NASA Contractor Report 4161 

Aircraft Flight Test 

Trajectory Control 

P. K. A. Menon and R. A. Walker 

Integrated Systems, Inc. 

Palo Alto, California 

Prepared for 

Ames Research Center 

Dryden Flight Research Facility 

under Contract NASZ-11877 

Nationa I Aeronautics 

and Space Administration 

Scientific and Technical 

Information Division 

1988 
TABLE OF CONTENTS 

1 . INTRODUCTION ........................ 1 

I 

1.1 Background ...................... 1 

1.2 Summary of Research Accomplished ........... 2 

1.4 Report Organization .................. 5 

1.3 Study Results ..................... 3 

2 . AIRCRAFT AND COMMAND AUGMENTATION SYSTEM (CAS) MODELS ... 6 

2.1 Aircraft Model .................... 7 

2.2 EngineModel ..................... 7 

2.3 Command Augmentation System (CAS) Model ........ 8 

2.4 Linear Model Summary ................. 13 

3 . MANEUVER MODELING ..................... 14 

3.1 Maneuver Modeling Overview .............. 14 

3.1.1 Constraints on Position Components ....... 15 

3.1.2 Constraints on Velocity Components and 

Altitude .................... 15 

3.1.3 Constraints on Combinations of Load. 

Speed and Altitude ............... 16 

3.2 Maneuver Modeling ................... 17 

3.2.1 Transient Trajectory .............. 18 

3.2.2 Level Acceleration/Deceleration ........ 20 

3.2.3 Pushover. Pullup ................ 21 

3.2.4 Zoom and Pushover ............... 24 

3.2.5 Excess Thrust Windup Turn ........... 28 

3.2.7 Constant Dynamic Pressure and Constant 

3.2.8 Constant Reynolds Number and Constant Load 

3.2.6 Constant Throttle Windup Turn ......... 29 

Load Factor Trajectory ............. 32 

Factory Trajectory ............... 35 

4 . MANEUVER AUTOPILOT DESIGN ................. 37 

4.1 The F-15 Flight Test System "Linear" Model ...... 38 

4.1.1 The F-15 Nonlinear Tabular Model ........ 38 

4.1.2 Linear Perturbation Controllers ........ 41 

4.1.3 Linear Time Varying Simulation ......... 45 

4.2 Linear Design Techniques ............... 47 

4.2.1 Eigenstructure Assignment Design ........ 47 

4.2.1.2 Decoupling Eigenassignment ...... 48 

4.2.1.3 Dominant Mode Eigenassignment ..... 49 

4.2.1.1 Minimally Restructured 

E i genass i gnment ............ 48 

4.2.1.4 Conclusions on Output Eigenstructure Assignment ......... 49 

i 
4.2.2 Minimum Error Excitation Output Feedback 

Design ..................... 50 

4.3 Nonlinear Flight Test Trajectory Controllers ..... 54 

5 . SIMULAITON AND EVALUATION ................. 57 

5.1 Maneuver Simulation .................. 57 

5.2 Maneuver Simulation Mechanization ........... 58 

5.3 Maneuver Simulation Results .............. 59 

5.3.1 Transient Trajectory .............. 59 

5.3.2 Level Acceleration ............... 62 

5.3.3 Pushover. Pul.lup. ............... 64 

5.3.4 Zoom and Pushover ............... 66 

5.3.5 Excess Thrust Windup Turn ........... 69 

5.3.6 Constant Thrust Windup Turn .......... 73 

Factor Maneuver ................ 77 

Factor Maneuver ................ 82 

5.3.7 Constant Dynamic Pressure - Constant Load 

5.3.8 Constant Reynold's Number - Constant Load 

6 . SUMMARY AND FUTURE WORK .................. 87 

6.1 System Modeling .................... 87 

6.2 Control Design Results ................ 87 

6.3 Control Design Technique Evaluation .......... 88 

6.4 Maneuver Autopilot Validation ............. 88 

6.5 Software Deliverables ................. 88 

6.6 Future Work ...................... 89 

REFERENCES ........................... 90 

APPENDIX A: 

APPENDIX A-I Paper Communicated to 1985 ACC. June 19.21. 

APPENDIX A-I1 Aircraft . CAS Linearized Model and FTTC Design 

Flight Test Trajectory Controller Synthesis with 

Constrained Eigenstructure Assignment ...... A.1 

Boston. MA ................... .A. 1.1 

10000' Altitude ................ .A.I 1.1 

Using Eigenstructure Assignment at 1.2 MACH. 

APPENDIX A-I11 Aircraft . CAS Linearized and FTTC Design Using 

Eigenstructure Assignment at 0.7 MACH. 10000' 

Altitude .................... .A.II 1.1 

the Synthesis ................. .A.I V.l 

A Demonstration Example for Exact Nonlinear 

Controller Design with Prelinearizing 

Transformation .................. B.l 

Linear Time Varying Simulation in SYSTEM BUILD . . D.1 

Minimum Error Excitation Output Feedback-Designs.E-1 

APPENDIX A-IV Desired Eigenvalues and Eigenvectors Used in 

APPENDIX B 

APPENDIX C Fortran Program for Maneuver Modeling ...... C.1 

APPENDIX D 

APPENDIX E 

ii 
I 

SECTION 1 

INTRODU CTI ON 

1.1 BACKGROUND 

Flight Test Trajectory Control (FTTC) is an emerging, pilot-aiding 

technology valuable for improving flight test results and applicable to 

maneuver control and guidance in a number of contexts. This technique has 

provided the means for flying maneuvers consistently, precisely, and 

repeatably from flight to flight. Two versions of these controllers have 

been used: a closed-loop automatic system and an open-loop system providing 

manual piloting information. A closed-loop system used to collect performance, pressures, and loads data from the highly maneuverable aircraft technology (HiMAT) vehicle is described in [I]. The application of the openloop system on the NASA F-111 transonic aircraft technology (TACT), F-15 

airframe/propulsion system interaction studies, and F-15 shuttle tiles test 

programs are given in [23. 

Originally, the open-loop flight- test- traj ectory guidance algorithms 

were developed on-line, in a piloted simulation using cut-and-try techniques 

that were not only man power intensive, but often produced less than optimal 

controllers. A closed-loop system designed using one-loop-at-a-time 

classical design approach is documented in [31. Full-state feedback 

approach for closed-loop system design using linear quadratic synthesis is 

described in [4]. 

methodology and controller complexity. 

Both these approaches have limitations in terms of design 

The work reported in reference [2] and more recent ongoing work have 

included detailed modeling of all the piloted aircraft subsystems for suitable control law development. This research has identified the maneuver 

modeling as a significant technology needing further clarification and development. The maneuver modeling aspect of trajectory guidance and control 

law design will be emphasi zed here in describing this developing technology. 

1 
In the present work, closed loop mechanization is carried out with the 

pilot in a supervisory role. The thrust here is in four areas 

1 . Maneuver modeling, 

2. Application of modern linear multivariable synthesis, 

3. Techniques for the development of reference command and gainscheduled perturbation controllers, and 

4. Exploratory investigation of the emerging nonlinear system design 

techniques via prel ineari zing transforms. 

1.2 SUMMARY OF RESEARCH ACCOMPLISHED 

Maneuver modeling for all the given flight test trajectories were 

successfully carried out. 

51 level turn trims at a load factor of 2, and 30 level turn trims at a load 

factor of 4 were employed. A 3-D linear interpolation was employed. Thus a 

flight condition is characterized by altitude, Mach number and load factor. 

Linearized aerodynamic coefficients were also stored. This data in conjunction with kinematic and some dynamic equations are then used to generate 

state and control histories for all the flight test maneuvers. 

In the present work 64 straight and level trims, 

Two multivariable synthesis techniques were used to obtain output 

feedback linear perturbation controllers viz. 

I) Constrained eigenstructure assignment (Shapiro & Andry, 1982) 

2) Minimum error excitation technique (Kosut, 1970) 

Out of these two, the constrained eigenstructure assignment technique was 

found to be the hardest one to iterate on primarily due to the current lack 

of understanding of the explicit relationship between eigenvectors and the 

desired time response. This difficulty, while not crucial in full state 

feedback design can become an extremely important element in output feedback 

2 
design. Due to its high sensitivity to the input eigenvectors, it was discarded after the initial research phase. A paper outlining these issues was 

presented at the 1985 American Control Conference (see Appendix A). The 

minimum error excitation suboptimal design was successful and is advocated 

as the main design approach. 

Exploratory research on nonlinear maneuver autopilot synthesis brought 

out the feasibility of generating the maneuver autopilots by employing singular perturbation theory in conjunction with prelinearizing transforms. 

The methodology is outlined, though the controllers have not been worked 

out. 

in Appendix B. 

be further pursued in future research. 

A simple example problem to serve as an illustrative example is given 

This technique appears to hold considerable promise and will 

The designs obtained were scheduled as functions of Mach number and 

load factor and were tested on a linear simulation. 

been found satisfactory within the validity of the linear model assumptions. 

In the next phase these designs will be tested on a nonlinear simulation of 

the F-15 aircraft. 

The performance has 

1.3 STUDY RESULTS 

Results from this work fall into four categories, (1) control design 

technique evaluations, (2) specific control analysis and design, 

(3) software developments for control law mechanization, and (4) specific 

control law validations. The various deliverables in these areas are: 

1) For control design technique evaluations 

a) A paper evaluating the eigenstructure assignment technique C61 

b) An assessment and extension of the nonlinear prelinearizing 

control technique C71, and demonstration on a simple 

example. (See Appendix B) 
2) For control analysis and design 

a) Development and clarification of maneuver modeling equations 

beyond the analysis in references [4, 101. 

b) Condensation of the F-15 aircraft nonlinear characteristics 

to a table of reference states and controls via nonlinear 

simulation trim values at approximately 145 conditions on 

the flight envelope. 

c) Decomposition of the 8 maneuvers over the flight envelope 

into 30 linear perturbation models (15 with fixed throttle 

and 15 with variable thrust). 

d) Solution of 30 output feedback gains which can be used with 

the 30 linear perturbation models to simulate maneuvers 

throughout the envelope. 

3) For software developments 

TM 

X a) Extensions t,pMIntegrated Systems, Inc. (ISI's) MATRIX 

SYSTEM BUILD package including a linear time varying 

FORTRAN block to give a genepic linear time varying simulation capability (such a capability can be easily mechanized 

to model linear time-varying simulations of engine and aircraft dynamics, for example). 

b) Development of a three-dimensional interpolation program 

which converts table look-ups in altitude, Mach, and load 

factor to a one-dimensional table look-up with respect to 

time for a specific maneuver, 

c) Documented command files Aircraft-CAS model building and 

Maneuver Auto Pilot (MAP) design in the MATRIXx language for 

the model generation, control law design, and simulation 

validation process, and 

4) Demonstration of the maneuver autopilot validations in a linear 

simulation. 

The linear control laws developed in this work are now ready for validation on a nonlinear batch simulation. With suitable algebraic equation 

manipulation, the nonlinear control laws can also be mechanized on a 

nonlinear simulation. 

4 
1.4 REPORT ORGANIZATION 

Section 2 describes the aircraft and command augmentation system (CAS) 

models, the choice of outputs useful for feedback and the procedure for 

obtaining linear models. Section 3 specifies the flight test maneuvers 

analyzed in terms of the maneuver objective and how that objective can be 

reduced to a set of equations which constrain a required set of outputs. 

Section 4 outlines the linear techniques evaluated and used for the design 

of perturbation feedback controllers as well as the nonlinear tracking feedback controller. Section 5 reviews the linear simulation mechanization and 

discusses the demonstration results for the eight required maneuvers. 

Conclusions are given in Section 6. The appendices describe in detail the 

issues in linear time-varying simulation, the evaluation of constrained 

eigenstructure assignment and the output error feedback controller designs, 

and nonlinear tracking control via prelinearizing transformations. 

5 
SECTION 2 

AIRCRAFT AND COMMAND AUGMENTATION SYSTEM (CAS) MODELS 

LINfRR DESIGN 

TtcnNlauEs 

W 

This section reviews the overall linear system model used to both 

develop and validate the linear perturbation control laws. 

shows the design process used. 

Figure 2-1 below 

rLlGHT CONDIT1OW 

NRSR-ORf 

RlRtRRfT LlNfRRlZRllON 

PROGRAM 

NRSR-Oflf 

'tOUIURLENT GAIN' 

CONSfN PROGRRM 

Figure 2-1. Use of Linear System Flight Test Trajectory Model 

The rest of the section describes the airframe, engine, and command 

augmentation (CAS) models and how they are integrated into the overall 

linear design and evaluation model. 

6 
2.1 AIRCRAFT MODEL 

As shown in Figure 2-1, the airframe model was obtained from the NASA 

Ames Dryden Flight Research Facility (ADFRF) LINEAR Program [ll], yielding 

the aircraft model in the standard form 

x = AX + Bu, 

y = HX + Fu 

where 

T 

x : CSV, ba, bq, be, 66, bp, 6r, 64, 6h1, 

The first nine outputs are required in the linearized CAS model. 

2.2 ENGINE MODEL 

An engine model of the form 

0.2 

s + 0.2 

was assumed. 

lead-lag compensator, the problem of a slow engine time constant was not 

However, since this lag can be compensated for with a simple 

7 
included in the overall design, assuming exact cancellation. The actual 

residual due to mismodeling can best be addressed with the nonlinear sirnulation. Figure 2-2 below shows the overall mechanization including the CAS 

states described next. 

COM?-lAKiED 

THROTTLE 

SETTING 

YiAn-LAG COXPEXLATION 

FOF, ENGINE DTNEL'ICS 

+ s + 0.2 - k 10 STATE I 

r 

s + 20 AIRCRAFT 

Figure 2-2. Linear Perturbation Flight Test Tr&jectory Zontrol System 

2.3 COMMAND AUGMENTATION SYSTEM (CAS) MODEL 

The CAS model is highly nonlinear with gain schedules and multiplicative and saturation nonlinearities. This was linearized with a gross 

linearization and an "equivalent gain" for the multiplicative nonlinearities 

determined by the NASA ADFRF CONSEN program. The CONSEN program simulates 

the actual CAS for several time steps at a given flight condition. When all 

the transients have decayed, the ratio of inputs and outputs are then used 

to compute the equivalent gains. 

8 
The CAS system has no access to ailerons in the aircraft under consideration and hence, all the flight test maneuvers will be accomplished using 

I throttle, elevator, rudder and differential tail. The maneuver autopilot 1 

outputs will be connected to the existing autopilot interface in the CAS, as 

shown in Figures 2-3 through 2-5. 

9 
++ 

-c 

c 

10 
d 

t 

I 

J f 

11 
A 

1 IO 

1 - 

Q 

0 

I2 

12 
1 2.4 LINEAR MODEL SUMMARY I 

I 

I 

The overall linear model has nine aircraft states, one engine state, 

and twenty-one CAS states. The outputs to be used in the maneuver autopilot, for all maneuvers, are 

I 

i 

Selection of these quantities for feedback are primarily dictated by the 

various flight test maneuvers to be executed. Thus, the altitude, Mach 

number and the angle of attack are to be either regulated about reference 

values or should be made to follow desired command histories. The choice of 

roll attitude feedback arises from a requirement to maintain wings level 

along symmetric flight test trajectories. In nonsymmetric flight test 

i 

1 

I 

I 

maneuvers, desired load factors can be sustained by an appropriate roll 

attitude command. For all flight test maneuvers, the angle of sideslip 

needs to be regulated about zero; this prompts the use of B feedback. An 

output feedback controller is envisaged in the present development. Since 

most output feedback synthesis approaches do not guarantee stability, it is 

decided to incorporate derivative feedbacks for as many output variables as 

feasible. This led to the selection of the body rates p, q, r to serve as 

derivative feedbacks for angle of attack, angle of sideslip and the roll 

attitude. The Mach number and the flight path angle serve as derivative 

feedbacks for altitude. Since most flight test maneuvers require angle of 

attack tracking, an integral feedback is incorporated in the angle of attack 

channel. While executing nonsymmetric flight test maneuvers, the roll 

attitude can sometimes be very close to 90°. At these conditions, small 

perturbations in the roll attitude can induce major changes in altitude and 

other states. Hence in order to provide a tight roll attitude control, two 

integral feedbacks are introduced for the roll attitude 4. 
SECTION 3 

MANEWER MODELING 

The desired trajectory or maneuver to be modeled governs the subsequent control law development in two ways. 

the equations of motion of the vehicle on the reference trajectory and 

secondly the maneuver regimes determine the linear perturbation equations 

about the commanded trajectory from which the linear control law is 

devel oped . 

First, the maneuver prescribes 

Aircraft flight test trajectories could be based on inertial reference 

(e.g., level-turn or 3-D guidance) or reference with respect to another 

vehicle or vehicles (e.g., in air-to-air combat). To place the flight test 

trajectory control design problem in a proper framework, the constraints 

which determine the equations of motion for various large classes of 

trajectories are described below. 

The subsections which follow give a general overview of maneuver 

modeling and the descriptions of the specified maneuvers analyzed in this 

work . 

3.1 MANEWER MODELING OVERVIEW 

We divide single vehicle flight paths into those which require continuous control along the trajectory and those that specify a final flight 

condition. In either case the flight test trajectory could be specified in 

terms of one of the following: 

1. Constraints on position components, 

2. Constraints on velocity components and altitude, 

3. Constraints on combinations of load, speed and altitude. 

14 
Combinations of these constraints could also be considered. The flight test 

maneuvers discussed in this report belong to the second and third 

categories. Reference [41 gives some early results in flight test 

trajectory modeling. 

3.1.1 Constraints on Position Components 

Examples of trajectories which involve position constraints along the 

flight path are 

1. 4-D guidance (x(t), y(t), h(t) are given functions of time). 

2. 3-D guidance (x, y, h are related to each other, e.g., fly along 

a hypothetical wire in space). Examples of 3-D guidance are 

approach to landing, terminal area flight paths and threat evasion for reconnaissance aircraft and bombers. This also includes 

straight and level flight and flights along predetermined paths. 

Examples of trajectories which specify position constraints at the final 

t r aj e ct or y point are: 

1. 4-D specification (arrive at a certain point, at a certain time, 

e.g., touchdown on the runway at a specified point). 

2. 3-D specification (fly-to-VOR, terrain following). 

Note that each of these trajectories requires position measurement. The 4-D 

guidance trajectory indirectly specifies velocity and acceleration components. Thus, 

constraint on 

for most test 

specification of position components is the most comprehensive 

the trajectory. Such a rigid constraint may be unnecessary 

maneuvers. 

3.1.2 Constraints on Velocity Components and Altitude 

While the horizontal position components do not, in general, affect 

aerodynamic variables, the altitude determines density and by itself affects 

dynamic pressure. Thus, it must always be considered as a possible variable 

15 
to be constrained. In fact, the altitude and dynamic pressure are so 

important that a majority of flight test trajectories will define the altitude profile (this includes maintaining constant altitude). 

Examples of this class of trajectories are: 

1. u(t), v(t), w(t) and h(t) [in other words, Mach number, dynamic 

pressure, B(t) and a(t>.l B(t) may be zero. 

2. Mach number, angle-of-attack and dynamic pressure (as in shuttle 

tile tests). 

Various other combinations of velocity components and altitudes could also 

be specified. 

Mach number, angle-of-attack and altitude constraints could also be 

desired at one point on the trajectory. For example, the zoom-and-pushover 

is a trajectory where angle-of-attack, Mach number and altitude are specified at one point on the trajectory. 

3.1.3 Constraints on Combinations of Load, Speed and Altitude 

The trajectory specifications could involve components of loads along 

the three axes, velocity components and altitude. The typical load specification will consist of desired vertical acceleration. The desired value of 

the lateral acceleration is usually zero. The total speed is often specified in lieu of the fore-and-aft acceleration. 

Many combinations of load, speed and altitude specifications are 

possible. Some examples are as follows: 

1. A constant load, constant Mach number level turn, 

2. A constant Mach number, constant altitude windup turn. 

Often, the desired flight trajectory for an aircraft depends upon the 

position and flight test trajectory of other vehicles. 

collision avoidance, air combat, or avoidance of air-to-air missiles. The 

specification is typically based on the position of a target aircraft with 

respect to the aircraft whose trajectory is being controlled. 

Typical examples are 
The next section gives a specific objectives and their analytical 

development for each of the maneuvers for which autopilots were designed. 

3.2 MANEUVER MODELING 

To summarize, the objective of maneuver modeling is to generate a consistent set-of state and control histories to serve as commands and open 

loop controls for the maneuver autopilot using a data base consisting of 

trim conditions. Two sets of trims have been found adequate for all the 

maneuvers discussed here, viz, straight and level trims and level turn 

trims. To the extent feasible, kinematic relationships are employed to 

generate the state-control histories. Whenever this is not possible, 

linearized aerodynamics and engine models are used. Note that the following 

development is not restricted a particular aircraft. 

The commands and the open loop controls consist of: 

Commands: Altitude, Mach number, angle-of-attack, flight path angle, 

roll attitude, angle of side slip, roll-pitch-yaw body 

rates. 

Openloop Controls: Throttle, elevator, rudder and differential tail. 

In the following, the maneuver modeling for individual flight test 

trajectories will be discussed in detail. It is important to note that all 

these maneuvers assume zero sideslip. 

17 
3.2.1 Transient Trajectory 

The objective of this maneuver is to transfer an aircraft flying 

straight and level at a Mach-altitude pair to another Mach-altitude pair at 

a desired flight path angle. This maneuver is normally employed as the 

initial-terminal transient to other flight test maneuvers and hence the 

name. 

The simplest way to mechanize this maneuver to assume a cubic polynomial in time for the altitude. Thus, 

h = h, + a,t + a,t2 + ast3; to - < t 5 tf 

from which 

2 6 = a, + 2a2t + 3a,t 

(3.1) 

(3.2) 

The requirement for a cubic polynomial arises from the constraints that one 

wishes to place at the two ends, i.e., initial and final altitudes are 

specified along with altitude rates at the two boundaries. 

development, constant acceleration along the flight path is assumed next, 

viz, 

To simplify the 

v = v, + ft, (3.3) 

where 

Vf - vo 

tf 

, V is the specified final speed and t the final f f’ v= 

time. 

18 
The flight path angle is readily computed from 

I 

I 

I 

I 

1 

I 

1 

I 

I 

1 

i 

I 

I 

I 

I 

I 

As noted elsewhere, a data base consisting of straight and level trims 

at several Mach-altitude pairs are available. 

Assuming that the path angle is small, such that 

Lift = weight 

along this maneuver, the angle of attack history can be generated by 

linearly interpolating between stored straight and level trim data at the 

Mach number - altitude given by the equations (3.1) and (3.3). Since the 

angle of attack will be close to the straight and level trim values, the 

aerodynamic surface setting at these trims can be used to generate approximate open loop control settings. Assuming next that under these conditions, 

the actual drag is close to the trim values, the required thrust may be 

computed as follows. 

Assuming that the aircraft thrust is aligned with the vehicle 

longitudinal axis, the acceleration along the flight path for symmetric 

flight is given by 

Tcosatrim-D 

m 

v= - g sinY (3.4) 

In the expression (3.41, T is the thrust, Y flight path angle, a the 

angle of attack, D the aerodynamic drag, rn the aircraft mass and V the 

velocity along the flight path. In order to compute the required thrust, 

the equation (3.4) can be manipulated to the form given below 

(t + gsinY)m + D 

cosa 

T= 

trim 

(3.5) 

19 
To compute the throttle setting, a linear thrust-throttle characteristic 

will be assumed. Thus, 

- Ttrim 

-- 

Tmax ‘trim ’ (3.6) 

is the straight and level throttle setting and Tmax is the thrust ‘trim 

corresponding to maximum throttle setting. 

(3.7) 

During this maneuver, one expects the body rates to be small. 

the commanded values of these quantities are zero. 

Consequently, 

If the throttle setting emerging from this analysis is greater than 

the maximum or less than the minimum, it indicates that the assumed time of 

flight for the maneuver is unrealistic or that the model is inadequate or 

both. 

maneuver feasible. 

This quantity has then to be changed appropriately to make the 

3.2.2 Level Accelerat i on/Decel erat i on 

This is a wings level, constant altitude maneuver with Mach number 

constant or changing at a specified rate. 

trajectory is essentially a subset of the trajectory 3.2.1. 

a,=a,=a,=O in (3.1) results in 

The maneuver modeling for this 

Putting 

h = h,, constant 

Y = 0, constant (3.8) 

and V = V, + it, with \i specified. 
The required thrust and the corresponding throttle can be computed as in 

3.2.1. The open loop control surface settings and the other commands are 

linearly interpolated from the straight and level trim data base. 

3.2.3 Pushover, Pullup 

This is a wings level, constant Mach number maneuver ,n whic.. the 

angle of attack is varied a specified increment about the trim value at sane 

specified rate. The maneuver is shown in Fig. 3.1. The corresponding angle 

of attack history is given in Fig. 3.2. 

Figure 3.1. Pushover - Pullup Flight Test Trajectory. 

21 
5 

Strai 

Figure 3.2. Angle of Attack History Along the Pushover - Pullup 

Flight Test Trajectory. 

The maneuver modeling for this flight test trajectory uses the assumption that pitch rate is close to zero. 

flight, the flight path angle Y can be calculated as the difference between 

the pitch attitude and 0 and the angle of attack a. 

Since the aircraft is in symmetric 

Thus 

Y=O-a, (3.9) 

and u is specified as a function of time, the flight path angle can be computed. Now, 

(3.10) 

22 
where C is the speed of sound specified as a function of altitude h and M 

is the desired Mach number. 

numerically integrated to yield the altitude history. 

may be computed from the following equations. 

Equation (3.10) can be analytically or 

The throttle setting 

Since Mach number is to remain constant throughout the maneuver, one 

V 

can differentiate the expression for Mach number (M = C) with respect to 

time and equate to zero to obtain 

substituting for from (3.10), one has 

v2 ac i = - -sinY c ah 

Equating expressions (3.12) and (3.41, one has 

- v2 ac TCosa - D - gsinY -sinY = c ah m 

From which 

v2 ac msinY D 

c ah Cosa Cosa T = [- - + g]-- + - 

23 

(3.11) 

(3.12) 

(3.13) 

(3.14) 
The drag at the commanded angle of attack can be computed from linearized 

drag coefficient specified as a function of a, the dynamic pressure and the 

reference area. As before, the linear throttle assumption is invoked to 

compute the throttle setting. 

“trim 

“actual Ttrim 

=To--- (3.15) 

3.2.4 Zoom and Pushover 

The zoom and pushover is a wings-level, thrust stabilized less than lg 

maneuver. The flight trajectory is a parabolic path with the target 

Mach/altitude/angle of attack point at the apex. An illustration of the 

zoom and pushover flight test trajectory is given in Fig. 3.3. The maneuver 

begins at 0 with straight and level flight conditions. A transient-maneuver 

is performed to transfer the aircraft to the point A with all controls 

active. At the point A, the throttle is fixed at a predetermined value and 

the aircraft executes the zoom and pushover trajectory. At the point B, the 

thrust control is reinstated and a transient trajectory transfers the 

aircraft back to straight and level flight conditions at point C. 

Le - - ’ - - - 

kV3riable throt tlc-bixed throt tl-Variable throttle 4 

Figure 3.3. Zoom and Pushover Flight Test Trajectory. 

24 
This is perhaps the most complicated of all the symmetric flight test 

maneuvers. This flight test trajectory consists of three phases. In the 

first phase, the aircraft is transferred from its straight and level flight 

condition to the beginning of the parabolic flight path. The second phase 

is the required zoom and pushover maneuver followed by the third phase which 

brings the aircraft back to its original straight and level flight conditions. The first and third phase maneuvers are essentially transients and 

the maneuver modeling discussed in 3.2.1 is directly applicable. The second 

phase will be analyzed in this section. 

A parabolic flight path has the following properties 

1. Horizontal acceleration is zero 

2. Vertical acceleration is constant 

3. Total energy is constant. 

Since the apex speed is specified, say V one has T’ 

T x = VcosY = constant = V 

Here, x is the down range. Note that since the aircraft is in 

symmetric flight, the cross range is zero. Thus, 

V -1 T Y = - + cos Cij-1 (3.16) 

a positive or negative sign has to be chosen based on whether the aircraft 

is flying towards the apex or away from it. 

From the given apex speed, angle of attack and altitude, the lift and 

drag can be computed using the straight and level trims data base using 

25 
linearized aerodynamic coefficients. From constant energy property, one 

has, at the apex of the parabola, 

TCosa = D 

T (3.17) 

From which, the throttle setting at the apex can be computed as follows 

‘trim 

‘actual Ttrim 

=TO- (3.18) 

with lift and thrust, the vertical acceleration at the apex can be comguted. 

Thus 

TsinaT + L 

m -g=ga (3.19) 

Note that ga should be a negative quantity, numerically less than the 

acceleration due to gravity g. The acceleration g has to remain constant 

through the parabolic path. To summarize, the aircraft trajectory approximates the path of a projectile in a uniform conservative force field. 

total energy of the aircraft in this field is given by 

a 

The 

= constant V2 E=h+-- (3.20) 

Expression (3.20) can be used to compute the speed along the parabolic path, 

given the altitude. 

26 
Next, given the altitude at which the parabola is to begin, and 

perhaps end, one can write 

ga h = h, + hot + -t2 2 

with h, computed using the following relations, 

V -1 T Yo = cos [-I 

vO 

and 

h, = V,sinY, 

(3.21 ) 

(3.22) 

The time of flight on this parabola is easily computed with equation (3.21) 

in the general case or with the following equation if the zoom and pushover 

parabola is symmetric about its axis. 

(3.23) 

Note that the throttle is to remain fixed at the value given by the equation 

(3.18). The angle of attack throughout the parabolic path is computed from 

T-a + Lo + Loa 

m 'a - gcosy = 

27 

(3.24) 
and small angle approximation for a has been used. ‘actual with T = Tmax 

The open loop control surface settings are assumed to be the interpolated values from the straight and level trim data base. 

3.2.5 Excess Thrust Windup Turn 

This is a maneuver with angle of attack linearly increasing from the 

wings-level trim condition to some specified final value at a specified 

rate. The maneuver is performed at constant altitude and constant Mach 

number. A schematic figure of this maneuver is shown in Fig. 3.4. 

Figure 3.4. Excess Thrust Windup Turn Flight Test Trajectory. 

As mentioned elsewhere in this report, the trim data base consists of 

straight and level trims at several Mach-altitude pairs along with level 
turns at several Mach-alti tude-load factor conditions. Since excess thrust 

windup turn can be considered to consist of several level turn trim conditions appropriately pieced together, this maneuver model is merely a 3- 

Dimensional interpolation using the trim data base. 

3.2.6 Constant Throttle Windup Turn 

This is a maneuver with angle of attack increasing linearly at a 

specified rate from trim to some specified final value. The maneuver is 

performed at a predetermined, constant thrust level. Mach number is maintained by trading potential for kinetic energy via an appropriate altitude 

rate. 

I 

I 

Since Mach number is constant, one can write as in equation (3.12) 

i=-- ’* sinY 

C ah (3.25) 

I 

! Further, since the altitude, Mach number and angle of attack at the initial 

point are known, one can compute 

I 

(3.26) 

from the trim data base. 

the actual thrust 

Let the throttle be fixed at a value nR. Thus, 

I = Tmax ‘R (3.27) 

2s 
If the actual thrust TR is greater than that required for level turn at the 

given Mach-altitude-angle of attack condition, the Mach number can be maintained constant only by a positive altitude rate. The reverse applies whenever T 

be 

is less than Ttrim. Let the excess thrust over the level turn trim R 

trim AT-T - R 

Now, one has 

v2 - ac AT Cosa - gsinY -sinY = C ah m 

from which , 

-1 ATCosa 

= sin i m(- C v2 - ah ac + g> I 

(3.28) 

(3.29) 

(3.30) 

Also, 

li = VsinY (3.31) 

The expression (3.31) can be nunerically integrated over one step to obtain 

the new altitude. The calculations may be repeated as many times as one 

wishes to obtain the trajectory. 

3c 
It is important to begin this trajectory at a high-g turn, since 

fixing the throttle at a straight and level condition can result in an 

initial acceleration or a high path angle climb/descent. Hence, in order to 

avoid confusion, this maneuver requires an initial and terminal maneuver. 

Thus three phases are required to execute this maneuver. 

1. A trajectory beginning at straight and level flight at the 

desired Mach-altitude pair and ending at a high-g level turn with 

all control surfaces and throttle active. 

2. Constant thrust windup trajectory. 

3. Terminal maneuver at constant altitude transferring the vehicle 

from the level turn at constant thrust conditions to straight and 

level flight. 

A typical constant thrust windup turn trajectory is given in Fig. 3.5. 

Figure 3.5. Constant Thrust Windup Turn Trajectory Descending Flight. 

31 
3.2.7 Constant Dynamic Pressure and Constant Load Factor Trajectory 

This maneuver is initiated at a predetermined load factor, Mach 

number, and dynamic pressure. Thus, the initiation of this maneuver is not 

necessarily the wings-level condition. This maneuver can be either an 

ascending or descending at a specified Mach number rate. Dynamic pressure 

and load factor are held constant throughout the maneuver. Altitude is 

gained or lost to maintain dynamic pressure with changing Mach number. 

As in the maneuver 3.2.6, this trajectory also requires three phases. 

The first is a level turn to achieve the required load factor from straight 

and level flight at the desired Mach-altitude pair. The third phase 

restores the aircraft to straight and level flight at the final Machaltitude condition. Both these maneuvers can be constructed by 

interpolating between the stored trim data points at the current Machaltitude-load factor conditions. In the following we shall discuss the 

constant dynamic pressure-constant load factor trajectory modeling. Note 

that this development does not depend explicitly on the load factor. Thus, 

this model is valid for all load factors including a constant unity load 

factor, wings level, constant dynamic pressure trajectory. 

The dynamic pressure, Q is given by 

1 

Q = T p(h) V2 (3.32) 

Differentiating the expression (3.32) with respect to time and using the 

altitude rate equation, with 6 = 0, one has 

- 1 - aP V3sinY + p(h)c = 0 

2 ah 

32 
or 

Since the Mach number is given by 

M = V/C(h), 

in order to maintain 8 desired Mach rate, one must have 

A=- t v2 ac 

- - sinY ah 

f r om ~ whi ch 

v2 ac t = k + c - sinY ah 

Equating the expressions (3.33) and (3.34), one has 

Y = Sin 

33 

(3.33) 

(3.34) 

(3.35) 
There are two important conclusions that can be drawn from the expression 

(3.35) 

aP ac 1. Since - and - are negative (density and speed of sound decrease 

with increasing altitude), a positive fi will result in a climbing 

trajectory while a negative fi will yield a descending path. 

ah ah 

are aP 2. This maneuver cannot be flown at altitudes where - and - ah ah 

nearly zero unless the desired Mach rate is also zero. 

As before, the altitude rate equation 

L = VsinY (3.36) 

may be numerically integrated over a small time step to obtain the new 

altitude. The actual throttle setting is again computed from the level turn 

trim thrust and throttle setting at the current Mach-altitude-load factor as 

follows . 

(3.37) 

(3.38) 

(3.39) 

34 
I 

The angle of attack and drag in the expression (3.38) are the interpolated 

values from the trim data base at the current Mach-altitude-load factor. 

The Mach number as a function of time is obtained from 

M = M, + it (3.40) 

The command body rates and open loop control surface deflections are again 

the interpolated values from the trim data base. 

3.2.8 Constant Reynolds Number and Constant Load Factor Trajectory 

This maneuver is initiated at a predetermined load factor, Mach number 

and Reynolds number. Thus, the initiation of this maneuver is not 

necessarily the wings-level condition. This maneuver can be either 

ascending or descending at a specified Mach number rate. Reynolds number 

and load factor are held constant throughout the maneuver. Altitude is 

gained or lost to maintain Reynolds number with changing Mach number. 

This maneuver model is different from 3.2.7 only in the way that one 

computes the flight path angle and the throttle setting. Hence only these 

two aspects will be discussed in the following. As in maneuver 3.2.7, the 

modeling does not depend explicitly on the load factor. Consequently, the 

following development is valid for all load factors including a wings level, 

unity load factor - Constant Reynold's number trajectory. 

The Reynold's number, Re is given by 

Re = VDp(h) , where p is the Viscosity of atmosphere (3.41) v(h) 

35 
Differentiating the expression (3.41) with respect to time and using the 

altitude rate equation, with Fie = 0; one has 

Equating the expression (3.42) to equation (3.34), 

(3.42) 

(3.43) 

As in the constant dynamic pressure, constant load factor flight test trajectory, we note that this maneuver cannot be flown at altitudes where ah, 

aC 

ah ah 

actual throttle setting can be computed from the level turn trim thrust and 

throttle setting at the current Mach-altitude-load factor as follows. 

aP 

- and - are nearly zero unless the desired Mach rate is also zero. The 

1 aP msinY D 

T= [(;%- - P -)V2 ah + g] - Cosa + - Cosa 

“trim 

“actual ‘Itrim 

=T- 

(3.44) 

(3.45) 

To facilitate easier computations, a FORTRAN program has been written 

to generate the required commands and open loop control settings given the 

appropriate data. A listing of this code is given in Appendix C. 
SECTION 4 

MANEWER AUTOPILOT DESIGN 

The previous section described the required maneuver modeling, whereby 

for eight chosen maneuvers, a subset of the outputs are constrained to 

prespecified time histories. For the control analysis and design done in 

this study to have any value when applied to the F-15 nonlinear simulation 

or the actual aircraft, consistent values for all of the dynamic states and 

corresponding control values along the trajectory must be found. There are 

at least two straightforward ways to generate the required reference states 

and controls. First, one could iteratively simulate with the nonlinear 

model until an open loop law approximates the desired output time histories. 

This could be done systematically with numerical nonlinear optimization, 

using a parameterization of the control surface and thrust time histories, 

Or secondly, one can trim the nonlinear simulation at a number of conditions 

close to the desired trajectory and approximate the dynamic reference trajectory from these trim values. The latter approach was chosen both because 

of the completeness and flexibility of this tabular representation of the 

nonlinear aircraft characteristics, and because of the connection with the 

perturbation trim controllers designed to work along with the reference trajectory commands. It should also be pointa out that this "static" approach 

eliminated the need for any nonlinear simulation during the control design 

stage, apart from execution of a linearizing program [ll] which contains the 

nonlinear F-l5 aircraft equations. 

The next subsection shows how a table of trim values can be used to 

develop a "linear model" of the entire F-15 flight test system for design 

and evaluation of the aircraft dynamic response in specified nonlinear 

maneuvers. A second subsection outlines two different linear control design 

techniques, evaluating their strengths and weaknesses, giving perturbation 

controller designs using these two techniques. A final subsection presents 

an assessment and extension of the nonlinear prelinearizing control 

technique [12], the application of which will be demonstrated in the next 

study phase, along with linear gain scheduled controllers, on the full 

nonlinear dynamic simulation. 

37 
4.1 THE F-15 FLIGHT TEST SYSTEM "LINEAR" MODEL 

It must be emphasized that while the control design approaches demonstrated in this work are based on linear models, the maneuvers desired are 

highly nonlinear. As discussed above, the nonlinear characteristics of the 

F-15 are condensed into a table of trim values, which when properly used 

with linear perturbation controllers gives a linear time varying simulation 

which accurately represents the nonlinear aircraft response through the 

nonlinear maneuvers. Adequate control through the nonlinear maneuvers 

requires not only consistent open loop reference commands but 1 inear 

perturbation gain-scheduled controllers as well. The gain-scheduled 

perturbation controllers are designed efficiently by decomposing the desired 

eight maneuvers into 15 linear design points for both the fixed and variable 

throttle cases. 

4.1.1 The F-15 Nonlinear Tabular Model 

The nonlinear dynamics of the F-15 can be represented over the 

envelope with a sufficient number of trim values ? and fi: 145 points were 

stored, distributed as shown on the altitude-Mach plane in Figure 4-1. 

A much coarser grid of flight conditions, than shown in this figure 

for the reference trajectory, was used for the linear perturbation models 

about the trajectory, and consequently in the linear control design stage. 

Table 4-1 shows the coarse discretization considered. Since it becomes 

increasingly difficult to trim the aircraft, particularly at high load 

factors, in the off-diagonal points in the altitude-Mach plane, only five 

conditions, the diagonal ones indicated in Table 4-1, were used with the 

three different load factors -- lg, 38, and 4g. The eight maneuvers were 

initiated at different conditions (see Section 3) to exercise controllers at 

various points on the flight envelope. 

38 
0.3 0.4 0.6 

AAAA 

/ 

/ 

, 

/ 

, 

I 

I 

I 

I 

I 

I 

I 

I 

I 

D 

1.0 1.2 1.4 

0 STRAIGHT AND EVE1 TRIM 

A EVE L WRY TRIM AT 2 WAD FACTOR 

0 EVE1 TLlRS TRIU AT 4 EA!) FACTOR 

Figure 4-1. F-15 Nonlinear TabGlar Model 

1.6 1.8 

Trim PointTABLE 4-1. F-15 LINEAR PERTURBATION WDEL AND DESIGN CONDITIONS 

h/M 0.6 0.8 1 .o 1.2 1.4 1.6 1.8 2.0 

50k X 

40k X 

30k X 

20k X 

1 Ok X 

39 
The trim conditions used in this work are those with - = 0, namely the 

net forces and moments are zero. Since altitude rate is zero, a load factor 

greater than one automatically pulls the aircraft into a level turn trim. 

The static nature of the trim points is overcome in constructing a 

dynamic reference trajectory by assuming a ramp in velocity for a specified 

time of transition between two trim points. This constant dV/dt yields the 

thrust adjustment necessary to add to the trimmed thrust for a "dynamic" 

reference thrust and - V command. 

adjusted - x command in all the states; however, the required computation and 

storage is large. Therefore, only the feedforward thrust (or throttle) command is adjusted as described here, and the linear perturbation controller 

generates the extra transient control commands necessary for all the states 

to transition between the trim points. This mechanization is discussed mme 

fully in the next subsection. 

Conceptually, one could solve for an 

In summary, the nonlinear F-15 model required for eight nonlinear 

maneuvers has been represented by a tabular description of trim values of 

the nonlinear simulation at discrete points on the flight envelope. The 

number of points used to represent the reference command is much finer than 

for the linear models because the reference command effectively contains the 

nonlinear behavior in it. Load factors greater than one, level turn trims, 

have been used to generate the asymmetric models. 

By keeping the linear perturbation model grid coarse, only a few perturbation controllers need be designed, reducing both linear model 

scheduling and gain scheduling requirements in the linear time-varying 

simulation validations. Nonlinear simulations in the next research phase 

will confirm whether or not the number of design points is sufficient. 

Accurately speaking, the ffcontroller", is not merely the linear perturbation 

gains, but the way they are mechanized along with the reference commands. 

40 
4.1.2 Linear Perturbation Controllers 

Currently, control systems for nonlinear plants are synthesized using 

perturbation models or the so called linearized plant models in conjunction 

with the powerful linear system design approaches. The controllers so 

obtained may be termed Linear Perturbation Controllers to denote the linear 

nature of the controllers and to indicate their function, viz, controlling 

perturbations about the reference condition. If the system is required to 

track a given command, the perturbation models need to be generated at 

several operating points along the command history and controllers designed. 

In highly nonlinear systems such as aircraft, these controllers can display 

significant variations, often requiring these to be scheduled as a function 

of the independent variable. 

The objective of the controller is to ensure that a given nonlinear 

system 

J = f(X,U) (4.1) 

follows a given trajectory - X(t). 

turbation models or the linearized models, some points along the desired 

trajectory - X(t> are selected, say -1 X ’ -2’ X -3 X . . . . A set of controls 

corresponding to these points, -1 U’ -2’ U -3 U . . . . are next computed such 

that 

Here X E En, U E Em. To obtain the perNote that this is not the only possible approach. If the desired trajectory 

satisfies - >i = f(X,v) for nonzero - >i, then it can be used in the subsequent 

development. 

41 
To derive the perturbation model with state perturbations 6X and control perturbations 6U, let 

(4.3) 

Expanding the nonlinear system (4.1) about -- X, U and retaining only the first 

order terms (this implies that the perturbation are small), one has 

6i = fX6X + fU6U (4.4) 

The subscripts denote partial derivative matrices. 

depend on -- X, U. 

which a controller of the form 

Note that f and fU X 

The expression (4.4) describes a linear dynamic system for 

6U = K6X (4.5) 

can be designed at the operating points X1, X 

points, the following procedure is setup. 

-2' ..- 

Next, to ensure that the system transits through these operating 

Assume that a linear interpolation scheme between X1, X . . . X -2 ' -n 

adequately describes the desired trajectory. Further, assume that at any 

42 
0. In this case, the 

interpolated reference flight conditions zj, in between Si, Xi+, : 

perturbation model (4.4) is given 

i = fX6X + fU6U - - ict 4.6) 

is a piecewise constant function and appears as a disturbance in 

the perturbation model. 

As noted earlier, the perturbation controller is designed with - >i = 0. 

Recalling that 

U=6U+U - (4.7) 

Using (4.7) in (4.6), the linearized equations can be put in the form 

or 

i = fXX + fUU - Z(t) 

where 

Z(t) = fx? + fG 

(4.8) 

43 
Similarly, the perturbation controller (4.5) becomes 

U 

> 

U - U = K(X-X) - - 

or 

__--_--__----- 

I I - T -_- H Nonlineer 

Sys tern 1 I 

I I - H 

I 

b = flH,U) 

U = K(X-X) + U (4.9) - - 

I 

I 

I. 

' 

IK 

Expression (4.9) indicates the implementation of the linear perturbation 

cc?:troller, given in Figure 4-2. for clarity. 

1 

REFERENCC I 

TRRJECTORV I 

GENERATOR I 

I 

7 

Oirlurbances 

Figure 4-2. Implementation of the Linear Perturbation Controller 

Note that the gain matrix K varies as a function of some scheduled 

variable. Expression (4.8) has an interesting interpretation, viz, when the 

perturbations about the desired path are small, it describes the nonlinear 

system dynamics with a high degree of fidelity. Hence, under closed loop 

control, the dynamics given by (4.8) will be very close to that of the 

44 
original onlinear system (4.1). This fact means that linear, time varying 

simulations can be mechanized in a straightforward way from the relationships and models discussed here. While section five reviews the concept of 

linear time-varying simulation before giving the validation results, it is 

convenient to continue our line of thought and go over the mechanization 

here in the maneuver autopilot design section. 

4.1.3 Linear Time Varying Simulation 

As noted elsewhere, for highly nonlinear systems such as aircraft, the 

gain matrix K in (4.9) would display large variations as a function of the 

flight condition. And hence, some type of scheduling strategy will be 

essential for the satisfactory operation of the control system. To evolve 

the scheduling strategy, it is desirable to have a simulation of the system 

which has lesser complexity than the original nonlinear system. 

Examining expression (4.8) in view of the above, one finds that the 

partial derivative matrices f 

computed at the controller design stage. The reference trajectory -- X, U is 

also known. Since this expression describes the nonlinear system dynamics 

adequately for small perturbations, it may be used to develop a linear time 

varying simulation to evaluate the controller scheduling. Figure 4-3 gives 

the formal structure of the linear time varying simulation. This block 

diagram is structurally similar to the linear perturbation controller 

implementation given in Figure 4.2. The essential difference between them 

is that the nonlinear aircraft model has been replaced by a linear timevarying model with disturbance inputs. 

- U are linearly interpolated between time points. 

mechanization, the partial derivative matrices f 

same time points and linearly interpolated. Thus, along a contemplated 

maneuver, three or four -- X, U, points are chosen and the corresponding fX, 

fU, matrices are stored. 

and fU as functions of -- X, U have already been X 

As mentioned in Section 4.1.2, - X and 

In order to simplify the 

and fU are stored at the X 

Note that in the present analysis the -- X, U, points 

45 
are chosen such that at an intermediate point f(Xj, U.) 1 0. 

of simulations, if it turns out that in order to track the required - X 

history, the total control required is greater than that available, it is 

indicative that either the assumed maneuver time is unrealistic or that the 

model is inadequate or both. 

In the course 

-J 

H = fxx + 1"U 

-(f" + 1 yl 

REFERENCE 

* TRRJECTORY 

GI NE R ATOR K 

+ 

TIME 

Figure 4-3. Linear Time Varying Simulation 

A description of how a generic time varying simulation can be implemented in the MATRIXTM SYSTEM BUILDTM model building and simulation program 

is given in Appendix D. 

46 
4.2 LINSAR DESIGN TECHNIQUES 

Given a small set of linear trim conditions, linear perturbation controllers can be designed to track desired reference commands. The simplest 

practical controllers, once the outputs are augmented with appropriate 

integrators for desirable steady state tracking, are output feedback 

controllers. 

eigenstructure assignment and the minimum error excitation technique. 

Two different output feedback ipproaches were evaluated, viz, 

4.2.1 Eigenstructure Assignment Design 

As described in Section 2, the model through which the desired 

maneuver autopilots (MAP) must work includes the fully augmented aircraft: 

This actually makes the control design task more difficult than dealing only 

with the bare airframe. The combination of CAS states and integral error 

states give a considerable number of eigenvalues that are slow, and therefore must be moved, and also exhibit a reasonable degree of state coupling. 

This coupling results in some sensitivity of results to the choice of 

eigenspace requested. The model used and specific results are given in 

Appendix A. Complete details of this eigenassignment approach can be found 

in 1133. It is perhaps of interest to note that in addition to the 

capability to handle controller structure constraints, this technique can be 

modified to accept partial specification of eigenvectors. This feature can 

be valuable in high order systems. In summary, to employ this synthesis 

approach, the following are required. 

(i) Based on the practical aspects of the problem, choose a minimal 

set of measurements which will permit the designer to achieve the 

desired performance. Introduce dynamic compensators such as 

integral feedbacks, lead-lag networks, etc., based on experience. 

(ii) Choose a set of desired eigenvalues and eigenvectors equal to the 

number of outputs. 

47 
While the selection of desired eigenvalues is often apparent in a 

given problem, the desired eigenvectors are difficult to select. Three 

approaches were developed to help guide this choice, viz, minimally restructured eigenassignment , decoupling eigenassignment and dominant mode 

ei genassi gnment . 

4.2.1.1 Minimally Restructured Eigenassignment 

Since it is known that the closed loop eigenvectors lie in a subspace 

spanned by the columns of (AiI-A) B , i = 1, . . . n, linear combinations of 

these vectors were used as desired eigenvectors. The weights to be used in 

generating these linear combinations were constructed from the additional 

information that unassigned eigenvectors should be close to their open loop 

values in a least square sense. 

-1 

4.2.1.2 Decoupling Eigenassignment 

Since we are interested in having the least cross axis coupling in the 

controller as possible, the desired eigenvectors in the longitudinal channel 

may be chosen so that the responses from lateral channels are blocked, i.e., 

select eigenvectors as 

n 

vd = 1 aivi 

i= 1 

I 0 

48 
V are the desired eigenvectors. a are selected using the same criteria as 

in the minimally structure case. 

to work at each flight condition by iterating on the desired eigenvalues, 

they failed to easily produce a set of acceptable eigenvectors which could 

be used at other flight conditions. Next, partial specification of desired 

eigenvectors was attempted using the dominant mode approach. 

d i 

Though these two approaches could be made 

4.2.1.3. Dominant Mode Eigenassignment 

According to ref C131, complete specification of desired eigenvectors 

are neither necessary nor desirable. Depending on the states that should or 

should not participate in a given mode, appropriate entries in the 

eigenvectors are made ones or zeros, leaving other entries free. With these 

eigenvectors, the desired eigenvalues are moved as far left from the 

imaginary axis as possible with least change in the location of unplaced 

eigenvalues. The desired eigenvectors so obtained appeared to work over 

most flight conditions. Note, however, that extensive iterations on the 

desired eigenvalues may often be required to produce a satisfactory design. 

4.2.1.4 Conclusions on OutDut Eiaenstructure Assianment 

Specific maneuver autopilot design results are discussed in Appendix 

A, along with the more complete description and evaluation of output 

eig€?nStrUCtWe assignment for application to flight test trajectory control, 

including difficulties in selecting desirable eigenvalues and eigenvectors. 

This approach demands several iterations to converge to a satisfactory 

design and does not appear to easily give suitable insight for output feedback design of high order multivariable systems which will be used at other 

operating points. If a rational method to generate an achievable set of 

eigenvectors is devised, this technique will be made more attractive. One 

possibility might be to generate gradients of the eigensystem between flight 

conditions and include this information in the single point design technique. The next two subsections describe two other techniques of output 

feedback gain solution. 
4.2.2 Minimum Error Excitation Output Feedback Design 

Following Kosut's [81 notation in describing his development of the 

minimum error excitation output feedback design method, for the linear 

sys tem 

x = AX + Bu, 

y = Hx, 

one first designs a full state LQ regulator 

m T T J = j (xH Q Hx + u Ru)dt, 20 Y 

with the optimal feedback control law u* = F*x. 

Both because algebraic output feedback methods do not guarantee 

stability and because of the presence of slow closed loop full-state 

feedback modes (either neutrally stable unobservable modes or other sluggish 

phugoid like, spiral or integral error modes), it is desirable to design 

with a guaranteed stability margin [14, 153. Specifying a stability margin 

of u ensures that the real parts 

This is equivalent to optimizing 

m 

J = 1 -2at e (xH 4 Hx 20 Y 

of all closed loop roots are less than -a. 

the performance index 

50 
and can be accomplished by destabilizing the open loop plant by = A + aI; 

solving for the corresponding optimal gains in the standard LQ problem, and 

using them with the original open loop dynamics matrix A. Costs were chosen 

according to Bryson's rule [153, the inverse of the squared deviation 

desired on inputs and outputs, with a scalar factor between Q and R to 

regulate the extent of high gain solution achieved. 

A minimum norm output error feedback law can be determined from 

u = C y, with 

Y 

+ 

C = H F*, Y 

+ 

where H is the pseudoinverse of H. This is merely a least squares projection of the full state gains onto the output subspace. 

The sensitivity of the projection described above can be minimized by 

doing a weighted least squares, where the weight is the closed loop state 

covariance excited by a unit error density. One merely solves the Lyapunov 

e quat ion 

(A + BF*)P + P(A + BF*)~ + I = 0, 

to obtain the constrained gain 

T T -1 F = F*PH (HPH ) H, or 

Y 

T T -1 C = F*PH (HPH ) Y 

The destabilized open loop plant 1 was used instead of 

equation solution to retain as much of the full-state law guaranteed 

stability margin as possible. 

A in the Lyapunov 

51 
Maneuver autopilots were designed at all flight conditions within two 

iterations using the above output feedback design procedure. The design 

criterion were 

Re(X.) <-.2 => a 5 second maximum time constant in tracking, and 

1 

6 > .7 => low overshoot. 

With the integral error states and general low frequency oscillatory poles 

pairs due to lateral/longitudinal coupling, the plant is not a simple one to 

control. Since Mach and load factor were chosen as the scheduling 

variables, 15 designs were generated - five designs at three load factors 

shown in Table 4-2. The state and control weighting matrices used in this 

synthesis is listed in Appendix E. 

are also given in this appendix. 

The corresponding output feedback gains 

Since the zoom-and-push-over and excess thrust windup turn require 

fixed throttle, 15 more designs were generated without this control, also 

shown in Table 4-2. 

Typical initial condition responses in a straight and level flight 

condition at 40K feet and Mach of 1.4 are shown in Figure 4-4. 

52 
TABLE 4-2. SLOWEST MODE BEHAVIOR AT ALL DESIGN CONDITIONS 

(Subscript 1 corresponds to the design with all controls, subscript 2 corresponds to the desigl without 

thr ot tie contr 0: ) 

- --- - - -- -- - 

h ,m 

LOA3 h = 10000' h = 20000' h = 3OOOO' h = 400001 h = 50;:' 

FACTOR M = 0.8 M = 1.0 M = 1.2 M = 1.4 M = 1.a 

A, = -0.277 A, = -0.289 A, = -0.3 A, = -0.308 A, = -0.29; 

5, = 0.996 5, = 0.97 51 0.95 51 = 0.945 5, = 0.94 

1 

A, = -0.28 A, = -0.257 A, = -0.234 52 = -0.2276 A2 = -O.?'s( 

52 = 1 5=1 52 = 1 52 = 1 52 = 1 

-__--. 

A, = -0.28 A, = -0.26 A, = -0.23 A, = -0.2274 A; = -0.22.~ 

5, = 0.99 5, = 0.94 5, = 0.9 5, = 0.2 A, = -c . F.=: a>-

2 

A, = -0.288 A2 = -0.268 A, = -0.254 A, = -0.276 A2 = -0.242 

E2 = 0.99 c2 = 0.947 Ez = 0.947 52 = 0.958 52 = 0.88 

AI = -0.267 A, = -0.23 A, = -0.209 A, = 0.17 A, = -0.116 

5, = 0.99 = 0.92 5, = 0.866 5, = 0.98 = 0.95 

A2 = -0.28 A, = -0.23 A, = 0.243 A2 = -0.2405 A, = -0.209 

52 = 0.99 52 = 1 52 = 0.95 52 1 E,2 = 0.895 

4 

53 
1 uo 

80 

60 

4a 

20 

0 

- za 1 .... I .... I .... I .... I _.._ I....t .... I ____ l....I... J 

0 3 10 15 20 25 50 35 40 45 50 

Figure 4-4. Initial Condition Response for the Output Feedback Minimum 

Error Excitation Perturbation Controller (Reference 

Condition H = 40k' , Mach = 1.4, Straight and Level Flight) 

Section 5 gives the tracking behavior of the output feedback minimum 

norm controllers in the eight maneuvers of interest. 

4.3 NONLINEAR FLIGHT TEST TRAJECTORY CONTROLLERS 

Research on nonlinear flight test trajectory control (FTTC) design was 

conducted with the goal of completely eliminating the need for gain 

scheduling. 

theoretical work can be applied to the FTTC problem before a demonstration 

problem is given to illustrate the approach. 

A brief literature review is given below, outlining how recent 

54 
This research deals with the synthesis of nonlinear flight test 

trajectory controllers using the recent results in prelinearizing transforms 

due to Meyer [17-333 and singular perturbation theory [34-363. The use of 

singular perturbation theory in this problem simplifies the command 

generation scheme in addition to providing a consistant approach for 

eliminating ignorable state variables. The prelinearizing transformations 

are more transparent in this formulation. The slow-fast computations are 

clearly separated and can be carried out at different rates on the flight 

control computer. It is interesting to note that in Ref. 21, even though 

the controller development did not make use of singular perturbation theory, 

the time-scale separation formed a basis for implementation on the flight 

control computer. A schematic block diagram of the slow-fast flight test 

trajectory controller is given in Figure 4-5. 

Flight test controller synthesis will be developed for the F-15 

fighter aircraft in the next contract phase. For the F-15 and most fighter 

aircraft, it can be assumed that the aircraft under consideration has the 

four usual controls: throttle, aileron, rudder and elevator. The objective 

of the flight test controller is to track the given commands in airspeed, 

angle of attack, angle of sideslip and altitude in presence of disturbances 

and modeling imperfections. It is clear that the commanded trajectory has 

to be executable by the aircraft under consideration. Note that the flight 

test control problem discussed here is distinct from those described by 

Meyer, et. al. [20-221 since, in their work, the trajectory to be followed 

consisted of the three position components specified as functions of time. 

Modeling and time-scale separation can be exploited to a considerable 

degree in this approach. The mechanization details of prelinearization and 

the slow-fast controller synthesis for general flight test maneuvers will be 

given in the next project phase based on the problem formulation developed 

her e. 

55 
Figure 4-5. Singular Pert UT ba I A 3n N on1 inear F 1 i ght Test Traj ect or y 

Controller. 

Slow states: V, a, 6, h 

Fast states: p, q, r, 8, Q 

Very fast states: CAS and Actuator states 

An illustrative example of this nonlinear controller approach is given 

in Appendix B. 

56 
SECTION 5 

SIMULATION AND EVALUATION 

The performance of perturbation controllers for initial condition 

errors was verified during the design phase. Some techniques for evaluating 

their tracking performance will be discussed in this section. Though it is 

clear that the actual performance of these controllers can only be assessed 

with the full nonlinear aircraft model, it is desirable to introduce an 

intermediate validation phase to ensure a smooth transition from 

perturbation controller design to full nonlinear simulation. It should be 

emphasized at the outset that the simulations discussed here are approximate 

and consequently the tracking performance will differ in the nonlinear 

simulation validation in the next study phase. 

5.1 MANEWER SIMULATION 

Originally, it was decided to carry out a linear time varying 

simulation of the systems using linearized aircraft models along the desired 

flight-test trajectory as discussed in Appendix D. For a two state variable 

model analyzed in Appendix B, the computing time was small. However, the 

aircraft and CAS system has 31 states and continuous interpolation was found 

to be extremely time consuming. I In view of the excessive computational 

effort required, and the limited value of the information obtained, it was 

then decided to switch models and controller gains along a desired flight 

test trajectory. This approach introduced artificial gain sui tching 

transients and also led to misleading conclusions. Hence whenever feasible, 

the simulations discussed here used one interpolated model, and gains based 

on the flight condition halfway through the maneuver. With this approach, 

the modeling inaccuracies will be almost equally distributed throughout the 

maneuver. 

57 
It should be emphasized that the simulations with linearized models 

will test only the feedback controller portion of the maneuver autopilot. 

The open loop control histories generated from maneuver modeling can be 

tested only during the full nonlinear simulation of aircraft and CAS. 

5.2 MANEWER SIMULATION MECHANIZATION 

The linear perturbation equations used for design have been discussed 

in Section 4. In this section, these will be modified to generate linear 

simulations. The linearized aircraft with CAS is of the form 

6x = F6x + C~U 

6y = H6x 

The output feedback perturbation controller is of the form 

6u = c 6y. 

Y 

Expanding the perturbation equations back out gives 

58 
from which the maneuver simulations can be mechanized. Note that can be 

viewed as a disturbance (and neglected) or as a part of the external 

reference command. 

cannot be ignored, but were computed numerically with a forward differencing 

of x. 

- 

For the high speed maneuvers simulated here - clearly 

- 

5.3 MANEUVER SIMULATION RESULTS 

As discussed earlier, - was computed numerically from - x and used as 

part of the reference command. Since - x has corners, there are jumps, or 

spurious step inputs due to the discontinuities in - i. 

quadratic or cubic spline fits to the trim points x will remedy this 

problem; however, the effects of the discontinuities can clearly be seen in 

the tracking trajectories in this subsection. 

Using a smooth - x from 

-i 

In the following, typical simulation results for each flight test 

trajectory will be presented. The flight conditions in these simulations 

are chosen so that the maneuver autopilot is exercised over nearly the 

entire aircraft envelope. Except where indicated, in the plots that follow, 

the dotted lines denote the commanded variables generated from the maneuver 

modeling program while the solid line represents the trajectory evolution 

from the linear simulation. 

5.3.1 Transient Trajectory 

In this simulation, a transient trajectory is setup to transfer the 

aircraft from straight and level flight conditions at 20000' altitude and 

Mach 0.8 to straight and level flight conditions at 30000' altitude and Mach 

1.2 in 60 seconds. The simulation results are presented in Figs. 5-1 

through 5-3. The altitude tracking is very good. However, the throttle 

history in Fig. 5-3 indicates that during the first five seconds, the 

controller commanded a negative throttle. This is caused by the nominimum 

phase behavior of the aircraft, clearly discernable in the Mach number and 

59 
angle of attack histories. In the actual implementation, the commanded 

altitude can be modified to include an initial descend leg in order to 

compensate for the nonminimum phase behavior. Alternately, the commanded 

Mach number can be modified to have an initial decreasing segment. In any 

case, mere command modification would fix the initial negative throttle 

difficulty. 

60 
cW 

w 

L 30000 

4: 2?0W !s 2 24000 

21000 

0 10 20 90 40 50 80 70 

TIME, SECONDS 

1.2 

.8 

0 10 ao so 40 60 

"ME, SECONDS 

60 70 

Fig.;!'e 5-1. Altituae and Mach N;unber Evolution Along the Transient 

T r a j e ct or y 

0 

.3 

10 20 30 40 50 

Ry&, SECONDS 

60 70 

0 10 20 30 40 a 

'R~E SECONDS 

60 70 

5-2. Angle of Attack and Flight Patrl Arigls Evolution Along the 

Trans i en t T r a j e c t or y 

61 
u 

w 

n 

LOO 

0 

-50 

- 100 

0 10 20 90 40 50 60 70 

TIyE, SECOND9 

E- -.012 

5 -.015 

0 10 ao 90 40 60 60 70 

RyB, SECONDS 

Figure 5-3. Throttle and Elevator Deflection Along the Trar.:-?::- 

Trajectory . 

5.3.2 Level Acceleration 

A level acceleration flight test trajectory at 3OOOO' altitude is 

considered here. 

Mach 1.2 in 60 seconds. The tolerence on altitude is 2 50' while Mach 

number error should be within 20.01. 

maneuver are presented in Figs. 5-4 through 5-6. The Controller was able to 

maintain the altitude within 20.1 feet while tracking the Mach number within 

the given specifications. 

Fig. 5-6 are well within the saturation levels. 

requirment i n t h i s maneuver, it appears that the maneuver time could be 

decreased by about 20 seconds. 

The aircraft is required to accelerate from Mach 0.9 to 

The simulation results for this 

The throttle and elevator deflections given in 

From the maximum throttle 

62 
I 

k 3OUool 

U' 

w 

L 

30000.05 P E 30000 

2 

ZBBQ9.85 

2c1989.9 

1.2 

2 1.1 

5 si1 

.9 

Figure 

0 10 20 30 40 50 60 70 

TIME, SECONDS 

0 10 20 40 60 60 70 

TIME, SECONDS 

Altitude and Mach Number Evolution Along the Level 

A ccel era t ion F 1 i gh t T es t Tr a j e ct ory . 5-4. 

0 10 20 3a 40 50 80 70 

m, SEZON9S 

.00004 

n 

2 .octo02 

9 

- 

zl 0 

-.00002 

--.00004 

0 10 ao 90 40 60 60 70 

TIyB, SECONDS 

Figure 5-5. Angle of Attack and Flight Path Angle Evolution Along 

the Level Acceleration F1 i ght Test Tr aj e ct ory . 

63 
om 

w 

P 

M) iL-0 !s 

*30 

20 

0 10 20 90 40 50 60 70 

"ME, SECOh'DS 

0 10 ao 99 40 60 60 70 

m, SECONDS 

Figure 5-6. Throttle and Elevator Deflection Along the Level 

A ccel er a t ion F1 i g ht T es t Tr aj e ct ory . 

5.3.3 Pushover, Pullup 

This maneuver was initiated at 3OOOO' altitude and 0.8 Mach straight 

and level flight condition. The objective is to track a piewise linear 

angle of attack history given in Fig. 5-8 while maintaining the Mach number 

constant at the initial value. The results of the maneuver simulation are 

given in Figs. 5-7 through 5-9. From Fig. 5-7, it can be seen that the Mach 

nunber error is within 0.001 of the commanded value. The angle of attack 

tracking error is less than 0.2O throughout the maneuver. The throttle 

history given in Fig. 5-9 shows a small negative region at the imitation of 

pullup at 30 seconds and is primarily due to the corner in the angle of 

attack command. Smoothing this corner in the final mechanization would 

eliminate this negative throttle region. 
Ew 

w 

L 

wi 

E 

30000 

2970 

29400 

29 100 

2eaoo 

0 20 40 80 80 10 120 

. =,SECONDS 

0 20 40 00 00 10 120 

"ME, SECONDS 

Altitude and Mach Number EVO~~iiOii ALUIIS the Pushover/Pui;up 

F1 i ght Test Trajectory 

Figure 5-7. 

0 20 40 60 80 LO 120 

m, SECONDS 

0 20 40 60 eo 100 la0 

RyE, SECONDS 

Angle of Attack and Flight Path Angle Evolution along the 

Pushover/Pullup Flight Test Trajectory 

Figure 5-8. 
0 20 4.0 60 IO 100 120 

"ME, SECONDS 

-.016 d 

rT 

ffi 

h 

8 -'Om 

-.02 

0 20 60 BO 100 120 

TIYE, SECONDS 

Figure 5-9. Throttle and Elevator Deflection along the Pushover/Pullup 

F1 i ght Test Trajectory . 

5.3.4 Zoom and Pushover 

As noted in the maneuver modeling, this flight test trajectory is 

executed in three phases. In the first phase the aircraft is transferred 

from straight and level flight conditions to the beginning of the zoom and 

pushover parabolic trajectory. 

pushover trajectory while the third phase restores the aircraft to the 

original straight and level flight condition. 

phases all the controls are active, but the throttle is fixed during the 

zoom and pushover trajectory. 

The second phase consists of the zoom and 

During the first and third 

66 
I 

! 

In the present case, the aircraft is first trimmed to fly straight and 

level at 30000' altitude and Mach 0.4. An initial transient trajectory is 

then executed with all controls active until the point marked A in Fig. 5- 

10. At this point, the throttle is fixed and the aircraft executes the 

parabolic zoom and pushover trajectory until the point B in this figure. 

The throttle is released at point B and the aircraft performs another 

transient maneuver to restore it to the original conditions. 

I 

Since the controller performance along the transient trajectory has 

already been investigated in Section 5.3.1, the tracking performance along 

the zoom and pushover trajectory will only be demonstrated here. The 

aircraft begins the zoom and pushover maneuver at about 28000' and Mach 0.45 

and completes the maneuver at about the same conditions. The controller 

performance is illustrated in Figs. 5-10 through 5-12. The conditions at 

the apex of the parabola is of particular interest in this maneuver. From 

Fig 5-11, it can be observed that the angle of attack at the apex is within 

0.005 radian of the required value. The altitude error is within 50' at the 

apex and the Mach number is within 0.05 of the required value. Initial 

transients in altitude and Mach number can be seen at point A in Fig. 5-10. 

These are essentially due to the availability of just one control variable, 

the elevator, to track three state variables: altitude, Mach number and 

angle of attack. Thus, an initial condition error on altitude would 

propagate to Mach number and angle of attack channels and vice versa. 

I 

L 

I 

67 
rw 

Lr. 

t: 3LOoo 

*\ 3OOOO Lr 

c 

f I I I I I I 1 

t I 1 /I I I I i 

27000 

0 20 40 m 80 loo 120 

SECONDS 

'MK, SECON9S 

Figure 5-10. Altitude and Mach Number Evclution along the Zoom and Pushover 

F1 i ght Test Tr a j e et ory . A: Thr tile Fixed 

B: Thr -:tle Released 

0 

.0 n 

U 

CI .3 

zo 

5 

4 

- .3 

- .6 

m, SECONDS 

Pushover Flight Test Trajectory. 

A: Throttle Fixed 

B: Throttle Released 

C: Apex of the Parabola 

FAz. 5-11. Angle of Attack and FlignL uOya Lr2g-= along the ? - 
0 

m, SECONDS 

Figure 5-12. Elevator Deflection along the Zoom and Pushover 

F1 ight Test Trajectory . 

5.3.5 Excess Thrust Windup turn 

The results for an excess thrust windup turn trajectory at 400001 

altitude and Mach 1.4 are given in Figs. 5-13 through 5-19. The altitude 

and Mach number were required to be constant throughout the windup turn 

trajectory, while tracking an angle of attack command as shown in Fig. 5-14. 

The aircraft roll altitude in this maneuver is close to 70° and results in a 

highly coupled model to be controlled by the maneuver autopilot. It can be 

observed that the maneuver autopilot maintained the altitude within +5l and 

the Mach number error is within - +0.002. 

of the maneuver, the angle of attack error is within 0.005 radian of the 

commanded value. For about 10 seconds during the beginning and end of the 

maneuver, the rudder deflection is close to 72O while in the high roll 

attitude region, an elevator deflection of nearly 17O was demanded. This 

indicates that at the present flight conditions, a less stringent maneuver 

should be attempted. 

- 

Except at the beginning and the end 

64 
e 

w 

E 4OOM 

n 

- 

0 

8 400M) 

39995 

0 10 20 30 40 50 60 70 ao 

TMEj SECONDS 

1.404 

1.396 

TIyEj SECONDS 

Figure 5-13. Altitude and Mach Number Evr,-;ltion along the Excess Thrust 

Windup Turn Flight Test Trajectory 

n .15 

U 

cc 

5 .1 

-06 

0 10 20 30 40 50 60 70 

TIME3 SECONDS 

.OQ 1 n 

4 

cc 

5 

-.001 

0 10 20 30 40 50 60 70 80 

m, SECONDS 

Figure 5-14 Angle of Attack and Flight Pacri kngie Evolution along the 

Excess Thrust dindup Turn Flight Test Trajectory 

70 
1.5 

n 41 

a 

g- .5 

0 

-.5 

0 10 20 30 40 50 60 70 ao 

"ME, SECONDS 

0 10 20 30 40 50 60 70 00 

m, SECONDS 

Figure 5-15. Roll Attitude and Angle of Side Slip Along the Excess Thrust 

Windup Turn Flight Test Trajectory 

.l 

cr) 

\ 2 -05 

[r 

a' 0 

-.E 

-. 1 

0 10 20 30 40 50 80 70 ao 

TIYE, SECONDS 

.1 

ro \ 

n 

d .05 

0' 

0 

0 10 20 30 40 50 60 70 00 

m, SECONDS 

Figure 5-16. Roll and Pitch Body hate Evolution Along the Excess Thrust 

Windup Turn Flight Test Trejectory 

71 
-04 

v) 

\ 

n 2 .02 

e0 

-.02 

0 10 20 30 40 50 60 70 80 

Figure 5-17. Yaw Body Rate Evolution along the Excess Thrust Nindup Turn 

F! i ght Test Trajectory 

0 

-. 1 

n 

a 

4 

E 

5 -3 --2 

- .4 

0 10 20 30 40 50 80 30 ao 

TIKE, SECONDS 

0 10 20 30 40 60 60 70 ao 

m, SECONDS 

Figure 5-18. Throttle and Elevator Deflecliorl alc:.g the Excess Thrusr 

Windup Turn Flight Test TrajFctory 

72 
I 

-0 1 

0 

..01 

0 10 20 30 40 50 00 70 80 

ITME, SECON3LC 

.2 n 

-.l 

- .2 

0 lo 20 30 40 50 60 70 80 

Ryg, SECONDS 

Figure 5-19. Differential Tail and Rudder Deflection along the 

Excess Thrust Windup Turn Flight Test Trajectory. 

5.3.6 Constant thrust windup turn 

A descending constant thrust windup trajectory will be illustrated in 

the following. The aircraft starting at straight and level flight condition 

enters a level turn with linearly increasing angle of attack, upto 30 

seconds in Fig. 5-21. At this point, the throttle is fixed and the constant 

thrust windup trajectory begins. At the end of the maneuver, the Angle of 

attack is gradually decreased to the straight and level trim values. During 

the constant throttle windup turn trajectory, the Mach number is to remain 

constant. The simulation results for this maneuver are given in Figs. 5-20 

through 5-26. From Fig. 5-20, it can be seen that the Mach number was 

maintained within +0.0075 while the angle of attack tracking error was 

within 0.01 radians. The control surface deflections were within the 

saturation limits except at the point where the constant throttle windup 

trajectory began. This is due to the corner present in the altitude and 

angle of attack command histories. By smoothing these corners in the 

commands, the control surface limit violation can be avoided. 

- 

73 
a 20 Ba EO 100 120 

TIyE, SECONDS 

32000 

1 .OM 

0 

r 1 

-996 

0 20 40 60 BO 100 120 

fDcg, SECONDS 

Figure 5-20. Altitude and Mach Number Evolution Along the Constant 

Throttle Windup Turn Flight Test Trajectory. 

2 C 

ri. 

[r 

e2 .l 

TIyg, SECON3S 

20 

8 -.l 

LT 

4- 

a 

x 

0 20 60 80 100 120 

'JIYB, SECONDS 

Figure 5-21. Angle of Attack and Flight Path Angle Evolution Along the 

Constant Throttle Windup Turn Flight Test Trajectory. 
1.5 

n u1 

0 

- .5 

20 40 60 BO 100 120 

mJ SECONDS 

0 20 60 eo 100 120 

m> SECONDS 

FIpd,-.? 5-22. Roll Altitude arid Angle of Side Slip Ev3luLA9n Along the 

Constant Throttle Windup Turn Flight Test Trajectory. 

Lo 

\ 

U 

E 

n 

n? 

.1 

0 

.1 

20 40 60 BO 100 120 0 

m, SECONX 

E0 l I I c" I...r...l...:*.> 1 

20 40 60 80 100 120 0 

m, SECONDS 

Figure 5-23. Roll and Pitch Body Race Lv;lcltion Along the Constant 

Throttle Windup Turn Flight Test Trajectory. 

75 
0 20 40 eo BO 100 120 

TIYE, SECON3S 

Figure 5-24. Yaw Body Rate Evolution along the Constant Throttle 

Windup Turn Flight Test Trajectory. 

0 20 4.0 60 80 LOO 120 

TIME, SECONDS 

0 20 40 60 80 100 120 

RYE, S E CON DS 

Figure 5-25. Throttle and Elevator Deflection along the Constant Throttle 

Windup Turn Flight Test Trajectory. 

75 
n 2 .4 

4- 

li 

.2 

n 

0 

0 20 60 eo 100 120 

Tn[g, SECONDS 

,6 

CI .4 

f!i 

g .2 

n 

a 

d 

3 

e 

0 

- .2 

0 20 80 BO 100 1u) 

m, SECONDS 

Figure 5-26. Differential Tail and Rudder Deflection along the Constant 

Throttle Windup Turn Flight Test Trajectory. 

5.3.7 Constant Dynamic Pressure - Constant Load Factor Maneuver 

As noted in the maneuver modeling section, this maneuver can be 

ascending or descending based on the required Mach rate. The controller 

performance along an ascending constant dynamic pressure-constant load 

factor trajectory is given in Figs. 5-27 through 5-34. This trajectory 

consists of three phases. The first phase begins with initial conditions 

chosen to obtain the desired dynamic pressure. Next the aircraft is placed 

in a turn to generate the required load factor. In the present case, a load 

factor of 4 was employed. Next, the desired Mach rate is initiated and the 

constant dynamic pressure-constant load factor trajectory is executed. In 

the present case, in order to achieve a Mach rate of 0.0067/second, the 

aircraft had to climb from 35000' altitude to 43000'altitude in 30 seconds. 

These histories are given in Fig. 5-27. The control surface deflections are 

well within the saturation limits except at the points where the altitude 

and Mach number commands contains corners. The dynamic pressure history 

given in Fig. 5-34 was maintained within 4% of the required value. 

77 
b 

w 

w 

L 

m 

45000 

42000 

39000 

36000 

33m 

TUX Y SECONDS 

1.2 

1 

Figj: . 

.15 

n 

d .I2 

4 

.bc 

'ID(g, SECONDS 

Angle of Attack and Flight Path Angle Evolution ;lor.g :'.a 

Constant Dynamic Pressure Constant Load Factor Flight Test 

Figure 5-28. 

Tr a j e ct or y 

78 
a 

e: 41 

Ei 

a .5 

0 

0 10 20 30 40 50 60 70 BO 90 100 

m, SECONDS 

.4 

5 .2 

go 

- .2 

- .4 

0 10 20 30 40 50 60 '70 w) 90 100 

m, SECONDS 

1 Figlire 5-29. Roll Attitude and Angle of Side Slip Evolution along the 

Constant Dynarr:~ Pressure Constant Load Factor Flight Test 

TraJ ectory ~ 

-.l 

0 10 20 30 U) 50 80 70 Bo 90 100 

m, SECONDS 

0 10 20 30 40 60 60 70 80 80 100 I m, SECOS 

Roll and Pitch Body Rates along the Constant Dynamic Pressure 

Constant Load Factor Flight Test Trajectory 

F-E:-F 5-39. 

7s 
- .3 

- .6 

nYg SECONDS 

Figure 5-31. Yaw Body Rate along the Constant Dynamic Pressure Constant 

Load Factor Flight Test Trajectory. 

CY 

w 

n 

120 

0 

.2 

P 

d d 

-0 

my SECONDS 

V .IJ .. ..I. .._ I.. ..I _. . .I.. .. I.. ..I ... .I 

0 10 20 30 40 50 60 70 80 90 100 

SECONDS 

Figure 5-32. Throttle and Elevator Deflection Along the Constant 

Dynamic Pressure Constant Load Factor Flight Test 

Trajectory. 

80 
c 

2 

P 

1 

.5 

0 

- .6 

-1 

TlyE, SECONDS 

0 10 20 30 44) 50 60 70 BO 80 100 

TM&, SECONDS 

Figure 5-33. Differential Tail ana kiiaaer Deflection along the CorLstait 

Dynamic Pressure Constant Load Factor Flight Test Trajectory 

t 

0 10 20 30 CO 50 60 70 BO 90 100 

RYE, SECONDS 

Figure 5-34. Dynamic Pressure along the Constant Dynamic Pressure Constant 

Load Factor Flight Test Trajectory 
5.3.8 Constant Reynold's Number - Constant Load Factor Maneuver 

This maneuver sequence is identical to the constant dynamic pressureconstant load factor trajectory. The controller performance for a negative 

Mach rate - Constant Reynolds' number-constant load factor trajectory is 

given in Figs. 5-35 through 5-42. 

Fig. 5-42 should be multiplied by the characteristic diameter to obtain the 

actual Reynold's number. From the performance results given in Fig. 5-40 

and 5-41, it can be seen that the control surface deflections momentarily 

violate the saturation limits at the points corresponding to the corners in 

the commanded values. Throughout the trajectory, Reynold's number is within 

1.5% of -?e reqilired value. 

Note that the Reynold's number given in 

0 10 20 30 4.0 50 60 '70 Bo 90 100 

m, SECONDS 

1.2 

0 1.15 

E si 1-05 

r 

1.1 

1 

0 10 20 30 44 60 60 70 BO BO 100 

m, SECONDS 

Figure 5-35. Altitude and Mach Number Evolution Along the Constan: 

Reynold's Number Constant Load Factor Flight Test 

T r a j e ct or y . 

82 
0 10 20 30 40 50 60 70 BO 90 100 

TIME, SECONDS 

0 10 20 30 40 60 60 70 80 80 llv) 

m, SECOKDS 

Figgre 5-36. Angle of Attack and Flight Path Ang?e Evolution along the 

Constant Reynold's Number Consta-.t Lo23 Factor Flight Test, 

Tr a j ect ory 

0 

0 10 20 30 4.0 50 60 70 BO 90 100 

m, SECONDS 

- 

Eo m 

-.I 

0 10 20 30 4.0 60 80 70 BO 80 100 

m, SECONDS 

Figure 5-37. Roll Attitude and Angle of Side Slip Evolution Along the 

Constant Reynold's Number Constant Load Factor Flight Test 

Trajectory. 

83 
TIYE, SECONDS 

. 

0 

Figure 5-38. Roll and Pitch Body Rates along the Constant Reynold's Number 

Constant Load Fact or F1 i ght Test Trajectory . 

m, SECONDS 

Figure 5-39. Yaw Body Rate along the Constant Reynold's Number Constant 

Load Factor Flight Test Trajectory. 

a4 
0 w6c: 

n 

z40 

E 

E 

20 

I... .I . I.. . .I ... .I. ... I...TI .... I.. .. I.. .. I... .I 

0 10 20 30 40 50 60 70 BO 90 100 

"ME, SECONDS 

- .4 

0 10 20 30 40 50 60 70 80 80 100 

=, sccoms 

Fig~re 5-?lrV. Throttle and Elevator Deflection along the Cor,stan: Aey"is1d's 

NmjSer Zcnstant io33 Factor Flight Test Trajectcry 

0 10 20 30 40 60 60 70 80 80 100 

m, SECONDS 

Figure 5-41. Differential Tail and Rir?dPr DeflectiQn Alcng the 7c~sta-it 

Reynold's Number Lo?" :a*+ T- Flight Test Trajectory. 

85 
0 10 20 30 M 50 BO 70 BO 90 10 

TIYE, SECONDS 

Figure 5-42. Reynold's Number Along the Constant Reynold's Number Constant 

Load Factor Flight Test Trajectory. 
SECTION 6 

SUMMARY AND FUTURE WORK 

This report summarizes the analysis done in the first phase of a study 

for developing and validating a flight test trajectory controller for eight 

maneuvers. System modeling, control design results, control design 

technique evaluation, validation and software deliverable results and 

conclusions are summarized here. This final section describes future work 

which will build upon this study in the further validation and flight 

testing of these flight test trajectory controllers. 

6.1 SYSTEM MODELING 

With newly developed linearization tools by NASA Ames Dryden Flight 

Research Facility, the linear airframe models are straight forwared to 

obtain. Command augmentation system CAS models are important and although 

they complicate the maneuver autopilot designs, they are essential since the 

maneuver autopilot must work through the CAS in the manned simulation or in 

the F-15 itself. 

The output analytical constrained relations developed in section 3 

which discusses maneuver modeling were found to be fundamental in achieving 

a consistent maneuver autopilot. The nonlinear aerodynamics can be 

effectively reduced to a reference command table by appropriate 

linearizations throughout the flight envelope. A small number of linear 

perturbation models can be used to decompose the eight desired maneuvers 

into a few sets of linear perturbation equations. 

6.2 CONTROL DESIGN RESULTS 

Output feedback controllers with appropriate integral aerostates were 

fou!?d to be the simplest form of feedback controllers. Integ-31 aerostates 

must be chosen carefully to avoi $ c-?t-?lI zt'! 1 i ty problems wk,i ch are 
problematic in the design stage. 

in the control design technique was found to be a powerful method when 

working with the high order augmented and coupled models. It was found that 

a five second response time with a minimal amount of overshoot was 

straightforward to achieve at all design points. 

The use of a guaranteed stability margin 

6.3 CONTROL DESIGN TECHNIQUE EVALUATIONS 

The eigenstructure assignment technique was found to work well on a 

bare airframe but with a complex augmented model was not useful as a design 

technique because there is no procedure to perform a converging iterative 

design approach. 

The minimum error excitation output feedback method with a preliminary 

guaranteed stability margin full state design worked well at all flight 

condi ti ons . 

6.4 MANEUVER AUTOPILOT VALIDATION 

The maneuver autopilots were validated in a linear simulation. The 

tracking response for reasonably high performance maneuvers were found to be 

quite acceptable, and except for exceeding the control authority at isolated 

points, is ready for further testing in a full nonlinear simulation. 

6.5 SOFTWARE DELIVERABLES 

The tools used in model development control design evaluation and 

design throughout the envelope as well as validation were made available to 

NASA Ames Dryden Flight Research Facility. 

simulation mechanized in ISI's MATRIXx SYSTEM - BUILD (see Appendix A). 

addition a stand alone program which performs a three dimensional 

These included a time varying 

In 
REFERENCES 

C11 

I: 21 

C31 

C43 

I: 51 

C61 

171 

C81 

C93 

t101 

E111 

C121 

Duke, E.L., "Automated Flight Test Maneuvers: The Development of a 

New Technique", Flight Testing Technology-A State-of-the-Art Review, 

pp. 101-119, Proceedings of the Society of Flight Test Engineers 13th 

Annual Symposium, New York, N.Y., September 1982. 

Duke, E.L., Swann, M.R., Enevoldson, E.K., and Wolf, T.D., !!Experience 

with Flight Test Trajectory Guidance", J. Guidance, Control, and 

Dynamics, pp. 393-398, Vol. 6., Sept.-Oct. 1983. 

"Final Report, HiMAT Maneuver Autopilot", TRA 29255-1, Teledyne Ryan 

Aeronautical, February 1981 . 

Walker, R.A., and Cupta, N.K., "Flight Test Trajectory Control 

Analysis", NASA CR 170395, February 1983. 

E.L., Duke, F.P. Jones, and R.B. Roncoli, "Development of a Flight 

Test Maneuver Autopilot for a Highly Maneuverable Aircraft ,I* AIAA 

Paper 83-0061, Jan. 1983. 

Menon, P.K.A., Saberi, H.A., Walker, R.A., and Duke, E.L., "Flight 

Test Trajectory Controller Synthesis with Constrained Ei genst ruct ure 

Assignment", 1985 Ameri can Control Conference, June 19-21 , 1985, 

Boston, MA. 

Menon, P.K.A., Badgett, M.E., Walker, R.A., and Duke, E.L., "Nonlinear 

Flight Test Trajectory Controllers for Aircraft", AIAA Guidance and 

Control Conference, Snowmass, Colo. August 19-21, 1985. 

Kosut, R.L., "Suboptimal Control of Linear Time-Invariant Systems 

Subject to Control Structure Constraints", IEEE Transactions on 

Automatic Control, Vol. AC-15, No. 5, Oct. 1970. 

Ly, Uy-Loi , tlApplication of the Finite Time Quadratic Performance 

Index to Robust Optimal Control Law Synthesis!', Ph.D. dissertation, 

Stanford University, 1982. 

Walker, R.A., Gupta, N.K., Duke, E.L., and Patterson, B., 

"Developments in Flight Test Trajectory Control ,I1 AIAA 84-0240, AIAA 

22nd Aerospace Sciences Meeting, Reno, Nevada, Jan. 9-12, 1984. 

Duke, E.L., Patterson, B.P., and Antoniewicz, R.F., "User's Manual for 

LINEAR, A Fortran Program to Derive Linear Aircraft Modelst1, NASA 

report in preparation. 

Meyer, G., and Cicolani, L., "Applications of Nonlinear System 

Inverses to Automatic Flight Control Design -- System Concepts and 

Flight Evaluations", Theory and Applications of Optimal Control in 

Aerospace Systems, P. Kant , ed., AGARDograph 251 , 1980. 

90 
interpolation in state variables and converts this through the interpolation ! 

1 into a 1D table as a filnction of time for a specific maneuvers was developed 

I in SYSTEM BUILD to mechanize a time varying simulation. Documented command 

I files were delivered for the model generation, control law design, and 

and also delivered to NASA. The 1-D interpolation in time can then be used 

- 

i 

I 

I maneuver. Command files for building, loading the data and executing a time 

I construction of linear simulation using a single model throughout the flight 

I varying simulation model were also provided. 1 

I 6.6 FUTURE WORK 1 

The next phase of this study will involve validation of flight test I 

trajectory control laws in a batch nonlinear simulation. The gain scheduled 

linear perturbation controllers developed in this study Are currently ready 

for validation in such a simulation. The fineness of the discretization 

both of the reference commands and linear perturbation models may need to be 

revised in the next phase. An open-loop simulation of the reference control 

values will give an initial check of the accuracy of our nonlinear tabular 

model. The nonlinear simulation provided by NASA Ames Dryden Flight 

Research Facility will include both the CAS model as well as the airframe 

I 

I 

I 

I 

I dynamics and Sensor models. The structure of the nonlinear control law, 

1 

~ 

while only outlined here, was developed to a point where with some syrnbolic 

or algebraic manipulation by hand, it can be mechanized on the nonlinear 

simulation. The advantage of this control law is that a single set of gains 

will work for all maneuvers throughout the envelope. Such a transformed 

linear system can be easily controlled and the resulting gains back 

transformed with the nonlinear equations to give a nonlinear control law. 

I 

, 

89 
I 

[l3] Shapiro, E.Y., Fredricks, D.A., and Rooney, R.H., "Suboptimal Constant 

Output Feedback and its Application to Modern Flight Control System 

Design", International Journal of Control, Vol. 33, No. 3, 1981, pp. 

505-51 7. 

[14] Anderson, B.D.O., and Moore, J.B., Linear Optimal Control, 

Prentice-Hall, Englewood Cliffs, N.J., 1971. 

[ 151 Bryson, A.E., "Kalman Filter Divergence and Aircraft Motion 

Estimators", SUDDAR-505, Dept. Aeronautics and Astronautics, Stanford 

University, Stanford, CA., July 1977. 

[16] Bryson, A.E. and Ho, Y.C., Applied Optimal Control, Hemisphere, 

Washington, D.C., 1975. 

[17] Krener, A.J., "On the Equivalence of Control Systems and Linearization 

of Nonlinear Systems", SIAM J. Control, Vol. 11, 1973, pp. 670-676. 

[18] Meyer, George; and Cicolani, Luigi S.: "A Formal Structure for 

Advanced Automatic Flight Control Systems", NASA TN D-7940, 1975. 

[19] Brockett, R.W., "Feedback Invariants for Nonlinear Systems'!, IFAC 

Congress, Helsinki, 1978. 

[20] Smith, G. Allan; and Meyer, George: "Total Aircraft Flight-Control 

System - Balanced Open- and Closed-Loop Control with Dynamic Trim 

Maps", Proceedings of the Third Digital Avionics Systems Conference, 

Ft. Worth, Texas, Nov. 1979, IEEE Catalog No. 79CH1518-0. 

[213 Cicolani, L.S., Sridhar, B. ; and Meyer, George: "Configuration 

Management and Automatic Control of an Augmentor Wing Aircraft with 

Vectored Thrust", NASA TP-1222, 1979. 

[22] Smith, G.A., and Meyer, G., ?'Total Aircraft Flight Control System 

Balanced Open- and Closed-Loop Control with Dynamic Trimmaps ,It AIAA 

3rd Digital Avionics Conference, Dallas, Texas, 1979. (Published in 

IEEE proceedings). 

[23] Wehrend, W.R., Jr., 'IPilot Control Through the TAFCOS Automatic Flight 

Control System", NASA TM-81152, 1979. 

[24] Smith, G.A., and Meyer, G., "Application of the Concept of Dynamic 

Trim Control to Automatic Landing of Carrier Aircraft", NASA TP-1512, 

1980. 

[25] Wehrend, W.R., Jr., and Meyer, G., "Flight Tests of the Total 

Automatic Flight Control System (TAFCOS) Concept on a DHC-6 Twin Otter 

Aircraft", NASA TP-1513, 1980. 
[26] Meyer, G., and Cicolani, L., "Applications of Nonlinear System 

Inverses to Automatic Flight Control Design -- System Concepts and 

Flight Evaluations", Theory arid Applications of Optimal Control in 

Aerospace Systems, P. Kant., ed., AGARDograph 251, 1980. 

C27] Meyer, C., "The Design of Exact Nonlinear Model Followers." Joint 

Automatic Control Conference, FA-3A, 1981. 

1283 Hunt, L.R., and Su, R., "Linear Equivalents of Linear Time-Varying 

Systems." International Symposium on Mathematical Theory of Networks 

and Systems, 1981, pp. 119-123. 

C291 

C 301 

C321 

L331 

C341 

E351 

I: 361 

E371 

Hunt, L.R., and Su, R., '!Control of Nonlinear Time-Varying Systems. 

IEEE Conference on Decision and Control", 1981, pp. 558-563. 

Hunt, L.R., Su, R., and Meyer, G., "Global Transformations of 

Nonlinear Systems". 

p. 24-31. 

IEEE Trans. on Autom. Control, vol. 28, 1982. 

Hunt, L.R., Su, R., and Meyer, G., "Multi-Input Nonlinear Systems", to 

appear in Differential Geometric Control Theory Conference, 

Birkhauser , Boston, Cambridge, Mass., 1982. 

- 

Meyer, C., Su, R., and Hunt, L.R., "Application of Nonlinear 

Transformations to Automatic Flight Control", - Automatica, Vol. 20, No. 

1, pp. 103-107, 1984. 

Dwyer, T.A., "Exact Nonlinear Control of Large Angle Rotational 

Maneuvers", IEEE Trans, Aut omati c Control, vol.. AC-29, No. 9, 

September 1984, pp. 769-774. 

Chow, J.H. and Kokotovic, P.V., "Two-time-scale Feedback Design of a 

Class of Nonlinear Systems", IEEE Trans. Automatic Control, vol. 

Ac-23, No. 3, June 1978, pp. 438-1143. 

Chow, J.H., and Kokotovic, P.V., "A Decomposition of Near-Optimum 

Regulators for Systems with Slow and Fast-Modes", IEEE Trans. Auto. 

Control, vol. 21, pp. 701-705, October 1976. 

Khalil, H.K., "Linear Quadratic Gaussian Estimation and Control of 

Singularly Perturbed Systems", in Singular Perturbations in Systems 

and Control, M.D., Ardema (Ed) Springer-Verlag, WIEN-New York CISM 

Courses and Lectures No. 280, 1984. 

Shah, S.C., Floyd, M.A., and Lehman, L.L., "MATRIXx: Control Design 

and Model Building CAE Capability", in Advances in Cmputer-Aided 

Control Systems Engineering, Eds. M. Jamshidi and C.J. Herget, North 

Holland, 1985. 
APPENDIX A 

FLIGHT TEST TRAJECTORY CONTROLLER SYNTHESIS 

WITH CONSTRAINED EIGENSTRUCTURE ASSIGNMENT 

A- 1 
I 1 INTRODUCTION : 

This Appendix gives the details of research conducted on the 

eigenstructure assignment. The material in the Appendix A-I was presented 

as a paper in the 1985 American Control Conference at Boston. 

and A-I11 give the aircraft - CAS models and the constrained eigenstructure 

designs at two flight conditions. 

eigenvalues and eigenvectors used in the synthesis. 

Appendix A-I1 

Appendix A-IV gives the desired 

The aircraft - CAS model used here is of the form 

x = FX + CU 

y = HX 

where: 

the state variables x are the perturbed values of 

V, total airspeed 

a, Angle of attack 

q, pitch body rate 

8, pitch attitude 

B, angle of sideslip 

p, roll body rate 

Y, yaw body rate 

4, roll attitude 

h, altitude 

engine actuator state and 21 CAS states 

U: the control vector consists of perturbed values of throttle, 

elevator, differential tail and rudder. 

I 
y: the measurement vector and consists of the perturbed values 

of 

p: roll angular acceleration 

An: normal acceleration 

q: pitch rate 

q: pitch angular acceleration 

p: roll rate 

A : y-body acceleration, not at the vehicle center of 

Y,i 

gr avi ty 

r: yaw angular acceleration 

r: yaw rate 

h: altitude 

M: mach number 

a: angle of attack 

n: load factor 

@: roll attitude 

8: pitch attitude 

1;: altitude rate 

u9: x-body axis velocity 

A nx : x-body axis acceleration at vehicle C.G. 

CONCLUSIONS: 

The constrained eigenstructure assignment design procedure in its 

present form demands several iterations to converge to a satisfactory design 

and does not appear to easily yield suitable insight for output feedback 

design of high order multivariable systems. If a rational method to 

generate an achievable set of desired eigenvectors is devised, this 

technique will be made more attractive. 

A- 3 c -a 
APPENDIX A-I 

PAPER PRESENTED AT 1985 ACC, JUNE 19-21 

BOSTON, M.A. 

A-1-1 
FLIGHT TEST TRAJECTORY CONTROLLER SYNTHESIS 

WITH 

CONSTRAINED EI GENSTRUCTURE ASSIGNMENT+ 

I 

P.K.A. Menon, H.A. Saberi, R.A. Walker 

Integrated Systems, Inc. 

101 University Avenue 

Palo Alto, CA 94301 

and 

E.L. Duke 

NASA Ames 

Dryden Flight Research Facility/OFDC 

Edwards, CA 93523 

ABSTRACT 

Flight test trajectory controller synthesis using constrained eigenvalue/ 

eigenvector assignment procedure is presented. 

and the difficulties encountered in employing this synthesis technique are 

highlighted. 

Associated modeling problems 

+ 

Research supported by NASA-Ames-Dryden Flight Research Facility under Contract NAS2-11877. 

A- I- 2 
1. - INTRODUCTION: - 

Flight test trajectory control is a technique designed to aid in the 

collection of large quantities of high quality data. This technique has 

provided the means for flying maneuvers consistantly, precisely, and repeatably from flight to flight. Two versions of these controllers have been 

used: a closed-loop automatic system and an open-loop system providing 

manual piloting information. A closed-loop system used to collect performance, pressures, and loads data from the highly maneuverable aircraft 

technology (HIMAT) vehicle is described in [l]. 

open-loop system on the NASA F-111 transonic aircraft technology (TACT), 

F-15 airframe/propulsion system interaction studies, and F-15 shuttle tiles 

test programs are given in [23. 

The application of the 

Ori ginall y , the open-1 oop f 1 i ght- tes t- traj e ct ory gui dance a1 gor i thms 

were developed on-line, in a piloted simulation using cut-and-try techniques 

that was not only man power intensive, but often produced less than desirable controllers. A closed-loop system designed using one-loop-at-a-time 

classical design approach is documented in [3]. Full-state feedback 

approach for closed-loop system design using Linear quadratic synthesis is 

described in [4]. Both these approaches have limitations in terms of design 

methodology and controller complexity. 

The research currently underway includes an exploration of various 

multivariable synthesis techniques for this problem. The first approach 

considered is that of constrained eigen valueleigen vector assignment [5]. 

A primary goal is to develop controllers based on output feedback so as to 

decrease controller complexity and to enhance robustness. The objective 

of this paper is to point out strengths and weakness of this technique when 

used in a, realistic, relatively a large problem such as required for flight 

test trajectory controllers . 

A-1-3 
2. THE FLIGHT TEST TRAJECTORY CONTROL PROBLEM: 

Flight test trajectories are flown to evaluate an aircraft within its 

known operational envelope and to explore the boundaries of its capabilities. This makes the flight test trajectory control a very demanding task. 

Control systems designed for this purpose must not only operate satisfactorily in terms of keeping the flight test variables within acceptable 

tolerance but should also be reasonably insensitive to model parameter 

variations. 

An approach t 0 cl osed-loop f 1 i ght test trajectory control 1 er synthesis 

consists of linearizing the aircraft model at several flight conditions 

about the flight test trajectory and desi gning mu1 ti variable controllers , 

which in some sense minimizes the deviations from the reference path. Time 

varying nature of the linearized aircraft model along the reference path 

brings about the need for scheduling gains as a function of time or as functions of some important flight variables such as dynamic pressure, Mach 

number etc. The gain scheduling aspects will not be pursued any further in 

this paper, and in all that follows, discussions will center around a 

linear-time-invariant aircraft model. Further, though the flight test trajectory controller is discrete, in this preliminary stage it will be assumed 

that the sampling rate is sufficiently high, permitting the application of 

continuous control design techniques. 

The aircraft under consideration is a high performance fighter with 

command augmentation system (CAS) engaged in all the three axes. The CAS is 

a hi ghly nonl inear system wi th saturations, mu1 tip1 icative nonl ineari ties 

and gain schedules. At a particular flight condition, this system can be 

approximated by a linear system with "equivalent" gains derived fran nonlinear simulations. The aircraft model at the same flight condition is 

obtained from a generic aircraft linearization code developed at NASA-Dryden 

Flight Research Facility [6]. The state variables used are total speed, 

angle of attack and angle of sideslip, pitch rate, yaw rate, roll rate, 

pitch attitude, roll altitude and altitude. Throttle, rudder, elevator and 

differential tail constitute the control variables. The engine dynamics are 

modeled as a first order lag. Thus, the aircraft together with CAS at a 

particular flight condition is a coupled 31-st order system with four controls, the aircraft model having 10 states. A block diagram of the system 

A-1-4 
with flight test trajectory controller (called the Maneuver AutoFilot here) 

included in the loop is shown in Figure 1. Note that MAP controls the aircraft through CAS. Due to this, even if the aircraft model is decoupled in 

longitudinal and lateral axes at a particular flight condition, the CAS introduces a strong coupling into the system. In concise terms, the chief 

objective of the present study is to synthesize the maneuver autopilot so as 

to effect satisfactory transient response. Six maneuvers have been 

analyzed, which are sketched belou. 

- - - 

2.1 Level Acceleration/Deceleration 

This is a wings-level, constant altitude maneuver with Kach number 

constant or changing at a specified rate. 

2.2 Pushover, Pullup 

This is a wings-level, constant Mach number maneuver in which angle of 

attack is varied a specified increment about the trim value at some specified rate. 

2.3 Zoom and Pushover 

The zoom and pushover is a wings-level, thrust stabilized less than lg 

maneuver. The flight trajectory is a parabolic path with the target 

Mach/altitude/angle of attack point at the apex. 

2.4 Excess Thrust Windup Turn 

This is a maneuver with angle of attack linearly increasing from the 

wings-level trim condition to some specified final value at a specified 

rate. The maneuver is performed at constant altitude and constant Mach 

number. 

A-1-5 
2.5 Constant Throttle Windup Turn 

This is a maneuver with angle of attack increasing linearly at a 

specified rate from trim to some specified final value. The maneuver is 

performed at a predetermined, constant thrust level. Mach number is maintained by trading potential for kinetic energy via an appropriate altitude 

rate. 

2.6 Constant Reynolds Number and Constant Load Factor Trajectory 

This maneuver is initiated at a predetermined load factor, Mach number, and dynaTic pressure. Thus, the initiation of this maneuver is not 

necessarily the wings-level condition. This maneuver can be either an 

ascending or descending maneuver at a specified Mach number rate. Reynolds 

number and load factor are held constant throughout the maneuver. Altitude 

is gained or lost to maintain Reynolds number with changing Mach number. 

The simplest of these is the level acceleration/deceleration trajectory and the MAP design for this maneuver will be used as the illustrative 

example in this paper. 

3. OUTPUT FEEDBACK DESIGN: 

As indicated earlier, output feedback is attractive because of simplicity. Further, gain scheduling will be essential in this situation and in 

order to minimize the amount of stored data, it is desirable to have a 

capability to impose control structural constraints. 

Currently, several approaches are available for output feedback 

design, see Refs [5, 7-10] for example. It is not the purpose of this paper 

to compare and contrast these, but to evaluate a specific technique from an 

application point of view. Constrained eigen valueleigen vector assignment 

technique of [5] was used in this particular application with the hope that 

it would permit the generation of designs based on the classical notions of 

poles, zeros and their relative location in the complex plane. 

-Additionally, the selection of this technique was motivated by the exarple 

in [5], viz, the lateral axis stability Augmentation System for the L1011 

A-1-6 
airplane. Salient features of this approach are outlined in the following 

for clarity of the subsequent discussions. 

3.1 Constrained Eigen Value Vector Assignment [53: 

Consider the linear time invariant systern 

x = AX + BU 

y = cx 

with 

compatible dimensions. 

X E Rn, U E. Rm, y E Rr and A, B, C are real constant matrices of 

It is desired to design a feedback controller of the form 

U = Fy 

with a structural constraint that some specified elements of F satisfy 

-s 

It is assumed that the system can be stabilized with given outputs y, 

and that any dynamic compensators required have been appended to the 

original system. 

The design problem is: given a self conjugate set of scalars 

{A ) i = 1, 2 ... r, and a corresponding self conjugate set of n vectors 

{vi] i = 1, 2 ... r, and a given controller structural constraint, find a 

real m x r matrix F such that r of the eigen values of A + BFC are 

"close1' to the set {A,) and the corresponding eigen vectors of A + BFC are 

"close" to the respective member of {vi). 

d 

a 

d 

d 

A-1-7 
Note that if there are no constraints on the feedback matrix, it is - 

feasible to place r desired eigen values exactly. 

However, one is not at liberty to place v,, v2 ... v by an arbitrary 

set of desired closed loop eigen vectors {v.) i - 1, 2 ... r since this set 

might not belong to the set of assignable closed loop eigenvectors. It then 

becomes necessary to find some approximation to the set {v.] which is 

assignable, yet q'close'q in some sense to the original {v. 1. It will be seen 

subsequently that this is the main difficulty in the application of this 

technique to a complex problem such as flight test trajectory controller 

design. 

r d 

1 

d 

1 d 

1 

Complete details of this approach can be found in [51. It is perhaps 

of interest to note that in addition to the capability to handle controller 

structure constraints, this technique can be modified to accept partial 

specification of eigen vectors. 

systems. In summary, to employ this synthesis approach, the following are 

required. 

This feature can be valuable ii high order 

closed loop eigenvectors v 

must lie in the subspace spanned by the columns of the matrix (XiI-A)B1. 

Further, the matrix C V, V - {v,, v2 . .. v 1 should be invertible. 

complete specification of desired eigenvectors are not feasible, a rough 

rule is to pick the entries in these eigenvectors as zeros or ones based on 

corresponding to the closed loop eigenvalues Ai i 

If the r 

I 

I whether a particular measurement needs to contribute to a particular mode or 

not. In any case, constructing a set of desired eigenvectors that are 

assignable constitutes the most difficult part of this design technique. 

(i) Based on the practical aspects of the problem, choose a minimal 

set of measurements which will permit the designer to achieve the 

desired performance. Introduce dynamic compensators such as 

integral feedbacks, lead-lag networks, etc. , based on experience. 

A-1-8 
4. FLIGHT TEST TRAJECTORY CONTROLLFR SYNTHESIS: -_____l_---------- 

As an illustrative exanple, constrained eigenstructure assignment 

technique is next used in the design of level accelerationldeceleration 

flight test trajectory. Alternate design approaches have been used for this 

maneuver and hence comparative evaluation was feasible. The level 

acceleration/deceleration maneuver requires the roll atti tude to be maintained zero and the altitude to be held constant. The mach nurnber shouid be 

maintained constant or should change at a specified rate. To ensure zero 

steady state errors, integral feedbacks are first introduced in the altitude 

and mach number channels. Since the mach number command can be a ramp, an 

additional integrator is required in this channel to decrease tracking 

errors. This, however, was not done because the tracking errors with a 

single integral feedback has been found to be within acceptable yalues. 

The desired eigenvalues to be used in the design are clear at the outset, based on the four modes for aircraft model, viz, the phugoid, short 

period, dutch roll and roll convergence. The controller structure constraint in this flight maneuver is that the errors in lateral channel will 

be corrected using rudder and differential tail, while the errors in longitudinal channel will be corrected using elevator and throttle. In view of 

the time varying nature of the model, one would like to pick a set of eigenvalues and eigenvectors at a particular flight condition through extensive 

design iterations and then attempt to use these at other flight conditions. 

The choice of desired eigenvectors is not clear at this point. Three 

approaches were tried with varying degrees of success. 

in the following. 

These are sketched 

4.1 Minimally Restructered Eigenassignment 

Since it is known that the closed loop eigenvectors lie in a subspace 

spanned by the columns of (XiI-A) 

these vectors were used as desired eigen vectors. The weights to be used in 

generating these linear combinations were constructed from the additional 

information that unassigned eigenvectors should be close to their open loop 

values in a least square sense. 

-1 B , i - 1, ... n, linear combinations of 

A- I- 9 
4.2 Decoupling Eigenassignment 

A variation to the above was attempted next. Since we are interested in 

having the least cross axis coupling in the controller as possible, the 

desired eigenvectors in the longitudinal channel may be chosen to that the 

responses from lateral channels are blocked, i .e. select eigenvectors as 

n 

vd = 1 aivi 

i= 1 

Vd are the desired eigenvectors. 

4.1. Though these two approaches could be made to work at each flight condition by iterating on the desired eigen values, they failed to easily produce a set of acceptable eigen vectors which could be used at other flight 

conditions. Next, partial specification of desired eigenvectors was 

attempted as follows: 

a are selected using the same criteria as i 

4.3 Dominant Mode Eigenassignment 

According to ref [SI, complete specification of desired eigenvectors are 

neither necessary nor desirable. Depending on the states should or should 

not participate in a given mode, appropriate entries in the eigenvectors are 

made ones or zeros, leaving other entries free. With these eigenvectors, 

the desired eigenvalues are moved as far left from the imaginary axis as 

possible with least change in the location of unplaced eigenvalues. The 

desired eigenvectors so obtained appeared to work over most flight conditions. Note, however, that extensive iterations on the desired eigenvalues 

may often be required to produce a satisfactory design. 

A-1-10 
The open-loop eigenvalues for the aircraft-CAS linearized modcl at 

Mach 1.2 and 10000' altitude is given in table 1. The system has a pole on 

the right half of s plane. Partial specification of the desired eigenvectors are constructed next, based on the states that should or should not 

participate in a given measurement. The desired eigenvectors used in the 

present example are given in table 2. It can be seen from this table that 

no restriction has been placed on states associated Kith CAS. A manlual 

iteration is now undertaken to deternine the desired eigenvalues. Since the 

designer is interested in producing a stable system with as high a speed of 

response as possible, the eigenvalues to be moved are the ones closest to 

the imaginary axis. Hence, these are moved as far to the left of the 

imaginary axis as possible with least change in the location of other eigenvalues. Table 3 shows a set of desired eigenvectors obtained from this 

exercise. The output feedback gains obtained from the constrained 

eigenvalue/eigenvector design technique is given in table 4. Table 5 gives 

the closed loop eigenvalues. Comparing this with table 3 shows that the 

achieved ejgen values are close to the desired ones. 

In Figures 2 and 3, the time response of the system for a ra-p msch 

number command are shown. A question that occurs naturally at this point is 

whether the design can be improved by further adjustment of desired eigen 

values and eigen vectors. Examination of the constrained eigen valueleigen 

vector approach yields no answer to this question. 

5. CONCLUSIONS 

Design of a maneuver autopilot for flight test trajectory control 

using constrained eigenvalue/eigenvector assignment was discussed. Diff iculties encountered in the generation of desired eigenvalues and eigenvectors were outlined. This approach demands several iterations to converge 

to a satisfactory design and does not appear to easily give suitable insight 

for output feedback design of high order multivariable systems which will be 

used at other operating points. If a rational method to generate an 

achievable set of eigenvectors is devised, this technique will be made more 

attractive. One possibility might be to generate gradients of the eigensystem between flight conditions and include this information in the single 

point design techni que. 

A-I- 1 1 
References : 

[l] Duke, E.L., "Automated Flight Test Maneuvers: The Development of a 

---- 

New Technique", Flight Testing Technology-A State-of-the-Art Review, 

pp. 101-119, Proceedings of the Society of Flight Test Engineers 13th 

Annual Symposium, New York, N.Y., September 1982. 

[2] Duke, E.L., Swann, M.R., Enevoldson, E.K., and Wolf, T.D., "Experience 

with Flight Test Trajectory Guidance", J. Guidance, Control, and 

DynaTics, pp. 393-398, Vol 6., Sept.-Oct. 1983. 

[3] "Final Fleport, HiMAT Flaneuver Autopilot", TRA 29255-1, Teledyne Ryan 

Aeronautical, February 1981. 

[4] Walker, R.A., and Gupta, N.K., "Flight Test Trajectory Control 

Analysis", NASA CR 170395, February 1983. 

[5] Shapiro, E.Y., and Chung, J.C., "Constrained Eigen ValueIEigen Vector 

Assignment - Application to Flight Control Systems", the 24th Israel 

Annual Conference on Aviation and Astronautics, Tel-Aviv, February 17, 

1982. 

[6] Duke, E.L., Patterson, B.P., and Antoniewicz, R.F., "User's Manual for 

LINEAR, A Fortran Propam to Derive Linear Aircraft Models", NASA 

report in preparation. 

[7] Kosut, R.L. , llSuboptimal Control of Linear Time-Invariant Systems Subject to Control Structure Constraints", IEEE Transactions on Automatic 

Control, Vol. AC-15, No. 5, October 1970, pp. 557-563. 

[8: Shapiro, E.Y., Fredricks, D.A., and Rooney, R.H., "Suboptimal Constant 

Output Feedback and its Application to Modern Flight Control System 

Design", International Journal of Control, Vol. 33, No. 3, 1981, pp. 

505-51 7. 

[g] Levine, W.S., and Athans, M., "On the Determination of the Optimal 

Constant Output Feedback Gains for Linear Multivariable Systems", IEES 

Transactions on Automatic Control, Vol. AC-15, No. 1, February 1970, 

pp. 44-48. 

[lo] Ly, Uy-Loi, llApplication of the Finite Time Quadratic Performance 

Index to Robust Optimal Control Law Synthesis", Ph.D. dissertation, 

Stanford Uni versi ty , 1 982. 

[ll] Liebst, B.S., Garrard, W.L., and Adams, W.M., "Eigenspace Design of an 

Active Flutter Suppression System", AIAA Guidance and Control 

Conference, paper 84-1867, Aug. 20-22, 1984, Seattle, WA. 

A-1-12 
I 

TABLE 1. OPEN-LOOP EICEN VALUES FOR AIRCRAFT-CA5 LINEAR17EI) FlODEL AT MAC3 

1 .? AN3 1000's ALTITUDE 

-1.2176DtOl +l.O192ptoli 

-1 . 217WOi -l,O192D+Oli 

-ZD3438p+o0 t5.311Ilptooi 

-2.9438M -5.3113D+#i 

-2.3186D-01 tl. 8504b-Oli 

-2.3186D-01 -1.8504D-Oli 

-3.4734Ddl -7. W0H9i 

-6.57MD-02 -8.2699P18i 

3.14- t4.3959D-63i 

3.1435D-M -4.3959D-03i 

-5.4174b-01 +O.OWi 

-5.oooOD-Ol +O.WD+OOi 

i. 4497p-03 +5. oooO~i 

-2.7WD-01 +O.OOOOptoOi 

-2.WOP-01 ~O.oooOp+ooi 

-1.oooO~ +Q.o6oo~i 

-1 92oOP+ol to. oooO~i 

A-1-13 
TABLE 2. PESI8ED EIGEN VECTORS AND THEIS INTEfiPRETATION "99" STANDS FOR 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99.99.99.99.99.99.99.99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 93. 1 89. 99. 99. 99. 99. 99. 99. 99. 

UNSPE 21 F IED COMPONENTS 

1 ' 

Heas ur enm t s 

0. 0. 0. 0. 99, i. 0. 0. Thrust 

99. 99. 99. 99. 99. 99. 99. 99. . 

Actuator 

A-I- 14 

States 

CAS States 
. TABLE 3. DESIRED EIGEN VALUES 

-2.ooOOM)l t2.WMHfi 

-2.oooOp-01 -2.oooOQ-Oii 

TABLE 4. OUTPUT FEEDBACK CONTROLLER 

TABLE 5. CLOSED LOOP EIGEN VALUES 

-4.1531h40 -l.WB+oli 

-4.1331MOO +1.4054Mli 

-3.202SBNO +4.0631Wi 

-3.2028p90 -4.0631Wi 

-1.0000k00 +O.oooOWi 

4.9439Wl +C.7315Wli 

-6.M39W1 5.731w11 

-5.oooOWl +O.oooOD9oi 

-4.776Swl d. oooOp9oi 

-3.4069Wl +1.1513D-l7i 

-1.s6ooW1-1.WW1i eigenvalues 

-2.70OOWl +1.3002D-l7i 

-2.5000W1 +1.3092&17i 

-2.0045wl -2.0051wli 

-2. 0045W1 42.0037W1 i 

-l.~Ml +l.OQQQMli 

-1.(000Mi -1.40WD-01i 

-1.(OoQWl *l.(ooOwli 

desired I 

A-I- 15 
__ CRS 1 flIRCRRFT 

21 

~ STATES 

10 STATES 

11 ir : 

1 RUTOPILOT 

6T 

TRACKING ,-I 

I 

I 

COMMANDS 

Figure 1. Flight Test Trajectory Control System 

A-1-16 
.24 

.21 

.1E 

.15 

0 

z 

. Of3 

.03 

0 

TIME 

Figure 2. Mach No. .vs Time Response for a Ramp Mach No. Command 

A-1-17 
Figure 3. Altitude vs. Time Response for a Ramp Mach No. Command 

A-1-18 
APPENDIX A-I1 

AIRCRAFT - CAS LINEARIZED MODEL AND 

FTTC DESIGN USING EIGENSTRUCTURE ASSIGNMENT 

AT 1.2 MACH, 10000 ALTITUDE 

A-11-1 
CUurN 1 TWY t 

- 7.3178D-02 -5.561111.01 

-2.¶5%@-05 -2.3342@*00 

7.3543D-03 -1.0458[*02 

O.OO@OO+W o.ooooD*oo 

o.oowb+00 O.O000D*~ 

0.0000D*30 0.0000Dt00 

o.woGo+w O.OOOOD100 

o.ooo6D+bc o.ooooD4w 

O.OOOOb+00 -1.2929P*03 

O.WWD+M 0.MK)CNOC 

1.B719D-01 6.7079P02 

0.0000o+w 0.0005bOQ 

0.ooOo~W 0.0000I*W 

o.oooob+oo 0.ooooD+00 

-3.743-?E-01 -1.3416hOJ 

1.8719D-02 6.7079NOl 

1 .B719D+W 6.70798*03 

o.oowD+oo o.ooooD+w 

O.ooWb00 o.ooo(\b+00 

O.#KK)bW o.oooob+00 

0. wo@D+w O.OOOOD+W 

O.WWD+W o.oooob+oc 

o.ooooD+w o.ooDlD+w 

o.w00D+oo O.ooOob+M 

0. oooo~00 0. OoOot*oc 

0.woOPIw 0.0000@+00 

0. oooob+w 0. DoODpIW 

o.ooooD+Oo 0.oooot.M 

o.ooooD+w O.oo00~ 

0.m)Cw o.OOOOD+OCL 

o.oooo)9o 0.oooopcoo 

COLM 

O.O003D+O@ 

0.0086D400 

6. WPOb*OC 

8. R)o!Jb+w 

-9.9993D-01 

9. A65D-01 

-1.0568WW 

1.1919-02 

O.WfrOD+W 

6. WOSDtM 

1.1649bM 

o.o0OoD+oo 

0.oooob+M 

0. WO(rD+M 

-2.3296b+w 

1.1649D-01 

1.1619D401 

0.ooDoWW 

O.O0~D+oO 

O.#KlbbOo 

-1.0533D+Ol 

8.8000b+01 

O.WM)D+W 

o.wwD+w 

o.wooD+w 

o.ooo@D+00 

0. w(roD+00 

O.#)OOD+W 

o.oo00D+w 

O.WWD+W 

O.#)OOb+#I 

-1.78931’ 17 

1.000OE*OD 

d.1574D*00 

l.OWOD90 

O.O0@@t90 

O.ODoDD90 

0.00001190 

0.0000D*00 

0.0WOD*00 

3.5929D42 

-5.16€7D*03 

0.0000PW 

-7.1859D92 

3.5929M1 

3.5929D93 

O.OWOD90 

o.OOooD+oo 

O.OOOOD90 

O.WOD90 

0.0000E40 

Q.OWOD90 

O.ooOoD90 

0.0000DW 

0.000OD90 

0.0WOD90 

O.OOOOP90 

O.OOOOB90 

O.O#K)D*OO 

O.ooOOt90 

o.oooor*0o 

o.oooorm 

-3.7144@+01 

-1.16?91#-06 

1.0534It-03 

0.0000D+W 

O.MO@OOD9@ 

0.0000Dt00 

0.0000Dt00 

O.OMOD40 

0.0Go0@*00 

O.OWGI40 

0.WOODNO 

0.0000D90 

-4.1316@62 

2.065BD43 

2.06SBD61 

0,WODW 

0.0000D90 

0.0000t90 

0.0000DW 

0. OooO@90 

0. WOOD40 

0.0000D90 

0.0000D90 

0.0003t90 

0. WOOW 

0.0000DW 

0.000dD90 

0.0000DW 

O.oooOD90 

i ,2929~03 

2.065~~2 

1.5440065 3.798ZDOl 

0.00001~90 -3.4886D+ 

-5.0352D68 2.913BD06 

0.0000D90 0.0000D90 

0.0000D90 0.0000D90 

O.OOOOD*oO O.OWOD+OO 

0.0000D90 0.0000D90 

0.0000DW O.OWOD+OO 

O.WD+OO -z.wr+i 

-1.032ODa -5.366M 

0.OOOO~ 0.OWODW 

0.000DPW 0.0000DW 

O.OOOOG90 O.OOOOD90 

2.WOD46 1.0733D-04 

-1.OYOD47 -5.M7D-M 

-1,032Ob45 -5.366lt-04 

o.ooooD+oo O.WOD40 

O.O#H)D(OO 0.ooOoDW 

O.ooOot90 O.#K)OW 

O.ooOoD40 0.oooODw 

0.OOOODW 0.0000DW 

O.OOOOD40 O.OOOOD90 

O.OOOOD90 0.0000DW 

O.OOOOD90 0.WODW 

O.OOOOD90 0.0000DW 

O.OWOD90 0.OWODW 

0.OOOODW 0.0000DW 

0.0000D90 O.OOOOD90 

0.0000D90 0.OOOODW 

0.oooODW 0.00000w 

o.oooor9o O.OOOODNO 

A-11-2 
F Matrix (Cont'dj 

0.6000~*00 

0.0000t+Oo 

0.00@0@*00 

0.6003PtOO 

0.00@0D*00 

o.ow0; too 

0.0000D40 

0.000999b 

0.OO00840 

0.00?3D+QO 

1.00@51~31 

0.0000D90 

O.OD@OD+OG 

0. OOOODdO 

-3.0000D41 

1. 000DE*OO 

1.0000062 

O.OOODD+OO 

0.0000DW 

0.0003t90 

O.OWOD90 

0.OOOODW 

0.000bD*00 

0.0WDD90 

0.000DD90 

O.WOL90 

0. OOOOFIOO 

O.oooOD40 

O.OWOD90 

O.OWOD90 

0.0000D100 

A-11-3 
F Matrix (Cont'd) 

I l*U 30 

o.oooobo@ 

o.ooDoD+ Dc 

O.ODOOD*O@ 

0.00002+W 

o.owoD+w 

O.DOOOD+OO 

0.0003t*00 

C.owI1.130 

0 * oD(1oD+oo 

0. ocm+OC. 

O.OWOE+W 

0.0000D+OG 

0.0000EW 

O.OWOD+W 

O.OWOD+W 

O.OoGO@+W 

o.oO0ob+w 

O.OOOOD+M 

O.ODoCI*oo 

o.o00ob+w 

O.oWOI*~ 

O.OWOD*W 

0.0000DW 

O.OWD1*W 

0.0300D+W 

O.O000D+W 

5.900DDN 

O.OWOD+W 

O.OWOD+W 

o.oooob+w 

o.oO0oD1oc 

0.0005F100 

0.00(133+00 

0.0050P90 

0. OMDlWD 

O.OOCiDP100 

O.OMCIDF100 

0.0009P40 

O.OD??F3D 

0.0D!0?100 

O.O3O3l*CD 

G.OM3P90 

0.0003c90 

0.W3C90 

O.OMOF+OD 

0. OOOOIMO 

0.0000f9D 

O.OOODD100 

0.WO~~ 

0.0005D100 

O.W3b+OO 

0.00001~90 

0 * OOoDI*40 

O.owDr40 

0.03005103 

0.0OGOh10D 

0.0000D*0d 

fi.9000P91 

1. WDD92 

0.00(13D100 

O.WOt90 

0.000Dt40 

A-11-4 
2. G-Matrix (31 x 41, 

O.OOOOD+OO 

0.0000DdO 

0.0000D+00 

O.OOOOD+OO 

0.0000D40 

O.O00OD+OO 

0.0000Dd0 

0. OOOODdO 

0.0000DMO 

0.0000D+00 

1.0000Ddi 

0.0000DNO 

0.0000D40 

0.0000D40 

0.0000Dt00 

1.0000D40 

1.0000D92 

0.0000DdO 

0.060OD90 

0.0000D4a 

0.0000DMO 

0.0000D40 

0.0000D90 

0.0000DtOO 

0.0000D90 

0.0000DtOO 

0.0000DNO 

0.0000DdO 

0.0000DMO 

0.0000DNO 

O.OOOOD40 

0. OOOODdO 

0.0000D+00 

0.0000D+00 

0.0000DtOO 

C.OorJOP40 

0.0000D40 

0.0000D90 

0.0000D40 

0.0000Di40 

0.0000DMO 

0.0000D+40 

0.0000DdO 

0.0000D40 

0.0000DdO 

0.0000DdO 

0.0060D90 

0.0M0D40 

0.0000DtOO 

1.0000D42 

0.0000D40 

0.0000DMO 

0,0000D90 

0.0000D40 

0.0000D40 

0.0000D40 

0.0000DM 

0.0M0D+oc) 

0.0000D90 

0.0000D40 

0.0000DW 

O.ooOOD90 

A-11-5 
3. H-Matrix (17 x 31) 

COLUWNS 1 THRU 6 

0.0000D+00 0.0000Dt00 

1,2151D-03 9.3810D+Ol 

O.OOOOD+OO 0.0000D+00 

7.35433)-03 -1.045ED+02 

0,001iODt00 0.0000Dt00 

O.OGOOD+OO 0.0000Dt00 

@. M00D+00 0.0000E+00 

O,OO?OD+M 0.0000Dt00 

0.00MDt00 0.0000DtW 

9.2815D-04 0.0000D+00 

o,oo00D+00 1.oO0oDtoo 

1.1892D-03 9.2813DMl 

O,0000D+00 0.0000D+00 

0.0000D+OO 0,000ODtOO 

0. ~00D+OO -1.2929DtO3 

9.9950-01 -1.5381D+Oi 

-2.2601W3 4.1083D-01 

caws 7iwu 12 

0.0000D+OO 0.0000D+00 

0.0000D+OO 0.0000D+00 

0.00OoDt00 o.OOooD+oo 

0.00oD+Oo o.oO0oD+oo 

0.0000D+00 0.oO00~00 

-1.0568MO 0.0000DtW 

1. 0000D+M 0.0000D+00 

0.0000D+00 0.0000Dt00 

0.00OoD+OO 0.0000DNO 

0.0000~00 O.OOOOD+00 

0,OOOODtOO 0.0000DtOO 

0.0600~00 1.0000Dt00 

o.ooooD+00 o.OOooD+OO 

0.0000~00 o.oO0oDtOO 

0.0600~00 0. ooooDt06 

o.ooooD+oo o.ooooD+oo 

9.9569-01 0.0000Dt00 

0.0000D+00 

0.0000Dt00 

1.000OD40 

-5.1574D40 

0.0000DMO 

0.0000Dt00 

0.3OOD40 

0.0000D90 

0.0000DMO 

0.0000D40 

0.0000D90 

0.0 00 ODMO 

0,OOOODMO 

0.0000DMO 

0.0000DMO 

0,0000D40 

o.oO0oD+oo 

0.0000DMO 

0. WOOD40 

0.0000DMO 

-5.0353w8 

0.0000D40 

0.0000DMO o.oO0oDw 

O.OOOOD+M 

9.7656D41 

0.0000DMO 

O.OOOOD40 

O,OOOOD90 

0.0000D+OO 

O.OOOOD+M 

O.oooOD+OO 

0.0000DMO 

4.7963D45 

O.OOOOD+OO 

-3.1062D-05 

0.0000DtOO 

1.0534D-03 

0. WOODM0 

0.0000DMO 

0.0000DMO 

0.0000DW 

0.0000D+OO 

0.0000D90 

0. 0000DW 

-1.2611D46 

0,0000DMO 

1 0000DW 

1. H29DtO3 

0.0000DNO 

8.83258+6 

0.0000DW 

-3.m-06 

0.0000DtOO 

2.9137D-06 

O.OoOOP90 

0.0000DW 

0.0000DMO 

0.0000DW 

O.#K)oD+OO 

0.0000D+QO 

0.0000D90 

-3.7834D06 

0.0000DtOO 

0,0000DW 

0,0000DW 

0.ooOODMO 

1.18051)42 

A-11-6 
COLUnHs 13 TtRU 18 

0.0000D+00 O.O#OD+M 

0,0000Dt00 0,0000Dt00 

O,0000D+00 0.0000D40 

OIOO@OD+OO OIOOOOD+OO 

OIOOOOD+OO 0.0000D+00 

0.0000Dt00 0.0000Dt00 

O.OOOOD+M 0.00#0D+00 

OI0000b+00 0.0000DtOO 

O.Moi1OD+00 0.0000Dt00 

O,0600D+00 0.0000D+00 

0 m oOOOD+M Om 0000D90 

0.06OoP00 o.o00oD+oo 

Om OOOOD+00 0.0000Dt00 

Om oOoODt00 O,0000D+00 

O.OOOOD+00 OIO000D+00 

0. WNDt 00 0.0000DW 

6. OOoObM Om 0000Dt00 

0.0000D40 

0,0000D40 

0. OOOODdO 

0.0000D40 

0.0000Dt00 

0.0000B+00 

0.0000D+00 

0,0000D40 

O.O#ODc00 

0.0000D40 

0.0000D40 

0.0000D40 

0,0000D90 

0.0000D40 

0.0000D40 

OIO#OD+OO 

0.0000D~0 

Om 0000D40 

OIOOOOD+OO 

0 m 0000D+00 

0 m 0000D40 

0.0000D90 

0.0000D+00 

Om 0000D40 

Om 0000D90 

0.0000DW 

0,0000D40 

0.0000D90 

0 m 0000D+00 

Om 0000D40 

Om 0000D90 

0.0000D40 

0.0000DW 

OmooOODtOO 

O,OO00D+00 

O.MOOD+OO 

0.0000D+00 

0.0000DtQO 

OIOOOOD+OO 

0.0000DtOO 

0.0000DMO 

0. MOOD40 

O,0000D+OO 

0.0000D40 

0.0000D+OO 

0.0000DtOO 

0.0000D40 

0.0000DtOO 

0.0000DtOO 

Om 0000D40 

Om00MD40 

0.0oO0~OO 

9.5783b+oQ 

0,~DtOO 

-4.5153DtOl 

0. ooooD+oo 

0, ooooD+OO 

0. ooooD+oo 

0. OOOoDtM 

0. ooooD+oo 

0. OoOoDtoo 

OmoooOD+~ 

9.581moo 

0 I ooooD+oo 

0; ooOoD40 

o.oOOoD+oo 

o.ooO0Dtoo 

-1.6616D-01 

A-11-7 
H-Matrix (Cont'd) 

caumts E TMU 30 

O.O000D+00 

0.0000D+00 

0.0000D+00 

O.OOQOD+OO 

0.0000D40 

0 m 0000D40 

0.0000D+00 

0. ooooDtoo 

0 m 0000DtM 

0.0000Dt00 

0.0000D40 

0.0000D+00 

Om 0000D+00 

0.0000D+00 

0.0000D+00 

0 m 0600Dt00 

0 m 0000D40 

Om 0000D40 

Om 0000D90 

O.OOQOD+OO 

0 m 0000D40 

Om 0000D90 

0.0000D90 

0, ooooi,9o 

0 . 0000DW 

0.0000D90 

O.OMOD40 

O.OMOD90 

O,OOOODtOO 

0 m 0000D90 

0 m 0000D40 

Om 0000DM0 

o.o0ooD+oo 

0. OOOOD~ 

0.00MD90 

0.OOOODMO 

O.oOM)D+OO 

0.0000D+OO 

0.0000D+OO 

0 m 0000Dt00 

0.0060DtOO 

Om0000DtOO 

0.0000DMO 

O.OOOOD+OO 

Om 0000DtOO 

O.OOOODtOO 

0.0000D+OO 

Om Dw)oD+OO 

0. 00ooDtoo 

Om0000D90 

0 m 0000Dt00 
Control 1 pi- Desi En 

U = GAIN * z 

where 

z are perturbed values of 

[c$ p h ax M IM lh] T 

and 

T 

u = r6T 6e 6a 

ap ap 

COLlNNS 1 TH?U 6 

-2.6551D-02 4.7021D-02 -1.2989D42 -1.14883+02 -1.0926DtO2 -le3532D+03 

1.3259b-01 4.1906b-01 5.9632D-42 2.9394042 -6,49(D+OO -4.9372Ml 

-6,0067DtOO -%1092P+oi 3.7164D-12 1.7632M -7.0812H2 6.166133-10 

2.6041D-01 -2.3116D+OO 1.067733-13 1.9321D-09 -4.5292p-13 7.29flb-ii 

COLMIS 7TtRu 8 

-1,38955b92 -2.7503D-03 

-7.9314DM 5.8963043 

-1,839W-11 3,6752D-13 

-1.0162D-12 4.1311D-14 

A-11-9 
Open-loop ei g~:iv31 ues: ------ 

Closed-loop eigenvalues: 

-1.3612kQ2 +7.9936klSi 

-1.OOOWO2 6.11 E9blCi 

-1.13)55D+OZ 42.4338E151 

-8.8194plOl +4.%66151 

-8.W*4D*Ol -5.7886DMl I 

-8.W4PlOl 6.zBbGDCcli 

-6.loBoD+Ol -2.1706Wll 

4.1080t91 t2.1706Wli 

-4.¶06%01 *1.0959blCi 

-2.6024D91 +6.4031D-l5i 

-1.SKK)blOl 40. ooooc9oi 

-5.418W +3.3073Wli 

d.4187DW -3.307SD91i 

-5. t713D4W -4.4sO6b+ol i 

d.071SD100 +4.4306D+Oli 

-4.1391bOO -1.4MWli 

-4.1391D90 41.4054Wli 

-3.2028D400 +4.0631DWi 

-3.2028p90 -4.0631Dcooi 

-1.oooOD+OO +O.#WK)Wi 

-6.UUWl *C.7315D-#li 

-6.U39Wl -6.73lsWli 

-5.oooOW1 +O.#WK)Wi 

-4.7765D-01 9.oooODWi 

-3.W9Wl +1.151SF17i 

-2.7OWE-01 +1.3002D-171 

-2.5DODwI +1.)092D-l7i 

-2.W5D-01 -2.0031wli 

-2.W5l+01 *2.0037Wli 

-1.5000Ddl -1.oooOWli 

-1.WOD91 +l.WOMli 

-l.(oooWl -1.(oooWli 

-I.(OOOWI +i.ummii 

A-11- 1 O 
.24 

.21 

.18 

.15 

0 - 

L 

B -12 

e09 

.06 

.03 

0 

CLOSED-LOOP TIME RESPONSE 

MACH NO - VS TIME FOR A RAMP MACH NO COM?IAND 

A-11-1 1 
15 

I@ 

5 

0 

-5 

-10 

-15 

-2D __ 0 5 ia t5 za 25 30 35 40 

TLME 

CLOS ED-LOOP TIME RESPONSE 

ALTITUDE VS TIME FOR A RAMP MACH NO COMMAND - 

A-11-12 
APPENDIX A-I11 

AIRCRAFT - CAS LINEARIZED AND 

FTTC DESIGN USING EIGENSTRUCTURE ASSIGNMENT 

AT 0.7 MACH, 10000 ALTITUDE 

A-111- 1 
caw 

.l. lneFO2 - 1.749?P-M 

7.6271D-04 

O.oooOD+OO 

5.3481b13 

-3.4192D-09 

-1.30?3P-10 

O.oooOD1M 

O.OOC41uOO 

o.oooob+OO 

1.03MD-01 

o.mD+w 

O.ooOoD+OO 

O.oooOD(OO 

-2.0759EOl 

l.OJBbD-02 

1.0380D+OO 

o.ooooD+w 

O.#KK)trW 

o.mt+OO 

l.lXlM9 

G.WOb+W 

O.oQooD+OO 

O.oooOD+OO 

o.ooOoD+w 

0.00006+#) 

0.oooOPW 

O.oooOD+OO 

O.ooop:D+W 

O.oooOb+OO 

O.#KK)D+W 

1 THRU 6 

2.2l?SDtOl 

-1.3?9?D+W 

-6.6234D4 

0.0000D40 

-1.71678 10 

8.9156P-10 

-4.9?08P-W 

@.oooODJW 

-7.54 1 ED42 

0.0000t100 

7.9T15pto2 

O.WD* 

O.#KK)W 

C.oooOM 

-1.5W3 

7.9775Ml 

7.9775w3 

0.00D9C 

O.#)OD+W 

O.WD+W 

-5.541SM7 

0.oooODW 

0.WWW 

0.00p40 

0.oooODW 

O.ooQoD+W 

0.00D+Oo 

0.oooODW 

O.oooOD+oo 

0.00W 

0.ooOODM 

muocs 7nac 12 

O.ooOoD+W O.#KK)D+W 

0. WD(W O.oooOD*OO 

O.WOD+OO 0.00D90 

o.ooooD+OO 0.ooooDW 

-3.9944D-01 4.2597D-02 

1. W47bOO -2.504 43-23 

-7.fBUD-01 3.2312.D-24 

3.3WD-02 O.OOOOD+oo 

O.ooOob+W 0.wm 

o.oooOD+OO o.oooOD+oo 

1.8073bOO 2.5849k23 

0. FoooD+ 00 0. mD+OO 

O.OO%P+O@ 0.OOOQDCOO 

O.oO@obOO 0.00D90 

-3.6146bW -5.1643L-23 

1.8073p-01 O.OOOOD+oo 

1.8073bOl -2.oWD-22 

0.mW 0.oOoQ~ 

o.oooOD+OO 0.00W 

o.ooooD+OO o.oooOp40 

-B.mD*#) %.059(W7 

8.8oooD+01 o.oooOD90 

0.oooObW o.wb9o 

O.rnhW 0.oooOW 

o.oooob+OO o.oooop1oo 

o.oooOD90 0.00D90 

0.oooOkOO o.mD90 

0.oOoGW 0.WWDa 

0.oooODa o.mp40 

..wow O.oooOb+W 

O.ooOob+W o.oooOp40 

-3.?144D+01 

0.0000D*OO 

1.0976b03 

0.0000D+OO 

0.0000D+OO 

0.WO~W 

0.ooOObtoQ 

O.oooOD1OO 

7.54181u02 

O.oooO31OO 

2.2495D-02 

O.WWD+OO 

0.0QWDcoo 

O.oooOD90 

-4.4991 W2 

2.2495W3 

2.2495Wl 

O.oooOD40 

0.oooOW 

o.oooOD1w 

O.oooO64a 

O.ooOoD+W 

O.WD+W 

0.WWW 

O.oooOD+OO 

0.ooOQW 

0.ooOoPW 

0.oooObW 

0.WWM 

O.oo00~ 

0.OOOOW 

0.00DW 

0.00DW 

0.0000M 

O.#KK)DW 

-2.OEWWl 

-4.3656m1 

1.0573NOl 

O.~DW 

0.WW 

0. ooooD+a 

-5.1onw1 

0.00DW 

0.ooOODW 

0.00DW 

1.0216D+o2 

-5.10nm 

-5.10np(o2 

0.00p40 

0.00DW 

0.00p+Oo 

2.97MpcO1 

0.00M 

0.00DW 

O.ooOOD+a 

0.00M 

O.ooo3W 

O.ooOoI*oQ 

0.00W 

O.oooOD90 

O.oooOD4Q 

O.ooOOD90 

A-I 11-2 
F Matrix (Cont'd) 

A-111-3 
0.0000D90 0.ooOODtW 

O.OOOOD90 0.ooOODtoO 

0.0000D46 O.oooOD+OO 

0.0000D40 -5.10TIw2 

0.0MOD90 -2.131OPOO 

O.OQOOD40 -5.145lbOO 

0.0000DW 0.oooOptoO 

0.0000b+00 0.0000F00 

o.ooooD4o o.ooOOM00 

0.0000Dt00 -2.4933W 

0.0000D+00 O.WD+OO 

0.0000~ o.wD+OO 

O.WD+00 O.oooOD+oO 

o.o0OoD+oQ 4.9866Dt00 

O.ooOOD+W -2.4933D-01 

O.oooOD+OO -2.493WOl 

o.oooob9o 0.oooo~00 

o.ooooD4o 0.wDtoO 

o.wD4o o.ooooD+oo 

O.ooOOb40 -6.4105RtOl 

o.ooooD4o 0.ooOoW 

o.ooooD9o o.oooO5+oo 

o.oooOD4o 0.ooooD.m 

0.ooOow 0.oooo~00 

O.oo00~ O.WD+oo 

o.ooooD9o 0.ooOoDtOO 

-1.ooooD+o2 0.ooOoDtOO 

4.8866D-01 -2.8oOOD+Ol 

O.oo00~ 0,ooOoDtOO 

0.OOoow o.ooOoB+oo 

0.0000Dt00 0.0000Dt00 

A-111- 4 
2. C-Matrix (31 x 4) 

A-111-5 
I 3. H-Matrix (17 x 31). 

CULUHNS 1 THRU 6 

-3.4B2D-09 9. M49D-10 

2.9388D-03 3.1117Dbi 

0.0000Dt00 O.WP90 

7.6271D-04 -6.6234W 

o.ooclbDto6 o.ooooD4o 

3.5240-11 -6.3682B-09 

-1.3023D-10 -4.92007WB 

o.oo66D+oo 0.0000w 

o,ooooD+oo O.oo00~ 

9.2815D-04 0.oooOpeoO 

0.6066D+OO 1,ooOo~ 

2.9309D-03 3.1042M1 

o.ooooD+00 0.6oOoD+oo 

0.0MODW O.OMOP40 

O.oooOD+oO -7.5418bt02 

9.9943D-01 -2.5208Ml 

-2.6793p-01 1.7293WO 

0. o0o0~00 

5.89190-05 

l.oooOP40 

-3.124aW 

O.ooobD+OO 

O.oooOb+oo 

0.0000DW 

0.oooODW 

O.oooOD+QO 

0.00OODcoo 

0.6ooODtOQ 

2.91470-05 

0. ooooD+oo 

0.ooOoDtOQ 

O.oo00~ 

0.6ooODcoo 

-2.1420D-M 

A-I 11-6 
A-111-7 
H-Matrix (Cont'd) 

A-11 1-8 
where 

z are perturbed vaiues of 

'6 p h a M JM JhIT 

X 

and 

CUW 1llUU 6 

-1.4mW2 -7.3096w2 -9.2850b-02 -1.7141H1 -1.3S9W2 -1.5352M3 

1.1924D-02 3,2934D-02 1.01W2 3.9151D-02 -2.W2DM -1,9619M1 

-2.5437DtOO -1.m1 2.W5D-11 4.3619D-11 7.2545wB J.#Roc 

1.410Jwl 9.3187l-02 -5.6683D-12 2.932Wll 5.98iOB-M -7.4-7 

cam 7lIRu 8 

-1.3916bCb2 -7.2697w3 

-2.4791DtOO i.OTJ2wJ 

-1.3702M 3.7197D-12 

-2.D3D-48 -i.l4bnk13 

A-III-g 
Closed-loo? eigenvalues -- 
CLOSED-LOOP TIME RESPONSE 

MACH NO VS TIME FOR A RAMP MACH NO - COMMAND 
CLOSED-LOOP TIME RESPONSE 

ALTITUDE VS TIME FOR A RAMP MACH NO COMMAND - 

A-I1 I- 12 
APPENDIX A-IV 

DESIRED EIGENVALUES AND EIGENVECTORS 

USED IN THE SYNTHESIS 

A-IV-I 
DESIRED EI GZN~LUES : 

> 

-2.WWf +2.oooOWli 

-2.0000M)i -2.0006D-Oii 

-1.5MOD-01 +l.oooOWli 

-1 . 5OOOD-01 -1 . WD-01 i 

-1.~D-01 +i.4000Wii 

-2.W)Wl +O.~p4oi 

-1.1(oOoW1 -1. WWf i 

-2.7000Wl +O.oooOp4oi 

States 

DESIRED ET GENVECTORS : 

Q 

Q. 

Om 

Om 

0. 

Om 

99. 

0. 

I. 

0. 

0, 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

P 

0. 

0. 

0. 

Om 

Om 

I. 

0. 

99. 

Om 

Om 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

h 

Om 

99. 

99. 

0. 

0. 

0. 

Om 

99. 

1. 

0, 

99. 

99, 

99, 

99, 

99. 

99, 

99. 

99, 

99. 

99. 

HE AS URE MZ NTS 

ha x M 

0. 99. 99. 99. 

99. I. 99. 0. 

99. 0. 0. 0. 

1. 99. Om 99. 

0, Om Om Om 

99. 0. 0. 0. 

0, Om Om 0. 

99. Om Om Om 

99, 99. Om 99. 

0, 99. 1. Om 

99. 99. 99. 99. 

99, 99. 99. 99. 

99. 99. 99. 99. 

99. 9% 99. 99. 

99. 99. 99. 99. 

99, 99. 99. 99. 

99, 99. 99. 99. 

99. 99. 99. 99. 

99. 99. 99. 99. 

99. 9s. 99, 99. 

jh 

Om 

99. 

99. 

Om 

0. 

0. 

0. 

Om 

99. 

Om 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99. 

99, 

99.99.99.99.99.99.99.99. 

99. 99. 99. 99, 99. 99. 99. 99. 

89. 99. 99. 99. 99. 99. 99. 99. 

99. s9. 99. 99. 99. 99. 99. 99. 

39. 99. 99. 99. 99. 99. 99. 99. 

39. 99. 99. 99. 99. 99. 99. 89. 

99. 99. 99. 99. 99. 99. 99. 99. 

99. 99. 99. 99. 99. 99. 99. 99. 

bga 99. 99. 99. 99. 99. 99. 89. 

39.99.99.99.99.99.99.99. 

39. 99. 99. 99. 99. 99. 99. 99. 4 

39, 99. Om Om 99. 99. 1. Om m 

99. 99. 99. 99. 0. 0. 0. 1. /h 
APPENDIX B 

A DEMONSTRATION EXAMPLE FOR EXACT NONLINEAR 

CONTROLLER DESIGN WITH PRELINEARIZING TRANSFORMATION 

€3- 1 
The theoi-y of prel incarizing transform.3tions for a class of nonl inc.ar 

feedback system design has already achieved maturity. Pioneering work on 

these transformations for flight control problems was carried out at NASA 

Ames research center by C. Meyer and others [l-143. 

are currently available, they deal with large dimension problems and consequently do not contain sufficient detail to serve as introductory example. 

Though several papers 

The objective of this technical memo is to illustrate the application 

of these transfornations with a simple but realistic example. The theory is 

excluded completely from the discussions since the transformation scheme for 

this example turns out to be direct. The performance of the exact nonlinear 

controller is compared with a gain scheduled linear perturbztion controller 

to bring out the advantages of the former in clear detail. 

---IILLUSTRATIVE EXAWLE : --- 

Consider a two state variable model of a transport aircraft landing in 

the presence of winds as follows 

nT-D - +any L mVcosY 

li = tanY (2) 

here, V is airspeed, h altitude, T maximum thrust, D aerodynamic drag, 

Y flight-path angle, and rl 1s the throttle setting. Y and are the control 

variables in this model and down range 1s the independent variable. Row, a 

hypothetical thrust and drag characteristics are assumed as 

T - To(V + Vw) 

2 D - eoBhD 0 (V + VW) 

E-2 
Note that ttw engine model is grossly incorrect at low speeds. 

It is required that the aircraft follow a given trajectory V (t), 

A 

hR(t). 

Since there are two state variables and two control variables in this 

problem, consider a transformation of the form 

x -V, x2=h 1 

nT-D = tanY 2 

u = ---- - tanY; u 1 mVcosY V 

such that the transformed system is the form 

1 x =u 1 

2 x =u 2 

B-3 

This model is in Brunovsky's canonical form according to Meyer [lo]. 

If the number controls are less than the number of states, a simple transformation such as the one given above is no longer possible. A transformation is still feasible under certain weak conditions but will involve 

additional algebra. 

To consider the regulator problem first, a controller of the form 
mVcosY D - [k V + k h + tanY] -- + - 11 12 V T ‘r (6) 

Expressions (5) and (6) give the nonlinear regulator for the aircraft 

problem. A block diagram of the plant with regulator is given in Fig. 1. 

I 

-1 Y Nonlinear , V 

tan (k2,V + kZ2h) 

fliircraft h 

[kllV + k12h + $tsny] . Model 

I 

Figure 1. Nonlinear Regulator with Aircraft 

The tracking problem is addressed next. Define the new state 

var i a bl es 

x - V‘VR 

Y - h-% 

ORIGINAL PAGE IS 

OF POOR QUALITY 
y - tanY-C 2 

where 

using the transformations 

x = x, x2 = y 1 

qT-D 

'1 mVcosY v - -- - tanv-C, 

u = tanY-C2 2 

one has the following 

x-u 11 

x2- u2 

which is in the Brunovsky's canonical form. 

As before, a feedback regulator can be synthesized for (7), (8) as 

shown below using linear design approaches. 

This controller can be transformed to original coordinates to give: 

E-5 
Expressions (9) and (10) give the nonlinear controller for trackin& the 

commands VR(t) and hR(t). 

gral feedbacks can be incorporated with very little difficulty. 

If a tighter control is desired, additional intehR * vR * 

Range h V 

feet feet fe;t/s 

0 10000 350 

150000 3000 280 

200000 0.0 200 

205000 0.0 60 

NONL I NE A R CONT R OL LE R E V A L Ll A T I ON 

c2 * * 

\j - - t; 

-0.04667 -0.0004 67 

-0.06 -0.001 6 

0.0 -0.02 

0.0 0.0 

To evaluate the nonlinear controller (9) and (10). a feedback gain 

matrix was computed first using linear quadratic regulator theo.ry with the 

following state and control weighting matrices. 

xx -[: 1:0] * RUU - C'a" 1 OI* o* 

These computations were carried out with the hypothetical data 

I -2 Y 

0 0 

- - 0.04025, 7 0.0136238 x 10 . m 

To serve as a standard for comparison. a gain scheduled linear perturbation controller was designed with the same data. Four design conditions 

were used which are given in Table 1. 

TABLE 1. LINEARIZATION POINTS FOR THE ILLUSTRATIVE 

EXAMPLE 

8-6 
K:>t c’ t ti-it t tit. 1 in(3ni-l znt ion p?i nt s W(*I-(> on 1 )IC corilii~ii~i~~d t i-ajc,ct ory. 

Tuo ~jniulat ions are n(.xt setup. The first one irnplcincBntjriE the nonIlncnr controller (9) and (10) and the second one using thc Enin schcduled 

linear perturbation controller. A disturbance in the form a wind shear was 

introduced in the simulations for addl tional real ism, (fig. 2). 

Figures 2-4 give the results. Note that the nonlinear controller has 

a slightly superior tracking performance. The real advantage of using this 

controller, however, is that it performs well using one set of gains 

throughout the operating region while the 1 inear perturbation controller 

requires a gain schedule to achieve acomparable performance. 

-- CONCLUS I Oh’s 

A simple illustrative example was discussed in this technical memo to 

demonstrate the power of the prelinearization transformations in 

synthesizing exact nonlinear controllers. 

gain scheduled linear perturbation controllers to show relative advantages. 

Comparisons were made against a 

E-7 
1 2 3 4 5 6 7 8 9 to 

WINDSPfXD(-/S) 

Figure 2. Wind Shear Used in Simulations 

8-8 
7 LOGO 

I moo 

2000 

6000 

2000 

a 

Figure 3. Comparison of the Performance of Linear and Nonlinear Controllers: Altitude vs. Range 

E-9 
w 

P 

3 

b 

EJ 

c 

w 

40tl 

350 

300 

250 

200 

t 5u 

50 

0 

194000 19- 198000 200099 20200b 264000 2b6000 

RAWGE 

Figure 4. Comparison of the Performance of Linear and Nonlinear 

Controllers in the Vicinity of Touchdown: Altitude vs. Range 

A: Nonlinear Controller 

B: Linear Perturbation Controller 

C: Commanded Trajectory 

E-10 
Figure 5. Comparison of the Performance of Linear and Nonlinear. Controllers: Airspeed vs. Range 

A : Nonl inear Controller 

5: Commanded Trajectory 

C: Linear Perturbation Controller 

B- 1 i 
r Ct' and Ci rwl ani , L ~igi f,.: A Fnrmrll Str-uct ur'e forAdvnr~ccd Automatic FliCht Control Systcrns. NASA TN D-79110, 1975. 

Smith, G. Allarl; and Meyet., Geor-gc: Total Aircraft Fligtlt-Cont rol 

System - B3lanced Open- and Closed-Loop Control With Dynamic Trim 

Maps. Proceedings of the Third Digital Avionics Systens Conference, 

Ft. North, Texas, Nov. 1979, IEEE Catalog No. 79CH1518-0. 

Cicolani, L.S.: Sridhar, E.: and Meyer, George: Configuration 

Manasenlent and Automatic Control of an Augmentor King Aircraft with 

Vectored Thrust. NASA TP-1222, 1979. 

Smith, G.A., and Meyer, G., Total Aircraft Flight Control System 

Balanced Open- and Closed-Loop Control with Dynamic Trimmaps. AIAA 

3rd Digital Avionics Conference, Dallas, Texas, 1979. (Published in 

IEEE proceedings.) 

Wehrend, W.R., Jr., Pilot Control through the TAFCOS Automatic Flight 

Control System. NASA TM-81152, 1979. 

Smith, G.P.., and Meyer, G., Application of the Concept of Dynamic Trim 

Control to Automatic Landing of Carrier Aircraft. NASA TP-1512, 1960. 

Wehrend, W.R., Jr., and Meyer, G., Flight Tests of the Total Automatic 

Flight Control System (TAFCOS) Concept on a DHC-6 Twin Otter Aircraft. 

NASA TP-1513, 1980. 

Meyer, G., and Cicolani, L., Applications of Nonlinear System Inverses 

to Automatic Flight Control Design - System Concepts and Flight 

Evaluations. Theory and Applications of Optimal Control in Aerospace 

Systems, P. Kant., ed., ACARDograph 251, 1980. 

Meyer, C., The Design of Exact Nonlinear Model Followers. Joint Automatic Control Conference, FA-3A, 1981. 

Hunt, L.R., and Su, R., Linear Equivalents of Linear Time-Varying 

Systems. International Symposium on Mathematical Theory of Networks 

and Systems, 1981, pp. 119-123. 

Hunt, L.R., and Su, R., Control of Nonlinear Time-Varying Systems. 

IEEE Conference on Decision and Control, 1981, pp. 558-563. 

Hunt, L.R., Su, R., and Meyer, G., Global Transformations of Nonlinear 

Systems, IEEE Trans. on Autom. Control, Vol. 20, 1982, pp. 24-31. 

Hunt, L.R., Su, R., and Meyer, G., Multi-Input Nonlinear Systems. To 

appear in Differential Geometric Control Theory Conference, 

Birkhauser, Boston, Cambride, Mass., 1982. 

Meyer, C., Su, R. and Hunt, L.R., "Application of NonJinear 

Transformations .to Automatic Flight Control", Automatics, VOl. 20, No. 

1, pp. 103-107, 1984. 

B-12 
APPENDIX C 

FORTRAN PROGRAM FOR WVEUVER MODELING 

c- 1 
Y) r 

E 

d 

I 

8 

..... 

rlNm*Y) 

. . . . . . . . . rlrUm*lnat-mO, 

I 4 N m 
Y 

s 

f d 

0 

-0 

is 

0 

uuuu U (Y d zuuuu 

I 

I 

uuu uu uuuuuu uuu uuuuuu uuu 

C-3 
d 0 

f 
d 4 

8 

d Suuu uu .+ I- uuu .+uu uuuu 

I 

E 

8 

t:: 

t 

d 

u uvu-uuuuuu uuuu u uuu uuuuuu uuu uuu 

c-5 
E 

0 

rl =luuvu uv U uuu I m 

E 

0 
0 

U rlm 

N 4 %uuuu 0 

UUU U U U N ... 

e- 7 
4 

CI I 

c( 

0 - 

0 

x U uuuuu uuu uuu guuuu ____ .. 

C-8 
. .- -a LDa . . . .e 

0 n 

B 

2 

uuu uuuuuuuuuuuuuuuuuuuuuuuuuuuuuu u uu , 

9 r(N d 0 VI d 0 N 00 44 

U UUU-UWV WUUUUUUUUUU UUU uwu 

c-9 
vu vuuvuuu vu uu uu uv vu uvuu 

0 

000 

'U v * m N u u v v L) u u u ugu u u vu u v u uv 

c-10 
I 

uu vu vu uv uv vu 3uuu uu I 

vu 

I 

i 

I 

I 

I 

1 

I 

~ 

1 

I 

! 
VU vu VU U vu UU vu I UUU UV 

0 

0 

0 m 0 

- m 

2 

, 

uu uu VU uu buuu V uuu vu vu 

c-12 
I 

I 

m 

huuu uv uu uv uu uu uu uv uu 

m 

8 

I 

huuu uu vu vu vu vu uu __ I uu 

C-13 
I uu uu vu vu 1 
I 

uu uu uuuu uu 

uu uuuu UU 

- 

C-15 
In 

0 

N d 

m 

- 

N 
x 

uu uuuu uu uu 

I 

uu uu .~ uu uu 

C-17 
I 

0 

0 

vu vu uuu uu 

vu uu uu 

C-18 
uv v' uv vu 

uv uuuv vu uv ~ 

c-19 
vu uu uu uv I 

> vu uu uuvu 

C-20 
vu vu vvvv vu 
I 

I 

uuuu uu uu 

t 
vv vu uu uv I 

uu uuuu vv uvuu ~ .- 

c-23 
I 

I vv vv vvv vu 

, 

vvvv vu vu 

c-24 
4 

I 0 

ur 00 0 

: V%S u uuuuuuuu uu Zuu uuuu m m m uuuu 

I 

.- . .m . . . om00 roo0 .4 ..4.4044.4 

0 

0 0 

uuuu uu..- - uzuuuuuuuuuuuuuu 4 

c-25 
00 

00 

00 

U U U U U U uuuu U4-4 uuuuuuuuuu u I 

0 

fo 

00 0 

f 2 su uu YU u. U U U U U 

C-26 
.. 

u,r 

rm 

In* 

g$ 

d0 

00 

00 

.. 

&- 

w4 

40 

00 

00 

I-4 

mm 

00 

00 

00 

.. 

.. 

P-N rm 

zz 

60;: 

222 

C0 O 

.. 

mtvr 

u,ma 

u,4* 

000 

000 

... 

E5 

=LN 

%Z 

En 

ad 

vi 

01 

vi 

u, 

0 

: 

0 

0 

0 

d 

In m 0 

d 

0 

s 

g% 

.\ 4Ln .u, 

4. 

.O 

4. .u, 

mu, dol

ru, 

.. 

FSO uu 

00 

uu uu uu uu Ud N uuuu u , 

0 

0 

uuuuu u uuuuuu uuuu uuu uuuuuu uuuu uuu uuuu u4 uuuu u 

c-27 
APPENDIX D 

LINEAR TIME VARYING SIMULATION 

IN SYSTEM BUILD - 

D- 1 
LINEAR TIME VARYING SIMULATION IN SYSTEM BUILD 

Linear time varying simulation can be built around a Fortran block 

evaluating the matrix equation 

y = A(t)X + B(t)U (D. 1) 

where y E El', X E Eln, U E Im. A and B are time varying matrices of compatible dimensions. To use this Fortran block, the A and B matrix 

elements must be arranged in the following tabular form and transferred to 

Matrix stack. 

X 

(D. 2: 

to All nn B1l B21 Bnl *** 

A21 ... Anl A12 A22...An2... A 

tl All A21 ... Anl A12 A22..An2 .... Ann 

t2 All A21 ... Anl A,2 A22..An2 .... Ann 

B1l B21 Bnl a** Brim 

B1l B21 Bnl *.* Brim .. 

.. 

Table = 

4 

in dependent 

var i ab1 e 

To build in the Fortran block, two distinct steps are necessary. The first 

one consists of building a dummy superblock with the name TABLE(N), where N 

could be any number from 0 to 9. The Table D.2 is built as a linear 

interpolation table block in this superblock, say at relative location M. 

Next, the superblock(s) including the time varying Fortran block may 

be built with the following responses for the queries 

c-2 
Dimension of RP & IP: 0,2 (no real parameters, two real parameters) 

enter IP : N,M (Dummy superblock Table number, relative 

locat ion) 

Note that to complete the time varying simulation, one has to define 

an independent string of R integrators. A schematic flow chart for building 

the time varying linear system with this FORTRAN block is given in Figure 

D.l. 

The SYSTEM - BUILDTM block diagram for linear time varying simulation 

for the aircraft example discussed in this Appendix is given in Figure D.l. 

The first block in this diagram computes the derivatives of the states using 

the Fortran block. These are then integrated in block 2. Block 3 

essentially routes various inputs and outputs. The reference trajectory is 

generated in Block 4 while the wind is generated in Block 5. 

ments the time varying gains or gain schedules, again using the Fortran 

block. 

Block 6 imple- 
TABLE AS ALINERR INlf RPOL AT ION BLOCK 

UllTH RELATILIE LOCATION 

M 

~~~ ~ 

BUILD THL TIME URRYING 

S IMULRTION SUPL RE1 OCKUfHllE BUILDING IN THE 

FORTRAN BLOCKS, ENTER 

DIMENSION Of 

RPDIP: 0,2 

IP: N,M 

BUILD OTHER BLOCKS G. 

CONNECT 

N: Any nuiribrt from 

0 to 9 

M: Any number from 

1 to6 

1 I 

Figure D. 1. Flow Chart for Building Linear Time Varying Simula ‘cr. on 

SYSTEM BUILD - 

D-4 
APPENDIX E 

MINIMUM ERROR EXCITATION OUTPUT FEEDBACK DESIGNS 

E- 1 
MINIMUM ERROR EXCITATION OUTPUT FEEDBACK DESIGNS 

This appendix gives the performance index specification for the 

control laws designed at each flight condition. 

description of section 4.2.2, these performance indeces completely specify 

how the 30 sets of output gains can be reproduced. 

Together with the algorithm 

LQR Perf ormance Index 

03 

-2at T T (x HA Hx + u Ruuu)dt, with 

Y 

J= I,e 

1 1 1 1 1 

(0.001)2’ (0.001)” (0.001)2’ (0.005)2’ (0.005)2’ 

1 

(0.005)‘ ’ 

T State weighting matrix, Rxx = k H A H, Y 

here: k is a scalar providing relative state-control weights 

H is the system output matrix 

The control weighting matrix R = diag’(B) uu 

Table D-1 lists the scalar weight 

function of the design flight condition. 

k and the state weighting vector B as a 

E-,? 
TABLE E-1. PERFORMANCE INDEX SPECIFICATION FOR EACH FLIBIT CONDITION 

(Subscript 1 corresponds to the design with all controls, 

subscript 2 corresponds to the design without throttle 

control. 1 

ALTITUDE AND MACH 

LOAD h = 10k h = 20k h = 30k h = 40k h = 50k 

FACTOR M * 0.8 M = 1.0 M = 1.2 M = 1.4 M= 1.8 

B1= C0.l 1 1 11 Bl COS1 1 1 11 Bl C0.l 1 1 13 B1 - CO.1 1 1 13 Bl = CO.1 1 1 11 

k, - 0.002 kl 0.002 kl 0.001 k1 0.001 k1 0.001 

1 

B, = [l 1 11 B2 = 11 1 11 B2 - C1 1 11 B2 = C1 1 11 B, = 11 1 11 

k, = 0.002 kz - 0.002 k, = 0.005 kz = 0.01 kz - 0.01 

Bl- C5 1 1 11 Bl = [5 1 1 11 Bi C5 1 1 11 Bl C5 1 1 11 B, 15 1 1 11 

k, - 0.001 k, = 0.001 k, - 0.001 k, = 0.001 kl - 0.002 

2 

B, L0.1 0.1 13 Bz = r0.1 0.1 11 B2 C0.1 0.1 11 B, C0.l 0.1 13 B, = CO.1 0.1 13 

k, = 0.0005 kz 0.001 k, = 0.001 k2 0.01 kz 0.00001 

Bl= 15 1 1 11 Bl = C5 1 1 11 Bi C5 1 1 11 B1 = [5 1 1 11 B1 = [5 1 1 13 k1 0.001 kl 0.001 kl 0.0005 

B, - C0.l 0.1 11 B2 C0.1 0.1 11 82 = C0.l 0.1 13 BZ [O.l 0.1 13 B, 10.1 0.1 I] 

kl = 0.0005 k 1 0.0001 2 

4 

k, 0.002 k2 0.002 k2 0.002 kz 0.002 kz * 0.00001 

E-3 
i 

Output Feedback Gains 

The output feedback Controller has the form 

u = cy 

where 

m 

1 

u = [&T beap &aap &rap] 

Flight condition: h=lOk ft, M=0.8, n=l 

(a) Throttle free 

COLUMNS 1"J 6 

6.1491D-04 1.9996D+O3 1.2553D+01 3.9128D+03 3.0025D-02 7.0i120-02 

-2.1757D-02 9.3094DlOO -4.8115D.01 -3.8109D+01 -5.8576D-01 -3.095i3-:1 

-8.57001)-05 2.7476D-02 -2.5024D-01 -1.2745D-01 5.8288D+01 -9.71123+01 

-3.4216D-06 1.3714D-03 -9.5302D-03 -5.2506D-03 6.184l.D-01 3.3190D*00 

COLUMNS 7mu 12 

-3.5007D-02 2.7833D+00 -6.4362D-03 -5.8824D+03 1.8056D-02 3.615OE-03 

-3.7222D-01 -2.5667D+01 1.6186D-01 -6.4055D+01 -3.1782D-01 -5.41632-02 

2.3275D*01 -1.5444D-01 2.3048Dl01 -2.4475D-01 3.2259D+01 5.48723.00 

8.139l.D-01 -7.543l.D-03 -4.3978D+00 -1.0375D-02 3.7226D-01 6.3560D-02 

(b) Fixed throttle 

COLUMNS 1mu 6 

O.OOOOD+OO O.OOOOD+OO 0.0000D+00 0.0000D+00 O.OOOOD+OO O.OOOOD+Oc 

-6.9831D-01 -1.7886D+04 -9.4324DlOl -2.5680D+02 -1.9276D-01 -2.7120D-01 

-1.9239D-03 -4.9332D*01 -2.6985D-01 -6.9729D-01 5.8289D+01 -9.7113D+01 

-1.5775D-05 -4.0474D-01 -2.277l.D-03 -5.6596D-03 6.18351)-01 3.3189D+00 

COLUMNS 7mu 12 

O.OOOOD+OO O.OOOOD+OO O.OOOOD+OO 0.0000D+00 0.0000D+00 0.0000D+00 

-1.5857D-01 -2.0762D*01 2.0846D-01 O.OOOOD+OO -9.49681)-02 -1.5752D-02 

2.3276D*01 -1.2968D-01 2.3048D+01 O.OOOOD+OO 3.2260Dl01 5.4873D+00 

8.13881)-01 -9.4264)-04 -4.3978D+00 O.OOOOD+OO 3.7223D-01 6.3555D-02 

E-4 
Flight Condition: h-1 Ok ft, M~0.8, n=2 

(a) Throttle free 

COLUMJS 1zHRU 6 

2.9627D-02 8.1168D+02 -2.85921)-02 -7.8192D-01 -1.3855D-01 -3.8065D-01 

-8.2939D-02 -1.7109D+03 -4.3457D.01 -4.2727DlOl -3.0855D+00 -5.8524D-01 

-6.9160D-03 -3.0315D+02 5.3834D+OO 3.8716D+01 4.1267D+01 -4.6182D+01 

-3.51561)-04 -6.7290D+01 3.0272DlOO 1.6798D+01 -9.9049D-01 1.2926D+01 

COLUMNS 7mu 12 

-8.12730-03 -3.1193D-01 6.5570D-02 7.703SD+01 -2.8132D-01 -8.7290D-02 

-1.623lD*00 -1.0912D+01 -2.1829D+00 -1.9868D+02 -6.1947D-01 -5.4515D-01 

1.7513D+01 1.8824D+OO -3.1690D+00 -4.6712Dl01 2.4901D+01 4.0680D+00 

-1.5747D-01 7.686I.D-02 -3.23YlD+OO -1.4266D*01 7.702I.D-01 -7.0858D-03 

(b) Fixed throttle 

COLUMNS 1lXRU 6 

0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.OOOCT -2C 

-9.4567D-01 -2.25%+04 -1.754lD+Ot -3.7794D+02 8.78220+01 -1.38093’32 

-5.4520D-02 -1.5096D+03 3.5769D+00 2.0487D+01 1.7270D.02 -6.21753.01 

-1.8692D-02 -5.2788D+02 2.5422D+00 5.5214D+OO 6.6557D+OO 5.6499D+00 

COLUMNS 7mu 12 

0.0000D+00 O.~OOOOD+OO 0.0000D+00 0.0000D+OO 0.0000D+00 0.0000D+00 

4.099lD+01 -1.282lD+02 9.6703D+01 -2.1918D+03 3.48461)*01 6.09219-CC S 397lD+01 1 74OlD+OO -1.994lD+OO -1.523BD+02 9.8190D+01 1.6852DlOl 

Flight Condition: h=lOk ft, M-0.8, n=4 

(a) Throt’le free 

OOLUMNS 1THRU 6 

2.2794D-03 7.1210D+01 4.024lD-01 -1.1510D+OO -6.1276D-02 -2.4384D-01 

-4.3918D-02 -9.4806D+02 -4.534lD*01 -8.3475D+OO -1.6230D+01 1.4902D+01 

-3.3214D-04 -2.5846D102 2.857OD+OO 5.711OD+01 3.9784D+01 -4.6713D+01 

2.0198D-03 -6.1265D+Ol l.O238D+OO 2.2273D+01 -2.9065D+00 1.5259D+01 

COLWS 7mu 12 

9.7857D-03 -8.7439D-02 1.9166D-02 1.8054D+01 -3.3040D-01 -1.2460D-01 

-1.7379D-01 -7.5154D*OC -1.8527D*OO -2.8085D+02 7.6036D+OO 6.2348D-02 

1.6825D+01 2.3431D+00 -3.2140D+00 -7.6799D.01 2.9976D*01 5.1782D.CC 

-6.3254D-01 2.4454D-01 -3.4999D*OO -2.0067D+01 2.1%71)+00 2.4978D-0; 

(b) Fixed throttle 

C’XJMNS 1 THRU 6 

0.000OD+OO O.OOOOD*OO 

-2.4868D- 01 -4. %BOD+ 03 

-1.3103D-02 -2.0613D+02 

1.96121)-03 -3.6559D+Ol 

COLWS 7mu 12 

0.0000D+00 0.0000D+00 

2.76151)+01 -1.6666D.02 

1.3028D*02 4.3779D100 

8.04870-01 -6.898l.D- 01 

0.0000D+OO 

-1.9039D+02 

1.2893D+00 

1.6883D-01 

O.OOOOD+OO 

4.0532D+01 

-2.27OlD+OO 

-2.30s 3- C e 

0.0000D+OO 

1.7878D+01 

1.4573D+01 

-1.8448D+O 2 

0.0000D+00 

-1.3112D*03 

-5.5957!2+0? 

-1.23462- 0 1 

2.44e6+00 

- .- 

3.OM3D+OO 

O.OOCCD+?O 

-6.6656D+01 

8.1336DiOO 

-3.5337D- Oi 

0.0000D+OO 

3.1686D+00 

2.6108D+01 

4.9332D-01 

E-S 
Flight Condition: h=20k ft, M=1.0, n-1 

(a) Throttle free 

C3LL1.3:S 1 %L' 6 

-3.OO28D-02 1.4232D+03 -1.4150D+02 1.2419D+04 1.56563-01 2.5076D-01 

-2.47071)-02 3.98928+00 -1.3192D+02 2.1845D+01 -2.2288D-01 -3.9225D-01 

-6.2268D-05 1.0190D-02 -2.5252D-01 6.6599D-02 5.6690Dl01 -l.O369D+02 

-1.5270D-06 3.3189D-04 -6.03951)-03 1.8372D-03 1.0355D+00 3.7093D+00 

C3LUMNS 7mu 12 

-3.17571)-02 -9.4699D+00 -2.85271)-02 -1.7748D+04 8.6553D-02 1.6153D-02 

-1.9513D-01 -2.3818D+01 2.9430D-01 -1.8270D+02 -1.22000-01 -2.0972D-02 

2.2984D+01 -1.95571)-01 2.5598D+01 -4.6193D-01 3.13663-Ci 5.33333-00 

1.0205D+00 -4.098l-D-03 -4.4825D+00 -1.2099D-02 6.0750D-01 1.0334D-01 

(b) Fixed throttle 

COLUMNS 1mU 6 

O.OOOOD+OO 0.0000D+00 O.OOOOD+OO O.OOOOD+OO 0.0000D+00 O.OOOOD+OO 

-1.4623D-01 -3.0862D.03 -1.3772D+02 -2.5546D.02 -2.3182D-01 -3.5661D-01 

-3.4606D-04 -7.3324D+OO -3.2365D-01 -5.9666D-01 5.6691D+01 -1.0369D.02 

-8.5753D-06 -1.8118D-01 -7.66131)-03 -1.4886D-02 1.0355D+00 3.7093D.30 

COLUMNS 7THRu 12 

0.0000D+00 0.0000D+00 0.0000D+00 O.OOOOD+OO 0.0000D+00 O.OOOOD-:O 

2.2965Dl01 -1.4990D-01 2.5599D+01 0.0000D*L0 5.3334:- . 

1.0205D+00 -4.2346D-03 -4.4824D+OO 0.0000D+CG ~:~~~~~ 1.0334; -:1 

-1 8998~-0i -2.1146~+01 2.7138~-01 O.OOOOD+OO -i.i~3~-oi -2.0058n--7 

Flight Condition: h=20k ft, M=1 .O, n=2 

(a) Tnrottle free 

COLUMNS 1mU 6 

6.9042D-03 1.9549D+02 1.5364D+OO -8.5917D-01 1.1123D-01 

-8.0054D-02 -1.6813D+O3 -9.5674D+Ol -3.7453D+01 -1.7710D+01 

-3.18111)-03 -1.9103D+02 7.1812Dl00 4.0158D+01 4.1578D+01 

7.3580D-04 -2.9358D+01 3.8067D*OO 1.8625D.01 -1.0029D+00 

COLUMNS 7mu 12 

1.17871)-01 -3.2178D-02 1.324l.D-01 1.8588D+Ol -1.5842D-01 

-1.2003D+01 -1.79471)+01 -8.0714D+OO -2.1281D+02 -7.5076D+00 

1.7896D+01 1.8398D+00 -2.7051D+00 -3.4723D+01 2.4838D+01 

8.7576D-03 -2.7686D-02 -3.0579D+00 -1.0897bO1 9.1879D-01 

-4.60720-01 

1.18871)+01 

-4.8085D+01 

1.2088D+O1 

-5.72231)-02 

-1.4183D+00 

4.0935D+00 

-1.1976D-02 

(b) Fixed throttle 

COLWS 1plRU 6 

O.OOOOD+OO O.OOOOD+OO 0.0000D+00 O.OOOOD+OO 0.0000D+00 0.0000D+03 

-2.5821.D-01 -5.1355D+03 -1.8856D+02 -2.4846D+02 2.3176D+01 -5.6307D+O; 

-3.3381D-03 -1.8932D+02 8.863U)+OO 3.0696D+01 2.0976D+02 -5.8322D+OL 

-1.4984D-03 -9.3411D+01 1.1209D+00 1.3272D+01 2.8245D+00 7.2034D+OG 

COLUMNS 7mu 12 

O.OOOOD+OO 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 

9.70171)+00 -1.1540D+02 2.1620D+01 -5.6373D+02 6.8088D+00 1.0118D+OG 

1.0345D+02 7.8005D+00 -3.3573D+OO -2.3474D+01 1.1937D+02 2.0505D+01 

1.2950D+00 -1.1666D+00 -2.2116D+00 -1.418l.D+01 2.4988D+00 3.6095D-01 

E-6 
Flight Condition: h = 20k ft, M=1.0, n=4 

(a) Throttle free 

COLUMNS 1mU 6 

-2.9947D-05 3.6599D+OO 5.5165D-01 -4.3026D-01 1.0464D-01 -1.4545D-01 

-2.4350D-02 -1.6104D102 -9.3784D+Ol -5.3770D+01 9.3674D+OO -2.0784D+Ol 

8.9437D-03 -5.3990D+01 5.8315D+00 6.7524D+Ol 3.8510D+01 -4.8181DSOl 

4.14911)-03 -1.8467D+O1 5.9942D-01 2.959lD+01 -3.6880D+00 1.6136D+01 

COLUMNS 7mu 12 

1.70351)-02 1.0704D-01 1.9226D-02 1.4165D+OO -3.8941D-02 -1.7625D-02 

3.76991)-01 -2.8745D+01 1.1755D+OO -9.7631DlOl 4.2783D-01 2.9496D-01 

1.6541D+01 1.7075D+00 -3.4304D+OO -2.9706D+01 2.9120D+01 5.0757D+00 

-1.9703D+OO -3.2254D-01 -4.4090D+00 -1.1588D+Ol 9.59021)-01 1.5582D-01 

(b) Fixed throttle 

- -.- 

0.0000D+00 O.OOOODLO0 0.0000D+00 O.OOOOD+OO 0.0000D+OO O.OOOOD+OC 

-6.0133D-02 -4.3196D102 -1.5452D+02 -1.3945D+02 1.1041D+01 -4.5528D+01 1.3187D-03 9.5875D+00 1.5789D+01 3.1355D+01 2.5958D+02 -6.5193D+01 3.5796D-03 -4.9691D-01 1.9333D+00 1.9762D+01 3.1445D-01 7.3005D+0@ 

COLUMNS 7mu 12 

O.OOOOD+OO 0.00006;OO 0.0000D+00 0.0000D+00 O.OOOOD+OO 0.0000D+00 

3.4082D+00 -1.4477D+02 3.1317D100 -2.7722D+02 -5.3801D+00 -1.2555D+00 1.2898D+02 1.7756D+01 -3.4610D+00 9.5732D+00 1.4989D*02 2.5871D+01 -4 7916D-01 2.1208D+OC -4.14233-00 -1.0696D+@O 1.6873D+00 2.9084D-01 

Flight condition: h = 30k ft, M11.2, n=! 

,&) Throttle free 

COLUMNS 1zHRU 6 

-4.6088D-02 1.6086D+O3 -4.0862D+01 7.7420D+03 5.0626D-01 2.2822D-01 

-5.7504D-02 -2.448lD+01 -1.9418D*O2 5.1413D*01 1.%22D+00 4.4702D-01 

-4.0537D-05 1.7568D-02 -2.0662D-01 1.4692D-01 4.2%2D+01 -7.5834D+Oi 

-7.3106D-08 1.2564D-04 -2.91041)-04 2.9682D-04 -2.2994D-01 8.08500*0: 

COLUMNS 7mu 12 

2.55311)-01 1.9677D+Ol -8.0973D-03 -6.7504D+03 2.9316D-01 5.1002D-02 

1.0298D+00 -5.9481D+OO 1.6625D-01 -2.3518D102 1.1180D+00 1.9185D-01 

1.6729D+01 -7.8761D-02 1.1728D*01 -3.0407D-01 2.3583Dl01 4.0053D+00 

6.7563D-01 -1.0035D-03 -5.54223+00 -7.55423-04 -4.145m-02 -4.9060D-03 

(b) Fixed Throttle 

COtUMNS 1mU 6 

O.OOOOD+OO O.OOOOD+OO 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 

-2.2035D-01 -4.5955D103 -2.4374D+O2 -4.8065D+02 1.45411)+00 -1.1746D-01 

-3.9297D-04 -8.2035D+00 -3.9043D-01 -8.4043D-01 7.7573D+01 -9.1042D+01 

-9.4818D-06 -1.9769D-01 -9.3204D-03 -2.0343D-02 1.9012D+00 2.6540D+00 

COLUMNS 7THRu 12 

0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 O.OOOOD+OO 0.0000D+00 

7.5137D-01 -1.7241D+01 2.9844D-01 0.0000D+00 8.25811)-01 1.419lD-01 

3.4197D*Oi -1.58163-01 2.8032D+01 0.0000D+00 4.3223D+01 7.3793D+OO 

1.3758D+00 -4.1546D-03 -5.1448D*00 O.OOCC3+00 l.O546D+OO 1.7917D-01 

E- 7 
Flight Condition: h=3Ok ft, K=1.2, n=2 

I 

COLUMNS 1T" 6 I 6.6424.D-03 2.0838D102 2.4550D+00 -4.6783D+00 2.7707D+OO -2.5482D+OC 

-7.9027D-02 -1.9271D+O3 -1.7763D+O2 3.9364.D+01 -1.1536D+O2 5.3945D+O: 

-2.56901)-03 -2.0704D+02 9.0405D+00 4.7758Dl01 4.4819Dl01 -3.8343D+Oi 

1.9215D-03 -2.3017D+OO 6.6459D.00 1.9007D+Ol 2.1634D+OO 7.7438D+OC 

I (a) Throttle free 

COLUMNS 7mu 12 

1.3539D+00 1.8265D-02 1.0172D+00 4.1165D+01 9.1485D-01 1.2343D-01 

-8.0765D+01 -1.9153D+01 -3.1558D+01 -4.4553D+02 -5.0057D+01 -7.8047D+00 

1.8876D+01 1.3449D+00 -3.428lD'OO -5.0063D+01 2.6362D+01 4.3945D+00 

2.45OlD+OO 5.2374D-01 -1.5152D+00 -3.9816D+00 3.1590D+00 2.56651)-01 

(b) Fixed throttle 

COLUMNS 1THRU 6 

0.0000D*00 0.0000D+OO 0.0000D+OO O.OOOOD+OO 0.0000D+00 0.0000D+OO 

-2.7603D-01 -5.6529D103 -2.9060D*02 -3.6612D102 5.2588D+01 -8.8793D.01 

-1.098OD-02 -4.1304D+02 3.7056D+00 4.2548D+01 2.1125D+02 -3.8144D+Ol 

-2.3OOl.D-03 -1.2616D+02 1.4133D+00 1.8791D+01 3.0150D+00 9.2470D.00 

rm.ims 7 . mil -_ 17 

0.0000D+OO O.OOOOD+OO 0.0000D+00 O.OOOOD+OO 0.0000D+OO 0.0000D+OO 

1.8824D+Ol -9.4044D01 4.0305D+01 -1.1392D+03 2.0379DiOl 3.5850D+00 

1.0450D+02 4.1414D+OO -7.5652Dl00 -8.7991D+01 1.2094D.02 2.0762Dl01 

Flight Conditions: h=3Ok ft, M=1.2, n=4 I 

(a) Throttle free 

COLUMU'S 1THRL' 6 

4.1769D-04 3.6498Dl01 1.4851D*00 -3.4841D+00 2.1042D+O!: -1.3394D*OC 

-6.61Ol.D-02 -1.0172D103 -2.1118D+02 -1.4182D+02 -1.1%2D+Ol -7.4184D+01 

6.4192D-03 -2.8674D102 4.5654D+OO 1.156l.D+02 2.78%D+01 -2.6873D+01 

4.5633D-03 -4.2698D+01 4.9646D'OO 4.3199D+01 -6.6763D.00 2.018lD+01 

COLUMNS 7mu 12 

6.363l.D-01 4.8008D-02 3.3416D-01 1.4069D*01 5.3312D-01 9.1566D-02 

-3.1037D+01 -2.6264D+Ol 1.7762D+00 -4.0761D+02 3.2%9D*00 7.1053D+00 

l.O313D+01 8.3152D-01 -4.94OOD+OO -1.1891D+02 2.7199Dl01 4.8492D+00 

-2.4702D+00 3.2870D-01 -4.1412D+00 -2.0022D+01 9.2925D-01 -2.3495D-02 

(b) Fixed throttle 

0.0000D+00 0.0000D+00 0.0000D+00 O.OOOOD+OO 0.0000D+OO 0.0000D+00 -1.4369D-01 -2.6046D+03 -2.7650D+02 -2.2663D+02 1.8778D+01 -6.0432DlOl -1.89251)-03 -1.9431D+02 2.7848D+OO 6.7997D+01 2.8567D+02 -3.9003D+Oi 2.3785D-03 -4.3929D+01 -1.3042D+00 2.6549D+01 -4.3513D+00 1.0744D+01 

COLUMNS 7mu 12 

O.OOOOD+OO 0.0000D+OO 0.0000D+OO O.OOOOD+OO O.OOOOD+OO O.OOOOD+OC 3.3267D+OO -1.323BD+02 2.5597D+Ol -1.0794D+03 -2.4847D+OO -7.8123D-01 1.4358D+02 5.409SD+00 -1.5@41D+01 -7.9803D+01 1.6880D+02 2.9097D+Oi -2.1208D*00 -8.7326D-01 -4.3539D+00 -1.9425D+01 2.8894D-01 4.0536D-CE-8 
Flight Condition: h=4Ok ft, rn=1.4, n=l 

(a) Throttle free 

COLUMNS 1pIRU 6 

-4.0290D-02 1.4738D+03 -7.8132D+01 1.2733D+04 5%2D+00 4.7039D-01 

-3.3645D-02 -3.4998D+01 -2.21041)+02 2.6056D+02 612lD+00 1.6425D+00 

-2.2849D-05 1.044l.D-02 -1.96761)-01 4.09651)-01 4.4062D+01 -5.9481D+01 

7.4382D-07 -2.6323D-04 6.3096D-03 -1.3187D-02 -1.5759D+00 1.1276D+01 

COLUMNS 7mu 12 

1.5059D+00 2.1803D+01 1.59951)-01 -7.5485D+03 1.4565Dl00 2.48431)-01 

4.9056D+00 -1.1218D+00 4.6368D-01 -3.3457D+02 4.7%9D+00 8.1530D-01 

1.8275D+Ol -7.3752D-02 6.8008D+00 -3.90801)-01 2.4078D+01 4.0875D+OO 

1.2100D-01 -1.61921)-03 -7.5773Dl00 1.2509D-02 -7.5866D-01 -1.24340-01 

(b) Fixed throttle 

C0i;lMh'S 1THRU 6 

0.0000D+00 O.OOOOD+OO 

-1.7781D-01 -1.0549D+04 

-3.63551)-04 -2.1593D+01 

-5.8903D-06 -3.4989D-01 

COLUMNS 7mu 12 

0.0000D+00 0.0000D+00 

6.0610D+00 -1.7085D+01 

4.7744D.01 -1.6007D-01 

1.7109D+00 -2.67491)-03 

O.OOOOD+OO 0.0000D+00 

-1.%243+02 -9.2410D102 

-3.6862D-01 -1.8757D+OO 

-5.9813D-03 -3.0397D-02 

Flight Condition: h=4Ok ft, m-1 .k, nz 2 

(a) Throttle free 

COLWNS 1ZXRU 6 

6.8518D-03 5.0683D+02 4.0818D*00 -5.6239D+00 

-4.4551D-02 -2.3251D+03 -1.8765D+02 -7.0742D+01 

-9.549l-D-04 -3.2593D+02 9.999l.D*00 7.9025D.01 

2.1000D-03 -1.5805D+Ol 9.018lD+OO 4.%55D+01 

COLUMNS. 7mu 12 

1.4281D+00 -1.3268D-01 7.72011)-01 8.0912D+01 

-6.5057D+Ol -1.5583D*01 -1.2903D+01 -4.1372D+O2 

1.80563+C1 1.2552D+00 -4.8808D+00 -6.2475D+01 

-6.8358D-01 5.66243-01 -5.3267D+OO -8.8420D+00 

(b) Fixed throttle 

COLWS 1mU 6 

O.OOOOD+OO 0.0000D+00 0.0000D+00 0.0000D+00 

-4.5737D-01 -2.480l.D+04 -7.2826D+02 -1.2147D+03 

-2.7317D-02 -2.0654D+03 -4.2632D+00 8.0166D+01 

-1.3OOl.D-02 -8.8348D+02 -1.3085D+01 l.lllOD+Ol 

COLUMNS 7mu 12 

0.0000D+00 O.OOOOD+OO 0.0000D+00 0.0000D+00 

2.0012D+02 -3.0735D+02 1.54340+02 -4.0026D+03 

2.5032D+02 7.0155D+00 -8.5866D+00 -3.3909D+02 

1.2102D+01 -1.0560D+Cl -6.1567D-01 -1.4738D+02 

1.1114D+O1 6.75308-01 

1.0197D+02 -7.4840D.01 

2.2552D+00 6.392OD+OO 

_~._ ._ 

1.2089D+00 2.0712D-01 

7.3992D-01 7.2793D-02 

-2.4749D+01 -1.9782D+00 

6.9321.D-01 -3.1452D-01 

2.5905D+01 4.2967D+GO 

E-9 
Flight Condition: h=40k ft, m=1.4, n-4 

(a) Throttle free 

~ __ 4.1124D-04 5.0036Di01 1.5631D+00 -3.1841D+00 1.3668D+00 -5.9777D-01 -2.3227D-02 -1.1%6D+03 -1.6878D+02 -5.7583D+01 -8.9560D+00 -1.7275D+01 4.3573D-03 -3.1217D+02 1.0447D+00 1.4759D+02 2.6414D+01 -2.7939D+01 3.9995D-03 -4.5419D+01 4.3753D+OO 7.62671)+01 -1.0108D+01 2.56541)+01 

cal"s 7mu 12 

3.5106D-01 -4.7050D-02 2.00389-01 1.547lD+01 5.5993D-02 1.5265D-02 

-3.3450D+01 -1.5111D+01 -8.9972D+00 -3.6314D+02 1.6913D+01 1.319lD+Ol 

1.0563D+01 6.9894D-01 -4.0%3D+OO -1.0337D+02 2.8564D+01 5.1314D+OO 

-3.8184D+OO 2.8959D-02 -6.45893+00 -1.7890D+01 1.5474DlOO 2.1119D-01 

(b) Fixed throttle 

COLUMNS 1THR'J 6 

0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+00 0.0000D+OO 0.0000D*00 

-7.4833D-02 -3.1628D103 -3.0725D+02 -2.2405D+02 5.2112D+01 -4.3588D+01 

-3.1778D-03 -2.5123D+02 -1.0373D+00 5.6016D+01 2.9213D+02 -4.4807D+Ol 

2.4617D-03 -5.0406D+01 -7.3654D-01 5.1439D+01 -1.3608D+01 1.6665D+01 

COLUMh'S 7 mu 12 

0.0000D+00 0.0000D*00 O.OOOOD+OO O.OOOOD+OO O.OOOOD+OO 0.0000D*00 

1.7193D+01 -1.2755D+C2 9.99493*00 -1.0546D+03 1.3909D+01 2.5306D.CO 

1.4775D+02 6.0470D+00 -6.7478D+00 -8.0806D+01 1.7152D+02 2.%lOD+Cl 

-6.2548DlOO -3.7364D-01 -8.1168D+00 -1.8416D+01 -2.9529D+00 -5.2277D-Cl 

Flight Condition: h=50k ft, m=1.8, n=l 

(a) Throttle free 

COLLWS 1 MRU 6 

-1.2428D-02 1.4946D+03 -1.7242D+02 1.86571)+04 6.688lD-01 2.9641.D-01 

-2.5982D-02 -1.2215D+01 -1.7927D*02 -1.7245D+01 8.9805D-01 1.0008D-01 

-1.5234D-05 4.55411)-03 -1.65581)-01 6.9925D-02 4.4274D+01 -4.7749D+01 

2.668lD-07 1.5175D-05 2.49381)-03 -1.1575D-03 -2.1426D+OO 1.2%0D+Ol 

COLVMNS 7mu 12 

3.1504D-01 -1.1959D+01 -7.9929D-02 -8.0056D+03 3.8123D-01 6.6208D-02 

4.4714D-01 -1.2251D+01 8.18140-02 -1.2989D.02 5.18171)-01 8.99771)-02 

1.9214D421 -6.8022D-02 4.8313D+OO -1.4003D-01 2.4148D+01 4.0977D+OO 

-1.8412D-01 -3.12923-04 -9.0948D+00 2.1152D-03 -1.05171)+00 -1.71691)-01 

(b) Fixed throttle 

CmUtW 1%L! 6 

O.OOOOD+OO O.OOOOD+OC 0.00003+00 0.0000D+OO 

-3.%88D-01 -3.8011D+04 -1.706lD+02 -3.6535D+03 

-8.5073D-04 -8.1553D+01 -3.4003D-01 -7.8179D.00 

-6.3987D-06 -6.13568-01 -2.5685D-03 -5.8789D-02 

CXXVMNS 7mu 12 

0.0000D+OO 0.0000D+00 0.0000D*OO O.OOOOD+OO 

1.4115D+02 -1.657l.D+01 -9.4218D+OO 0.0000D+OO 

4.8944D+01 -1.4756D-01 1.4088D+01 0.0000D+OO 

1.458OD+OO -1.1025D-03 -1.1752D+01 O.OOOOD+OO 

0.0000D+OO 0.0000D+OO 

1.4298D+02 2.4462D*Cl 

5.6839D+Ol 9.7199D+OO 

7.77991)-01 1.3455D-01 

E-10 
Flight Condition: h=50k ft, m-1.8, n=2 

(a) Throttle free 

COLUMNS 1THF.J 6 

3.5959D-03 4.6794D+02 3.5335D+00 -1.1105D+Ol 1.8649D+00 -8.03959-01 

-2.8089D-02 -1.8184D+O3 -1.3708D+02 -1.4276D+02 3.3696Dl01 -3.5020D+01 

-2.72271)-03 -8.1445D+02 8.1206D+00 1.32171)+02 2.2062D+01 -2.0562D.01 

-6.32171)-04 -4.5145D+02 7.5518D+00 1.0742D+02 -9.6043D+00 2.2742DlOl 

COLUMNS 7mu 12 

4.6028D-01 -3.51621)-01 2.6473D-01 7.7890D+01 1.0763D-01 -1.1116D-01 

-1.6152D+01 -6.41890+00 -2.1512D.00 -3.1546D+02 3.0005D+01 7.06433-00 

7.47513+00 1.2381D+00 -1.7457DlOO -1.4860D+02 1.4924D+01 2.3107L)*OO 

-4.0962D.00 6.7878D-01 -7.0306D+00 -8.6670Dl01 -1.4925DlOO -7.20273-51 

(b) Fixed throttle 

COLUMNS 1”RU 6 

0.0000D+00 O.OOOOD+OO 0.0000D*00 0.0000D+00 0.0000D+00 0.0000D*0( 

-6.0982D-02 -4.1785D+03 -2.511lD+02 -2.8238D102 -6.99911)+01 -7.16992+0; 

-3.6150D-03 -5.4820D+02 -9.6721D-01 3.8862D+01 4.1818D+Ol -4.1017D*ti 

-1.40511)-03 -2.8559D+02 4.3810D-01 3.2729D+01 -7.6396D+00 6.2683D+OC 

COLUMNS 7mu 12 

O.OOOOD+OO 0.0000D+00 O.OOOOD+OO 0.0000D+00 0.0000D+00 0.0000D+00 

-7.2926D.01 -3.0622D+01 1.9139D+01 -7.2199D102 -3.7145D401 -5.9716DC’ 

1.89!XD+01 6.32651)-01 2.6529D*00 -9.4700D+01 2.4460D+01 4.1671D*C; 

1.5019D+OO -9.5266D-02 -1.8004D+OO -5.0892D101 -1.3886D-01 -1.6@lZT’-C: 

Flight Condition: h=50k ft, m=’ .e. n=4 

(2‘ Throttle free 

mLmS IIMRU ~ 6 

-8.7236D-06 3.3564D;Ol 1.34071)+00 -2.8980D+00 5.884l.D-01 -2.19089-01 

-1.47061)-02 -9.3343D+02 -1.4934D+02 -3.1412D3-01 3.4538D+01 -2.749G+C1 

9.5740D-03 -2.8197D.02 -1.1045D+00 2.3251D.02 1.35071)+01 -2.1212Dl01 

6.8283D-03 -8.9545Dl01 -2.31771)+00 1.4395D+02 -1.5042D+01 3.3745D+01 

COLUMNS 7mu 12 

3.9985D-02 -6.0302DI02 5.04771)-02 1 .llllD+Ol -1.1816D-01 -8.0972D-03 

-7.0423D+00 -1.6189D+00 -5.6683D+00 -3.0429D+02 5.0324D+01 1.7827D.01 

5.1013D+00 7.2874D-01 -3.3023D-01 -1.0164D+02 2.4240D+01 4.4423D+OC 

-4.8778D+00 2.63071)-03 -7.9269DlOO -3.6055D+01 8.3040D-01 4.2546D-01 

(b) Fixed throttle 

COLUMNS 1mU 6 

0.0000D+00 0.0000D+00 O.OOOOD+OO O.OOOOD+OO 0.0000D+00 O.OOOOD+OO 

-1.7721D-02 -9.5730D+02 -1.7563D+02 -9.2676D+01 5.9326D+01 -2.8669D.01 

1.09312)-03 -8.27171)+01 -2.1945D+00 4.2145D+01 4.4064D+01 -6.5929D+01 

1.2398D-03 -4.0076DlOl -3.3685D+OO 3.5458D+01 -7.1816D+OO 7.5126D+00 

COLUMNS 7mu 12 

0.0000D*00 0.0000D+00 O.OOOOD+OO O.OOOOD+OO 0.0000D+00 0.0000D+00 

8.9326D+OO -1.3331D+01 9.2995D+00 -3.4701D+02 3.1186D+01 8.2989D100 

1.9945D+01 4.71411)-01 6.5720D*00 -2.9380D+01 2.7616D+01 4.7279D+00 

-4.1092D-01 -4.00341)-01 -1.8719D+00 -1.5270D+01 -2.1025D+00 -1.6154D-01 

E-1 1