guidance, flight mechanics and trajectory optimization

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Equation (2.4.131) governing the evolution of the matrix S is nonlinear and, hence, difficult to solve. However, the matrix S need not be explicitly evaluated to determine the optimal solution which from (2.4.130) to (2.4,135) depends only on the terms SX and z. It will be shown next that these terms satisfy a linear equation and can be evaluated rather easily. Let P be the n-dimensional vector Substitution of this variable into (2.4.130) to (2.4.134) and using the state equation for x provides with the boundary conditions u= Q,-’ G ‘P 2 p’ = -A; +ZQ,/u i G Q;'G; -Ax- 2 (2.4.135) (2.4.136) (2.4.13’7) (2.4.138) x = %o AT t=c$ (2.4.139) Note that the new equations in p and x are linear and that the two-point boundary problem as represented in Eqs. (2.4.137) to (2.4.140) can be solved directly (i.e., without iteration). The optimal control is then evaluated using EYQ. (2.4.136). The method will be illustrated next on the simple attitude control problem of Section (2.4.8). 95 (2.4.140)
The governing equations for the attitude controller are with the initial conditions 2, = x2 i2 = u %, ( to ) -- z,o %- cr,) = x2 0 (2.4.141) (2.4.142) and with no terminal conditions imposed; that is, all the y in Q. (2.4~18~) are identically zero. The matrices A, Q1, Q2 and G for this'problem are Hence, Eqs. (2.4.138) to (2.4.140) become (2.4.143) (2.4.144) Since no terminal constraints are imposed, it can be shown from Eqs. (2.4.117), (2.4.1331, and (2.4.140) that at the terminal point The solution to this system can be written 96 (2.4.145) (2.4.146)
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NASA OI 0 0 P; U 4 cd 4 z . . _ -.
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c FOREWORD This report was prepared
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SECTION PAGE 2.5 Dynamic Programmin
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2.0 STATE OF THE ART 2.1 Developmen
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There are several ways to accomplis
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2.2 Fundamental Concepts and Applic
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II. Although the previous problem w
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I,. Citg B. C optimum cost Path for
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The optimum path can be found by st
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The integral in Equation 2.2.1 can
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2.2.2.1 Shortest Distance Between T
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The cost of the allowable transitio
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In a,completely analogous manner th
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2.2.2.2 Variational Problem with Mo
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Ii _ 6 5 9 3 I 8 I I The cost integ
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2.2.2.3 Simple Guidance Problem As
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This process continues in the same
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The algorithm for solving this prob
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The reader, no doubt, has a reasona
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are applied to Min (fl), the range
Page 45 and 46:
Note that each diagonal corresponds
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The value of X can be found by empl
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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
Page 83 and 84: The boundary condition to be satisf
Page 85 and 86: 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
Page 95 and 96: 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
Page 101: where S(t) is some n x n symmetric
Page 105 and 106: y 2.4.11 Dynamic Programming and th
Page 107 and 108: 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
Page 113 and 114: with and with the boundary conditio
Page 115 and 116: 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
Page 127 and 128: variance characterizing Ilt) can be
Page 129 and 130: The solution takes the form with s
Page 131 and 132: Since V is positive definite for f
Page 133 and 134: where the first the second with exp
Page 135 and 136: Taking the limit and using the expr
Page 137 and 138: Since V is positive definite for t
Page 139 and 140: 2.5.3 The Treatment of Terminal Con
Page 141 and 142: F while Equation (2.5.87B) requires
Page 143 and 144: ._ __. ..- . . - which will equal t
Page 145 and 146: 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 - ._ .-_
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4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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