LIDS-P-1932
NOVEMBER 1989
Application of Modern Control Theory on
a Model of the X29 Aircraft
Munther A. Dahleh
David M. Richards
October 22, 1989
Abstract
In this report, two different control strategies for a model of the
X29 are presented. The first is based on the 11 design methodology
and the second is based on the HE design methodology. The purpose
here is to show that these methodologies make the design process
quite systematic, and that the resulting controllers conform with our
intiuition.
1
X29 Model
The model for the X29 was taken directly out of [8]. A diagram of a
forward swept wing aircraft is shown in Fig. 1. The model describes
the use of the canard and flaperon to control the pitch angle and angle
of attack. The pitch angle 8 is the angle of the nose of the aircraft
with respect to horizontal, and the angle of attack a is the angle of
the nose with respect to the direction of the aircraft's velocity. The
flight path angle -7 is 0.- a and is the angle of the velocity with respect
to horizontal. These quantities are all shown in Fig. 2.
Three different systems are presented in [8]. The first has the
canard deflection as input and the angle of attack as output. The
second has pitch attitude as output, and the third system uses both
the canard and the flaperon to control the pitch attitude and the angle
of attack. In this work, only the first case is treated.
--
--
I
-----------~--~
- ----------- 1
~"I~-~-~~---
~
----
t/ 1 _____
canard
flaperon
stFake
-
on
a
Figure 1: Diagram of the X29 Control Surfaces
2
plane axis
ity
0
r
horizon
Figure 2: Pitch, Flight Path, Angle of Attack
2
11 Control Design
The challenge of the control design in the case of the X29 is to meet the
various design specifications. For the first design case with the canard
controlling the angle of attack, the specifications are that 10% error
or less is required in command following for 0.01 < w < 1 rad/sec,
disturbances must be attenuated by at least a factor of 1.5 for w < 1
rad/sec, and the system must be robust in the face of multiplicative
errors. Multiplicative errors are introduced from ignoring the high
frequency wing torsion mode and from the scaling.
A final objective is to avoid exciting the wing bending mode at
about 60 rad/sec. To accomplish this, it is desired to have the system
bandwidth less than 60 rad/sec. This will cause attenuation of any
excitation of the bending mode. In the design, this spec is approximated by requiring the crossover frequency of GK to be below 60
rad/sec.
All together, these specs require that the sensitivity be small and
that the closed loop transfer function rolls off sufficiently at high
frequency to guarantee robust performance. We will employ the l1
methodology to achieve this. For details on the 11 problem consult
(1,3,4,5,6,9,101. To meet these requirements, a good strategy is to
minimize the sensitivity function, particularly in the range w < 1.
With no weighting of the sensitivity, the design specifications were
not quite met. The results are shown in Figs. 3-4. The sensitivity
was sufficiently small in the appropriate range at low frequencies, but
the bandwidth of the system was too high and did not satisfy the
3
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Figure 3: Sensitivity
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Figure 5: Optimal Sensitivity With Weight 1+
robustness requirements.
Infact the requirements were not met even with several weights
which were used, an example of which is depicted in Fig. 5. The
weighting was the discretized equivalent of 1 Compared to Fig. 3,
Fig. 5 demonstrates the effect of the higher weighting at low frequency.
The low frequency sensitivity has been decreased at the expense of the
high frequency sensitivity as expected.
After using several weighting functions, it was determined that
the requirements could not be met (or would be difficult to meet)
using only the sensitivity function. The next step was to use the
complementary sensitivity GK(I - GK)- 1 in the objective function.
The sum of the sensitivity and the complementary sensitivity wras used
as the objective.
The results of the new optimization are shown in Figs. 6-8. The
low frequency sensitivity has increased slightly and GC crosses the
0db point at about 40 rad/sec. This new system meets the design
specs. Various weights were used to try to tune the system better,
5
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Complelentary Sensitivity·~r~· ~ prde
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lts
wereII found11 wich
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specif·icain cnieeie
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problem~1···
directly. Bas··icly
wereuie hesnstvy
to be small
low~~~··· < w frequency·
(0··_<
1),adtecmlmnaysniii ob
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at···
hih
frequency
Afesevera iteains h wih o
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Figure
8:
--~~~··
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]····
·
11111
I···· Complelentary
r~l( z 1111
l)2 'Sensitivity
I Lt·
To-us ~ereetI algorihm(·ha~ involvesoving ~w icai q
t none
i
ns[],i
necessary which
I(·o· produced
include 11he··
control
~rnseresults.
fncin n
but
wereisfound
improved
objective.final
Hence
obetieis~
"he muchinmze~e~nomo
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Control Design
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I ~·I
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i s s m a1·· l l · I nubr hersingcoe
lornse uc
CTinspecifications·
ar
hw
nFgrs
-01
eo.I
i
paet~a
are me~
·· .·
· · ~ theI· mixed
To meet the above
specification,
we )I·I
considered
sensitivity
problem directly. Basically, we require the sensitivity to be small at
tionsfrequency
are shown
Figures
9-10-11
below. It is apparent
thatto the
are
esensitivity
low
(0 in
<met.
uw
<~~·specifications
1), and
the complementary
be
small at high frequency. After several iterations, the weight on the
sensitivity function was choosen as
100
( + 1)2'
To use the recent algorithms that involve solving two Riccati equations (71, it is necessary to include the control transfer function in the
objective. Hence the final objective is to minimize the H,, norm of
pKS~
where p is a small number. The resulting closed loop transfer func-
m
a
10
t
U
-IO
r-------------T-~--1-'~I~ii
-----Figure 10: GK
e
a
·
--- ---
,__4~.
0
I
.
,
, ,
11
(
1
III
,
.............
(,
1,I
,I
II,~~~~~~~~~~~~~t'-
--
I
Figure 9: Sensitivity
4~~~~~~~~~0~~~
frequency I in rad/sec
I
i
10
io
a
g
0
e
ri,,
fI~~ r i~~IiequencyI~
.
,
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Figure
,,,,
.
.
1 i
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i Ifrequency in rad/sec
References
(11
M.A. Dahleh and Y. Ohta. "A necessary and sufficient condition
for robust BBO stability," Systems and Control Letters, Vol. 11,
pp. 271-275, 1988.
(21 M.A. Dahleh and J.B. Pearson. a'I optimal feedback controllers
for discrete-time systems,
1986.
[31
Proceedings A CC, Seattle, WA, June
M.A. Dahleh and J.B. Pearson. 'I' optimal feedback controllers
for MMO discrete-time systems," IEEE Trans. A-C, Vol AC-32,
April 1987.
(41
M.A. Dahleh and J.B. Pearson. "Ll optimal feedback compensators for continuous-time systems," IEEE Trans. A-C, Vol AC32, October 1987.
[51
M.A. Dahleh and J.B. Pearson. "Optimal rejection of persistent
disturbances, robust stability and mixed sensitivity minimization," IEEE Trans. Automat. Contr., Vol AC-33, pp. 722-731,
August 1988.
[61 M.A. Dahleh and J.B. Pearson. "Minimization of a regulated
response to a fixed input," IEEE Trans. Automat. Contr., Vol
d
AC-33, pp. 924-930, October 1988.
[71 J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis. "State
space solutions to standard H 2 and H,o control problems," IEEE
Trans. A-C, Vol AC-34, pp. 831-847, August 1989.
[81 W. Quinn. 'Multivariable control of a forward swept wing aircraft,' Laboratory for Information and Decision Systems Report
LIDS-TH-1530, MIT, 1986.
[9] D.M. Richards. '1I-optimal control: Solution software and design ezamples,' M.S. Thesis, LIDS-TH-1906, MIT, Cambridge,
August 1989.
[101 J.S. Shamma and M.A. Dahleh. 'Time varying vs. time invariant
compensation for rejection of persistent bounded disturbances
and robust stability," to appear in IEEE Trans. A-C.
10