Decentralized Adaptive Control for a Class of Large-Scale Nonlinear Systems with Unknown Interactions
|
|
- Baldric Cole
- 6 years ago
- Views:
Transcription
1 Decentrazed Adaptve Contro for a Cass of Large-Scae onnear Systems wth Unknown Interactons Bahram Karm 1, Fatemeh Jahangr, Mohammad B. Menhaj 3, Iman Saboor 4 1. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: bahram-karm@aut.ac.r. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: jahangr_ftm@aut.ac.r 3. Amrkabr Unversty of echnoogy, ehran, Iran E-ma: menhaj@aut.ac.r 4. Center of Advanced Computatona Integence, Amrkabr Unversty of echnoogy, ehran, Iran E-ma: saboor@aut.ac.r Abstract: hs paper presents a decentrazed adaptve controer for a cass of arge-scae nonnear systems wth unknown subsystems and nteractons. A drect adaptve controer s devsed based on Lyapunov stabty anayss so that the stabty of the cosed oop system s guaranteed by ntroducng a sutaby drven adaptve rue. he adaptve controer proposed n ths paper can guarantee the stabty of the cosed-oop system wthout knowng the sgn of the controer coeffcent. o show the effectveness of the proposed decentrazed adaptve controer, a nonnear system s chosen as a case study. Smuaton resuts are very promsng. Key Words: Adaptve contro and Decentrazed nonnear system 1 IRODUCIO In the past decades, there has been an ncreased attenton n the deveopment theores for arge-scae systems. For arge-scae systems, decentrazed contro can often provde better performance over centrazed contro []. For ths reason, an ncreasng amount of attenton s beng drected towards decentrazed contro to extend the cass of systems to whch t s appcabe [], [3]. he dffcuty and uncertanty n measurng parameter vaues wthn a arge-scae system may ca for adaptve technques. Snce these restrctons ncude a arge group of appcatons, a varety of decentrazed adaptve technques has been deveoped. Mode reference adaptve contro (MRAC based desgns for decentrazed systems have been studed n [4] [6] for the contnuous tme case. Decentrazed adaptve controers for robotc manpuators were presented n [7] and [8], whe a scheme for nonnear subsystems wth a speca cass of nterconnectons was formuated n [9]. Knowng that most of physca arge-scae systems are nonneary couped to the dynamcs of the processes, the researchers are st tryng to contro these systems [1], [13]. Mosty, they ether nvestgate subsystems whch are near n a set of unknown parameters.e. [13], or they consder soated subsystems to be known,.e. [1, [14]. For most practca appcatons, the near contro synthess s appcabe to nearzed modes of arge-scae systems. However, ths ony guarantees stabty n a regon about the operatng pont and possby degradaton n performance and nstabty over a arge doman of operaton. In [1] - [13], some decentrazed nonnear controers for arge-scae systems have been deveoped. For exampe, decentrazed adaptve controers are desgned under the assumpton that the soated subsystems s known n [11], [1] and n [13], the subsystems wth unknown parameter were consdered near n a set of unknown parameters. It has been shown that the adaptve controer proposed n [13] can guarantee the stabty of the cosed-oop system when the sgn of the controer coeffcent s known. Our objectve s to present adaptve controers for a cass of decentrazed systems wth unknown nonnear subsystems and unknown nteractons. he stabty of the cosed-oop system s guaranteed by Lyapunov s stabty theory and the proposed decentrazed adaptve contro scheme ensures that a sgnas n the cosed-oop system be bounded and the trackng error goes asymptotcay to zero wthout the requrement that the sgn of the controer coeffcent shoud be known n advance. In [15] we presented an adaptve controer for a cass of affne nonnear decentrazed arge-scae systems. hs paper consders a more genera cass of these systems n whch constrants of the subsystems nteractons have been reaxed. hs paper s organzed as foows: In Secton, the detas of the probem statement and dervaton of the error dynamcs for the decentrazed system are descrbed. Secton 3 presents the man resuts of decentrazed adaptve contro for each subsystem usng ony oca nformaton and stabty anayss for composte system. An ustratve exampe s then used n Secton 4 to demonstrate the effectveness of the decentrazed adaptve technque and fnay Secton 5 concudes the paper /1/$6. c 1 IEEE 57
2 PROBLEM FORMULAIO AD DERIVAIO OF HE ERROR DYAMICS A arge-scae nonnear system comprsed of nterconnected subsystems, s consdered. he th subsystem (S, whch s assumed to be snge-nput snge-output, s gven as x = ( x1,..., x + g ( x u, y = h( x1,..., x, = [ x, x,..., x ], 1 = [,,..., ] where x,1,, n s state vector, x x1 x x s fu state of the overa n system, (., (. g, h (., are unknown smooth functons, u and y are respectvey the nput and output of the th subsystem. If each subsystem has strong reatve degree m, then the output dynamcs may be rewrtten as [16] ( m = α, k, k + +Δ 1 ( m y s the y f ( x u ( x,..., x ( where m th tme dervatve of y and α, k, are unknown parameters. o desgn the controer, n Eq. ( we do not need to know the sgn of n a subsystems. Assumpton 1. he nterconnecton Δ ( x1,..., x s bounded by Δ ( x1,..., x ρ, where ρ s unknown and ρ >. ow defne the trackng error e = r y for S, where r s the desred output trajectory and y s the output of th subsystem. Our objectve s to desgn an adaptve contro system for each subsystem whch w cause the output y to track the desred output trajectory r n the presence of nterconnectons and, usng ony oca measurements. hs requrement eads to the foowng assumpton. Assumpton. he desred output trajectory and ts ( m dervatves r,..., r for the th subsystem S are measurabe and bounded. Let the output error vector for the th subsystem be defned ( m 1 by [,,..., e = e e e ] and wrte the tme dervatve of e as ( ( [,,..., e = e e e ] (3 he error dynamcs may be expressed as ( m ( m ( m e = r y (4 It s desred that the output error of the th subsystem ( ( 1 foows m m m, a, =, where m the coeffcents are chosen so that each L ( s = s + (1 m 1 a, m 1... s + + a, has ts roots n the open eft-haf pane. ow use ( and (4 to obtan ( m ( m e = r α, kf, k( x u Δ (5 3 DECERALIZED ADAPIVE COROLLER AD SABILIY AALYSIS hs secton presents an adaptve controer for ( wth unknown nterconnecton functons. An adaptve agorthm s defned to estmate α, k wth α, k, wth, and ρ wth ρ and compensate the unknown nteractons. Defne a controer whch compensates for the dynamcs of each subsystem. For the th subsystem, the controer s defned by v u = (6 where the sgna v s defned as ( v a e + a e + a e a e ( m 1 (,,1,, 1 m ( m α, kf, k( x + r + ρsgn( epb, where b and P are gven n (13 and (14 respectvey. Substtute (6 nto (5 to obtan ( m ( m = α, k, k( Δ (7 e r f x v (8 he thrd term,, n (8 s equa to: = 1 = 1 (9 where the s defned as =. Use (8 and (9 to obtan: ( m ( m e = r α, kf, k( x v + v Δ(1 From (7 and (1, equaton (1 becomes: ( m e = α f ( x,, ( m 1 ( a, e + a,1 e a, m 1e k k ρ sgn( e Pb + v Δ, (11 where the α, k s defned as α, k = α, k α, k. Substtutng (11 n (3, the error dynamcs (3 can be wrtten n a matrx form as: 1 Chnese Contro and Decson Conference 571
3 e = A e + b ( α f ( x, k, k ρ sgn( e Pb + v Δ (1 where the Hurwtz matrx A and vector b are A..... =, a, a,1 a,... a, m 1 b = [... 1]. (13 Snce A s Hurwtz, a unque postve defnte souton P to the foowng Lyapunov equaton exsts A P + PA = Q (14 where the matrx Q s postve defnte. Consder the foowng update aws: α, k = γ α f, ( k x epb (15 / = γ e Pbv (16 ρ = γ ρ e Pb (17 where γα, γ, γ ρ > are constant desgn parameters. he man resuts of ths paper are now summarzed n the foowng theorem. heorem: Gven the error dynamca system (1 for the decentrazed system (1 wth a reference mode satsfyng Assumpton and the nteracton between subsystems satsfyng Assumpton 1, then the contro aw (6 wth adaptaton aws (15 (17 makes the trackng error asymptotcay converge to zero and a sgnas n the cosed oop system bounded. Proof: Consder the foowng Lyapunov functon V = ( V,1 + V, (18 = 1 Wth V = e Pe (19,1 α k 1, k = ρ V, = + + γα γ γ ρ ( where α, k = α, k α, k, = and ρ = ρ. ρ Use the defnton of the error dynamc (1 to wrte the tme dervatve of V as = 1 ( V = e A P + PA P α, kf, k + e b ( x + e Pb ρ sgn( e Pb + v Δ + V, (1 From (14, we have V = eqe = 1 P α, kf, k + e b ( x + e Pb ρ sgn( e Pb + v Δ + V, ( Furthermore, knowng that Δ ( x1,..., x ρ, we can obtan the foowng upper bound for the tme dervatve of V : V eqe+ epb α, kf, k( x = 1 e Pb ρ + ( e Pb v + e Pb ρ + V,, eqe+ ( epb α, kf, k( x = 1 (3 (, e Pb ρ + e Pb v + V, From (18, ( and (3, we can wrte, k V α eqe + α, k ( e Pbf, k ( x + = 1 γ α e Pbv ρ + ( + ρ( e Pb γ γ ρ (4 Use the parameters adaptve rues (15 (17, to obtan V = 1 e V eqe (5 Usng Barbaat s emma, convergence of the trackng error to zero s guaranteed. hs w make the convergence of the update aws, equatons (15 (17, possbe. hs competes the proof. Q.E.D. Remark 1: he proposed method gven n secton 3 s apparenty more genera than the subsystems as consdered n [15]. Resuts of the proposed controer to the nverted 57 1 Chnese Contro and Decson Conference
4 penduum probem are fuy presented n [15]. As we w show n next secton, the controer exhbts faster response compared wth that of [15], whch confrms superorty of the ths controer. 4 A ILLUSRAIVE EXAMPLE In ths secton, an nverted penduums connected by a sprng [13], s used as a case study to ustrate the capabty of the proposed decentrazed adaptve contro. Each penduum may be postoned by a torque nput u apped by a servomotor at ts base. It s assumed that both θ and θ (anguar poston and rate are avaabe to the th controer for = 1,. he nonnear equatons whch descrbe the moton of the penduums are defned by (6 where x1,1 = θ1 and x,1 = θ are the anguar dspacements of the penduums from vertca. he parameters m1 = kg and m =.5kg are the penduum end masses, j1 =.5kg and j =.65kg are the moments of nerta, /m s the sprng constant of the connectng sprng, r =.5m s the penduum heght, =.5m s the natura ength of the sprng, and g = 9.81 m / s s gravtatona acceeraton. he dstance between the penduum hnges s b=.4m. A, b< ndcates that the penduums repe one another when both are n the uprght poston [17]. Here we w attempt to reguated the anguar d postons to zero, so that e = θ [.e., x =, = 1, ]. x 1,1 x1, = mgr 1 kr kr x 1, = sn ( x1,1 + ( b j1 4j1 j1 u1 kr + + sn ( x,1 j1 4 j1 y = x 1 1,1 x,1 = x, mgr kr kr x, = sn ( x,1 ( b j 4j j u kr + + sn ( x 1,1 j 4 j y = x,1 (6 o show the effectveness of the proposed method, two controers are studed for the purpose comparson. We w frst demonstrate how a smpe decentrazed proportona pus ntegra (PI controer 1 t u = e + edτ, = 1, (7 woud contro the system. We fnd that the penduums exhbt undesrabe response wth reatvey arge oscatory behavor due to the ack of dampng as shown n Fgs 1 -. x1,1 x, me(sec. Fg. 1. PI controer for the frst subsystem ( x 1,1 = θ me(sec. Fg.. PI controer for the second subsystem ( x,1 = θ decentrazed adaptve controer proposed n Secton 3 s then apped to ths system. he controer s taken as 1 ( m 1 u = ( a, e + a,1 e a, m 1e (8 ( m α, kf, k( x + r + ρsgn( epb, where, α, k, ρ are updated by adaptve rues (15 (17. he controer parameters are taken as γα = γ 1 =, γ ρ =. Fgures 3-4 show the smuaton resuts for the desgned controer and ustrate that, after a short transent perod, the states track the gven trajectores very cosey. Comparng the resuts n Fgs. 1- and 3-4, t can be seen that the proposed decentrazed adaptve controer presents desrabe performance whch confrms fast convergence of the adaptve parameters. he parameters α 1,1, α1,, 1, α,1, α,, and ρ 1, ρ, the adaptve parameters, are depcted n Fgs. 5-7 respectvey. Fgures 8-9 are aso shown the hstory of the contro nput u, = 1,. From fgures 5-9, t s nterestng to note that (8 mantans a robust performance to a wde cass of perturbatons n the system dynamcs, as ong as the nteractons are bounded. In ths sense, the controers guarantee robustness aganst modeed dynamcs naccuraces. 1 Chnese Contro and Decson Conference 573
5 x 1,1 (rad me(sec Fg. 3. Decentrazed adaptve controer for the frst subsystem ( x1,1 = θ1 x,1 (rad..1 Updated Interactons Upper Bound me(sec. Fg. 7. Convergence of the nteractons upper bounds ( ρ, ρ u me(sec. Fg. 4. Decentrazed adaptve controer for the second subsystem x = θ Frst Subsystem Updated Parameters (, me(sec. Fg. 5. Convergence of the frst subsystem adaptve parameters ( α, α, Second Subsystem Updated Parameters 1.5 1,1 1, me(sec. Fg. 6. Convergence of the second subsystem adaptve parameters ( α, α,,1, u mes(sec. Fg. 8. Contro nput u 1 for the frst subsystem me(sec. Fg. 9. Contro nput u for the second subsystem 5 Concuson hs paper ntroduced a decentrazed adaptve controer for a cass of arge-scae nonnear systems. hs cass conssts of a such systems wth unknown SISO subsystems and unknown nteractons. he proposed adaptve controer ensured the cosed-oop stabty and convergence of the trackng errors asymptotcay to zero. he stabty anayss was performed usng the Lyapunov s theory. he smuaton resuts approved the vadty of the proposed controer. REFERECES Chnese Contro and Decson Conference
6 [1] D. D. Sjak, Decentrazed Contro of Compex Systems. Mathematcs n Scence and Engneerng,vo. 184, Academc Press, San Dego, [] C. Wen, Y. C. Soh, Decentrazed Adaptve Contro Usng Integrator Backsteppng, Automatca, vo. 33, no. 9, , [3] M. C. Han, Y. H. Chen, Decentrazed contro desgn: uncertan systems wth strong nterconnectons, Int. J. Contro, vo. 61, no. 6, , [4] P. A. Ioannou, Decentrazed adaptve contro of nterconnected systems, IEEE Automat. rans. Contr., vo. 31, no. 4, pp , Apr [5] D.. Gave and D. D. Sjak, Decentrazed adaptve contro: Structura condtons for stabty, IEEE rans. Automat. Contr., vo. 34, pp , Apr [6] A. Datta, Performance mprovement n decentrazed adaptve contro: A modfed mode reference scheme, IEEE rans. Automat. Contr., vo. 38, pp , ov [7] L.-C. Fu, Robust adaptve decentrazed contro of robot manpuators, IEEE rans. Automat. Contr., vo. 37, pp , Jan [8] H. Seraj, Decentrazed adaptve contro of manpuators: heory, smuaton, and expermentaton, IEEE rans. Robot. Automat., vo. 5, pp , [9] S. Shekhoesam and C. A. Desoer, Indrect adaptve contro of a cass of nterconnected nonnear dynamca systems, Int. J. Cont., vo. 57, no. 3, pp , [1] C. Wen, Decentrazed adaptve reguaton, IEEE rans. Automatc Contr., vo. 39, pp , Oct [11] L. Sh and S. K. Sngh, Decentrazed adaptve controer desgn for arge-scae systems wth hgher order nterconnectons, IEEE rans. Automat. Contr., vo. 37, pp , Aug [1] Y. Guo, Z. P. Jang, and D. J. H, Decentrazed robust dsturbance attenuaton for a cass of arge-scae nonnear systems, Syst. Contro Lett., vo. 37, pp , [13] Y. ang, M. omzuka, G. Guerrero, and G. Montemayor, Decentrazed robust contro of mechanca systems, IEEE rans. Automat.Contr., vo. 45, pp , Apr.. [14] P. Krshmanurthy, and F. Khorram, Decentrazed contro of arge-scae nonnear systems n generazed output-feedback canonca form, In Proceedngs of the 4th IEEE conference on decson and contro, Orando, pp , 1. [15] B. Karm, M.B. Menhaj, A. Afshar, and I. Saboor, A Decentrazed Drect Adaptve Controer for a Cass of Large-Scae Interconnected onnear Systems, IEEE Internatona Symposum on Integent Sgna Processng, Madrd Span, Oct.,3-5, 7. [16] J.. Spooner and K. M. Passno, Adaptve contro of a cass of decentrazed nonnear systems, IEEE rans. Automat. Contr., vo. 41,pp. 8 84, Feb [17] J.. Spooner, and K. M Passno, Decentrazed adaptve contro of nonnear systems usng rada bass neura networks, IEEE ransactons on Automatc Contro, 44(11, 5 57, Chnese Contro and Decson Conference 575
Research on Complex Networks Control Based on Fuzzy Integral Sliding Theory
Advanced Scence and Technoogy Letters Vo.83 (ISA 205), pp.60-65 http://dx.do.org/0.4257/ast.205.83.2 Research on Compex etworks Contro Based on Fuzzy Integra Sdng Theory Dongsheng Yang, Bngqng L, 2, He
More informationOptimal Guaranteed Cost Control of Linear Uncertain Systems with Input Constraints
Internatona Journa Optma of Contro, Guaranteed Automaton, Cost Contro and Systems, of Lnear vo Uncertan 3, no Systems 3, pp 397-4, wth Input September Constrants 5 397 Optma Guaranteed Cost Contro of Lnear
More informationLECTURE 21 Mohr s Method for Calculation of General Displacements. 1 The Reciprocal Theorem
V. DEMENKO MECHANICS OF MATERIALS 05 LECTURE Mohr s Method for Cacuaton of Genera Dspacements The Recproca Theorem The recproca theorem s one of the genera theorems of strength of materas. It foows drect
More informationON AUTOMATIC CONTINUITY OF DERIVATIONS FOR BANACH ALGEBRAS WITH INVOLUTION
European Journa of Mathematcs and Computer Scence Vo. No. 1, 2017 ON AUTOMATC CONTNUTY OF DERVATONS FOR BANACH ALGEBRAS WTH NVOLUTON Mohamed BELAM & Youssef T DL MATC Laboratory Hassan Unversty MORO CCO
More informationNetworked Cooperative Distributed Model Predictive Control Based on State Observer
Apped Mathematcs, 6, 7, 48-64 ubshed Onne June 6 n ScRes. http://www.scrp.org/journa/am http://dx.do.org/.436/am.6.73 Networed Cooperatve Dstrbuted Mode redctve Contro Based on State Observer Ba Su, Yanan
More information[WAVES] 1. Waves and wave forces. Definition of waves
1. Waves and forces Defnton of s In the smuatons on ong-crested s are consdered. The drecton of these s (μ) s defned as sketched beow n the goba co-ordnate sstem: North West East South The eevaton can
More informationA finite difference method for heat equation in the unbounded domain
Internatona Conerence on Advanced ectronc Scence and Technoogy (AST 6) A nte derence method or heat equaton n the unbounded doman a Quan Zheng and Xn Zhao Coege o Scence North Chna nversty o Technoogy
More informationChapter 6. Rotations and Tensors
Vector Spaces n Physcs 8/6/5 Chapter 6. Rotatons and ensors here s a speca knd of near transformaton whch s used to transforms coordnates from one set of axes to another set of axes (wth the same orgn).
More informationDynamic Analysis Of An Off-Road Vehicle Frame
Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS Dnamc Anass Of An Off-Road Vehce Frame ŞTEFAN TABACU, NICOLAE DORU STĂNESCU, ION TABACU Automotve Department,
More informationExample: Suppose we want to build a classifier that recognizes WebPages of graduate students.
Exampe: Suppose we want to bud a cassfer that recognzes WebPages of graduate students. How can we fnd tranng data? We can browse the web and coect a sampe of WebPages of graduate students of varous unverstes.
More informationDistributed Moving Horizon State Estimation of Nonlinear Systems. Jing Zhang
Dstrbuted Movng Horzon State Estmaton of Nonnear Systems by Jng Zhang A thess submtted n parta fufment of the requrements for the degree of Master of Scence n Chemca Engneerng Department of Chemca and
More informationMARKOV CHAIN AND HIDDEN MARKOV MODEL
MARKOV CHAIN AND HIDDEN MARKOV MODEL JIAN ZHANG JIANZHAN@STAT.PURDUE.EDU Markov chan and hdden Markov mode are probaby the smpest modes whch can be used to mode sequenta data,.e. data sampes whch are not
More informationAssociative Memories
Assocatve Memores We consder now modes for unsupervsed earnng probems, caed auto-assocaton probems. Assocaton s the task of mappng patterns to patterns. In an assocatve memory the stmuus of an ncompete
More informationResearch Article H Estimates for Discrete-Time Markovian Jump Linear Systems
Mathematca Probems n Engneerng Voume 213 Artce ID 945342 7 pages http://dxdoorg/11155/213/945342 Research Artce H Estmates for Dscrete-Tme Markovan Jump Lnear Systems Marco H Terra 1 Gdson Jesus 2 and
More informationSPATIAL KINEMATICS OF GEARS IN ABSOLUTE COORDINATES
SATIAL KINEMATICS OF GEARS IN ABSOLUTE COORDINATES Dmtry Vasenko and Roand Kasper Insttute of Mobe Systems (IMS) Otto-von-Guercke-Unversty Magdeburg D-39016, Magdeburg, Germany E-ma: Dmtr.Vasenko@ovgu.de
More informationRemark: Positive work is done on an object when the point of application of the force moves in the direction of the force.
Unt 5 Work and Energy 5. Work and knetc energy 5. Work - energy theore 5.3 Potenta energy 5.4 Tota energy 5.5 Energy dagra o a ass-sprng syste 5.6 A genera study o the potenta energy curve 5. Work and
More informationA NOVEL DESIGN APPROACH FOR MULTIVARIABLE QUANTITATIVE FEEDBACK DESIGN WITH TRACKING ERROR SPECIFICATIONS
A OVEL DESIG APPROACH FOR MULTIVARIABLE QUATITATIVE FEEDBACK DESIG WITH TRACKIG ERROR SPECIFICATIOS Seyyed Mohammad Mahd Alav, Al Khak-Sedgh, Batool Labb Department of Electronc and Computer Engneerng,
More informationNeural network-based athletics performance prediction optimization model applied research
Avaabe onne www.jocpr.com Journa of Chemca and Pharmaceutca Research, 04, 6(6):8-5 Research Artce ISSN : 0975-784 CODEN(USA) : JCPRC5 Neura networ-based athetcs performance predcton optmzaton mode apped
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More information3. Stress-strain relationships of a composite layer
OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. tress-stran reatonshps of a composte ayer Introducton
More informationAdaptive sliding mode reliable excitation control design for power systems
Acta Technca 6, No. 3B/17, 593 6 c 17 Insttute of Thermomechancs CAS, v.v.. Adaptve sldng mode relable exctaton control desgn for power systems Xuetng Lu 1, 3, Yanchao Yan Abstract. In ths paper, the problem
More informationTHE GUARANTEED COST CONTROL FOR UNCERTAIN LARGE SCALE INTERCONNECTED SYSTEMS
Copyrght 22 IFAC 5th rennal World Congress, Barcelona, Span HE GUARANEED COS CONROL FOR UNCERAIN LARGE SCALE INERCONNECED SYSEMS Hroak Mukadan Yasuyuk akato Yoshyuk anaka Koch Mzukam Faculty of Informaton
More informationPredicting Model of Traffic Volume Based on Grey-Markov
Vo. No. Modern Apped Scence Predctng Mode of Traffc Voume Based on Grey-Marov Ynpeng Zhang Zhengzhou Muncpa Engneerng Desgn & Research Insttute Zhengzhou 5005 Chna Abstract Grey-marov forecastng mode of
More informationMULTIVARIABLE FUZZY CONTROL WITH ITS APPLICATIONS IN MULTI EVAPORATOR REFRIGERATION SYSTEMS
MULTIVARIABLE FUZZY CONTROL WITH I APPLICATIONS IN MULTI EVAPORATOR REFRIGERATION SYSTEMS LIAO QIANFANG Schoo of Eectrca and Eectronc Engneerng A thess submtted to the Nanyang Technoogca Unversty n parta
More informationMultispectral Remote Sensing Image Classification Algorithm Based on Rough Set Theory
Proceedngs of the 2009 IEEE Internatona Conference on Systems Man and Cybernetcs San Antono TX USA - October 2009 Mutspectra Remote Sensng Image Cassfcaton Agorthm Based on Rough Set Theory Yng Wang Xaoyun
More informationAdaptive Architectures for Distributed Control of Modular Systems
Adaptve Archtectures for Dstrbuted Control of Modular Systems ansel Yucelen and Jeff S. Shamma Abstract We consder the dstrbuted control of large-scale modular systems,.e., systems consst of physcally
More informationNumerical Investigation of Power Tunability in Two-Section QD Superluminescent Diodes
Numerca Investgaton of Power Tunabty n Two-Secton QD Superumnescent Dodes Matta Rossett Paoo Bardea Ivo Montrosset POLITECNICO DI TORINO DELEN Summary 1. A smpfed mode for QD Super Lumnescent Dodes (SLD)
More informationLimit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems
Lmt Cycle Generaton for Mult-Modal and 2-Dmensonal Pecewse Affne Control Systems atsuya Ka Department of Appled Electroncs Faculty of Industral Scence and echnology okyo Unversty of Scence 6-3- Njuku Katsushka-ku
More informationAndre Schneider P622
Andre Schneder P6 Probem Set #0 March, 00 Srednc 7. Suppose that we have a theory wth Negectng the hgher order terms, show that Souton Knowng β(α and γ m (α we can wrte β(α =b α O(α 3 (. γ m (α =c α O(α
More informationSimulations and Trajectory Tracking of Two Manipulators Manipulating a Flexible Payload
Smuatons and rajectory racng of wo Manpuators Manpuatng a Fexbe Payoad Peng Zhang Coege of Communcaton Engneerng Jn Unversty Changchun, Chna Yuan-chuan L Coege of Communcaton Engneerng Jn Unversty Changchun,
More informationCOEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN
Int. J. Chem. Sc.: (4), 04, 645654 ISSN 097768X www.sadgurupublcatons.com COEFFICIENT DIAGRAM: A NOVEL TOOL IN POLYNOMIAL CONTROLLER DESIGN R. GOVINDARASU a, R. PARTHIBAN a and P. K. BHABA b* a Department
More informationDistributed Exponential Formation Control of Multiple Wheeled Mobile Robots
Proceedngs of the Internatonal Conference of Control, Dynamc Systems, and Robotcs Ottawa, Ontaro, Canada, May 15-16 214 Paper No. 46 Dstrbuted Exponental Formaton Control of Multple Wheeled Moble Robots
More informationOff-policy Reinforcement Learning for Robust Control of Discrete-time Uncertain Linear Systems
Off-polcy Renforcement Learnng for Robust Control of Dscrete-tme Uncertan Lnear Systems Yonglang Yang 1 Zhshan Guo 2 Donald Wunsch 3 Yxn Yn 1 1 School of Automatc and Electrcal Engneerng Unversty of Scence
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationD hh ν. Four-body charm semileptonic decay. Jim Wiss University of Illinois
Four-body charm semeptonc decay Jm Wss Unversty of Inos D hh ν 1 1. ector domnance. Expected decay ntensty 3. SU(3) apped to D s φν 4. Anaytc forms for form factors 5. Non-parametrc form factors 6. Future
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationChapter 6: Dynamic Simulation Environment
Chapter 6: Dnac Suaton Envronent Prevous neatc and dnac anass has deonstrated the snguart-free worspace and the hgh force-bearng characterstcs of the Wrst. Ths chapter copetes the dnac ode of the Carpa
More informationCyclic Codes BCH Codes
Cycc Codes BCH Codes Gaos Feds GF m A Gaos fed of m eements can be obtaned usng the symbos 0,, á, and the eements beng 0,, á, á, á 3 m,... so that fed F* s cosed under mutpcaton wth m eements. The operator
More informationk p theory for bulk semiconductors
p theory for bu seconductors The attce perodc ndependent partce wave equaton s gven by p + V r + V p + δ H rψ ( r ) = εψ ( r ) (A) 4c In Eq. (A) V ( r ) s the effectve attce perodc potenta caused by the
More information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
More informationNeuro-Adaptive Design II:
Lecture 37 Neuro-Adaptve Desgn II: A Robustfyng Tool for Any Desgn Dr. Radhakant Padh Asst. Professor Dept. of Aerospace Engneerng Indan Insttute of Scence - Bangalore Motvaton Perfect system modelng s
More informationNeuro-Adaptive Design - I:
Lecture 36 Neuro-Adaptve Desgn - I: A Robustfyng ool for Dynamc Inverson Desgn Dr. Radhakant Padh Asst. Professor Dept. of Aerospace Engneerng Indan Insttute of Scence - Bangalore Motvaton Perfect system
More informationXin Li Department of Information Systems, College of Business, City University of Hong Kong, Hong Kong, CHINA
RESEARCH ARTICLE MOELING FIXE OS BETTING FOR FUTURE EVENT PREICTION Weyun Chen eartment of Educatona Informaton Technoogy, Facuty of Educaton, East Chna Norma Unversty, Shangha, CHINA {weyun.chen@qq.com}
More informationQuantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry
Quantum Runge-Lenz ector and the Hydrogen Atom, the hdden SO(4) symmetry Pasca Szrftgser and Edgardo S. Cheb-Terrab () Laboratore PhLAM, UMR CNRS 85, Unversté Le, F-59655, France () Mapesoft Let's consder
More informationDelay tomography for large scale networks
Deay tomography for arge scae networks MENG-FU SHIH ALFRED O. HERO III Communcatons and Sgna Processng Laboratory Eectrca Engneerng and Computer Scence Department Unversty of Mchgan, 30 Bea. Ave., Ann
More informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationApplication of Particle Swarm Optimization to Economic Dispatch Problem: Advantages and Disadvantages
Appcaton of Partce Swarm Optmzaton to Economc Dspatch Probem: Advantages and Dsadvantages Kwang Y. Lee, Feow, IEEE, and Jong-Bae Par, Member, IEEE Abstract--Ths paper summarzes the state-of-art partce
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCOMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN
Transactons, SMRT- COMBINING SPATIAL COMPONENTS IN SEISMIC DESIGN Mchae O Leary, PhD, PE and Kevn Huberty, PE, SE Nucear Power Technooges Dvson, Sargent & Lundy, Chcago, IL 6060 ABSTRACT Accordng to Reguatory
More informationNONLINEAR SYSTEM IDENTIFICATION BASE ON FW-LSSVM
Journa of heoretca and Apped Informaton echnoogy th February 3. Vo. 48 No. 5-3 JAI & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 NONLINEAR SYSEM IDENIFICAION BASE ON FW-LSSVM, XIANFANG
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationNote 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2
Note 2 Lng fong L Contents Ken Gordon Equaton. Probabty nterpretaton......................................2 Soutons to Ken-Gordon Equaton............................... 2 2 Drac Equaton 3 2. Probabty nterpretaton.....................................
More informationwe have E Y x t ( ( xl)) 1 ( xl), e a in I( Λ ) are as follows:
APPENDICES Aendx : the roof of Equaton (6 For j m n we have Smary from Equaton ( note that j '( ( ( j E Y x t ( ( x ( x a V ( ( x a ( ( x ( x b V ( ( x b V x e d ( abx ( ( x e a a bx ( x xe b a bx By usng
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationNested case-control and case-cohort studies
Outne: Nested case-contro and case-cohort studes Ørnuf Borgan Department of Mathematcs Unversty of Oso NORBIS course Unversty of Oso 4-8 December 217 1 Radaton and breast cancer data Nested case contro
More informationRobust Fuzzy Control of Electrical Manipulators
J Intell Robot Syst (21) 6:415 434 DOI 1.17/s1846-1-943-y Robust Fuzzy Control of Electrcal Manpulators Mohammad Mehd Fateh Receved: 1 September 29 / Accepted: 19 Aprl 21 / Publshed onlne: 22 June 21 Sprnger
More informationSupplementary Material: Learning Structured Weight Uncertainty in Bayesian Neural Networks
Shengyang Sun, Changyou Chen, Lawrence Carn Suppementary Matera: Learnng Structured Weght Uncertanty n Bayesan Neura Networks Shengyang Sun Changyou Chen Lawrence Carn Tsnghua Unversty Duke Unversty Duke
More informationOptimum Selection Combining for M-QAM on Fading Channels
Optmum Seecton Combnng for M-QAM on Fadng Channes M. Surendra Raju, Ramesh Annavajjaa and A. Chockangam Insca Semconductors Inda Pvt. Ltd, Bangaore-56000, Inda Department of ECE, Unversty of Caforna, San
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationQuantitative Evaluation Method of Each Generation Margin for Power System Planning
1 Quanttatve Evauaton Method of Each Generaton Margn for Poer System Pannng SU Su Department of Eectrca Engneerng, Tohoku Unversty, Japan moden25@ececetohokuacjp Abstract Increasng effcency of poer pants
More informationOutput Feedback Adaptive Robust Control of Uncertain Linear Systems With Disturbances
Bn Yao 1 School of Mechancal Engneerng, Purdue Unversty, West Lafayette, IN 47907 e-mal: byao@purdue.edu L Xu Research and Advanced Engneerng, Ford Motor Company, Dearborn, MI 48124 Output Feedback Adaptve
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationController Design of High Order Nonholonomic System with Nonlinear Drifts
Internatonal Journal of Automaton and Computng 6(3, August 9, 4-44 DOI:.7/s633-9-4- Controller Desgn of Hgh Order Nonholonomc System wth Nonlnear Drfts Xu-Yun Zheng Yu-Qang Wu Research Insttute of Automaton,
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationarxiv: v1 [physics.comp-ph] 17 Dec 2018
Pressures nsde a nano-porous medum. The case of a snge phase fud arxv:1812.06656v1 [physcs.comp-ph] 17 Dec 2018 Oav Gateand, Dck Bedeaux, and Sgne Kjestrup PoreLab, Department of Chemstry, Norwegan Unversty
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationGlobally Optimal Multisensor Distributed Random Parameter Matrices Kalman Filtering Fusion with Applications
Sensors 2008, 8, 8086-8103; DOI: 10.3390/s8128086 OPEN ACCESS sensors ISSN 1424-8220 www.mdp.com/journa/sensors Artce Gobay Optma Mutsensor Dstrbuted Random Parameter Matrces Kaman Fterng Fuson wth Appcatons
More informationNonlinear Optimal Line-Of-Sight Stabilization with Fuzzy Gain-Scheduling
Word Academy of Scence, Engneerng and echnoogy Internatona Journa of Eectrca and Computer Engneerng Vo:5, No:8, Nonnear Optma Lne-Of-Sght Stabzaton wth Fuzzy Gan-Schedung A. Puras rueba, J. R. Lata García
More informationControl of Uncertain Bilinear Systems using Linear Controllers: Stability Region Estimation and Controller Design
Control of Uncertan Blnear Systems usng Lnear Controllers: Stablty Regon Estmaton Controller Desgn Shoudong Huang Department of Engneerng Australan Natonal Unversty Canberra, ACT 2, Australa shoudong.huang@anu.edu.au
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationIntroduction. - The Second Lyapunov Method. - The First Lyapunov Method
Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February 2009 1 Introducton Stablty s the most promnent characterstc of
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationDevelopment of whole CORe Thermal Hydraulic analysis code CORTH Pan JunJie, Tang QiFen, Chai XiaoMing, Lu Wei, Liu Dong
Deveopment of whoe CORe Therma Hydrauc anayss code CORTH Pan JunJe, Tang QFen, Cha XaoMng, Lu We, Lu Dong cence and technoogy on reactor system desgn technoogy, Nucear Power Insttute of Chna, Chengdu,
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationA MIN-MAX REGRET ROBUST OPTIMIZATION APPROACH FOR LARGE SCALE FULL FACTORIAL SCENARIO DESIGN OF DATA UNCERTAINTY
A MIN-MAX REGRET ROBST OPTIMIZATION APPROACH FOR ARGE SCAE F FACTORIA SCENARIO DESIGN OF DATA NCERTAINTY Travat Assavapokee Department of Industra Engneerng, nversty of Houston, Houston, Texas 7704-4008,
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationMonica Purcaru and Nicoleta Aldea. Abstract
FILOMAT (Nš) 16 (22), 7 17 GENERAL CONFORMAL ALMOST SYMPLECTIC N-LINEAR CONNECTIONS IN THE BUNDLE OF ACCELERATIONS Monca Purcaru and Ncoeta Adea Abstract The am of ths paper 1 s to fnd the transformaton
More informationThe line method combined with spectral chebyshev for space-time fractional diffusion equation
Apped and Computatona Mathematcs 014; 3(6): 330-336 Pubshed onne December 31, 014 (http://www.scencepubshnggroup.com/j/acm) do: 10.1164/j.acm.0140306.17 ISS: 3-5605 (Prnt); ISS: 3-5613 (Onne) The ne method
More informationOn Event-Triggered Adaptive Architectures for Decentralized and Distributed Control of Large- Scale Modular Systems
Unversty of South Florda Scholar Commons Mechancal Engneerng Faculty Publcatons Mechancal Engneerng 26 On Event-Trggered Adaptve Archtectures for Decentralzed and Dstrbuted Control of Large- Scale Modular
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationNeural-Based Decentralized Robust Control of Large-Scale Uncertain Nonlinear Systems with Guaranteed H Performance
Proceedngs of the 45th IEEE Conference on Decson & Control Manchester Grand Hyatt Hotel an Dego CA UA December 3-5 26 Neural-Based Decentralzed Robust Control of Large-cale Uncertan Nonlnear ystems wth
More informationLower bounds for the Crossing Number of the Cartesian Product of a Vertex-transitive Graph with a Cycle
Lower bounds for the Crossng Number of the Cartesan Product of a Vertex-transtve Graph wth a Cyce Junho Won MIT-PRIMES December 4, 013 Abstract. The mnmum number of crossngs for a drawngs of a gven graph
More informationDecentralized Adaptive Regulation for Nonlinear Systems with iiss Inverse Dynamics
Vol. 3, No. ACTA AUTOMATICA SINICA February, 8 Decentralzed Adaptve Regulaton for Nonlnear Systems wth ISS Inverse Dynamcs YAN Xue-Hua, XIE Xue-Jun, LIU Ha-Kuan Abstract Ths paper consders the decentralzed
More informationA General Column Generation Algorithm Applied to System Reliability Optimization Problems
A Genera Coumn Generaton Agorthm Apped to System Reabty Optmzaton Probems Lea Za, Davd W. Cot, Department of Industra and Systems Engneerng, Rutgers Unversty, Pscataway, J 08854, USA Abstract A genera
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationOutput Group Consensus for Heterogeneous Linear Multi-Agent Systems Communicating over Switching Topology
Output Group Consensus for Heterogeneous Lnear Mut-Agent Systems Communcatng over Swtchng Topoogy Jahu Qn Qchao Ma We Xng Zheng Department of Automaton Unversty of Scence and Technoogy of Chna Hefe 37
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationPolite Water-filling for Weighted Sum-rate Maximization in MIMO B-MAC Networks under. Multiple Linear Constraints
2011 IEEE Internatona Symposum on Informaton Theory Proceedngs Pote Water-fng for Weghted Sum-rate Maxmzaton n MIMO B-MAC Networks under Mutpe near Constrants An u 1, Youjan u 2, Vncent K. N. au 3, Hage
More information