guidance, flight mechanics and trajectory optimization

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(In what follows, the symbol 4 will frequently be used to denote 4 &,t).) Since H is a Prm matrix, it is necessary that Q be a P* P symmetric matrix. Also, since PI is positive semi-definite, it follows.that Q is also positive semi-definite. The reason for this assumption as to the form of Eq. (2.5.88) will be made clear subsequently and it will be shown that Eq. (2.5.88) is physically consistent with the terminal con- straints of Eq. (2.5.87). Following the usual procedure of the Calculus of Variations, the problem of minimizing the functional subject to a terminal tc I 41 constraint on the quantity is equivalent to minimizing the modified functional where ._ (2.5.90 where A is a pxr constant diagonal matrix of Lagrange multipliers (recall that H is a pzn matrix), and selected so that the specified terminal variance condition is satisfied. The particular form of the matrix A will depend on the particular' terminal constraint which is imposed [i.e., Equation (2.5.87A) or (2.5.87B) or (2.5.87c)]. For example, if X is a six dimensional vector and H is the matrix H= I, 0, 0, 0, 0, o 0, 1,0, 0, 0, O the terminal constraint of Eq. (2.5.878) becomes and the quantity to be adjoined to Eq. (2.5.85) to form (2.5.91) is This form is equivalent to provided quantity to be adjoined to Eq. (2.5.91) is 135 (2.5.87B) is imposed, the
._ __. ..- . . - which will equal to + tat tf q] In any event, whatever the form of the matrix H , if the terminal constraint is to satisfy one of the conditions in Eq. (2.5.87), then the problem can be handled as is indicated in Eq. (2.5.91). Using the definition in Eq. (2.5.88) and noting that the performance index can be written as One further simplification is necessary before proceeding with the optimization problem. Let (2.5.93) Thus, differentiation of this expression with respect to time and using Eqs. (2.583) and (2.5.89) provides with the boundary condition 2 = Hd'G,, fffv (2.5.94) (2.5.95) A Now, since XO is a Gaussian random variable with mean X, and covariance VO it follows from (2.5.95) that Co is Gaussian with mean and covariance given by 136 (2.5.96)
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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
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 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: F while Equation (2.5.87B) requires
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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