CONTROLO 6 7h Poruguese Conference on Auomaic Conrol Insiuo Superior Técnico, Lisboa, Porugal Sepember -3, 6 AN OPTIMAL CONTROL PROBLEM FOR SPLINES ASSOCIATED TO LINEAR DIFFERENTIAL OPERATORS Rui C. Rodrigues, F. Silva Leie, Insiuo Superior de Engenharia de Coimbra Deparameno de Física e Maemáica 33-99 Coimbra, Porugal and Insiuo de Sisemas e Robóica Universidade de Coimbra - Pólo II 33-9 Coimbra, Porugal ruicr@isec.p Deparameno de Maemáica Universidade de Coimbra 3-454 Coimbra, Porugal and Insiuo de Sisemas e Robóica Universidade de Coimbra - Pólo II 33-9 Coimbra, Porugal fleie@ma.uc.p Absrac: We formulae an opimal conrol problem, where he dynamics are given by an inpu-oupu equaion, in order o emphasize he opimal properies of some inerpolaing curves, known in he lieraure as L-splines. We presen he soluion of he opimal conrol problem and illusrae he heory wih some examples. Keywords: Opimal conrol, Linear sysems, Splines.. INTRODUCTION In many engineering applicaions one has o find rajecories ha are consrained o pass hrough some specific poins wih prescribed derivaives and some required degree of smoohness. Examples of hese are he cubic splines ha also minimize acceleraion and oher polynomial funcions, such as quinic polynomials ha minimize jerk. Parially suppored by PRODEP/3 (Medida 5, Acção 5.3), Projec NCT4 (FCT) and ISR-Coimbra. Parially suppored by Projec NCT4 (FCT) and ISR- Coimbra. One common feaure o hese examples is ha he number of inerpolaion condiions, as well as he number of smoohness condiions, is he same a each poin (excep possibly he firs and he las). Less known are he curves named L-splines. These are obained by joining smoohly pieces of soluions of cerain linear differenial equaions. Conrary o he previous examples, hey allow he prescripion of uneven condiions a each poin. In his aricle we demonsrae he opimal behavior of L-splines, by showing ha hese curves
can be realized as he opimal oupu of a cerain opimal conrol problem wih piecewise conrols. The formulaion of his problem appears in secion. The dynamics are given by an appropriae inpu-oupu equaion, in opposiion o he usual classical formulaion, where he dynamics appear in sae-space form. The connecion wih L-splines is presened in secion 3, and he properies of hese curves are used, in secion 4, o presen he soluion of he proposed opimal conrol problem. An equivalen sae-space formulaion appears in secion 5. Finally, in he las secion, we include hree examples o illusrae he heory. This paper highlighs several aspecs of he opimal behavior of splines associaed o linear differenial operaors reaed in (Rodrigues and Silva Leie, ) and correcs some inaccuracies and imprecise saemens conained in his aricle.. FORMULATION OF THE OPTIMAL CONTROL PROBLEM Le : a = < < < m = b be any pariion of he ime inerval [a, b] R, p N, and Z = (z, z, z,...,z m, z m ), wih z = z m = p and z i p, any incidence vecor for. L and L c are wo linear differenial operaors wih real consan coefficiens given by L D p + a p D p + + a D + a, L c c p D p + + c D + c. We do no impose c p and define σ as σ = maxj : c j }. We assume ha he characerisic polynomials associaed o L and L c, have no common facors. The admissible conrols are piecewise coninuous funcions, bounded over [a, b], which are also C p+σ -smooh in each subinerval [ i, i+ ]. As usual, he se of all admissible conrols is denoed by U. We consider he following opimal conrol problem, denoed by (P c ). b a L y() = L c u(), (L c u()) d u U he inerpolaion condiions y (k) ( i ) = α k i, i =,...,m, k =,...,z i, he boundary condiions y (k) ( ) = η k, y(k) ( m ) = η k m, k =,...,p, min, and he smoohness condiions y (k) ( i ) = y (k) ( i + ), i =,...,m, k =,...,p z i, where α k i, ηk and η k m are prescribed real numbers. This is no a classical formulaion of an opimal conrol problem since he dynamics are described by an inpu-oupu equaion and he opimal oupu is required o saisfy a cerain number of uneven condiions a he kno poins i. Noe ha here is a direc connecion beween he incidence vecor Z and he number of condiions prescribed a each kno poin of he pariion. Indeed, here are z = p iniial condiions a he ime insan, z m = p final condiions a he ime insan m, and z i inerpolaion condiions a i, i =,...,m. Taking ino accoun he number of prescribed smoohness condiions, one ges a oal of p condiions a each i, i =,...,m, and p condiions a and m. For deails concerning linear conrol sysems see, for insance, (Kailah, 98), (Ribeiro, a), (Ribeiro, b) and (Rugh, 996). 3. CONNECTIONS WITH L-SPLINES In his secion we show ha he opimal oupu of he problem (P c ) is a curve named L-spline, known in he lieraure for many years (Schulz and Varga, 967). Esablishing he connecion beween our opimal conrol problem and hese curves, will highligh he opimal behavior of L- splines already demonsraed in (Rodrigues and Silva Leie, ). This complemens wha has already been done for oher kinds of splines, namely for he so called generalized splines (see (Ahlberg e al., 964)), in (Zhang e al., 997) and (Rodrigues e al., 999). The opimal conrol problem (P c ) has a naural variaional formulaion as: J(y( )) = b a (Ly()) d y Ω subjec o he inerpolaion condiions min, y (k) ( i ) = α k i, i =,...,m, k =,...,z i, he boundary condiions y (k) ( ) = η k, y(k) ( m ) = η k m, k =,...,p, and he smoohness condiions y (k) ( i ) = y (k) ( i + ), i =,...,m, k =,...,p z i, where Ω is he se of funcions which are C p - smooh in each subinerval [ i, i+ ].
We denoe by Ω A he se of all admissible curves for his variaional problem, and by L he adjoin of he operaor L, ha is, L ( ) p D p + ( ) p a p D p + +a. Nex, we sae a necessary and sufficien condiion for an admissible curve o minimize he given funcional, which can be derived from resuls in (Schulz and Varga, 967). Theorem y Ω A is a soluion of he variaional problem above if and only if y is a soluion of he linear differenial equaion L L x = in every subinerval [ i, i+ ]. Remark () The differenial equaion in he previous heorem is nohing bu he Euler-Lagrange equaion associaed o he variaional problem. () I urns ou ha here is only one curve which is a soluion of his differenial equaion (of order p) on each subinerval and saisfies he prescribed inerpolaion, boundary and smoohness condiions. Thus, he variaional problem has one and only one soluion. This soluion is an L-spline (of ype I) ha can be easily compued by solving a consisen se of pm linear algebraic equaions in pm unknowns. 4. SOLVING THE OPTIMAL CONTROL PROBLEM We now discuss how o solve he opimal conrol problem (P c ) using he facs presened in he previous secion. Theorem When σ, problem (P c ) has several soluions, which are piecewise coninuous funcions composed by m segmens, each one depending on σ free parameers. When σ =, problem (P c ) has a unique soluion. Proof. Since he variaional problem has a soluion, i is clear ha problem (P c ) has, a leas, a soluion, which generaes an L-spline curve as opimal oupu. Le ỹ denoe his L-spline curve, which is he unique soluion of he variaional problem, and le u denoe any soluion of problem (P c ). Since ỹ is composed by m segmens (which are C p -smooh) and y and u are such ha L ỹ() = L c u() in inerval [a, b], hen u is a piecewise funcion composed by m segmens (which are C p+σ -smooh). Define u i and ỹ i o be he segmens of u and ỹ in he inerval [ i, i+ ]. I is clear ha u i is a soluion of he complee linear differenial equaion L c x = φ i where φ i = L ỹ i. The segmen u i is an elemen of he affine space v + kerl c (where v is any paricular soluion of he differenial equaion L c x = φ i ) and hus u i depends on σ free parameers. Therefore, when σ problem (P c ) has several soluions wih he saed properies. Clearly, here is only one soluion oherwise. Remark () As seen, when σ problem (P c ) has many soluions. To compue a specific soluion one has o fix a oal of mσ real consans. When σ =, he opimal conrol is given by ũ() = c L ỹ(), [a, b]. () In general he opimal soluions are no coninuous in [a, b]. However, when σ = and he incidence vecor Z = (p, z,...,z m, p) is such ha z i < p, he unique opimal soluion is always coninuous. The nex resul shows an imporan propery of he opimal soluions. Theorem 3 If u is an opimal conrol for problem (P c ), hen each segmen u i belongs o he kernel of he linear differenial operaor L L c. Proof. This follows immediaely from he fac ha L L ỹ i = and L ỹ i = L c u i. In paricular, when σ =, ũ i is a soluion of L x =. 5. STATE-SPACE APPROACH We now presen an alernaive formulaion of our opimal conrol problem, which is classical in he sense ha he dynamics are in sae-space form. In his formulaion one expecs o find he opimal conrol wihou previous knowledge of he corresponding oupu. Wih his approach, previous resuls abou he opimal behavior of generalized splines follow easily. The sae-space represenaion (inernal represenaion) corresponding o he inpu-oupu equaion (exernal represenaion) L y() = L c u() is ẋ() = Ax() + Bu(), y() = Cx() + du(), where [a, b] and x() X = R p. This single inpu single oupu linear ime invarian sysem is a minimal realizaion since he operaors L and L c have no common facors. Therefore, wihou loss of generaliy, we may assume ha he marices A, B, C and scalar d are he following:
. A =..... a a a p p p B =. C = [ c a c p c p a p c p ] p d = c p. p This paricular sae-space represenaion is clearly conrollable and observable. Remark 3 If x is he firs componen of he sae vecor, he following can be easily derived from he saespace represenaion above: L x () = u(), y() = L c x (). The inpu-oupu equaion L y() = L c u() follows from here immediaely. sense ha hey share he same se of soluions and also produce he same opimal oupu. () We poin ou ha if he operaors L and L c are no relaively prime, hen all he resuls in he previous secion are sill valid. However, in his case, he problems (P c ) and (P cs ) are no longer equivalen. Indeed, he se of opimal conrols of he problem (P cs ) is a proper subse of he se of opimal conrols of he problem (P c ). In spie of his discrepancy, boh problems produce he same oupu, even when he operaors L and L c have common facors. For he paricular case when he incidence vecor is Z = (p,,...,, p) and σ =, he resuls coincide wih hose obained in (Zhang e al., 997) and (Rodrigues e al., 999). Now, le (P cs ) denoe he following opimal conrol problem: b a (L c u()) d u U ẋ() = Ax() + Bu(), y() = Cx() + du(), he inerpolaion condiions y (k) ( i ) = α k i, i =,...,m, k =,...,z i, he boundary condiions y (k) ( ) = η k, y (k) ( m ) = η k m, k =,...,p, and he smoohness condiions min, y (k) ( i ) = y (k) ( i + ), i =,...,m, k =,...,p z i, where α k i, ηk and ηk m are prescribed real numbers. Since we prescribe inerpolaion condiions on he oupu, i would be naural o assume ha he sysem is oupu conrollable, ha is, he marix Γ = [ CB CAB CA p B d ] (p+) has full rank. Neverheless, in our case his follows from he conrollabiliy and observabiliy of he sysem. This opimal conrol problem may be solved direcly by applying he Ponryagin Maximum Principle. Remark 4 () As formulaed, boh opimal conrol problems (P c ) and (P cs ) are equivalen in he 6. EXAMPLES We consider hree examples for p =, he pariion : a = < / < < < 9/4 < 3 = b, and he associaed incidence vecor Z = (,,,,, ). We prescribe he inerpolaion condiions y(/) = 3/, ẏ(/) =, y() =, y() = /, ẏ() = /, y(9/4) = /6, and he boundary condiions y() = 3, ẏ() =, y(3) =, ẏ(3) =. The smoohness condiions are immediaely clear from he incidence vecor above. The fis example is adaped from he opimal conrol problem The cheapes sop of a rain conained in (Agrachev and Sachkov, 4). Example 3 ÿ() = u() (u()) d min, and all he inerpolaion, boundary and smoohness condiions saed above.
Noice ha he conrol sysem is given in saespace form by ẋ = x, y = x. ẋ = u, This opimal conrol problem has a unique soluion (because σ = ). Each segmen of he opimal u is a sraigh line and u is clearly coninuous a = and a = 9/4. The opimal oupu is a cubic polynomial in each subinerval [ i, i+ ], alhough no a classical cubic spline on [, 3]. Example 3 ( u() + u()) d min, ÿ() + 36y() = u() + u() and all he inerpolaion, boundary and smoohness condiions saed above. The conrol sysem is given in sandard form by ẋ = x, y = x + x. ẋ = 36x + u, The soluion of he problem is no unique, since σ =. The opimal soluion presened bellow corresponds o a paricular choice of a free parameer in each subinerval. The opimal oupu funcion is in each subinerval a linear combinaion of ψ () = cos(6), ψ () = cos(6), ψ 3 () = sin(6) and ψ 4 () = sin(6). Example 3 3 (ü() u() u()) d min, ACKNOWLEDGEMENT We would like o hank E. Marques de Sá for helpful hins and criicisms ha helped us o clarify some of he ideas conained here. REFERENCES Agrachev, Andrei A. and Yuri L. Sachkov (4). Conrol heory from he geomeric viewpoin. Vol. 87 of Encyclopaedia of Mahemaical Sciences. Springer-Verlag. Berlin. Ahlberg, J. H., E. N. Nilson and J. L. Walsh (964). Fundamenal properies of generalized splines. Proc. Na. Acad. Sci. U.S.A. 5, 4 49. Kailah, T. (98). Linear sysems. Prenice-Hall. Prenice-Hall Informaion and Sysem Sciences Series. Ribeiro, Maria Isabel (a). Análise de sisemas lineares. Vol. of Colecção ensino da ciência e da ecnologia. IST Press. Ribeiro, Maria Isabel (b). Análise de sisemas lineares. Vol. of Colecção ensino da ciência e da ecnologia. IST Press. Rodrigues, Rui C. and F. Silva Leie (). L- splines a manifesaion of opimal conrol. IMA J. Mah. Conrol Inform. 9(3), 33 34. Rodrigues, Rui C., F. Silva Leie and C. Simões (999). Generalized splines and opimal conrol. In: Proceedings of he European Conrol Conference, ECC 99. Karlsruhe, Germany. CD-ROM paper F59.PDF. Rugh, Wilson J. (996). Linear sysem heory. nd ed.. Prenice-Hall. Schulz, M. H. and R. S. Varga (967). L-splines. Numer. Mah., 345 369. Zhang, Zhimin, John Tomlinson and Clyde Marin (997). Splines and linear conrol heory. Aca Appl. Mah. 49(), 34. ÿ() + 9ẏ() y() = ü() u() u() and all he inerpolaion, boundary and smoohness condiions saed above. The conrol sysem is given in sandard form by ẋ = x, y = 8x x +u. ẋ = x 9x + u, The soluion of he problem is no unique, since σ =. The opimal soluion presened bellow corresponds o a paricular choice of free parameer in each subinerval. The opimal oupu funcion is in each subinerval a linear combinaion of exponenial funcions. The opimal curves for hese problems are presened a he end of he paper. 3 y().5.5.5.5.5 3 Fig.. Firs example - opimal oupu
u().5.5 3 Fig.. Firs example - opimal inpu u() 6 5 4 3.5.5 3 Fig. 6. Third example - an opimal inpu 3 y().5.5 3 Fig. 3. Second example - opimal oupu 6 u() 8 4.5.5 3 Fig. 4. Second example - an opimal inpu 3 y().5.5.5.5.5 3 Fig. 5. Third example - opimal oupu