guidance, flight mechanics and trajectory optimization

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subject to the state equations 1 = HtpGU t H$q and a terminal constraint on the quantity Jz (q zf', - The initial state t. is a Gaussian variable with mean and covariance given in Eq. (2.5.96). (2.5.115) Let 2 denote the expected value of t and P its covariance; that is, 2 = E (Zj =H@E(X) P (&s - 2, = bflp& (X-3 (X-3 4THT (2.5.116) Now, the expected value and covariance of X were calculated for the perfectly inobservable treatment given in Section (2.5.22) [see Eq. (2.5.36)] . Substituting these expressions into (2.5.116) provides with the boundary conditions Also, letting it follows that 3: = H&U ; =h~x~‘w Z= ;+z E (z) =E (2 f--j= 0 t’ = WV Thus, substituting the value for 7 given in (2.5.119) into (2.5.114) reduces the performance index to 141 (205.117) (2.5.118) (2.5.119) (2.5.120) (2.5.12oA)
(2.5.I.21) Thus, the control is to be selected to minimize the quantity inside the first set of brackets in Eq. (2.5.X1) (the quantity in the second bracket does not depend on U ), and the stochastic problem has been reduced to deterministic form. Then, using the Dynamic Programming approach, it follows that with the solution 0 = ktlN ~‘Q~+LL~Q~u+~~ + u(t) -z The optimal control is given by (2.5.122) @.5.=3) (2.5.124) u=- Q;lGri 'ff rSi @.5.=5) To determine the value of h for which the terminal constraint is satisfied, note that 142
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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
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But (2.4.10) since the function on
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Now,. noting that the second MIN op
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where v 'is determined from and wit
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NOW, combining these two expression
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The situation is pictured to the ri
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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
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characteristics associated with Eqs
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2.4.8 The Problem of Bolza The prec
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The initial position, velocity and
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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
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
Page 147: of the p and $ equations (i.e., Eqs
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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