guidance, flight mechanics and trajectory optimization

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TABIE OF CONTENTS SECTION PAGE 1.0 2.0 STATEMENT OF THE PROBLEM. . . . . . . . . . . . . . . . . . . 1 STATE OF THE ART. . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Development of Dynamic Programming .......... 2.2 Fundamental Concepts and Applications. ........ 2.2.1 Simple Multi-Stage Decision Problem. ..... 2.2.2 Simple Applications to Calculus of Variations. 2.2.2.1 Shortest Distance Problem ...... 2.2.2.2 Variational Problem with a Movable Boundary. .......... 2.2.2.3 Simple Guidance Problem ....... 2.2.3 Maximum Minimum Problem. ........... 2.2.4 Equipment Replacement Problem. ........ 2.3 Computational Considerations ............. 2.3.1 Merits of Dynamic Programming over Conventional Analysis. ............ 2.3.1.1 Relative Extrema. .......... 2.3.1.2 Constraints ............. 2.3.1.3 Continuity. ............. 2.3.2 Dynamic Programming versus Straightforward Combinational Search ............. 2.3.3 Difficulties Encountered in Dynamic Programming. ................. 2.3.3.1 Curse of Dimensionality ....... 2.3.3.2 Stability and Sensitivity ...... 2.4 Limiting Process in Dynamic Programming. ....... 2.4.1 Recursive Equation for the Problem of Lagrange 2.4.2 Recursive Equation in Limiting Form. ..... 2.4.3 An Example Problem .............. 2.4.4 Additional Properties of the Optimal Solution. 2.4.5 Lagrange Problem with Variable End Points. .. 2.4.6 N-Dimensional Lagrange Problem ........ 2.4.7 Discussion of the Problem of Lagrange. .... 2.4.8 The Problem of Bolza ............. 2.4.9 Bellman Equation for the Bolza Problem .... 2.4.10 Linear Problem with Quadratic Cost ...... 2.4.11 Dynamic Programming and the Pontryagin Maximum Principle. .............. 2.4.12 Some Limitations on the Development of the Bellman Equation ............. -v- . 3 . 7 . 10 . 14 . 17 . 23 . 27 . 35 . 41 . 48 . 49 . 49 . 49 . 50 . 50 . 51 . 52 . 60 . 61 . 61 . 64 . 67 . 69 . 71 . 75 . 78 . 82 . 87 . 92 . 98 . 105
SECTION PAGE 2.5 Dynamic Programming and the Optimization of Stochastic Systems. . . . . . . . . . . . . . . . . . . . 108 2.5.1 2.5.2 2.5.3 Introduction. .................. 108 Problem Statement ................ 110 2.5.2.1 Perfectly Observable Case. ....... 113 2.5.2.2 Perfectly Inobservable Case. ...... 118 2.5.2.3 Partially Observable Case. ....... 124 2.5.2.4 Discussion ............... 130 The Treatment of Terminal Constraints ...... 132 2.5.3.1 Perfectly Observable Case. ....... 137 2.5.3.2 Perfectly Inobservable Case. ...... 140 2.5.3.3 Partially Observable Case. ....... 143 2.5.3.4 Discussion ............... 147 3.0 RECOMMENDED PROCEDURES .................... 148 4.0 REFERENCES .......................... 149 'vi
Page 1 and 2: NASA OI 0 0 P; U 4 cd 4 z . . _ -.
Page 4: c FOREWORD This report was prepared
Page 10 and 11: 2.0 STATE OF THE ART 2.1 Developmen
Page 12 and 13: There are several ways to accomplis
Page 14 and 15: 2.2 Fundamental Concepts and Applic
Page 16 and 17: II. Although the previous problem w
Page 18 and 19: I,. Citg B. C optimum cost Path for
Page 20 and 21: The optimum path can be found by st
Page 22 and 23: The integral in Equation 2.2.1 can
Page 24 and 25: 2.2.2.1 Shortest Distance Between T
Page 26 and 27: The cost of the allowable transitio
Page 28 and 29: In a,completely analogous manner th
Page 30 and 31: 2.2.2.2 Variational Problem with Mo
Page 32 and 33: Ii _ 6 5 9 3 I 8 I I The cost integ
Page 34: 2.2.2.3 Simple Guidance Problem As
Page 37 and 38: This process continues in the same
Page 39 and 40: The algorithm for solving this prob
Page 41 and 42: The reader, no doubt, has a reasona
Page 43 and 44: are applied to Min (fl), the range
Page 45 and 46: Note that each diagonal corresponds
Page 47 and 48: The value of X can be found by empl
Page 49 and 50: The initial condition is: K.&P : P(
Page 51 and 52: it is seen that (for a two stage pr
Page 53 and 54: The optimal policy can now be found
Page 55 and 56:
2.3 COMPUTATIONAL CONSIDERATIONS So
Page 57 and 58:
If classical techniques were to be
Page 59 and 60:
2.3.3.1 The Curse of Dimensionalitv
Page 61 and 62:
Next, fl is evaluated for all allow
Page 63 and 64:
% Xl + 5 = 2 (A2 = 2) 5+X2=3 (A2=3.
Page 65 and 66:
that minimizes f. This interchange
Page 67 and 68:
2.3.3.2 Stability and Sensitivity I
Page 69 and 70:
Let the lower limit of integration
Page 71 and 72:
But (2.4.10) since the function on
Page 73 and 74:
Now,. noting that the second MIN op
Page 75 and 76:
where v 'is determined from and wit
Page 77 and 78:
NOW, combining these two expression
Page 79 and 80:
The situation is pictured to the ri
Page 81 and 82:
Hence, the boundary condition OAI f
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The boundary condition to be satisf
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2.4.7. Discussion of the Probltsi o
Page 87 and 88:
characteristics associated with Eqs
Page 89 and 90:
2.4.8 The Problem of Bolza The prec
Page 91 and 92:
The initial position, velocity and
Page 93 and 94:
where I is the moment of inertia, F
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or, as . -... .-. . .._-_--- to ind
Page 97 and 98:
Since R(t, x(t) ) is the minimum va
Page 99 and 100:
2.4.10 Ljnear Problem with Quadrati
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where S(t) is some n x n symmetric
Page 103 and 104:
The governing equations for the att
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y 2.4.11 Dynamic Programming and th
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M Since 3t does not depend on u exp
Page 109 and 110:
with The P vector for the system is
Page 111 and 112:
where With the control known as a f
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with and with the boundary conditio
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2.5 DYNAMIC PROGRAMMING AND THE OPT
Page 117 and 118:
2.5.2 Problem Statement Let the sys
Page 119 and 120:
Now, introducing the variables qs2,
Page 121 and 122:
Thus, the performance index takes t
Page 123 and 124:
#R where tr denotes the trace of th
Page 125 and 126:
The minimum value of the performanc
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variance characterizing Ilt) can be
Page 129 and 130:
The solution takes the form with s
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Since V is positive definite for f
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where the first the second with exp
Page 135 and 136:
Taking the limit and using the expr
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Since V is positive definite for t
Page 139 and 140:
2.5.3 The Treatment of Terminal Con
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F while Equation (2.5.87B) requires
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._ __. ..- . . - which will equal t
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is determined. Let p(z,t') be given
Page 147 and 148:
of the p and $ equations (i.e., Eqs
Page 149 and 150:
(2.5.I.21) Thus, the control is to
Page 151 and 152:
Note, as in.Section (2.5.2.3), the
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.6 t -(h.(S~tj~fitr~f-‘~~~~n~) =
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3.0 RECOMMENDED PROCEDURES - ._ .-_
Page 157 and 158:
4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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