Advances in Experiment Design for High Performance Aircraft

NASA/TM-

/ c/_,_ _ 207994

lit" - o ._ - i I_/

Paper No. 8

Advances in Experiment Design

for High Performance Aircraft

Eugene A. Morelli

MS 132

NASA Langley Research Center

Hampton, VA 23681 - 0001

NATO Defense Research & Technology Organization

Symposium on System Identification for

Integrated Aircraft Development and Flight Testing

May 5 - 7, 1998 / Madrid, Spain

ADVANCES IN EXPERIMENT DESIGN FOR HIGH PERFORMANCE AIRCRAFT Eugene A. Morelli Dynamics and Control Branch MS 132, NASA Langley Research Center Hampton, Virginia 23681- 2199 USA jth input amplitude constraint /aj SUMMARY ith discrete measurement noise vector v(i) A general overview and summary of recent advances in experiment design for high performance aircraft is np-dimensional parameter vector presented, along with results from flight tests.

jth model parameter Oj General theoretical background is included, with some Cram&-Rao bound for the jth parameter trj discussion of various approaches to maneuver design.

Flight test examples from the F-18 High Alpha lateral stick equivalent time delay, sec % Research Vehicle (HARV) are used to illustrate rudder pedal equivalent time delay, sec T r applications of the theory. Input forms are compared using Cramtr-Rao bounds for the standard errors of kth output amplitude constraint estimated model parameters. Directions for future V for all, for every research in experiment design for high performance Tr trace aircraft are identified.

Superscripts OF SYMBOLS LIST T transpose linear accelerations, g's ay, a z -1 matrix inverse expectation operator

e{ }

J cost function Subscripts L,M,N body axis aerodynamic moments o average or trim value M information matrix N total number of sample times s stability axis body axis angular velocities, rad/sec p,q,r discrete noise covariance matrix R 1, INTRODUCTION S(i) output sensitivity matrix at time iAt Aircraft flight tests designed to collect data for T maneuver duration, sec modeling purposes are generally motivated by one or u n;dimensional control vector more of the following objectives: V airspeed, ft/sec 1. The desire to correlate aircraft aerodynamic x n,-dimensional state vector characteristics obtained from wind tunnel no-dimensional output vector Y experiments and aerodynamic calculations with y(i) output vector at time iAt flight test data.

measured output vector at time iAt 2. Refinement of the aircraft model for control z(i) system analysis and design.

Y,Z body axis aerodynamic forces O_ angle of attack, rad 3. Accurate prediction of the aircraft response using the mathematical model, including flight

13 sideslip angle, rad

simulation and flight envelope expansion.

Kronecker delta 4. Aircraft acceptance testing.

At sampling interval, sec The design of an experiment to achieve any of the aileron deflection, rad _a above objectives involves specification of the rudder deflection, rad instrumentation, the signal conditioning, the flight _r test operational procedure, the inputs for the flight test stabilator deflection, rad _s maneuver, the model structure, and the data analysis ¢,0,_ Euler angles, rad methods. In this work, the maneuver design - specifically, design of flight test input signals - will longitudinal stick deflection rle be studied independently of the other aspects which lateral stick deflection rla impact the success of the flight test.

rudder pedal deflection Or R-1 Designing an input that excites the aircraft dynamic The flight test maneuver (equivalently, the flight test response as much as possible when modal frequencies input) has a major impact on the quality of the data for are imperfectly known, while simultaneously modeling purposes. Designing an input for accurate satisfying practical constraints, is a difficult problem.

model parameter estimation requires rich excitation of Several researchers have studied the problem of finding the system, which is frequently at odds with various optimal inputs for aircraft parameter estimation 2"11.

practical constraints. One such practical constraint is The most serious obstacles to using the results of the requirement that output amplitude excursions (e.g., these studies in flight have been practical in angle of attack or sideslip angle) about the flight test condition be limited in order to assure the validity implementation issues. These include unrealizable of an assumed model structure. Input amplitudes must optimal input forms, and failure to account for be constrained for the same reasons, and to avoid closed-loop control, actuator dynamics, or constraints nonlinearities such as mechanical stops and rate on input and output amplitudes. Computationally, the difficulties have been selection of an appropriate limiting when the model is linear. These practical constraints translate to amplitude constraints on the optimality criterion, inadequate numerical optimization techniques for finding global optimal inputs and outputs during the flight test.

solutions, and difficulties associated with multiple Tests for high performance aircraft often involve flight input design.

at high angles of attack, sometimes using drop Recent research 12"16 has produced an optimal input models. In these cases, flight test time is extremely design technique which addresses the above issues.

limited due to rapid altitude loss, and it is imperative The technique generates square wave inputs which are that information content in the data per unit of flight globally optimal in the sense that information content time be maximized for effective use of expensive in the data is maximized for a fixed flight test time, flight test time. Such considerations highlight the or, alternatively, specified parameter accuracy goals are importance of optimizing the flight test inputs.

achieved in minimum flight test time.

In general, an aircraft model contains multiple response variables, multiple aircraft model parameters, The global optimal square wave input design technique has been shown to be theoretically sound l''13, has and one or more inputs. The overall goal is to design been validated in flight for aerodynamic model a maneuver that produces data from which model parameter estimation experiments using pilot parameters can be estimated accurately. This translates implementation, including demonstrated higher into exciting the system modes so that the parameter accuracies compared to compound doublet sensitivities of the model outputs to the parameters are inputs 14, has been used successfully to specify flight high and correlations among these sensitivities are test maneuvers for closed loop model identification at low. Frequency sweep inputs t can be used to do this, high angles of attack 15, has shown improved requiring little more than knowledge of the frequency parameter accuracy in comparison to doublet and range of interest for the modeling. This technique is 3-2-1-1 inputs in flight tests 16, and has compared restricted to moving a single input at a time, so that favorably to other techniques in the literature for a off-axis responses or coupled motions are generally standard test problem 17. In Ref. [17], the global not well modeled from frequency sweep data.

optimal square wave input produced the lowest value Frequency sweeps also require relatively long maneuver times (i.e., 1-2 minutes) to run through the of the sum of estimated parameter variances, even though the maneuver time allotted for this design was frequency range of interest. Low frequency the smallest of any of the techniques studied (see Table components of the frequency sweep contribute to long maneuver times, and also increase the tendency for the 3 of Ref. [17], p. 281). This fact, though not pointed out by the authors of Ref. [17], demonstrates the aircraft to depart from the desired flight test condition.

effectiveness of the global optimal square wave input For high performance aircraft, limited flight test time, design technique.

multiple control effectors, and flight conditions such as high angle of attack make the frequency sweep The purpose of this work is to give an overview of approach difficult to use and expensive.

NASA research on optimal input design for high performance aircraft, and to present relevant flight test An alternate approach is to take advantage of a priori results. The next section outlines the theory involved knowledge about the dynamics of the aircraft to focus in optimal input design, and discusses the choices the input energy at frequencies near the system modes.

made in developing the global optimal square wave An a priori model can be assembled using wind tunnel input design technique. Next, the F-18 High Alpha aerodynamic data and knowledge of rigid body dynamics and the control system. With the a priori Research Vehicle (HARV) test aircraft and some model, a short flight test maneuver can be designed to details of flight test procedure are described.

produce data with high information content.

Following this, results from selected flight tests are Resulting flight test data can be analyzed using a presented and discussed.

variety of methods in the time and frequency domains.

A paradox occurs here, in that very good inputs will be designed when the a priori model is very good; however, in this case the experiment is less needed.

Obviously, the input design technique must be robust to errors in the a priori model.

8-2 2, THEORETICAL DEVELOPMENT I yk(t) l < _k Vt k e (1, 2..... n o) (7) Airplane dynamics can be described by the following where/.tj and {k are positive constants.

linear model equations: Some researchers have implemented practical flight :t(t) = Ax(t) + nu(t) (I) test constraints using an energy constraint on the input, x(O) = x o (2) (8) y(t) - Cx(t) + Ou(t) (3) _Tu(t)T u(t) dt = E z(i)= y(i)+ v(i) i=1,2, .... N (4) where E is some fixed value of the allowable input energy, chosen by experience or intuition. This Linear models are used in Eqs. (1) and (3) because of constraint is intended to limit input and output the common practice of estimating stability and amplitudes, but it is also chosen for convenience in control derivatives from flight test data collected at a the optimization. Input energy is typically introduced chosen flight condition. Elements of the system as a constraint on the input form, while the cost matrices A, B, C, and D contain stability and control function quantifying achievable model parameter derivatives, which are the unknown model parameters accuracy based on the data is optimized using to be estimated from flight test data. If the maneuver variational calculus to arrive at an optimal input is designed for small perturbations of the inputs and design. In practice, there is no direct constraint on the outputs about a chosen flight condition, the stability amount of input energy which can be applied during and control derivatives can be assumed constant.

the flight test, since neither the pilot nor the control system have inherent energy limitations. The Measurement noise v(i) is assumed Gaussian with practical flight test situation dictates that the constraints be directly on the amplitudes of both the E{v(i)} =O and EIv(i)v(j)T_=RSiJt J (5) input and the output variables, as given by Eqs. (6) and (7), respectively. The constraint in Eq. (8) limits Eqs. (1)-(5) can be used to characterize bare airframe the input and output amplitudes indirectly with an dynamics, where the inputs are control surface integral expression.

deflections and the outputs can include air data When estimating model parameter values from (V, o_,fl), body axis angular velocities (p, q, r), Euler measured data, the minimum achievable parameter angles (_, 0, IV), and translational accelerations standard errors using an asymptotically unbiased and efficient estimator (such as maximum likelihood) are ( a x, ay, a z ). The same general model structure can called the Cramtr-Rao lower bounds 12'js'lg. The be used to characterize closed loop dynamics, where Cramtr-Rao lower bounds for the parameter standard the inputs are pilot stick and rudder deflections and the errors are computed as the square root of the diagonal outputs are selected from the same list as before. For elements of the dispersion matrix 1912jsa9. The closed loop modeling, the input includes a pure time dispersion matrix is defined as the inverse of the delay z, called the equivalent time delay, to account for information matrix M, the latter being a measure of phase lag effects from sources such as high order the information content of the data from an control system dynamics, digital sampling delay, and experiment. The expressions for these matrices are actuator dynamics. For either bare airframe or closed N loop modeling, longitudinal and lateral cases are M = E S(i)T R-Is(i) (9) treated separately, with the linear model structure shown above resulting from the usual small i=l perturbation assumptions.

D = M -1 (10) Constraints arising from practical flight test considerations can be represented as limits on all input where S(i) is the matrix of output sensitivities to the amplitudes and selected output amplitudes. Input amplitudes are limited by mechanical stops, flight parameters, control software limiters, rate limits, or linear control effectiveness. Selected output amplitudes must be S(i) = Oy(i) (11) limited to avoid departure from the desired flight test 30 _b condition and to ensure validity of the assumed linear model structure. In addition, constraints may be and 0 denotes the parameter vector estimate.

required on aircraft attitude angles for flight test operational considerations, such as flight safety and The information matrix can be loosely interpreted as maintaining line of sight from the downlink antenna signal-to-noise ratio for multiple output, multiple aboard the aircraft to the ground station. These parameter linear systems. In this interpretation, the constraints are specified by signal is the sensitivity of the outputs to the parameters. If these sensitivities are large relative to (6) luj(t)l<.lt j Vt j=l, 2 ..... ni the noise level (3 to 1 ratio or greater) and are Q "1 uncorrelated with one another, then the output Similarly, the dispersion matrix g_ depends dependence on the parameters is strong and distinct for nonlinearly on the states, which are often the same as each parameter. Parameter values can then be the outputs. Therefore, output amplitudes must be estimated with high accuracy when adjusting each of comparable if an input design comparison is to be the parameters so that model outputs match measured focused only on the merits of the input forms. For outputs in a least squares sense. Elements of the this reason, as well as to ensure validity of the information matrix also depend on the measurement assumed model structure, the inputs should be sampling rate and the measurement noise designed to produce comparable output amplitudes. If characteristics, which are determined when specifying the maneuver duration, input amplitudes, and output the instrumentation system.

amplitudes are not the same for all input designs being The output sensitivities for thejth parameter appear as compared, it is possible to arrange matters so that the jth column of the sensitivity matrix, and are almost any chosen input form will appear to be the computed from best, based on a criterion function that depends on g_.

For the global optimal square wave input design, the flight test maneuver duration T = NAt might be fixed _t = A +_x +_u (12) aoj aoj a priori due to practical time constraints of the flight test or an analysis of the rate of decrease of the Cramtr-Rao bounds with increasing maneuver time (13) using the optimized input. For the flight test examples included in this work, the cost function to be minimized for a fixed maneuver duration was the ay _ax ac aD (14) sum of squares of the Cramtr-Rao bounds for the

ooj aoj aoj

parameter standard errors, np for j = 1,2, .... np. Eqs. (12)-(14) follow from J=_ay=TdM-l]=Tr[_ ] foragivenT (15) differentiating Eqs. (1)-(3) with respect to Oj, j=l combined with the assumed analyticity ofx in the model equations. Note that it is necessary to have Another formulation of the cost can be defined to nominal (a priori) values for the model parameters to design the input for minimum flight test time to solve Eqs. (12)-(14). The output sensitivities S(i) achieve specific goals for the Cram&-Rao bounds 12.

can also be computed using finite difference This is a minimum time problem, so that the cost is perturbations from the nominal parameter values and given by Eqs. (l)-(3).

J=T whenO'k<_k Vk=l,2 ..... np (16) From Eqs. (9)-(14), it is clear that the information matrix elements (and therefore the Cramtr-Rao For the flight test examples included here, the optimal bounds) depend on the input through the sensitivity input applied to the dynamic system described by equations (12)-(14). The input u influences the Eqs. (1)-(5) minimized the cost function in Eq. (15), sensitivities both directly as a forcing function in the subject to the constraints in Eqs. (6) and (7).

sensitivity equations and indirectly as an influence on the states, which also force the sensitivity equations.

The dependence of the Cramtr-Rao bounds on the 3. OPTIMAL INPUT SOLUTION input is nonlinear in the input amplitude, regardless of whether or not the system equations (1) and (3) are The optimization problem posed in the last section is linear, because of the nonlinear character of Eqs. (9) difficult to solve in general. For the particular and (10).

problem of optimal input design for aircraft parameter estimation, there are good reasons to restrict the Eq. (9) is a discrete approximation to a time integral allowable input form to full amplitude square waves over the maneuver duration T = NAt. Therefore, only. Among these are analytical work on a similar when comparing the effectiveness of various input problem 6, which indicated that the optimal input designs using some function of the dispersion matrix should be "bang-bang" (i.e., a full amplitude _9 as the criterion for comparison, the input designs switching input). Square wave inputs are simple to being compared should have the same maneuver implement for either an onboard computer or the pilot.

duration, and in light of the last paragraph, also the Finally, several flight test evaluations {°,t4,16,17 have same allowable maximum input amplitude. This demonstrated that square wave inputs were superior to approach contrasts with comparisons presented in sinusoidal and doublet inputs for parameter estimation previous works 9'10"11'17, which were based on constant experiments, largely due to richer frequency spectra.

input energy. If only constant input energy is imposed on all inputs, a comparison among the inputs For the above reasons, and to make the optimization using a criterion which is a function of 19 is problem tractable, input forms were limited to full inherently unfair because a wide range of values for amplitude square waves only; i.e., only full positive, maximum allowable input amplitude and maneuver full negative, or zero amplitude was allowed for any duration can give the same input energy.

input at any time. Full input amplitude was used in R_zl.

feedback control system still operating. The pilot held order to excite the system as much as possible.

stick and rudder deflections constant at the trimmed Choice of the pulse timing and having zero amplitude available gave the optimizer the ability to use full values until the maneuver was complete. The maneuver could be disengaged manually by the pilot input amplitudes without exceeding output amplitude constraints. With the above restrictions on the input toggling the engage/disengage button, or form, the problem becomes a high order combinatorial automatically by the research flight control system, problem involving output amplitude constraints, based on g-limits, etc. The pre-programmed square which is well-suited to solution by the method of wave perturbation inputs were standard 3-2-1-1 inputs, doublets, or square waves obtained from the optimal dynamic programming.

input design technique described above.

Dynamic programming is essentially a very efficient Various downlink data transmission rates were method for doing a global exhaustive search.

Arbitrary dynamics such as control surface actuator employed on the F-18 HARV aircraft, but all of the dynamics, feedback control, and general nonlinear data used for analysis was converted to a common models can therefore be included inside the sampling rate of 40 Hz. Corrections were applied to optimization without difficulty. The result obtained is the angle of attack, sideslip angle, and linear accelerometer measurements to account for sensor a globally optimal square wave input obtained in a single pass solution. The technique includes offsets from the center of gravity, and the angle of provisions to adjust the input possibilities at certain attack measurement was corrected for upwash. Data compatibility analysis 21 revealed the need for a scale times in order to account for practical limitations on frequency content of the input, such as avoiding factor correction on the angle of attack and sideslip structural resonance frequencies. The dynamic angle measurements from the wing tip vane, and small bias error corrections on the measurements from the programming solution smoothly handles the multiple input problem, since this just changes the number of rate gyros and accelerometers.

square wave input possibilities. Keeping the system responses within the output space for which the assumed model structure is valid can be handled $, FLIGHT TEST RESULTS directly with dynamic programming by discarding any For bare airframe short period longitudinal dynamics, input sequence whose output trajectory exceeds the the state vector x, input vector u, and output vector y constraint limits. More details on the dynamic in Eqs. (1)-(4) are defined by programming solution method can be found in Refs.

[12] and [13].

x=[ot q]T u=[t_s I]T y=[a q az]T (17) System matrices containing the model parameters are: 4, AIRCRAFT AND TEST PROCEDURES The F- 18 High Alpha Research Vehicle (HARV) is a (18) modified F/A-I 8 fighter 2°. The flight test inputs were A=IZa M a Mq J =LMts M o implemented by the pilot, and also by a computer- controlled On-Board Excitation _.ystem (OBES).

The pilot initiated each maneuver by first trimming 0 the aircraft at the specified flight condition. For the 0 (19) piloted square wave inputs, the pilot signaled the ground controllers to send the square wave input sequence via the uplink from a computer on the az o ground. The maneuver was described by required stick and rudder deflections, represented to the pilot as For bare airframe lateral-directional dynamics, movements of a cockpit indicator in real time.

Accurate implementation of the input was achieved if x=[fl p r O] T u=[t_ r t_a 1] T (20) the pilot accurately tracked the indicator movement with his own stick and rudder inputs. Using this procedure, the pilot was able to produce a high fidelity (21) y=[fl p r O ay] r realization of the desired square wave inputs.

For square wave inputs implemented by the OBES, System matrices A, B, C, and D contain the model the pilot first selected a pre-programmed maneuver parameters; using buttons on a Digital Display Interface (DDI) inside the cockpit. The aircraft was then brought to Yr - cos oto _ cos 0 o Yfl Yp + sint_ o the desired trimmed flight condition and an

Vo

engage/disengage button on the DDI was pressed to

L r 0

A= Lfl Lp (22) initiate the maneuver. Square wave perturbation inputs from the OBES were added directly to the

Nr 0

Nfl Np appropriate control surface actuator commands (for 0 1 tan 0 o 0 bare airframe modeling) or to pilot stick and rudder commands (for closed loop modeling), with the I 0 0 0" r_ a 11o" Y6_ 0 1 0 0 L_, L8 a Lo B_ (23) l_ _r N8 a No 0 1 0 C= 0 (30) o % 0 0 0 1 1 0 0 O" 0 l 0 0 0 0 l 0 C__ (24) 0 0 0 0 0 0 1 0 0 0

Vo

0 0 0 0 0 0 (31) 0 0 0 0 0 0 0 0 0 0 0 0 D_ (25) 0 0 0 The L 0 and N O terms are present in the closed loop

Vov Vov

lateral-directional model equations because the control T _a aYo law used bank angle feedback for gravity compensation to coordinate stability axis rolls. Closed loop model For closed loop lateral-directional dynamics in parameters are in general different from the bare stability axes, the model is airframe parameters, because the closed loop model parameters include the dynamics of the control system (26a) X=[fl Ps rs d?] T in addition to the bare airframe. Model equations could also be written using non-dimensional parametersls.

(26b) u=[rlr(t-_) rla(t-_)1] T A priori linear models used for the input design cases included here were derived from a nonlinear batch simulation of the F-18 HARV 22, which uses a wind (27) Y=[fl Ps rs dP ay] T tunnel database for the aerodynamics. Noise variance estimates for the a priori models were obtained from previous flight test data records using an optimal - Y# Yp Yr - 1 gcos 0 o Fourier smoothing technique 23. The models used for Vo parameter estimation from flight test data were Lfl Lp L r L¢ identical in structure to the a priori models, except that A_ (28) the apriori models did not include linear accelerometer Nfl Np N r N 0 outputs.

All flight test data analysis was done using output cost o sin r o 0 0 error maximum likelihood parameter

cosO o cosO o

estimation 1s,19,24. For the closed loop modeling, the equivalent time delays were estimated as the pure time delay from pilot input to control surface deflection.

- Y'or Yrla Yo" The equivalent time delay can be estimated very accurately this way because the signals involved have Lnr LTI,_ Lo very low noise levels and the pilot inputs were square B_ (29) waves. Equivalent time delay was then held fixed at NOr Nrl a No this estimated value during the maximum likelihood estimation. The Cram6r-Rao bounds for the parameter 0 0 _Po standard errors were the square root of the diagonal elements of the dispersion matrix 19 computed from Eq. (10). In the time domain, a correction for colored output residuals from maximum likelihood estimation is necessary if the Cram6r-Rao bounds are to accurately represent the error in the parameter R-6 estimates 19,25. The correction was notapplied tothe run in immediate succession on the same flight. With the model structure held fixed for the data analysis on

time domain results given here, because theinput

comparison results were unaffected by it, and also each maneuver, any differences in the resulting model because the Cramtr-Rao bounds used in the optimal parameter accuracies can be attributed to effect of the input design assumed white Gaussian measurement input form.

noise. The same white noise assumption is made in Parameter estimation results for the OBES computing the Cramtr-Rao bounds from flight test lateral-directional 3-2-1-1 and optimal inputs at data using output error maximum likelihood 5 degrees angle of attack are given in Table I.

estimation. For flight test data analysis in the Column 1 in Table 1 lists the model parameters, frequency domain, the correction is not necessary.

column 2 contains the a priori values of the Refs. [19] and [25] address this issue in detail. Model parameters used for the input design, column 3 structure was held constant for the compared contains parameter estimates and Cramtr-Rao lower maneuvers, so that the number of parameters estimated bounds for the parameter standard errors using the from each data record was identical. All data analysis 3-2-1-1 input. Column 4 contains the corresponding and parameter estimation was done using angle results for the optimal square wave input. The dashed measurements in radians, but the plots were made lines on the right side of Figures 1 and 2 are the model using degrees.

responses computed using the measured inputs and the The first input design was a bare airframe estimated model parameters from columns 3 and 4 of lateral-directional case using the OBES to implement Table 1. The match is very good in both cases.

sequential rudder and aileron inputs. The flight Values in column 5 of Table 1 are the percent change condition was 5 deg angle of attack, Much 0.6, and in the Cram_r-Rao bound for each model parameter altitude of approximately 25,000 ft. The model was standard error for the optimal input maneuver given by Eqs. (1)-(5) and (20)-(25). Perturbation compared to the 3-2-1-1 maneuver, based on the input and output amplitude constraints resulting from 3-2-1-1 value. The optimal input reduced parameter various practical flight test constraints were: standard errors (equivalendy, increased parameter accuracy) by an average 20%, with lower parameter

la, l_<4.0deg laa 1_<2.5 deg (32 )

standard errors for every estimated parameter.

Parameter estimates in columns 3 and 4 of Table 1 are It I -< 5.0 deg I O ] _ 32. 0 deg (32b) generally in good agreement.

The percent error of the a priori parameter values The 3-2-1-1 input form has been shown to be very relative to the parameter values estimated from flight effective for aircraft parameter estimation in previous test data (computed as the average of values in flight test investigations 9,1°, so this input was chosen columns 3 and 4 of Table 1) varied from 4.2% to to compare with the globally optimal square wave 65.1%, with an average value of 24.2%.

input design. Standard 3-2-1-1 inputs and globally Nevertheless, both input design methods based on the optimal square wave inputs were designed using the a priori model produced experimental data with same input amplitude constraints in (32a), the same excellent information content, as evidenced by the low maneuver duration, the same a priori model, and the standard error bounds in Table 1.

same output amplitude constraints in (32b).

Symmetric stabilator input designs implemented by The 3-2-1-1 inputs were designed by matching the OBES for longitudinal model identification are shown frequency of the "2" pulse to the frequency of the in Figures 3 and 4. In this case, the distortion of the dominant oscillatory mode for the a priori model, and input forms by the feedback control was accounted for adjusting amplitudes and control sequence timing so in the a priori model by including a linear model of that the chosen output amplitude constraints were the feedback control identified from the nonlinear satisfied. Optimal inputs were designed with a simulation. The same a priori design model was used computer program that implemented the optimal input to design both inputs shown in Figures 3 and 4. The design procedure described above 12. The duration of flight condition was again 5 deg angle of attack, each maneuver was 24 seconds.

Mach 0.6, and altitude of approximately 25,000 ft.

Figures l and 2 show the input and output time The model used for the parameter estimation is given histories measured in flight for the OBES lateral- by Eqs. (1)-(5) and (17)-(19). The same methods were directional 3-2-1-1 and optimal inputs at 5 degrees used for the input designs and the data analysis, except angle of attack. The solid lines on the left side of that the optimal input design was allowed a higher Figures 1 and 2 are the commanded inputs from the input amplitude than the 3-2-1-I input. This was OBES, and the dashed lines are the actual measured done to investigate the capability available with the control surface positions. The desired input forms optimal input design routine to use higher input were distorted by the feedback control system, as can amplitudes for increased parameter accuracies while be seen in the figures. The distortion of the input maintaining the same output amplitude constraints.

forms by the lateral-directional feedback control Such flexibility is not available with the 3-2-1-1 input system was not accounted for in the design process for because of its fixed form. Perturbation input and either input design. Figures 1 and 2 show that the output amplitude constraints were: maximum input and output amplitudes for these two maneuvers were very nearly the same, and the length of each maneuver was the same. The maneuvers were

forthe3-2- 1-1input [ r/e [ < 1.0 in (34a)

(33a)

fortheoptimal input

I tx [ < 4.5 deg (34b)

I_[ < 3.0 deg (33b)

Each maneuver lasted 14 seconds. The left side of Figure 6 shows the pilot longitudinal stick deflection Each maneuver lasted 26 seconds, and the maneuvers for the doublet sequence. The right side of Figure 6 were run in immediate succession on the same flight.

shows the target optimal square wave input (dashed The left sides of Figures 3 and 4 show the significant line) with the pilot's realization of that target input in distortion of the stabilator commands resulting from flight (solid line). The pilot's realization of the the longitudinal feedback control. Parameter optimal square wave input is highly accurate in estimation results are given in Table 2 using the same frequency, but somewhat inaccurate in amplitude. The format as Table 1. The parameter accuracies are now inputs shown in Figure 6 have similar maximum improved by an average 72% using the optimal input input amplitudes. Table 3 contains the results of compared to the 3-2-1-1 input. The optimal input maximum likelihood parameter estimation using the maneuver produced larger t_ perturbations than the same longitudinal model structure as before.

3-2-1-1, although maximum a amplitude was the Compared to the doublet sequence input, the optimal same for both inputs in the design phase using the square wave input maneuver produced lower parameter a priori model. The reason for this discrepancy was standard errors (higher accuracy) for every model that the control law removed most of the "3" pulse for parameter, despite distortion in the pilot's the 3-2-1-1, and this effect was not well modeled in implementation of the optimal square wave input.

the a priori model. The optimal input used shorter The average improvement was 47%, based on the pulses in general, and thus was less affected. The standard error value from the doublet sequence dashed lines on the right sides of Figures 3 and 4 maneuver. This example demonstrates that the indicate a good match between the measured outputs optimal square wave input design is robust to and the model responses using the measured inputs and distortion, and can be successfully implemented by a the estimated model parameters from columns 3 and 4 pilot in flight.

of Table 2. The estimates of pitching moment The next optimal input design example is a closed parameters in columns 3 and 4 of Table 2 do not loop modeling case executed using the OBES at agree. Lower parameter standard error bounds for the 60 deg angle of attack, Mach 0.25, and average optimal input indicate that the pitching moment altitude approximately 24,000 ft. At this high angle parameter estimates from the optimal input should be of attack, the aircraft sink rate was approximately more accurate. To check this, a different maneuver at 150 feet per second, as altitude dropped from the same flight condition was used to investigate the 25,500 feet to 21,900 feet during the 24 second prediction capability of the models using the maneuver. Perturbation input and output amplitude parameters in Table 2. Figure 5 shows measured and constraints imposed for the input design were: predicted pitch rate response using the model parameters from Table 2 with the same model

I 1 80 Ibf [ r/a 1<2.5 in (35a)

structure used before. The stabilator input (not shown) was a perturbation input with amplitude It ] < 5.0 deg ] _ ] < 20. 0 deg (35b) approximately :!:5 deg from the trim value of 2 deg.

The stabilator input was applied to both models to produce the prediction responses plotted with the The solid lines in Figure 7 show the measured input measured response in Figure 5. The prediction using and output time histories from flight for the the parameters estimated from the 3-2-I-1 input lateral-directional optimal square wave inputs at 60 deg (shown on the left side of Figure 5) was less accurate angle of attack. Since the square wave perturbation than the prediction using the parameters from the inputs were implemented by the OBES, the inputs optimal input (shown on the right side of Figure 5), realized in flight matched desired optimal inputs both in frequency and amplitude. This result gives exactly; therefore, only one trace is shown for each confidence that the parameters estimated from the input on the left side of Figure 7.

optimal input maneuver are indeed more accurate, as The data analysis was done using output error indicated by the computed Cram_r-Rao bounds.

maximum likelihood parameter estimation in the Next, two longitudinal maneuvers flown at 20 deg frequency domain 24. The model used for the parameter angle of attack, Mach 0.4, and approximately 25,000 estimation is given by Eqs. (1)-(5) and (26)-(31).

feet altitude are studied to compare the optimal square Results for the OBES lateral-directional optimal wave input design to a sequence of doublets. The inputs at 60 degrees angle of attack are given in maneuvers were implemented by the pilot in this case, Table 4. Column 1 in Table 4 lists the closed loop using the procedure described in section 4 above. The parameters, column 2 contains the a priori parameter objective was accurate modeling of the bare airframe values, and column 3 shows the parameter values short period dynamics. Perturbation input and output estimated from flight test data. Column 4 contains amplitude constraints were: the Cramtr-Rao lower bounds for the parameter standard errors using flight test data from the optimal R-R

square wave input maneuver. Thedashed lines onthe

[ r/r 1< 102 lbf [ r/a 1< 1.5 in (36a) fight side of Figure 7 indicate the model responses using the estimated closed loop model parameters from column 3 of Table 4 and the measured inputs. The

I I -< 6.0 deg I ¢ I -< 20. 0 deg (36b)

match is good considering that the aerodynamic dependencies are generally nonlinear at this flight Table 5 contains the maximum likelihood estimation condition. The input and output amplitude constraints results for the piloted optimal input, obtained in the imposed during the input optimization restrained these same manner and presented in the same format as nonlinearities throughout the maneuver, so that the before for the OBES closed loop optimal input assumed linear model structure could be used.

maneuver. The dashed lines on the left side of Figure 9 show the target optimal input, and the solid Comparing columns 2 and 3 of Table 4, the a priori lines indicate the pilot's implementation in flight.

model parameters were significantly different from the The pilot inputs are again highly accurate in frequency parameter values estimated from the flight test data.

(i.e., the square wave switching times were reproduced The percent error of the a priori parameter values well), with some error in the amplitudes. The dashed relative to the parameters estimated from flight test lines on the right side of Figure 9 are the model data varied from 3.1% to 226.5%, with an average responses using estimated closed loop model value of 41.9%. Nevertheless, the optimal input parameters from column 3 of Table 5 and the measured design based on the a priori model produced pilot inputs. As in the OBES closed loop optimal experimental data with excellent information content, input case, the match is good despite higher order as evidenced by the low standard error bounds in dynamics and nonlinearities in the physical system.

column 4 of Table 4. All pairwise correlations The high accuracy of the estimated parameters shown between estimated parameters were less than 0.8, with in column 4 of Table 5 indicates that the optimal most below 0.2. This example demonstrates the robustness of the optimal input design technique to input design technique is robust to errors both in the a priori model, and in the implementation of the errors in the a priori model, as well as the optimal square wave input form.

applicability of the technique to closed loop flight test maneuver design at high angles of attack.

The parameter standard errors for this 30 ° t_ pilot implementation case were lower that those seen for the Figure 8 shows a prediction case at roughly the same flight condition as the last maneuver. The solid lines computer-implemented optimal input at 60 ° ct, mainly due to the severe amplitude distortion by the pilot. If in Figure 8 represent measured flight test data. The the distortion in the implementation of the optimal dashed lines in Figure 8 were generated using the input (by the pilot or the feedback control system) can measured flight test inputs in Figure 8 and the closed be characterized by linear dynamics (an excellent loop model estimated from the flight test data of assumption in the case of the feedback control system Figure 7 (i.e., model parameters from column 3 of distortion), then the effect of the input distortion on Table 4). The plots in Figure 8 show the excellent the flight test results is similar to the effect of errors prediction capability of the closed loop model in the a priori design model.

estimated from flight test data generated by the optimal square wave inputs. This result gives confidence that the linear model can be usefully 6. CONCLUDING REMARKS employed at high angles of attack, and moroaver that such models can be accurately estimated from short The expense associated with flight testing high data records using optimized square wave inputs.

performance aircraft dictates that flight test data for Figure 9 shows flight test data for a lateral-directional modeling purposes be collected as efficiently as optimal square wave input design implemented by the possible. This work reviewed some recent research in pilot. The maneuver was flown at 30 deg angle of maneuver design for high performance aircraft, attack, Mach 0.28, and average altitude of including examples from F-18 HARV flight tests.

approximately 24,000 ft. The inputs were optimized Single and multiple input design cases were studied for with the square wave pulses constrained to be integer bare airframe and closed loop modeling over a range of multiples of 0.5 second, and including a constraint angles of attack, including fair comparisons of global that one second of zero input separate the rudder pedal optimal square wave inputs to conventional 3-2-1-1 and lateral stick input square waves (see Figure 9).

and doublet input forms. The impact of the different These constraints were included to help the pilot input forms on estimated parameter accuracy was accurately realize the optimal input form using the quantified through these investigations. For a flight rudder pedals and the lateral stick in sequence. Such test comparison done on an equal basis, the optimal constraints can be easily incorporated into the dynamic square wave input decreased estimated parameter programming optimization for the optimal square standard errors (equivalently, increased estimated wave input design. Ref. [121 describes this feature of parameter accuracy) by an average 20% compared to the optimal input design technique in detail.

the 3-2-1-1 input. The decrease in estimated parameter Perturbation input and output amplitude constraints standard errors improved to an average 72% using imposed for the piloted optimal input design were: higher input amplitudes in the optimal input design while maintaining flight condition. Compared to a compound doublet sequence, the optimal input R°O decreased estimated parameter standard errors by an 3. Mehra, R. K., "Optimal Input Signals for average 47% in a piloted flight test. For all the Parameter Estimation in Dynamic Systems - comparisons, every individual parameter was estimated Survey and New Results", IEEE Transactions on more accurately using the optimal square wave input.

Automatic Control, AC-19, 6, December 1974, These results were obtained with optimal square wave pp.753-68.

inputs implemented successfully by both the pilot and 4. Gupta, N.K. and Hall, W.E., Jr., "Input Design an onboard computer system.

for Identification of Aircraft Stability and Control The results of this investigation indicate that a Derivatives", NASA CR-2493, February 1975.

properly designed 3-2- I- 1 input can give good 5. Mehra, R. K. and Gupta, N. K., "Status of Input performance relative to the optimal square wave.

Design for Aircraft Parameter Identification", Optimal square wave input designs demonstrated AGARD-CP-172, May 1975, paper 12.

increased data information content in all cases studied, but the optimal input design technique is perhaps 6. Chen, R. T. N., "Input Design for Aircraft most valuable because of its ability to address practical Parameter Identification: Using Time-Optimal design issues. Examples include an automated ability Control Formulation", AGARD-CP-172, May to limit output amplitude excursions during the flight 1975 paper 13.

test maneuver, good robustness to errors in the 7. Gupta, N.K. and Hall, W.E., Jr., "Model a priori model and to distortions in the realized input Structure Determination and Test Input Selection form, and the design flexibility to investigate the for Identification of Nonlinear Regimes", Office impact of changes in the conditions or constraints of of Naval Research, Arlington, VA, the input design, such as available maneuver time, Report ONR-CR215-213-5, February 1976.

control surface rate limits, or input/output amplitude constraints. Such changes can be evaluated in terms 8. Gupta, N.K., Mehra, R.K. and Hall, W.E., Jr., of estimated parameter accuracies, using the single "Application of Optimal Input Synthesis to pass global optimizer in the optimal input design Aircraft Parameter Identification", Journal of procedure. Some of these capabilities were Dynamic Systems, Measurement and Control, demonstrated in this work using flight test results.

98, 2, June 1976, pp. 139-45.

In the future, optimal design of maneuvers to collect 9. Plaetschke, E. and Schulz, G., "Practical Input data for dynamic modeling purposes should move off Signal Design", AGARD-LS-104, November the engineer's workstation and onto the aircraft. This 1979, paper 3.

is possible because of increasing capabilities of flight 10. Plaetschke, E., Mulder, J.A., and Breeman, J.H., control computers and improved understanding of the "Flight Test Results of Five Input Signals for important aspects of the input optimization. Initial Aircraft Parameter Identification", in Proceedings studies in this area are already underway _. In of the Sixth IFAC Symposium on Identification addition, research in the area of optimal input design and System Parameter Estimation, Pergamon for model parameter estimation should influence real Press, Vol. 2, June 1982, pp. 1149-1154.

time parameter estimation schemes that are required for adaptive and reconfigurable control. More areas to 11. Mulder, J. A., Sridhar, J.K., and Breeman, J.H., explore include optimal input design for nonlinear "Identification of Dynamic Systems - models, unsteady aerodynamic effects, and structural Applications to Aircraft, Part 2: Nonlinear dynamics.

Analysis and Manoeuvre Design", AGARD-AG-300, Vol. 3, Part 2, May 1994.

12. Morelli, E. A., "Practical Input Optimization for 7, ACKNOWLEDGMENTS Aircraft Parameter Estimation Experiments", Discussions with Dr. Vladislav Klein of the George NASA CR-191242, May 1993.

Washington University contributed to the work 13. Morelli, E. A. and Klein, V., "Optimal Input presented here. Flight tests were carried out at NASA Design for Aircraft Parameter Estimation Using Dryden Flight Research Center, with Ed Schneider as Dynamic Programming Principles", MAA paper pilot.

90-2801, AIAA Atmospheric Flight Mechanics Conference, August 1990.

8. REFERENCES 14. Morelli, E.A., "Flight Test Validation of optimal Input Design using Pilot Implementation", in 1. Williams, J.N., Ham, LA., and Tischler, M,B., Proceedings of the 10th IFAC Symposium on "Flight Test Manual, Rotorcraft Frequency System Identification, Danish Automation Domain Flight Testing", U.S. Army Aviation Society, Vol. 3, July 1994, pp. 43-8.

Technical Test Center, Edwards AFB, CA, 15. Morelli, E.A., "Optimal Input Design for Closed AQTD Project No. 93-14, September 1995.

Loop Modeling at High Angles of Attack", 2. Stepner, D.E. and Mehra, R.K., "Maximum AIAA paper 96-3418, AIAA Atmospheric Flight Likelihood Identification and Optimal Input Mechanics Conference, July 1996.

Design for Identifying Aircraft Stability and Control Derivatives", NASA CR-2200, March 1973.

g-lO 16. Morelli, E.A. "Flight Test Validation of Optimal Input Design and Comparison to Conventional Inputs", AIAA paper 97-3711, AIAA Atmospheric Flight Mechanics Conference, August 1997.

17. van der Linden, C.A.A.M., Mulder, J.A., and Sridhar, J.K., "Recent Developments in Aircraft Parameter Identification at Delft University of Technology - Optimal Input Design", Aerospace Vehicle Dynamics and Control, Clarendon Press, Oxford, 1994, pp. 259-84.

18. Maine, R.E. and Iliff, K.W., "Application of Parameter Estimation to Aircraft Stability and Control - The Output-Error Approach", NASA RP 1168, June 1986.

19. Morelli, E.A. and Klein, V., "Determining the Accuracy of Maximum Likelihood Parameter Estimates with Colored Residuals", NASA CR 194893, March 1994.

20. Kempei, R. "F-18 High Alpha Research Vehicle Description", Internal Document, NASA Dryden Flight Research Facility, Edwards, CA.

2 I. Klein, V., and Morgan, D.R. "Estimation of Bias Errors in Measured Airplane Responses using Maximum Likelihood Method", NASA TM-89059, January 1987.

22. Messina, M.D., et al., "Simulation Model of the F/A-18 High Angle-of-Attack Research Vehicle Utilized for the Design of Advanced Control Laws", NASA TM-110216, May 1996.

23. Morelli, E.A., "Estimating Noise Characteristics from Flight Test Data using Optimal Fourier Smoothing", Journal of Aircraft, 32, 4, July-August 1995, pp. 689-695.

24. Klein, V. "Aircraft Parameter Estimation in Frequency Domain", AIAA paper 78-1344, AIAA Atmospheric Flight Mechanics Conference, August 1978.

25. Morelli, E.A. and Klein, V., "Accuracy of Aerodynamic Model Parameters Estimated from Flight Test Data", Journal of Guidance, Control, and Dynamics, 20, 1, January-February 1997, pp. 74-80.

26. Morelli, E.A. "In-Flight System Identification", AIAA paper 98-4261, AIAA Atmospheric Flight Mechanics Conference, August 1998.

Table 1 Maximum Likelihood Results Table 2 Maximum Likelihood Results for Lateral-Directional OBES Maneuvers for Longitudinal OBES Maneuvers F-18 HARV, 0.6 / 25K, a = 5 ° F-18 HARV, 0.6 / 25K, tx= 5 ° a priori 3-2- !- 1 Optimal a priori 3-2-1-1 Optimal Parameter Estimate Estimate Estimate Std. Error Parameter Estimate Estimate Estimate Std. Error ±Std. Error ±Std. Error Percent ±Std. Error ±Std. Error Percent Chan_e Change Yfl -0.1316 -0.0970 -0.0859 Za -0.5832 -0.5940 -0.6050 ±0.0013 ±0.0012 -7.2 ±0.0126 ±0.0047 -62.8 I/8, 0.0285 0.0304 0.0327 Zq 0.0 0 0 t ±0.0009 ±0.0008 -14.1 Z6 s -0.1093 -0.0378 -0.0789 Y8 a 0.0053 0 0 t ±0.0063 ±0.0032 --49.3 Ma -2.2600 -4.543 -2.195 Lfl -11.56 -11.376 -10.764 ±0.048 ±0.037 -22.8 ±0.080 ±0.012 -85.1 Lp -1.592 -1.8120 -1.7998 Mq -0.2927 -4.746 -1.341 ±0.0070 ±0.0055 -21. I • 0,109 ±0.014 -86.8 Lr 0.5462 0.3396 0.1727 M8 s -6.0380 -5.482 -4.597 ±0.0224 ±0.0200 -10.4 ±0.104 ±0.024 -76.4 /.8, 1.910 2.3074 1.8768 1"= parameter dropped in model structure determination ±0.0398 ±0.0316 -20.7 L_ a -15.81 -19.480 -17.470 ±0.0623 ±0.0441 -29.3 Nfl 2.139 1.2807 1.3120 ±0.00_9 ±0.0028 -27.9 Table 3 Maximum Likelihood Results Np -0.0085 0 0 for Longitudinal Pilot Maneuvers F- 18 HARV, 0.4 / 25K, t_ = 20 ° Nr -0.0940 -0.1027 -0.0436 ±0.0021 ±0.0019 -11.7 Compound Optimal N8 r -1.223 -1.3924 -1.3450 Doublet ±0.0056 ±0.0043 -23.5 Parameter Std. Error Std. Error Std. Error __N6a 0.2444 0.1738 0.2383 Percent -26.6 ±0.0038 ±0.0028 Chanl_e t = parameter dropped in model structure determination Za 0.0069 0.0027 -61.4 Zq 0.0044 0.0028 -35.1 Z8 s 0.0038 0.0031 -17.0 Mot 0.0114 0.0033 -70.6 Mq 0.0086 0.0039 -54.0 M6 s 0.0094 0.0052 -45.2 8-12 Table 4 Maximum Likelihood Results Table 5 Maximum Likelihood Results for Lateral-Directional Closed Loop OBES Maneuvers for Lateral-Directional Closed Loop Pilot Maneuvers F-18 HARV, 0.25 / 24K, tx= 60 ° F-18 HARV, 0.28 / 24K, oc= 30 ° Optimal Optimal Parameter A Priori Estimate Std. Error Parameter A Priori Estimate Std. Error Value Value Yfl --0.062 -0.044 0.0018 Yfl -0.043 -0.102 0.004 rp 0.002 0 _" Yp 0.018 0.044 0,002 Yr -1.059 --0.989 0.001 Yr -1,015 -0,918 0,004 Y0r 0.336 -0.00252 0.00012 YOr 0.142 -0.00080 0.00015 YOa --0.796 --0.1908 0.0034 Yrla -1.121 -0.2818 0.0084 Lfl -1.705 -2.103 0.070 Lfl -2,800 -7,708 0.296 Lp -0,730 -0,643 0,024 Lp -1.669 -2.519 0.0817 Lr 0.309 0.684 0.048 Lr -0.101 0 t L¢ -0.005 0 t L¢ 0.080 0.895 0.079 Lr/r -3.022 -0.0387 0.0044 LOr -20.36 --0.3415 0.0096 L0a 26.99 7.293 0.146 L% 81.16 11.66 0.42 Nfl 2.435 4.066 0.040 N/3 1.638 6.331 0.124 Np 0.122 0.389 0.027 Np 0.142 0 t Nr -1.293 -1.334 0.032 Nr -1.703 -2,600 0.066 N¢ 0,083 0 t N¢ 0.140 -0.515 0.039 NOr 17.88 O. 176 0.004 NOr 13.18 0.2104 0.0057 N0a --4.876 -3.579 0.103 NOa -7.183 1.671 0.182 Zr 0 0.037 * Zr 0 0.046 * "t'a 0 0.038 * _a 0 0.022 * t = parameter dropped in model structure determination t = parameter dropped in model structure determination * = fixed parameter * = fixed parameter Q lO, i0 Maneuver 372e I I Maneuver 372e I flight I .... i ' i ' ' i ' ' 1 command 4_.._ ' '.A .... I .... I 1 model ......... position 3 ......................... i................. !. .........

6r 0 (deg) -5 p 10 .............. _ .............. i........ :............. - .............

' _ (dps) 0 .....i.... ' ......... i i -10 ' " ........ i L I I I 1 1 i L I I I I I i I I i I J L I i i .............

-10 .... f ........ _ .... ! ....

............... i................................... _, ....... _...... i...............

3 ..... "., --"............. ": ................. ": ................................ T.............

3 ............... ; ................. _ ................. 2--* ....... 1_......_- ..............

: Is Ii : r 2 ............... ,: ......... _ ......................... .: ..............

(dps) 1 ........ i ..................

............... _ ................. _ ...................... ..' ...............

6. o '% ''*,. : .ol ..__ J ]Jll_ .... *'""

-2 -3 ....... ' If' i II li' ' J I ' _ '' =I (_g) _ ] -2 ............... _ ................. _ ........... _ ...............

60.,,,, I I i I ] I I I l I , 1 1 I l I I -3 ................................. _ ................................... _- .............. 4°I............. i!i .............. HI .............. i................. ii! i, iiii -4 i J I l l i i I i I I I I I I J i ili II"

-50

5 0 15 20 25 Time (sex) 0 5 10 15 20 25 Time (sex) Figure 1 Lateral-Directional OBES 3-2-1-1 Input F-18 HARV, 0.6 / 25K, a = 5 ° Maneuver 372f I , , , , i , , , , i , , , , i i o-z.

_ _ .] ......... morea ! i [ ......... position Maneuver 372f l[ fll,ht J .... ! .... !'' ] command I ...... ...._ ....... ,_-. !_ ............... :................. :...............

iiiiiiii!ii5 ......

(deg) o • J , :1 e . : • ", 1 _1 -l 6r 0 • - _..:,._:: . .':, • :,,,. ',.:. : . , . , ". . 7o. -2 (c_g) I 320_ .... I .... I .... .......

-5 0 ..............".................i............ _ ............... - .........

...............................',_-.-_- ..............................i...............

p _o ............. i............. i.......... i ............ i ........

: i (dry) o ...... 1 ....... ! . .i .......... i.

-10 -]0 i ..... - 20 ,, , , , ,, , , , ,, i ,, ,, i,, ,, ................................................... i................. i..............

4_ ........ i .... ! .... ,, _ _2 ZZIZIiZT:-IZIZZ-ZI .... all ..... iZi!ZZ.!ii;i 1 ............. i.............. j............

(dps) o ...... i........ : ..........

-1 ................ "................. i..............

6a 0 -3_i'2 . ._ ................. ] ................. J ..............

(deg) _ ] -2 60 ..... I ............ I''''4 -3 -4 -,,,,I,,,,I,,, ..... I_,,, -5 5 10 1 20 25 Time (sex) Time (sex) LateraI-Dirextional OBES Optimal Input F-18 HARV, 0.6 / 25K, a= 5 ° R_IA Maneuver 372c [ I .... I .... I .... 1' '1 . night .... I ..... I ......... position Maneuver 372c I --command ' i ............ i ............................. i ........ I " ....... model

I

6 ............ i............................. +..-._ .......... i.............. _ ............

3 ............ i.............. ' .............. :.............. i.............. i...........

(deg) i i, _ i I I I I I F I I I I t i i i i i I f i i i i i i i.

-2 t ......... !T_....... ]....................... ;] .............. i............

q 2 k 'i i I I I I * [ 1 I I I J I I [ I I t I 7L,IL:L,, (dps) -4 0 5 10 15 20 25 30 -2 Time (sex) -4 - ........... +: .............. - ............. _.............. +.............. ,L...........

" .... i .... i .... I ........ J .... -- -6 0 5 10 15 20 25 30 Time (sex) Longitudinal OBES 3-2-1-1 Input o_= 5 ° F-18 HARV, 0.6 / 25K, Maneuver 372d ......... model I command I Maneuver 372d ......... position -- m_t I

8: .... ,+"_':'+: ,: : .... . .:!i'"i' '!'i[ .... -

_..LI.L].'..LLLLL_.'LLL_ ..............•.......... __

'- l: : l + ............

(deg)

a+ 2 3

(deg) 0 _ :-'+",. -i ..... _.........

i ....... I .... I .... I .... I .... + -2 _ .................. ..., ..........._ ............ _ ........... ;..+ .......... -..

-4 ........ "++ ........ ,-++°• +,, ................ , :i+ i..........

q 2 -6 (rips) 0 5 25 30 10 15 20 -2 Time (sec) -4 -6 0 5 10 15 20 25 30 Time (sex) Longitudinal OBES Optimal Input 0_= 5 ° F-18 HARV, 0.6 / 25K, ml Maneuver 274e II 274e I -- flight optimal input model ]13 ........

Maneuver i ......... 3-2-1-1 input model 11 i -- flight 10 ........

t, ,m ..... i ....

,I :I ++ ;I '.I q ° 0 - q !-_ .._I.-I. + ....

0 "-"+-- IM ' I (%s)

:I it

+q + ° (dps) _._+ M--.._ 'i .... i ..........

-5 " i J i I t I i I I I 1 I I I I 1 I I / I I I I I I I I I I I .... I ......... -113 0 5 10 15 _,o °......... ,5 .....

10 15 Time (sex) Time (sex) Longitudinal Prediction Case F-18 HARV, 0.6 / 25K, o_ = 5 ° 8- 15 pilot input I | Maneuver 153a ......... optimal input Maneuver 168f [ _ doublet input I 1.5 1.5 I | ..................... T ........... i ............ ! ........ ......... i ..........

0.5 0.5 r/s .... ' .... i ........... _............. _ (deg) ......... i- . i ..... __.-i ................ -.....................

-0.5 -0.5 ....... i.......... /:: ...........L .... i.........

-I -1 i - -1.5 ll,lj,_l,,,l,,llll,I ...... - -1.5 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 Time (sec) Time (sec) Longitudinal Pilot Inputs F-18 HARV, 0.4 / 25K, _= 20 ° Maneuver 329s 12 .... M a?e,uve[ 329_, , I flight | so_,f'F' 'F I' "_ .... ' .....

6o:-:- ...... I rt!i ........ ':i 1 ......... :i ................. i.............. ---__

'_ _iii_iiii_ __ ii '_ii_ _ _i_'_ iiiii,!ii

...; .... ._..

40 ----- _--. ! .............. i ".............. t i............... --

=o---.....1 ........ II......l..IJ.....I ......... J. i............. -

((leg) _ - ........... i " _........... _:, ...... /" 0 ................................. ] f Ob0 i i -

_o_,,, ,. ........ ,..,,,,,, ,_

._o .......... .......................... i................. i..............

............. ,: ................. _ ................. _ ............. !..............

-40 - -.................................. _ ................. ._..............

5 ........... _ .............. _............ ,.' ..... : .............

-60 ......................................................

-_ j i i -

(@s) _o _ .... ...... _iiii " "iiiiiiii....

-80-' ..... I .... t _ _ , _- -10 ......

15_''''1 .... i .... I .... I ....

10 _ ............. i............. .,.-i--...-r_ ......... _. ................. _ .............. _.

!

(dps) rs O_ti_ _- 5 ........... !......... !............ 4 ....................... - 0 -10 . . -............ i................. i......

(in)

.1_-_ _, _i;i -

-i 20 ,,,,_ .... t .... I ........

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: -2 '": "._: :-ii_ii?" ..... i -3 ((leg) 0 5 10 15 20 25 -_o ' 'i................. i.......... ': Time (sec) 0 5 10 15 20 25 Time (sec) Lateral-Directional Closed Loop OBES Optimal Input F- 18 HARV, 0.25 / 24K, ot = 60 °, TV Mode 8-16 Maneuver 358e _ h,_t._s..+., , , , , , , ...... , I0__ .... .... .... II ......... model !Maneuver! 358e !11 flight I 4 iii iiiiiii::i 7ii!i:::iiiiiii:i:":::: .... +::::: Or 20 (Ibf) 0 -4 -20

3o_ .... i .... _.... _.... _'"'_

-40

::i: ::::: :::::::i::: ::::::::::: i::::::::: :-

-60

- ' ,i ,,,i .... i .... -

-80 1111 IIII 11111111111111 u ............. -: .............. ..' ................. _ ................. t............. -..

I (dos) rs .7_i_i_i5 ............ _ ..................... i.............. " .......... - rla -10 G-_ - - -i .............. -- (in)

-15 __',,-,, !

-1 :;:::;.12:;:::::

__F..._._.:..._...i...:..:...:...:..!...:.._..._

-2 20_-.............. _-.._-.._ ........... i................. !--_4

.... i-.J

10 ............." ......... i................. _f-" ...........

.... I .... I .... I .... I ....

o ............ 4 ................ ',' .............

-3 0 5 10 15 20 25 Time (see) -20 "" :

.3ok_ ; :_ ,_

0 5 10 15 20 25 Time (see) Lateral-Directional Closed Loop OBES Prediction Case F-18 HARV, 0.25 / 24K, a -- 60 °, TV Mode

Maneuver 273h I II , ,Maneuver 273h , I m_,, ]

1501- .... I .... I ' [ pilot | 6 _, , ! ' ' ' ! ' I ......... model 100 .............. ÷ ........... ,:--,-.! ........ • ]7 2 ' ', '', , ,: i 7 i i I ......... target| 4_ (IbO 10 _ i i I r/r 500 _ _i'-! '-"' .......... }i ...................... i.................. (deg) 0.4"2 -50 ........................................

-100 (d_) o ......

_150 I- ................ -I I I I _ I I - 8. .... I .... I .... I ....

1.5 rs 4 ............................... i......................i..................

02 ......... :::: :::::::-::!::: ........... , ............

(dos) -2 ...... ] rla 0.5 (in) 0 -0.5 25_ .... I .... I .... I ' ' '.'.._ 20 ...................:.....................,......................,.............. ""

l,_.................. i..................... _ ........ .,,_ _ .......... t-_

-1 lo .................. i..................... i.......... i........

-1.5 (deg) 5 .................. i.................. _ .... _ ......

:,,,,i,,,,l,,,,i,,,, -2 (3 5 10 15 20

_

Time (see) 0 5 10 15 20 Time (see) Lateral-Direetional Closed Loop Pilot Optimal Input F-18 HARV, 0.28 / 24K, _ = 30 °, TV Mode R=17