quasi-optimal control for the reentry and return

5th INTERNATIONAL CONFERECE ON LAUNCHER TECHNOLOGY - Madrid 2003- (S5.1)

QUASI-OPTIMAL CONTROL FOR THE REENTRY AND RETURN FLIGHT OF AN RLV Josef KLEVANSKI, Martin SIPPEL, Space Launcher Systems Analysis (SART), DLR, Cologne, Germany [email protected], [email protected]

The estimation methods for the optimal flight control for reentry and return flight to the launch site by preliminary design of a multistage reusable launch vehicle (RLV) are considered. The problems and the difficulties by application of the classic optimization methods based on the gradient method are discussed: the shortage of reliable data in the preliminary design and complexity of the cost function definition which simultaneously satisfies all flight constraints (load factors, dynamic pressure, heating flux limitation e.g.) and robustness of the found solutions. The use of terminal control and the practical advantages of the traditional engineering methods for preliminary design (socalled "quasi-optimal") methods are substantiated: namely, simplifying of the mathematical model and ease of finding the practicable robust solution. The suggested approach is discussed in more details: the analysis of the whole flight and consideration of the separated flight phases, the tasks of the control systems and its structure, control laws and the character of mathematical simulation. The application of these "quasi-optimal" methods is demonstrated with the example of reusable launch vehicle: liquid fly-back booster.

Nomenclature D Drag H altitude L Lift M Mach-number P Thrust S distance T Throttle V velocity W weight k, K control system coefficients m mass n load factor q dynamic pressure angle of attack α control surface deflection δc γ flight path angle ε part signal of control bank angle σ υ pitch angle ω pitch rate ψ λ φ

azimuth longitude latitude

1

N m N kN m m/s N

INTRODUCTION

The search of optimal flight control for re-entry and return flight to the launch site is a very important task in the preliminary design of a multistage reusable launch vehicle (RLV) [1]. Classic optimization methods based on the gradient method are usually applied for such problems [2]. However, the complete solution of the whole optimization task is quite difficult due to the shortage of reliable data in the preliminary design and due to the complexity of the cost function definition which simultaneously satisfies all flight constraints (load factors, dynamic pressure, heating flux limitation etc.). Furthermore, even the solution of the optimization task found will unavoidably be changed as a result of data modification and therefore is not robust.

kg, t Pa Rad, deg Rad, deg Rad, deg Rad, deg Rad, deg Rad/s, deg/s Rad, deg Rad, deg Rad, deg

Because of the above-stated difficulties, we opt for the use of traditional engineering methods, or socalled "quasi-optimal" methods, for preliminary design, which search for the best suitable solution for each of the flight phases. Application of such methods allows to simplify the mathematical model and to easily find the practicable solution.

Subscripts, Abbreviations 3 DOF three degrees of freedom GLOW Gross Lift-Off Weight LFBB Liquid Fly-Back Booster MECO Main Engine Cut Off RLV Reusable Launch Vehicle RCS Reaction Control Systems sep Separation

These "quasi-optimal" methods include the preliminary definition of the optimal mode for each flight phase: for example, the flight mode with the minimal fuel consumption per range. The flight conditions found can then be reformulated into a control law (algorithms) and the entire optimization process can be performed as a terminal control task. 1

nX , nZ GNC System:

nz , q,Q − lim λ , ϕt arg

Engines:

ψ t arg = f (λ , ϕ cur , λ , ϕ t arg ) M , H opt = f ( m)

α& set = f ( M , H opt , n z _ lim )

T

r  Feng  r   Meng  =    fr 

σ& set = f (λ , ϕ cur ,ψ t arg )

λ,ϕcur

T&set = f ( M , H opt ) δ& = f (α ω ) set ,

c

F

Forces and Moments:

y

Atmosphere:

V, H  M

 ρ  =  T q  

Move Equations:

M     V ,T  ρ   

Aerodynamics:

δc

r  Faer   r = M   aer 

f(V,H);

α

r r F,M

r r r  F   F AER   FENG   r ;  r = r M M  + M     AER   ENG  F F n X = X ; nZ = Z ; mg mg



FV ,δ ; q 

c

 α&     ω& y   γ&     V&   H&  =    σ&   ψ&     λ&   ϕ&   

α  ωy γ  V H σ  ψ  λ ϕ 

F

   r F r M , m; r I    

 

α,ωY ,γ ,V, H,σ,ψ,λ,ϕ

FIG. 1: Primary Block-Structure of the Completed Mathematical Model

The mathematical model for the simulation uses the dynamical nonlinear equations in CAUCHYForm:

r

Ballistic phase, the re-entry glide descent and turn: the distance from the launch site should be minimized while remaining within stress limitation boundaries. Cruise flight phase - return flight to the launch site with the air breathing engines: the return flight fuel consumption should be minimized.

•

(1)

r

where: x - is the state vector; u -is the control r vector and y - the output vector. The mathematical model is modular (s. FIG. 1) and includes the following modules: • • • • • •

Separation

100

Ballistic Flight, Reentry and Turn

Ascent

GNC System: Guidance, Navigation and Control System Atmosphere Model Vehicle Aerodynamics Engines: Model of Air breathing engines All Forces (Accelerations) and Moments Equations of Motion: 3DOF – point mass motion + solid body rotation equations

80 60 40 20

-53

Cruise Return Flight 0

-52

5

-51

λ [°]

-50 -49

FIG. 2: Flight Phases of RLV

For the application of traditional engineering methods (also known as the "quasi-optimal" methods) instead of the classic optimization methods, 2

4.5 4

[°]

r r r x& = F(x, u) r r r y = φ(x, u)

•

H [km]

MATHEMATICAL MODEL

ϕ

2

the whole task should be divided into stages with a clear defined optimization task according to the flight phases after separation (s. FIG. 2):

3

The control system uses the integral control law and applies during re-entry the following partial algorithm:

BALLISTICAL PHASE, REENTRY GLIDE, DESCENT AND TURN

if (n z cur > n z lim ) : α& set = K nz ⋅ (n z lim − n z cur )

We now continue to consider the features of the ballistic, re-entry and turn flight phase in more detail. The initial conditions for this flight phase correspond to the booster separation conditions during ascent. These flight conditions are chosen such that the maximal payload can be achieved (ascent optimization). Therefore they are common for all variants of the return flight and will not be considered in this paper.

(3)

α& set = 0

if (n z cur ≤ n z lim ) :

where α set is the assigned value of the angle of attack, n z lim - is the maximal admissible value of the normal acceleration, n z cur - is the current value of the normal accelera-

The possibilities of the trajectory control in the ballistic part of the flight are highly restricted in any case. The control system provides in this phase the stabilization and satisfies the flight limitations and constraints.

tion and K nz is the control systems coefficient.

After separation, while flying along the ballistic curve the control system assures that the program value of the angle of attack αreentry is reached and maintained.

The angle of attack reduction results almost directly in the reduction of the normal acceleration. However, as the above described control law has a dynamic inaccuracy, the assigned value of the normal acceleration is chosen to be slightly below the stress limitation n z lim .

The corresponding part of the control law is:

The necessary deflection of the control surface is:

α& set = K α ⋅ (α reentry − α cur )

δ&c = K α ⋅ (α cur − α set ) + K ω ⋅ ω y

(2)

(4)

y

where α set is the assigned value of the angle of

where α set is the assigned value of the angle of

attack, α cur is the current value of the angle of

attack, α cur - is the current value of the angle of

attack, ω y is the pitch angle rate, and K α set , K ω y

attack and K α is the control systems coefficient.

are the control systems coefficients.

This task is fulfilled at first via use of the RCS (Reaction Control Systems) as long as the dynamic pressure remains low. A gradual switching to aerodynamical control, as the dynamic pressure increases during re-entry, is then performed.

This signal is used as an input signal by a simple mathematical model of the control surface actuator.

ALFA [deg]

During re-entry very important limitations should be respected: the normal g-load factor nz≤nz lim, the admissible dynamical pressure q≤q lim (the structure design stress limitations) and the heat flux limitations. Since the normal acceleration nz will be determined by the aerodynamical data of the LFBB as a function of the angle of attack, Mach number and dynamical pressure

NZ_LOAD [g0]

40

4

35

3.5

30

3

25

2.5

20

2

15

1.5

10

1

5

NZ_LOAD [g0]

ALFA [deg]

0.5

0 0

200

400 600 TIME [s]

800

0 1000

FIG. 3: Angle of Attack and Dynamical Pressure Reentry History

n z = C L (α , M ) ⋅ q ⋅ S ref / m

it is natural to entrust the control system with the limiting of its values via α - angle of attack control. It is significant to note that normal acceleration nz can be easily measured in real time with the satisfactory precision.

The typical re-entry history of α and nz is shown in the FIG. 3

3

DYNPRESSURE [Pa]

25000

Max. Dynamical Pressure [Pa]

Turn Begin Time [s]

20000 160

15000

180

10000

200

5000

220

0 0

200

400 600 TIME [s]

800

25000 20000 15000 10000 5000 0 150

1000

170

190

210

230

Turn Begin Time [s]

FIG. 4: Dynamic Pressure Reentry Histories by Different Turn Begin Times

FIG. 7: Maximum of Dynamic Pressure by Different Turn Begin Times

The dynamical pressure cannot be simultaneously directly controlled, but its peak values depend on the chosen turning time and on the limitation of the bank angle applied to the turn (s. FIG. 4, 5).

The turn on the return course will be performed according to the following control law:

σ& = Kψ (ψ t arg −ψ cur ) + n z ⋅ sin σ σ ≤ σ max

Turn Begin Time [s]

SIGMA [deg]

5 0 -5 0 -10 -15

200

400

600

800

where σ -is the assigned bank angle. ψ t arg - is the return flight target azimuth calculated

1000

by navigation system, considered below in detail, ψ cur - is the current value of the flight azimuth,

160

-20 -25 -30 -35

180

n z - normal acceleration, Kψ - transfer coefficient of the control system,

200

-40 -45

220 TIME [s]

σ max - the bank angle limitation.

FIG. 5: Bank Angle Reentry Histories by Different Turn Begin Times

After decreasing of the maximum normal acceleration and after transit through the maximum dynamical pressure the longitudinal channel of the control systems provides descent with the maximal lift-drag ratio until the rated return flight altitude.

On the one hand, an early turn begin can result in too high peak values of the dynamic pressure at reentry while, on the other hand a late turn begin unnecessarily increases the return flight distance (s. FIG. 6).

4 Turn Begin Time [s]

LATITUDE [deg]

6

5

(5)

RETURN CRUISE FLIGHT WITH AIRBREATING ENGINES

The return flight fuel consumption will be minimal if two conditions are satisfied:

160 180

4

•

200 220

3 -53

-52

-51 -50 -49 LONGITUDE [deg]

-48

•

-47

FIG. 6: Turn Tracks by Different Turn Begin Times

The flight route will be along the shortest way to the target point. The current fuel consumption per range is always at the achievable minimum.

If these conditions are satisfied in each point of the return flight, an optimal return flight mode is achieved.

That means that compromise solution which satisfies the dynamical pressure limitation and at the same time provides the minimal return flight distance, can be properly found by performing variation at the time of turn begin (FIG. 7).

The calculation algorithm of the return flight azimuth that provides the shortest return flight route – the flight along the "orthodrome", a segment of the 4

great circle, is presented below. The great circle includes three points (s. FIG. 8):

a x = Ycur ⋅ Z t arg − Z cur ⋅ Yt arg

1. Current coordinates of the vehicle described by current altitude Hcur, current geographical longitude λcur and latitude φcur 2. Target point coordinates, described by geographical longitude λtarg and latitude φtarg 3. Center of the earth ellipsoid.

a z = X cur ⋅ Yt arg − Ycur ⋅ X t arg

a y = Z cur ⋅ X t arg − X cur ⋅ Z t arg 2

2

a = ax + a y + az

(9)

2

the inclination angle i of the great circle plane can be calculated:

a i = arccos z  a

   

(10)

Since the azimuth ψ = 0 is defined in the East direction, the target azimuth can be calculated using:

 cos i    cos ϕ 

ψ t arg = arccos

In the special case, in which the vehicle has the same geographical longitude λcur as target longitude λtarg only the target azimuth direction should be defined: ψ t arg = ±90°

FIG. 8: Current position and Target on the "Orthodrome"

This algorithm with the permanent refinement of the target azimuth ψtarg can be applied to unlimited return flight distance (from few kilometers to many thousand kilometers) and can guarantee the return flight by the shortest way (along an "orthodrome")

The Cartesian coordinates of the current position and the target point can be calculated according to: 1 ⋅ ( R E + R P + ( R E − RP ) ⋅ cos 2λcur ) + H cur 2 = Rcur ⋅ cos λcur ⋅ sin ϕ cur

Rcur = X cur

Ycur = Rcur ⋅ cos λcur ⋅ cos ϕ cur

(6)

The above described control law provides the navigation:

Z cur = Rcur ⋅ sin ϕ cur

σ& = Kψ (ψ t arg −ψ cur ) + n z ⋅ sin σ and

σ ≤ σ max 1 ⋅ ( RE + RP + ( RE − RP ) ⋅ cos 2λt arg ) 2 = Rt arg ⋅ cos λt arg ⋅ sin ϕ t arg

Rt arg = X t arg

Z t arg = Rt arg ⋅ sin ϕ t arg

where RE and RP are the earth equator radius and earth polar radius respectively. The Cartesian coordinates of the earth center are: X O = 0;

YO = 0;

Z O = 0;

(12)

The detailed consideration of the optimal return flight mode expects, that aerodynamical data (lift coefficient CL drag coefficient CD and moment coefficient Cm as a function of angle of attack α, Mach number M and control surfaces deflection δc) and the primary air breathing engine data (thrust P [kN] and specific fuel consumption sfc [g/kNs] as a function of Mach number M, flight altitude H and throttle T) are known.

(7)

Yt arg = Rt arg ⋅ cos λt arg ⋅ cos ϕ t arg

(11)

The preliminary calculation of the aerodynamical data was gained using the DLR program cac. The refinement of the aerodynamical data was made using CFD – methods and wind tunnel tests (s. FIG. 9)

(8)

After calculation of a x , a y , a z and module a

5

To satisfy the trimmed flight conditions

8 6

0.4

4 CL/CD

1.2 2

0 -2

0

10

20

30

40

50

4 6

-4 -6 Alfa [°]

fcr =

FIG. 9: CL/CD as a function of Angle of Attack and Mach number.

sfc ⋅ P( M , H , T ) V

(15)

The best solution can be searched for amongst the set of obtained stationary solutions once these have been sorted out by fuel consumption per range (s. FIG. 11). This solution corresponds to the optimal cruise flight mode for a given flight mass and center of gravity position.

The calculation of the air breathing engines data was made using the DLR program abp on basis of published data (s. FIG. 10). Altitude [km]

1 0.8 0.6 0.4

0

1

2

0.95

4

0.9

5

0.85

fcr rel.

1.2

6

0.2

Altitude [km] 0.50 1.00 1.50

0.8

2.00

0.75

2.50

0.7

0 0

0.1

0.2

0.3

0.4

0.5

0.6 0.25

Mach Number

3.50 0.3

0.35

4.00

0.4

4.50

Mach Number

FIG. 10: Relative Thrust as a function of Mach number and Altitude.

FIG. 11: Fuel consumption per range as a function of Mach number and altitude.

The steady flight conditions (trimmed flight conditions) were defined for several flight mass values of the LFBB for a series of altitudes H and Mach numbers M. Horizontal flight conditions are defined as stationary solutions of the dynamical equations of motion:

It is significant to note, that although the gained angle of attack is very close to the value, providing the maximal lift-drag ratio, it does not coincide with it exactly. This is the result of the fact that the selection and adjustment of the air breathing engines with airframe was made on basis of the design reasons (minimal engine mass and dimensions). The variety of the optimal flight conditions H opt , M opt = f (m) will be set in the control system as the optimal flight profile (s. FIG. 12).

(13)

Mach

Mach Number

Fz  g V γ& = m ⋅ V − cos γ ⋅  V − R      My ω& y = I yy   α& = ω y − γ&  V& = Fx − g ⋅ sin γ  m  H& = V ⋅ sin γ  

3.00

0.65

0.6

where

Fx = f (α , δ c , M , H , T )

Altitude

0.4

5

0.38

4

0.36

3

0.34

2

0.32

1

0.3 0.9

Fz = f (α , δ c , M , H , T ) M y = f (α , δ c , M , H , T )

0.95 1 Relative Mass

Altitude [km]

P rel.

(14)

the control state vector variables and control parameters α , δ c , M , H , T were varied. At the same time the necessary thrust P and the specific fuel consumption sfc were gained, which can be easily transformed into the fuel consumption per range:

0.8

2 -10

γ& = 0; ω& y = 0;α& = 0;V& = 0; H& = 0

Mach

0 1.05

FIG. 12: Optimal cruise flight Mach number and altitude as a function of vehicle mass.

are the corresponding forces and moment

6

During the return cruise flight this profile will be followed using the following integral control law:

α& set = K H ⋅ ( H opt − H cur )

5 •

For the use of the traditional engineering methods the target problem, criteria and limitations should be clearly formulated in technical terms for each operation phase.

•

The mathematical model of the vehicle for the simulation should be complemented by a mathematical model of the control systems with relevant control laws.

•

The use of the presented traditional engineering "quasi-optimal" method by preliminary design allows the best suitable solution which satisfies all constrains, to be easily found.

•

The "quasi-optimal" solution found is nearoptimal, i.e. very close to the optimal solution, which can be found using classic optimal methods.

•

The used terminal control mode can noticeably reduce the necessary processing time.

•

The presented method also acts as a first test of the prototype control laws.

(16)

T& = K T ⋅ ( M opt − M cur )

where α set is the assigned angle of attack value,

T is the assigned engine throttling, H opt = f (mcur ) , M opt = f (mcur ) - are the optimal altitude and Mach number for the current mass value, H cur , M cur are current values of the of the altitude and Mach number respectively and K H , K T are the control systems coefficients. After the here above described preparatory work the numerical integration for the cruise flight phase needs to be performed only once. This noticeably reduces the necessary processing time in comparison with the classical optimization methods. The analysis of the mathematical model, which includes the control system allows dimensioning of the RCS and the aerodynamical control surfaces in the preliminary design phase. Including of a mathematical model in the closed loop simulation allows the solution to be found even for an instable vehicle through a created artificial stability and improved damping for all phases. The time-dependent displacement of the aerodynamic center of pressure and vehicle center of gravity is taken into account.

6

20

10

10

5

0 -10

200

400

600

-20 -30

800

0 1000 -5 -10

TIME [s]

[3] Klevanski, A. Herbertz, J. Kauffmann, V. Schmid. Aspekte der Stabilität und Steuerbarkeit in der Flug- und Separationsphase unsymmetrischer Träger-Konfigurationen. DGLR JT 2000

DELTA_CONTR [deg]

GAMMA [deg]

15

0

[2] D. Iranzo-Greus, F. Deneu, M. Malassigné, N. Berend, C. Jolly. Preliminary Scenarios for the Return to the Launch Site of the First Stage of a TSTO RLV.

DELTA_CONTR [deg]

30

REFERENCES

[1] M. Sippel, U. Atanassov, I. Klevanski, V. Schmid. First Stage Design Variations of Partially Reusable Launch Vehicles. J. Spacecraft, V. 39, No. 4, pp. 571-579, JulyAugust 2002.

The typical histories of the flight path angle γ and of the control surfaces deflection δc for the vehicle, which is instable in the subsonic flight phase, are shown in the FIG. 13. GAMMA [deg]

CONCLUSIONS

[4] A.L. Greensite. Analysis and Design of Space Vehicle Flight Control Systems. Spartan Books, New York, Washington, 1970

-15

FIG. 13: Pitch Angle and Control Surface Deflection History

7