Random-search algorithm employs local and global properties to solve two-point boundary value problem in Pontryagin maximum principle for either fixed or variable end-time problems. Mixed boundary value problem is transformed to an initial value problem. Mapping between initial and terminal values utilizes hybrid computer

Brocker, D. H.

Kavanaugh, W. P.

Stewart, E. C.

English

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December 1970	 Brief 70-10637 
NASA TECH BRIEF 
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nology Utilization Division, NASA, Code UT, Washington, 4D.C. 20546. 
Global Search Algorithm for Optimal Control 
_____________ Analog _______________________  
Computation	 lit 4	 Digital Computation 
I	 I	 I 
	
w ADC	 Computation[ 
i k 
Memi X(t)	 X 1 I 	 Metric
	
_I^L, 
p(t) 	 pk(T)	 jk[xk(T).Pk(T)] 
= f(x,u.t)  
= g(p,x,u.t)	 I	 Algorithm Logic I	 fk+1 (E 1 E2Ekjojl 
u = u(x.p)
11+ 
Guassian Source	 :	 I 
X(0)
	
p(0)
	 p k E k	 Standard	 k 
	
DAC	 Deviation, a	 I 
Mean m=O	 I	 I 
L_------------
 
Memory M -------J 
The Pontryagin Maximum Principle provides a 
powerful theory for determining the optimal control of 
dynamic systems describable by nonlinear differential 
equations with bounded control. When applied to a 
high-order system, the Maximum Principle yields 
much information about the nature of the solution, 
but the actual solution is seldom obtained, because a 
difficult, mixed boundary-value problem is invariably 
encountered. In such instances, the known boundary 
conditions are divided between initial and terminal 
values, and the mapping from the initial to terminal 
boundary values is not, in general, explicitly known.
Adaptive Loops 
Most current computer algorithms which treat this 
problem are based on local-seeking methods and the 
assumption of a single minimum. Such methods are of 
limited value when the hypersurface of the perform-
ance function being studied is multipeaked or dis-
continuous. 
To resolve some of these difficulties, a method based 
on random techniques has been devised and used 
successfully. The method is based on an adaptive 
random-search algorithm employing both local and 
global properties to solve the two-point boundary-
value problem in the Maximum Principle for either 
(continued overleaf) 
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fixed or variable end-time problems. The mixed-bound-
ary-value problem is transformed to an initial-value 
problem, and the required mapping between initial 
and terminal values is constructed with the aid of a 
hybrid computer. 
For simplicity, the essence of the method is illus- 
trated for the fixed-time case. On the left-hand side 
of the figure, the set of equations to be solved which 
result from an application of the Maximum Principle 
is shown. The solution of these equations determines 
the optimal control function which minimizes an 
arbitrary performance function. The complete set 
of boundary conditions for these equations is generally 
not known. On the right-hand side of the figure, the 
algorithm used to determine the unknown initial 
boundary conditions is illustrated. The algorithm is 
based on: (1) introducing a vector metric which meas-
ures the vector distance by which the desired terminal 
boundary conditions are not satisfied; and (2) reduc-
ing this vector metric iteratively by searching for the 
unknown initial boundary conditions from a Gaussian 
probability density functiOn. However, the pure ran-
dom search is generally unsatisfactory because of ex-
cessively long convergence times. The convergence 
properties are improved by making the system adap-
tive by varying the mean M and standard deviation 
a as a function of the past performance as indicated 
by the adaptive loops and the algorithm logic 
f(E', E2 E3 J0 , J I ,	 •Jk) Iteration stops when the 
vector metric J is small enough to satisfy the problem 
requirements. 
Several variations of the function f of the algorithm 
were investigated. The most useful characteristic 
given to the function f was based on the notion of a 
creeping, expanding and contracting search. The 
creeping character of the search is provided by varying 
the mean of the distribution to equal the initial condi-
tion of the adjoint vector on the last successful itera-
tion. The expansion and contraction character of the 
search is provided by varying the standard deviation 
so that the search is localized when successful, but 
expanded geometrically when not successful. This type
search has desirable local and global search proper-
ties; "hanging up" on a local minimum, as with 
gradient methods, is avoided. Furthermore, it will 
not matter if the boundary surface is discontinuous or 
how many peaks or valleys there are. 
The hybrid computer is ideally suited to implement-
ing the above algorithm. By comparison, an all-digital 
implementation is about 1,000 times slower. The 
requirement for solving the equations of motion a 
large number of times is fulfilled most efficiently on an 
analog device, and the memory and computation 
capability required by the algorithm are best imple-
mented on a digital device. The amount of apparatus 
required increases linearly with the order of the system. 
The feasibility of the method has been demonstrated 
experimentally on several highly nonlinear systems 
of an order as large as six. 
Notes: 
1. The following documentation may be obtained 
from: 
Clearinghouse for Federal Scientific 
and Technical Information 
Springfield, Virginia 22151 
Single document price $3.00 
(or microfiche $0.65) 
Reference: 
NASA-TN-D-5642 (N70-18083), An Adaptive 
Random-Search Algorithm for Implementa-
tion of the Maximum Principle 
Requests for further information may be directed 
to:
Technology Utilization Officer 
Ames Research Center 
Moffett Field, California 94035 
Reference: TSP70-10637 
Patent status: 
No patent action is contemplated by NASA. 
Source: E. C. Stewart, W. P. Kavanaugh, 
and D. H. Brocker 
Ames Research Center 
(ARC-10359)
S 
0 
Brief 70-10637	 Category 09


Global search algorithm for optimal control

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