Optimization of Supersonic Transport Trajectories

APPENDIX A - NOMENCLATURE D = drag E = mechanical energy F = normalized tangential force g = gravity h = altitude H = Hamiltonian d = cost functional L = lift m = mass M = Mach number q = dynamic pressure T = thrust v = velocity z = range a = angle of attack /3 = fuel flow rate c = small parameter 7 = flight path angle A = adjoint variable _b = cost function p = air density

APPENDIX B - NUMERICAL INTEGRATION OF STATE EQUATIONS

In this appendix we give the algorithms by which the state equations are integrated within ACSYNT.

Two cases are of interest. First, the integration of the trajectory when pairs of altitude and energy (equivalently altitude and speed or Mach number) are given; (E0, h0), (El, hl), " ", (El, hy). This is sometimes called path following. Second, the integration of the trajectory when the normal load factor N is held constant.

Path Following

It is assumed that all variables are known at step n- 1. These values are sought at step n, knowing only En and hn. Of particular interest are the values of tn, ran, and Xn. We start with equations (7), with + = 0, written in finite difference form from step n - 1 to step n: Am

--A-T = -_

Az = _COS_ 031) 'h-'f AE _' -_-T = g---_ - Ah = _sin_ At

o = co v)

V where F I = T cos a - D and, if Q is any variable,

nQ = On - Qn-1

Qn + Qn-1

Q=

From the first of equations (B 1), (B2) mn = ran-1 - -_At so that ¢9 = ran-1 -- -_At Substituting this into the third of equations 031) and solving for At, At = ran-1 033) Note that g, AE, -h, and _ are all known (the latter if throttle is fixed). The only quantity not known in equation 033) is F, which depends on a. If At were known, the fourth of equations 031) gives "7: = sin_ 1 ( _xh '] (B4)

\FET]

The algorithm may now be statedas follows: 1. Guess an.

2. Compute At from equation 033).

3. Compute _ from equation 034).

4. Check to see if the fifth of equations 031) is satisfied to a suitable degree of accuracy. If not, select a new an by a suitable one-dimensional search procedure and return to step (2). If satisfied, continue.

5. Compute mn from equation 032) and tn and Xn from: tn = tn-1 + At Xn = Xn-1 + _Atcos_ Phillips 35 has proposed an alternative integration scheme as follows. The third of equations (B 1) is now averaged directly AE 1 + Vn-lFn_ 1 At 29 \ mn ran-1 This is then combined with equation 032) to give This is a quadratic equation to be solved for At, and replaces equation (B3) in the numerical procedure.

As the integration step size tends to Zero, these two integration schemes become equivalent.

Constant Normal Load Factor (N) Paths In this case, AE is not a suitable integration variable because it may happen that AE _< 0, which causes serious numerical problems. Alternative choices are At and AT. Because the choice At results in an algorithm with three nested iterations, we follow Phillips 35 and choose A,,/. For this integration we do not neglect _'.

Because N, A,,/, and ";'n are now known, -K = g( N - cos 7) is a known constant. Thus the finite difference form of the last of equations (8) is w A? K At g Use this andequation 032) to eliminate mn and At from the rest of equations 031). The result is hn = hn_l + (ATs__._n') (Vn +2n-1)2 (B5) AE = (B6) The algorithm is as follows: (1) Guess Vn.

(2) Solve for hn from equation 035).

(3) ComputeEn=hn+ 1 2 andAE En En-1 2-..-_ v n -- _ .

(4) Guess an.

If (5) Check to see if equation 037) is satisfied. If not, select a new an and repeat this step.

satisfied, continue.

If (6) Check to see if equation 036) is satisfied. If not, select a new Vn and return to step (2).

satisfied, continue.

(7) Compute all other quantities of interest.

Thus this algorithm requires a nested two parameter search, whereas the path following routine required a one parameter search. From equations 035)-037) it is seen that K = 0 (N = cos 7) is not allowed.

Should this happen, one solution is a At integration but, as mentioned earlier, this involves a three parameter search.

Phillips 35 has proposed an alternative method of constant load factor integration with A,y as integration variable. This approach holds all variables constant at the previous step but does a second order integration of the altitude state equation. The increments At and Av may be now directly computed from the third and fifth of equations (3): Vn-1 A,,/ At = g(N - cos%_l) Av = g(Fn-1 - sin'Yn-1) At Differentiating the fourth of equations (3): = l_ sin "/+ v'}' cos "/

Using the third and fifth of equations(3) this becomes

h = 9(Fn_l sin")'n-1 + NcosTn-1 - 1) = C Integrating twice: h = 1Ct2 + Cl t The constant of integration Cl is determined from _tn_ 1 -- vn_ 1 sin _'n-1 = Cl SO that 1 2 hn = Vn-lAtsin_/n-1 + _gAt (Fn-1 sin"fn-1 + Ncos_'n-1 - 1) with At determined as above. Reference 35 shows that this gives good numerical results.

APPENDIX C - NECESSARY CONDITIONS FOR FAST DYNAMICS The state equations of the fast-dynamics are the last two of equations (54):

1 = A v7 + Yv g(F- 7)

(C1) -_ = g-(N- 1) v with m, x, and E (the slower states) all known constants; the control variable is a.

From equation (23) the Hamiltonian is H = -K: - K2_ + )_z v + AEvF + Af [fh v7 + fv g(F - 7)] (C2) i=1 where the constraints equations (1 i) are now state constraints and must be adjoined to H with multipliers vi ; the si are assumed to be written as functions of f and E, the latter a known constant. The adjoints Az and AE are known constants from the slower dynamics solutions, equations (43) and (51). From equation (24) the adjoint equations are A f = K2j3f - Az vf - AEvfF- )_EVFZ - A f [ (fh)fV7 + fh v f7 + fv gFy (C3) _7 = -Af fh v + Af fvg In these equations, if Q(h, v) is any where the notations of equations (5)and (40) have been used.

function then (C4) Qf = Qhhf + Qvvf Assuming an unbounded optimal control, conditions (a) and (b) of the maximum principle give (C5) -K: - K2/3 + ikz v + AEVF + Af[fhv7 + fvg(F -7)] + A'rg( N- 1) = 0 From equations (2) explicit forms for Fa and Na are F_- - -- (Ta cos a - T sin a - Da) mg (C6) _h_ = u (Ta sin a + T cos a + La) mg Equations(C1), (C3), and(C5) areusedto modeltransitionsfrom aninitial condition to the energy dynamics solution (energyclimb path, or ECP), from the ECP to a terminal condition, and between different branchesof the ECP if the ECP is discontinuous.In what follows, the first case,transition from an initial conditionto the ECP will be considered for the purposeof illustration.

For this case,the boundaryconditionson equations (C1) and(C3) are

/(o) =/o

-_(o) : "_o

(C7)

K1 + K2#0 - Azvo- AEvoFo- ATo_(No - 1)

_:(o) =

AoVO'yO + Ao g(Fo -_o)

A._(0) = A-r0 selected to match with ECP where the second of equations (5) was used and where all quantities are known except _'_o. In summary, equations (C1) and (C3) are to be integrated with control given by the first of equations (C5) subject to initial conditions equations (C7).

The fast dynamics equations depend on the nature of the ECP solution because this solution determines the choice of variable f. If the ECP solution is an unbounded optimum, singular perturbation theory states that the ECP solution will be an equilibrium point of the fast dynamics, 7,8 and the goal is to find a solution of the fast dynamics such that the solution approaches the ECP as t _ oc. If, on the other hand, the ECP is on a constraint, then the fast dynamics solution may reach the ECP in finite time. 36 Some examples will nov, be given. If the ECP is on a terrain limit,

f = y(h, v) = h - hm

In this case the transformation (h. v) ---, (E, f) and its inverse are given by 1 2 E=h+_gV f=h-hm so that g fh = l, fv=O, h/= l, vf =-- U and from equation (C4) Q.f = Qh - g-Qv V

Putting theseresultsinto equations(C1) and(C3)

]=V'y

,_= g (N-1)

v ,_, = -Afv and into equations (C5) AEvFa + )_TgN_ = 0 1) -K1 - K2_ + AxV + AEvF + Afv.,/ + A-rg(N The initial conditions for the integration of equations (C7) are as follows: f(O) = fo

-y(o)= "_o

K1 +//'2/30 - Azvo - AEvOFO - ATo g(N0 - 1) v

_j(o) =

vO'YO A.;,(O) = A_, o , selected to match with ECP If the ECP is on a dynamic pressure limit, the transformation (h, v) ---* (E, f) is E=h+lv 2 2g 1 2

f= _ -qm

with p = p(h) so that

h = :ph v2 L = p_

The inverse transformation is implicit. Taking differentials and using the fact that E = const.: dE = dh + V dv = 0 g 1 2 elf = -_ph v dh + pv dv Combining these equations gives

_'- v(___)

hf = ½PhV 2 _ Pg Then from equation (C4) Qh Qv This gives, for example, Ph pf = ½PhV2 -- pg l p v 2 hh Ph v = + . Ph v + P

Fh

+

Zh Zv

+

_: _v___ v(___)

Equations (C1) and (C3) become 1 3 g

-_= v(N-1)

1 3 _'7 -- --/_f_Ph v + Afpvg and equations (C5) become

),E.vF_ + AvpvgF,_+ A.yg No, = 0

f

1) 1 3 + pvg(F-'7)]+ g - Initial conditions equations (C7) become f(O) = fo

:,(0) = ;.o

h'l + K2_o - A_vo- AE, voFo - A-yo _(No - l)

A f(0)

_pho@YO + povog(Fo - "yo)

AT(0) = _70 selected to match with ECP In all of these equations, quantities such as v f, Ff, and (fh)f are to be determined from the equations derived above.

Finally, consider the case for which the ECP is an unbounded local optimum. From equation (52), f in this case is f=Pv-v-Ph g where 19 is given by equation (50). Thus q_

.fh= Pvh - _Phh

1 v

Iv- - - Shv

Because E = const., dE = dh + V dv = 0 g df = fhdh + fvdv so that vf =

_ _ __&

g Let ¢ = ¢(h, v) be defined as

_fh

¢=f_---

g Then 1 2Vp v 2 ¢ = Pvv -- gPh - g hv +-_Phh and equation (C4) becomes This explains how to compute quantities such as fly, Fy, and (fh)f in equations (C3). Equations (C1), (C3), (C5), and (C7) will not be written out explicitly for this case.

REFERENCES 1. Myklebust, Arvid; and Gelhausen, P.: Putting the ACSYNT on Aircraft Design. Aerospace America, Sept. 1994, pp. 26-30.

2. Calise, A.: Singular Perturbation Methods for Variational Problems in Aircraft Flight. IEEE Transactions on Automatic Control, vol. AC-21, June 1976, pp. 345-353.

3. Calise, A.: Extended Energy Management Methods for Flight Performance Optimization. AIAA Journal, vol. 15, no. 3, Mar. 1977, pp. 314-321.

4. Calise, A.: A New Boundary Layer Matching Procedure for Singularly Perturbed Systems.

IEEE Transaction on Automatic Control, vol. AC-23, no. 3, June 1978.

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6. Kelley, H.; and Edelbaum, T.: Energy Climbs, Energy Turns, and Asymptotic Expansions.

Journal of Aircraft, vol. 7, Jan. 1970, pp. 93-95.

7. Ardema, M.: Solution of the Minimum Time-to-Climb Problem by Matched Asymptotic Expan- sions. AIAA Journal, vol. 14, no. 7, July 1976.

8. Ardema, M.: Singular Perturbations in Flight Mechanics. NASA TM X-62,380, Aug. 1974.

9. Shinar, J.: On Applications of Singular Perturbation Techniques in Nonlinear Optimal Control.

Automatica, vol. 19, no. 2, 1983, pp. 203-211.

10. Bryson, A.; Desai, M.; and Hoffman, W.: Energy-State Approximation in Performance Opti- mization of Supersonic Aircraft. Journal of Aircraft, vol. 6, no. 6, Nov.-Dec. 1969.

11. Kelley, H.; Cliff, E.; and Weston, A.: Energy State Revisited. Optimal Control Applications & Methods, vol. 7, 1986, pp. 195-200.

12. Ardema, M.; Bowles, J.; Terjesen, E.; and Whittaker T.: Approximate Altitude Transitions for High-Speed Aircraft. Journal of Guidance, Control, and Dynamics, vol. 18, no. 3, May-June 1995, pp. 561-566.

13. Ardema, M.; and Rajan, N.: Selection of Slow and Fast Variables in Three-Dimensional Flight Dynamics. American Control Conference, WP7-3:30, June 1984.

14. Ardema, M.; and Rajan, N.: Slow and Fast State Variables for Three-Dimensional Flight Dynamics. Journal of Guidance, Control, and Dynamics, vol. 8, no. 4, July-Aug. 1985.

15. Erzberger, H.: Automation of On-Board Flight path Management. NASA TM 84212, Dec.

1981.

16. Erzberger, H.: Theory and Applications of Optimal Control in Aerospace Systems. AGARDo- graph No. 251, July 1981.

17. Erzberger, H.; and Lee, H.: Algorithm for Fixed-Range Optimal Trajectories. NASA TP 1565, July 1980.

18. Breakwell,J.: Optimal Flight-Path-Angle Transitions in Minimum-_me Airplane Climbs. Jour- nal of Aircraft, vol. 14, no. 8, Aug. 1977.

19. Breakwell, J.: More about Flight-Path-Angle Transitions in Optimal Airplane Climbs. Journal of Guidance and Control, vol. 1, no. 3, May-June 1978.

20. Weston, A.; Cliff, E.; and Kelley, H.: Altitude Transitions in Energy Climbs. Automatica, vol. 19, Mar. 1983, pp. 199-202.

21. Shinar, J.; and Fainstein, V.: Improved Feedback Algorithms for Optimal Maneuvers in Vertical Plane. AIAA Paper 85-1976, 1985.

22. Ardema, M.; and Yang, L.: Interior Transition Layers in Flight-Path Optimization. Journal of Guidance, vol. 11, no. 1, Jan.-Feb. 1988.

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Journal of Guidance, Control, and Dynamics, vol. 8, no. 2, Mar.-Apr. 1985.

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26. Jacobson, E.; Lele, M.; and Speyer, J.: New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints. Journal of Mathematical Analysis and Applica- tions, vol. 35, Aug. 1971, pp. 255-284.

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29. Tihonov, A.: Systems of Differential Equations Containing Small Parameters Multiplying Some of the Derivatives. Math. Sb., vol. 73, no. 3, N.S. (31), 1952 (in Russian).

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4O

Table1. Characteristics of supersonic transport

753,500Ib

Grosstake-off weight

5500ft2

Wing planform area

137.35ft

Wing span

48 deg

Leadingedgesweep

3.43

Aspectratio

314ft

Body length

Payload

first classpassengers

coachclasspassengers

flight crew

flight attendants

2.4

Maximum Mach number

1000psf

Maximum dynamicpressure

h

loft ceiling

lift :oeffici ¢ _ dynamic pressure

/

I M

terrain Sketch of constraints defining flight envelope.

Figure 1.

1.20 1.15 1.10 1.05 1.00 n .95 .90 .85 .80 .75 I I 1 J .70 I v 20 40 60 80 100 -80 -60 -40 -20 0 tl Time, se¢ t2 Load factor history during altitude transition for a high performance aircraft.

Figure 2.

hi

/

h2 v V v2 Sketch of an altitude transition.

Figure 3.

1.40e+05

I

130e+05 Leqend • Specific Energy 35,234 ft • Specific Energy 37,437 ft 12.0e+05 • Specific Energy 39,640 ft • Specific Energy 41,844 ft 1.10e+05 Specific Energy 44,047 ft Specific Energy 46_.50 ft 1.00e+05 Thrust lb. 9.ooe+o4 8.00e+04 7.00e+04 6.00e+04 5.00e+04 4.00e+04 0.60 0.70 0.80 0.q0 1.00 1.10 120 130 MACH No.

Figure 4. Thrust vs. Mach number in the transonic region.

950O0 90000 Legend Specific Energy 35,234 ft 85000 Specific Energy 37,437 ft Specific Energy 39,640 It 80000 Specific Energy 41,844 tt e Specific Energy 44,047 ft Q Specific Energy 46,250 ft 750OO )ecific Drag lb. 70000 65000 • i 60000 55000 .................. k 50000 45000 0.60 0.70 0.80 0.90 1.00 1.10 1 2.0 130 MACH No.

Figure 5. Drag for N = 1 vs. Mach number in the transonic region.

6OOOO loft ceiling 5OOOO . ._cos(alpha)=D) 40OOO i Altitude ft. 3OOOO 20O0O q% stall 'x_ : '. max dynamic pressure 10000 o 0.0 0.5 1.0 1.5 2.0 2.5 MACH No.

Figure 6. Optimum cruise points in the flight envelope for minimum fuel.

0.035 0.030 Lain X 0,025 0.020 0.015 : 0.010 0.G 0.8 1.0 1.2 1.4 1.6 1.8 2.0 22 2.4 02 0.4 MACH No.

Figure 7. )_z vs. Mach number.

8O e 6o ft/s 4O 2O -20 0£)0 0,70 0.80 030 1,00 1.10 12.0 130 MACH No.

Figure 8. Energy rate vs. Mach number in the transonic region.

600oo I [ ,,_nd [ Q Min. Fuel, no max q bound | _a_eouooe_ 50000 [ O Min. Fuel. max a = 1000 psflt ....................... !............................... !"'¢ ......................

........... _................. q ............. II : : 4O0O0 Altitude ft

°°°°iioooo iiiiiii

10000 .....

0 i 2.5 0.0 0,5 1.0 1.5 2.0 MACH No.

Figure 9. Energy climb path with and without dynamic pressure constraint.

60000 Leqend 500O0 ...... [3 Min.Fuelwith LambdaX 0 Min.Fuelwithout LambdaX 40OOO Altitude ft 3o000 2OO0O 1OO0O o 0.0 o5 1.o 1.5 2.0 2.5 MACH No.

Figure 10. Energy climb path with and without )_z.

10 Ib ...._.690,683 lb _ ____ 657,3 52,481 t_ I ....... I-- altitude _.- 753,498 Ib I 349 nm 836 nrn range Figure 11. Sketch of trajectories with and without Az.

6O0OO 500O0 Min. Time 0 Min. Fuel ,_ Min. Cost 40000 Altitude It 300OO 20OOO 1OOOO 0.0 05 1.0 1.5 2.0 2.5 MACH No.

Figure 12. Energy climb path for minimum fuel, minimum time, and minimum cost.

60000 Energy climb Path, min fuel Legend I 50000 .......................... Altitude Transition ....................................... _ ;"" _"':"" __" //!

Q i /_# 4OOO0 ................................................ :....... : ......................................... ) ............. !...,;_,...., ............................

: ¢: : ,_ : :,6 : Altitude r.

............ __ ............ ,........... , ............. : ............. ......................... _ ..........................

ft 30000 _. : _ ............ ............. '_,./_i i, .'_ 20000 i t "_" ( : : • _ 10000 ............. ".- ............................ :_-2' 0.2. 0,4 0.6 018 1,0 12. 1.4 1.6 1.8 2.0 22. 2.4 MACH No.

Figure 13. Transonic altitude transition for N1 = 0.97, N2 = 1.05, nonlinear determination of _ and 7.

35OOO Leg.end Energy Climb Path, rain. fuel Altitude Transition ........... i ...................................... : 3OOOO Altitude ft 25O00 20000 0.9 1.0 1.1 12. 1.3 MACH No.

Altitude transition in the transonic region.

Figure 14.

35000 /4, [ Legend ; ! , ] Energy Climb Path, rain. fuel | ,- ""t"_ I "' • I Altitude Transition • f , • .----- ............ . ............

.m . _ . : 3000O .................. ; ...................... • ............ : ...................................... t ..................

Altitude :: i _ , i ft i i •_ i - ,%, !

• ............................ : ...................................... ; .............................. _",,.....i ..................................

25O00 20000 i i 1.3 0,9 1.0 .1 12 MACH No.

Figure 15. Altitude transition, linear determination of g and 7.

35000 rl i ,'_" •', I Legend | ,'" _"_, I Energy Climb Path, min. fuet I ." ,_'_, Altitude Transition | r _ _ : • i •4 !

3OO0O ' ; "% : i Altitude i i •• :: ft ; i % i ! _ : 25OOO ,e i

, ! ,."

2OOOO • o " i ' 0,9 1,0 1.1 1 2. 1,3 MACH No.

Figure 16. Altitude transition for 2V 1 = 0.5, N2 = 1.5.

P

t

ft/sec 2O .............................................................................. r .................. _ .........................................................

g 0.90 0-q5 1.00 1.05 1.10 1.15 1.20 12.5 1.30 MACH No.

Figure 17. Energy rate during altitude transition, N1 = 0.97, N2 = 1.05.

2.0 1.0 0.0 -1.0 Gamma -2 .O deg -3.0 -4.0 -5.O -6.0 0.90 0.95 1.O0 1.05 1.10 1.15 125 130 MACH No.

Figure 18. Flight path angle during altitude transition, N1 = 0.97, N2 = 1.05.

" ................................................................ =. .................. =..................... _...................... :......................

25 - ......................... :................................................................................. '..................... "_ P ft/sec ..................... ' ..................... 7 ..................... : ...... _..................... _..................... "_ 15 ° o -5 ......... i ........ '....... !........ i .......... .......... i ....... ! ......... ......... ! .......... i ........... '.......... ..........

-10 1,00 1,05 1.10 1.15 12.0 12.5 130 0.95 MACH No.

Figure 19. Energy rate during altitude transition, N1 = 0.5, N2 = 1.5.

5O -5 Gamma deg -lO -15 -2O -25 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 MACH No.

Figure 20. Flight path angle during altitude transition, N1 = 0.5, N2 = 1.5.

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REPORT TYPE AND DATES COVERED 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE Technical Memorandum March 1998 5. FUNDING NUMBERS 14. TITLE AND SUBTITLE Optimization of Supersonic Transport Trajectories

t27o

522-41-42 6. AUTHOR(S) Mark D. Ardema,* Robert Windhorst,* and James Phillips 8. PERFORMING ORGANIZATION 7. PERFORMING b'R_;A"NI'ZATION NAME(S) AND ADDRESS(ES) REPORT NUMBER Ames Research Center A-98-09997 Moffett Field, CA 94035-1000 10. SPONSORING/MONITORING 9. SPONSORING/MONITORING AGENCY NAME(S) ANDADDRESS(ES) AGENCY REPORT NUMBER National Aeronautics and Space Administration NASA/TM--1998-112223 Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES Point of Contact: James Phillips, Ames Research Center, MS 237-1 l, Moffett Field, CA 94035-1000 (650) 604-5789 *Santa Clara University, Santa Clara, California 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified -- Unlimited Subject Category 08 13. ABSTRACT (Max' mum 200 words) This paper develops a near-optimal guidance law for generating minimum fuel, time, or cost fixed-range trajectories for supersonic transport aircraft. The approach uses a choice of new state variables along with singular perturbation techniques to time-scale decouple the dynamic equations into multiple equations of single order (second order for the fast dynamics). Application of the maximum principle to each of the decoupled equations, as opposed to application to the original coupled equations, avoids the two point boundary value problem and transforms the problem from one of a functional optimization to one of mul- tiple function optimizations. It is shown that such an approach produces well known aircraft performance results such as minimizing the Brequet factor for minimum fuel consumption and the energy climb path.

Furthermore, the new state variables produce a consistent calculation of flight path angle along the trajec- tory, eliminating one of the deficiencies in the traditional energy state approximation. In addition, jumps in the energy climb path are smoothed out by integration of the original dynamic equations at constant load factor. Numerical results performed for a supersonic transport design show that a pushover dive followed by a pullout at nominal load factors are sufficient maneuvers to smooth the jump.

15. NUMBER OF PAGES 14. SUBJECT TERMS Supersonic aircraft trajectories, Singular perturbations, 16. PRICE CODE Transonic altitude discontinuity A04 20. LIMITAT'_ON OF ABSTRACT 19. SECUR'ITY CLASSIFICATION 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION OF ABSTRACT OF REPORT OF THIS PAGE Unclassified Unclassified Standard Form 298 (Rev. 2-89) _ISN 7540-01-280-5500 Prescribed by ANSI Std. Z3e-18 298-102