General, closed-form, analytical solutions are determined for certain classes of C-minimax control problems, several alternative mathematical theories are derived, and a controller design theory is developed to give optimal control in the presence of unmeasureable external disturbances

Johnson, C. D.

English

NASA Technical Reports Server

June 1970
	 Brief 70-10315 
NASA TECH BRIEF H 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage commercial 
application. Tech Briefs are available on a subscription basis from the Clearinghouse for Federal Scientific and Technical 
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be directed to the Technology Utilization Division, NASA, Code UT, Washington, D.C. 20546. 
Chebyshev Minimax Control Theory 
In a detailed study of Chebyshev Minimax (C-Mini-
max) control theory, general, closed-form, analytical 
solutions are determined for certain classes of C-Mini-
max control problems; several alternative mathemati-
cal theories are derived; and a controller design theory 
is developed to give optimal control in the presence 
of unmeasurable external disturbances. Results of 
this study may be applied to the design of control 
systems in many areas of technology, such as aircraft 
flight control, marine navigation through disturbances 
from ocean currents or atmospheric winds, control 
of interest rates in a business economy, allocation of 
raw materials among competing processes, or process 
control in the chemical and pharmaceutical indus-
tries. 
The search for general solutions to certain classes 
of C-Minimax problems was successful, in the sense 
that several classes were solved explicitly. However, 
each solution contained some form of degeneracy, 
and the classes solved were not representative of the 
problems encountered in practical applications. 
Several alternative formulations of C-Minimax 
problems were developed. These theories may lead 
to more effective computing techniques and will 
certainly permit a more rational approach to the 
design of simplified "almost optimal" C-Minimax 
controls. This domain involving approximate solu-
tions to C-Minimax problems, will also be aided by 
the verification theorems developed, which can be
used to test the degree of optimality of conjectured 
control laws. 
The C-Minimax control problem in the discrete-
time domain was formulated and solved. This has 
extended the range of C-Minimax theory to include 
problems in economics, operations research, and 
other areas where decisions are made in discrete steps. 
The disturbance absorption theory, by which opti-
mal feedback control may be achieved in the presence 
of unmeasurable external disturbances, is believed 
to be a new, highly important result in the theory of 
dynamic systems. The theory is particularly attractive 
because it results in solutions that can be implemented 
with available control hardware. 
Note: 
Requests for further information may be directed to: 
Technology Utilization Officer 
Marshall Space Flight Center 
Huntsville, Alabama 35812 
Reference: B70-10315 
Patent status: 
No patent action is contemplated by NASA. 
Source: C. D. Johnson of 
General Dynamics/Convair Division 
under contract to 
Marshall Space Flight Center 
(M FS-20639) 
Category 03 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
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Chebyshev minimax control theory

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