Abstract
The computed solutions for a realistic model of a hypersonic space vehicle demonstrate the feasibility of simultaneously optimizing the stage separation and the flight path by an indirect method. This is enabled by using state of the art optimization codes such as multiple shooting and a generalization of the necessary conditions of optimal control theory. The later is necessary in order to include in the problem formulation simultaneously such important features as piecewise defined model functions and state constraints.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Abbreviations
- v:
-
velocity
- γ:
-
path inclination
- χ:
-
azimuth inclination
- h:
-
altitude
- Λ:
-
geographical latitude
- θ:
-
geographical longitude
- m:
-
mass
- x:
-
=(v, γ, ..., m)T
- CL :
-
lift coefficient
- μ:
-
bank angle
- δ:
-
mass flow
- ɛ:
-
thrust angle
- u:
-
=(C L, μ, δ, ɛ)T
- a:
-
speed of sound
- D, L:
-
drag and lift force
- C D, C L :
-
drag and lift coefficient
- f:
-
right hand side of o.d.e.
- F:
-
reference area
- g:
-
gravitational acceleration
- I sp :
-
specific impuls
- J:
-
performance index
- m 0 :
-
prescribed initial total mass
- m fuel,I, m fuel,II :
-
fuel mass of stage I / stage II
- m structure,I, m structure,II :
-
structure mass of stage I / stage II
- m payload :
-
payload
- M:
-
mach number
- ψI, ψII :
-
structure mass model
- \(\hat \psi _1 ,\hat \psi _2 ,\tilde \psi\) :
-
boundary and interior point conditions
- \(\bar \psi\) :
-
staging conditions
- q:
-
dynamic pressure
- r o :
-
Earth's radius, R = r o + h
- S:
-
state and control constraints
- T:
-
thrust force
- t:
-
time
- t e :
-
switching time of engines (stage I)
- t s :
-
separation time of stage I and stage II
- t f :
-
final time
- ϱ:
-
atmospheric density
- ω:
-
angular velocity
References
A.E. Bryson; Y.-C. Ho: Applied Optimal Control, 2nd ed. Hemisphere Publishing Corp., Washington, D.C. (1975).
R. Bulirsch: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report der Carl-Cranz-Gesellschaft e.V., Oberpfaffenhofen (1971); Nachdruck: Mathematisches Institut, TU München (1985).
R. Bulirsch; K. Chudej: Guidance and Trajectory Optimization under State Constraints. Preprints of the 12th IFAC Symposium on Automatic Control in Aerospace — Aerospace Control '92. Ottobrunn, Eds. D.B. DeBra, E. Gottzein. Düsseldorf: VDI/VDE-GMA (1992) 533–538.
P. Hiltmann: Numerische Lösung von Mehrpunkt-Randwertproblemen und Aufgaben der optimalen Steuerung mit Steuerfunktionen über endlichdimensionalen Räumen. Dissertation, Mathematisches Institut, TU München (1990); Report No. 448, DFG-Schwerpunktprogramm: Anwendungsbezogene Optimierung und Steuerung, Mathematisches Institut, TU München (1993).
E. Högenauer: Raumtransporter. Z. Flugwiss., 11 (1987) 309–316.
D.H. Jacobson; M.M. Lele; J.L. Speyer: New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints. J. Math. Anal. Appl., 35 (1971) 255–284.
C. Jänsch, K. Schnepper, K.H. Well: Trajectory Optimization of a Transatmospheric Vehicle. Proc. of the American Control Conference, Boston, Massachusetts (1991) 2232–2237.
H. Kuczera; P. Krammer; P. Sacher: Sänger and the German Hypersonics Technology Programme — Status Report 1991. 42nd IAF-Congress, Montreal, Kanada, Paper-No. IAF-91-198 (1991).
H. Maurer: Optimale Steuerprozesse mit Zustandsbeschränkungen. Habilitation. Mathematisches Institut, Universität Würzburg (1976).
H. Maurer: Differential Stability in Optimal Control Problems. Appl. Math. Optim., 5 (1979) 283–295.
H. Maurer: On the Minimum Principle for Optimal Control Problems with State Constraints. Report No. 41, Rechenzentrum der Universität Münster (1979).
A. Miele: Flight Mechanics I, Theory of Flight Paths. Addison-Wesley, Reading, Massachusetts (1962).
D.O. Norris: Nonlinear Programming Applied to State-Constrained Optimization Problems. J. Math. Anal. Appl., 43 (1973) 261–272.
H.J. Oberle: Numerische Berechnung optimaler Steuerungen von Heizung und Kühlung für ein realistisches Sonnenhausmodell. Habilitation, Institut für Mathematik, TU München (1982); Report TUM-Math-8310, Institut für Mathematik, TU München (1983).
G. Sachs; W. Schoder: Optimal Separation of Lifting Vehicles in Hypersonic Flight. Proc. of the AIAA Guidance, Navigation and Control Conference. New Orleans, Louisiana, Paper-No. AIAA-91-2657 (1991) 529–536.
U.M. Schöttle: Flug-und Antriebsoptimierung luftatmender aerodynamischer Raumfahrtträger. Dissertation, Institut für Raumfahrtsysteme, Universität Stuttgart (1988).
U.M. Schöttle; H. Grallert; F.A. Hewitt: Advanced air-breathing propulsion concepts for winged launch vehicles. Acta Astronautica, 20 (1989) 117–129.
G.-C. Shau: Der Einfluß flugmechanischer Parameter auf die Aufstiegsbahn von horizontalstartenden Raumtransportern bei gleichzeitiger Bahn-und Stufungsoptimierung. Dissertation, Fakultät für Maschinenbau und Elektrotechnik, TU Braunschweig (1973).
J. Stoer; R. Bulirsch: Numerische Mathematik 2, 3. Auflage, Springer, Berlin (1990).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag
About this paper
Cite this paper
Chudej, K. (1994). Optimization of the stage separation and the flight path of a future launch vehicle. In: Henry, J., Yvon, JP. (eds) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, vol 197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035498
Download citation
DOI: https://doi.org/10.1007/BFb0035498
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-19893-2
Online ISBN: 978-3-540-39337-5
eBook Packages: Springer Book Archive