Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 FLIGHT TEST OF A TRAJECTORY CONTROLLER USING LINEARIZING TRANSFORMATIONS WITH MEASUREMENT FEEDBACK Robert F. Antoniewicz*and Eugene L. Duke* NASA Ames Research Center Dryden Flight Research Facility Edwards, California and P.K.A. Menon** Georgia Institute of Technology Atlanta, Georgia Abstract The design of nonlinear controllers has relied on the use of detailed aerodynamic and engine models that must be associated with the control law in the flight system implementation. Many of these controllers have been applied to vehicle flightpath control problems and have attempted to combine both innerand outer-loop control functions in a single controller. This paper presents an alternate approach to the design of outer-loop controllers. The approach simplifies the outer-loop design problem by separating the inner-loop (stabilization and control) f;om the outerloop (guidance and navigation) functions. Linearizing transformations are applied using measurement feedback to eliminate the need for detailed aircraft models in outer-loop control applications. Also discussed is an implementation of the controller. This implementation was tested on a six-degree-of-freedom F-15 simulation and in flight on an F-15 aircraft. Proof of the concept is provided by flight test data which is presented and discussed. CAS command augmentation sytem CLC control law computer FTTC flight test trajectory control HIDEC highly integrated digital electronic control PCM total automatic flight control system WATR Western Aeronautical Test Range body axis normal accelerometer output x-axis body accelerometer output y-axis body accelerometer output z-axis body accelerometer output total aerodynamic drag error signal gravitational acceleration altitude altitude rate linear control law gains total aerodynamic lift Mach number pulse code modulation Copyright 0 1990 by the American Institute of Aeronautics and .4stronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. T h e U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. TAFCOS an Acronyms augmentor wing jet STOL research aircraft short takeoff and landing Symbols Nomenclature AWJSRA STOL JAA. 518 rn mass P Q body axis roll rate body axis pitch rate 7 body axis yaw rate T t V thrust; linearizing transformations velocity v velocity rate V linear control or pseudocontrol W inverse linearizing transformations XT thrust along the x-body axis Y total sideforce time Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 thrust along the z-body axis measurement form angle of attack proportional control term angle of attack rate plant angle of sideslip reference trajectory angle of sideslip rate function of input vector commanded incremental load factor function of state vector engine orientation angle angle of attack throttle body reference frame Euler roll angle rlCOm commanded throttle initial condition; zero altitude 8 body reference frame Euler pitch angle e Introduction body reference frame Euler pitch angle rate b body reference frame Euler roll angle 4 body reference frame Euler roll angle rate The problem of aircraft trajectory control is generally thought of as an “outer-loop’’problem, that is, the aircraft is assumed to have a stabilizing, “inner-loop” command augmentation system. Often, the inner-loop problem is called the control problem, while the outerloop problem is called the guidance problem. For simple trajectories, outer-loop control may be provided by an autopilot that maintains altitude and velocity or performs somewhat more complex tasks such as automatic landing. Vectors and Matrices error signal vector plant vector function input vector function measurement vector function The fundamental objective of trajectory control research at NASA Ames Research Center, Dryden Flight Research Facility (Ames-Dryden),is to define a generic approach for designing guidance laws that will command an aircraft to fly precise flight test maneuvers to obtain accurate, repeatable flight data. The intent is to use modem control theory design techniques to establish a method of designing flight test trajectory controllers for various aircraft at Ames-Dryden to improve data collection performance. nonlinear control vector state vector state derivative vector linear system state vector i linear system state derivative vector Y observation vector; reference trajectory State z measurement vector A linear system plant matrix B C I linear system input matrix Previous work at Ames-Dryden justifies the usefulness of outer-loop controllers which command the aircraft to fly precise trajectories.’ In addition, having trajectory tracking capabilities available allows the researcher to design experiments to test other outer-loop functions such as optimal trajectories, intercept trajectories, navigation functions, or even automated approach and landing flightpath control. linear system observation matrix identity matrix Subscripts corn commanded D derivative control term e st estimated h altitude I k integral control term M Mach number In flight test trajectory control the requirement is to provide precise, repeatable control of an aircraft during maneuvers used to gather aerodynamic, structural, propulsion, and performance data. For conventional flight regimes these maneuvers are fairly straightforward and, in general, easily modelled. Typical of these maneuvers are level-accelerations/decelerations, pushover/pullups, and windup turns. kinematic 519 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 Numerous approaches to the flight test trajectory problem were taken including cut-and-try techniques on a nonlinear simulation? linear optimal contro~,~ optimal cooperative c o n t r o ~ eigenstructure ,~ a~signrnent,~ and classical control theory.’ Although both open-loop and closed-loop systems were successfully developed and utilized in flight, each of these approaches had limitations. The use of nonlinear control simplifies many of these problems, although some cutand-try tuning may still be required because of modelling assumptions. Meyer and Cicolani5 describe a formal structure for advanced flight control systems that provides the basis for the results described in this paper. This formal structure, the total automatic flight control system, has been applied in detailed nonlinear simulations of the de Havilland C-8A Buffalo augmentor wing jet STOL research aircraft (AWJSRA),6 a vertical attitude takeoff and landing aircraft? and others. Excellent results were achieved in all of these studies, and the claim made by Meyer, et al.* that “the method is effective for a large class of dynamic systems that require multiaxis control and that have highly coupled nonlinearities, redundant controls, and complex multidimensional operational envelopes” can be taken as thoroughly validated. jectory design. Since the trajectories to be flown are standard flight test maneuvers, time scale separation will be achieved through gain selection. To do this, the aircraft plant is analyzed for the slowest dynamic characteristics in the flight regime over which the guidance law is to be used. Gains can then be selected using any appropriate linear design technique so that the outer-loop dynamics are sufficiently separated in time scale from the slowest plant dynamics. The plant can then be considered by the outer-loop controller as having no apparent dynamics. A flight test trajectory controller (FTTC) was designed using this approach and was tested on a realtime, six-degree-of-freedom, full-envelope, nonlinear simulation of an F-15 aircraft and subsequently in flight on a preproduction F-15 aircraft. Flight testing covered a subset of the maneuvers tested in the simulation. The results are presented and discussed. An Alternate Approach to the Nonlinear Control Problem This paper presents the theory for and an application of an approach that provides a solution to the flight test trajectory control problem meeting the stated objective. This approach eliminates the need for inverse aerodynamic models by using measurement feedback. The elegance of the approach is that no inherent aerodynamic models are used, so it is easy to apply the controller to different aircraft. The use of nonlinear inverse transformations and time scale separation presents a single linear system covering a large portion of the envelope to the designer. Only simple linear proportional-integral or proportional-integralderivative guidance laws need be designed for the outer loop. The nonlinear inverse transformationsmap the linear trajectory command to the nonlinear input required by the plant. Time scale separation provides the justification to ignore the plant dynamics which are fast relative to the outer-loop commands. Put simply, the dynamics of the outer loop are designed to be slow enough so that the plant achieves the commanded input prior to the guidance law computing a new command. This can be done through gain selection or tra- 520 The use of detailed aerodynamic models in an outerloop, nonlinear controller can be eliminated by using aircraft accelerometer, rate gyro, and other sensor outputs. This simplifies the nonlinear controller in two ways: by eliminating aerodynamic coefficient calculations, and by making the nonlinear controller “model independent” given the same command paths into the inner-loop control system. While both of these features provide the potential for major design, implementation, and execution time benefits, the model-independence feature is perhaps the most significant contribution. It offers the possibility of designing a generic, general-purpose outerloop guidance law that, for similar types of aircraft, is independent of the aircraft to which it will be applied. Obviously, complete model independence can only be achieved by representing vehicles with identical models. Linear and Nonlinear System Formulations The plant around which the FTTC is designed includes the aircraft, the flight control system, and the instrumentation system. This plant is modelled by the equation putes the real input to the nonlinear system. The T-map shown in the same figure is the mapping from the plant states to the linear system outputs described in previous formulation^.^ In this formulation it is the identity matrix. Implicit in equation (9) is the assumption that gm(z) r f o . Given this plant model, it is desired to develop a system of the form Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 in the Brunovsky canonical form. This system is known as the apparent linear plant. Development of this system is achieved throughlinearizing transformations and time scale separation The state vector x contains all necessary states of the nodinear system state vector, %, as well as all necessary integral states. Linear Guidance Law Design In the formulation of the FITC, the linear control law is a simple tracker as opposed to the explicit model follower used by Meyer et aL9 The linear control law or pseudocontrol receives its input, e , from the feedback of the linear system, y , and the reference or command trajectory, ysef. so that Derivation of the Linearizing 'handormation The input to the apparent linear system is derived from equation (2) to be % = V e = Yref - Y (3 ) For the tracker, the first step of the design is to determine the nature of the command to be used (that is, step, ramp, parabola, and so forth). The designer can then determine the number of integrators required to achieve adequate tracking performance. Once this has been established, the pseudocontrol gains can be determined using appropriate modem or classical control techniques. This equation and equation (1) are then combined so that v = fp(x) + & ( X ) U (4) This is the linearizing transform for state feedback. Note that the number of states and controls are the same so that gp( x) is a square matrix function. Next, the terms fp(x) and gp(x) in equation (1) must be transformed into terms that are functions of the measurement vector z , which includes all necessary accelerometer, rate gyro, airdata, and so forth, measurements. fp(x) = fm(z) ( 5) g p w = gm(z) ( 6) ( 10) Nonlinear System Equations For the F- 15 aircraft on which this approach is to be tested, the aircraft can be represented by a nonlinear system equation with the form so that equation (1) becomes It is this transformation which eliminates the need for aerodynamic models in the linearizing transformations of systems developed previously. The linearizing transformation as a function of measurements is given by v = fm(z) + gm(z)u ( 8) The controls for this system are the inputs to the innerloop controller, and are defined for this application tobe [ ;;][ ] A%, The real control input into the plant can then be determined by solving equation (8) for the nonlinear input~ sothat u= = (13) where A%- = a, -1 ( 14) The nonlinear system state equations used are the standard wind axis equations of motion which corre- This equation represents the inverse linearizing transform, referred to as the W-map (Fig. 2). which com- 521 spond to the states in the state vector and can be found in Ref. 10. They are ti= Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 &= AX 1 = -{cos am P[gm(a, cos a - a,, sin a) - XTestcos a1 +gma,sinp+ XTCOSCYCOS~ + -mg(cosacospsinO - sin p sin 4 cos 6 - sin a c o s pcos +cos e ) } 1 -[-Dcosp+Ysinp+XTcosacosp m + ZT sin a c o s p - mg( cos a c o s /3 sin 0 - sin p sin 4 cos 8 - sin a c o s pcos COS e ) ] (28) 1 [ - L + ZT cos a - XT sin a v m cos p + mg(cosacos4cose + sin a s i n e ) ] +g - tan p ( p cos a + r sin a) &= 1 v cos p [ -g( an COS + a, sin aces 4cos e - sin a sin e ) ] +q-tanp(pcosa+ rsina) -COS h = a, sin 0 - a y sin 4 cos 0 +a,cos &cos e - 1 1 i;= -[ sin e(xT- D cos a + L sin a) Equations (17), (18), and (28) through (30) are used to derive the inverse transformations for the guidance laws. If we define that a maneuver begins at the trim condition t = t o = 0 so that x = 0 , then equation (11) can be written as Equations (17) and (18) are already in the form needed since all of the terms can be measured on the aircraft. Equations (15), (16), and (19) need to be transformed to functions of observed variables. This is done using the equations relating accelerometers and states also found in Ref. 10. They are 1 a,&= - ( x T - D c o s a + gm - gmsin 0 ) ay,k = X(0) = o = f ( x ( 0 ) ) and as we also define that for the initial trim condition Ui(0) = O f o r a = 1,3,thenitfollowsthat f(x(0)) = 0 Lsina Guidance Laws 1 -(Y+ grnsindcosd) The guidance laws for the nonlinear FITC are driven by error signals obtained by computing the difference between the reference state vector prescribed by the desired aircraft trajectory and the aircraft measurement feedback. The output of these linear control laws are pseudocontrols which are transformed, using the W-map, into commands for the aircraft/innerloop system. gm 1 az,k = -( ZT - Dsin a - Lcos a gm + gm cos 4 cos e) a,,k = a, - sin 0 ay,k = ay + sin +cos 0 a,,k = a, + g(x(O))u(O) The guidance laws used for the implementation discussed later in this paper are derived here. There are three command loops available for control by the FITC. These are pitch, which is controlled by incremental load factor command, A%-; roll, which is controlled by roll rate command, pcm; and thrust, which is controlled by throttle command, qcm. + cos 4 cos 8 The measurement feedback form of equations (IS), (16). and (19) are 522 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 Three different trajectories were desired to be flown: a level acceleration maneuver, a pushover/pullup maneuver, and an excess-thrust windup turn maneuver. The level acceleration maneuver is defined as a wings level, constant altitude maneuver with Mach number increasing (or decreasing) at a specified rate. The pushover/pullup maneuver is a wings level, constant throttle maneuver with angle of attack varying at a specified rate from trim to a minimum value, then to a maximum value, then back to trim. The excessthrust windup turn is a constant altitude Mach maneuver, with angle of attack increasing at a specified rate to a specified value. computed using the inverse linearizing transformation g, + tan P(pc0s a! + r sin a!) 1 --[ - g( a,, cos a! + a, sin a!) v cos p = iu, + g(cos a!cos 4cos e + sin a sin 0) ] (34) The pitch rate command qcm is then used to determine the A &. using the following inverse lineanzing transformation: An,. The linearizing inverse transformations are also derived from Ref. 10. = -1 v cos p + a,, + (Pcm g cos a - Q) ( 35) These two transfornations could be combined into one transformation; however, the pitch rate command generated by equation (34) must be filtered because of noise and system delays. This effect is discussed in the Implementation Experience section. Pitch Guidance Laws The pitch command for the F?TC is computed using two different guidance laws and transfornation equations. The level-acceleratioddeceleration maneuver uses reference altitude and altitude rate to compute a vertical acceleration command; the excess-thrust windup turn and the pushover/pullup maneuver use reference angle of attack to compute an angle-of-attack rate command. Roll Guidance Laws The lateral commands for the FTTC are computed using two different guidance laws and transformations. For the level-acceleratioddeceleration and pushover/pullup maneuvers, the reference roll attitude brei is known and dCm is computed using the linear control law Vertical acceleration is computed in the altitude command pitch loop using the following linear guidance law: where with ej, = href - h and p , , is computed using the inverse linearizing transformation equation eh = href - h Then A%,, is obtained from hCmusing the inverse linearizing transformation equation A%,,, = -1 + h,,,-a,sinB+a,sin+cosB+ COS e cos 4 pCm= (bco,-tan8(qsin++ = k,,e, + k,, 6’ e,dt ( 33) where 4,,f e , = a,,f - a From ,ut ( 37) The guidance law computation for the excess-thrust windup turn requires the use of two layers of outerloop controllers. Each outer loop must be slower than its inner loop. For this control loop, the outer-loop linear guidance law computes an h,, using the same linear control law shown in equation (31). Roll attitude command is then computed using the following inverse linearizing transformation equation: 1 ( 32) The second pitch control is the angle-of-attack command loop. The linear control law for this loop is dr,, rcos+) an intermediate pitch rate command is 523 = -cos-1 a, sin 8 - hcm/g - ( cosq- This intermediate command is used in the linear control law and to describe drawbacks which had or were expected to have noticeable impacts on the results. Flight Test System Description The F-15 highly integrated digital electronic control (HIDEC) aircraft and the remotely augmented vehicle (FL4V) system'' shown in Fig. 1 were used to flight test the guidance laws. The F-15 HIDEC is a preproduction F-15 aircraft with Pratt & Whitney 1128 engines, a digital command augmentation system (CAS), and a digital electronic engine control. The RAV system has both downlink and uplink capabilities, and has a ground-based minicomputer in which control laws may be implemented for flight test. This system eliminates the need for large computational capabilities onboard the aircraft. from which p,, is computed using the inverse linearizing transformation shown in equation (37) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 Thrust Guidance Laws Two throttle commands were required for the maneuvers defined. The first command holds the throttle position constant at the current or at some predefined position, qc, = qref. The throttle command for all other maneuvers is a Mach number controller. The commanded Mach rate is computed using the linear control law hi,, = kMDen;r + k M p e M + 6' kMIeMdt The RAV system is designed so that software developed and tested on the Ames-Dryden simulation system may be transported directly and run on identical computers in the RAV system. This enables the engineer to cheaply and easily perform verification and validation tests on the simulation before using the aircraft and RAV system. (39) where e& = eM &ref - n;i = Mref - M which is used to establish a thrust command T,, from the inverse linearizing transformation equation Tc, The drawback to the RAV system is that an additional 80 to 100 msec of pure delay are added to the control loop. The time added to an aircraft system delay as a result of the asynchronous systems of up to three frames (or up to 150 msec) can make the control task much more difficult. For the outer-loop guidance problem, this delay was expected to cause a noticeable problem in the lower dynamic pressure ranges. = {maucm -gm[ ay sin /3 + cos p( a, cos a - a, sin a)] - XT cos CY cos p + mg( cos a c o s psin 6 - sin psin C#J cos 6 - sin COS /3 COS COS e)}/ The allowable command ranges for the controller were set at *150.0 dedsec in roll rate, -0.75 to 4.0 g in incremental load factor (0.25 to 5.0 g normal acceleration), and full range for both throttles. The only problem foreseen resulting from the command range limits was in the incremental load factor command. (cos E cos a cos p + sin E sin a cos /3) (40) with E being the angle between the x-body axis and the direction of thrust. The actual throttle setting is determined by using a model of the thrust-throttle relationship and inverting it so that Flight Test Pajectory Control Software Program The linear control laws and nonlinear inverse transformations described in previous sections, along with a command generator to produce the reference time histories of the states to be tracked, were implemented in FORTRAN in a computer program called the flight test trajectory controller or FTTC (Fig. 2). An operator interface to the controller was developed to input maneuver specifications, and to monitor the F " C and flight test system. The thrust term XT in equation(40) is obtained using the same model with a simple lag to emulate the engine lag. Discussion These sections describe the system and software used to flight test this approach. No effort is made to give an in-depth description of the system used. Instead, descriptions are given to provide adequate information regarding how the experiment was performed, The basic frame time of the program is 12 msec. However, only the maneuver selection logic and the 524 to 240 msec were examined over a matrix of maneuvers simulated at Mach 0.75 and 1.20 and at altitudes from 10,OOO to 40,OOO ft. The controller performed acceptably at all conditions. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 linearizing transformations run at this rate. The command generator and pseudocontroller run every other frame. This is done because the command generator and the pseudocontroller represent the outerloop or slow controller, while the linearizing transformations and the aircraft CAS represent the innerloop controller. The FTTC was flown between May and August 1989. During these flights, pseudocontroller gains were adjusted and minor changes were made in the guidance law to improve performance of the Controller. Four of the maneuvers flown are presented in Figs. 3 through 6 and are discussed here. The frame time was selected because it was the fastest that the software could run without getting frame overruns. A slower frame time would have added to the overall loop delay of the system, so it was desirable to minimize the frame time. However, the controller sees constant values of sensed data between downlink data frames which have the same rate as the aircraft system (50 msec). This could be a problem during very dynamic maneuvers. The first of the four maneuvers shown in Fig. 3 is a level acceleration maneuver from Mach 0.70 to Mach 1.20 at a rate of 0.01 Mach numbedsec. Roll attitude is held to within f2.0" of wings level. Mach number increases at a constant rate except for a short time around Mach 1.0. This effect is caused by the aircraft pitching rapidly to correct altitude error resulting from the shock wave forming at the tip of the noseboom. The effect is also evident in the altitude, altitude rate, and commanded incremental load-factor plots. Results Note that in the nonlinear guidance laws defined by equations (31) through (41), none of the equations use aerodynamic terms and, therefore, none of these guidance laws require aircraft-specific aerodynamic models. In fact, no aircraft-specific models are required except for the simple thrust-throttle mapping shown in equation (41). These results are significant in their differences from the results reported by previous authors. The pushover/pullup maneuver shown in Fig. 4 was flown at Mach 0.90 and 10,OOOft with angle of attack varying between 0" and 2.5" at a rate of 0.5 deg/sec. Tracking of reference angle of attack is affected noticeably by sensor noise. This effect is discussed in more detail later in this paper. During simulation and flight test, controller performance was measured by the ability to keep aircraft parameters within specified tolerances. These tolerances are AO.01 Mach number, 6100 ft altitude, and f0.3" angle of attack. In addition, the controller must have smooth transitions from the capture of initial conditions to the beginning of the maneuver itself. Also, when parameters to be tracked are ramped, the controller must maintain the tolerances while ramping those parameters smoothly. The large tolerance in angle of attack is caused by the noise and accuracy problems associated with the measurement of that parameter. Figures 5 and 6 show excess thrust windup turns. The first windup turn is flown at Mach 0.65 and 25,000 ft, while the second one is flown at Mach 1.20 and 25,000 ft. The maneuver at Mach 0.65 tracks altitude poorly with an error of approximately 150 ft before rolling out to straight and level. This maneuver also has a difficult time tracking Mach number with an error of approximately Mach f0.025. This is partially caused by a 1.5-sec delay built into the throttle command to eliminate the problem of the throttle toggling between core power and afterburner. The second excess thrust windup turn performed at Mach 1.20 does not show this problem. Altitude is kept to within f25 ft within Mach f0.008. Here, however, angle of attack is not ramped smoothly to the desired value of 5", and even though the commanded incremental load factor is at its maximum value, the angle of attack measured is just over 3". The problem of the unknown scaling effect in the flight control system occurred in both the pitch and roll command paths. Simulation evaluation was conducted for all three maneuvers across a Mach number range of 0.60 to 1.20 and across an altitude range of 10,OOOto 40,000 ft with acceptable results, except at the high and slow conditions. While the results at the high and slow conditions did not fall within the specified tolerances, the controller was still able to perform the maneuver. Tests were also conducted in the simulation with varying amounts of pure delay in the uplink and downlink emulation. Total system delays ranging from 120 The effect during flight test which the pilots found least acceptable, and in some cases unacceptable, was the effect termed roll ratcheting which appears at the 525 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 not modelled in the simulation. The problem of roll ratcheting warrants further investigation. beginning of the roll into the turn. This effect, seen in Figs. 5 and 6 , vaned in magnitude and frequency depending on the conditions of the turn. In the slower or higher turns, the effect was not noticeable to the pilot. The origin of this effect is unclear. However, one possible explanation is that the CAS seemed to scale the command generated by the inverse transformation. The gains in the linear control law were driven up to account for this instead of simply scaling up the commands out of the inverse transformations. This may have caused the linear gains to be too high in some flight conditions, thus causing the ratcheting. This problem does point out, however, that such scaling effects and nonlinearities in the inner-loop control system need to be accounted for in the inverse transformations. The transformations derived herein only account for aerodynamic nonlinearities and the nonlinearities in the equations of motion. Conclusions Research is ongoing at the Ames-Dryden Flight Research Facility to define a technique for designing flight test trajectory controllers. Many techniques have been investigated. The most promising technique of the previous research involved the use of linearizing transformations using state feedback. An approach to the outer-loop trajectory control problem for developing nonlinear guidance laws using linear controls with nonlinear transformations and measurement feedback was presented. A controller was designed which applies this approach specifically to the problem of flight test trajectory control. Testing was accomplished on a full-envelope, nonlinear, six-degree-of-freedom simulation and in flight using an F-15 aircraft. Flight test results are presented. The origin of the noise in the angle of attack and commanded incremental load factor seen in the pushover/pullup maneuver can be explained, at least in part, by these two maneuvers. During both maneuvers, the commanded incremental load factor reaches its maximum limit and is held constant for a short time. However, noise in the angle-of-attack trace is significant (up to approximately 0.3") and is probably a result of either measurement noise, turbulence, or both. Angle of attack provides the feedback for the linear control law that computes the linear system input of angle-of-attack rate (equation (33)), as well as heavily influencing the inverse transformation shown in equations (34) and (35). The commanded incremental load factor computation seems to amplify the noise on angle of attack. While not a divergent process, filtering sensed angle of attack is necessary to provide smoother tracking of commanded angle of attack. Simulation and flight test results indicate that the flight test trajectory controller is capable of performing an acceptable job of controlling an F-15 aircraft through level accelerations, pushover/pullups, and windup turns over a large portion of the flight envelope. Some problems are encountered at high and slow (low dynamic pressure) conditions where time scale separation between the fast and slow controllers disappears as aircraft performance decreases. Another problem for this controller was a ratcheting motion of the aircraft that occurred when entering windup turns at high dynamic pressures. This effect may be cleared up by running the controller at a slower rate than the tested rate of 83.3 Hz. Also in this application, certain nonlinearities were not accounted for in the linearizing transformations. It is likely that these nonlinearities contributed to the problem of roll control. The results of the flight test of the FTTC indicate that the approach of using linearizing transforms with measurement feedback is a valid approach for developing outer-loop controllers for nonlinear systems. Tracking of reference trajectories is acceptable over a large range of altitudes and Mach numbers using only a single set of gains. The approach for designing outer-loop controllers for high-performance aircraft by using inverse linearizing transformations with measurement feedback has been validated by the results in simulation and flight testing. With the exception of a simplified engine model, this approach does not require implicit or explicit aircraft models in the design. It may be easily applied to other aircraft requiring similar inputs to the control system. In general, the flight test data matched the simulation data with the exception of sensor noise, which was 526 References Smith, G. Allan, and Meyer, George, Application of the Concept of Dynamic Trim Control to Automatic Landing of Carrier Aircrafr, NASA ”-1512, 1980. ‘Duke, Eugene L., Jones, Frank P., and Roncoli, Ralph B., Development and Flight Test of an Experimental Maneuver Autopilot f o r a Highly Maneuverable Aircraft, NASA TP-2618, 1986. Smith, G. Allan, and Meyer, George, “Application of the Concept of Dynamic Trim Control and Nonlinear System Inverses to Automatic Control of a Vertical Attitude Takeoff and Landing Aircraft,” AIAA 812238,1981. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 *Duke, Eugene L., Swann, Michael R., Enevoldson, Einar K., and Wolf, Thomas D., “Experience with night Test Trajectory Guidance,” Journal of Guidance, Control, ana‘ Dynamics, vol. 6 , no. 5, Sept.Oct. 1983, pp. 393-398. 8Meyer, G., Hunt, R.L., and Su, R., Design of a Helicopter Autopilot by Means of Linearizing Transformations, NASA TM-84295, 1982. 3Menon, P.K.A., Badgett, M.E., Walker, R.A., and Duke, E.L., “Nonlinear night Test Trajectory Controllers for Aircraft,” Journal of Guidance, Navigation, and Control, vol. 10, no. 1, Jan.-Feb. 1987. 9Meyer, George, “The Design of Exact Nonlinear Model Followers,” Proceedings of 1981 Joint Automatic Contro! Conference, Charlottesville, Virginia, 1981, pp. FA3A. 4Gary, Sanjay, and Schmidt, David K., Optimal Cooperative Control Synthesis of Active Displays, NASA CR-4058,1987. “Duke, Eugene L., Antoniewicz, Robert E, and Krambeer, Keith D., Derivation and Definition of a Linear Aircrajlt Model, NASA RP-1207, 1988. ’Meyer, George, and Cicolani, Luigi, A Formal Structurefor Advanced Automatic Flight-Control Systems, NASA TN D-7940,1975. “Petersen, Kevin L.,“night Experience With a Remotely Augmented Vehicle Test Technique,” AIAA 81-2417, NOV.1981. Radar antenna 900283 Fig. 1 Remotely augmented vehicle system used to flight test the F?TC. 527 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 Y, Yo Y , V Command generator Pseudocontroller WT-map Fig. 2 Structure used for the FTTC. 528 i = f(x) + g(x)u $7, U Aircraft plant Pcom 1 deglsec 0 -10 I I I I I I I I Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 4- /'" 0 -4 ~ Mach 21 -' I 0 40 h, ftlsec 0 =- I -40 25,100 h9 ft 25,000 I I I I I ; 24,900 0 10 30 Time, sec 20 40 60 50 900265 Fig. 3 Flight data of a level acceleration maneuver; Mach 0.70 to1.20,25,000 ft. 529 - -0 -10 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 10 - Pcom 9 deglsec * I I I I 0- I / 1 I I I I 1 - com 8 g's 40 Mach 0 Time, sec 900266 Fig. 4 Flight data of a pushover/pullup maneuver; Mach = 0.9, 10,000ft, between a! = 0" and a = 2.5". 530 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 Start maneuver End maneuver P, deglsec -40 100 - Pcom? deglsec -100 25,200 I I I I I I I h’ 24,800 ft 24,400 * com 9 g‘s llLzlzk -4 Mach 2oo PLA deg corn ’ 100 0 4 8 12 16 Time, sec 20 24 28 900267 Fig. 5 Flight data of an excess thrust windup turn maneuver; Mach = 0.65,25,000 ft, from 531 atrim to cy = 12’. Start maneuver 4:Ft"-c.r, I -100 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3373 deglsec PI i i i i i i -40 Pcom deglsec , -100 25,040 2 25,000 24,960 l -4 l*L 1 Mach 1.20 1.19 com 9 100 deg 8o0 4 8 12 Time, 16sec 20 24 28 900268 32 Fig. 6 Flight data of an excess thrust windup turn maneuver; Mach = 1.2,25,000 ft, from CYtrim to CY = 4'. 532

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