One of the factors that influences the fidelity of an aircraft digital simulation is the sampling rate. As the sampling rate is increased, the calculated response of the discrete representation tends to coincide with the response of the corresponding continuous system. Because of computer limitations, however, the sampling rate cannot be increased indefinitely. Moreover, real-time simulation requirements demand that a finite sampling rate be adopted. In view of these restrictions, a study was undertaken to determine the influence of sampling rate on the response characteristics of a simulated aircraft describing short-period oscillations. Changes in the calculated response characteristics of the simulated aircraft degrade the fidelity of the simulation. In the present context, fidelity degradation is defined as the percentage change in those characteristics that have the greatest influence on pilot opinion: short period frequency omega, short period damping ratio zeta, and the product omega zeta. To determine the influence of the sampling period on these characteristics, the equations describing the response of a DC-8 aircraft to elevator control inputs were used. The results indicate that if the sampling period is too large, the fidelity of the simulation can be degraded

Howard, J. C.

English

NASA Technical Reports Server

, . . 
NASA Technical Memorandum 84354 
NASA-TM-84354 1983001 4996 
Influence of Sampling Rate on the 
Calculated Fidelity of an 
Aircraft Simulation 
James C. Howard 
April 1983 
NI\SI\ 
National Aeronautics and 
Space Administration 
LIBRARY opy 
L; .~" ? - 1983 
}.ANGLEY RESEARr.H CENTER 
LIBRARY, NASA 
HAMPTON, VIRGINIA 
https://ntrs.nasa.gov/search.jsp?R=19830014996 2020-03-21T04:47:33+00:00Z
NASA Technical Memorandum 84354 
Influence of Sampling Rate on the 
Calculated Fidelity of an 
Aircraft Simulation 
James C. Howard, Ames Research Center, Moffett Field, California 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field , California 94035 
ABSTRACT 
INFLUENCE OF SAMPLING RATE ON THE CALCULATED 
FIDELITY OF AN AIRCRAFT SIMULATION 
James C. Howard 
Flight Systems and Simulation Research Division 
NASA Ames Research Center 
Moffett Field, California 
One of the factors that influences the fidelity of an aircraft digital simulati on i s the 
sampling rate. As the sampling rate is increased, the calculated response of the discrete 
representation tends to coincide with the response of the corresponding continuous system. 
Because of computer limitations, however, the sampling rate cannot be increased indefinitely. 
Moreover, real-time simulation requirements demand that a finite sampling rate be adopted. 
In view of these restrictions, a study was undertaken to determine the influence of sampling 
rate on the response characteristics of a simulated aircraft describing short-period oscil-
lations. Changes in the calculated response characteristics of the simulated aircraft 
degrade the fidelity of the simulation. In the present contex t, fidelity degradation is 
defined as the percentage change in those characteristics that have the greatest influence 
on pilot opinion : short-period frequency w, short-period damping ratio ~ , and the product 
~w . To determine the influence of the sampling period on these characteristics, the equa-
tions describing the response of a DC-8 aircraft to elevator-control inputs were used. The 
results indicate that if the sampling period is too large, the fidelity of the simulation 
can be degraded. For example, it was determined that the motion characteristics of the simu-
lated aircraft, represented by ~ and ~w , as well as the corresponding fidelity, decreased 
linearly with sampling period and vanished when the period reached 640 ms. 
I NTRODUCTION 
The response of a dynamical system to control inputs may be stable or unstable, dependin ~ on 
the characteristics of the system. When the response is being calculated by a digital com-
puter, the dynamical characteristics are modified by the need to use a discrete representa-
tion of the system. This involves the choice of a sampling rate, which influences the 
fidelity of the simulation and determines the feasibility of simulating large aeronautical 
systems, such as helicopters. 
One way to determine the influence of a specif ic parameter on the r esponse characteristic s 
of a continuous dynamical system is to map the eigenvalues on t he complex plane and note 
their migrations, in each mode, as the parameter is varied (1 ) . When the loop is closed 
around an open-loop system , and the feedback gain is varied, the conventiona l r oot loci are 
obtained (2). These loci reveal the stability character istics of the system, as the fre-
quency and the damping ratio vary with changes in the feedback gain. 
By using the same procedure for a discrete repr esentation of t he system, t he i nfluence of 
sampling rate on the response characteristics can be determined. Whereas a l inea r , cont inu-
ous dynamical system is described in terms of the Laplace transform , the discrete represen-
tation requires a formulation in terms of the Z-transform. When the Z-transfer function 
corresponding to a given integration algorithm has been obtained as a function of the sam-
pling period, and the discrete eigenvalues determined, these can be plotted on the complex 
z-plane to ascertain their location relative to the unit circle and thus determine the pos-
sibility of instability. To facilitate the interpretation of results, the discrete eigen-
values were transformed from the z-plane to the s-plane. By proceeding in this manner, and 
using the equations describing the response of a DC-8 aircraft to elevator-control inputs, 
the influence of sampling rate was determined. 
I 
ANALYSIS 
Equations of Motion 
The differential equation describing an aircraft's pitching motion. which is assumed to be 
linear and time invariant. is (from Ref. 3) 
(d~Y + A3 d3y + A2 d2y + A dY + AOY\ _ (d2 U + B dU + B U\ dt~ dt3 dt 2 1 dt / dt2 1 dt 0 ) (1) 
where Y is the pitch angle. U is the elevator-control input. and Ai. Bj are cons t ants . 
The formulation is simplified by specifying a state variable X which satisfies the differ-
ential equation 
d4 X + A d3X + A d2X + A dX dt~ 3 dt3 2 dt2 1 dt + AoX = U 
The remaining state variables can then be assigned as follows: 
X m Xl 
Xl a X2 
X2 - X3 
X3 a X4 
Using matrix notation. these equations assume the conventional form; that is, 
where 
X -
A -
BT _ 
and 
From Equation (4). 
x ~ AX + BU 
y = CTX 
(Xl X2 Xl X4 ) T 
0 1 0 
0 0 1 
0 0 0 
-AO -A1 -A2 
(0 0 0 K) 
0 
0 
1 
-A3 
( 2) 
(3) 
(4 ) 
( 5) 
(6) 
(7) 
(8) 
(9) 
X(t) a eA(t-t O)X(t O) + it eA(t-T)BU(T)dT 
to 
(10) 
When the input is sampled with a zero-order hold after the sampler . the input is held con-
stant for one sampling period (see Figure 1). 
During this period the limits of integration are t m (n + l)T and to m nT, and Equa-
tion (4) becomes (from Ref. 4) 
2 
Therefore, 
By omitting the T from the arguments of X and U, this equation assumes a more compact 
form, namely, 
X(n + 1) • ~x(n) + BoU(n) 
(11) 
(12) 
where ~ - eAT, and BD is the discrete form of the control distribution matrix. That is, 
BD - A-l (~ - I)B (13) 
It should be noted that Equation (13) is valid only if A is nonsingular. 
Z-Transforms and Z-Transfer Functions 
Assuming zero initial conditions, the Z-transform of the state vector is obtained from 
Equation (12): 
The corresponding Z-transform of the output is 
(14) 
The Euler integration algorithm uses only the first two terms of the transition matrix eAT: 
eAT - ~ t (I + AT) (16) 
Substituting this value into Equation (13) gives the Euler form of the control distribution 
matrix BD' 
BD - A-l(I + AT - I)B - TB (17) 
Substituting this result into Equation (IS) gives the Z-transform of the output correspond-
ing to the Euler integration algorithm: 
Y(Z) - TCT[(Z - 1)1 - ATj-1BU(Z) (18) 
The Z-transfer function relating the output Y(Z) to the input U(Z) is 
Y(Z)/U(Z) - TCT[(Z - 1)1 - AT)-lB (19) 
By introducing matrix A from Equation (7) and the column and row vectors from Equations (8) 
and (9), respectively, the required form of the transfer function is obtained: 
where 
RESULTS 
T - sampling period 
Po c BoT2 - B1T + 1 
Pl = B1T - 2 
qo 2 AoT" - A1T3 + AzT z - A3T + 1 
qz = AzT z - 3A3T + 6 
q3 D A3T - 4 
(20) 
The following values of the elements of the system dynamics matrix A and the control dis-
tribution matrix B were obtained from Teper (3). 
3 
Ao - 0.07006953 
Al - 0.09712526 
A2 - 2.6813872 K - 1.338 
A3 - 1. 700522 
Bl - 0.5955 
Bo - 0.032675 
When these values were substituted into Equation (20), the open-loop eigenvalues correspond-
ing to the phugoid and the short-period modes were obtained as functions of the sampling 
period T. To determine the influence of the sampling rate on the stability of the short-
period motion, the discrete eigenvalues were plotted as a function of the sampling period, 
and their locations relative to the unit circle were noted. The results are plotted in 
Figure 2. It is seen that as the sampling period is increased, the eigenvalues remain 
inside the unit circle until T - 640 ms. At this point they lie on the unit circle, and 
further increases in T cause them to move into the region of instability. 
In order to determine the changes in frequency and damping as functions of the sampling 
period, the discrete eigenvalues were transformed from the Z-plane to the s-plane. The 
migrations of the eigenvalues in the s-plane give a good indication of the influence of the 
sampling period on the frequency and the damping of the simulated system. The required 
transformation can be effected by using the equation of definition: 
s - (logeZ)/T (21) 
Equation (21) was used to calculate the T-locus of open-loop eigenvalues for the folloWing 
range of sampling periods: 
0.02 ~ T ~ 0.80 s (22) 
The results are plotted in Figure 3, where it is seen that the locus reaches the imaginary 
axis when the sampling period is 640 ms. This figure should be compared with Figure 2, where 
the locus of discrete eigenvalues reaches the unit circle for the same value of T. More-
over, the data used to plot Figure 3 can be used to calculate the modal frequency and the 
damping ratio as functions of the sampling period. The T-locus of f requency is plotted in 
Figure 4, and the locus of the corresponding damping ratios is shown in Figure 5. When T 
is increased from 20 to 640 ms, it appears that the damping ratio decreases almost linearly 
from its initial value to zero. The undamped natural frequency is shown plo tted as a func-
tion of the damping ratio in Figure 6. The range of this curve indicates that as the sam-
pling period is increased, unsatisfactory regions of the flying-qual i ties plot will be 
encountered (5). When simulation data become available, boundaries of acceptable flying 
qualities will be established. These boundaries assume the form shown in Figure 7, where 
acceptable and unacceptable flying qualities are related to the shor t -period frequency and 
to the short-period damping ratio (6). 
Because the response of a simulated dynamical system to control input s is calculated by a 
digital computer, the motion characteristics of the discrete representation are functions of 
the sampling rate; consequently, simulation fidelity is degraded. In the present context, 
fidelity degradation is defined as the percentage change in those cha racteristics that have 
the greatest influence on pilot opinion: short-period frequency w, short-period damping 
ratio ~ , and the product ~w . For example, the fidelity loss owing to changes in the damp-
ing ratio is defined as 
(23) 
where f~ is the fidelity loss, ~ c is the continuous damping ratio, and ~D is the damping 
ratio of the discrete representation. The f~ term is plotted as s function of the sampling 
period in Figure 8. As might be expected, f~ has a functional form similar to that of ~ , 
as shawn in Figure 5. 
The loss of fidelity that results from natural fr equency changes is defined similarly: 
(24) 
4 
where fw is the fidelity loss, Wc is the natural frequency of the continuous system, and 
olD is the natural frequency of the discrete representation. As shown in Figure 8, · the 
influence of natural frequency changes on system fidelity, as measured by Equation (24), is 
not so pronounced as the influence of the damping ratio. The third measure considered 
assesses the loss of fidelity owing to the characteristic ~w. The loss of fidelity caused 
by changes in ~w is defined as 
(25) 
where again the subscripts c and D denote continuous and discrete representations, respec-
tively. This characteristic brings out the influence of the real component of the natural 
frequency vector on simulation fidelity. Since that component determines the damping ratio, 
the loss of fidelity, as defined by Equation (25), is seen to follow the same trend as f~. 
The results are plotted in Figure 9. 
CONCLUSIONS 
One of the factors that influences the fidelity of an aircraft simulation is the sampling 
rate. As the sampling rate is increased, the calculated response to control inputs , of the 
discrete representation, tends to coincide with the response of the continuous system. 
Because of computer limitations, however, the sampling rate cannot be increased indefinitely, 
and simulation fidelity is degraded. In the present context, fidelity degradation is 
defined as the percentage change in those characteristics that have the greatest influence 
on pilot opinion: short-period frequency w, short-period damping ratio ~,and the product 
~w . In order to determine the influence of the sampling period on these characteristics, the 
equations describing the response of a DC-8 aircraft to elevator-control inputs were used. 
Results indicate that if the sampling period is too large, the fidelity of the simulation can 
be degraded significantly. For example, it was determined that the motion characteristics of 
the simulated aircraft, represented by ~ and ~w , as well as the corresponding fidelity, 
decreased linearly with sampling period and vanished when the sampling period reached 640 ms. 
REFERENCES 
1. McRuer, Duane; Ashkenas, Irving; and Graham, Dunstan: Aircraft Dynamics and Automatic 
Control. Princeton University Press, Princeton, N.J., 1973. 
2. Truxal, J. G.: Control System Synthesis. McGraw-Hill Book Company, Inc., New York, 
1955. 
3. Teper, Gary L.: "Aircraft Stability and Control Data," Report 176-1, Systems Technology, 
Hawthorne, Calif., Apr. 1969. 
4. Gupta, C. S.; and Masdorff, L.: Fundamentals of Automatic Control. John Wiley & Sons, 
Inc., New York, 1970. 
5. Kolk, W. R.: Modern Flight Dynamics. Prentice-Hall, Inc . , Englewood Cliffs, N.J., 1961. 
6. O'Hara, F.: "Handling Criteria," J. Royal Aero. Soc., Vol. 71, No. 676, 1967, 
pp. 271-291. 
5 
~ T 
X- AX + BU 
Figure 1 - St ate Variable Sampled- Data System • 
. 6 
-.S 
Fi gure Z - T-Locus fo r Solution of Equation (20) by Eul e r In t eg r a t i on . 
360 Jw 
360 
260 340 400 
240\ 280 I 
220 " 180 \ \ 320 ir 100 -... -160 300 60 ' 120 20 mile 80 
-;- 40 1.4 
"'" OPEN·LOOP PO LES FOR 
CON TI NUOUS SYSTEM 
1.0 .8 .6 .4 .2 0 .2 a 
1.4 
1.8 
Fi gure 3 - T-Locus of Open- Loop Poles fo r t he Shor t - Period Mode . 
6 
1.9 
1.8 
w 1.7 
STABLE REGION 
300m_ 
I 
400 maec 
I SOOmsoc 
I 
\
" 20m •• c 
1.6 OPEN.LOOP NATURAL 
o 
FREOUENCY FOR THE 
CONTINUOUS CASE 
100 200 300 400 500 600 
TIME. moec 
Figure 4 - T-Locus of Natural Frequencies for the Short-Period Mode . 
20 m.oc 
.5 
.4 
.3 
.2 
.1 
o 100 200 300 400 
TIME. msoc 
/ 500m.oc 
500 700 
Figure 5 - T-Locus of Open-Loop Damping Ratios for the Short-Period Mode. 
7 
z 
o (; 
w 
a: 
w 
..J 
III 
~ 
t; 
Z 
:::l 
700 
1.8 
~ ~ 1.7 
>' 
~ 
w 
:> 
o 
~ 1.6 
Yo 
.j 
<I: 
a: 
::;) 
t-
<I: 
~ 1.6 
w 
... 
~ 
o 
z 
::;) 
1.4 
o 
600 7' .. c 400 moec 600 msec ~_____ I 
I 
2oom .. c 
/ 
.1 .2 .3 .4 
DAMPING RATIO. t 
20 moec 
,/ 
.5 .6 
Figure 6 - I - Locus of Undamped Na t ural Frequency versus Damping Ratio . 
6r---------------,-------r------,---,--,-,----, 
INITIAL RESPONSE 
FAST. TENDENCY TO 
OSCILLATION AND 
TO OVERSHOOT 
LOADS 
UNACCEPTABLE 
INITIAL RESPONSE FAST. 
OVERSENSITIVE. LIGHT 
STICK FORCES 
..,...---, 
\ 
\ 
\ 
SLUGGISH . LARGE 
STICK MOTION 
AND FORCES 
VERY SLOW RESPONSE. 
LARGE CONTROL MOTION 
TO MANEUVER. DI FFICULT 
TOTRIM 
o~--------------~------~----~--~--~~--~ 
.1 .6 1 2 3 4 6 
DAMPING RATIO. t 
Figure 7 - Flying- Qualities Boundaries . 
8 
, , 
-. 
w 
~ 
z 
< J: 
(.) 
-.c 
-20 
-40 
~f ~ _IWC-WDI xl00 W Wc 
-80 
-80 
-,ooLo------2-o~0------4-00~----~~r----8~OO~-----1-000~----~1-2~OO~--~1~400 
TIME, ms8C 
Figure 8 - Loci of Fidelity Measures for DC- 8 During Landing Approach . 
o 
-20 
2- -40 
w' 
~ 
Z 
< 
:r 
(.) 
-.c -60 
-80 
We CONTINOUS NATURAL FREQUENCY 
DISCRETE NATURAL FREOUENCY 
CONTINOUS DAMPING RATIO 
DISCRETE DAMPING RATIO 
_100L--------L------~--------~~----~ 
o 200 400 800 800 
TIME, msee 
Figur e 9 - Locus of Fi delity Measures fo r DC- 8 Aircraft in Approach Condition. 
9 
- -~~-----~ 
1. Report No. 2. Government Accession No. 
NASA TM-84354 
4. Title and Subtitle 
INFLUENCE OF SAMPLING RATE ON THE CALCULATED FIDELITY OF AN 
AIRCRAFT SIMULATION 
7. Author(s) 
James C. Howard 
9. Performing Organization Name and Address 
NASA Ames Research Center 
Moffe tt Field , Calif. 94035 
12. Sponsori ng Agency Name and Address 
National Aeronau tics and Space Administration 
Washing t on, D. C. 20546 
15. Su pplementary Notes 
3. Recipient's Catalog No. 
5 . Report Date 
April 1983 
6. Performing Organization Code 
8. Performing Organization Report No. 
A-9293 
10. Work Unit No. 
T- 523 2 
11 . Contract or Grant No . 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
505- 35-31 
Presented at the ISA 14th Annual Modeling and Simulation Conference, April 21-22, 1983 , 
Pittsburg, Mass. Point of Contact: James C. Howard, NASA Ames Research Center, MS 243-9, 
Moffett Field , Calif. 94035 (415) 965-5165 or FTS 448- 5165 . 
16. Abstract 
One of the factors that influences the fide lity of an aircraft digital simulation is the 
sampling rate . As the sampling rate is increased, the calculated response of the discrete 
representa t ion tends to coincide with the response of the corresponding continuous system . 
Because of computer limitations , however, the sampling rate canno t be increased indefinitely . 
Moreover, real-time simulation requirements demand that a finite sampling r ate be adopted. 
In view of these restrict ions, a study was undertaken t o determine the influence of sampling 
rate on the response characteristics of a simulated aircraf t describing short-period oscil-
lations . Changes in the calculat ed response characterist ic s of the simulated air c r af t 
degrade the fidel it y of the simulation. I n the presen t context, fidel i ty degrad ation is 
defined as the percentage change in those charac t eristics that have the gr eatest influence 
on pilot op inion: short- period frequency w, s hort-period damping ra tio ~ , and the pr odu c t 
~w. To determine the influence of the sampling period on these characteristics, the equa -
tions describing the response of a DC-8 aircraf t t o elevator- control inputs were used . The 
results indicate that if the sampling period is t oo large, the fidelity of the simulation 
can be degraded . For example , it was determined that the motion characteristics of the simu-
lated aircraft, represented by ~ and ~w , as well as the corresponding f ideli t y, decreased 
linearly with sampling period and vanished when the period reached 640 ms . 
17. Key Words (Suggested by Author(s)) 
Simulation 
Sampling rate 
Simulation fidel ity 
19. Security Classif. (of this report) 
Unclass if ied 
18. Distribution Statement 
Unl imited 
20. Securi ty Classif . (of this page) 
Unclass if ied 
Subject Category - 01 
21 . No. of Pages 
12 
"For sale by the National Technical Information Service, Springfield, Virginia 22161 
22. Price" 
A02 
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Influence of sampling rate on the calculated fidelity of an aircraft simulation

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