guidance, flight mechanics and trajectory optimization

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To reduce (2.4.109) to a partial differential equation, one must assume that all second derivatives of R with respect to t <strong>and</strong> x are bounded; that is, Under this assumption,R(z!+df,t(~ tAf))has the series expansion where T denotes transpose dx dt Substituting (2.4.111) into (2.4.109) along with the values for k from (2.4.108), provides In the limit asdt + 0 this expression becomes (2.4.110) (2.4.113) which is a first-order, partial differential equation <strong>and</strong> will be referred to as the Bellman equation for the Problem of Rolza. The boundary condition which R(t, x(t)) must satisfy, will be considered next. 89
Since R(t, x(t) ) is the minimum value of the performance index for the solution which starts at the point (t, x(t)), it follows that R must satisfy the terminal condition However, in addition, the terminal point (tf, x (t,)) must satisfy the terminal constraints ?$(f&) =o AT t’=$ ; j =I,@? (2.1+.115) Hence, the boundary condition on R becomes Analogous to the development of IQs. (2.4.63B) from the boundary condition (2.4.63A) for the problem of Lagrange, the above expression can be reworked to yield the equivalent condition where/C is the vector P = = 1, h (2.4.117) If the final time tf is itself not specified, then the additional boundary condition must hold. 90 (2.4.118) (2.4.119)
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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
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
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: or, as . -... .-. . .._-_--- to ind
Page 99 and 100: 2.4.10 Ljnear Problem with Quadrati
Page 101 and 102: where S(t) is some n x n symmetric
Page 103 and 104: The governing equations for the att
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
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of the p and $ equations (i.e., Eqs
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(2.5.I.21) Thus, the control is to
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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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