4.1 The Gap Between Theory and Practice: The Digital Computer The underlying purpose of this investigation has been to map existing theoretical concepts into synthesis techniques, computational algorithms,and slight theoretical extensions which apply to the solu- tion of a specific engineering problem. The author feels that the greatest advantage of this approach has been that even though the
results are supported by a theoretical foundation, they are at the same time grounded to practical engineering reality. Consequently the diffi- culties encountered have been the traditional ones which separate theory from practice.
One of the primary difficulties in applying optimization theory to engineering problems is the specification of the system performance criterion. Often the final engineering design should represent the best compromise among a myriad of conflicting goals, but i t is usually difficult to interpret and properly weight all the factors in terms of a mathematical expression of performance. For example, i t is often ar- gued that a guidance system must consume a minimum amount of fuel. But considering the fact that the Mariner midcourse guidance systems have typically carried on-board five times the fuel required to correct the maximum expected initial velocity de vi at ion, i t would appear that efforts to minimize only the guidance fuel consumption would be somewhat wasted.
Other considerations such as system accuracy, simplicity, and imple- mentation should certainly be stressed in the specification of the final
design.
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The first attempt in this study has been to make the problem formulation as realistic as possible, and only then to seek a solution.
There has been virtually no attempt to force the problem into a form in which there already exists an analytic solution, and then turn about
and try to justify the desired formulation. Indeed, the solutions and synthesis techniques presented in this study have t ime and again re- lied heavily on the use of one of the most poHerful design tools in modern technology, the digital computer. There can be little doubt that the computer allows the solution of problems which •vould other- wise be deemed impossible. Yet while i t is very short sighted to ig- nore its capabilities in favor of gross simplifi cations and approxi- mations, i t is equally undesirable to al low the computer to inspire laziness and poorly conceived solution algorithms. In the MTV gui - dance problem the computer has been employed to the maximRm possible advantage, but only after the mathematical development has been carried out as far as possible.
4.2 Stochastic Optimal Control
It is popular nowadays to w-rite off stochastic optimization theory as a somewhat futile endeavor. The mainstream of discontent apparently comes from the immense difficulties involved in solving even \.,rhat seem to be the easiest examples. There are also those vTho claim that the differences between a deterministic design and a sto- chastic design vil l not usually be very great, and therefore the extra design effort vill hardly be •-mrth i t . 'rhese arguments may be vTell taken in many instances but thei r general veracity cannot be asserted
without becoming rather arbitrary about the scope of problems being considered. For instance, i t has been made clear that the MTV gui- dance problem has no deterministic analog; therefore a stochastic de- sign is essential. And even though the exact solution has no·t been attained, i t has been seen that the approximate stochastic system has a great deal to offer. In this way, the results establish at least one concrete example of the practical benefits of stochastic optimal control.
4.3
Extensions and Future EffortsIt is usually quite tempting to try to extend a once successful idea beyond its original point of application. While this is essen- tially a very good idea, i t is also tempting to claim far more gen- erality than is varranted. For the present problem there are tva areas vhere the results vould appear to have clear application. The first is the more general class of powered flight guidance problems;
e.g. , booster guidance. These other targeting guidance problems would possess essential similarities to low-thrust, interplanetary gui-
dance in the aspects of bounded control levels, stochastic disturbance inputs, and similar dynamic behavior. The second area is the space vehicle attitude-stabilization problem in vhich i t is desired to min- imize the total nunilier of stepper-motor actions, or reaction-jet firings, on any given mission. The solution of this problem vould seem to be straightforvard in view of the identical dynamic response (i.e., purely inertial), and also the built-in capability of the M'l'V guidance method to yield the svi tching probabilities for each control configuration.
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Future effort in this area could certainly branch into many
different directions. In each instance there is really no way of telling initially to ~-rhat extent any given idea wil l be significant. It :ls
the contention of the author that rather than document a long list of alternative pathways, i t is perhaps more effective to l eave the reader unbiased in choosing new and interesting branches to explore.
Appendix A
SOLUTION OF THE MINIMUM TIME PROBLD4
Consider the dynamical equations
X
. =
AX+ b(t)u 1~1 !{ k (A.l)where X is an n-vector, b is an n x m matrix, and k and u are m-vectors. Also
X(t )
=
X- 0 ~
(tf is minimum)
The Hamiltonian for this problem is
The optimal u minimizes the Hamiltonian. Hence
u* = K(-sgn(bT(t)l)) K =
0
where the sgn function is defined as
sgn(y)
~{
+l-1
0
• k m
y > 0
y < 0
(A. 2)
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Applied to a vector, the sgn function acts on each component. Thus
The equations of motion are
X
=
= AX - b(t) Ksgn(bT(t)~)
:\ =
(A. 3)The transversality condition yields
(A. 4)
Equations A.3, with conditions A.2 and A.4, yield a two-point boundary value problem that must be solved in order to obtain the optimal con- trol.