Sensitivities of any arbitrary system are calculated using general purpose digital computer with available software packages for transfer function analysis. Sensitivity shows how element variation within system affects system performance. Signal flow graph illustrates topological system behavior and relationship among parameters in system

Chan, S. P.

Munoz, R. M.

English

NASA Technical Reports Server

June 1971	 Brief 71-10164 
NASA TECH BRIEF 
Ames Research Center 
- 
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 National Technical Information 
Service, Springfield,' Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D. f. 20546. 
A Topological Approach to Computer-Aided Sensitivity Analysis 
A mathematical development using the theory of 
linear oriented signal flow graphs proves in a rel-
atively simple way that the sensitivities of any 
arbitrary system can be calculated using a general 
purpose digital computer with available software 
packages for transfer function analysis. 
Functional systems consist of interrelated com-
ponents. (In electronic circuits, for example, the com-
ponents include elements such as resistors and capaci-
tors that are described by impedance, admittance, 
and other parameters.) The systems may also be 
considered as entire subsystems consisting of a num-
ber of individual elements connected together as a 
functioning group and described by a transfer func-
tion. The concept of sensitivity remains unchanged, 
however, regardless of whether impedance, admittance, 
or transfer functions are used to express subsys-
tem relationships; and this concept may be applied 
to systems as diverse as sociological systems, where 
human beings are the subsystem components, and 
economic systems, where business and production 
organizations are the subsystem components. The im-
portance of sensitivity as an analytic tool in under-
standing systems operation is largely due to its 
ability to show how the variation of an element 
within the system affects system performance. 
One method of measuring system sensitivity in-
volves calculating the partial derivative of some 
system performance criterion with respect to the var-
iation of a parameter (i.e., a quantitative subsystem 
characteristic, such as a resistor value). Sensitivity, 
as thus defined, is a measure of the incremental 
change in a system performance criterion due to 
the change in a performance parameter when all 
other system parameters are held constant. A simple 
xample would be the variation of the gain of an
amplifier circuit due to a change in the value of one 
resistor within the circuit. Further development of the 
sensitivity concept, from one which provides a meas-
ure of the absolute change of a system performance 
criterion to one which provides the relative change 
of a performance criterion with respect to the rel-
ative change in a system parameter, leads to the 
derivation of a logarithmic sensitivity which can be 
expressed as the percentage change in a performance 
criterion due to a percentage change in the param-
eter. 
The computation of sensitivity is a very tedious 
operation, prone to error if done manually, especially 
for large systems; digital computation offers a great 
advantage, but is performed only with difficulty. Since 
digital computers do not conveniently differentiate the 
complex algebraic expressions representing system 
performance, a systematic procedure must be struc-
tured around the more common numerical capabil-
ities of computer operations. Sensitivity and logarith-
mic sensitivity can be determined systematically by 
computing certain transfer functions. Since many 
highly refined transfer-function analysis programs 
have been developed, it would be valuable if these 
programs could be used directly in the study of 
sensitivities. 
It has been proven that this is possible by using the 
theory of signal flow graphs and by developing 
the topological relationships that express sensitivity. 
The signal flow graph is ideal, as it illustrates 
topologically the behavior of a system and the re-
lationship among parameters within the system. The 
performance criterion in signal flow graph terminology 
is identical to transmittance, T, which was selected 
for study. Any transmittance, T, of an arbitrary 
connected signal flow graph (and the system modeled 
(continued overleaf) 
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 ownedrights.
https://ntrs.nasa.gov/search.jsp?R=19710000163 2020-03-17T02:25:06+00:00Z
by it) can be expressed as a function of the gain 
of any edge, P, within the graph corresponding to 
the value of a parameter in the system. By an 
appropriate differentiation, an expression can be 
obtained for partial derivatives which permits their 
computation by transfer functions alone. This expres-
sionis	 4)T TT 
l2 ,7 p
where T 1 is directly interpretable as the transfer 
function from the input to the parameter in ques-
tion. For example, consider any system voltage trans-
fer function where the sensitivity with respect to a 
resistor in the system is sought. In this case, T1 
is the transconductance from the system input to the 
resistor; i.e., the resistor current divided by the sys-
tem voltage which produces that current. T 2 is 
similarly interpretable as the transfer function from 
a measurable output of the parameter in ques-
tion to the output of the system. In this case, T2 
is the ratio of output voltage to the voltage across 
the resistor. The logarithmic sensitivity can be found 
simply by multiplying (ôT/P) by P and l/T (the 
reciprocal of the network transmittance). This form 
of computation is satisfactory for practical use in
a computer if an algorithm for,
 polynomial arith-
metic is available. 
Reference: 
Munoz, Robert M., and Chan, S. Park: An Optimal 
Strategy for Topological Analysis of General Net-
works by Computer. Presented at the Joint Con-
ference on Mathematical and Computer Aids to 
Design, sponsored by ACM/SIAM/IEEE, October 
27-31, 1969, Anaheim Convention Center, Anaheim, 
California. 
Note: 
Requests for further information may be directed 
to:
Technology Utilization Officer 
Ames Research Center 
Moffett Field, California 94035 
Reference: TSP71- 10164 
Patent status: 
No patent action is contemplated by NASA. 
Source: Robert M. Munoz and S. Park Chan

Ames Research Center

(ARC-10214) 
Brief 71-10164	 Category 02


A topological approach to computer-aided sensitivity analysis

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