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APPLICATION OF NONLINEAR FEEDBACK CONTROL THEORY TO SUPERMANEUVERABLE AIRCRAFT FINAL REPORT SEPTEMBER 21, 1988 TO DECEMBER 21, 1991 NASA GRANT NAG - 1 - 821 PRINCIPAL INVESTIGATORS: DR. WILLIAM L. GARRARD DR. DALE F. ENNS DEPARTMENT OF AEROSPACE ENGINEERING & MECHANICS UNIVERSITY OF MINNESOTA MINNEAPOLIS, MN 55455 NASA TECHNICAL MONITOR DR. BART BACON NASA LANGLEY RESEARCH CENTER HAMPTON, VA 23665 ('A"A-C._-]_'033c,) AopLIC._TI_,,_ ,_;_: ,;G_tLfhFAc. r N92-Z 5_;4o ,:_-'.I-:...'_CK C_.),'._IO::;lThFQ_-y TO '_,L_PLRMA_FUVF-R_PLE J_'c. 19o[ (;'innes:)to ,Jniv.) 10 D APPLICATION OF NONLINEAR FEEDBACK CONTROL THEORY TO SUPERMANEUVERABLE AIRCRAFT INTRODUCTION Controlled flight at extremely high angles of attack, far exceeding the stall angle, and/or at high angular rates is sometimes referred to as supermaneuvering flight. During supermaneuvers, transient angles of attack may reach 90 degrees. Studies have shown that fighter aircraft which have the capability of supermaneuverability may have tactical advantages over aircraft which are not capable of being controlled during high angle of attack maneuvers. The unaugmented flying qualities of aircraft at high angles of attack can be quite different from those at low angles of attack. The dutch-roll mode becomes less well damped and the spiral mode becomes less stable. The natural frequency of the phugoid increases with the low speeds which are typical of high angle of attack operations and the coupling between phugoid and short period longitudinal modes is more pronounced than at lower angles of attack. Effectiveness of the control surfaces is reduced making the aircraft less responsive to pilot inputs. The rudder, which is critical in controlling sideslip and providing yaw damping becomes almost totally ineffective at angles of attack near stall. Modern combat aircraft are usually designed to be statically unstable in the longitudinal mode in order to minimize trim drag. Longitudinal short period instability combined with lightly damped dutch-roll makes it virtually impossible for a pilot to maintain control of the aircraft without a closed loop flight control system to provide stability augmentation. The importance of turn coordination is also increased at high angles of attack since excessive asymmetry in the airflow over the wing can lead to spin. These effects make it difficult to maintain control of the unaugmented aircraft during supermaneuvers, and thus it is necessary to have a flight control system which can maintain predictable dynamic response characteristics throughout the extended flight envelope.
The objective of this study was to examine methods for design of control laws for aircraft performing supermaneuvers. Since the equations which govern the motion of aircraft during supermaneuvers are nonlinear, this study concentrated on nonlinear control law design procedures. The two nonlinear techniques which were considered were Nonlinear Quadratic Regulator (NLQR) theory and nonlinear dynamic inversion. A conventional gain scheduled proportional plus integral ( P + I ) controller was also developed to serve as a baseline design typical of current control laws used in aircraft.
At the time the research was initiated, no data base for an aircraft operating at high angles of attack was available to the investigators, so a mathematical model of a generic supermaneuverable aircraft similar to the X-31A was developed from data obtained from the literature. This aircraft had aileron, rudder, and canard aerodynamic control surfaces and longitudinal and lateral thrust vectoring control (TVC). The longitudinal dynamics of the aircraft were statically unstable. The mathematical model contained nonlinearities due to (1) the aerodynamics, (2) kinematics, and (3) nonlinear inertial coupling.
A detailed computer simulation of the aircraft model was developed. This simulation allowed us to fly proposed supermaneuvers and was used to (1) evaluate the performance of the control law designs and (2) generate linearized models of the aircraft at different flight conditions. The control laws were tested with numerous simulations.
Since the open loop aircraft was statically unstable, it was necessary to design a baseline controller in order to be able to fly various maneuvers using the simulation. This was the P + I PRECEDING PAGE BLANK NOT FILMED control law described above. The P + I control law was developed based on linearized models of the aircraft at various flight conditions ( velocity and angle of attack ). Conventional frequency response methods were used. The details of this design are given in Ref. 1. NLQR theory was applied to the design of longitudinal control law for the aircraft and compared with the P + I controller [2-3]. Similar performance was obtained with both control laws. Lateral control laws were designed with the NLQR technique but the results were no better than those obtainable with Linear Quadratic Regulator theory (LQR). Since the NLQR appeared to yield results which were no better than those obtainable with linear theories, this methodology was not pursued in any further. Nonlinear dynamic inversion was applied to design of both lateral and longitudinal control laws and yielded excellent results [4 - 10]. Most of the research effort concentrated on this methodology.
MATHEMATICAL MODEL The mathematical model used for this study was based on data collected from the literature. It is representative of a modern fighter type aircraft but does not represent any particular aircraft although the general configuration is that of the X-31A. The aircraft is modeled by 6 degree of freedom, nonlinear, rigid body dynamics. The model has 12 states. Six states give the position and velocity of the center of mass in space and six states specify the angular orientation and angular velocity. The wind axis angles of heading, flight path angle, and bank angle about the velocity vector were used instead of the body-axis Euler angles. The standard roll, pitch and yaw rates give the angular velocity, and the translational velocity of the center of mass is given by its magnitude, the angle of attack and the side slip angle. A flat non-rotating earth and uniform gravitational field are used. The standard Euler equations of motion are used to model the rigid body motions of the aircraft. The forces and moments acting on the aircraft are due to gravity (force only), aerodynamics, and the propulsive system. The aircraft aerodynamics were obtained from graphs and tables taken from various sources in the literature and are representative of an aircraft similar to the X-31A. Although in general the aerodynamic coefficients are nonlinear functions of the all of the system states as well as the control deflections, the literature from which the data used in this study were taken modeled the aerodynamic and control coefficients as functions of the angle of attack only. This simplifies the modeling considerably as it allows modeling of the aerodynamic coefficients as polynomial functions of the angle of attack only. The aerodynamic forces are affine in all other variables except velocity which appears quadratically in the dynamic pressure. The aircraft has an unstable longitudinal static margin of 4.65%.
Aerodynamic control moments are provided by ailerons, a single rudder, and canards. Moment analyses of high angle of attack maneuvers reported in the literature indicate that thrust vectoring control (TVC) is necessary for some high angle of attack maneuvers since aerodynamic control surfaces become ineffective at high angles of attack. Lateral TVC is provided to augment the rudder, since the rudder is ineffective for angles of attack greater than 40 degrees. Longitudinal TVC is provided to enhance the canards at high angles of attack. No facility is provided to allow TVC to generate rolling moments.
CONTROL LAW DESIGNS The control laws designed in this study consisted of a maneuver generator which was used to simulate pilot commands, outer loop control laws which were designed to control aircraft attitude angles, and inner loop control laws which were designed to control attitude rates.
Maneuver Generator
The supermaneuverswhich were simulated in this study were obtained from optimization
studiesreportedin the literature. Thesestudieswere basedon a threedegreeof freedom,point
mass modelof the aircraft. The outputof this model is an optimalvelocityvector which results
in a minimumtime maneuver. The outputsof the optimizationstudiesare (1) the magnitudeof
the velocity vector, (2) the flight path angle and (3) the heading angle. Our six degree of freedomsimulationrequiredas inputsimulatedpilot commandswhich consistedof thrust level,
angle of attack,and bank angle aboutthe velocityvector. A methodfor convertingthe optimal
velocityvectorto simulatedpilot commands was requiredbeforecontrol lawscould be tested in
simulatedsupermaneuvers.Thesepilot commandswere generatedby a maneuvergenerator
which solved the inverse problemof computingthe thrust, angle of attack and bank angle
required to producethe rate of change of velocity vector necessary to achieve the optimal
trajectory. The maneuvergeneratoruses three control loops which comparethe commanded
velocitymagnitude,headingangle, andflight path angle from the optimizationstudieswith the
actual values of these variables obtained from the simulation. These differences are are
consideredto be proportional to the desiredaccelerations and are multipliedby feedbackgains The outputsof the gains are the desiredaccelerations which are fed into the inversioncontroller
which uses the point mass equationsof motionof the aircraftto calculatecommandedthrust
level,angleof attackandbankangleaboutthe velocityvector. The maneuvergeneratordoesnot
form any part of the onboard,flight controlsystem of the aircraft but is used simplyto allow
supermaneuvers to be "flown"on the computer. The onboardflight controlsystemsare usedto
produceaccuratetrackingin the angleof attack,bank anglerate aboutthe velocityvector,and
side slip (in this studycommanded side slip is zero).
Gain-Scheduled P + I Controller
A baseline, gain scheduled P + I controller similar to those used on existing aircraft is
implementedfor comparisonwith the other control laws developed. This P + I controller is
designedusinglinear,frequencyresponsetechniques. P + I elementsare usedto providegood
tracking and desensitizationto modelingerrors. Schedulinggains with angle of attack and
dynamic pressure is important because the wide range of operating conditions result in
significantchangesin the aerodynamic propertiesduring supermaneuvers. Desensitization to
modelingerrors is beneficialbecausethe aircraftcannotbe modeledprecisely. The design is
somewhatunconventional in that gains are scheduledwith angle of attack, a rapidly changing
variable,as well as with the moreconventionaldynamicpressure,a slowly changingvariable.
To track a desired trajectory accurately, the aircraft must respond precisely to the pilot
commands for angleof attack,sideslipangleandbankangle. Two outerloopswereimplemented
to controlangleof attackandbank angle. The loops had relativelylow bandwidthsof 1 rad/sec
and 2 rad/sec,respectively. Becausetherewas no trackingrequirement on side slip (it was to
be maintainedat zero), it was regulatedby one of the inner loops. In additionto the outer
loops,there are three inner loops eachhavinga crossover frequencyof 10 rad/sec. Theseare
usedto augmentthe stabilityof the fast dynamicsassociatedwith the longitudinalshort period,
dutch roll and roll subsistencemodes. The inner loops interfacedirectly with the nonlinear
aircraft dynamics and are used to control three regulatedvariables which characterizethe
aircraft motion. One of the inner loops controlsthe unstable longitudinaldynamics and the
other two control the lateral-directionalmodes. The crossoverfrequency for the inner loops
wasselectedas 10 rad/secso thatthe loopgain wasas highas possiblein orderto achievegood
trackingand desensitization but with low enoughgain at higherfrequenciesto allow the effects of unmodeled structuralandactuatordynamics to be neglected.
To allow frequencyresponsemethodsto be applied,a set of linearizedmodelsof the aircraft
were derived at various flight conditions. A linear P + I control law was designedfor each
flight condition. The controllergains for valuesof angleof attackbetweenthe designconditions
were obtainedby linear interpolation. This producedcontrol law gain scheduleswhich were
piece-wise linearfunctions of angleof attack. The gainschedules alsohadan inversedependence
on dynamicpressureto accountfor the reductionof loop gain resultingfrom decreasedcontrol
surfaceeffectiveness at lowspeeds.
The flight control system used the canard to control the longitudinal motions. Thrust was
controlleddirectlyby the maneuvergeneratorandwas not usedas part of the feedbacksystem.
The enginedynamicsare slowand not suitablefor usewithincontrolloops havingbandwidths as
high as 10 rad/sec. If necessaryan outer loop could be added for thrust control. Since
longitudinal TVC was not used, a single regulatedvariable,rv, was defined. The controlof rv
providedthe requiredstabilityand responsecharacteristics.The longitudinalrv consistedof a
blendof low-passed normalacceleration, nz, and speed,V, combinedwith pitch rate,q.
(1)
rv = Kqq + [3/( s + 3 )][nz + Kv(Vc - V)]
The regulated variable was designed to allow accurate control of nz at steady state and provide feedback for gust rejection. It contains a term in q with a constant gain to provide for pitch rate damping. A small angle of attack dependent gain, KV, multiplying the velocity error is used to stabilize the phugoid mode.
Normal acceleration is used in the regulated variable because it is usually one of the primary outputs that the pilot wishes to control. It is directly proportional to the structural loads on the aircraft and can be used to limit these loads to safe levels. Furthermore, normal acceleration gives an immediate measure of aerodynamic loads due to wind gusts, which makes it ideal to use in the gust rejection loop. Finally normal acceleration can be used to estimate angle of attack and can be measured by accelerometers which have a proven record for accuracy and reliability.
This may not be true for direct angle of attack sensors. Pure nz feedback is not desirable because (1) high frequency elastic modes may be excited, (2) a non-minimum phase transfer functions may result, and (3) accelerometers are inherently noisy at high frequencies. These problems can be alleviated by low passing the output of the accelerometer and combining it with a pitch rate signal as shown in Eq. 1. The regulated variable also includes a low passed term in speed error, which is required to stabilize the phugoid mode. The gain on the speed error, K v, is critical and must be scheduled with angle of attack in order to stabilize the phugoid mode. The P + I compensator is implemented with the following transfer function k(s) = [Coc/hrv][(s + 3)Is] (2) The zero in the P + I controller cancels the pole in the low pass filter in Eq. 1, (oc is the desired cross over frequency, and hrv is proportional to dynamic pressure. Selection of the various gains and parameters in the control law defined by Eqs. 1 and 2 must be verified for each design flight condition. Thus the design process can be time consuming.
Lateral-directional controllers are designed in much the same manner as the longitudinal control laws; however, there are three controls, aileron, rudder, and lateral thrust vectoring, and two outputs, roll rate and lateral acceleration. Thus the lateral-directional controller design problem is multiple input multiple output (MIMO), and is more difficult than the longitudinal controller design. A MIMO, P + I control law is developed which requires considerable gain scheduling.
The control laws described above are for the inner loops. Outer loop control laws with lower bandwidths are then formulated for a tracking of angle of attack, bank angle, and side slip commands ( a zero side slip command is assumed ). Outer loop designs are performed using classical frequency response methods.
NLQR Controller NLQR theory is an extension of LQR theory to nonlinear systems. Nonlinear systems equations are used and control laws are obtained from the approximate solution of the Hamilton-Jacobi-Bellman partial differential equation of optimal control theory. The resulting control laws are nonlinear functions of the system state. As the nonlinearities approach zero, the NLQR control laws approach the standard LQR control laws and have the same desirable robustness properties. The NLQR control laws are nonlinear functions of the angle of attack, which is equivalent to scheduling gains with angle of attack.
The NLQR technique was applied to the design of an inner loop longitudinal control law for the mathematical model of the aircraft described above. The control law was designed to track normal acceleration commands by minimizing a quadratic performance index which consisted of the integral of the square of the normal acceleration and the control deflection. The control laws obtained from application of the NLQR methodology were very similar in structure to the gain scheduled P + I control laws and most of the gains resulting from the two methods were close to one another. The NLQR procedure was also applied to the design of lateral-directional control laws.
Nonlinear Dynamic Inversion Control Laws Exact dynamic inversion is based on an intuitively simple idea which can easily be demonstrated by a scalar example. Consider the first order system ;: = f(x) + g(x)u (3) This system can be given any desired dynamics by suitable choice of the control u. For example the stable first order dynamics given by E_i. 4 might be chosen.
(4) ;c = = mc (Xc - x) The required control can then be computed by inverting Eq. 3 Here co c is the desired bandwidth.
to give (5) u = g(x)-l(x d _ f(x)) Substitution of Eq. 5 into Eq. 3 clearly yields the desired dynamics of Eq. 4. The method of dynamic inversion can be extended to higher order systems provided g(x) is invertible. In aircraft control problems, g(x) may be invertible if there are sufficient control effectors; however, there will often be conditions where g(x) is nearly singular. This would result in excessively large commands and saturation of control effectors. The near singularity of g(x) is due to the fact that the control moment effectors produce very small forces and thus provide very little direct control of attitude angles. Thus it is difficult to use dynamic inversion directly for aircraft with more or less standard control effectors. If direct lift and side force effectors are available, dynamic inversion may be directly applicable; however, such effectors are not included in the mathematical model used in this study.
The problems associated with the invertibility of g(x) was overcome by separating the dynamics into fast and slow subsystems each having only three states. The fast subsystem corresponds to the body axis angular rates and the slow subsystem corresponds to the angle of attack, side slip angle, and bank angle about the velocity vector. An exact inner loop inversion was carried out by using the five control effectors, canards, ailerons, rudder, lateral TVC, and longitudinal TVC. Since there were more controls than states it was possible to select the controls in such a way as to minimize the norm of the control vector. The desired inner loop dynamics were P + I with a cross over frequency of 10 rad/sec.
Once the inner loop control law was formulated, an outer loop controller was designed using an approximate inversion. It was assumed that the dynamics of the fast states, the body axis angular rates, were so much faster than the slow states that the fast states reached their steady state values essentially instantaneously and could be used as control inputs for the slow states.
The approximation used in the outer loop inversion neglected the effects of forces produced by control surface deflections. The desired dynamics for the outer loops were also P + I but with cross over frequencies of 2 rad/sec.
In fact the full order rigid body dynamics are decomposed into four subsystems. These are denoted as fast, slow, very slow and extremely slow. The the extremely slow states are the components of the position vector of the center of mass of the aircraft relative to the earth.
These states are controlled by the translational velocity vector. The very slow states are the magnitude and direction of the translational velocity vector. These states are controlled by the pilot or maneuver generator which produces cockpit commands for angle of attack, thrust level, side slip angle and bank angle rate. The purpose of the onboard flight control system is to cause the aircraft to respond to cockpit commands in a desirable manner. This is accomplished by control of the slow states, the attitude angles, and the fast states, the body angular rates, using dynamic inversion as described above.
RESULTS FOR SIMULATED SUPERMANEUVERS A number of maneuvers were flown using a digital simulation which included all of the dynamic, kinematic, and aerodynamic nonlinearities. These maneuvers consisted of minimum time reversals in direction with various final conditions imposed. During these maneuvers all of the state variables underwent very large changes. For example transient angles of attack of over 80 degrees were observed. The P + I gain scheduled controller performed well in maneuvers in which the angular rates were small, however, the performance deteriorated when the aircraft was subjected to high angular rates at high angles of attack. Specifically, sideslip and lateral acceleration were significantly higher than expected. Gyroscopic coupling was found to account for part of this degradation. Significant coupling between roll rate and lateral acceleration was observed at large values of angle of attack.
In a vertical turn in which only longitudinal dynamics were simulated, the NLQR control law was shown to perform about the same as the P + I gain scheduled controller. Since there was no clear performance gain in using the NLQR control law over more conventional designs, this methodology was not considered further in the study.
It was observed that in all of the maneuvers, the response of the gain scheduled controller was more oscillatory in the angular rates than was the dynamic inversion system. This is due to the fact that the inner loops of the dynamic inversion system control the angular rates directly so that the responses of these states accurately track the inputs. Accurate control of side slip is very important in post-stall maneuvering. The dynamic inversion control is more successful in providing precise control of both bank angle rate and side slip angle than the gain scheduled control law, particularly at high angles of attack. This suggests that, in the gain scheduled control law, the directional control afforded by the inner loop controlling the lateral acceleration needs to be improved.