Underwaer vehicles: The iniu ie proble M. Chyba Deparen of Maheaics 565 McCarhy Mall Universiy of Hawaii, Honolulu, HI 968 Eail: chyba@ah.hawaii.edu H. Sussann Deparen of Maheaics Rugers Universiy, Piscaaway, NJ 8854 Eail: sussann@hilber.rugers.edu H. Maurer, G. Vossen Insiu für Nuerische Maheaik Wesfälische Wilhels-Universiä Münser Einseinsrasse 6, D-4849 Münser, Gerany Eail: aurer@ah.uni-uenser.de Absrac We consider he iniu ie proble for a class of underwaer vehicles. We focus on he siuaion of iniial and final configuraions a res saisfying x x f, z = z f, θ = θ f =. We suppleen our heory wih a nuerical sudy of opial bang bang and singular soluions and include a discussion on a possible Fuller like phenoenon. I. INTRODUCTION In his paper we pursue he analysis of he iniu ie proble for a special class of underwaer vehicles, see [5], [6] for previous works. More precisely, he fully acuaed conrolled echanical syse we consider here describes he equaions of oion for an ellipsoidal, neurally buoyan, uniforly disribued vehicle suberged in an infiniely large volue of incopressible, irroaional and inviscid fluid a res a infiniy. Moreover, we resric he vehicle o ove in he plane (x, z) and we assue i is fully acuaed by he use of hrusers. Boundedness on he power of each hruser is assued. The priary goal is o undersand he srucure of he ie opial rajecories. Our copuaions are based on he Ponryagin axiu principle. Many quesions have o be answered: local and global bounds on he nuber of swichings, opialiy of singular arcs, ec. In [6] we sudy he -singular exreals while heir opialiy is discussed in [5]. We focus here our aenion on he following special siuaion of iniial and final configuraions boh a res: x x f, z = z f, θ = θ f =. We conjecure ha in his case an opial rajecory us conain oally bang arcs as well as u 3 -singular arcs. The heoreical conjecures are suppored by a nuerical sudy of opial soluions for differen values of he oen of ineria I. For higher values of I we obain Research suppored in par by NSF gran DMS-364 a purely bang-bang conrol in all hree coponens wih a large oal nuber of swiching ies. Specifically, for I = we ge a oal nuber of 9 swiching ies. The srucure of he bang-bang conrol is deerined by applying nonlinear prograing echniques o he discreized conrol proble using a fine grid. We hen proceed o a refineen of he soluion by direcly opiizing he swiching ies. This approach enables us o perfor a nuerical es of second-order sufficien condiions [], [], [], [], [3] which shows ha he copued soluion is indeed a srong local iniu. For a saller and ore realisic value of I (here I =.), he nonlinear prograing approach wih. grid poins yields a raher coplicaed cobinaion of bang-bang and singular arcs ha suppor our hypohesis of a chaering conrol exhibiing a Fuller-like phenoenon [9], [4]. However, opialiy of he copued bang-singular soluion canno be shown since applicable sufficien opialiy condiions are no available in he lieraure. Clearly our assupions in he odel are idealized o a high degree and we canno draw any pracical conclusion fro our resuls. However, our siplified odel raises challenging heoreical and nuerical quesions and could be a good saring poin o sudy a ore coplee odel aking ino accoun viscous effecs. II. THE MODEL For ore deails on he odel of underwaer vehicles we consider here, see for insance [5]. We resric our underwaer vehicle o oions in a plane. The variables x and z denoes respecively he absolue horizonal and verical posiions of he vehicle, while θ represens is orienaion. The horizonal and verical velociies of he vehicle in he body frae coordinaes are denoed v, v 3.
Ω is he scalar angular rae in he plane. We assue he body-fluid ass ers in he body horizonal and verical direcions, no o be equal, while I is he body-fluid oen of ineria in he plane. Under few assupions on he fluid and on he vehicle and neglecing viscous effecs, he equaions of oion for he conservaive syse are given by ẋ ż θ v v 3 Ω = cos θv + sin θv 3 cos θv 3 sin θv Ω v 3 Ω 3 v Ω v v 3 I We assue he vehicle acuaed wih hrusers. More precisely, we consider he following inpus: u is a force in he body -axis, u is a force in he body -axis and u 3 is a pure orque in he plane. I follows ha he equaions of oion of our fully acuaed underwaer vehicle are described by an affine conrol syse: ẇ = f(w) + 3 g i (w)u i, i= where he drif f is given by he equaions of oion of he conservaive syse described above, he g i are he following consan vecor fields: g = (,,,,, ), g = (,,,,, ), g 3 = (,,,,, I ), and he conrols u i are easurable bounded funcions. Moreover, o reflec he fac ha he hrusers have liied power, we assue he following consrains on he inpus: u i, i =,, 3. Our syse belongs o he class of conrolled echanical syses for which a sudy on he iniu ie proble wih bounded conrols was iniiaed in [7]. III. CONJECTURE Our previous resuls are based on he axiu principle which gives necessary condiions for a rajecory o be opial. For he conrol syse we consider here he Hailonian funcion akes he for: H(w, λ, u) = λ f(w) + λ 4 u + λ 5 u + λ 6 I u 3 where he λ i s are he adjoin variables. Fro he axiu principle, he adjoin variable are soluions of λ = H w which can be wrien as: λ =, () λ =, () λ 3 = λ (v sin θ v 3 cos θ), +λ (v cos θ + v 3 sin θ) (3) λ 4 = λ cos θ + λ sin θ λ 5 Ω λ 6 v 3 α, (4) λ 5 = λ sin θ λ cos θ + λ 4 Ω λ 6 v α, (5) λ 6 = λ 3 + λ 4 v 3 λ 5 v, (6) where α = 3 I is a nonzero consan. A soluion of he axiu principle is called an exreal. The axiizaion condiion iplies ha u i 3 () = sign(λ i ()) if λ i (), i = 4, 5, 6. In oher words, he srucure of he ie opial pah is governed by he zeroes of he adjoin variables λ i+3 = λ g i (w). We call λ 4, λ 5, λ 6 he swiching funcions. An exreal is said o be u i bang if he conrol u i akes i value in {, +} for alos every ie and is said o be oally bang if i is u i bang for all conrols. A consequence is ha an exreal is u i bang if he corresponding swiching funcion does no vanish alos everywhere. If here exiss a nonepy inerval such ha a given swiching funcion is idenically zero, no inforaion on he conrol u i is direcly provided by he axiu principle. We hen say ha he exreal is u i singular on ha inerval. An exreal is oally singular if i is u i singular for all conrols a he sae ie. A ie s such ha u i is no alos everywhere consan on any inerval of he for ] s ε, s + ε[, ε >, is called a swiching ie for u i. I is well known ha for every fully acuaed conrolled echanical syse here is no oally singular exreal, see [5]. In [7] he singular exreals are analyzed (i.e. exreals such ha conrols are singular a he sae ie) and i is proved ha along such exreals here is a os one swiching for he nonsingular conrol. The singular exreals such ha he singular conrols are idenically zero correspond o oions ha are horizonal (u = u 3 ) or verical (u = u 3 ) ranslaions in body frae coordinaes or pure roaions (u = u ). Noice ha if here is a swiching ie along a exreal, hen he nonsingular conrols have o be idenically zero. Opialiy of hese rajecories is discussed in [5]. I is proved ha he horizonal ranslaion in body frae coordinaes beween wo configuraions a res wih a u swiching a half ie is no ie opial. Generalizaion of his resul o verical ranslaions in body frae coordinaes and o pure roaions, boh cases for iniial and final configuraions a res and a half ie swiching on he nonsingular conrols, uses siilar arguens. Noice ha he rajecories described above are he fases beween he se of singular exreals joining wo posiions a res (indeed i iplies ha we need a leas
one swiching ie). Then, a ie opial rajecory beween wo posiions a res canno be a singular exreal. I is an easy verificaion ha he arguens used in [5] sill hold wihou he res condiion on he iniial and final configuraions. In oher words, we have he following. PROPOSITION 3.: Horizonal and verical ranslaions in body frae coordinaes as well as pure roaions are no ie opial rajecories. In paricular, singular exreals wih one swiching ie on he nonsingular conrol are no ie opial. REMARK 3.: Since we sudy he proble of joining wo posiions a res a conrol has o be eiher idenically zero or has o conain a leas one swiching. I follows ha an opial rajecory beween wo posiions a res canno be a singular exreal or be fored by concaenaions of singular exreals. The siulaions show an opial rajecory ha does no conain any singular arc. x final posiion iniial posiion z Fig.. Iniial and Final configuraions Le us now resric o he siuaion when he iniial and final configuraions are respecively w = (,,,,, ) and w f = (x f,,,,, ), see Fig.. Nuerical soluions are presened in Secion IV. Fro Reark 3. we know ha an opial rajecory for his choice of configuraions is no a concaenaion of singular arcs. We hen have o find a way o go faser. Le us assue for he nex copuaions ha he bounds on he conrol u 3 are very big, in oher words assue we can apply a pure orque alos as big as we wan. Now, insead of resricing ourself o an horizonal oion in body fixed frae using only full power on he force in he -axis direcion we assue he vehicle o be roae in such angle so ha we use boh forces in he and axis direcion. Le us copue his angle and show ha we go faser. Since we do no wan any verical oion along our pah, we us saisfy ż which is, in oher words: cos θv 3 () sin θv () = (7) for each. If we ake he axiu value for he wo forces u, u, we ge v () = and v 3 () = (we focus only on he firs half of he rajecory since i is syeric). I follows ha equaion (7) becoes cos θ + sin θ = which can be wrien as sin θ cos θ = an θ = (8) (i is obvious ha he opial soluion will no be π ). Using now he fac ha he previous equaion leads o sin θ = + 3 and ha ( 3 ẋ() = cos θv + sin θv 3 = sin θ + ) we have: ( 3 ẋ() = + + ). (9) 3 We clai ha he coefficien uliplying in equaion (9) is sricly bigger han. Indeed, his coefficien is ( 3 + ) + 3 = 3 + = + >. 3 In oher words, if θ is deerined such ha equaion (8) is verified, hen we have ẋ() >. This is he angle a which we can ove he fases along he sraigh line since i uses he axiu power on boh hrusers u, u. Noice ha due o he consrain Ω = i has o be a sric u 3 singular arc. I sees reasonable o believe ha an opial rajecory has o be syeric. Then, since our copuaions on he previously described u 3 singular arc hold if θ is replaced by π θ a plausible conjecure for an opial rajecory would be ha i is a concaenaion of oally bang and sric u 3 singular arcs as follows. We begin and end wih oally bang arcs (o ove he vehicle o he righ orienaion), while a hird oally bang arc would connec he wo u 3 singular pieces. The nuerical copuaions, see Secion IV, show ha for our choice of values of I,, when he bounds on he conrol are aken o be equal o one hen an opial rajecory is sill fored by oally bang arcs as well as u 3 singular arcs while if he bounds are very sall hen here is no longer a u 3 singular arc. IV. NUMERICAL BANG-BANG AND SINGULAR SOLUTIONS For nuerical copuaion we choose he values = 3. and = 5.6 and he boundary condiions x() =, x( f ) = and z() = z( f ) =. We sudy he behavior of he opial conrols w.r.. he paraeer I. For higher values of I, he opial conrols are bang-bang in all coponens, whereas for saller values of I, he conrol coponen u 3 conains singular arcs while u, u are bang bang. The copuaions for a wide range of values I show ha he opial conrol u is bang bang wih one swiching ie f / and he conrol sequence u () =. Resuls for he following wo cases were obained by discreizing he opial conrol proble and hen using he AMPL/LOQO prograing and opiizaion environen [8], [6].
Case I = : Using grid poins and Heun s inegraion ehod for ODEs, AMPL/LOQO gives he purely bang bang conrols u, u 3 and he corresponding swiching funcions λ 5, λ 6 shown in Fig.. u u 3.5.5.5.5 λ 5 4 4 6 λ 6.3.....3 Fig.. I = : opial conrols u (), u 3 () and swiching funcions λ 5 (), λ 6 (). v.5.4.3.. Fig. 3. v 3..5.5. Velociies v () and v 3 (). Fig. shows ha here is a oal of 9 swiching ies for all hree conrol coponens. To copue he swiching ies =: < <... < 9 < := f and he opial final ie = f wih higher precision han he grid size, we use he conrol package NUDOCCCS of Büskens [3], [4] in cobinaion wih an arc paraerizaion ehod described in Kaya, Noakes []. In addiion, his approach will allow us o verify he second order sufficien condiions (SSC) for bang bang conrols which have recenly been developed in [], [], [3]. The opial conrol proble is reforulaed as a finie-di. opiizaion proble using he opiizaion vecor z := (ξ,..., ξ ), ξ k := k k, k =,...,, where ξ k denoes he arc duraion of he k-h bang-bang arc. The resuling scaled proble (cf. proble forulaion (PM) in [], p. 8) is given by in f = k= ξ k s.. ẇ = ζ(f(w) + g (w)u + g (w)u + g 3 (w)u 3 ), ζ() := ξ k for I k := [ k, k ], k =,..,, w() = (,,,,, ), w() = (,,,,, ). The values of u() in I k, he arc duraions ξ k and he swiching ies k are lised in Table, Fig. 4. k ξ k k u Ik u Ik u 3 Ik.786.786 -.389466.9599 - - 3.3449999 3.374598-4.37987736 3.7539378 - - 5.6396 4.8789738-6.663853 5.884868-7.37894378 6.63646 - - 8.34453 8.6747876 - - - 9.37937 8.954578 - -.7887 9.636866 - Fig. 4. Table - Swiching ies and opial conrol srucure The copued iniial values for he adjoin variables are λ () =.933, λ () =., λ 3 () =.5879, λ 4 () =.566, λ 5 () =.4336, λ 6 () =.96. The swiching funcions λ i (), i = 4, 5, 6, are nonzero in he inerior of he bang bang arcs and saisfy he conrol law u i () = sign(λ i+3 ()), i =,, 3; cf. Fig.. Furherore, he sric bang-bang propery in [] is saisfied in all swiching ies k. Boh conrols u and u 3 swich a 5 = f / due o syerie. Hence, he assupion in [], [] ha differen conrol coponens have differen swiching ies is no fulfilled here. However, one ay argue ha due o syery his assupion can be dropped here. The es for SSC in [], [], [], [], [3] requires furher o copue he Hessian of he Lagrangian associaed wih he above opiizaion proble. Then one has o show ha he Hessian is posiive definie on he kernel of he Jacobian of he erinal consrains. Here he Hessian is a arix while he Jacobian is 6 arix. The projeced Hessian is a (4 4)-arix ha has salles eigenvalue.65 and, hence, is posiive definie. Thus he soluion in Table provides a srong local iniu. Moreover, based on SSC a sensiiviy analysis of opial soluions and swiching ies w.r.. variaions of syse paraeers can be perfored along he lines of he approach in []. Deailed nuerical resuls for he paraeric sensiiviy derivaives will be given elsewhere. Case I =. : Here, we use a very fine grid wih 4 grid poins and again Heun s ehod for
AMPL/LOQO. The opial ie is copued as 9.54699 while he opial conrol candidaes u (), u 3 () and he corresponding swiching funcions λ 5 (), λ 6 () are depiced in Fig. 5. A zoo on Fig. 5 shows ha he conrol u 3 conains in fac five singular arcs. I is ineresing o reark ha along hese arcs, eiher here is no swiching for u, u or boh swiching funcions λ 4, λ 5 have o vanish a he sae ie. For insance, on he hird u 3 singular arc, here is a u swiching a f / while he swiching funcion λ 5 vanishes (bu does no change sign). I can be explained wih he adjoin equaions. Indeed, i is a consequence of he fac ha in he equaion for λ 6 we have he wo ers λ 4 v 3 λ 5 v while λ 3 is absoluely coninuous. A ore general resul can be saed. PROPOSITION 4.: Le i, j, k be all disincs. Le γ be a u i singular arc and assue his arc o be u j, u k bang. Then, along γ he swiching funcions corresponding o he bang conrols u j, u k have o vanish a he exacly sae ies. Noice ha if a swiching funcion vanishes i does no iply ha here is a swiching ie for he corresponding conrol. Anoher ineresing poin o noice fro he picures u.5.5 λ 5 5 5 5 This could no have happened along a oally bang bang arc. Indeed if we assue he exreal o be u and u bang, hen he condiion Ω iplies v v 3 ( ) = u 3. If u 3 is ±, hen we us have v v 3 consan. Assue Ω. Then, since u, u are bang we have v and v 3 consans which iplies ha v and v 3 are linear. Naely, wo linear funcions such ha heir produc is consan have o be consan. In our case i would ean ha u and u are zero which is a conradicion. If Ω is a consan ha is d no zero we can wrie he equaion for d (v v 3 ) = and replace v and v by funcions depending on v 3 only. This gives us a polynoial of order 4 for v 3, u v3 4 Ω + v3 3 v 3 u γ + γ Ω =, where γ, γ are nonzero coefficiens, which iplies ha v 3 has o be consan if u and u are. The nuerical opiizaion resuls raise anoher ineresing quesion. I concerns he juncion beween bang and singular arcs. Fig. 5 suggess ha he conrol u 3 is chaering a hese juncions. This is no surprising since he order of he singular arc is q = (cf. [4]) which eans ha along a sric u 3 -singular arc we need o copue he fourh derivaive of λ 6 o obain an expression for he singular conrol. Moreover, he nuerical resuls show ha he srenghened Legendre Clebsch condiion holds and he conrol is disconinuous a all juncions. Corollary in McDanell and Powers [4] hen iplies ha he conrol is chaering hus providing anoher ineresing exaple of a Fuller like phenoenon. u 3.5.5 λ 6.3.....3 θ.5.5 Ω 3 3 Fig. 5. I =. : opial conrols u (), u 3 (), heir swiching funcions λ 5 (), λ 6 () and θ(), Ω(). is ha if Ω is consan hen i is along a u 3 singular arc.
- "!#$&%('*)+",.-/.3.....3.5..5.5. =?>A@B5CED.5... =?>A@IBKJ..3.... =?>NCBKD.3 Fig. 6. =?>NCB.4.. 354687:9 ;9/<.4.5..5.5. =F>G@BHD.5... =F>G@BML..3.... =?>OCPBHQPD.3 =F>AD Hooopy wih respec o he value I REMARK 4.: The siulaions show ha for each choice of bounds α 3 < < β 3 for he conrol u 3 : α 3 u 3 β 3, here exiss a value I a which we have he ransiion beween oally bang-bang soluions and bangsingular soluions, see Fig. 6. Equivalenly, if we fix I and vary he bounds on he conrol here exiss α 3 < < β 3 a which he ransiion occurs. For insance for he value I =. he soluion is oally bang-bang if we consider he bounds u 3.8 insead of, he ransiion value being.7. I eans ha he power we allow on he orque influence he srucure of he ie-opial rajecories for our choice of iniial and final configuraions a res. Based on he picures for θ and Ω a possible explanaion of his phenoenon could be ha along a u 3 singular exreal Ω needs o be alos consan. This eans ha u 3 has o balance he er v v 3 ( ) and if here is no enough power on he orque i is ipossible. REFERENCES [] A.A. Agrachev, G. Sefani and P.L. Zezza, Srong opialiy for a bang bang rajecory, SIAM J. Conrol and Opiizaion, vol. 4, No. 4,, pp. 99 4. [] C. Büskens, J.-H.R. Ki, H. Maurer and Y. Kaya, Opiizaion ehods for he verificaion of second order sufficien condiions for bang bang conrols, subied [3] C. Büskens and H. Maurer, Sensiiviy analysis and real ie opiizaion of paraeric nonlinear prograing probles, in: Online Opiizaion of Large Scale Syses, M. Gröschel e al., eds., Springer Verlag, Berlin,, pp. 3 6. [4] C. Büskens and H. Maurer, SQP ehods for solving opial conrol probles wih conrol and sae consrains: adjoin variables, sensiiviy analysis and real ie conrol, J. of Copuaional and Applied Maheaics, vol.,, pp. 85 8. [5] M. Chyba, A suprising non opial pah for underwaer vehicle, Proc. of he 4nd IEEE Conf. on Decision on Conrol, 3. [6] M. Chyba, N.E. Leonard and E.D. Sonag, Opialiy for underwaer vehicles, in Proc. of he 4nd IEEE Conf. on Decision on Conrol, pp. 44 49,. [7] M. Chyba, N.E. Leonard and E.D. Sonag, Singular rajecories in uli-inpu ie-opial probles. Applicaion o conrolled echanical syses, Journal on Dynaical and Conrol Syses, vol. 9 (), pp. 73 88, 3. [8] R. Fourer, D.M. Gay and B.W. Kernighhan, AMPL: A Modeling Language for Maheaical Prograing, Duxbury Press, Brooks- Cole Publishing Copany, 993. [9] A.T. Fuller, Sudy of an opiu nonlinear conrol syse, J. Elecronics Conrol, vol. 5, 963, pp. 63-7. [] C.Y. Kaya and J.L. Noakes, Copuaional ehod for ie-opial swiching conrol, J. of Opiizaion Theory and Applicaions, vol. 7, 3, pp. 69 9. [] J.-H.R. Ki and H. Maurer, Sensiiviy analysis of opial conrol probles wih bang-bang conrols, in Proc. of he 4nd IEEE Conf. on Decision and Conrol, Maui, USA, Dec. 3, pp. 38 386. [] H. Maurer and N.P. Osolovskii, Quadraic sufficien opialiy condiions for bang bang conrol probles, Conrol and Cyberneics, vol. 33, 3, pp. 555 584. [3] H. Maurer and N.P. Osolovskii, Second order sufficien condiions for ie opial bang bang conrol probles, SIAM J. Conrol and Opiizaion, vol. 4, 4, pp. 39 63. [4] J.P. McDanell and W.F. Powers, Necessary condiions for joining opial singular and nonsingular subarcs, SIAM J. on Conrol, vol. 9, 97, pp. 6 73. [5] E.D. Sonag and H.J. Sussann, Tie-opial conrol of anipulaors, in Proc. IEEE In. Conf. on Roboics and Auoaion, 986, pp. 69 697. [6] R.S. Vanderbei and D.F. Shanno, An inerior poin algorih for nonconvex aheaical prograing, Cop. Opi. Appl., vol. 3, 999, pp. 3 5.