A Wavelet Collocation Method for Optimal Control. Problems

Transcription

1 A Wavelet ollocation ethod or Optimal ontrol Problems Ran Dai * and John E. ochran Jr. Abstract A Haar wavelet technique is discussed as a method or discretizing the nonlinear sstem equations or optimal control problems. he technique is used to transorm the state and control variables into nonlinear programming NLP parameters at collocation points. A nonlinear programming solver can than be used to solve optimal control problems that are rather general in orm. Here general Bolza optimal control problems with state and control constraints are considered. Eamples o two inds o optimal control problems continuous and discrete are solved. he results are compared to those obtained b using other collocation methods. Kewords Haar Wavelet ollocation Discrete Optimal ontrol Nonlinear Programming * Ran Dai Department o Aerospace Engineering Auburn Universit Auburn AL aviator_dai@hotmail.com John E. ochran Jr. Department o Aerospace Engineering Auburn Universit Auburn AL jcochran@eng.auburn.edu.

2 Introduction Recentl wavelet theor has attracted considerable attention due to the advantages wavelets have over traditional Fourier transorms in accuratel approimating unctions that have discontinuities and sharp peas. Wavelets have been applied in signal processing multi-scale phenomena modeling and pattern recognition. he Haar wavelet unction was introduced b Alred Haar in 9 in the orm o a regular pulse pair []. Ater that man other wavelet unctions were generated and introduced. hose include the Shannon Daubechies and Legendre wavelets. Among those orms Haar wavelets have the simplest orthonormal series with compact support. hese characteristics maes Haar wavelets good candidates or application to optimal control problems. he wavelet applications in dealing with dnamic sstem problems especiall in solving partial dierential equations with two-point boundar value constraints have been discussed in man papers [-5]. B transorming dierential equations into algebraic equations the solution ma be ound b determining the corresponding coeicients that satis the algebraic equations. Some eorts have been made to solve linear optimal control problems b using wavelet collocation [6-9]. But when the sstem equations become comple and highl nonlinear it is necessar to see some useul tools to solve those inds o problems. he nonlinear programming solver NLPS SNOP [ ] which uses a Sequential Quadratic Programming algorithm to solve the nonlinear problems seems to be an eicient tool or this tas. At each major iteration o the NLPS algorithm the solver inds the search direction or the current nominall optimized points. his process is repeated until convergence occurs. hen the parameterized objective unction can be optimized with sstem equation constraints satisied.

3 he collocation methods developed to solve nonlinear optimal control problems generall all into two categories local collocation[3] and global orthogonal collocation [4-7]. In local collocation methods such as trapezoidal Hermite-Simpson and Runge-Kutta methods the time interval considered is divided into a series o subintervals within which the integration rule must be satisied. hese local collocation methods were introduced in direct collocation and nonlinear programming DNLP and have wide application [8-]. In recent ears more attention has been ocused on global orthogonal collocation methods such as Legendre hebshev Gauss method and some others. B epanding the state and control variables into piecewise-continuous polnomials the derivative o the state variables can be approimated b combinations o these interpolating polnomials and their derivatives. hen the objective unction and sstem constraints are all converted into algebraic equations with unnown coeicients. he orthogonal collocation methods are generall better than local collocation methods in achieving ast convergence rate and high accurac. here are three major classes o orthogonal unctions the orthogonal polnomials lie Legendre hebshev etc. the sine-cosine unctions in Fourier series and the constant basis unction lie Haar bloc-pulse etc.. he irst two classes o orthogonal unctions have been widel applied in collocation as mentioned above. But apparentl no attempts have been made to appl constant basis orthogonal unctions in collocation. hus it is o interest to see how this new collocation method wors. In this paper we irst introduce the Haar wavelets theor and properties including the Haar wavelets basis and its integral operational matri. hen we will assume that the control variables and derivatives o the state variables in the optimal control problems ma be epressed in the orm o Haar wavelets and unnown coeicients. he state 3

4 variables can be calculated b using the Haar operational integration matri. hereore all variables in the nonlinear sstem equations are epressed as series o the Haar amil and its operational matri. Finall the tas o inding the unnown parameters that optimize the designate perormance while satising all constraints is perormed b the NLPS. o demonstrate the applicabilit o this new collocation method we consider two eample optimal control problems one with a smooth control and one with a non-smooth control. Haar Wavelets. Haar Functions he basic and simplest orm o Haar wavelet is the Haar scaling unction that appears in the orm o a square wave over the interval t [ as epressed in Eq. and illustrated in the irst subplot o Fig. : t < φ t elsewhere he above epression called Haar ather wavelet is the zeroth level wavelet which has no displacement and dilation o unit magnitude. orrespondingl there is a Haar mother wavelet to match the ather wavelet which is described as φ t t < t < elsewhere 4

5 he Haar mother wavelet is the irst level Haar wavelet and its graph is given in the second subplot o Fig.. his mother wavelet can also be written as the linear combination o the Haar scaling unction with translation and compression to hal o its original interval φ t φt φt 3 Similarl the other levels o wavelets can all be generated rom φ b the operations o translation and t compression. For eample the third subplot in Fig. is ormed b compression φ to let hal o its original t interval and the orth subplot is the same as the third one plus translating to the right side b /. In general we can write out the Haar wavelet amil as φi t +.5 t [ m m t [ m m elsewhere 4 here m is the level o the wavelet assume the maimum level resolution is integer J then m equals to j j K J ; the translation parameter K m. he series inde number i is deined b m and and i m +. For an ied level m there are m series o φ i to ill the interval [ corresponding to that level and or a provided J the inde number i can reach the maimum value J + when including 5

6 all levels o wavelets. Each Haar wavelet is composed o a couple o constant steps o opposite sign during its subinterval and is zero elsewhere. hereore the have the ollowing relationship j φi t φl t dt i l i l j + 5 his relationship shows that Haar wavelets are orthogonal to each other and thereore constitute an orthogonal basis. his allows us to transorm an unction square integrable on the interval time [ into Haar wavelets series.. Function Approimation b Haar Wavelets We just pointed out that a square integrable unction can be epressed in terms o Haar orthogonal basis on interval [. However beore the procession to this transer it is necessar to uni the time interval. B using a linear transormation the actual time t can be epressed as a unction o via t [ t t + t ] 6 where t is the initial time and t is the inal time in a square integrable unction t. he objective is to write this t in orm o wavelet epansion series with coeicients that is t a φ t i i i A Ψ t 7 6

7 where the coeicient vector A ] [ a a K a and Ψ t [ φ t φ t K φ t]. In vector A each coeicient ai is determined b j ai t φi t dt 8 and it is epected to approimate unction t with minimum mean integral square error ε deined as ε t A Ψ t dt 9 Obviousl ε should reduce when the level gets larger and it should go close to zero when approaches ininite. I we set all the collocation pic point t s at the middles o each wavelet then t s is deined as t s s.5 / where s K. With these chosen collocation points the unction is discretized into a series o nodes with equivalent distances. he vector Ψ t can also be determined at those collocation points. Let the Haar matri H be the combination o Ψ t at all the collocation points. hus we get H [ Ψ t Ψ t K Ψ t ] φ t φ t φ t φ t φ t φ t L L O K φ t φ t φ t 7

8 or eample [ Ψ t Ψ ] H t and [ Ψ t Ψ t Ψ t Ψ ] H t 3 3 hereore unction t is approimated as ts H..3 Haar Operational Integration atri In the solution o optimal control problems we alwas need to deal with equations involving dierentiation and integration. I the sstem unction is epressed in Haar wavelets the integration or dierentiation operation o Haar series cannot be avoided. he dierentiation o step waves will generate pulse signals which are diicult to handle while the integration o step waves will result in constant slope unctions which can be calculated b the ollowing equation ' ' Ψ t dt PΨ t 4 where P is the operational integration matri and is given b Gu and Jiang [] or how to calculate this matri 8

9 P P H / / H / P [ ] 5 3 Formulation o Optimal ontrol Problems he objective o an optimal control problem is to ind the histor o the control variables that will maimize or minimize a given perormance inde while satising the sstem constraints. he sstem constraints include irstorder ordinar dierential equations subject to initial and inal boundar conditions and some additional constraints on the states and controls. he dierential equations are written here as & tu 6 where is an n vector o states is an n vector o continuous dierential unctions t is the time and u is an m vector o controls. he states are subject to prescribed initial conditions and inal boundar conditions ψ t where ψ is a p vector o unctions t is the initial time and the terminal time t mabe ied or ree. Some problems also include etra path constraints which are unctions o state and control variables ormulated as g tu 7 9

10 where g is a q vector. In this paper we consider problems o the Bolza [] tpe which are ocused and minimizing the scalar perormance inde o the orm J φ t t + t L t u dt 8 where φ is a scalar unction o the inal time t and inal state variables and L t u is a scalar unction o the time state and control u. 4 Direct ollocation and Nonlinear Programming 4. Haar Discretization ethod In the discussion o Haar wavelets we have alread addressed how to approimate a unction in Haar wavelets and its corresponding operational integration matri. We are epecting to appl this methodolog in optimal control problems so that Haar discretization is used in direct collocation. hus the continuous solution to a problem will be represented b state and control variables in terms o Haar series and its operational matri to satis the dierential equations. he standard interval considered here is denoted as [ with collocation points set as.5 / K 9 where is the number o nodes used in discretization and also is the maimum wavelet inde number. Note that the magnitude o is in the power o so that the number o collocation points is also increasing in that power.

11 All the collocation points are equall distributed over the entire time interval [ with as the time distance to adjacent nodes. We assume that the derivative o the state variables & and control variables u can be approimated b Haar wavelets with collocation points i.e. & Ψ u Ψ u where [ K ] u [ K u u u ] B using the operational integration matri P deined in Eq. 5 the state variables can be epressed as & ' d ' + Ψ ' d ' + PΨ + 3 As stated in Eq. the epansion o the matri Ψ at the collocation points will ield the Haar matri H h h K h ]. It ollows that [ & h u h Ph u K 4

12 From the above epression we can evaluate the variables at an collocation point b using the product o its coeicients vector and the corresponding column vector in the Haar matri. 4. NLP Solver: SNOP 6. he NLPS used to solve the NLP problem considered in this wor is based on a Sequential Quadrature Programming SQP algorithm and is called SNOP. SNOP can be used to solve problems lie the ollowing: inimize a perormance inde J subject to constraints on individual state and/or control variables L U 5 constraints deined b linear combinations o state and/or control variables: b A L b U 6 and/or constraints deined b nonlinear unctions o state and/or control variables: c c L c U 7 When the Haar collocation method is applied in the optimal control problems the NLP variables can be set as the unnown coeicients vector o the derivative o the state variables and control variables together with initial and inal times that is [ K u u K u t t ] 8

13 3 he objective unction in Eq. 8 is then restated as φ d P L t t t J u Ψ + Ψ + 9 Since the Haar wavelets are epected to be constant steps at each time interval the above equation can be simpliied as Ψ + Ψ + u P L t t t J φ 3 with path constraints ormulated as Ψ + Ψ u P g 3 Substituting & u and in Eq. 6 with the Haar wavelets epression o Eq. 4 separatel ields: u P t t Ψ + Ψ Ψ 3 he sstem equation constraints and path constraints are all treated as nonlinear constraints in NLP solver. he boundar constraints need to be paid more attention. Since the irst and last collocation points are not set as the initial and inal time the initial and inal state variables are calculated according to / / & & + 33

14 In this wa the optimal control problems are transormed into NLP problems in a structured orm. 5 Eamples In this section we consider two optimal control problems that have nown solution and see how well the results o the NLP with direct collocation can approimate eact solution. 5. Brachistochrone Problem he Brachistochrone Problem is that o inding the shape o the curve down which a weight classicall a bead acted upon b the orce o gravit will descend rom rest and accelerate to a desired point in the least time assuming there is no riction and the ground orce is uniorm. athematicall this problem is described as inding the minimum time t or a bead rom the starting point [ ] to the inal point [ ] while satising the sstem equation constraints & & g g cosu sin u 34 where angle u is the slope o the curve and is treated as control variable o this problem. Using the Haar wavelets collocation method with collocation points the objective is to minimize J t with sstem equation constraints h h t t g g Ph Ph + cos h + sin h u u K 35 4

15 and boundar constraints Ph + Ph + h / h / Ph + Ph + h / h / 36 where u and t are all NLP variables solved at the collocation points. he analtical solution or this problem is epressed as a ccloid generated b a circle o diameter R that rolls through angle θ rom the vertical and described mathematicall b θ R θ sinθ θ R cosθ 37 he analtical optimal control epressed in terms o time is t π u t g / R 38 For the numerical solution we assume the starting point is [ ] and the ending point is [ 5] use 6 and 3 collocation points separatel or the Haar wavelets discretization method. he initial guess o the solution is the straight line connecting the starting and ending points. hereore the initial control and NLP variables are estimated according to this coarse initial guess. he trajector and control histor results using this new method is shown in Fig. and 3 separatel and compared to the analtical results. Also there are results o using the hebshev Pseudospectral P collocation method to solve the same problem with the same number o collocation points. 5

16 From Fig and 3 it can be seen that the results generated b the Haar wavelets collocation method made good approimation to the analtical solution. When the number o collocation points increases the accurac is improved and urther increase will mae the inal trajector converge to the eact solution. he absolute error o the objective unction between wavelet solution J and analtical solution * J with respect to the number o nodes is shown in Fig Discrete Optimal ontrol Problem he discrete optimal control problem considered here is to use Bang-Bang ontrol with maimum and minimum bounds to minimize the cost unction J t 39 subject to simple sstem equation constraints & & u 4 and initial state variable constraints t t 3 4 where u is the control and is constrained so that u t. he analtical solution or this problem is [6] 6

17 t / + t + t t / t t + t / t + t < t t t t t > t t < t u t t > t t < t t > t 4 where the switching time is calculated at t and the optimized inal time t. t For the DNLP solutions we assume that the trajector starts rom [ 3] and ends at ] [ again use 6 and 3 nodes in Haar wavelets collocation and the initial guess o the state variables are straight line connecting the starting and inal points. hen the results to analtical and P methods are compared in Fig. 4 or state variables and Fig. 5 or controls. It s obvious to see that Haar wavelet collocation has an advantage in Bang-Bang optimal control problems when the switching time is unnown. Instead o the slope solution with long and unstead time interval beore and ater the switching point in P method the wavelet solution provides Bang-Bang control wave which switch values ver close to the eact analtical switching time. he convergence rate or the wavelet solution is shown in Fig. 7. When the number o nodes increases it is epected the wavelet solution will generate the optimal control solution with error close to zero. Besides the state variables o the wavelet converge to the analtical solution. 7

18 6 onclusion A new collocation method is applied to solve two well nown optimal control problems using a nonlinear programming solver. he sstem equations are all epressed in Haar wavelets with unnown coeicients and solving those unnown coeicients while optimizing the perormance inde becomes the tas o this ind o parameterized problems. his new method produces results similar to other collocation methods or the continuous optimal control problem and shows advantages in discrete optimal control problems when switching time is unnown. Reerences. Boggess A. and Narcowich F. J. A First ourse in Wavelets with Fourier Analsis Prentice-Hall Inc... Bertoluzza S. and Naldi G. A Wavelet ollocation ethod or the Numerical Solution o Partial Dierential Equations Applied and omputational Harmonic Analsis 3 No pp Lepi U. Numerical Solution o Dierential Equations Using Haar Wavelets athematics and omputers in Simulation No. 68 pp Belin G. On Wavelet-Based Algorithms or Solving Dierential Equations in Wavelets: athematics and Applications R Press 994 pp Xu J.. and Shann W.. Galerin-Wavelet ethods or wo-point Boundar Value Problems Numerical athematic Vol. 63 pp hen. F. and Hsiao. H. Wavelet Approach to Optimizing Dnamic Sstems IEE Proc.-ontrol heor Appl. Vol. 46 No. pp. 3-9 arch

19 7. Karimi H. R. aralani P. J. oshiri B. and Lohmann B. Haar Wavelet-Based ontrol o ime-varing State-Delaed Sstems: A omputational ethod International Journal o omputer athematics Razzaghi. and Yousei S. Legendre Wavelets ethod or onstrained Optimal ontrol Problems athematical ethod in he Applied Sciences 5 pp Hsiao. H. and Wang J. W. Optimal ontrol o Linear ime-varing Sstems via Haar Wavelets Journal o Optimization heor and Application Vol. 3 No. 3 pp Holmstrom K. Goran A.O. and Edvall. User s Guide or OLAB /SNOP omlab Optimization Inc. 5 Gano S. E. Perez V.. and Renaud J. E. Development and Veriication o A ALAB Driver For he SNOP Optimization Sotware AIAA Paper -6.. Hargraves. R. and Paris S. W. Direct rajector Optimization Using Nonlinear Programming and ollocation J. o Guidance ontrol and Dnamics Vol. No. 4 pp Jul- August Betts J.. and Human W. P. Path-onstrained rajector Optimization Using Sparse Sequential Quadratic Programming Journal o Guidance ontrol and Dnamics Vol. 6 No. pp Fahroo. and Ross I.. Direct rajector Optimization b a hebshev Pseudospectral ethod Journal o Guidance ontrol and Dnamics Vol. 5 No. pp Pietz J. A. Pseudospectral ollocation ethods or the Direct ranscription o Optimal ontrol Problems.A. hesis Dept. o omputational and Applied athematics Rice Universit Houston X Benson D. A Gauss Pseudospectral ranscription or Optimal ontrol Ph.D. hesis Department o Aeronautics and Astronautics I Nov. 4. 9

20 7. Huntington G.. Benson D. A. and Rao A. V. A comparison o Accurac and omputational Eicienc o hree Pseudospectral ethods in Proc. o the AIAA Guidance Navigation and ontrol onerence Hilton Head S August Horie K. and onwa B. A. Optimal Aeroassisted Orbital Interception Journal o Guidance ontrol and Dnamics Vol. No. 5 pp Geiger B. R. Horn J. F. DeLullo A.. AND Long L. N. Optimal Path Planning o UAVs Using Direct ollocation with Nonlinear Programming AIAA Guidance Navigation and ontrol onerence and Ehibit Kestone olorado AIAA Herman A. L. and Spencer D. B. Optimal Low-hrust Earth-Orbit ransers Using Higher-Order ollocation ethods Journal o Guidance ontrol and Dnamics Vol. 5 No. pp Gu J. S and Jiang W. S he Haar wavelets operational matri o integration international Journal o Sstems Science vol.7 no. 7 pp Brson. A.E. and Ho Y.. Applied Optimal ontrol Optimization Estimation and ontrol Hemisphere publishing corporation 975. List o Figures Fig. Graph o Haar Function. Fig. rajector or Brachistochrone Problem Fig. 3 ime Histor o ontrol or Brachistochrone Problem Fig. 4 Objective Function Error or Brachistochrone Problem

21 Fig. 5 ime Histor o States or Bang-Bang Problem Fig. 6 ime Histor o ontrol or Bang-Bang Problem Fig. 7 Objective Function Error or Bang-Bang Problem Fig. Graph o Haar Function.

22 Fig. rajector or Brachistochrone Problem - - analtical solution wavelet with 3 nodes wavelet with 6 nodes cp with 6 nodes

23 Fig. 3 ime Histor o ontrol or Brachistochrone Problem control u rad analtical solution wavelet with 3 nodes wavelet with 6 nodes cp with 6 nodes time sec 3

24 Fig. 4 Objective Function Error or Brachistochrone Problem..8 J-J * Number o Nodes 4

25 Fig. 5 ime Histor o States or Bang-Bang Problem and analtical solution wavelet with 3 nodes wavelet with 6 nodes cp with 6 nodes time sec 5

26 Fig. 6 ime Histor o ontrol or Bang-Bang Problem.5 analtical solution wavelet with 6 nodes wavelet with 3 nodes P with 6 nodes control u time sec 6

27 Fig. 7 Objective Function Error or Bang-Bang Problem J-J * Number o Nodes 7