assgns the optmal coutaton topology. Fnally, ths new approach s llustrated through smulaton of two 3DOF helcopters. ACKOWLEDGMET The authors would lke to thank the atural Scences and Engneerng Research Councl of Canada (SERC) for fundng ths project. REFERECES [] D. Q. Mayne, J. B. Rawlngs, C. V. Rao, P. O. M. Scokaert, Constraned model predctve control: Stablty and optmalty, Automatca, 36,, pp [] H. Chen, F. Allgower, A quas-nfnte horzon nonlnear model predctve control scheme wth guaranteed stablty, Automatca, 34, 998, pp.5-7. [3] C. Scheel, B. McInns, Parallel processng of optmal control problems by dynamc programmng, Informaton Scences, 5, 98, pp [4] T.S. Chang, X.X. Jn, P.B. Luh, X. Mao, Large scale convex optmal control problems: tme decooston, ncentve coordnaton, and parallel algorthm, IEEE Trans. Auto. Contr., vol. 35, no., 99, pp [5] S.C. Chang, T.S. Chang, P.B. Luh, A herarchcal decooston for large scale optmal control problems wth parallel processng structure, Automatca, vol. 5, no., 989, pp [6] S.Y. Ln, A hardware lementable recedng horzon controller for constraned nonlnear systems, IEEE Trans. Auto. Contr., vol. 39, no. 9, 994, pp [7] H. Kopetz, Real-tme systems: Desgn prncples for dstrbuted embedded applcatons, Chapters 9 and, Sprnger, 997. [8] T. Kevczky, F. Borrell, G. J. Balas, Decentralzed recedng horzon control for large scale dynamcally decoupled systems, Automatca, 6 [9] A. Azm, B.W. Gordon, C.A. Rabbath, Dynamc schedulng of recedng horzon controllers wth applcaton to multple unmanned hovercraft systems, Proc. of the Amercan Control Conference, July 7, pp [] A. Azm, B.W. Gordon, C.A. Rabbath, Dynamc schedulng of decentralzed recedng horzon controllers on concurrent processors for the cooperatve control of unmanned systems, Proc. of the 46th IEEE Conference on Decson and Control, Dec., 7, pp [] A. Azm, B.W. Gordon, C.A. Rabbath, Dynamc schedulng of multple decentralzed recedng horzon controllers subject to coutatonal delay, Proc. of the Amercan Control Conference, June 8. [] M. Mercangoz, F.J. Doyle III, Dstrbuted model predctve control of an expermental four-tank system, Journal of Process Control, vol. 7, 7, pp [3] K. Schttkowsk, LPQLP: A Fortran lementaton of sequental quadratc programmng algorthm wth dstrbuted and non-monotone lne search user's gude, verson., Bayreuth Unversty, Dec. 4. [4] P.E. Gll, W. Murray, M.A. Saunders, User s gude for SOPT verson 7: software for large-scale nonlnear programmng, Feb. 6. [5] Quad-core Intel Xeon processor 53 seres datasheet, Sept. 7. [6] H.A. Izad, B.W. Gordon, C.A. Rabbath, Communcaton bandwdth allocaton for decentralzed recedng horzon control of multple vehcles, Proc. of the 8 IEEE/ASME Internatonal Conference on Advanced Intellgent Mechatroncs, Chna, July 8, pp