AIAA 99-2185 Optimal Mixture-Ratio Control for a Single-Stage-To-Orbit Rocket L. Casaiino and D. Pastrone Politecnico di Torino Torino, Italy

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AIAA-99-2185

OPTIMAL MIXTURE-RATIO CONTROL FOR A SINGLE-STAGE-TO-ORBIT ROCKET

Lorenzo Casalino: and Dario Pastronet Politecnico di Torino, 10129 Torino, Italy

Abstract

An indirect procedure is used to optimize the performance of a Single-Stage-To-Orbit rocket which uses liquid oxygen and hydrogen as propellants. A simplified model for the ascent trajectory is con- sidered to highlight the effects of the propellant mixture-ratio control. By assuming that the tank mass is proportional to the propellant volume, either the gross mass or the dry mass are minimized for an assigned payload. The results quantify the advan- tage that can be obtained by using a high mixture ratio (i.e., high propellant bulk density) in the ini- tial phase of the trajectory and by exploiting a low mixture ratio (i.e., high specific impulse) in the final phase. Because of the ascent losses, the control law is different to a deep space operation, where gravi- tational and aerodynamic forces are not present. A higher mixture ratio is used in the initial phase to obtain a slightly higher thrust level and a faster ve- hicle mass reduction. The continous mixture-ratio control provides a slight benefit in terms of propel- lant consumption compared to the constant mixture- ratio operation, while a larger benefit is obtained in terms of payload, due to the tank mass reduction.

Nomenclature

At = throat area *

LD - characteristic velocity 5 discharge coefficient

C~,,C~, = coefficients; see Eq. (19) CFV = vacuum thrust coefficient CFo &I = coefficients; see Eq. (20)

*Researcher, Dipartimento di Energetica. Member AIAA. t Associate Professor, Dipartimento di Energetica, Senior

Member AIAA.

Copyright 01999 by the American Institute of Aeronautics and Astronautics, Inc. AII rigths reserved.

D = drag I-- = drag factor H = Hamiltonian; see Eq. (11) md = dry mass me = engine mass mh = exhaust hydrogen mass mt = tank mass mu = payload mP = propellant mass flow P = see Eq. (12) P = pressure

G = radius = switching function

t = time T = thrust U = vertical velocity component V = horizontal velocity component v, = relative velocity a - oxygen/fuel mixture ratio aD, aF = coefficients; see Eqs. (19) and (20) AK3 = ideal velocity increment E = nozzle expansion area ratio

:: = angle above horizon = adjoint variable

P = density 6 = longitude r1 = vertical ascent time-length 4J = performance index; see Eq. (17) or (18) W = Earth’s angular velocity

Subscripts

0 = initial value a = air

; = combustion chamber = final value

h = hydrogen max = maximum value mean = mean value min = minimum value 0 = oxygen

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P

iL

= propellant = structure = sea level

must be considered to be only a preliminary assess- ment of the benefit provided by the optimal control.

Introduction

The development of a reusable Single-Stage-To- Orbit (SSTO) h 1 ve ic e is one of the most challenging near-future objectives with the aim of reducing the satellite launch costs. Any improvement in structure technology and engine performance is advisable. As far as the propulsion system is concerned, the control of the propellant mixture ratio (oxidizer/fuel mass ratio) appears to be one of the most promising means to increase the performance of a rocket which uses liquid oxygen (LOX) as the oxidizer and liquid hy- drogen (LH2) as the fuel. Because of the great dif- ference in the bulk densities of the propellants, the tank mass can significantly be reduced by enhancing the oxidizer consumption.

Martin and Manski’ have analyzed the ascent trajectory of a rocket whose engines can be operated with two different mixture-ratio values. They have shown the benefit that can be obtained by shifting to a lower mixture ratio during the ascent. In their arti- cle, attention has been mainly given to the influence of the design parameters on the hardware weights, whereas attention is here focused on the mixture- ratio control, when the design parameters have been fixed. The authors have already applied the optimal control theory to rocket operation in the absence of gravitational and aerodynamic forces, either by con- sidering a continuous variation of the mixture ratio2 or by dividing the operation into phases with dif- ferent constant values of the mixture ratio.3 In the present paper the mixture-ratio control is coupled to the optimization of the ascent trajectory that the authors analyzed in a previous paper.4 To highlight the effect of the mixture ratio, a very simple model for the ascent trajectory is now adopted, that is, an equatorial trajectory with thrust parallel to the rel- ative velocity; no constraints are imposed on the ac- celeration and dynamic pressure, even though they could be easily introduced. Either the gross mass or the dry mass for the assigned payload are minimized; the influence of the mixture ratio on the propellant flow rate, on the thrust level and therefore on the gravitational, misalignment and aerodynamic losses is considered. The paper is aimed at investigating the main features of the problem and the theoreti- cal reasons of the optimal strategy; numerical results

Statement of the problem

The rocket is assumed as a point mass; for the sake of simplicity, its motion is assumed to occur in the equatorial plane under the influence of a radial inverse-square gravitational field, and the Earth ro- tation is taken into account. The rocket position in an inertial equatorial reference frame is given by the radius and longitude; the rocket velocity in a local reference frame is given by the vertical and horizon- tal components. The rocket mass and the ejected hydrogen mass are two further state variables. Be- sides gravity, the thrust and aerodynamic drag act on the rocket; these are considered to be parallel during the whole ascent trajectory. All the variables are normalized by assuming the Earth radius, the corresponding circular velocity and the rocket initial mass as reference values. The equations of motion are

dr iii = ‘u

du T-D -=-L-+;+- dt m.

sin y

dw uv T-D w=--..-- dt

r +ycoq

dmh tip -=- dt 1+(Y

(4)

The initial conditions are TO = 1, do = 0 and zero relative velocity, i.e., uc = 0 and ~0 = w; the initial masses are rn.0 = 1 and rnhc = 0. The tra- jectory is split into phases. The first phase is the vertical ascent/vertical thrust (y = 0) phase with an assigned time-length (71 = 0.016, i.e., about 13 seconds). In this phase, the drag is assumed to be vertical, since the misalignment between the relative

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velocity and the vertical direction is negligible. Af- ter this phase, an instantaneous rotation (kick) of the rocket velocity takes places; the kick angle is optimized by the procedure, whereas no propellant is considered to be required by this maneuver. Fi- nally, the rocket performs a zero-lift ascent phase with thrust parallel to the relative velocity

siny = u v-wr

cos y =- K v, (7)

where V, = 29 + (II -wry. In the case of op- eration with constant values of mixture ratio, each constant-a: phase constitutes a separate arc. Oc- casionally, the introduction of a coast arc (T = 0) may be required. Because of the Earth’s rotation, the misalignment losses appear. These losses can be reduced using a higher acceleration level, since the thrust-velocity misalignment decreases as the rocket velocity is increased and the ascent time is short- ened.

A fixed-geometry engine with constant chamber pressure is assumed. The trajectory is controlled by the mixture ratio which determines the characteris- tic velocity, the discharge coefficient, and, together with the nozzle expansion area ratio, the vacuum thrust coefficient. The propellant flow rate is

and the thrust magnitude, which depends also on the altitude via the external pressure p,, is

Optimization

The indirect optimization procedure which is used in the present paper, is based on the theory of optimal control; an adjoint variable is associated to each equation and the Hamiltonian is defined

where

P=X,siny+X,cosy (12)

The problem is autonomous and H is piecewise- constant during the optimal trajectory (it may be discontinuous at the arc junctions). One should note that the propellant flow rate +, influences the per- formance, whereas time can be eliminated in a deep space operation3 and mp can be assumed as the in- dependent variable. This,fact is related to the pres- ence of the ascent losses which, in turn, depend on the thrust level and mass flow rate.

The differential’equations for the adjoint vari- ables are provided by the Euler-Lagrange equations

-2E d& dt axi

T = PeAt(CFv - EP~/P~) (9) which are omitted for the sake of conciseness.

An additional control is given by the possibility of switching off the engine; this is done by considering the thrust factor p,At either to be null or assume an assigned (maximum) value.

The normalized drag magnitude is

D=ICp”.V,2 PSL

The constraining conditions at the final bound- ary are rf = p, uj = 0 and vf = l/p, where p is the assigned value of the final orbit.

3

The optimal controls maximize the Hamiltonian during the whole trajectory. As far as the thrust factor is concerned, the maximum-thrust and null- thrust arcs alternate according to the sign of the switching function

(14

in agreement with Pontryagin’s Maximum Principle.

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The trajectory is also controlled by the mixture ratio cr, which determines cFV and co. In the case of a continuously variable mixture ratio, the optimal control is obtained by enforcing

l3H --=I) da (15)

On the contrary, if the trajectory is a succession of constant-a phases, the trivial equation dcu/dt = 0 and the corresponding adjoint variable X, are added;3 the optimal values of the mixture ratio and time-length for each phase are found by the proce- dure.

The problem is completed with the optimal boundary conditions; the procedure assumes the switching structure of the trajectory which is split into subintervals with an assigned control strategy. Under this assumption, a general expression for the optimality conditions can be found.4 Once the prob- lem has been solved, the trajectory is checked in the light of Pontryagin’s Maximum Principle and even- tually modified.

The state variables are assigned at the initial boundary and the adjoint variables are free. The ve- locity adjoint variable (usually called primer vector) is parallel to the velocity just before and after the kick. The Hamiltonian continuity is prescribed at the internal boundaries; when the thrust is switched on or off, this condition is equivalent to null S. In the case of operation with piecewise-constant-a, the con- dition is equivalent to imposing the switching func- tion continuity at the internal boundaries where the mixture-ratio value is changed; in this case, )L, = 0 is also prescribed. At the final boundary, one obtains X$ = 0, H = 0 and two further conditions

& = 2

which assume different expressions according to the performance index C$ which is maximized. The gross mass minimization for an assigned payload is equiva- lent to the payload maximization, as the initial mass is unit, and the performance index is

where the tank mass is assumed to be proportional to the propellant volume via the structural density ps. The parameter m, includes, beside the engine mass, all the other masses that are independent of the consumed propellant.

On the other hand, if the the dry mass md per unit payload is minimized, the payload/dry-mass ra- tio

$fL$=l+~

Ps Ps ( >

--- _ k’h PO

mhf+Ps+me PO

md (18)

is the performance index that is maximized.

The boundary value problem which arises from the application of the theory of optimal control is solved by means of a procedure5 based on Newton’s method, which finds the unknown initial values of the adjoint variables and the engine switching times.

Numerical examples

The characteristics of the rocket are presented in Table 1. One should note that the thrust factor peAt is assigned. The thrust level is provided by the pro- cedure through the thrust coefficient which depends on the mixture ratio. The discharge and thrust co- efficients have been computed by assuming chemical equilibrium in the combustion chamber at the as- signed pressure and one-dimensional frozen flow in an ideal nozzle. For the sake of simplicity, the depen- dence on the mixture ratio has been approximated using parabolic laws

CD = CD, + CD, (a - aDj2 (19)

Table 1 Rocket characteristics

engine mass me 0.05 drag factor Ii 10 thrust factor PA 0.8 structural density PS 16 kg/m3 nozzle expansion area ratio E 80 chamber pressure PC 200 bar

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4 5 6 7 8

M ixture ratio a

Fig. 1 Engine performance.

CFv = CF, - cF~(a- ffF)'

3.8 v”

E 3.6 3

8

3.4 F 4 2 .H IQ

3.2

3.0

(20)

where CD, = 3.23611, CD, = 2.033 x 10m2, ‘3ug = 3.1, c& = 1.95023, CF, = 2.035 x l@-3, ffF = 8.3. A comparison is shown if Fig. 1. Under this as- sumption, the optimal mixture ratio is obtained by solving Eq. (15), h h w ic results to be a third-degree equation. A numerical solution would be used for a.ny different expressions of CD(o) and CF~(CY). An exponential law is assumed for the air density and pressure

pa/ps~ = exp (-1938.0 (T - 1)l.15’) (21)

p,/ps~ = exp (-1839.6 (r - 1)l.lr8) (22)

According to Pontryagin’s Principle, the opti- mal trajectory is constituted by two burn arcs joined by a coast arc, for the final radius in the range of practical interest (rf = 1.03 + 1.05, that is, for or- bits with a 180 + 300 km altitude). In this range, the optimal control law is almost independent of the final orbit radius. A final radius rf = 1.03 is used in the examples. The final burn presents the charac- teristics of an apogee burn, with an extremely short time-length.

Figure 2 presents the optimal mixture ratio, as a function of the propellant mass, for two differ-

5

4 0 .2 .4 .6 .8 1.0

Propellant mass mP

Fig. 2 Comparison to deep space operation.

ent operations maximizing the payload: The as- cent trajectory and the deep space operation2 (in the absence of losses) with the same ideal veloc- ity increment A& = s,“’ T/m dt = 1.0849. One should note the higher mixture ratio during the initial phase of the ascent trajectory; because of gravitational and misalignment losses, it is conve- nient to have higher acceleration levels. This is ob- tained by increasing the discharge coefficient at low altitudes, to quickly reduce the vehicle mass while slightly increasing thrust. A mixture ratio close to the value corresponding to the maximum thrust co- efficient (LY = QF) is used at liftoff. The possibil- ity of reducing the losses increases the advantage given by the mixture-ratio control during an ascent trajectory compared to a deep space operation; the mixture-ratio control provides a 2.5% payload in- crement compared to constant-a operation for the ascent trajectory, but only 1 % for the deep space

Table 2 Comparion of constant-a and variable-~ trajectories

No. of phases: 1 co A%i 1.0882 1.0849

aerodynamic losses 0.0020 0.0021 misalignment losses 0.0831 0.0806 gravitational losses 0.0764 0.0755

Cmean 0.5661 0.5649

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0 .2 .4 .6 .8 1.0

Propellant mass mp

Fig. 3 Comparison to piecewise-constant-o operation.

operation. One should note that the deep space case provides a higher payload since the mixture- ratio control does not need to search for a reduction of the ascent losses. This better result is obtained using a lower (Y,,,, and a higher mean specific im- pulse; the payload is higher despite the higher tank mass, because less propellant is required.

Figure 3 compares the control law, in the case of continuously variable mixture ratio, to the oper- ation with one or more constant-a phases, for max- imum payload and same final orbit. Tables 2 and 3 summarize the performance. Table 2 shows that the mixture-ratio control allows a reduction of the ascent losses and a lower AI& is required. This benefit is related to higher acceleration levels. The greatest benefit is obtained in terms of misalignement losses.

Table 3 Performance of maximum payload trajectories

No. of phases: 1 2 3 03

mu .0548 .0558 .0560 .0562

m P .8537 .8535 .8535 .8535 mt .0415 .0407 .0405 .0404

~tnax 5.198 6.138 6.538 7.358 %nean 5.198 5.377 5.414 5.446

amin 5.198 4.804 4.650 4.346

0 .2 .4 .6 23 1.0

Propellant mass mP

Fig. 4 Mixture-ratio comparison for different performance indexes.

Gravitational losses are also reduced while aerody- namic losses are slightly increased. The propellant saving is not very large since the mean effective ex- haust velocity c,,,, is lower. The mixture-ratio con- trol in fact imposes a higher (Y,,~~ to reduce the tank mass as in a deep space operation3 (Table 3). The mean mixture ratio and the payload increase with the number of constant-a phases. Most of the ben- efit of the mixture-ratio control is obtained by split- ting the trajectory into only two constant-o phases (1.8 % payload increment compared to constant-a operation).

Figure 4 shows the mixture-ratio history in the cases of minimum gross mass (i.e., maxi- mum payload) and minimum dry mass (maximum payload/dry-mass ratio). In the latter case, a higher

Table 4 Performance of maximum payload/dry-mass ratio trajectories

No. of phases: 1 2 3 cm

mu/md .3771 .3836 .3850 .3861

m P .8561 .8561 .8560 .8560 mt .0396 .0387 .0385 .0384

an-lax 5.650 6.705 7.155 8.099 amean 5.650 5.876 5.923 5.961

amin 5.650 5.142 4.953 4.573

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.058

.056

.052

m

n4ny. .“I”

.80 .82 .84 .86 .88 .90

PC 4

Fig. 5 Optimal payload and engine mass for different thrust factors.

mixture ratio is exploited to obtain a further re- duction of the tank mass,. while the propellant consumption is slightly greater. Table 4 summa- rizes the performance of trajectories with minimum dry mass for different control laws. The benefit of the mixture-ratio control compared to the constant- (Y operation is again about 2.5%.

As already stated, the losses during ascent can be reduced by increasing the acceleration level. This can be obtained also by increasing the engine size, i.e. the thrust factor p,At. On the other hand, the engine mass increment must be taken into account; as an example, m, is assumed to be related to the thrust factor pcAt according to the simple relation

me (23)

Figures 5 and 6 show the influence of the thrust fac- tor on the optimal payload and payload/dry-mass ratio. An optimal engine size exists for each of per- formance indexes. The corresponding liftoff acceler- ation is 1.26 g to obtain the maximum payload and 1.10 g to obtain the maximum payload/dry-mass case.

The payload increment provided by the mixture- ratio control is some percent of the engine weight;

7

5

.385 " * .'.'* l " " '. " " " " .75 .76 .77 .78 .79 .804

PC 4

Fig. 6 Optimal payload/dry-mass ratio for different thrust factors.

the improvement should be compared to the addi- tional engine weight and complexity which are re- lated to the necessity of varying the mass flow of the propellants.

Conclusions

An indirect optimization method has been ap- plied to investigate the effects of the propellant mixture-ratio control during the ascent trajectory of an SST0 rocket. The results show ,that the benefit that can be obtained is greater for this kind of tra- jectory compared to deep space operations, as the possibility of reducing the ascent losses by exploit- ing a high mixture ratio (i.e., a high acceleration level) during the initial phase of the ascent adds to the tank mass reduction. The overall advantage is however limited, and other aspects of the problem should be considered before deciding on the useful- ness of the mixture-ratio control. The method can easily be extended by adding the possibility of choos- ing the optimal thrust direction and by introducing acceleration and dynamic pressure constraints.

References

1 Martin, J. A., and Manski, D., “Variable- Mixture-Ratio and Other Rocket Engines for Ad-

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vanced Shuttles,” Journal of Propulsion and Power, Vol. 7, No. 4, 1991, pp. 549-555.

2 Colasurdo, G., Pastrone, D., and Casalino, L., “Mixture-Ratio Control to Improve Hydrogen-Fuel Rocket Performance,” Journal of Spacecraft and Rockets, Vol. 34, No. 2, 1997, pp. 214-217.

3 Casalino, L., “Optimization of a Dual-Mixture- Ratio Hydrogen-Fuel Rocket,” Journal of Spacecraft and Rockets, Vol. 34, No. 4, 1997, pp. 574-576.

4 Colasurdo, G., Pastrone, D., and Casalino, L., “Optimization of Rocket Ascent Trajectories Using an Indirect Procedure,” Proceedings of the AIAA Guidance, Navigation, and Control Con.fer- ence, AIAA, Washington, DC, 1995, Part 3, pp. 1375-1383. (AIAA Paper 95-3323).

5 Colasurdo, G., and Pastrone, D., “Indirect Opti- mization Method for Impulsive Transfers,” Proceed- ings of the AIAA/AAS Astrodynamics Conferen.ce, AIAA, Washington, DC, 1994, pp. 441-448. (AIAA Paper 94-3762).

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