(c)l999 American Institute of Aeronautics & Astronautics or p&kk&MttrpEEiwsion of author(s) and/or author(s)’ sponsoring organization.

A99933908 AMA 99-4171

Optimizing Boost Phase of Rocket Using Lyapunov Optimal Feedback Control

Fayyaz Ahmed Lohar* Faculty of Engineering

International Islamic University Malaysia, Kuala Lumpur, Malaysia Tariq Masood**

College of Electrical and Mechanical Engineering National University of Sciences and Technology, Rawalpindi, Pakistan

Syed Firasat Ali? Department of Aerospace Engineering

Tuskegee University, Tuskegee, AL 36088

Nomenclature A = constant in decent function, 20; b =l/scale height; B- &scDro

2% ,2

CD = drag coefficient;

G = 4p d

z;

D = aerodynamic drag force, ~C&i’~V2 ;

m = instantaneous mass of vehicle; m, = initial mass of vehicle; r = radial distance from the earth’s center; 6 = radial distance Corn the center to the surface of the earth; S = surface area; t = time; T =thrust;

T u =-* 2’ wJJr*

%m = 2.4 maximum; V = velocity of vehicle;

a w =-- d- P/r,3

3

XI = r/y,;

x2 =&;

x3 = mlm,

Greek Symbols

,u = gravitational constant (3.98601.3 km3/sec2); p = density; p,, = atmospheric density at reference altitude, 1.25 kg/m3; - z=t$; d G

-0 = spin of the Earth, 72.92x lo4 rad/s

Subscript

= initial values; J = final values;

Abstract This paper addresses the problem of developing a guidance law using Lyapunov optimal feedback control for the boost phase of the futuristic rocket capable of throttling for the launch of satellite and missile. The decent function is derived and optimal control inputs are computed in the absence and presence of density fluctuations for the boost phase of the rocket. The results show that the guidance algorithm based on Lyapunov optimal

* Assistant Professor, member ANA ** Undergraduate student t Associate Professor, member AIAA Copyright 0 1999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.

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feedback control satisfies the necessary conditions.

Introduction various missions for advance space transportation systems, such as the orbital transfer vehicle (OTV), involve transfer from a high Earth orbit (HEO) to a low Earth orbit (LEO)‘“. For example, a vehicle might transfer a payload to a geosynchronous orbit and the rendezvous with either a shuttle or a space station. These missions will involve aeroassisted orbital maneuvering, one of the principal technical difficulties in the use of aeroassisted orbit transfer is the design of a guidance algorithm capable of compensating for large unpredictable fluctuations in atmospheric density so called “potholes in the sky”. This problem has been addressed by Byoung Soo Lee and Walter J. Grantham in 198g4. They used Lyapunov optimal feedback control technique for optimizing the trajectory of OTV, the control input used by them was lift coefficient. In this paper, Lyapunov optimal feedback control technique is applied to develop guidance algorithm of a futuristic throttleable rocket for missile and satellite launch, which uses liquid propellant.

The paper first time addresses the problem of developing a guidance algorithm based on Lyapunov optimal feedback control to optimize the boost phase trajectory of rocket. There could be many problems solved based on Lyapunov optimal feedback control technique such as, wind speed uncertainty, aerodynamic modeling uncertainty etc. However, in this paper the technique used for compensating large unpredictable fluctuations in the atmospheric density. The thrust is taken as control input, this investigation shows that Lyapunov optimal feedback controller can simplify guidance and control problems in the launch of missile and satellite.

Figure 1 shows a typical ICBM and satellite launch trajectory. It can be divided into three phases.

iii) Boost phase: the boost phase in which the vehicle is boosted by one or more rocket stages through the atmosphere until final shut down or cut

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off. The rocket after vertical lift keeps going vertically then gets a kick angle to preset the trajectory.

iv) Free Flight Phase: covering the major part of the range and at the end of which the vehicle either intersect the desired orbit in the case of satellite or in the case of missile enters the atmosphere.

For a satellite 9 The final thrust impulse can be

applied when the vehicle intersects the desired orbit to put the satellite in the orbit.

For a missile v> The Reentry Phase: it is the

subsequent passage downwards through the atmosphere until the missile hits the desired location at the surface of the Earth.

In this study, single stage rocket is considered for the launch, and it is also assumed that the aerodynamic drag coefficient remains constant during the boost phase. Furthermore, the part of the boost phase within the sensible atmosphere will be optimized. After the sensible atmosphere, the desired kick angle related single impulse could be applied to put the satellite into desired orbit or the missile in free flight or ballistic phase so that the missile hit any desired location on the surface of the Earth.

Equations of Motion It is assumed that the flight of the missile

is vertical, i.e., flight-path angle = 90°, we have equations of motion as 5

dr -V z- dv=F,-cL+~2r dt m r2

. . . . . . . . . . . . . . . . . . . . . . . . . . (1)

dm T -=-- dt dsp

where FT =T-D. We have nondimensionalized equations of motion in the state space form

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. x, = x.2 . u x =-- 2

Bx22e-bc(x’ -‘I + x w2 _ 1 1 . ...(2)

x3 x3 . u * =-- 3 yr

where (*) means d&and ‘u, the control input, is thrust with specific control constraints u E u = {uI 1~1 I urnax}. A discussion on parabolic drag polar is not given here, the interested reader is referred to Ref. 5.

Lvapunov Optimal Feedback Control Our design objective is to drive the

system ~=~.016,0.2too3.0.3].

final target state, A “decent function”

w[x(t)]may be thought of as being entirely analogous to “distance” to the target, but in general it can be any candidate of Lyaptmov type function, i.e., any function such that if dwCx(Ol

dt < Ocould be achieved then x(t)

would move to the target. A differential scalar function W(X) is a

decent function if and only if the following conditions hold

vi) W(x)>0 for all x f f vii) W(i) = 0, viii) d W(X)/ dx +O for all x ;t 2 .

A Lyapunov optimal feedback control function U*(X) is computed as u*w = arg(ri[x,u,,1H[x,u"(x)U,..(3) u where

awed

x = .&&

/I /I

,,$ :;,, . . . . . . . . . . . . . . . . . . . (4)

ax

and u” (x) denotes the solution to ()= JJw, 4

ad . . . . . . . . . . . . . . . . . . . . . . (5)

(the expressions for Eqs. (4) and (5) are omitted as they are very lengthy)4B6. Equation (5) will yield a quadratic polynomial as O=p2zi2 +p,u+p, . . . . . . . (6).

The above polynomial is solved numerically during the simulation studies.

Obviously, it has two roots but in our case the Lyapunov optimal control input is unique and with positive real part. ! :;. We-have decent function as

W(x) = [xl - 1.016 ‘I1 ‘12 I 1 PI2 P22 . ..(7) lx1 - 1.016 x3 -0.3]+[x2 -0.2031 where

p,,=A2sin2~+B2cos2+

p12 =si.n~cos~(B2 -A2) . . . . . . . . . . . . . . . . . (8)

p2,=B2sin2$+A2sin2t$ A=20, B=2 and +O. The following vehicle parameters are used: m, = 155,000 kg, CD = 0.5 , Is+, ~300s) r,=6378 km, surface area S = 2~10~ km2. The atmospheric density fluctuations considered are up to 20%.

Results and Discussion The Lyapunov optimal control problem

was solved in such a way that the equations of motion (Eqs. 2) were solved numerically using a Fehlberg Fourth-Fifth order Runge Kutta method with local error controlled to less than 1.0x 10m6 percent. The control input was determined by finding the roots of Eq. (6) by using Newton Raphson method. Equation (6) was derived using MapleV release 4. In all the cases studied, it was po&ible to get one of the roots with positive real part for solving the problem. However, the sign of the real parts of the roots needs an in depth future study.

Figure 2 shows that the missile boost phase satisfies the given conditions when there, are no atmospheric fluctuations, i.e., up to altitude of 200 km the velocity is 1.6 km/s, 70 % mass is consumed, and the maximum control input u is considered as 6 and that is not violated. The kick angle related single impulse could be applied put a satellite or a missile on desired free flight phase.

Figure 2 also reveals the robustness of the Lyapunov optimal control that the rocket boost phase satisfies the given conditions even in the presence of 20% atmospheric density fluctuations. The results shows that due to atmospheric density fluctuations the time of the boost phase is increased and at the end of boost phase the thrust becomes slightly higher

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than in the case of without atmospheric density fluctuations. This is logical, since it is less expensive in terms of fuel consumption to accelerate at high altitudes where gravitation pull is less and aerodynamic drag is approximately zero.

Conclusion The problem of developing a guidance

algorithm for the boost phase for throttleable rocket using Lyapunov feedback control has been considered. The decent function and necessary conditions were derived analytically and the boost phase trajectory was determined numerically. The results show that optimal Lyapunov feedback control successfully satisfies all the desired final conditions at the end’ of the boost phase. Furthermore, in addition to its practical on board application for launching a missile or satellite, this method is considerably simpler for optimizing the boost phase for preliminary analysis than the conventional methods, such as Pontryagin’s maximum principle. In the future studies, the problem should be extended to multi-input problem, which should also include wind speed fluctuations, and modeling uncertainty, such as uncertainty in drag coefficient.

1. References 2. Walberg, G. D., “A Survey of Aeroassisted Orbit Transfer”, Journal of Spacecraft and Rockets, Vol. 22, No. 1, Jan- Feb., 1985, pp. 3-18. 3. Mease, K. D., and Vinh, N. X., “Minimum Fuel Aeroassisted Coplanar Orbit Transfer using Lift-Modulation”, Journal of Guidance, Control, and Dynamics, Vol. 8, No. 1, Jan. 1995, pp. 134-141. 4. Lohar, F. A., Khan, A, and Misra, A. K., “Optimal Transfer Between Coplanar Elliptical Orbits Using Aerocruise”, Paper No. 96-3594, presented at AIAMAAS Astrodynamics Specialist Conference, San Diego, July 29-3 1, 1996. 5. Lee, B. S., and Grantham, W. J., “Aeroassisted Orbital Maneuvering Using Lyapunov Optimal Feedback Control”, Journal of Guidance, Control, and Dynamics, Vol. 12, No. 2, 1989, pp. 237-242.

6. Vinh, N. X., “Optimal Trajectories in Atmospheric Flight”, Elsevier Scientific Publishing Company, New York, 1981. 7. Lee, B. S., “‘Aeroassisted Orbital Maneuvering Using Lyapunov Optimal Feedback Control”, Master’s thesis, Department of Mechanical Engineering, Washington State University, Pullman, Washington, USA, 1987.

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free flight phase

sensible ,atmospheric shell

missile impact site

(a> Free flight after kick

/ atmosphere

Figure 2: (a) A typical trajectory of ICBM, (b) a satellite launch trajectory.

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1. 1.4 1.2

1

1 0.8 0.6 0.4 ___.-.

time (set)

_Ce___./ -------

_..---

, .__ .- .--”

0.2 /

Ob 50 100 150

time (set) 200 250

7 u=tmst/(initid weight of vehicle)

0 50 1M 150 200 250

Figure 2: Time variation of state variable and control input; ‘-’ for no density fluctuations, ‘- -’ with 20% density fluctuations.

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