6.1990-3371

AIM-90-3371-CP
Multivariable Control Law for Flat-Turn Strafing Maneuver
by A Supermaneuverable Aircraft
Chien Y. Huang
Coprate Research Center
Grumman Corporation, MS AO8-35
Bethpage, NY 11714
(PIIMF) control law that allows an advanced fighter
aircraft to perform flat-turn sWing maneuver.
Flat-turn strafing is used in terrain-following
scenario to quickly acquire, track, and maybe destroy
targets on the ground. This is a difficult task since it
requires an airplane to yaw from left to right (and back)
without gaining side velocity (i.e., with zero sideslip)
while keeping the nose in level position. Side velocity
(lateral acceleration) can be avoided by rolling the airplane
such that the lift vector is tilted to cancel the side force,
however, doing so leads to rapid banking maneuvers,
which compromises pilot's ability to keep track of the
target. An equally desirable companion maneuver is to be
able to command side-velocity alone without yawing the
airplane.
Implicit model-following technique has been very
successfully applied to the control of aircraft [7] and has
been demonstrated to possess inherent reconfiguration
properties [8]. It also has been shown to have good
robustness against sensor noise and plant disturbance and
reformulated for attitude pointing purpose [9]. Recently
there is study to incorporate the model-following
technique into the HOOrobust control framework [lo].
The organization of this paper is as follows.
Section I1 formulates the control strategy and examines
some of the properties. Additional robustness using loop
transfer recovery is also described. Section I11 details the
target supermaneuverable aircraft while Section IV
presents the simulation results. Comments on additional
control power involved and complexity of control
algorithm are also made. Summary and conclusions are
offered in Section V.
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ABSTRACT
A control law that allows a supermaneuverable aircraft to
perform flat-turn strafing maneuver is described. It is
based on proportional-integral model-following concepts,
which has inherent advantage of quickily obtaining a
design that has satisfactory performance. The control law
is applied to an advanced fighter aircraft and simulation
results show that flat-turn strafing maneuvers are possible
with the use of the model-following control law.
I.
Introduction
Future aircraft are expected to meet higher
performance and combat supeliority requirements, which
has been characterized as supermaneuverability [ 11.
Superagility or supermaneuverabilityindicates an aircraft's
capability to perform maneuvers in post-stall or highangle-of-attack (or high-alpha) flight regimes or in
situations where there is substantial cross-coupling
between the modes. In the case of high-alpha maneuvers,
the problems are loss of control effectiveness and
nonlinear dynamics, which are being addressed by thrust
vectoring and specialized control strategies. In the case of
cross-coupled maneuvers, the problem is absence of
dynamic modes separation, which requires control surfaces
to be simultaneously deflected in an unusual combination
and setting. Performing unconventional maneuvers in
otherwise conventional flight regimes is a difficult task
for a pilot, who is accustomed to aircraft that are
controlled as single-input single-output (SISO)systems.
Moreover, he may be hard-pressed to keep up with the
associated fast-changing dynamics, resulting in a highworkload environment. All these factors point to the need
of control laws that are truly multivariable in nature to
carry out these multi-input multi-output. Yet there is
additional handling quality requirement on the control law
to achieve an apparent "feel" of a SISO system.
Studies on controls for supermaneuverable
aircraft have concentrated mostly in high-alpha vehicles
using strategies such as nonlinear control [2], variable
gain control [3], and robust control using loop-transfer
recovery [41 and H=.optimization techniques [51. They
are relatively few studies on highly dynamic maneuvers,
one being a study of using nonlinear control to rapidly
change attitudes [ 6 ] . The objective of this paper is to
apply proportional-integral implicit model-following
11.
Control Law Formulation
The proportional-integral model-following
(PIIMF) control law is reviewed here. For more details,
refer to [8]. Let the plant be described by the usual (A, B,
C, D) matrices and the model to be described by system
matrix Am, namely
Y
= Ax+Bu
= CX+DU
Xm
= Amxm
X
Presented at 1990 AIAA GNC Conference, Copyright Q 199Oby Grumman Corp. Published by AIAA with permission
499
(3)
Implicit model-following incorporates the model
in the quadratic cost function by weighting the difference
between the actual and model state rates, that is,
The state variable is augmented to include 6,
JT
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= 1/2
[X
T
T
Q x + ~ xM U + U Ruldt
(3
0
T
T
where Q = (A-Am) Qi(A-Am), M = (A-Am) QiB,
and the cost function becomes
T
R = B QiB, and Qi is a weighting matrix.
It can be shown that if B is square and invertible,
then perfect model matching can always be achieved. If
fact, the necessary feedback gain K is a simple algebraic
relation given by
K
= B
CT
J = 1/2
T
T
[AXa Z A X ~ + ~ ASAU+AU
X~
RAuIdt (14)
0
- 1 (A-Am)
weighting on 6. The (linear quadratic) control obtained by
optimizing Eq. 11 is
with this gain, the closed-loop system becomes
Ac = A - B K = Am
0
Typically B is not square; in this case the
feedback giin wili depend on the structure of the control
matrix and the weighting matrix Qi,
Expandingand reatrangingEq. 15, we get
T
-1 T
where BwL = (B QiB) B Q; and is the weighted left
pseudo-inverse of B, while P is the solution of the
Riccati equation associated with optimization of Eq. 5.
Although the PIIMF control law possesses
inherent sensitivity to plant parameter variations, we can
improve its low-frequency disturbance-rejection
characteristicsand provide zero steady-state error command
response by adding proportional-integra1
compensation. Let Yd be the desired equilibfiumoutputs;
then the perturbations around the
POint
(XO, UO) are:
The second term on the right side provides
proportional feedback of the error, while the third term
provides the integral effect being sought. The dimension
of the commanded input yd is chosen to equal the
dimension of u, reflecting the fact that the number of
controllable outputs is equal to the number of the controls
available [12]. Eq. 16 is cast into incremental form in
actual implementation to eliminate trim requirements and
measurement bias.
PIIMF control law as it stands if robust against
parameter variations; however, its robustness against
unmodeled dynamics is not well-known. Recently, the
implicit model-following control law has been adapted to
take incorporate loop-transfer recovery (LTR)[lo]. The
objective of LTR is to force the loop-transfer function
G K , where G(s) = C(sI-A)-lB and K is the feedback
gain (in the forward loop), to a desirable shape. The steps
involved in deriving K(s) are the design of a Kalman-Bucy
filter, which in essence is the desired loop shape, and then
Define output error integral 4 to be
500
CRCA is a generic advanced fighter aircraft intended for,
among other things, studying the effects of damages and
benefits of control reconfiguration. The aircraft is
statically unstable (-12% mean aerodynamic chord) to
achieve good transonic performance and to reduce
supersonic trim drag. Thrust vectoring is available but
not used here. Nine control surfaces exist. There are two
canards with 30Qdihedral angle. The dihedral creates
directional instability, but it provides an additional yaw
power which is useful for control reconfiguration; and for
our case, it allows further performance enhancement in
lateral-directional axis. CRCA also has six trailing-edge
("E) surfaces, which are independently controlled to
achieve both roll and pitch. Finally, a single rudder
provides further directional control.
The linear model used is obtained by perturbation
using nonlinear CRCA model in terrain-following terrain
avoidance (TFI'A) mode (altitude = sea level, Mach
number = 0.9). Only the lateral-directional model is
needed here as we use only anti-symmetric control. The
state vector x of the model consists of sideslip angle (p.
rad), roll rate (p, radsec), yaw rate (r, rad/sec), roll an le
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the design of a Linear Quadratic regulator, which
"recovers" the loop shape. Model-followingLTR will
assume that the model transfer function (Cm(S) = C(s1A m j l B ) , modified by any robustness considerations, is
the desirable loop shape, which if achieved, will naturally
lead to robust model-following. This procedure is briefly
described below.
The first step is to design a Kalman-Bucy filter
(i.e., find the filter gain Kf), so that its transfer function,
-1
Gkf = C(sI-Am) K f , is Gm. The standard LTR
approach calls for finding fictitious noise and disturbance
matrices to make the equality true. Although this process
can be carried out, it is by no means straight forward.
Instead, the easiest way to obtain a filter gain Kf, which
must be equal to B, is to simply assume that it is given.
That is, if augment the plant
=
[;: A",]
5.
(Q,rad), and yaw angle (w,rad); that is, x = [P p r Q w]
The control surfaces (Fig. 1) are left and right canards (Cl,
Cr, deg), left and right trailing-edges-#l (Tell, Telr, deg),
left and right trailing-edge-#2 (Te21, Te2r. deg), and left
and right elevators (El, Er, deg), and rudder (R, deg).
These surfaces are further geared into three groups:
canards, trailing-edge surfaces, and rudder. That is 6 = [Cn
T
Te R] . With these vectors defined, the system and
control matrices are found to be,
and set the KaIman-Bucy filter gain to be
Kf
=
[-I
Then
-0.2538
-66.9342
8.2821
0
0
Gkf = Ca(sI-Ad"Kf
C(SI-Am)-'B
The second step is to "recover" the loop shape
obtained by the formulation of Kalman-Bucy filter via a
linear quadratic regulator. This is done by first setting
T
state weighting Q (e.g., to C aCa> and control weighting
R (typically to 91) and decreasing the conrrol weighting.
As p -+ 0, GK approaches Gm.
The approach outlined above can also be extended
to Hco formulations. Details can be found in [lo].
Although both LTR and Hoo model-following design
methods are viable, their properties are not wellunderstood. They will be further examined in a future
paper. We will confine here to use PIIMF to achieve the
flat-turn maneuvers.
HI.
B
=
[
0.0155
-5.4612
-0.0299
1.OOOO
0
-0.9990 0.0320 0
1.3049
0
-1.2709
0
0.0155
0
0
1.0000
0
0
0.0024 0.0020 0.0036
0.2336 2.0074 0.3737
0.lr 0
.
r
-0.y]
(23)
Note that very little power is available to impart side-force
0)despite canted canards and a large rudder.
The Supermaneuverable Aircraft Model
The proportional-integra1 implicit modelfollowing control law is applied to ControlReconfigurable Combat Aircraft (CRCA) [I31 (Fig. 1).
50 1
IV.
Simulation Results
requires simultaneous control surface movements, which
can only be achieved with a truly multivariable control
law. Proportional-integral model-following approach was
used due to its formulation, which permits satisfactory
performance to be quickly obtained. Simulation results
show that flat-turn strafing maneuvers are possible with
the use of the model-following control law.
Several comments are in order. Simulations
shown that flat-turn strafing maneuvers require
considerable amount of control power. This is true for
our aircraft since both canards and rudder produce sideslip
as by product of commanding yaw and vice versa.
Therefore, more surface deflections are needed to achieve
necessary cancellations. With a different aircraft layout, it
may be possible to reduce considerably the required control
effort (for example, if a pure side-force control or thrust
vectoring is available). This may be a design requirement
for supermaneuverable aircraft. No attempt has been made
to optimize further the design. One area for improvement
is the model itself, which was not derived with modem
aircraft in mind. One may accept less stability for
performance gains. Finally, the control law appears to
work rather well for linear models. It remains to be seen
if this is true when it is tested in a nonlinear simulation.
The model to be followed is given by
Am
=[
-0.7430
-1.oooO 0.0586 0
0
5.8700 0.0400 -0.5070
0
0
1.oooo
0
0
0
0
0
1.Oooo
0
0
0
-1O.oooO -4.oooO 0.8650
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Because there are three inputs, up to three
controls can be specified. The natural choice in this case
is yd = [p 4 yIT, which means the observation matrix is
To achieve an acceptable response, following
weighting matrices are used:
Qi = O.l*I5
Qg
= diag(200,50,10)
(26)
Output responses and control efforts for a
T
command of 0.2 rad (1 1.5 deg) yaw (Le., yd = [O 0 0.21
are shown in Figs. 2 and 3. As can be seen, the control
power required is large. In fact, rudder deflection is limited
to +3@ (while the canards are limited to +60Qand -30*).
To obtain more reasonable control deflections, the
response must be slowed down. This can be accomplished
by changing the integral weighting matrix Qc. In fact, if
Qg is set to be diag(200,50,0.05), then the control effort
will be within the surface limits, as shown in Figs. 4 and
5.
Alternatively, we can command pure sideslip
maneuvers without acquiring yaw and roll. Output
responses for a command of 0.2 rad (11.5 deg) sideslip
T
(i.e., yd = [0.2 0 01 are shown in Figs. 6. In this case,
it is difficult to null out roll and yaw completely.
However, the maximum roll and yaw angles are no larger
than 0.5 degrees. The control effort (not shown) for this
case is also excessive. However, this can be reduced by
lowering the desired sideslip angle (0.2 rad of p for this
flight envelope is roughly equivalent to 220 fvsec).
A final simulation showing a series of flat-turn
maneuvers is shown in Fig. 7. In this scenario, yaw
angles of 0.2 rad is commanded for 5 sec before switching
to -0.2 rad. As shown by the plots, the command
following is quite good.
V.
REFERENCES
1. Herbst, W. B., "Future Fighter Technologies," J. of
Aircraft, Vol. 17, No. 8, Aug. 1980.
2. Garrard, William, Enns, Dale F., Snell, S. Antony,
"Nonlinear Longitudinal Control of a
Supermaneuverable Aircraft," Proc. of ACC,
Pittsburgh, PA, June, 1989.
3. Ostroff, Aaron, "A Variable-Gain Output Feedback
for High-Alpha Control," AIAA Guidance,
Navigation, and Control Conference, Paper 89-3576CP, Boston, MA, Aug., 1989.
4. Looze, Douglas, and Freudenberg, James, "Linear
Quadratic Design for Robust Performance of a Highly
Maneuverable Aircraft," AIAA Guidance,Navigation,
and Control Conference, AIAA paper 89-3457-CP,
Boston, MA, Aug.. 1989.
5. Voulgaris, Petros, and Valavani, Lena, "High
Performance Linear Quadratic and Hoo Designs for a
'Supermaneuverable' Aircraft," AIAA Guidance,
Navigation, and Control Conference)AIAA paper 893456-CP, Boston, MA, Aug., 1989.
6 . Snell, S. Antony, Enns. Dale F., Garrard, William,
"Nonlinear Control of a SupermaneuverableAircraft,"
AIAA Guidance, Navigation, and Control Conference,
AIAA paper 89-3486-CP, Boston, MA, Aug., 1989.
7. Kreindler, E., and Rothschild, D., "Model-Following
in Linear Quadratic Optimization," AIM Journal,
Vol. 14, no. 7, July 1976, pp. 835-842.
8. Huang, Chien.Y., and Stengel, Robert F.,
"Restructurable Control Using Proportional-Integral
Summary and Conclusions
A control law that allows a
supermaneuverable aircraft to perform flat-turn strafing
maneuver is described. The maneuver is difficult as it
502
Iinplicit Model-Following," J . of Guidance. Control,
and Navigarion, Vo1.3, No. 2. March-April, 1990.
9. Iluang. Chien Y.,"A Methodology for KnowleclgeBased Restructurable Control to Accorriinodate
System Failures," P h . D . Thesis. Princeton
Univcrsity, 1989.
10. Iluang, Ctiien Y.,"Application of Robust MotlclFollowing Concepls to Aircraft Control," Proc. of
ACC. San Diego, CA. May, 1990.
11. Asseo. S . . "Applicalion of Optinial Control lo
Perfect Model-Following." J . o/Aircra/r. Vol. 7, No.
- 313.
12. Stengel, R., Stochastic Optimal Conrrol: Theory and
Application. John Wiley & Sons, Inc., New York.
1986.
13. Mcrcadaiite, et at., "Control Recnnfigurable Combal
Aircraft Piloted Simulation Development," Proc. o/
NAECON, Dayton, 011, May, 1988. pp. 512 - 519.
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4, July-Ailg., 1970. pp. 308
Fig. 2 Conlrol Reconfigurable Combal Alrcrafl (CRCA)
503
response to yaw command
12
10
8
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6
4
2
0
-2
(
1
2
3
4
5
6
7
time, sec
Figure 2 Output responses to 0.2 rad yaw command
response to yaw command
6
time, sec
Figure 3 surface response to 0.2 rad yaw command
504
7
response to yaw command
15
10
-
M
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4
+T2
.r(
5-
d
8
0
I
0
1
2
3
4
5
6
7
time, sec
Figure 4 output response to 0.2 rad yaw command with less control power
time, sec
Figure 5 surface response to 0.2 rad yaw command with less control power
505
-
-
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-
time, sec
Figure 6 Output responses to 0.2 rad sideslip command
response to flat-turn yaw strafing maneuver
15
1C
5
3
a
$ 0
c,
.r(
d
8
-5
-10
-15
0
5
15
10
20
time, sec
Figure 7 Output responses to 0.2 flat-turn strafing maneuver
506
25