guidance, flight mechanics and trajectory optimization

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I- savings offered by Dynamic Programming makes soluble some problems that are physically impossible to attempt with a straightforward combinational search because the exorbitant number of computations. In order to see the relative merits of Dynamic Programming in a small problem,consider the problem of finding the optimum path from point A to point B in the following sketch. Decision Points (Stages) The brute force method of solving this problem would be to evaluate the cost of each of the 20 possible paths that could be taken. Since there are six segments per path, there will be five additions per path or a total of 100 additions and one search over 100 numbers for a complete solution. The same problem can be solved by Dynamic Programming (see Section 2.2.1) by performing two additions and one comparison at each of the nine points where a decision was needed and one addition at the re- maining six points. This approach results in 24 additions and six comparisons as opposed to 100 additions and one search which were necessary with the brute force method. This comparison can be performed for an n stage process (the previous example was a six stage process). The expressi2n for the number of additions for the Dynamic Programming approach is s+?? . The brute force method involves(n-r)n!/(f!)' additions. Using these expressions, the merits of Dynamic Programming begin to become very evident as n increases. For instance, the 20-stage process would require 220 additions using Dynamic Programming as opposed to 3,510,364 additions by the brute force method. 2.3.3 DIFFICULTIES ENCOUNTERED IN DYNAMIC PROGRAMMING It should not be assumed that because Dynamic Programming overcomes the difficulties discussed in Section 2.2.1, that it is the answer to all optimization difficulties. To the contrary, many problems are created by its use. The following section discusses some of the difficulties encountered when Dynamic Programming is applied to multi-dimensional optimization problems. 51
2.3.3.1 The Curse of Dimensionalitv In Section 2.2.2.3 a simple guidance problem is presented. It is pointed out in that section that the number of computations involved was quite large because of the four dimensional nature of the state space. In general, the,number of computation points increases as an, where a is the number of increments in one dimension and n is the number of dimen- sions in the space. With the limited storage capabilities of modern digital computers,it is not difficult to realize that a modest multi- dimensional problem can exceed the capacity of the computer very easily, even with the methods of Dynamic Programming. This impairment does not prevent the solution of the problem; however, it means that more sophisti- cated techniques must be found in order to surmount this difficulty. Although this field has had several important contributions, it is still open for original research. One of the more promising techniques that can be used to overcome dimensionality difficulties is the method of successive approximations. In analysis,this method determines the solution to a problem by first assuming a solution. If the initial guess is not the correct solution, a correction is applied. The correction is determined so as to improve the previous guess. The process continues until it reaches a prescribed accuracy. The application of successive approximations to Dynamic Programming takes form as an approximation in policy space. The two important unknown functions of any Dynamic Programming solution are the cost function and the policy function. These two equations are dependent on each other, i.e., one can be found from the other. This relation is used to perform a successive approximation on the solution of the policy function by guessing at an initial solution and iterating to the correct solution. (This tech- nique is called approximation in policy space.) It should be noted that such a procedure sacrifices computation time for the sake of reducing storage requirements. The use of approximation in policy space will be illustrated via an allocation problem. Mathematically, two dimensional allocation problem can be stated as finding the policy that minimizes subject to the condition 52 3 '0 k; 20 (2.3.1) (2.3.2) (2.3.3)
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NASA OI 0 0 P; U 4 cd 4 z . . _ -.
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c FOREWORD This report was prepared
Page 7 and 8: SECTION PAGE 2.5 Dynamic Programmin
Page 10 and 11: 2.0 STATE OF THE ART 2.1 Developmen
Page 12 and 13: There are several ways to accomplis
Page 14 and 15: 2.2 Fundamental Concepts and Applic
Page 16 and 17: II. Although the previous problem w
Page 18 and 19: I,. Citg B. C optimum cost Path for
Page 20 and 21: The optimum path can be found by st
Page 22 and 23: The integral in Equation 2.2.1 can
Page 24 and 25: 2.2.2.1 Shortest Distance Between T
Page 26 and 27: The cost of the allowable transitio
Page 28 and 29: In a,completely analogous manner th
Page 30 and 31: 2.2.2.2 Variational Problem with Mo
Page 32 and 33: Ii _ 6 5 9 3 I 8 I I The cost integ
Page 34: 2.2.2.3 Simple Guidance Problem As
Page 37 and 38: This process continues in the same
Page 39 and 40: The algorithm for solving this prob
Page 41 and 42: The reader, no doubt, has a reasona
Page 43 and 44: are applied to Min (fl), the range
Page 45 and 46: Note that each diagonal corresponds
Page 47 and 48: The value of X can be found by empl
Page 49 and 50: The initial condition is: K.&P : P(
Page 51 and 52: it is seen that (for a two stage pr
Page 53 and 54: The optimal policy can now be found
Page 55 and 56: 2.3 COMPUTATIONAL CONSIDERATIONS So
Page 57: If classical techniques were to be
Page 61 and 62: Next, fl is evaluated for all allow
Page 63 and 64: % Xl + 5 = 2 (A2 = 2) 5+X2=3 (A2=3.
Page 65 and 66: 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
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with The P vector for the system is
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where With the control known as a f
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with and with the boundary conditio
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2.5 DYNAMIC PROGRAMMING AND THE OPT
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2.5.2 Problem Statement Let the sys
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Now, introducing the variables qs2,
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Thus, the performance index takes t
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#R where tr denotes the trace of th
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The minimum value of the performanc
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variance characterizing Ilt) can be
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The solution takes the form with s
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Since V is positive definite for f
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where the first the second with exp
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Taking the limit and using the expr
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Since V is positive definite for t
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2.5.3 The Treatment of Terminal Con
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F while Equation (2.5.87B) requires
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._ __. ..- . . - which will equal t
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is determined. Let p(z,t') be given
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of the p and $ equations (i.e., Eqs
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(2.5.I.21) Thus, the control is to
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Note, as in.Section (2.5.2.3), the
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.6 t -(h.(S~tj~fitr~f-‘~~~~n~) =
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3.0 RECOMMENDED PROCEDURES - ._ .-_
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4 Section 2.5 (2.5.1) (2.5.2) (2.5.
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