guidance, flight mechanics and trajectory optimization

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2.2.3 Maximum - Minimum Problem In order to demonstrate the more analytical applications of Dynamic Programmin g, a simple Maxima-Minima Problem will be examined. The procedure utilized to formulate a problem for the application of Dynamic Programming is not always immediately obvious. Many times the problem formulation for a Dynamic Progr amming solution is quite different from any other approach. The following problem will be attacked in a manner such that the Dynamic Programming formulation and method of attack can be seen. The problem is to minimize the expression subject to the constraints K, + A-- f x3 = /o (2.2.22) (A problem similar to this is often used by Dr. Bellman to introduce the concepts of Dynamic Programming). At first glance, the methods of Dynamic Programming do not seem to apply to this problem. However, if the problem is reduced to several smaller problems, the use of Dynamic Programming becomes apparent. Consider the minimization of the following three functions: $ = xf (2.2.23) 4 = ( +2*; (2.2.24) Applying the constraintsix = 10, x. 1 Otto th e first function give3 the trivial result xl = 10. T k is resulg is not so helpful. However, if the constraints x, = A I 0 4 A, L /o x, 2 0 35 (2.2.25)
are applied to Min (fl), the range of Min (f ) can be found for various admissible values of xl. This step can be thought of as the first stage of a three stage decision process, where the various choices for xl can be as many as desired within the limits 0=X, 4/O. Now that all the choices for the first decision have been investigated, the second decision must be considered. Again, care must be taken in the specification of the constraint equations. The optimal value function for the second decision is The constraints on 3 and x2 are chosen to be $=(-+2X; (2.2.26) X, + x2 = AZ OLAz g 10 (2.2.27) For each value of A2 that is to be investigated, there are many combinations of x, and x2 to be considered. More precisely, the number of combinations of "I and x2 to be considered is the same as the number of AlIs that are less than or equal to the A being considered, e.g., if Al was investigated for comb~a;i~~s5~u,'I A ;de&=tiedis 5 being considered, then the following A - %, - x2 0 5 5- 0 The third decision does not require as much computation as the others in this case because of the original constraint equations. ;“, + xz f z 3 XL ZO Since the first %and +x "1 2' two decisions were investigated for many possible it is only necessary to consider values of = /o x, f x2 + x3 = IO because the various choices for will specify 5 + x2 = 10 - 3 and these possibilities have already been 3 ' vestigated. (2.2.28) The arithmetic solution will now be shown so that the previous discussion will be clear. For simplicity, only integers will be considered for the allowable values for xl, x2, and x3. A table can be constructed for the first decision as follows: 36
Page 1 and 2: NASA OI 0 0 P; U 4 cd 4 z . . _ -.
Page 4: 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: The reader, no doubt, has a reasona
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 and 58: If classical techniques were to be
Page 59 and 60: 2.3.3.1 The Curse of Dimensionalitv
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
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where S(t) is some n x n symmetric
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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
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
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._ __. ..- . . - which will equal t
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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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