guidance, flight mechanics and trajectory optimization

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Since the quantity Q(-k+) is independen$ of the control action (see (2.5.117)), a*cons%raint on -e(+Z,) is equivalent to a constraint on the quantity ff ;,' .' Let then from (2.5.117) and (2.5.125) with w(t)= a7 (2.5.126) % = 2, ;: (2.5.127) Thus A is to be selected so that the simultaneous solution of the 5 and w equations, which satisfies the boundary conditions of Eq. (2.5.124) and (2.5.LZ'7), provides a value of WJ(t,) which satisfies the terminal constraint. As in the previous case, the solution will usually require iteration. However, the matrix A is again negative semi-definite and this condition will aid in the iteration process. 2.5.3.3 Partially Qbservable Case The problem is to select the control a to minimize the functional subject to the state equation E (J) = E {l:fTQ t + UrQ, U) dt + S&L 2+ (2.5.128) 0 2 = H#GU + n4f and a termi,nal constraint on E (z+ $1 In this case, however, observations of the state variable x are made continuously as represented by the observation equation Y' hdx+JL (2.5.129) where q is a Gaussian white noise with zero mean and variance r(t) ; that is, & (qj = O 143 (2.5.130)
Note, as in.Section (2.5.2.3), the performance index can be written as E (J)=$ {& [J/Y]} Thus, letting and it follows that ; ct) = ' (' Ct)/ 9 @)) ; ; (f) = & ( x cd/;1 (4)) (2.5.132) (2.5.133) (2.5.134) But the quantities s and V are given in Eqs. (2.5.63A) and (2.5.63B). Thus, using these expressions provides . L H&Gil + ff 4 r/d+-‘( Y- M;) (2.5.135) Note that these two equations contain the mean and covariance of the vector X . This fact will not effect the znalysis since the matrix does not depend on the control. Thus, if X is evaluated at any point, the corresponding value of 2 can be readily determined. Finally, let and observe that z=;+z (2.5.136) 144
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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
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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 and 148: of the p and $ equations (i.e., Eqs
Page 149: (2.5.I.21) Thus, the control is to
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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