guidance, flight mechanics and trajectory optimization

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In order to proceed with the solution for the general problem given in Eqs. (2.5.1) to (2.5.8), the feedback or observation loop must be specified. The reason for this is that the averaging process, that is, the expectation operator in Eq. (2.5.8), varies as the type and quantity of the observational data varies. As indicated in the introduction, three different types of observability will be considered and these are treated in the three succeeding sections. 2.5.2-l %f.ectly Observable Case In the perfectly observable case, it is assumed that the entire output of the system (i.e., all the components of the vector x ) can be determined exactly at each instant of time. This type of situation is represented in the sketch below. The state equations are -Gk Control a z Logic System Dynamics ic = AxtGu+k Sketch (2.5.1) >i, A%+Gui-c where 5 Is a Gaussian white noise with * E(e) = o (Disturbing forces) . = z CtJ d(t-T) I Sensors - - x (Output) But, since the system is perfectly observable, it is assumed that the initial state of the system is known exactly with (2.5.9) (2.5.10) x = Coat t = to (2.5.11) 113
Thus, the performance index takes the form where the variable F has been placed under the expectation operator to indicate that the ltaveraging' is to be conducted over this particular variable, the only random element appearing in the problem. To determine the solution to this problem, the Principle of Optimality can be employed essentially as it was in the deterministic case. Let R(x,t, denote the minimum value of the performance index for the system which starts in state X at time .! ; that is, Now, this expression can be rewritten as R&t) = MIAJ. f U (7) &T) t G-r Lt* LbTQ,U)dt + / 44 (XrQ,X+Ur’hU) dt+ %;A% t+At (2.5.14) But, since the first term in the square bracket on the right of (2.5.14) does not depend onU(r,) or f(Tz)for ttAtl% 6 ii , Eq. (2.5.14) can be written in the form 114
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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
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 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: Now, introducing the variables qs2,
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 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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