guidance, flight mechanics and trajectory optimization

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TECH LIBRARY KAFB, NM llnllllllln~lllnlll llClbflbb8 NASA CR- 1009 GUIDANCE, FLIGHT MECHANICS AND TRAJECTORY OPTIMIZATION Volume X - Dynamic Programming By A. S. Abbott <strong>and</strong> J. E. McIntyre Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Issued by Originator as Report No. SID 66-1678-2 Prepared under Contract No. NAS 8-11495 by NORTH AMERICAN AVIATION, INC. Downey, Calif. for George C. Marshall Space Flight Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific <strong>and</strong> Technic01 Information Springfield, Virginia 22151 - CFSTI price $3.00 tk-. - ~~.
Page 1: NASA OI 0 0 P; U 4 cd 4 z . . _ -.
Page 6 and 7: TABIE OF CONTENTS SECTION PAGE 1.0
Page 8: 1.0 STATENI3JT OF THE PROBLEN This
Page 11 and 12: The extension of Dynamic Programmin
Page 13 and 14: The purpose of this monograph is to
Page 15 and 16: "brute forcel' method (examining al
Page 17 and 18: 2.2.1 Multi-Stage Decision Problem
Page 19 and 20: The the The previous computations a
Page 21 and 22: 2.2.2 Applications to the Calculus
Page 23 and 24: where x is an n dimentional state v
Page 25 and 26: In order to keep the problem manage
Page 27 and 28: I (5, 6, 7) COST .._I 8.366 7.071 8
Page 29 and 30: W,O,Z 6 401 / - + -- - / / c 0, 0)
Page 31 and 32: The circle q-m is the classical sol
Page 33 and 34: z I-M J J-M E, E-I E-J E-K E 'F-I F
Page 36 and 37: Ii-- From this sketch, the followin
Page 38 and 39: equations 2.2.13, 2.2.14, 2.2.10 an
Page 40 and 41: As mentioned previously, the mass o
Page 42 and 43: 2.2.3 Maximum - Minimum Problem In
Page 44 and 45: o 12 3 4 5 6 7 8 910 f10 14 9 16 25
Page 46 and 47: once again due to Dynamic Programmi
Page 48 and 49: 2.2.4 auipment Replacement Problem
Page 50 and 51: la - p=:-= -. _-. _ - . _ .- .~.. -
Page 52 and 53:
AGE a 0 1 2 3 4 5 6 7 8 9 10 Fl (a)
Page 54 and 55:
AGE I 9 ~ 10 F, ((~1 K R : R F, (a)
Page 56 and 57:
2.3.1 MELTS 0~ mwac PR~ORAMMINO The
Page 58 and 59:
I- savings offered by Dynamic Progr
Page 60 and 61:
I. -. In order to give an appreciat
Page 62 and 63:
x1=0,x2=0 x1 = 0, x2 = 1 xl =1,x,=0
Page 64 and 65:
So far, the principle of optimality
Page 66 and 67:
Initialize A = Ai for fii~ Initiali
Page 68 and 69:
2.4 LIMITING PROCESS IN DYNAMIC PRO
Page 70 and 71:
Using this notation, Eq. (2.4.3) ca
Page 72 and 73:
Again, the problem under considerat
Page 74 and 75:
The combined solution of (2.4.24) a
Page 76 and 77:
thus, verifying that the solution i
Page 78 and 79:
Collecting the results of this sect
Page 80 and 81:
Alternately, for two neighboring po
Page 82 and 83:
2.4.6 N-Dimensional Lagrange Proble
Page 84 and 85:
conditions, wh?ich are essentially
Page 86 and 87:
Substituting (2.4.70) into (2.4.68)
Page 88 and 89:
efficient to develop the solution b
Page 90 and 91:
where the final time tf may or may
Page 92 and 93:
with the initial condition and the
Page 94 and 95:
2.4.9 Bellman Equation for the Bolz
Page 96 and 97:
To reduce (2.4.109) to a partial di
Page 98 and 99:
I- Collecting the results of this s
Page 100 and 101:
of the control vector u. Also, the
Page 102 and 103:
Equation (2.4.131) governing the ev
Page 104 and 105:
X@,rb) = where 7, is the 4 x 4(matr
Page 106 and 107:
s(3) the boundary conditions on the
Page 108 and 109:
But from (2.4.155B) it follows that
Page 110 and 111:
must also hold. Let then From Eq. (
Page 112 and 113:
(2.4.12) Some Limitations on the De
Page 114 and 115:
Now if the Maximum Principle is use
Page 116 and 117:
can be determined simply by integra
Page 118 and 119:
The noise vector, 3 , which appears
Page 120 and 121:
In order to proceed with the soluti
Page 122 and 123:
(XrQ,%fU&&) 4t + o(bt) + R(x+bx, t+
Page 124 and 125:
I - Performing the minimization ind
Page 126 and 127:
Note that since there is no feedbac
Page 128 and 129:
I ---. thus, substituting the expre
Page 130 and 131:
It is interesting to compare the va
Page 132 and 133:
Let Y tt) denote the observations t
Page 134 and 135:
Then, substituting (2.5.66) into (2
Page 136 and 137:
II while the optimal control takes
Page 138 and 139:
e achieved by a partially observabl
Page 140 and 141:
to satisfy an inequality condition
Page 142 and 143:
(In what follows, the symbol 4 will
Page 144 and 145:
The developments in the preceding p
Page 146 and 147:
and the boundary conditions The opt
Page 148 and 149:
subject to the state equations 1 =
Page 150 and 151:
Since the quantity Q(-k+) is indepe
Page 152 and 153:
Substituting this value for Z into
Page 154 and 155:
The matrix A is to be selected so t
Page 156 and 157:
Section 2.1 4.0 REFERENCES (2.1.1)
Page 158:
Section 3.0 REFERENCES (3.1) McGill
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