A ph mesh refinement method for optimal control

Transcription

1 OPTIMAL CONTROL APPLICATIONS AND METHODS Optm. Contro App. Meth. (204) Pubshed onne n Wey Onne Lbrary (weyonnebrary.com)..24 A ph mesh refnement method for optma contro Mchae A. Patterson, Wam W. Hager 2 and An V. Rao, *, Department of Mechanca and Aerospace Engneerng, Unversty of Forda, Ganesve, FL 326, USA 2 Department of Mathematcs, Unversty of Forda, Ganesve, FL 326, USA SUMMARY A mesh refnement method s descrbed for sovng a contnuous-tme optma contro probem usng coocaton at Legendre Gauss Radau ponts. The method aows for changes n both the number of mesh ntervas and the degree of the approxmatng poynoma wthn a mesh nterva. Frst, a reatve error estmate s derved based on the dfference between the Lagrange poynoma approxmaton of the state and a Legendre Gauss Radau quadrature ntegraton of the dynamcs wthn a mesh nterva. The derved reatve error estmate s then used to decde f the degree of the approxmatng poynoma wthn a mesh shoud be ncreased or f the mesh nterva shoud be dvded nto subntervas. The degree of the approxmatng poynoma wthn a mesh nterva s ncreased f the poynoma degree estmated by the method remans beow a maxmum aowabe degree. Otherwse, the mesh nterva s dvded nto subntervas. The process of refnng the mesh s repeated unt a specfed reatve error toerance s met. Three exampes hghght varous features of the method and show that the approach s more computatonay effcent and produces sgnfcanty smaer mesh szes for a gven accuracy toerance when compared wth fxed-order methods. Copyrght 204 John Wey & Sons, Ltd. Receved 6 September 203; Revsed 29 January 204; Accepted 30 January 204 KEY WORDS: optma contro; coocaton; Gaussan quadrature; varabe-order; mesh refnement. INTRODUCTION Over the past two decades, drect coocaton methods have become popuar n the numerca souton of nonnear optma contro probems. In a drect coocaton method, the state and contro are dscretzed at a set of appropratey chosen ponts n the tme nterva of nterest. The contnuoustme optma contro probem s then transcrbed to a fnte-dmensona nonnear programmng probem (NLP), and the NLP s soved usng a we-known software [, 2]. Orgnay, drect coocaton methods were deveoped as h methods (e.g., Euer or Runge Kutta methods) where the tme nterva s dvded nto a mesh and the state s approxmated usng the same fxed-degree poynoma n each mesh nterva. Convergence n an h method s then acheved by ncreasng the number and pacement of the mesh ponts [3 5]. More recenty, a great dea of research has been carred out n the cass of drect Gaussan quadrature orthogona coocaton methods [6 20]. In a Gaussan quadrature coocaton method, the state s typcay approxmated usng a Lagrange poynoma where the support ponts of the Lagrange poynoma are chosen to be ponts assocated wth a Gaussan quadrature. Orgnay, Gaussan quadrature coocaton methods were mpemented as p methods usng a snge nterva. Convergence of the p method was then acheved by ncreasng the degree of the poynoma approxmaton. For probems whose soutons are smooth and we-behaved, a Gaussan quadrature coocaton method has a smpe structure and converges at an exponenta rate [2 23]. The most we-deveoped Gaussan quadrature methods are those that *Correspondence to: An V. Rao, Department of Mechanca and Aerospace Engneerng, Unversty of Forda, Ganesve, FL 326, USA. E-ma: anvrao@uf.edu Copyrght 204 John Wey & Sons, Ltd.

2 M. A. PATTERSON, W. W. HAGER AND A. V. RAO empoy ether Legendre Gauss ponts [9, 3] or Legendre Gauss Radau (LGR) ponts [4, 5, 7] or Legendre Gauss Lobatto ponts [6]. Whe h methods have a ong hstory and p methods have shown promse n certan types of probems, both the h and p approaches have mtatons. Specfcay, achevng a desred accuracy toerance may requre an extremey fne mesh (n the case of an h method) or may requre the use of an unreasonaby arge-degree poynoma approxmaton (n the case of a p method). In order to reduce sgnfcanty the sze of the fnte-dmensona approxmaton, and thus mprove computatona effcency of sovng the NLP, hp coocaton methods have been deveoped. In an hp method, both the number of mesh ntervas and the degree of the approxmatng poynoma wthn each mesh nterva s aowed to vary. Orgnay, hp methods were deveoped as fnte eement methods for sovng parta dfferenta equatons [24 28]. In the past few years, the probem of deveopng hp methods for sovng optma contro probems has been of nterest [29, 30]. Ths recent research has shown that convergence usng hp methods can be acheved wth a sgnfcanty smaer fnte-dmensona approxmaton than woud be requred when usng ether an h or a p method. Motvated by the desre to mprove computatona effcency when sovng the NLP whe provdng hgh accuracy n the dscrete approxmaton of the optma contro probem, n ths paper, we descrbe a new ph Gaussan quadrature coocaton method for sovng contnuous-tme nonnear optma contro probems. Here, we have deberatey changed the order from hp to ph because our scheme tres to acheve the prescrbed error toerance by ncreasng the degree of the poynomas n the approxmatng space, and f ths fas, by ncreasng the number of mesh ntervas. The method s dvded nto three parts. Frst, an approach s deveoped for estmatng the reatve error n the state wthn each mesh nterva. Ths reatve error estmate s obtaned by comparng the orgna state varabe to a hgher-order approxmaton of the state. Ths reatve error estmate s used to determne f the degree of the poynoma approxmaton shoud be ncreased or f mesh ntervas shoud be added. The poynoma degree s ncreased f t s estmated that the ensung mesh requres a poynoma degree that s ess than a maxmum aowabe degree. Otherwse, the mesh s refned. Ths process s repeated on a seres of meshes unt a specfed accuracy toerance s met. The decson to ncrease the poynoma degree or refne the mesh s based on the rato of the maxmum reatve error and the accuracy toerance and s consstent wth the known exponenta convergence of a Gaussan quadrature method for a probem whose souton s smooth. Varous mesh refnement methods empoyng drect coocaton methods have been descrbed n recent years [5, 29 3]. Reference [3] descrbes a method that empoys a dfferentaton matrx to attempt to dentfy swtches, knks, corners, and other dscontnutes n the souton, and uses Gaussan quadrature rues to generate a mesh that s dense near the end ponts of the tme nterva of nterest. Reference [5] empoys a densty functon and attempts to generate a fxed-order mesh on whch to sove the probem. References [32] and [33] (and the references theren) descrbe a dua weghted resdua method for mesh refnement and goa-orented mode reducton. The dua weghted resdua method uses estmates of a dua mutper together wth oca estmates of the resduas to adaptvey refne a mesh and contro the error n probems governed by parta dfferenta equatons. References [29] and [30] descrbe hp adaptve methods where the error estmate s based on the dfference between an approxmaton of the tme dervatve of the state and the rghthand sde of the dynamcs mdway between the coocaton ponts. It s noted that the approach of References [29] and [30] creates a great dea of nose n the error estmate, thereby makng these approaches computatonay ntractabe when a hgh-accuracy souton s desred. Furthermore, the error estmate of References [29] and [30] does not take advantage of the exponenta convergence rate of a Gaussan quadrature coocaton method. Fnay, n Reference [3], an error estmate s deveoped by ntegratng the dfference between an nterpoaton of the tme dervatve of the state and the rght-hand sde of the dynamcs. The error estmate deveoped n Reference [3] s predcated on the use of a fxed-order method (e.g., trapezod, Hermte Smpson, and Runge Kutta) and computes a ow-order approxmaton of the ntegra of the aforementoned dfference. On the other hand, n ths paper, an estmate of the error on a gven mesh s obtaned by varyng the degree of the poynoma n the Gaussan quadrature approxmaton. Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

3 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL The method of ths paper s fundamentay dfferent from any of the prevousy deveoped methods due to the fact that t takes advantage of the exponenta convergence propertes of a Gaussan quadrature. Specfcay, n the dscretzaton used n ths paper, the state s approxmated usng a pecewse poynoma, and the dynamcs are coocated at the LGR quadrature ponts n each nterva. Ths eads to a fnte-dmensona mathematca programmng probem that s soved to obtan estmates for the contro and the state at the coocaton ponts. A reatve error estmate s then derved that uses the dfference between an nterpoated vaue of the state and an LGR quadrature approxmaton to the ntegra of the state dynamcs. Ths reatve error estmate remans computatonay tractabe when a hgh-accuracy souton s desred and reduces sgnfcanty the number of coocaton ponts requred to meet a specfed accuracy toerance when compared wth the methods of Reference [29] or [30]. Furthermore, dfferent from the error estmate deveoped n Reference [3], the error estmate n ths research utzes the LGR quadrature ponts n both the nterpoaton of the state and the ntegraton of the state aong the nterpoated souton. Consequenty, the approach of ths research enabes the use of a Gaussan quadrature method to estmate the errors, thereby achevng convergence usng fewer coocaton ponts than may be necessary usng an h method wth the error estmate of Reference [3]. It s noted, however, that n the case of an h LGR method (.e., a method whose order s the same n a mesh ntervas), the error estmate n ths paper s smar to the estmate derved n Reference [3] wth the excepton that the approach of ths paper st empoys a hgher-order ntegraton method than the approach of Reference [3]. Whe the mesh refnement method presented n ths paper s based on ony the state error and seems to gnore the error n the costate as we as gnore the contro mnmum prncpe, n earer work (such as References [4 6]), a cose connecton has been estabshed between the frst-order optmaty condtons for the mathematca program and the contnuous frst-order optmaty condtons (.e., the system dynamcs, the costate equaton, and the mnmum prncpe). Specfcay, t s observed that at a souton of the mathematca programmng probem, the mnmum prncpe hods at the coocaton ponts. Thus, by sovng the mathematca program, the mnmum prncpe s satsfed exacty. As has been shown n References [4 6], the dscretzaton of the state equaton eads to an nduced dscretzaton for the costate equaton that s expressed n terms of the mutpers of the mathematca programmng probem. Hence, when the mathematca programmng probem s soved, n essence a two pont boundary-vaue probem s soved n whch the contro has been emnated through the mnmum prncpe. In ths paper, we deveop a mesh refnement method that s based entrey on the estmaton of the error n the state. Whe the error n the nduced costate approxmaton coud be montored usng the same approach as s used to montor the state error, one fnds that the errors connected wth the system dynamcs and the nduced costate equaton are tghty couped n that the nduced costate error s arge durng ntervas where the state error s arge. As a resut, the beneft of montorng the error n the state and the nduced costate s margna when compared wth montorng the error n ony the state. Thus, t s more effcent and smper to focus entrey on the error n the state equaton. The effectveness of our mesh refnement strategy s studed on three exampes that have dfferent features n the optma souton and, thus, exercse dfferent benefts of the new ph approach. It s found that the mesh refnement method deveoped n ths paper s an effectve yet smpe way to generate meshes and to reduce computaton tmes when compared wth fxed-order h methods. The sgnfcance of ths research s threefod. Frst, we deveop a systematc way to estmate the error n the Radau dscretzaton of an optma contro probem usng the fact that Radau coocaton s a Gaussan quadrature ntegraton method. Second, based on the derved estmate of the error, we provde a smpe yet effectve ph mesh refnement method that aows both the degree of the approxmatng poynoma and the number of mesh ntervas to vary. Thrd, we show on three nontrva exampes that the ph mesh refnement method deveoped n ths paper s more computatonay effcent and produces smaer meshes for a gven accuracy toerance when compared wth tradtona fxed-order h methods. Ths paper s organzed as foows. In Secton 2, we provde a motvaton for our new ph method. In Secton 3, we state the contnuous-tme Boza optma contro probem. In Secton 4, we state the ntegra form of the ph LGR ntegraton method [4 6] that s used as the bass for the ph mesh refnement method deveoped n ths paper. In Secton 5, we deveop the error estmate and our Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

4 M. A. PATTERSON, W. W. HAGER AND A. V. RAO new ph-adaptve mesh refnement method. In Secton 6, we appy the method of Secton 5 to three exampes that hghght dfferent features of the method. In Secton 7, we descrbe the key features of our approach and compare our method to recenty deveoped hp-adaptve methods. Fnay, n Secton 8, we provde concusons on our work. 2. MOTIVATION FOR NEW ph-adaptive COLLOCATION METHOD In order to motvate the deveopment of our new ph-adaptve Gaussan quadrature coocaton method, consder the foowng two frst-order dfferenta equatons on the nterva 2 Œ ; C : dy d D f./ D cos./; y. / D y 0 ; () 8 dy < 0; 6 < =2 2 d D f 2./ D cos./; =2 6 6 C=2; ; y : 2. / D y 20 : (2) 0; C=2 < 6 C The soutons to the dfferenta equatons () and (2) are gven, respectvey, as y./ D y 0 C sn./; (3) 8 < y 20 ; 6 < =2; y 2./ D y 20 C C sn./; =2 6 6 C=2; (4) : y 20 C 2; C=2 < 6 C: Suppose now that t s desred to approxmate the soutons to the dfferenta equatons () and (2) usng the foowng three dfferent methods that empoy the LGR [34] coocaton method as descrbed n varous forms n References [4 7, 9]: () a p method where the state s approxmated usng an N th degree poynoma on Œ ; C and N k k s aowed to vary; () an h method usng K equay spaced mesh ntervas where K s aowed to vary and a fxed fourth-degree poynoma s empoyed wthn each mesh nterva; and () a ph method where both the number of mesh ntervas, K, and the degree of the approxmatng poynoma, N k, wthn each mesh nterva are aowed to vary. Usng any of the aforementoned approxmatons (p, h, orph), wthn any mesh nterva ŒT k ;T k, the functons y./;. D ; 2/ are approxmated usng the foowng Lagrange poynoma approxmatons: y.k/ NX C./ Y.k/./ D D Y.k/ `.k/./; `.k/./ D NY C D.k/.k/.k/ ; (5) where the support ponts for `.k/./; D ;:::;N k C, are the N k LGR ponts [34]..k/ ;:::;.k/ N / on ŒT k ;T k such that.k/ D T k and.k/ N C D T k s a noncoocated pont that defnes the end of mesh nterva k. Wthn any partcuar mesh nterva ŒT k ;T k Œ ; C, the approxmatons of y.k/./; D ; 2, are gven at the support ponts.k/ C ;D;:::;N,as NX y.k/.k/ C Y.k/.k/ C D Y.k/ C I.k/ f.k/ ;. D ; 2/;. D ;:::;N/; (6) D where Y.k/ s the approxmaton to y..k/.k/ / at the start of the mesh nterva and I.; D ;:::;N k / s the N k N k LGR ntegraton matrx (see Reference [4] for detas) defned on the mesh nterva ŒT k ;T k. Suppose now that we defne the maxmum absoute error n the souton of the dfferenta equaton as ˇ E D max y ˇˇˇ.k/ ;. D ; 2/: 2Œ;:::;N k k2œ;:::;k ˇY.k/ Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

5 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL Fgure (a) and (b) show the base-0 ogarthm of E as a functon of N for the p method and as a functon of K for the h method. Frst, t s seen that because y./ s a smooth functon, the p method converges exponentay as a functon of N whe the h method converges sgnfcanty more sowy as a functon of K. Fgure (c) and (d) show E 2. Unke y, the functon y 2 s contnuous but not smooth. As a resut, the h method converges faster than the p method because no snge poynoma (regardess of degree) on Π; C s abe to approxmate the souton to equaton (2) as accuratey as a pecewse poynoma. However, whe the h method converges more qucky than does the p method when approxmatng the souton of equaton (2), t s seen that the h method does not converge as qucky as the p method does when approxmatng the souton to equaton (). In fact, when approxmatng the souton of equaton (), t s seen that the h method acheves an (a) (b) (c) (d) (e) Fgure. Base-0 ogarthm of absoute errors n soutons of equatons () and (2) at Lagrange poynoma support ponts usng p, h, andph methods. Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

6 M. A. PATTERSON, W. W. HAGER AND A. V. RAO error of 0 7 for K D 24, whereas the p method converges exponentay and acheves an error of 0 5 for N D 20. As a resut, an h method does not provde the fastest possbe convergence rate when approxmatng the souton to a dfferenta equaton whose souton s smooth. Gven the aforementoned p and h anayss, suppose now that the souton to equaton (2) s approxmated usng the aforementoned ph Radau method (.e., both the number of mesh ntervas and the degree of the approxmatng poynoma wthn each mesh nterva are aowed to vary). Assume further that the ph method s constructed such that the tme nterva Œ ; C s dvded nto three mesh ntervas Œ ; =2, Œ =2; C=2, andœc=2; C, and Lagrange poynoma approxmatons of the form of equaton (5) of degree N, N 2,andN 3, respectvey, are used n each mesh nterva. Furthermore, suppose N, N 2,andN 3 are aowed to vary. Because the souton y 2./ s a constant n the frst and thrd mesh ntervas, t s possbe to set N D N 3 D 2 and vary ony N 2. Fgure (e) shows the error n y 2./, E 2ph D max y 2 Y 2 usng the aforementoned three-nterva ph approach. Smar to the resuts obtaned usng the p method when approxmatng the souton of equaton (), n ths case, the error n the souton of equaton (2) converges exponentay as a functon of N 2. Thus, whe an h method may outperform a p method on a probem whose souton s not smooth, t s possbe to mprove the convergence rate by usng a ph-adaptve method. The foregong anayss provdes a motvaton for the deveopment of the ph method descrbed n the remander of ths paper. 3. BOLZA OPTIMAL CONTROL PROBLEM Wthout oss of generaty, consder the foowng genera optma contro probem n Boza form. Determne the state, y.t/ 2 R n y, the contro u.t/ 2 R n u, the nta tme, t 0, and the termna tme, t f, on the tme nterva t 2 Œt 0 ;t f that mnmze the cost functona J D.y.t 0 /; t 0 ; y.t f /; t f / C g.y.t/; u.t/; t/ dt (7) t 0 subect to the dynamc constrants dy D a.y.t/; u.t/; t/; (8) dt the nequaty path constrants c mn 6 c.y.t/; u.t/; t/ 6 c max ; (9) and the boundary condtons b mn 6 b.y.t 0 /; t 0 ; y.t f /; t f / 6 b max : (0) The functons, g, a, c,andb are defned by the foowng mappngs: W R n y R R n y R! R; g W R n y R n u R! R; a W R n y R n u R! R n y ; c W R n y R n u R! R n c ; b W R n y R R n y R! R n b; where a vector functons of tme are treated as row vectors. In ths presentaton, t w be usefu to modfy the Boza probem gven n equatons (7) (0) as foows. Let 2 Œ ; C be a new ndependent varabe such that t D t f t 0 C t f C t 0 : () 2 2 The Boza optma contro probem of equatons (7) (0) s then defned n terms of the varabe as foows. Determne the state, y./ 2 R n y, the contro, u./ 2 R n u, the nta tme, t 0, and the termna tme, t f, on the tme nterva 2 Œ ; C that mnmze the cost functona J D.y. /; t 0 ; y.c/; t f / C t f t 0 2 Z tf Z C g.y./; u./; I t 0 ;t f /d (2) Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

7 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL subect to the dynamc constrants dy d D t f t 0 a.y./; u./; I t 0 ;t f /; (3) 2 the nequaty path constrants and the boundary condtons c mn 6 c.y./; u./; I t 0 ;t f / 6 c max ; (4) b mn 6 b y. /; t 0 ; y.c/; t f 6 bmax : (5) Suppose now that the tme nterva 2 Œ ; C s dvded nto a mesh consstng of K mesh ntervas S k D ŒT k ;T k ; k D ;:::;K,where.T 0 ;:::;T K / are the mesh ponts. Themesh K[ K\ ntervas S k.k D ;:::;K/have the propertes that S k D Œ ; C and S k D;, whe the mesh ponts have the property that D T 0 <T <T 2 < <T K DC.Lety.k/./ and u.k/./ be the state and contro n S k. The Boza optma contro probem of equatons (2) (5) can then be rewrtten as foows. Mnmze the cost functona J D y./. /; t 0 ; y.k/.c/; t f C t f t 0 KX Z Tk g y.k/./; u.k/./; I t 0 ;t f d; 2 T kd k.k D ;:::;K/; (6) subect to the dynamc constrants dy.k/./ D t f t 0 a y.k/./; u.k/./; I t 0 ;t f ;.k D ;:::;K/; (7) d 2 the path constrants c mn 6 c y.k/./; u.k/./; I t 0 ;t f 6 c max ;.k D ;:::;K/; (8) and the boundary condtons b mn 6 b y./. /; t 0 ; y.k/.c/; t f 6 b max : (9) Because the state must be contnuous at each nteror mesh pont, t s requred that the condton y.t k / D y.t C k /;.k D ;:::;K / be satsfed at the nteror mesh ponts.t ;:::;T K /. kd kd 4. LEGENDRE GAUSS RADAU COLLOCATION METHOD The ph form of the contnuous-tme Boza optma contro probem n Secton 3 s dscretzed usng coocaton at LGR ponts [4 7, 9]. In the LGR coocaton method, the state of the contnuoustme Boza optma contro probem s approxmated n S k ;k2 Œ;:::;K,as y.k/./ Y.k/./ D N k C X D Y.k/ `.k/./; `.k/./ D N k C Y D.k/.k/.k/ ; (20) where 2 Œ ; C ; `.k/./; D ;:::;N k C, s a bass of Lagrange poynomas,.k/ ;:::;.k/ N k are the LGR [34] coocaton ponts n S k D ŒT k ;T k /,and.k/ N k C D T k s a noncoocated pont. Dfferentatng Y.k/./ n equaton (20) wth respect to, we obtan dy.k/./ d D N k C X D Y.k/./ : (2) d d`.k/ Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

8 M. A. PATTERSON, W. W. HAGER AND A. V. RAO The cost functona of equaton (6) s then approxmated usng a mutpe-nterva LGR quadrature as J Y./ ;t 0; Y.K/ N K C ;t f C KX N X k kd D t f t 0 2 w.k/ g Y.k/ ; U.k/ ;.k/ I t 0 ;t f ; (22) where w.k/. D ;:::;N k / are the LGR quadrature weghts [34] n S k D ŒT k ;T k, k 2 Œ;:::;K, U.k/ ; D ;:::;N k, are the approxmatons of the contro at the N k LGR ponts n mesh nterva k 2 Œ;:::;K ; Y./ s the approxmaton of y.t 0 /,andy.k/ N K C s the approxmaton of y.t K / (where we reca that T 0 D and T K DC). Coocatng the dynamcs of equaton (7) at the N k LGR ponts usng equaton (2), we have where N k C X D D.k/ Y.k/ D.k/ t f t 0 2 d`.k/..k/ D d / a Y.k/ ; U.k/ ;.k/ I t 0 ;t f D 0;. D ;:::;N k /; ;. D ;:::;N k ; D ;:::;N k C /; are the eements of the N k.n k C / LGR dfferentaton matrx [4] D.k/ assocated wth S k ;k2 Œ;:::;K. Whe the dynamcs can be coocated n dfferenta form, n ths paper, we choose to coocate the dynamcs usng the equvaent mpct ntegra form (see References [4 6] for detas). The mpct ntegra form of the LGR coocaton method s gven as Y.k/ C Y.k/ t f t 0 2 N X k D I.k/ a Y.k/ ; U.k/ ;.k/ I t 0 ;t f D 0;. D ;:::;N k /; (23) where I.k/ ;.D;:::;N k; D ;:::;N k ;k D ;:::;K/s the N k N k LGR ntegraton matrx n mesh nterva k 2 Œ;:::;K ; t s obtaned by nvertng a submatrx of the dfferentaton matrx formed by coumns 2 through N k C : h I.k/ D D.k/ 2 D.k/ N k C : It s noted for competeness that I.k/ D.k/ D (see References [4 6]), where s a coumn vector of ength N k of a ones [4 6]. Next, the path constrants of equaton (8) n S k ;k2œ;:::;k, are enforced at the N k LGR ponts as c mn 6 c Y.k/ ; U.k/ ;.k/ I t 0 ;t f 6 c max ;. D ;:::;N k /: (24) The boundary condtons of equaton (9) are approxmated as b mn 6 b Y./ ;t 0; Y.K/ N K C ;t f 6 b max : (25) It s noted that contnuty n the state at the nteror mesh ponts k 2 Œ;:::;K s enforced va the condton Y.k/ N k C D Y.kC/ ;.k D ;:::;K /; (26) where the same varabe s used for both Y.k/ N k C and Y.kC/. Hence, the constrant of equaton (26) s emnated from the probem because t s taken nto account expcty. The NLP that arses from the LGR coocaton method s then to mnmze the cost functon of equaton (22) subect to the agebrac constrants of equatons (23) (25). Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

9 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL 5. ph-adaptive MESH REFINEMENT METHOD We now deveop a ph-adaptve mesh refnement method usng the LGR coocaton method descrbed n Secton 4. We ca our method a ph method because we frst try to adust the poynoma degree to acheve convergence, and f ths fas, we adust the mesh spacng. The ph-adaptve mesh refnement method deveoped n ths paper s dvded nto two parts. In Secton 5., the method for estmatng the error n the current souton s derved, and n Secton 5.4, the p-then-h strategy s deveoped for refnng the mesh. 5.. Error estmate n each mesh nterva In ths secton, an estmate of the reatve error n the souton wthn a mesh nterva s derved. Because the state s the ony quantty n the LGR coocaton method for whch a unquey defned functon approxmaton s avaabe, we deveop an error estmate for the state. The error estmate s obtaned by comparng two approxmatons to the state, one wth hgher accuracy. The key dea s that for a probem whose souton s smooth, an ncrease n the number of LGR ponts shoud yed a state that more accuratey satsfes the dynamcs. Hence, the dfference between the souton assocated wth the orgna set of LGR ponts and the approxmaton assocated wth the ncreased number of LGR ponts shoud yed an estmate for the error n the state. Assume that the NLP of equatons (22) (25) correspondng to the dscretzed contro probem has been soved on a mesh S k D ŒT k ;T k ; k D ;:::;K, wth N k LGR ponts n mesh nterva S k. Suppose that we want to estmate the error n the state at a set of M k D N k C LGR ponts O.k/.k/ ;:::;O M k,whereo.k/ D.k/ D T k,andthato.k/ M k C D T k. Suppose further that the vaues of the state approxmaton gven n equaton (20) at the ponts O.k/.k/ ;:::;O M k are denoted Y.O.k/.k/ /;:::;Y.O M k /. Next, et the contro be approxmated n S k usng the Lagrange nterpoatng poynoma N X k U.k/./ D D U.k/ Ò.k/./; Ò.k/./ D N Y k D.k/.k/.k/ ; (27) and et the contro approxmaton at O.k/ be denoted U.O.k/ /, 6 6 M k. We use the vaue of the rght-hand sde of the dynamcs at.y.o.k/ /; U.O.k/ /; O.k/ / to construct an mproved approxmaton of the state. Let OY.k/ be a poynoma of degree at most M k that s defned on the nterva S k.ifthe dervatve of Oy.k/ matches the dynamcs at each of the Radau quadrature ponts O.k/ ;66M k, then we have OY.k/ O.k/ DY.k/. k /C t M f t 0 X k 2 O I.k/ D a Y.k/ O.k/ ; U.k/ O.k/ ; O.k/ ; D2;:::;M k C ; (28) where I O.k/ ;;D;:::;M k,sthem k M k LGR ntegraton matrx correspondng to the LGR ponts defned by O.k/.k/ ;:::;O M k. Usng the vaues Y.O.k/ / and OY.O.k/ /, D ;:::;M k C, the absoute and reatve errors n the th component of the state at.o.k/.k/ ;:::;O M k C / are then defned, respectvey, as E.k/ O.k/.k/ D ˇ OY O.k/ Y.k/ O.k/ ˇˇˇ ; E.k/ e.k/ O.k/ O.k/ D ;:::;Mk C ; : (29) D ˇˇˇ; D ;:::;n y ; ˇ C max ˇY.k/ O.k/ 2Œ;:::;M k C Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

10 M. A. PATTERSON, W. W. HAGER AND A. V. RAO The maxmum reatve error n S k s then defned as e.k/ max D max 2Œ;:::;ny 2Œ;:::;M k C e.k/.o.k/ /: (30) 5.2. Ratonae for error estmate The error estmate derved n Secton 5. s smar to the error estmate obtaned usng the modfed Euer Runge Kutta scheme to numercay sove a dfferenta equaton Py.t/ D f.y.t//.thefrst- order Euer method s gven as y C D y C hf.y /; (3) where h s the step sze and y s the approxmaton to y.t/ at t D t D h. In the second-order modfed Euer Runge Kutta method, the frst stage generates the foowng approxmaton Ny to y.t C=2 /: Ny D y C 2 hf.y /: The second stage then uses the dynamcs evauated at Ny to obtan an mproved estmate Oy C of y.t C /: Oy C D y C hf. Ny/: (32) The orgna Euer scheme starts at y and generates y C. The frst-stage varabe Ny s the nterpoant of the ne (frst-degree poynoma) connectng.t ;y / and.t C ;y C / evauated at the new pont t C=2. The second stage gven n equaton (32) uses the dynamcs at the nterpoant Ny to obtan an mproved approxmaton to y.t C /. Because Oy C s a second-order approxmaton to y.t C / and y C s a frst-order approxmaton to y C, the absoute dfference Oy C y C s an estmate of the error n y C n a manner smar to the absoute error estmate E.k/.O.k/ /.D ;:::;M k C / derved n equaton (29). The effectveness of the derved error estmate derved n Secton 5. can be seen by revstng the motvatng exampes of Secton 2. Fgure 2(a) and (b) show the p and h error estmates, respectvey, E p and E h, n the souton to equaton (); (c) and (d) show the p and h error estmates, respectvey, E 2p and E 2h, n the souton to equaton (2); and (e) shows the ph error estmates, E 2ph, n the souton to equaton (2). It s seen that the error estmates are neary dentca to the actua error. The reatve error estmate gven n equaton (30) s used n the next secton as the bass for modfyng an exstng mesh Estmaton of requred poynoma degree wthn a mesh nterva Suppose agan the LGR coocaton NLP of equatons (22) (25) has been soved on a mesh S k ;kd ;:::;K. Suppose further that t s desred to meet a reatve error accuracy toerance n each mesh nterva S k ;kd;:::;k. If the toerance s not met n at east one mesh nterva, then the next step s to refne the current mesh, ether by dvdng the mesh nterva or ncreasng the degree of the approxmatng poynoma wthn the mesh nterva. Consder a mesh nterva S q ;q2œ;:::;k,wheren q LGR ponts were used to sove the NLP of equatons (22) (25), and agan, et be the desred reatve error accuracy toerance. Suppose further that the estmated maxmum reatve error, e max,.q/ has been computed as descrbed n Secton 5. and that e max.q/ > (.e., the accuracy toerance s not satsfed n the exstng mesh nterva). Fnay, et N mn and N max be user-specfed mnmum and maxmum bounds on the number of LGR ponts wthn any mesh nterva. Accordng to the convergence theory summarzed n [35, 36], the error n a goba coocaton scheme behaves ke O.N 2:5 k /,wheren s the number of coocaton ponts wthn a mesh nterva and k s the number of contnuous dervatves n the souton [2, 22]. If the souton s smooth, then we coud take k D N. Hence, f N was repaced by N C P, then the error bound decreases by at east the factor N P. Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

11 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL (a) (b) (c) (d) (e) Fgure 2. Base-0 ogarthm of absoute error estmates n soutons of equatons () and (2) at ponts O2 ;:::;O Mk usng p, h,andph methods. Based on these consderatons, suppose that nterva S q empoys N q coocaton ponts and has reatve error estmate e max.q/ that s arger than the desred reatve error toerance ; to reach the desred error toerance, the error shoud be mutped by the factor =e max..q/ Ths reducton s acheved by ncreasng N q by P q where P q s chosen so that N P q q D =e max.q/ or, equvaenty, Ths mpes that N P q q P q D og Nq D e.q/ max :! e max.q/ : (33) Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

12 M. A. PATTERSON, W. W. HAGER AND A. V. RAO Because the expresson on the rght sde of (33) may not be an nteger, we round up to obtan &!' e max.q/ P q D og Nq : (34) Note that P q > 0 because we ony use (34) when e max.q/ s greater than the prescrbed error toerance. The dependence of P q on N q sshownnfgure p-then-h strategy for mesh refnement Usng equaton (34), the predcted number of LGR ponts requred n mesh nterva S q on the ensungmeshs QN q D N q C P q, assumng e max.q/ has not reach the specfed error toerance. The ony possbtes are that QN q 6 N max (that s, QN does not exceed the maxmum aowabe poynoma degree) or that QN q >N max (.e., QN exceeds the maxmum aowabe poynoma degree). If QN q 6 N max,thenn q s ncreased to QN q on the ensung mesh. If, on the other hand, QN q >N max, then QN q exceeds the upper mt and the mesh nterva S q must be dvded nto subntervas. Our strategy for mesh nterva dvson uses the foowng approach. Frst, whenever a mesh nterva s dvded, the sum of the number of coocaton ponts n the newy created mesh ntervas shoud equa the predcted poynoma degree for the next mesh. Second, each newy created subnterva shoud contan the mnmum aowabe number of coocaton ponts. In other words, f a mesh nterva S q s dvded nto B q subntervas, then each newy created subnterva w contan N mn coocaton ponts and the sum of the coocaton ponts n these newy created subntervas shoud be B q N mn. Usng ths strategy, the number of subntervas, B q, nto whch S q s dvded s computed as & '! N Qq B q D max ;2 ; (35) where t s seen n equaton (35) that 2 6 B q 6 d QN q =N mn e. It s seen that ths strategy for mesh nterva dvson ensures that the same tota number of coocaton ponts s the same regardess of whether the poynoma degree n a mesh nterva s ncreased or the mesh nterva s refned. Second, because the number of LGR ponts n a newy created mesh nterva s started at N mn,the method uses the fu range of aowabe vaues of N. Because of the herarchy, the ph method of ths paper can be thought of more precsey as a p-then-h method where p refnement s exhausted pror to performng any h refnement. In other words, the poynoma degree wthn a mesh nterva s ncreased unt the upper mt N max s exceeded. The h refnement (mesh nterva dvson) s then performed after whch the p refnement s restarted. N mn (a) (b) Fgure 3. Functon that reates the ncrease n the degree of the approxmatng poynoma to the rato e max = and the current poynoma degree N. Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

13 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL It s mportant to note that the ph method deveoped n ths paper can be empoyed as a fxedorder h method smpy by settng N mn D N max.thehverson of the method of ths paper s smar to an adaptve step-sze fxed-order ntegraton method, such as an adaptve step-sze Runge Kutta method, n the foowng respect: In both cases, the mesh s refned, often by step-havng or step-doubng [37], when the specfed error toerance s not met. A summary of our adaptve mesh refnement agorthm appears next. Here, M denotes the mesh refnement teraton, and n each oop of the agorthm, the mesh number ncreases by. The agorthm termnates n step 4 when the error toerance s satsfed or when M reaches a prescrbed maxmum M max. 6. EXAMPLES In ths secton, the ph-adaptve LGR method descrbed n Secton 5 s apped to three exampes from the open terature. The frst exampe s a varaton of the hypersenstve optma contro probem orgnay descrbed n Reference [38], where the effectveness of the error estmate derved n Secton 5. s demonstrated and the mproved effcency of the ph method over varous h methods s shown. The second exampe s a tumor ant-angogeness optma contro probem orgnay descrbed n Reference [39], where t s seen that the ph method of ths paper accuratey and effcenty captures a dscontnuty n a probem whose optma contro s dscontnuous. The thrd exampe s the reusabe aunch vehce entry probem from Reference [3], where t s seen that usng the ph method of ths paper eads to a sgnfcanty smaer mesh than woud be obtaned usng an h method. Ths thrd exampe aso shows that aowng N mn to be too sma can reduce the effectveness of the ph method. When usng a ph-adaptve method, the termnoogy ph-.n mn ;N max / refers to the ph-adaptve method of ths paper where the poynoma degree can vary between N mn and N max, respectvey, whe an h-n method refers to an h method wth a poynoma of fxed degree N.Forexampe, a ph-.2; 8/ method s a ph-adaptve method where N mn D 2 and N max D 8, whe an h-2 method s an h method where N D 2. A resuts were obtaned usng the optma contro software GPOPS II [40] runnng wth the NLP sover IPOPT [4] n second dervatve mode wth the mutfronta massvey parae sparse drect sover MUMPS [42], defaut NLP sover toerances, and a mesh refnement accuracy toerance D 0 6. The nta mesh for a ph-.n mn ;N max / or h-n mn method conssted of 0 unformy spaced mesh ntervas wth N mn LGR ponts n each nterva, whe the nta guess was a straght ne between the known nta condtons and known termna condtons for the probem under consderaton wth the guess on a other varabes beng a constant. The requred frst and second dervatves requred by IPOPT were computed usng the Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

14 M. A. PATTERSON, W. W. HAGER AND A. V. RAO but-n sparse frst and second fnte-dfferencng method n GPOPS II that uses the method of Reference [9]. Fnay, a computatons were performed on a 2.5 GHz Inte Core 7 MacBook Pro runnng Mac OS-X Verson (Lon) 6 GB of 333 MHz DDR3 RAM and MATLAB verson R202b. The CPU tmes reported n ths paper are 0-run averages of the executon tme. Exampe : hypersenstve probem Consder the foowng varaton of the hypersenstve optma contro probem [38]. Mnmze the cost functona subect to the dynamc constrant and the boundary condtons J D 2 Z tf 0 x 2 C u 2 dt (36) Px D x C u (37) x.0/ D :5 ; x.t f / D ; (38) where t f s fxed. It s known that for suffcenty arge vaues of t f, the souton to the hypersenstve probem exhbts a so-caed take-off, cruse, and andng structure where a of the nterestng behavors occur near the take-off and andng segments whe the souton s essentay constant n the cruse segment. Furthermore, the cruse segment becomes an ncreasngy arge percentage of the tota traectory tme as t f ncreases, whe the take-off and andng segments have rapd exponenta decay and growth, respectvey. The anaytc optma state and contro for ths probem are gven as where c x.t/ D c exp.t p 2/ C c 2 exp. t p 2/; u.t/ DPx.t/ C x.t/; D p p c 2 exp. t f 2/ exp.tf 2/ (39) p :5 exp. tf 2/ p : (40) :5 exp.t f 2/ Fgure 4(a) and (b) show the exact state and contro for the hypersenstve probem wth t f D 0000 and hghght the take-off, cruse, and andng features of the optma souton. Gven the structure of the optma souton, t shoud be the case that a mesh refnement method paces many more coocaton and mesh ponts near the ends of the tme nterva when t f s arge. Fgure 4(c) shows the evouton of the mesh ponts T k whe Fgure 4(d) shows the evouton coocaton (LGR) ponts on each mesh refnement teraton usng the ph-.3; 4) scheme. Two key reated features are seen n the mesh refnement. Frst, Fgure 4(c) shows that mesh ntervas are added on each refnement teraton ony n the regons near t D 0 and t D t f, whe mesh ntervas are not added n the nteror regon t 2 Œ000; Second, Fgure 4(d) shows that after the frst mesh refnement teraton, LGR ponts are aso added ony n the regons near t D 0 and t D t f and are not added n the nteror regon t 2 Œ000; Ths behavor of the ph-adaptve method shows that error reducton s acheved by added mesh and coocaton ponts n regons of t 2 Œ0; t f where ponts are needed to capture the changes n the souton. Fnay, for comparson wth the ph-adaptve method, Fgure 4(e) and (f) show the souton obtaned usng an h-2 method. Unke the ph-.3; 4/ method, where mesh ponts are added ony where needed to meet the accuracy toerance, the h-2 method paces many more mesh ponts over much arger segments at the start and end of the overa tme nterva. Specfcay, t s seen that the mesh s qute dense over tme ntervas t 2 Œ0; 3000 and t 2 Œ7000; 0000, whereas for the ph-.3; 4/ method, the mesh remans dense over the smaer ntervas Œ0; 000 and Œ9000; Admttedy, the ph-.3; 4/ does add LGR ponts n the regons t 2 Œ000; 3000 and t 2 Œ7000; 9000, whereas the h-2 method adds more mesh ntervas, but the mesh obtaned usng the ph-.3; 4/ s much smaer (273 coocaton ponts) than the mesh.k/ Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

15 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL (a) (b) (c) (d) (e) (f) Fgure 4. Exact souton to Exampe wth t f D 0000 and mesh refnement hstory when usng the ph-.3; 4/ and h-2 methods wth an accuracy toerance D 0 6. obtaned usng the h-2 method (672 coocaton ponts). Thus, the ph method expots the souton smoothness on the ntervas Œ000; 3000 and Œ7000; 9000 to acheve more rapd convergence by ncreasng the degree of the approxmatng poynomas nstead of ncreasng the number of mesh ntervas. Next, we anayze the quaty of the error estmate of Secton 5. by examnng more cosey the numerca souton near t D 0 and t D t f. Fgure 5(a) and (b) show the state and contro n the regons t 2 Œ0; 5 and t 2 Œ9985; 0000 on each mesh refnement teraton aongsde the exact souton usng the ph-.3; 4/ method, whe Tabe I shows the estmated and exact reatve errors n the state and the exact reatve error n the contro for each mesh refnement teraton. Frst, t s seen n Tabe I that the state and contro reatve error on the fna mesh s qute sma at 0 9 for the state and 0 8 for the contro. In addton, t s seen from Fgure 5(a) and (b) show that the state and contro approxmatons mprove wth each mesh refnement teraton. Moreover, the Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

16 M. A. PATTERSON, W. W. HAGER AND A. V. RAO (a) (b) (c) (d) Fgure 5. Souton near end ponts of t 2 Œ0; t f for Exampe wth t f an accuracy toerance D 0 6. D 0000 usng the ph-.3; 4/ and Tabe I. Estmated reatve state error, ex max, exact reatve state error, emax x;exact, and exact reatve contro error, eu;exact max, for Exampe wth t f D 0000 usng a ph-.3; 4/ wth an accuracy toerance D 0 6. M e max x e max x;exact e max u;exact 3: : : : : : : : : : : : error estmate shown n Tabe I agrees quatatvey wth the soutons on the correspondng mesh as shown n Fgure 5(a) and (b). It s aso nterestng to see that the state reatve error estmate s approxmatey the same on each mesh teraton as the exact reatve error. The consstency n the reatve error approxmaton and the exact reatve error demonstrates the accuracy of the error estmate derved n Secton 5.. Thus, the error estmate derved n ths paper refects correcty the ocatons where the souton error s arge and ph-adaptve method constructs new meshes that reduce the error wthout makng the mesh overy dense. Fnay, we provde a comparson of the computatona effcency and mesh szes obtaned by sovng Exampe usng the varous ph-adaptve and h methods descrbed n Secton 5. Tabe II shows the CPU tmes and mesh szes, where t s seen for ths exampe that the ph-.3; 4/ [shown n bod n Tabe II] and h-2 methods resut n the smaest overa CPU tmes (wth the ph-.3; 4/ beng sghty more computatonay effcent than the h-2 method). Interestngy, whe the ph-.3; 4/ and h-2 methods have neary the same computatona effcency, the ph-.3; 4/ produces a sgnfcanty smaer mesh (N D 293 LGR ponts, neary the smaest among a of the methods) whe the h-2 mesh produced a much arger mesh (N D 672 LGR ponts, by far the argest among Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

17 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL Tabe II. Mesh refnement resuts for Exampe usng varous ph-adaptve and h methods. Inta mesh Mesh refnement Tota Constrant N mn N max CPU tme (s) CPU tme (s) CPU tme (s) N K M Jacoban densty (%) 2 2 0:34 3:0 3: : :33 6:93 7: : :33 6:77 7: : :33 6:2 6: : :33 5:7 5: : :33 6:33 6: : :33 3:28 3: : :32 3:89 4: : :32 3:43 3: : :33 3:79 4: : :33 3:49 3: : :34 3:86 4: : :33 3:65 3: : :33 3:37 3: : :33 3:69 4: : :33 4:3 4: : :33 4:95 5: :43 a of the dfferent methods). In fact, Tabe II shows for ths exampe that, for any fxed vaue N mn, the ph-.n mn ;N max / methods produced smaer mesh szes than the correspondng h-n mn method. Thus, whe an h method may perform we on ths exampe because of the structure of the optma souton, the ph method produces the souton n the most computatonay effcent manner whe smutaneousy producng a sgnfcanty smaer mesh. Exampe 2: tumor ant-angogeness optma contro probem Consder the foowng tumor ant-angogeness optma contro probem taken from Reference [39]. The obectve s to mnmze J D y.t f / (4) subect to the dynamc constrants y.t/ Py.t/ D y.t/ n ; h y 2.t/ Py 2.t/ D q.t/ b dy 2=3.t/ Gu.t/ ; wth the nta condtons the contro constrant y.0/ D Œ.b /=d 3=2 =2; y 2.0/ D Œ.b /=d 3=2 =4; (42) (43) 0 6 u 6 u max ; (44) and the ntegra constrant Z tf 0 u./d 6 A; (45) where G D 0:5, b D 5:85, d D 0:00873, D 0:02, u max D 75, A D 5,andt f s free. A souton to ths optma contro probem s shown usng the ph-.3; 0/ method n Fgure 6(a) and (b). Upon coser examnaton, t s seen that a key feature n the optma souton s the fact that the optma contro s dscontnuous at t 0:2. In order to mprove the accuracy of the souton n the Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

18 M. A. PATTERSON, W. W. HAGER AND A. V. RAO (a) (b) (c) (d) Fgure 6. Souton and mesh refnement hstory for Exampe 2 usng the ph-.3; 0/ method wth an accuracy toerance of 0 6. Tabe III. Mesh refnement resuts for Exampe 2 usng varous ph-adaptve and h methods. Inta mesh Mesh refnement Tota Constrant N mn N max CPU tme (s) CPU tme (s) CPU tme (s) N K M Jacoban densty (%) 2 2 0: 0:72 0: : :0 0:88 0: : :0 :02 : : :0 0:75 0: : :0 0:74 0: : :0 0:69 0: : :7 0:55 0: : :7 0:7 0: : :7 0:86 : : :7 0:62 0: : :7 :56 : : :6 0:90 : : :6 0:70 0: : :6 0:96 : : :6 0:88 : : :6 : : : :6 :0 : :40 vcnty of ths dscontnuty, t s necessary that ncreased numbers of coocaton and mesh ponts are paced near t D 0:2. Fgure 6(b) shows the contro obtaned on the fna mesh by the ph-.3; 0/ method. Interestngy, t s seen that the ph-.3; 0/ method concentrates the coocaton and mesh ponts near t D 0:2. Examnng the evouton of the mesh refnement, t s seen n Fgure 6(c) and (d) Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

19 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL that the mesh densty ncreases on each successve mesh teraton, but remans unchanged n regons dstant from t 0:2. The reason that the mesh s modfed near t D 0:2 s because the accuracy of the state s owest n the regon near the dscontnuty n the contro. In order to mprove souton accuracy, addtona coocaton and mesh ponts are requred near t D 0:2. Thus, the ph method performs propery when sovng ths probem as t eaves the mesh untouched n regons where few coocaton and mesh ponts are needed, and t ncreases the densty of the mesh where addtona ponts are requred. Next, Tabe III summarzes the CPU tmes and mesh szes that were obtaned by sovng Exampe 2 usng the varous ph and h methods descrbed Secton 5. Whe for ths exampe the CPU tmes are qute sma, t s st seen that computatona effcency s ganed by choosng a ph method over an h method. Specfcay, t s seen that the ph-.3; 0/ method [shown n bod n Tabe III] produces the owest CPU tme wth an h-3 method beng sghty ess effcent than the ph-.3; 0/ method. More mportanty, Tabe III shows the sgnfcant reducton n mesh sze when usng a ph method. For exampe, usng a ph-.3; N max / or ph-.4; N max /, the maxmum number of LGR ponts (a) (b) (c) (d) (e) (f) Fgure 7. Souton to Exampe 3 usng the ph-.2; 4/ method wth an accuracy toerance of 0 6. Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

20 M. A. PATTERSON, W. W. HAGER AND A. V. RAO s N D 99, whereas the owest number of LGR ponts usng ether an h-2 or h-3 or h-4 method s N D 6. Moreover, whe the ph-.3; 4/ and h-2 methods have neary the same computatona effcency, the ph-.3; 4/ produces a sgnfcanty smaer mesh (N D 293 LGR ponts, neary the smaest among a of the methods) whe the h-2 mesh produced a much arger mesh (N D 672 LGR ponts, by far the argest among a of the dfferent methods). Thus, whe an h method may perform we on ths exampe because of the structure of the optma souton, the ph method produces the souton n the most computatonay effcent manner whe smutaneousy producng the smaest mesh. Exampe 3: reusabe aunch vehce entry Consder the foowng optma contro probem from Reference [3] of maxmzng the cross range durng the atmospherc entry of a reusabe aunch vehce. Mnmze the cost functona subect to the dynamc constrants Pr D v sn Pv D D m ; P D v cos sn r cos g sn ;P D L cos mv and the boundary condtons J D.t f / (46) g v v r cos ; P D v cos cos ; P D L sn r mv cos ; v cos sn tan C ; r (47) r.0/ D r 0 ;r.t f / D r f ;.0/D 0 ;.t f / D Free;.0/ D f ;.t f / D Free ;v.0/dv 0 ;v.t f / D v f ; (48).0/ D 0 ;.t f / D f ;.0/D 0 ;.t f / D Free: It s noted that the mode and the numerca vaues.r 0 ;r f ; 0 ; f ;v 0 ;v f ; 0 ; f ; 0 / are taken from Reference [3] wth the excepton that a quanttes n Reference [3] are gven n Engsh unts whe the vaues used n ths exampe are n Système Internatona (SI) unts. A typca souton of ths probem s shown n Fgure 7(a) (f) usng the ph-.2; 4/ method. It s seen that the souton to ths exampe s reatvey smooth; athough there seems to be a rapd change n the ange of attack n Fgure 7(e) near t D 2000, the tota defecton s at most one degree. As a resut, one mght hypothesze that t s possbe to obtan an accurate souton wth a reatvey Tabe IV. Mesh refnement resuts for Exampe 3 usng varous ph-adaptve and h methods. Inta mesh Mesh refnement Tota Constrant N mn N max CPU tme (s) CPU tme (s) CPU tme (s) N K M Jacoban densty (%) 2 2 0:76 :56 2: : :76 :55 2: : :76 2:22 2: : :75 :33 2: : :3 :53 2: : :3 :09 2: : :3 :6 2: : :3 0:78 2: : :32 0:78 2: : :32 0:78 2: : :9 0:90 4: : :20 0:80 4: : :9 0:68 3: : :20 0:43 3: : :2 0:44 3: : :9 0:43 3: :025 Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)

21 A ph MESH REFINEMENT METHOD FOR OPTIMAL CONTROL (a) (b) Fgure 8. Mesh refnement hstory for Exampe 3 usng the ph-.2; 4/ method wth an accuracy toerance of 0 6. sma number of coocaton and mesh ponts when compared wth an h method. Ths hypothess s confrmed n Tabe IV where severa nterestng trends are observed. Frst, t s seen that the ph-.2; 4/ and ph-.2; 6/ methods [shown n bod n Tabe IV] are the most computatonay effcent. In partcuar, the ph-.2; 4/ and ph-.2; 6/ methods are 30%, 44%, and 6% faster, respectvey, than the h-2, h-3, andh-4 methods. Aso, t s seen that the ph-.2; 4/ and ph-.2; 6/ methods produce smaer meshes (wth a tota of 70 coocaton ponts) when compared wth the h-2, h-3, or h-4 methods (where the tota numbers of coocaton ponts are 486, 26, and88, respectvey). Next, Fgure 8(a) and (b) show the evouton of the meshes for the ph-.2; 4/ method, where t s seen that the number of mesh ntervas ncreases from that of the nta mesh ony at the very end of the traectory because of the rapd change of the fght path ange near t D t f. As a resut, for the vast maorty of the souton, the argest decrease n error s obtaned by usng a arger poynoma degree n each mesh nterva and usng fewer mesh ntervas. In ths case, the fact that the ph-.2; 4/ and ph-.2; 6/ methods outperform the other methods (n partcuar, outperform the h methods) s consstent wth the fact that the souton to ths probem s smooth, havng ony reatvey sma oscatons n the attude and fght path ange. Thus, as stated, the accuracy toerance can be acheved by usng reatvey few mesh ntervas wth a hgh-degree poynoma approxmaton n each nterva. 7. DISCUSSION Each of the exampes ustrates dfferent features of the ph-adaptve mesh refnement method deveoped n Secton 5. The frst exampe shows how the computatona effcency of the ph-adaptve mesh refnement scheme s smar to the computatona effcency of an h method whe generatng a much smaer mesh for a gven accuracy toerance than s requred when usng an h method. Ths frst exampe aso demonstrates the effectveness of the error estmate derved n Secton 5.. The second exampe shows how the ph-adaptve method can effcenty capture a dscontnuty n the souton by makng the mesh more dense near the dscontnuty whe smutaneousy not pacng unnecessary mesh and coocaton ponts n regons dstant from the dscontnuty. Furthermore, smar to the resuts obtaned n the frst exampe, the second exampe shows the sgnfcanty smaer mesh that s generated usng the ph-adaptve method when compared wth the mesh generated usng an h method. Next, the thrd exampe demonstrates how the ph-adaptve method does not unnecessary add mesh ntervas when t s ony necessary to ncrease the degree of the poynoma approxmaton to acheve a desred accuracy toerance. Next, the method of ths paper takes advantage of the fact that n regons where the souton s smooth, t s possbe to gan sgnfcant accuracy by ncreasng the degree of the poynoma approxmaton. On the other hand, f the estmated poynoma degree exceeds a gven threshod (.e., N exceeds the upper mt N max ) wthout satsfyng the accuracy toerance, then a further Copyrght 204 John Wey & Sons, Ltd. Optm. Contro App. Meth. (204)