Goddard s Problem Steven Finch

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1 Goddard s Problem Steven Finch June 12, 2015 Arocketliftsoff vertically at time =0. Let ( ) be the mass of the rocket (payload and fuel) and ( ) be the altitude. We wish to choose the thrust ( ) and a final time such that the altitude ( ) is maximized [1]. It is assumed that ( )(0) = 3 5, thatis,60% oftherocketispayloadand40% is fuel. For convenience, let 0 = (0), = ( ) and = ( ). It is further assumed that the earth is flat and the atmosphere is negligible, thus = 1 2 ( ) ( ) (0) = 0 =0 =0 = 1 ( ) where is the (constant) acceleration due to gravity. Normalize to be 1. Fuel consumption is proportional to thrust; set =1 2. Integrating = 2 ( ) we obtain = ln ( ) hence 0= ln and = ln = Integrating again, we have [2, 3] = Z ln ( ) = 0 ln 12 2 = 2 2 ln = Copyright c 2015 by Steven R. Finch. All rights reserved. 1

2 Goddard s Problem 2 since, by the calculus of variations, it is optimal to select ½ if =0 ( ) = if 0 In words, the rocket will reach maximum altitude if the thrust ( ) is an impulse at =0(a special case of a bang-bang control). All fuel is used instantaneously; the rocket achieves maximum velocity immediately. For consistency with [4, 5], define ( ) = 1 0 = if =0 ln if 0 and 0 = lim ( ) =ln = If there is non-negligible aerodynamic drag, then an interesting tradeoff occurs. High velocity achieved at low altitudes (by an impulsive start) will confront great resistance. It appears that a better strategy would be to save some fuel for intermediate altitudes, but determination of exactly how to execute this is non-trivial. Replace the first ODE by 2 = 1 ( ) " ( ) exp ( ( )) # 2 where =310=(1 2)(620) and =500. In words, air density decreases exponentially with altitude but drag increases quadratically with velocity. Although we cannot solve this nonlinear equation in the same manner as previously, it is remarkable that closed-form expressions for certain quantities even exist. Note that =310 (3 0 5) = (1550 3) 0 The following discussion is due to Tsien & Evans [4], with follow-on work by Leitmann [5, 6, 7]. Let 1 be the burnout time, that is, the end of powered flight. The optimal 1 is 0 for travel in a vacuum; 1 0 if there is significant drag. The rocket continues to coast upward, without fuel, until time. Of course ( 1 )=. Let 1 = ( 1 ) and 1 = ( 1 ). Let = = p (1 ) 2 +8 ( ) =Ei µ 2 2 exp( ) ( ) = 2 +(1 ) 2 +(1 )+ ( ) = 2 +(1 ) 2 2 +(1 ) 2 ( ) = (1 ) 2

3 Goddard s Problem 3 where Ei is the exponential integral [8]. Here is a system of five simultaneous equations, arising from the calculus of variations, that enable us to solve for 1, 1,, 0, 1 : 1 2 = 2 exp µ2 2 exp( 1) [ ( ) ( 1 )] (1 + 1 )=exp( 1 ) 1 = ln ( 1 ) + 3+ ln ( ( 1 0 ) ( 0 ) 2 1 =ln ln ( 1 ) + 1+ ln ( ( 1 0 )) ( 0 ) 2 0 = (1 ) 0 2 µ [ ( 0 ) ( 1 )] exp ( 0 )+exp Given the prescribed parameter values, we obtain 1 =062642, 1 =005085, =013579, 0 = and 1 = In particular, is smaller than the final altitude computed for a vacuum and 1 is considerably larger than 0. Finding the thrust at =0is equivalent to computing 1 lim ( ) = 0 exp( 0 )= that is, approximately 9 8% of the rocket mass is expended at the start. Mass at any time 0 1 can be found via replacing ( 0 ) in the right-hand side of the fifth equation by ( ( )), and then multiplying the whole by exp( ( ) ). Finding velocity, given 0 1, is done by substituting 1, 1 everywhere in the fourth equation by, and then solving for. Thetrickiestpartiscalculating, for which no analogous equation seems to be available. By call to a numerical ODE solver: + 2 exp ( ( )) + =0 ( 2 1 )= 1 = 1 = 1 we obtain = at which vanishes. This, again, is smaller than the final time computed for a vacuum. We also confirm numerically that ( )=. The earth is, in fact, round let its radius be 1 thereforeadistance ( ) = ( )+1 separates the rocket and earth s center. Replace the first ODE by " = 1 # 2 ( ) exp ( ( )) 2 ( ) ( +1) 2

4 Goddard s Problem 4 Figure 1: Histories of optimal flight characteristics for decreasing values of. where and are as before. Suppose that 0 =1. Additional realistic constraints on thrust and dynamic pressure 0 ( ) 7 2 ( ) =1 2 0 exp ( ( )) 2 10 make the optimization more difficult, where the parameter 0 = is air density at sea level. A substantial literature exists on the numerical solution of this problem [9, 10, 11, 12, 13, 14, 15, 16, 17]; the optimal final time is = and the optimal final distance is = +1= Figure 1 constitutes relevant Matlab graphical output [18, 19], where is a penalty parameter. The phase between initial thrust =3 5 and final thrust =0is known as the singular arc [20]. See also [21] for informal history and [22, 23, 24] for more examples and techniques. We mention finally control problems involving a missile moving obliquely in a vertical plane, maximizing the horizontal range covered [25] or a spacecraft attempting to make a soft landing on the moon, minimizing fuel consumption [26].

5 Goddard s Problem Acknowledgements. I am thankful to Paul Malisani for his Matlab code [18, 19], which provided Figure 1, and for his kind correspondence. References [1] R. H. Goddard, A Method of Reaching Extreme Altitudes, Smithsonian Instit., 1919, pp. 5 11; [2] F. J. Malina and A. M. O. Smith, Flight analysis of the sounding rocket, J. Aeronautical Sciences, v. 5 (1938) n. 5, [3] F.Y.M.Wan,Introduction to the Calculus of Variations and its Applications, 2 nd ed., Chapman & Hall, 1995, pp ; MR (97a:49001). [4] H. S. Tsien, and R. C. Evans, Optimum thrust programming for a sounding rocket, J. Amer. Rocket Society, v. 21 (1951) n. 5, [5] G. Leitmann, Optimum thrust programming for high-altitude rockets, Aeronautical Engineering Rev., v. 16 (1957) n. 6, [6] G. Leitmann, A calculus of variations solution of Goddard s problem, Astronautica Acta 2 (1956) 55 62; MR (18,685h). [7] G. Leitmann, A note on Goddard s problem. Astronautica Acta 3 (1957) [8] S. R. Finch, Euler-Gompertz constant, Mathematical Constants, Cambridge Univ. Press, 2003, pp [9] H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems, J. Guidance, Control, and Dynamics 13 (1990) [10] P. Tsiotras and H. J. Kelley, Drag-law effects in the Goddard problem, Automatica J. IFAC 27 (1991) ; MR [11] P. Tsiotras and H. J. Kelley, Goddard problem with constrained time of flight, J. Guidance, Control, and Dynamics 15 (1992) ; [12] H. Seywald and E. M. Cliff, Goddard problem in presence of a dynamic pressure limit, J. Guidance, Control, and Dynamics 16 (1993)

6 Goddard s Problem 6 [13] K. Graichen and N. Petit, Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions, Proc. 17 th World Congress, International Federation of Automatic Control, Seoul, 2008, ed. M. J. Chung and P. Misra, pp ; [14] E. Cristiani and P. Martinon, Initialization of the shooting method via the Hamilton-Jacobi-Bellman approach, J. Optim. Theory Appl. 146 (2010) ; arxiv: ; MR (2011g:49050). [15] M. S. Aronna, J. F. Bonnans and P. Martinon, A shooting algorithm for optimal control problems with singular arcs, J. Optim. Theory Appl. 158 (2013) ; arxiv: ; MR [16] P. Malisani, F. Chaplais and N. Petit, An interior penalty method for optimal control problems with state and input constraints of nonlinear systems, Optim. Control Appl. Methods, to appear; [17] M. Gerdts, Optimal Control of ODEs and DAEs, de Gruyter, 2012, pp. 4 5; MR [18] P. Malisani, Goddard problem Matlab code, 2012, [19] P. Malisani, Pilotage dynamique de l énergie du bâtiment par commande optimale sous contraintes utilisant la pénalisation intérieure, Ph.D. thesis, Ecole Nationale Supérieure des Mines de Paris, 2012, /. [20] A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, rev. ed., Hemisphere, 1975, pp ; MR (56 #4953). [21] R. D. Rugescu, Goddard s 85 years optimal ascent problem finally solved, Proc. 57 th International Astronautical Congress, Valencia, 2006, pp ; History of Rocketry and Astronautics, Proc. 40 th History Symposium, International Academy of Astronautics, ed. M. Freeman, 2012, pp ; [22] H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl. 18 (1976) ; MR (53 #12011). [23] G. Fraser-Andrews, Numerical methods for singular optimal control, J. Optim. Theory Appl. 61 (1989) ; MR

7 Goddard s Problem 7 [24] G. Vossen, Switching time optimization for bang-bang and singular controls, J. Optim. Theory Appl. 144 (2010) ; MR (2011a:49075). [25] G. Leitmann, The Calculus of Variations and Optimal Control: An Introduction, Plenum Press, 1981, pp. 3 5, , , , , ; MR (84m:49002). [26] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, 1975, pp , 28 33, 35 37; MR (56 #13016).

INVERSION IN INDIRECT OPTIMAL CONTROL

INVERSION IN INDIRECT OPTIMAL CONTROL François Chaplais, Nicolas Petit Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 35, rue Saint-Honoré 7735 Fontainebleau Cedex, France,