A nonlinear feedback control law for the F/A-18 airplane that provides time-optimal or agile maneuvering of the half-loop maneuver at high angles of attack is given. The feedback control law was developed using the mathematical approach of singular perturbations, in which the control devices considered were conventional aerodynamic control surfaces and thrusting. The derived nonlinear control law was used to simulate F/A-18 half-loop maneuvers. The simulated results at Mach 0.6 and 0.9 compared well with pilot simulations conducted at NASA

Stalford, Harold

English

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FINAL TECHNICAL REPORT 
HIGH ALPHA FEEDBACK CONTROL FOR AGILE HALF-LOOP 
MANEUVERS OF THE F-18 AIRPLANE 
NASA GRANT NAG- 1-873 
APRIL-DECEMBER 1988 
Submitted to 
P. Douglas Arbuckle 
(804) 865-2132 
Mail Stop 489 
GCD-Analytical Methods Branch 
NASA Langley Research Center 
Hampton, VA 23665-5225 
Harold Stalford 
(703) 961-6740 
Aerospace Engineering 
Virginia Polytechnic Institute and State University 
Blacksburg, Virginia 24061 
(NASA-CR-181300) HJCH A L F H A  € € E L L A C K  N89-15106 CClDTPCL €OB A C l L E  HALF-LOOF L L b E C V E B S  OP 9HE 
P-18 AIlrELABIB Final T e c h n i c a l  Beport, Apr- - 
oec. 198E [Virgiria PolytecEnac hst .  and Unclas 
s t a t e  u n i v - )  I C  F CSCL O l C  63/05 0183290 
https://ntrs.nasa.gov/search.jsp?R=19890005735 2020-03-20T04:52:23+00:00Z
HIGH ALPHA FEEDBACK CONTROL FOR AGILE HALF-LOOP 
MANEUVER OF THE F-18 AIRPLANE 
Harold Stalford 
(703) 961-6740 
Aerospace Engineering 
Virginia Polytechnic Institute and State University 
Blacksburg, Virginia 24061 
ABSTRACT 
We developed a nonlinear feedback control law for the F/A- 18 airplane that provide time-optimal 
or agile maneuvering of the half-loop maneuver at high angles of attack. Our feedback control law 
was developed using the mathematical approach of singular perturbations, in which the control 
devices considered were conventional aerodynamic control surfaces and thrusting. The derived 
nonlinear control law was used to simulate F/A-18 half-loop maneuvers. The simulated results at 
Mach 0.6 and 0.9 compared well with pilot simulations conducted at NASA. 
I. INTRODUCTION 
Modem high performance combat aircraft are being designed to provide an increased level of dy- 
namic maneuverability-supermaneuverability . The performance requirements for such aircraft call 
for capabilities to: 
(1) rapidly accelerate or decelerate 
(2) turn tightly and quickly 
(3) change maneuver conditions rapidly 
These require high levels of a&ty and maneuverability at high angles of attack (alpha). High 
performance aircraft in high alpha maneuvers experience signtficant nonlinearities and undergo large 
roll, pitch and yaw rates in performing combat maneuvers in least time. Aircraft and pilot limits 
are easily exceeded. The resulting impact is the need for automated nonlinear feedback control to 
provide combat maneuvers with optimal performance within the limitations of aircraft and pilot. 
We seek the development of such control laws for the F/A-18 airplane in its performance of some 
basic maneuvers at high alpha: half-loop, split-S, Herbst's, level turn, 360" roll, unloading g's, etc. 
Our approach is a four-step procedure in the development of a nonlinear feedback control law for 
a combat maneuver. It consists of (1) singular perturbation analysis, (2) Pontryagin's maximum 
principle analysis, (3) closed-loop synthesis of optimal open-loop control and Fliess' nonlinear 
feedback regulation concepts and (4) simulation analysis. 
Our research procedure, therfore, consists of the following goals and approaches. 
A. Singular Perturbation Analysis 
Goal: To construct automatic nonlinear feedback control laws for agile maneuvering of 
the F/A-18 airplane at high alpha. 
Approach: Apply singular perturbation techniques to derive outer and transition layers of 
nonlinear feedback control laws. 
1 
B. Pontryagin’s Maximum Principle Analysis 
Goal: To generate a family of open-loop controls for optimal agile (i.e., minimum time) 
maneuvering of the F/A- 18 airplane at high alpha. 
Approach: Apply Pontryagin’s maximum principle and singular arc conditions of optimal 
control theory to obtain candidates for optimal open-loop controls. 
C. Closed-Loop Synthesis 
Goal: To construct nonlinear feedback laws for optimal agile maneuvering of the 
F/A- 18 airplane at high alpha. 
Synthesize the family of open-loop controls obtained using Pontryagin’s 
maximum principle into a nonlinear feedback control law and employ Fliess’ 
nonlinear feedback regulation concepts to construct nonlinear feedback control. 
Approach: 
D. Simulation and Comparison Analysis 
Goal: 
Approach: Simulate combat maneuvers using feedback control laws and compare the 
To establish relative merits of feedback control laws 
performance of the optimal feedback control law against that of the ”bench mark” 
singular perturbation technique. 
Singular perturbation analysis provides a direct method for constructing a nonlinear feedback con- 
trol law. Fast and slow variables together with equilbrium equations for the fast variables permit 
construction of an ”outer layer” feedback control law in terms of the slow variables. Transition 
feedback control is derived to transfer the state to and from the outer layer. This approach while 
not always providing a feedback control that is globally optimal it does give relative simple non- 
linear expressions that yield insight into optimal maneuvering and that serve as a bench mark in the 
evaluation of optimal open-loop controls. 
Pontryagin’s maximum principle provides an indirect method for constructing a nonlinear feedback 
control law. Since this principle is a necessary condition to be met by an optimal open-loop control 
2 
additional conditions are required to establish optimality. The maximum principle together with 
transversality conditions yield a two-point boundary valued problem for generating candidate op- 
timal open-loop controls. By varying the initial and final states (e.g., Mach number, altitude) this 
technique provides a family of open-loop controls that span a region of the state space. 
The parametrized family of open-loop controls obtained by the maximum principle analysis is to 
be synthesized into a closed-loop control law. The adjoint state variables and the switching func- 
tion of the parametrized family are utilized to construct a closed-loop control. The final step is to 
implement the nonlinear feedback control laws into computer code, to simulate state time histories 
and to compare the responses. 
We have carried out parts of the above four-step procedure in the development of nonlinear feed- 
back control laws for the half-loop maneuver of the F/A- 18. Our present research work to derive 
nonlinear feedback control for high angle of attack maneuvers started with an application to the 
T-2C aircraft, Refs. 1 and 2. Therein, using the singular perturbation technique, Refs. 7-14, and 
the maximum principle, Ref.15, together with numerical optimization software, Refs. 16 and 17, 
we derived nonlinear control laws for a constant speed model of the T-2C aircraft to perform a 
pitch-up maneuver. Since then our research work has focused on applying the above four-step 
procedure to the F/A- 18 airplane. 
11. RECENT F-18 RESEARCH RESULTS 
The application proposed in Ref. 3 of the above described four-step procedure to the F/A- 18 fighter 
airplane for the half-loop maneuver is given partially in Refs. 4,5 and 18. A nonlinear feedback 
control law for agile half-loop maneuvers of the F-18 is derived in Ref. 4 using the singular per- 
turbation technique. The derived control rotates the velocity vector (ie., the flight path angle) 
through 180" at a maximum equilibrium pitch rate with a nearly constant 38" angle of attack (i.e., 
near maximum lift conditions). The half-loop simulations for entry Mach numbers 0.6 and 0.9 at 
15,000 feet altitude, Refs. 4 and 5, yield 9 seconds and 13 seconds, respectively, for the longitudinal 
3 
portion of the half-loop maneuver. Those results compare well with NASA's pilot simulated 
half-loop maneuvers which gave 15 seconds and 22 seconds, respectively, for the same entry con- 
ditions. 
The nonlinear control law derived in Ref. 4 is unconstrained with respect to pilot load factors. That 
is, such load factors were not enforced in its development. But pilot load factors were computed 
in the executions of the derived nonlinear control law to determine the regions in which such pilot 
limitations may be violated. The normal load factor exceeded 7.5 g's only in the first second of the 
0.9 Mach entry case. At such Mach numbers the control could be limited to prevent violations of 
a prescribed maximum load factor. The modified control would add at most another second onto 
the execution time of the half-loop maneuver. 
The derived nonlinear control law does produce a relatively fast half-loop maneuver that may be 
useful in practice. Also, it provides a good baseline for half-loop maneuver studies of the F/A-18 
airplane. It is particularly useful as a bench mark for time optimal studies. The derived law is 
feedback control and is nonlinear. It has been coded into a subroutine module of table look-up 
values, Ref. 4. We have just recently reduced the table of look-up values to three simple 
polynomial equations. The derived control law consists of three parts. The first and third portions 
of the nonlinear feedback control law are transitions solutions which minimize the time to bring the 
aircraft from the initial trim values to the outer layer (;.e., second portion) and from the outer layer 
to the final values of the half-loop maneuver. The stabilator angle feedback control is bang-bang 
in these first and third portions. 
The second portion is an outer layer solution of singular perturbation theory in which the stabilator 
angle is a nonlinear function of Mach number, fight path angle and altitude. The angle of attack 
is nearly constant at the stall maximum lift value (i.e.,36" to 38" ). The pitch rate , which is a 
function of Mach number, fight path angle and altitude, is the value that maximizes the rate of 
change of the fight path angle while providing equilibrium to the angle of attack and the pitch rate 
differential equations of motion. This second portion of the control law is defined by surfaces (Le., 
4 
table look-up values) for the outer layer values of angle of attack, pitch rate and stabilator angle. 
Three polynomial equations are used by the feedback subroutine to determine the outer layer values 
for Mach number, flight path angle and altitude states. The data covers angle of attack from 0" to 
90", Mach numbers from 0.05 to 0.9, flight path angles from 0" to 180" and altitudes from sea level 
to 60,000 feet. The feedback subroutine uses the three polynomial equations to compute the outer 
layer values of the feedback control law value of the stabilator angle for the outer layer. 
111. F-18 RESEARCH WORK IN PROGRESS 
The nonlinear feedback control law derived using singular perturbation analysis, while not being 
necessarily optimal, provides a baseline for time optimal half-loop maneuver studies. We are cur- 
rently using the maximum principle approach to generate candidate solutions for time optimal 
half-loop maneuvers of the F- 18 airplane, Ref. 18. We are also using the maximum principle to 
pitch-up the F-18 airplane to 90" angle of atttack in minimum time for the Herbst's maneuver. 
Furthermore, we are using the singular perturbation technique to develop a nonlinear feedback 
control law for the split4 maneuver of the F-18 airplane. This work will be completed and docu- 
mented over the next six months. Specifrcally, we are addressing the following problems: 
PROB 1. Use the maximum principle to derive time optimal half-loop maneuvers and compare 
with singular perturbation control law time responses. 
PROB 2. Use singular perturbation analysis to develop a nonlinear feedback control law for the 
split-S maneuver. 
PROB 3. Use the maximum principle to optimize the pitch-up to 90" angle of attack for the 
Herbst's maneuver. 
5 
VI. REFERENCES 
1. Stalford,H. and Hoffman, E., "On minimum time to point, maneuver, and shoot: Singular 
perturbation feedback law in head-on-pass engagement," 16th Congress of the International 
Council of the Aeronautical Sciences (ICAS), Jerusalem, Israel, August 28-September 2, 1988. 
2. Hoffman,E. and Stalford,E., "On minimum time to point, maneuver and shoot: Maximum 
principle solution in head-on-pass engagement," AIAA Guidance, Navigation and Control Con- 
ference, Minneapolis, MN, August 15-17, 1988. 
3. 
proposal funded under NASA Grant NAG-1-873, April 1988. 
Stalford,H., "High alpha feedback control for agile half-loop maneuver of the F- 18 airplane," 
4. Garrett Jr.,F.E., "Fast half-loop maneuvers for the F/A-18 fighter aircraft using a singular 
perturbation feedback control law," M.S. thesis, Aerospace Engineering Department, Virginia 
Polytechnic Institute and State University, Blacksburg, Virginia, September, 1988. 
5.  Garrett Jr.,F.E. and Stalford,H.L., "Fast half-loop maneuvers for an high alpha fighter air- 
craft using a singular perturbation feedback control law," AIAA 27th Aerospace Sciences Meeting, 
Reno, Nevade, January 9-11, 1988. 
6. 
tember 1987. 
Arbuckle,P.D. and Buttrill,C.S., "NASA F/A-18 fighter escort model, computer code, Sep- 
7. Kelley,H.J., "Flight-path optimization with multiple time scales," Journal of Aircraft, Vol. 
8, April 1971, pp.238-240. 
8. Kelley,H.J., "Aircraft maneuver optimization by reduced-order approximation," Control and 
Dynamic systems, Vol. 10, edited by C.T. Leondes, Academic Press, New York, 1973, pp.131-178. 
6 
9. 
Transactions on Automatic Control, Vol. AC-21, No.3, 1976, pp.345-353. 
Calise,A.J., "Singular perturbation methods for variational problems in aircraft flight," IEEE 
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Singular Perturbations in Systems and Control, edited by M.D. Ardema, Springer Verlag, Vienna, 
1982, pp.1-92. 
1 1 .  
Journal of Guidance, Control and Dynamics, Vo1.18, No.2, March-April 1985, pp.225-278. 
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12. 
Automatica,Vol. 16, pp.99- 104, 1980. 
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13. Ardema,M.D., "Singular perturbations and the sounding rocket," JACC, 1979. 
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Keller,H.B., "Numerical solution of two point boundary value problems," J.W. Arrowsmith 
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7 
18. Stalford,H. and Hoffman E., ”Maximum principle solutions for the half-loop maneuver of 
an high alpha fighter aircraft,” to be presented at the 1989 American Control Conference, 
Pittsburgh, Penn, June 21-23, 1989. 
8 


High alpha feedback control for agile half-loop maneuvers of the F-18 airplane

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