DIRECT collocation methods for solving a continuous

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1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY Adaptve Mesh Refnement Method for Optmal Control Usng Decay Rates of Legendre Polynomal Coeffcents Abstract An adaptve mesh refnement method for solvng optmal control problems s descrbed. The method employs orthogonal collocaton at Legendre Gauss Radau ponts. Accuracy n the method s acheved by adustng the number of mesh ntervals, the polynomal degree wthn each mesh nterval, and, when possble, reducng the mesh sze. The decson to ncrease the degree of the polynomal wthn a mesh nterval or to create new mesh ntervals s based on the decay rate of the coeffcents of a Legendre polynomal approxmaton of the state as a functon of the ndex of the Legendre polynomal expanson. The polynomal degree n a mesh nterval s ncreased f the Legendre polynomal coeffcent decay rate exceeds a user-specfed threshold. Otherwse, the mesh nterval s dvded nto subntervals. The method developed n ths bref s then demonstrated on two examples, one of whch s a practcal problem n arcraft performance optmzaton. It s found that the approach developed n ths bref s more effcent, s smpler to mplement, and requres the specfcaton of fewer user-defned parameters when compared wth recently developed adaptve mesh refnement methods for optmal control. Index Terms Collocaton methods, Legendre polynomals, mesh refnement, optmal control. I. INTRODUCTION DIRECT collocaton methods for solvng a contnuous optmal control problem are mplct smulaton methods where the state and control are parameterzed and the constrants n the contnuous optmal control problem are enforced at a specally chosen set of collocaton ponts. The approxmaton obtaned usng collocaton results n a fntedmensonal nonlnear programmng problem (NLP) [1] and the NLP s then solved usng a well-nown software []. Tradtonal drect collocaton methods tae the form of an h method (for example, Euler or Runge Kutta methods), where the doman of nterest s dvded nto a mesh, the state s approxmated usng the same fxed-degree polynomal n each mesh nterval, convergence s acheved by ncreasng the number and placement of the mesh ponts [1]. Manuscrpt receved October 4, 015; revsed May 3, 016 and January 6, 017; accepted Aprl 1, 017. Date of publcaton June 6, 017; date of current verson June 11, 018. Ths wor was supported n part by the U.S. Offce of Naval Research under Grant N , n part by the U.S. Natonal Scence Foundaton under Grant CBET , Grant DMS-1569, and Grant CMMI-15635, and n part by the U.S. Ar Force Research Laboratory under Contract FA D-0108/0054. Ths paper was recommended by Assocate Edtor A. Serran. (Correspondng author: Anl V. Rao.) F. Lu s wth the Department of Mechancal and Aerospace Engneerng, Unversty of Florda, Ganesvlle, FL 3611 USA (e-mal: clu11@ufl.edu). W. W. Hager s wth the Department of Mathematcs, Unversty of Florda, Ganesvlle, FL 3611 USA (e-mal: hager@ufl.edu). A. V. Rao s wth the Department of Mechancal and Aerospace Engneerng, Unversty of Florda, Ganesvlle, FL 3611 USA (e-mal: anlvrao@ufl.edu). Color versons of one or more of the fgures n ths paper ares avalable onlne at Dgtal Obect Identfer /TCST Fengn Lu, Wllam W. Hager, and Anl V. Rao In contrast to an h method, n recent years so called p methods have been developed. In a p method, the number of ntervals s fxed and convergence s acheved by ncreasng the degree of the polynomal approxmaton n each nterval. In order to acheve maxmum effectveness, p methods have been developed usng collocaton at Gaussan quadrature ponts [3] [5]. For problems whose solutons are smooth and well-behaved, Gaussan quadrature collocaton converges at an exponental rate [6] [10]. Gauss quadrature collocaton methods use ether Legendre Gauss [4], Legendre Gauss Radau (LGR) [5], or Legendre Gauss Lobatto [3] ponts. Varous h or p drect collocaton methods have been developed prevously. Gong et al. [11] descrbe a p method that used a dfferentaton matrx to dentfy potental dscontnutes n the soluton. Zhao and Tsotras [1] develop an h method that uses a densty functon to generate a sequence of nondecreasng sze meshes on whch to solve the optmal control problem. Betts [1] develops an error estmate for the state usng a low-order h method based on the dfference between the ntegraton of the dynamcs and the ntegraton of the tme dervatve of the state. Dfferent from all of ths prevous research where the order of the method s fxed and the mesh can only ncrease n sze, n the method of ths bref, the degree of the polynomal approxmaton s vared and the mesh sze can be reduced. Although h methods have been used extensvely and p methods are useful on certan types of problems, both the h and p approaches have lmtatons. In the case of an h method, t may be requred to use an extremely fne mesh to mprove accuracy. In the case of a p method, t may be requred to use an unreasonably large degree polynomal to mprove accuracy. In order to reduce sgnfcantly the sze of the fntedmensonal approxmaton, and thus mprove computatonal effcency of solvng the NLP, n recent years, the new class of hp collocaton methods has been developed for solvng an optmal control problem. In an hp method, both the number of mesh ntervals and the degree of the approxmatng polynomal wthn each mesh nterval are allowed to vary. Whle hp adaptve methods can be developed usng classcal dscretzatons (for example, usng a Runge Kutta method where the order of the method n a mesh nterval can be vared), employng Gaussan quadrature has advantages over classcal approaches. Frst, exponental convergence can be acheved by ncreasng the degree of the polynomal approxmaton n segments where the soluton s smooth. Second, Runge phenomenon (where the error at the ends of a mesh nterval becomes very large as the polynomal degree s ncreased) s elmnated usng a Gaussan quadrature. Thrd, less mesh refnement needs to be performed usng a Gaussan quadrature when compared wth a classcal method, because the mesh IEEE. Translatons and content mnng are permtted for academc research only. Personal use s also permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 1476 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 018 only needs to be refned n segments where smoothness s lost. Whle hp methods were orgnally developed as fnte-element methods for solvng partal dfferental equatons [13], n the past few years, hp methods have been extended to optmal control and a convergence theory for these methods has been establshed [6] [10]. In ths bref, a novel hp-adaptve mesh refnement method for optmal control s developed. The novelty of the method descrbed n ths bref les n the result that was proven n [14]. Specfcally, [14] proved that the error n a Legendre polynomal approxmaton of an analytc functon decays as a functon of the polynomal degree at the same rate as the Legendre polynomal coeffcents used n the approxmaton (t s noted that the result of [14] was ntroduced n [15], whle an hp method for solvng ellptc partal dfferental equatons was developed n [16] based on the result of [15]). The hp mesh refnement method of ths bref s then developed usng the followng approach that employs the rgorously proven result of [14]. Frst, usng the result n [14], the decay rate of the state wthn a mesh nterval s estmated to be the decay rate of the Legendre polynomal coeffcents wthn the mesh nterval. If the decay rate wthn the mesh nterval s larger than a user-specfed threshold decay rate, then the state n the mesh nterval s assumed to be an analytc functon and the degree of the approxmatng polynomal s ncreased. If, on the other hand, the decay rate wthn the mesh nterval s smaller than the user-specfed threshold decay rate, then the state wthn the mesh nterval s assumed to be a non-analytc functon and the mesh nterval s dvded nto subntervals. The novelty of the method descrbed n ths bref les n the fact that t enables utlzng the exponental convergence of a Gauss quadrature when the state s estmated to be an analytc functon and creates new mesh ntervals only when the state s estmated to be non-analytc. Moreover, the method s straghtforward to mplement, mang t usable n practcal optmal control problems. To demonstrate ts utlty, the method s appled to two examples where the second example s a nontrval mnmum-tme supersonc arcraft clmb optmal control problem. It s found that the approach developed n ths bref s more effcent, s smpler to mplement, and requres the specfcaton of fewer user-defned parameters when compared wth recently developed adaptve mesh refnement methods for optmal control. Ths bref s organzed as follows. Secton II descrbes the bass of the new hp-adaptve method based on the decay rate of the Legendre polynomal coeffcents. Secton III descrbes the optmal control problem n Bolza form, whle Secton IV descrbes the LGR collocaton method. Secton V descrbes the new hp-adaptve method developed n ths bref based on the approach developed n Secton II. Secton VI provdes the steps that comprse the hp-adaptve algorthm based on the components descrbed n Secton V. Secton VII demonstrates the hp-adaptve method on two examples from the open lterature and compares the results obtaned wth ths method wth the results obtaned usng the hp-adaptve methods n [17] and [18]. Secton VIII provdes a dscusson of the results. Fnally, Secton IX provdes conclusons on ths research. II. FOUNDATION OF hp-adaptive MESH REFINEMENT METHOD In ths secton, we provde the bass for the new hp-adaptve mesh refnement method descrbed n Secton V. The new hp-adaptve method developed n ths bref s motvated by the comparson study of hp-adaptve methods for ellptc partal dfferental equatons. In partcular, n [14], t has been shown that the decay rate of a Legendre polynomal approxmaton of a pecewse smooth functon can be estmated from the coeffcents of the Legendre polynomal approxmaton. The relatonshp establshed n [14] provdes a way to determne f the functon beng approxmated s smooth or nonsmooth. If the coeffcents decay suffcently fast as the polynomal degree s ncreased, the functon s regarded as beng smooth and the approxmaton error s most effectvely decreased by ncreasng the degree of the polynomal approxmaton. On the other hand, f the decay rate of the Legendre polynomal coeffcents s suffcently slow, the functon s estmated to be nonsmooth and the error n the approxmaton s most effectvely decreased by dvdng the nterval nto subntervals and usng a pecewse polynomal approxmaton. In the remander of ths secton, the decay rate of the error n a Legendre polynomal approxmaton s derved. A. Legendre Polynomal Approxmaton of Functons Let y(τ) be a pecewse smooth bounded functon on the nterval τ [ 1, +1]. Suppose that t s desred to approxmate y(τ) wth a polynomal Y (τ) of degree N. Arbtrarly, the polynomal approxmaton can be expressed n terms of a bass of N +1 Legendre polynomals P (τ), ( = 0,...,N) as y(τ) Y (τ) = N â P (τ). (1) In [15], t has been dscussed that the decay rate of the coeffcents â, ( = 0,...,N) can be used to estmate the smoothness of the functon y(τ). In ths secton, t s shown that the decay rate of the Legendre coeffcents â, ( = 0,...,N) gven n (1) s the same as the decay rate of the upper bound on the error. Because y(τ) s a pecewse smooth bounded functon, t can be represented by the nfnte Legendre polynomal seres y(τ) = a P (τ). () Then, the absolute error n the approxmaton Y (τ) of (1) satsfes the nequalty e = y(τ) Y (τ) N = a P (τ) â P (τ) N = a P (τ) + (a â )P (τ) a P (τ) + N (a â )P (τ) (3)

3 LIU et al.: ADAPTIVE MESH REFINEMENT METHOD FOR OPTIMAL CONTROL 1477 where f = f, f 1/ s the norm nduced by the nner product f, g = 1 1 f (τ)g(τ)dτ. (4) Suppose now that the frst and second terms n the nequalty of (3) are denoted as e t and e a, respectvely, that s, e t = a P (τ), e N a = (a â )P (τ). (5) The quanttes e t and e a represent the estmates on the truncaton error and the alasng error, respectvely, and are each estmated as follows. Frst, usng the defnton of the nner product n (4), the Legendre polynomals satsfy the orthogonalty property 1 P m, P n = P m (τ)p n (τ)dτ = 1 n + 1 δ mn (6) where δ mn s the Kronecer delta functon. Usng (6) together wth (5), e t and e a are gven, respectvely, as [ ] e t = a P (τ) = a 1/ + 1 [ N N ] 1/ e a = (a â )P (τ) = (a â ). (7) + 1 It has been noted n [19] that often a â, n whch case e a s neglgble compared wth e t. Neglectng e a leads to a =â, ( = 0,...,N). Next, under the assumpton that y(τ) s analytc n a neghborhood of the nterval τ [ 1, +1], thas been shown n [14] that the Legendre polynomal coeffcent values a decay le c10 σ, σ > 0. Note, however, that there exsts a slghtly smaller value σ > 0, such that c10 σ c10 σ. Because σ dffers from σ only slghtly and the analyss that follows s smpler and more convenent usng the more conservatve exponental upper bound, n ths bref, the decay rate of the form c10 σ s used. As a result, the Legendre polynomal coeffcents can be approxmated usng an exponental least-squares ft of the form [15] A = c10 σ, σ > 0 (8) to estmate the coeffcents a,( = 0,..., ) as a functon of, where σ n (8) approxmates the exponental decay rate n the Legendre polynomal coeffcents [14]. Moreover, estmates of a,( > N), can be obtaned from (8) as [ e < + 1 a ] 1/ < [ a ] 1/ = c10 σ(n+1) 1 10 σ where we have used the fact that a s a geometrc seres wth a geometrc rato r = 10 σ. The upper bound on the error e n (3), denoted ê, s then gven as ê = c10 σ(n+1) (10) 1 10 σ and s used as the error estmate. It s seen from (8) and (10) that the coeffcents â decrease at the same rate as a functon (9) of as the error decreases as a functon of N. Consequently, the Legendre coeffcents â, ( = 0,...,N) can be used to estmate the decay rate σ of the error as a functon of the degree of the polynomal approxmaton gven n (1). B. Assessng Functon Smoothness The decay rate of the Legendre polynomal coeffcents descrbed n Secton II-A provdes a way to estmate f a functon s smooth or nonsmooth. Based on ths estmate, f σ s suffcently large, t wll be preferable to approxmate the functon usng a sngle polynomal, whle f σ s suffcently small, t may be necessary to employ a pecewse polynomal approxmaton. Assumng a threshold of sgnfcance σ, the functon s consdered to be smooth f σ > σ and s consdered nonsmooth otherwse. Moreover, when σ > σ, the approxmaton s mproved by ncreasng the polynomal degree, whle f σ σ, the approxmaton s mproved by dvdng the tme nterval τ [ 1, +1] nto subntervals and usng a dfferent polynomal approxmaton n each subnterval. In ths secton, a bref study s provded that demonstrates how the decay rate of the Legendre polynomal coeffcents provdes an assessment of the smoothness of a functon and further demonstrates the agreement of the Legendre polynomal coeffcent decay rate and the upper bound of the approxmaton error. C. Demonstraton of Decay Rate Derved n Secton II-A In order to demonstrate the effectveness of the error estmate derved n Secton II-A, consder the followng two functons on τ [ 1, +1]: { 1, τ 0 y 1 (τ) = exp(τ), y (τ) = (11) 0, τ < 0. It s seen that y 1 (τ) s smooth, whle y (τ) s dscontnuous. Suppose now that y 1 (τ) and y (τ) are each approxmated by a Legendre polynomal seres of the form gven n (). In ths bref, the Legendre polynomal coeffcents are constructed by transformng a Lagrange polynomal approxmaton N+1 y(τ) Y (τ) = Y l (τ), l (τ) = =1 N+1 l=1 l = τ τ l τ τ l (1) where the N + 1 support ponts of the Lagrange polynomals l (τ), ( = 1,...,N + 1) are the N LGR ponts [5] (τ 1,...,τ N ) on [ 1, +1] plus the pont τ N+1 =+1. 1 Usng the Lagrange nterpolaton gven n (1), the Legendre coeffcents (â 0,...,â N ) are obtaned n terms of the Lagrange coeffcents as 1 â 0 P 0 (τ 1 ) P 1 (τ 1 )... P N (τ 1 ) â 1. = P 0 (τ ) P 1 (τ )... P N (τ ) â N P 0 (τ N+1 ) P 1 (τ N+1 )... P N (τ N+1 ) Y 1 Y.. (13) Y N+1 1 It s noted from the property of Lagrange polynomals that Y (τ ) = Y, where τ s a support pont of the Lagrange bass gven n (1),

4 1478 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 018 III. BOLZA OPTIMAL CONTROL PROBLEM Wthout loss of generalty, consder the followng general optmal control problem n Bolza form. Determne the state y(τ) R n y and the control u(τ) R n u on the doman τ [ 1, +1], the ntal tme t 0 and the termnal tme t f that mnmze the cost functonal Fg. 1. Base ten logarthm of the Legendre polynomal coeffcent magntudes of polynomal approxmatons of functons gven n (11). (a) log 10 â versus of y 1 (τ). (b)log 10 â versus of y (τ). J = M(y( 1), t 0, y(+1), t f ) +1 + t f t 0 L(y(τ), u(τ), t(τ, t 0, t f )) dτ (14) 1 subect to the dynamc constrants dy dτ t f t 0 a(y(τ), u(τ), t(τ, t 0, t f )) = 0 (15) the nequalty path constrants c mn c(y(τ), u(τ), t(τ, t 0, t f )) c max (16) Fg.. Base ten logarthm of exact errors and upper bounds of error estmates of polynomal approxmatons of functons gven n (11). (a) log 10 ê and log 10 e versus N of y 1 (τ). (b)log 10 ê and log 10 e versus N of y (τ). Fg. 3. Decay rates of Legendre polynomal coeffcent magntudes of polynomal approxmatons of functons gven n (11). The coeffcents â,( = 0,...,N) are then used to construct a least-squares exponental ft of the form gven n (8). Fg. 1 shows log 10 â as a functon of and Fg. shows log 10 ê and log 10 e as a functon of N for the two Legendre polynomal approxmatons. Several features can be seen n the results. Frst, Fg. shows that the error estmates are n close proxmty to the actual error. Second, t s seen that the error estmate of the smooth functon y 1 (τ) converges rapdly as a functon of the polynomal degree. On the other hand, the error estmate of the dscontnuous functon y (τ) converges slowly usng a sngle polynomal. Furthermore, Fg. 1 shows the decay rates of the Legendre polynomal coeffcents for dfferent types of functons. The Legendre coeffcents of the approxmaton of the smooth functon (that s, the Legendre coeffcents of Y 1 (τ)) decay much more rapdly than the coeffcents of approxmaton Y (τ)) ofthe dscontnuous functons. Fnally, as shown n Fgs. 1 and, the decay rate of the Legendre polynomal coeffcents of any of the functons s n close proxmty to the decay rate of the error n the correspondng functon approxmaton. Specfcally, the decay rates of the errors n y 1 (τ) and y (τ) n Fg. are 1.17 and 0.03 respectvely, whch match closely the decay rates σ 1 = 1.09 and σ = 0.0 for the Legendre coeffcents wth N = 15,asshownnFg.3. and the boundary condtons b mn b(y( 1), t 0, y(+1), t f ) b max. (17) It s noted that the tme nterval τ [ 1, +1] can be transformed to the tme nterval t [t 0, t f ] va the affne transformaton t t(τ, t 0, t f ) = t f t 0 τ + t f + t 0. (18) In order to dscretze the optmal control problem usng an hp method, the doman τ [ 1, +1] s parttoned nto a mesh consstng of K mesh ntervals S =[T 1, T ], = 1,...,K,where 1 = T 0 < T 1 <... < T K =+1. The mesh ntervals have the property that K =1 S =[ 1, +1]. Let y () (τ) and u () (τ) be the state and control n S.Usng the transformaton gven n (18), the Bolza optmal control problem of (14) (17) can then be rewrtten as follows. Mnmze the cost functonal J = M(y (1) ( 1), t 0, y (K ) (+1), t f ) + t f t 0 K T L(y () (τ), u () (τ), t) dτ (19) T 1 =1 subect to the dynamc constrants dy () (τ) dτ t f t 0 a(y () (τ), u () (τ), t) = 0, ( = 1,...,K ) (0) the path constrants c mn c(y () (τ), u () (τ), t) c max, ( = 1,...,K ) (1) and the boundary condtons b mn b(y (1) ( 1), t 0, y (K ) (+1), t f ) b max. () Because the state must be contnuous at each nteror mesh pont, t s requred that the condton y(t ) = y(t + ), ( = 1,...,K 1), be satsfed at the nteror mesh ponts (T 1,...,T K 1 ).

5 LIU et al.: ADAPTIVE MESH REFINEMENT METHOD FOR OPTIMAL CONTROL 1479 IV. LEGENDRE GAUSS RADAU COLLOCATION The multple-nterval form of the contnuous-tme Bolza optmal control problem n Secton III s dscretzed usng collocaton at LGR ponts as descrbed n [5], [17], and [0]. In the LGR collocaton method, the state of the contnuoustme Bolza optmal control problem s approxmated n S, [1,...,K ], as y () (τ) Y () (τ) = N l () +1 (τ) = l=1 l = N +1 =1 τ τ () l τ () τ () l Y () l () (τ) (3) where τ [ 1, +1], l () (τ), ( = 1,...,N + 1), s a bass of Lagrange polynomals, (τ () 1,...,τ() N ) are the LGR [5] collocaton ponts n S =[T 1, T ),andτ () N +1 = T s a noncollocated pont. Dfferentatng Y () (τ) n (3) wth respect to τ gves dy () (τ) dτ = N +1 =1 Y () dl () (τ). (4) dτ Defnng t () = t(τ (), t 0, t f ) usng (18), the dynamcs are then approxmated at the N LGR ponts n mesh nterval [1,...,K ] as N +1, t () ) = 0, D () Y () t f t 0 a ( Y (), U () =1 ( = 1,...,N ) (5) where D () = dl () (τ () )/dτ, ( = 1,...,N ), ( = 1,...,N + 1), are the elements of the N (N + 1) LGR dfferentaton matrx [5] n mesh nterval S, [1,...,K ]. The LGR dscretzaton then leads to the followng NLP. Mnmze J M ( Y (1) 1, t 0, Y (K ) N K +1, t ) f K N + =1 =1 t f t 0 w () L ( Y (), U (), t () ) (6) subect to the collocaton constrants of (5) and the constrants c mn c ( Y (), U (), t () ) cmax, ( = 1,...,N ) (7) b mn b ( Y (1) 1, t 0, Y (K ) N K +1, t ) f bmax (8) Y () N +1 = Y(+1) 1, ( = 1,...,K 1) (9) where N = K =1 N s the total number of LGR ponts and (9) s the contnuty condton on the state and s enforced at the nteror mesh ponts (T 1,...,T K 1 ) by treatng Y () N +1 and Y (+1) 1 as the same varable n the NLP. Fnally, t s noted that the mesh refnement method developed n ths bref requres an estmate of the soluton error on the current mesh. In ths bref, the approach for estmatng the soluton relatve error s taen from [17]. The relatve error approxmaton derved n [16] s obtaned by comparng two approxmatons to the state, one wth hgher accuracy. The ey dea s that for a problem whose soluton s smooth, an ncrease n the number of LGR ponts should yeld a state that more accurately satsfes the dynamcs. Hence, the dfference between the soluton assocated wth the orgnal set of LGR ponts and the approxmaton assocated wth the ncreased number of LGR ponts should yeld an approxmaton of the error n the state. The detals of ths relatve error estmate are beyond the scope of ths bref and the reader s referred to [17] for the detals. V. hp-adaptive MESH REFINEMENT METHOD After solvng the NLP arsng from the aforementoned LGR collocaton method on mesh M, the maxmum relatve error estmate s computed n each mesh nterval usng the method mentoned at the end of Secton IV and as descrbed n [17]. If the maxmum relatve error n any mesh nterval exceeds a user-specfed accuracy tolerance, ɛ, then the mesh nterval s modfed ether by ncreasng the degree of the approxmatng polynomal n the mesh nterval or by dvdng the mesh nterval nto smaller ntervals. In ths bref, the crtera for modfyng the mesh s based on the approach descrbed n Secton II. In partcular, f the coeffcents of a Legendre polynomal approxmaton of the state decay at a rate faster than a user-defned decay rate, then the polynomal degree s ncreased, as descrbed n Secton V-A. On the other hand, f the Legendre polynomal coeffcents decay at a rate slower than the user-defned decay rate, then the mesh nterval s dvded nto subntervals as descrbed n Secton V-B. In addton to ncreasng the sze of the mesh, usng the procedure n [18], the mesh sze can be reduced f the maxmum relatve error s less than the mesh refnement accuracy tolerance n one or more adacent mesh ntervals. A bref descrpton of the mesh sze reducton procedure s gven n Secton V-C. A. Method for Increasng Polynomal Degree Suppose now that the error tolerance n a gven mesh nterval S has not been met and that the decay rate of Legendre coeffcents s greater than σ. In ths case, the soluton n the mesh nterval s regarded as smooth n S and, f possble, the degree of the polynomal approxmaton used on mesh M + 1 s ncreased n order to reduce the soluton error. Let e (M) denote the error on nterval S of mesh M. Treatng the upper bound of the error estmate n (10) as e (M) gves the relatonshp ( ) e (M) c = 1 10 σ 10 σ N (M) +1. (30) Furthermore, assume for the ensung mesh M + 1thatts desred to acheve a maxmum relatve actual error accuracy ɛ. Agan, usng the upper bound of the error estmate n (10) as the actual error estmate gves ( ) c ɛ = 1 10 σ 10 σ N (M+1) +1. (31) Equatons (30) and (31) can then be solved for N (M+1) N (M+1) ( (M) ) e log = N (M) 10 ɛ + σ to gve (3)

6 1480 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 018 where the value of σ s computed usng the method n Secton II. Now, n order to obtan a strct ncrease n the number of collocaton ponts n S on mesh M + 1, the result of (3) s replaced wth ( (M) ) e log N (M+1) = N (M) 10 ɛ + (33) σ where replaces the argument wth the next hghest nteger. B. Method for Dvdng a Mesh Interval Assume now that the decay rate of Legendre coeffcents s less than σ on the current mesh and that the mesh nterval needs to be dvded. If the decay rate of the Legendre coeffcents σ s close to zero, whch ndcates that the error converges very slow, the small σ wll result n a large mesh sze on mesh M + 1. To avod creatng an unnecessarly large mesh, the threshold σ s used to determne the number of subntervals on the next mesh. The sum of the number of collocaton ponts n the newly created mesh ntervals should equal the predcted polynomal degree for the next mesh n (3) usng σ. The sum of the number of collocaton ponts n the newly created mesh ntervals s obtaned as ( e (M) ) log Ñ = N (M) 10 ɛ +. (34) σ Each newly created subnterval on mesh M + 1 contans the same number collocaton ponts on mesh M. Usng ths strategy, the number of subntervals, H, nto whch S s dvded s computed as H = Ñ N (M). (35) C. Mesh Sze Reducton In addton to the two strateges descrbed n Sectons V-A and V-B for ncreasng the sze of the mesh, the mesh refnement method n ths bref also allows for reducng the sze of the mesh usng the method n [18]. As descrbed n [18], the mesh sze can be reduced by ether reducng the number of collocaton ponts n a mesh nterval or by reducng the number of mesh ntervals. Refer to [18] for complete detals of the mesh reducton method employed n ths bref. VI. hp-adaptive ALGORITHM The hp-adaptve mesh refnement method that arses from Secton V s summarzed n Fg. 4, where M denotes the mesh refnement teraton. The algorthm termnates n Step 4 when the error tolerance s satsfed or when a prescrbed maxmum number of mesh refnement teratons, M max, s attaned. VII. EXAMPLES The hp-legendre mesh refnement method descrbed n Secton V s now appled to two examples and the performance of ths method s compared wth the prevously developed ph and hp methods descrbed n [17] and [18], respectvely. The followng termnology s used n the forthcomng Fg. 4. hp-legendre mesh refnement method. analyss. Frst, M denotes the number of mesh refnement teratons (M = 0 corresponds to the ntal mesh), whle N and K denote the total number of LGR collocaton ponts and the number of mesh ntervals, respectvely. All results were obtaned wth the MATLAB optmal control software GPOPS II [1] usng the NLP solver SNOPT [] and an optmalty tolerance of 10 7 wth NLP solver dervatves computed usng the MATLAB algorthmc dfferentaton tool AdGator []. No upper lmt s mposed on the allowable polynomal degree n a mesh nterval, but the number of mesh refnements s lmted to 5. In each example, the ntal mesh conssts of ten unformly spaced mesh ntervals and four collocaton ponts per mesh nterval, and the ntal guess for all examples s a straght lne for varables wth boundary condtons at both endponts and s a constant for varables wth boundary condtons at only one endpont. All computatons were performed on a.3-ghz Intel Core 7 MacBoo Pro runnng MAC OS-X verson (Yosemte) wth 16-GB 1600-MHz DDR3 of RAM and MATLAB Verson R014b (buld ) and the CPU tmes reported are ten-run averages (excludng the tme requred to solve the NLP on the frst mesh). Example 1 (Mnmum-Tme Reorentaton of a Robot Arm): Consder the followng mnmum-tme reorentaton of a robot arm taen from [3]. The obectve s to move an arm from an ntal poston wth zero velocty to a fnal poston wth zero velocty whle mnmzng the tme taen to perform the maneuver. Mathematcally, the problem s stated as follows. Mnmze J = t f subect to ÿ 1 = u 1 /L, ÿ = u /I θ, ÿ 3 = u 3 /I φ u, ( = 1,, 3) (36) and the boundary condtons (y 1 (0), y 1 (t f )) = (9/, 9/), (ẏ 1 (0), ẏ 1 (t f )) = (0, 0) (y (0), y (t f )) = (0, π/3), (ẏ (0), ẏ (t f )) = (0, 0) (y 3 (0), y 3 (t f )) = (π/4,π/4), (ẏ 3 (0), ẏ 3 (t f )) = (0, 0) (37) where L = 5, I φ = ((L y 1 ) 3 + y 3 1 )/3, and I θ = I φ sn (y 5 ). It s nown for ths problem that the control components u 1, u,andu 3 tae the value +1 on the ntervals [.86, 6.855], [0, 4.570], and [.87, 6.385], respectvely, and tae the value 1 otherwse [that s, the control s dscontnuous at

7 LIU et al.: ADAPTIVE MESH REFINEMENT METHOD FOR OPTIMAL CONTROL 1481 TABLE I PERFORMANCE OF THE hp-legendre, ph, AND hp METHODS FOR EXAMPLE 1 Fg. 5. Mesh refnement hstory for Example 1 usng the hp-legendre method ( σ = 0.5) wth an accuracy tolerance ɛ = 10 7.(a)hp-Legendre ( σ = 0.5) mesh pont hstory. (b) hp-legendre ( σ = 0.5) collocaton pont hstory. t = (.86,.87, 4.570, 6.385,6.855)]. Fg. 5(a) shows the evoluton of the mesh ponts for ths example usng the hp-legendre method wth σ = 0.5. It s seen that on the frst mesh teraton (M = 1), a few mesh ponts are added to the ntervals that contan the dscontnutes, and the frst, thrd, and ffth dscontnutes are accurately located. From the second mesh teraton (M = ) to the fnal mesh teraton, more mesh ntervals are added n the neghborhoods of the second and fourth dscontnutes and no mesh nterval s added to the segments [0, ] and [7, t f ], where the soluton s smooth. The manner n whch the mesh evolves demonstrates that the decay rate of the Legendre polynomal coeffcents (used as the bass of the method) s effectve n dentfyng the regons of smoothness and regons where dscontnutes n the soluton exst. For ths example, the decay rate of the Legendre polynomal coeffcents, σ, s less than the threshold σ = 0.5 n the neghborhoods of the dscontnutes and s greater than the threshold σ = 0.5 n the segments [0, ] and [7, t f ], where the soluton s smooth. Because the decay rate of the Legendre polynomal coeffcents s below the threshold near the dscontnutes, n these segments, the soluton error s reduced by creatng new mesh ntervals. Contrarwse, because the decay rate of the Legendre coeffcents s larger than the threshold n the segments [0, ] and [7, t f ], the error n these segments s reduced by ncreasng the degree of the polynomal approxmaton. Thus, the hp-legendre approprately ncreases the degree of the approxmatng polynomal or refnes the mesh n the regons where the soluton s smooth and nonsmooth, respectvely. Correspondng to the estmated decay rate, the mesh ponts are placed more sparsely n segments where the soluton s smooth and are placed more densely n regons where the soluton s nonsmooth. Next, the computatonal effcency and mesh sze of the hp-legendre method are compared wth the computatonal effcency and mesh sze of the ph method n [17] and the hp method n [18]. Table I shows the performance of the varous hp-legendre methods alongsde the ph method and the hp method for ɛ = (10 7, 10 8, 10 9 ). Frst, t s seen that usng the ph method, the number of mesh teratons requred to meet the mesh refnement accuracy tolerance ɛ = 10 7 s much greater (where M = 4) than that usng the hp-legendre method [where M = (4, 5, 6, 6) for σ = (0.5, 0.5, 0.75, 1), respectvely]. In the ph method, the error s decreased most often by ncreasng the polynomal degree as opposed to creatng new mesh ntervals. In ths example, however, the error s the largest near the dscontnutes n the control. As a result, ncreasng the polynomal degree n these segments results n a slow decrease n the error. Thus, the CPU tme requred to solve the problem usng the hp-legendre methods s less than usng the ph method. Moreover, as the tolerance gets tghter [ɛ = (10 8, 10 9 )], the ph method fals to solve the problem. Next, t s seen from Table I that for ɛ = 10 8, the number of mesh teratons requred to meet the mesh refnement accuracy tolerance usng the hp method (where M = 5) s the least among all the methods, but the CPU tme s the largest. The CPU tme s the largest usng the hp method, because the number of collocaton ponts n a sngle mesh nterval obtaned from the hp method s extremely large (reachng values as large as 10), because the upper bound on the LGR convergence rate (used as the bass of the hp method) results n an extremely large estmate for the requred polynomal degree. On the other hand, the growth n the polynomal degree usng the hp-legendre method s much slower than that of the hp method, because the new value of the polynomal degree computed from (33) wll be sgnfcantly smaller than the value computed by the hp method. Furthermore, t s seen for ths example that n all the cases, the fnal mesh sze obtaned by the hp-legendre method s smaller than the fnal mesh sze obtaned by the hp method, whle the number of mesh teratons requred to meet the mesh refnement accuracy tolerance, ɛ, s approxmately the same for ether method. Interestngly, t s also seen that the hp-method s more computatonally effcent than the hp method as ɛ decreases. Thus, the CPU tme usng the hp-legendre method grows more slowly as the demand for accuracy ncreases when compared wth the hp method. Example (Mnmum-Tme Supersonc Arcraft Clmb): Consder the followng problem of mnmzng the tme requred to transfer a supersonc arcraft from taeoff to a termnal alttude and speed. Ths optmal control problem, taen from [4], whch uses the model n [5], s stated as

8 148 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 018 Fg. 7. Mesh refnement hstory for Example usng the hp-legendre method ( σ = 0.5) wth an accuracy tolerance ɛ = 10 7.(a)hp-Legendre ( σ = 0.5) mesh pont hstory. (b) hp-legendre ( σ = 0.5) collocaton pont hstory. Fg. 6. Soluton of Example usng the hp-legendre method ( σ = 0.5) wth an accuracy tolerance ɛ = 10 7.(a)y 1 (t) versus t. (b)y (t) versus t. (c) y 3 (t) versus t. (d)u(t) versus t. follows. Mnmze J = t f subect to ḣ = v sn γ, v = T D m the boundary condtons g sn γ, γ = g u cos γ v (h(0), h(t f )) = (0, 0) m (v(0), v(t f )) = (0.193, 0.951) m s 1 (38) (γ (0), γ (t f )) = (0, 0) rad (39) and the state nequalty path constrants h(t) 0, < γ(t) π/4 (40) where h s the alttude, v s the speed, γ s the flght path angle, T = T (h, M) s the thrust force, D = D(h, M) s the drag force, u s the load factor (whch s the vertcal component of the lft), g s the acceleraton due to gravty, M = v/a s the Mach number, and a = a(h) s the speed of sound. Ths example s a challengng real-world applcaton where the thrust and the drag are obtaned from two-dmensonal polynomal fts of tabular data and the speed of sound s obtaned from a one-dmensonal polynomal ft of tabular data. It s noted that expressons for T, D, anda are obtaned from the data gven n [5], and the soluton to ths problem s shown n Fg. 6. It s shown n Fg. 6 that the soluton s least smooth n the segments where the one or both of the state nequalty constrants are actve. As a result, one would expect the hp-legendre method to place mesh ponts near the swtches n the actvty of path constrants. Examnng Fg. 7(a), where the hp-legendre method s mplemented wth ɛ = 10 7 and σ = 0.5, t s seen that the hp-legendre method progressvely ncreases the number of mesh ponts n the regon near the path constrant actvty. Frst, t s seen that on the frst mesh teraton (M = 1), mesh ponts are added to the segment t [0, 40], where the two aforementoned path constrants are actve. As the mesh refnement proceeds, more mesh ponts are added to the segment where the path constrants are actve and the decay rate of the Legendre polynomal coeffcents drops below σ = 0.5, because the soluton n the resultng mesh ntervals becomes nonsmooth. Now, although at the start of the mesh refnement a few mesh ponts are added n the segment t [50, 70] (where the soluton s smooth), from the second mesh teraton (M = ) onward, no more mesh ponts are placed n ths regon. Second, t s seen that the mesh ponts are placed sparsely n the segment t [70, 110] (agan, where the soluton s smooth), and new mesh ntervals are not created n ths segment. As a result, n the segment t [70, 110], the error n the soluton s decreased only by ncreasng the polynomal degree n the exstng mesh ntervals. The computatonal effcency and mesh sze of hp-legendre mesh refnement method s now compared aganst the prevously developed ph method n [17] and the hp method of [18], and Table II summarzes the results usng mesh refnement accuracy tolerances ɛ = (10 7, 10 8, 10 9 ).Frst,tsnoted that the ph method fals to solve ths example wth all the values of ɛ. Second, t s seen from Table II that for ɛ = 10 7, the hp method requres eght mesh refnement teratons to acheve the requred accuracy, whle the hp-legendre method requres less than eght mesh refnement teratons for all values of σ.furthermore,ɛ = (10 8, 10 9 ),andthehp method fals because the maxmum polynomal degree obtaned from the hp method s 814 n at least mesh nterval, whle the maxmum polynomal degree for the hp-legendre method s only. Thus, whle the hp s an mprovement over the ph method, the hp-legendre method performs sgnfcantly better than the hp method. VIII. DISCUSSION The examples llustrate dfferent features of the hp-legendre mesh refnement method. The frst example shows that the method accurately locates dscontnutes due to the predcton of nonsmoothness n the soluton from the slow decay rate of the Legendre polynomal coeffcents. The second example shows that the hp-legendre mesh refnement strategy can detect nonsmoothness of the problem that ncludes actve state path constrants. In addton, n the examples, the largest polynomal degree attaned usng the hp-legendre method was 5, whch s qute reasonable and made t unnecessary to place an upper lmt on the polynomal degree. In fact, the ncrease n the polynomal degree, defned by the term log 10 (e/ɛ)/σ n (33), wll never be very large. On the other hand, because the polynomal degrees obtaned by the ph and hp methods were larger polynomal than those of the hp-legendre method, the ph method was unable to

9 LIU et al.: ADAPTIVE MESH REFINEMENT METHOD FOR OPTIMAL CONTROL 1483 TABLE II PERFORMANCE OF THE hp-legendre, ph, AND hp METHODS FOR EXAMPLE attan the requred mesh refnement accuracy tolerance or dd so whle requrng a polynomal degree as large as 183. Next, whle the number of mesh ntervals created by the hp and hp-legendre methods was smlar when nonsmoothness was detected, the hghest polynomal degree attaned by the hp method was an extremely computatonally neffcent value of 814. Fnally, t s noted that the hp-legendre method s much smpler to mplement than the hp method, because t does not requre ntalzaton by comparng the soluton on two meshes and s based on a smple computaton of a decay rate of Legendre polynomal coeffcents. IX. CONCLUSION An adaptve mesh refnement method for solvng optmal control problems has been descrbed. The method modfes the mesh ether by ncreasng the polynomal degree wthn a mesh nterval or by dvdng a mesh nterval nto subntervals. The decson to ncrease the degree of the polynomal wthn a mesh nterval or to create new mesh ntervals s based on the decay rate of the coeffcents of a Legendre polynomal approxmaton of the state as a functon of the ndex of the Legendre polynomal expanson, where t has been shown that the decay rate of these coeffcents s the same as the decay rate on the upper bound of the state approxmaton error. In addton, the method allows for mesh sze reducton usng a prevously developed mesh reducton technque. The foundaton of the method has been provded, the ey components of the method have been descrbed, and the method has been demonstrated on two examples. It s found that the approach developed n ths bref s more effcent, smpler to mplement, and requres the specfcaton of fewer user-defned parameters when compared wth recently developed adaptve mesh refnement methods for optmal control. REFERENCES [1] J. T. Betts, Practcal Methods for Optmal Control and Estmaton Usng Nonlnear Programmng, nd ed. 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