guidance, flight mechanics and trajectory optimization

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(XrQ,%fU&&) 4t + o(bt) + R(x+bx, t+bt) .(2-5.15) Finally, since the first term on the right of (2.5.15) does not depend on fty:) for t &7; 6 ttdt (2.5.16) Equation (2.5.16) is essentially a mathematical statement of the Principle of Optimality for the problem at hand. It indicates that the minimum average value for the functional is achieved by an optimum first control decision followed by an optimal sequence of control decisions which are averaged over all possible states resulting from the first decision. Note that R( X+LIX, f+ Ad) has the, expansion Using the expression for 2 and 5 in (2.5.9) and (2.5.10) and taking the expected value of~(Y+A~,icAt)over ftr;) fort c t;A t+At provides 115
#R where tr denotes the trace of the matrix C(f) ~7 . This last term is derived from the expected value of the'quantity E f(T) f'T'f+4t f f'rLt*At (2.5.18) The Dirac delta appearing in the variance expression for F in Eq. (2.5.10) causes this term to reduce to first order in dt . Substitution of Eq. (2.5.18) into (2.5.16) and taking the ,limit as At goes to zero provides the final result The boundary condition on Rtz,t) is easily developed from the definition of R given in (2.5.13). Thus, or alternately Rx,+) =;rTA;r (2.5.20‘) Eq. (2.5.19) is similar to that developed in the deterministic case [ see Equation (2.4.113)] , the only difference being the appearance of the term $(r;$) . This, however, is a major difference. Wh;Lle the Bellman equation is a first order partial differential equation and can be solved in a straightforward manner using the method of characteristics, this equation is a second order equation of the diffusion type. As a general rule, diffusion processes are rather difficult to solve. Fortunately, Eq. (2.5.19) solves rather easily. 116
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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
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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(
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it is seen that (for a two stage pr
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The optimal policy can now be found
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2.3 COMPUTATIONAL CONSIDERATIONS So
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If classical techniques were to be
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2.3.3.1 The Curse of Dimensionalitv
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Next, fl is evaluated for all allow
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% Xl + 5 = 2 (A2 = 2) 5+X2=3 (A2=3.
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that minimizes f. This interchange
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2.3.3.2 Stability and Sensitivity I
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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 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: Thus, the performance index takes t
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
Page 153 and 154: .6 t -(h.(S~tj~fitr~f-‘~~~~n~) =
Page 155 and 156: 3.0 RECOMMENDED PROCEDURES - ._ .-_
Page 157 and 158: 4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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