Research on flight control systems is reviewed. Investigations were conducted to evaluate and test a recent method of calculation of the performance index, application to various example designs, and programming for the discrete data control system case

Whitaker, H. P.

English

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ANNUAL PROGRESS REPORT
NASA GRANT NGL 22 - 009--548 4
"Flight Control Systems Research"
1 September 1974 to 31 August 1975
Principal Investigator: Professor H. Philip Whitaker
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Aeronautics and Astronautics
Measurement Systems Laboratory
Cambridge, Massachusetts 02139
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ACXNOWLEDGEMENT
This report was prepared under DSR Project 73598 sponsored
by the NASA-Edwards Flight Research Center, through National
Aeronautics and Space Administration grant number NGL-22-009-548.
The publication of th'_s report does not constitute approval
by the NASA of the findings or conclusions contained herein. It
is published only for the exchange and stimulation of ideas.
Annual Progress Report
NASA Grant NGL 22-')09-548
Prepared by
H. P. White.ker
During this reporting period under the step funding provision
of the Grant, manpower was reduced and rea:hed a level of 25% nf
the principal investigator in February 1975. There has been a
corresponding reduction in output. The main emphasis of the
activity has been placed upon evaluating and testing the most re-
cent method of calculation of the performance index, application
to various example designs, and programming for the discrete data
control system case. The speciric items are summarized in the
paragraphs that follow. The detail descriptions are to be con-
tained in a Contractor Report, the writing of which will form a
major part of the activity under the Grant in the coming year.
1. Computation of the performance index:
Due to the inaccuracies that were introduced by digital word
length round-off and by the matrix inversion routines, an alter-
native method of computing the performance index was developed.
This involves the direct integration of the performance index
1
and makes use of the partial fraction expansion of the Laplace
transform of the excitation input to the error state. This
technique requires accurate knowledge of the poles and zeroes
of that transform so that accurate values o:' residues can be
obtained	 This in turn madem d'-it necessary  to spend effort o f	 ,
fying that portion of the program, and finally a program package
for finding the eigenvalues of matrices prepared at the Argonne
Laboratory was adopted. In th ,_ examples investigated to this
date, the performance index calculation has exhibited acceptable
accuracy. A brief development of the new computational techni-
que is presented in Appendix A.
2. Corresponding changes were made to the computer program for
the discrete design case. That program has not been tried out
to the extent that the program for the continous systems has.
There remain several areas that need to be modified. It is not
anticipated that these involve major programming efforts, but
rather involve changes that are needed to expedite the manner in
which data is presented in the input description. Since both con-
tinuous and discrete sub-systems need to receive input data
description, the input requirements should be simple and easy to
implement to avoid input data errors.
The discrete program has been checked out Using the same
configuration that was evolved for an analog C O' system. Figure 1
presents a mathematical block diagram for the system. The air-
plane and elevator servo are analog, and the rest of the control
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Fig, 2, Step C* Command Response
system configuration is provided by the digital section. The
C* error signal is formed and fed to the elevator. Inner loop
feedbacks of pitch rate, 6, and normal acceleration, a z , are
provided to give a closed loop static sensitivity of unity.
Figure 2 presents the step function C* response. The optimiza-
tion of the continuous systent indicated that the pitch rate feed-
hack sensitivity ratio was insensitive, and for this preliminary
check of the digital program that gain was kept constant. The
variable parameters were chosen to be the forward loop static
sensitivity, Sc,C*63, the acceleration feedback, S c[az,6] , and
the integrator sensitivity, S i . The model -esponse shown in
Figure 2 is that of the discrete model which differed from the
continuous model by less than 58 due to the difference between
an analog and digital representation. The model corresponds to
a second order lag with a)n = 9.5 rad/sec and ^ = 0.5. Table 1
summarizes the parameter values that resul+.A . The optimization
was terminated
Table 1
Parameter	 Analoo System
S c(C*,6] (Deg/g)
	
2.02
Sc(a,,6](Deg/g)	 -0.89
S.	 sec-1	0.5041
when the parameters become insensitive.
Discrete System
1.84
-1.?3
0.504
It may be that even
closer agreement would be found at the mathematical optimum.
3
i	 1
Figure 2 indicates that such variation is insignificant however
in a practical design.
3. Search algorithm: The algorithm for finding the minimum of
the performance index is a straightforward gradient search. Some
control of step size is incorporated, and effort was expended in
improving the convergence characteristics as more design examples
have been investigated. No algorithm seems to be optimum for
all cases, but the present version appears to be a practical com-
promise. Appendix B presents a summary of the algorithm.
4. Lateral control system studies: As an example of a more com-
plicated control system configuration, an investi gation of a
rudder coordination system for the F-8 was undertaken. One can
select several different indications of desired coordination:
(a) the minimization of the excitation of the lateral oscillation
for aileron inputs, (b) minimizing sideslip by causing the yaw
stability axis component of angular velocity to be proportional
to roll angle during the entry into a constant bank angle turn,
or (c) minimizing the average squared sideslip during turn entries.
Cases (a) and (b) involve different model choices, and case (c)
involves a slightly different performance index. Since the pre-
sently used version of the performance index computation evaluates
an expression of the form
( p I) = J [i (t) ] 2dt
4
where the Laplace transform of i(t) is known in pole-zero form,
the use of
(PI) a . IIG(t)I?dt
is a minor modification requiring only that S rather than i be
taken as the output quantity. Mien case (b) was examined, the
closed-loop zeroes of the yaw rate transfer function apparently
cause a large initial transient effect that leads to numerically
	
i
high values of the minimum performance index. This seems to be
unsatisfactory and needs to be investigated more thoroughly.
Otherwise all three examples gave satisfactory results.
Planning
The next year of this Grant will be devoted to writing the
final report and to more thorouyh examination of the discrete
design case. During the latter, emphasis will be placed upon the
effect of sampling frequency, and preliminary work on this has
already begun.
5
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Appendix A. Computation of the Performance Index
Reference:
Palsson, T., Parame:.er Uncertainties in Control System
Design, Measurement Systems Laboratory Report TE -46,
M.I.T., May 1971.
Section 3.5 of the referenced report shows that the Model
Performance Index is directly equivalent to
(PI) = J (i (t) ] 2dt
where i(t) is the excitation input to the dynamic model of the
error states. Referring to figure A-1, the error state can be
obtained in two ways, giving the two expressions
Ly = y - ym = (G - Gm ) u
Solving for i
G n+k
i -	 G - 1 •u,	 i(p) = E Ri/(p-pi)
m	 i=1
where pi are the (n + k) poles obtained from the system poles
and the model zeroes; and R  are the corresponding residues.
•	 a n+k n+k
(PI) = I E	 E R,epitR epjt
	
°j-1 i-1
	 j
n+k n+k R. R.
(PI) 	 E	 I: - 1
-j-1 i-1 pi+pj
Real (p i ) < 0
Thus by obtaining the partial fraction expansion of i(p)
one can evaluate the performance index explicitly. This also
requires having the eigenvalues of the augmented system.
A-2
l	 i	 l	 i	 l
Appendix B. Currently Used Search Algorithm for the Parameter
Optimization Program 7 August 1975
This memo surmnarizes the algorithm used to search for the
minimum of the performance index as of the modification made up
to 22 July 1975.
1. The changes made to the parameters at the end of an
iteration are proportional to the gradient of the performance
index. The proportionality factor, SGRD, is initially taken to
be 10.0. The predicted change to the performance index is cal-
culated using the parameter increments, and a correction factor,
CORR, is calculated which is the ratio of the desired fractional
change in the performance index to the predicted fractional
change. In effect the proportionality factor is then corrected
by multiplying by CORR. If the first order prediction were
accurate, the actual fractional change in the performance index
on the next iteration would be equal to the desired change.
2. If a parameter has reached a point at which its gradient
changes sign #
 its contribution to the reduction of the perfor-
mance index is reduced by the step factor, STF, wn,^ h is generally
equal to or less than one. The gradient step change is multiplied
by STF. When a parameter's gradient changes sign, its step fac-
tor is reduced to 0.6 of its previous value. Thus if a parameter
is alternately stepping across its optimum value, its permitted
change decreases rapidly. To allow for the case for which the
B-1
0i
gradient changes sign as a result of other parameters reaching a
more favorable point in parameter space, the step factors are
allowed to recover by increasing them by 15^ when success ­ .,,.
iterations result in the same sign of the gradient. The increase
is not applied if the step factor is 1.0 or greater.
3. When the gradients are small, the predicted change in
parameters needed to obtain a desired change in the performance
index may be large. The correction factor is further modified
so that no parameter will change by more than 200% in any given
iteration.
4. After the resulting parameter changes have been cal-
cula	 . the parameters are updated, and any parameter con-
straints applied.
5. The search algorithm also provides for automatic adjust-
ment of the called for change in the performance index on any
iteration. This takes place by modifying the parameter DELTA
whose initial value is read as part of the input data. DELTA is
the desired fractional change in (PI) in a given iteration. Since
(PI) seldom varies linearly with a parameter value, the linear
prediction is good only far from the optimum point where the slope
changes slowly. By modifying DELTA one can slow the parameter
change as the minimum is approached and cut down the tendency to
overshoot the minimum. The success ratio, SP, is defined as the
ratio of the actual change in (PI) to the predicted change.
Typically a success ratio of 0.5 is good. The program increments
B-2
!^	 I	 I	 I	 I
DELTA by ±20% whe.iever SR is outside the range 0.4 to 0.8. This
is done in the OPT1 subroutine.
6. When the minimum is approached, large changes in (PI)
are not possible. If one has specified a value other than zero for
the reference (PI) value, DELTA is further changed so that the
called for change in (PI) would not be greater than one that
changes (PI) L-) the reference value. This again prevents excessive
parameter changes near the optimum point. It may not be necessary
however.
7. Further .logic affectii,g the search is provided in the OPT1
subroutine. If the parameter change causes the system to become
unstable, all parameter changes are decreased tc, 40% and the itera-
tion re-started. If tr parameter changes cause the performance
index to increase over the value on the previous iteration, the
changes are reduced by 70%. If this still results in increased
(PI), the maximum parameter change is reduced to 5%. If this still
results in passing the minimum, the search terminates since a 58
change in the parameters should be within practical tolerance
bounds on the design parameters. One is cautioned however that if
one parameter (or more perhaps) is very insensitive in this region
of parameter space at which the search terminates, a larqe change
in the insensitive parameter may take the process to a region at
which further optimization of the other parameter can result in a
significantly lower value of the (PI). The current program leaves
it to the engineer to use his judgement to restart the parameter
B-3
if he feels Further optimization is necessary rather than letting
the gradient search proceed in very small increments using ex-
cessive computation time.
B. Various stopping conditions are used in the OPT1 sub-
routine. One can specify (PIREF) in the input data as a value of
(PI) that is acceptably small and stop the search wht it is
reached. If the change in (PI) is lesE than 5% on three neccessive
iterations, the search terminates on the assumption that the DELTA
modifications are indicating that the minimum point has been
reached. It also terminates on reaching the specified maximum
number of iterations. This can be used to calculate (PI) for
given parameters values by inputting ITMAX = 0. If the normalized
slope is less than 10% of the specified reference (PI), it would
take a ten-fold parameter change to make a change in (PI) equal to
the reference (PI). That is an indication of acceptably small
gradients and hence that the minimum has been found, so the search
is terminated.
I3-4


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