Output model-following control synthesis for an oblique-wing aircraft

e- NASA Technical Memorandum 100454

, Output Model-Following Control

Synthesis for an Oblique-Wing

Aircraft

Joseph W. Pahle

i April 1990 j,- w N90-I924I MODEL-FOLLOWING (NASA-TM-IO0454) OUTPUT OBLIQUE-WING CONTROL SYNTHESIS FOR AN CSCL OiC AIRCRAFT (NASA) 30 p Unclas 0271809 G3/OB National Aeronautics and Space Administration i _ g sq I w _ I i NASA Technical Memorandum 100454

" Output Model-Following Control

Synthesis for an Oblique-Wing

Aircraft

Joseph W. Pahle Ames Research Center, Dryden Flight Research Facility, Edwards, California National Aeronautics and Space Administration Ames Research Center Dryden Flight Research Facility Edwards, California 93523-0273 S CONTENTS SUMMARY 1 NOMENCLATURE 1 INTRODUCTION 2 PROBLEM DEFINITION 3 OUTPUT ERROR LINEAR QUADRATIC REGULATOR 3 OBLIQUE WING RESEARCH AIRCRAFT DESCRIPTION 6 OBLIQUE WING RESEARCH AIRCRAFT DESIGN EXAMPLE 7 RESULTS 8 APPENDIX 12 REFERENCES 14 iii PRECEDING PAGE BLANK NOT FILMED SUMMARY A decoupling control law synthesis technique is presented that integrates stability augmentation, decoupling, and the direct incorporation of desired handling qualifies into an output feedback controller. The proposed design technique uses linear quadratic regulator (LQR) concepts in the framework of explicit model following (EM_ and proportional plus integral (PI) feedback. An idealized model is used as a part of the controller for robustness and handling qualities requirements. The PI error feedback is used to balance model-following performance with surface activity and to force steady-state errors to zero. The output feedback gains are then computed from a projection of the full-state gains. Closed-loop performance is shown by application of the control laws to the linearized equations of motion, and a six-degree-of-freedom simulation of an oblique-wing aircraft. Model foUowing is shown to be quite good, but with some significant time delay in both linear and nonlinear evaluations. Decoupling of the longitudinal and lateral-directional axes is excellent, with some small oscillations evident in the nonlinear responses.

NOMENCLATURE A, B, H, F state space quadruple defining the model state and output equations an normal acceleration, 9 lateral acceleration, 0 ay C error selection matrix DFBW digital fly by wire EMF explicit model following identity matrix of dimension n i

v':-r

J quadratic cost function state feedback gain matrix Kv output feedback gain matrix LQR linear quadratic regulator M Mach number OWRA oblique wing research aircraft PI proportional plus integral roll rate, deg/sec P q quadratic state weighting matrix pitch rate, deg/sec q R quadratic control input weighting matrix 7" yaw rate, deg/sec S quadratic cross-weighting matrix for states and inputs control vector U W output vector weighting matrix X state vector output vector

Y

output error vector

Y_

Ot angle of attack, deg angle of sideslip, deg fag left aileron defleclion, deg right aileron de//ection, dcg left horizontal tail deflection, deg tShL 6hR right horizontal tail deflection, deg 6R rudder deflection, deg roll angle, deg Subscripts e error _re integral error model plant P Superscripts T complex conjugate transpose INTRODUCTION Early wind-tunnel and theoretical studies have shown that a variable-skew oblique wing offers a substantial aero- dynamic advantage over conventional or symmetric variable-sweep aircraft for missions that require both subsonic loiter and supersonic dash or cruise (Nelms, 1976 and Wiler, 1985). The NASA Ames Research Center, Dryden Hight Research Facility AD-I flight program successfully demonstrated the oblique-wing concept and explored the low subsonic handling qualities and performance envelope (McMurtry, 1981). Because most of the aerodynamic advantage occurs at transonic and supersonic speeds, a high-speed flight researchprogramwas jointly sponsored by Ames-Dryden and the U.S. Navy. The oblique wing research aircraft (OWRA) was to be based on a modified F-8 digital fly-by-wire (DFBW) aircraft with a composite wing and wing-pivot mechanism substituted for the existing - high wing structure (fig. 1). _ --= _ __ ....

A major challenge in the OWRA program is the design and implementation of a control system architecture that will provide stabilization, decoupling, and acceptable flying qualifies across the Mach number (M), angle-of- attack, and wing-skew envelope. Because of the vehicle dynamic cross-coupling, the control system design must be applied to at least five (rigid body) degrees of freedom simultaneously, rather than separating the problem into the more typical two or three degrees of freedom in the longitudinal- and lateral-directional modes. The OWRA configuration provides an excellent opportunity to apply modern control methodologies to the problems associated with an oblique-wing configuration.

Model following has been shown to be an effective method for dccoupling and stabilization of an initial OWRA configuration in previous work by Alag and others (1986). Implicit and explicit model-following (EMF) techniques were used successfully in a variety of controller structures to decouple the aircraft dynamics, but control surface activity was normally high, and steady-state model-following errors degraded the control system performance over a period of time.

V'mcent (1984) proposed an integrated model-following technique where dynamic elements are introduced into the constant gain controller to reduce steady-state model-following errors, and a nonopfimal output-feedback strategy is used to implement the resulting control law in a realizable form. In this way, handling quality requirements and trajectory tracking can be directly incorporated into a linear quadratic regulator (LQR) synthesis technique. Output error weighting is used instead of state weighting in the LQR cost functional, which gives the designer a better physical feel for the parameters being minimized.

This paper presents a form of the integrated model-following technique as applied to the OWRA model. The resulting controller is structured as a proportional plus integral (PI) output-feedback control law. The effectiveness of the controller is shown by closed-loop time responses from linearized equations of motion as well as nonlinear six-degree-of-freedom simulation results at a given high subsonic flight condition and wing-skew position.

PROBLEM DEFINITION Model-following is a useful technique when the dynamic characteristics for an ideal model can be specified.

In EMF, the dynamic model is an integral part of the controller itself. The objective of EMF control is to force the aircraft to respond as the model would to a given pilot command. The model-following problem can be stated as follows: Given the linearized plant dynamic equation = + Bpup (1) and the linearized model dynamics (2) Xra = .¢4mXra + BmUra where xp, x,_, %, and u,n are real vectors, and all matrices Ap, Am, Bp, and B,n are of suitable dimensions, find the control u t, that will force the aircraft states xp to approximate the model gates x,_. Although sufficient conditions exist for guaranteeing "exact" or perfect model following, the conditions cannot be met by a physical system (Tyler, 1964). This problem definition can be put in the context of the LQR, where the cost functional can be expressed in terms of the plant and model states or outputs. Therefore, the squared error between the plant and the model would be minimized, forcing the plant to follow the model states or outputs. An output error LQR formulation was used in this investigation primarily to simplify the task of selecting cost function weighting matrices and to give the designer better insight into the parameters being minimized.

OUTPUT ERROR LINEAR QUADRATIC REGULATOR Consider the aircraft plant and model dynamic equations given above in equations (1), (2), and the out- put equations yp= Hpxp + Fpup (3) (4) Ym = Hraxm + Fraura The most common form of the quadratic cost is a function of the state and control vectors of the system, written as (5) J = xTQx + uTRu dt f0 ° or in output formulation J = yXQy + uXRu dt (6) f0 ° where Q > 0 and R > 0 are the quadratic weighting matrices. Define the output error as Ye = Yp - Ym tO determine the cost as a function of the error between the plant and model outputs. Integral error can be added by defining the error state equation: xze = eye (7) where C is a selection matrix operating on the output error vector. Thus the state and output equations can be augmented to form the following system: (8) x1. = CHp 0 -CH,. x1_ + CFp -CF,,, up

[x l l 1[ 1

x,. 0 0 A,. x,. 0 B_ Y = YIe = 0 I 0 Xle + 0 0 up (9) Ym 0 0 Hm xm 0 F,_ u_ J Since the output equation (9) is a linear function of the states and controls, the cost functional equation (6) can then be restated as 3 r = xTQx + xTSu + RTsTx + uT/_u dt (10)

/J

with _ > 0 and/_ _> 0, o _ > 0, and given by 0 -HpTQeHm HpT QeHp (1I) Or_ o 0 H,nTQ_H,n --HmTQeHo HpT QeFp - H ;,TQ . F,.

0 (12) rS= 0 H ,. T Q _ F,.

--HmT QeFp ]_=[ FPT Qe-FP --FpTQeFm] _FmT QeFp F, nT QeFm + R (13) where Q_ is the weighting matrix for the output error terms, and Qre is the weighting matrix for the integral error terms. If (A, B) is stabilizable, and (H, A) is observable, the linear steady-state optimal control that minimizes J equation (10) is found by solving the steady-state Riccati equation (Bryson, 1975), and is of the form u = -K:_x (14) The resulting optimal control law isa function of all thestates in the augmented system. A projection from the state space to theoutput spaceis possible without changing the closed-loop cigcnvalues (appendix) if thenumber of uniqueoutputs is equaltoor greater thanthenumber of states in the augmcnted system.The resulting cigenvectors are notguaranteed tobc unchanged,butthemodel-following control structure will dominatetheresponse behavior by driving the error and integral error to zero. Thisprojection can be stated as follows: Define a weighted output vector _"= Wy =/_/'x + Fu where H = W H, and _' = W F.

Since/I is assumed full rank, and (_r_) > 0, this implies that (f/T//)-1 exists. The output equation can then be rewritten in terms of the system states as x = (HTA r)-t Arr_ _ (ArT_) -I ArT/_u (15) The optimal control can therefore be written in the output form (16) u = - [I- Kx(BTB)-IBT:']-tK=(BTB)-IBT J K, where K v can be partitioned as [ Kp Kx_ K,n] (see fig. 2).

It can be shown that if jr = x (this implies H = W = I and F = 0), it is possible to reduce the above equation to the state form of the optimal control law equation (14). W and R can be chosen in such a way that only the desired output feedback paths will have nontrivial gain values. Thus, all the feedback loops to the model can be eliminated, preserving the desired closed-loop eigenvalues of the idealized model. The control system design procedure can be stated as follows: 1. Determine idealized model dynamics, 2. Choose quadratic cost matrices Qe, R, Qte, 3. Calculate optimal state feedback gains Kx from steady-state Riccati equation, 4. Choose output weighting matrix W so that only the desired feedback loops have nontrivial gains, 5. Determine output feedback gains K v from state gain projection, and 6. Iterate steps two through five until desired system performance is achieved.

Figure 2 shows a block diagram of the final PI feedback controller. The resulting control law is nonopdmal in a strict sense because of the gain projection, but the design technique incorporates some important features.

1. The output error formulation allows the designer to balance model-following performance with control surface activity, both in surface deflection and rate (if the actuator dynamics are modeled in the plant), 2. The integral error feedback forces steady-state model-following errors to zero, thus making the control law more robust to modeling errors in the aircraft plant dynamics, 3. The idealized dynamic feedforward model allows the incorporation of lead-lag networks, stick prefilters, or general command shaping in the forward loop for handling qualities requirements, and 4. Theaugmented controlsystem structure, a dynamic prefiltcrwithPI errorfeedback, is similartoclassical PI controllers. Therefore, designers canapply knowledge andintuitionfrompast design experience, animportant factorin ahighlyinterrelated problem.

Asin mostmethodologies, thissynthesis technique hassome drawbacks.

1. Thenumber of resulting gains, evenintheoutput errorformulation, isprohibitively high,andgainscheduling between flightconditions is a complex problem.

2. Althoughthefull state LQRhasguaranteed phase andgainmargins, theprojection fromstate tooutputspace negates anyrobustness guarantees except forclosed-loop stability.

3. Theselection of theproper weighting matrices Qe and R is not a straightforward process. This problem is made more difficult because the desired time or frequency response behavior not specified in the model cannot be easily related to the weighting matrices. Additionally, great care must be taken when implementing a linear idealized model (perturbation model, valid only for small angle maneuvers, instantaneous response to inputs, etc.) in a nonlinear, six-degree-of-freedom flight system.

It is hoped that future research can overcome these drawbacks and allow this technique to be applied to a broader class of problems.

OBLIQUE WING RESEARCH AIRCRAFT DESCRIPTION The F-80WRA design configuration was based on a feasibility design study done by Rockw¢l! International (White and others, 1984) which had its roots in the initial work done at NASA Ames Research Center in the early 70s (Graham and others, 1985; Smith and others, 1975 and 1976). The F-80WRA is basicall_ _e _-8 D_ aircraft with the 357 ft 2 wing replaced by a 200 ft2 composite wing and pivot structure (fig. 1). Wing skew is right wing forward, from 0 (unskcwcd) to 65 °. = The OWRA aerodynamic model was based on a combination of wind-tunnel studies, computational fluid dy- namics, and the simulation aerodynamic database of the F-8 DFBW aircraft. Much of the data from allthree sources had to be linearized, interpolated, or modified in some way to form a cohesive full-envelope aerodynamic model.

This modified aerodynamic model was implemented in a six-degree-of-freedom, nonlinear, fixed-base simulation used for piloted evaluation and control systems development. Linear models used in the control system design were obtained from this nonlinear aerodynamic model using the program LINEAR (Duke, 1987). A comparison of linear and nonlinear open-loop time histories showed excellent agreement indicating that the linear models could be used even at the maximum wing-skew conditions.

The engine model for the F-80WRA is based on the simulation engine model of the F-8 DFBW. The engine is equipped with an afterburner and is modeled in the simulation as two main rotational elements (compressor and fan) co-located along the thrust line in the vehicle x-z plane.

Different types of trim exist because of the unique aerodynamics of oblique-wing =configurations. Trim is defined as the conditions necessary to set the derivative of the state vector (x) to zero. When the wing is skewed, a nonzero sideslip angle/5 or roll angle $ is required to keep the trim $ =/_ = 0 simultaneously. For this reason, level trim is defined as wings !eye! (__- 0)_varying/5 as trim requires, _dsidesl!p trim is defin_ as co_t_t _= (nominally [ zero) varying $'as required. Figure 3 shows the relationship between trim/5 and trim _ at a single flight condition.

Although many potential trim strategies exist, the ailerons are used to trim the roll axis, symmetric horizontal tail is used to trim the pitch axis, rudder is used to trim the yaw axis, and the throttle is used to obtain a steady-state velocity. Inboard flaps are used for unskewed powered approach only. Each surface moves independently, allowing the controls designer greater flexibility in surface management which becomes more important at the high skew conditions, where the ailerons are effective (but highly coupled) controllers in all three axes.

OBLIQUE WING RESEARCH AIRCRAFT DESIGN EXAMPLE The OWRA design problem is well suited to the EMF design technique because a realistic model exists with the desired time response characteristics and closed-loop eigenvalues. The model used was a six-state approximation to the closed-loop dynamics of the OWRA at zero wing skew (table 2). A simple rate and acceleration feedback control system was designed for the unskewed aerodynamics and then implemented in a six-degree-of-freedom fixed-base simulation. The feedback gains for this simple control system were modified until satisfactory pilot comments were obtained. Linear models were then generated to approximate the closed-loop performance of this augmented system.

The six states used were the minimum number required to incorporate four of the five classical rigid-body dynamic modes: short period, dutch roll, spiral, and roll mode. It was not necessary to include the open-loop phugoid mode in the desired model dynamics because it remains stable and relatively fixed throughout most of the flight envelope.

The use of the zero-skew aircraft as a model has an additional benefit because the desired closed-loop dynamics of the aircraft remained constant over the wing-skew envelope. Thus, the pilot would not have to adapt to new characteristics as the wing is skewed.

For the OWRA design example, the model and plant state vector is given by O/ angle of attack, rad sideslip angle, rad roll angle, rad X _ (17) P roll rate, rad/sec pitch rate, tad/sec q 7" yaw rate, rad/sex: and the control vector 6hL left horizontal tail deflection, rad 6hR right horizontal tail deflection, rad 6aL (18) left aileron deflection, tad 6aR right aileron deflection, rad 6R rudder deflection, rad Actuators were not modeled to simplify the design process and to reduce the number of feedback gains required.

Because of this, only surface deflection could be directly minimized in the design, as no cost was incurred because of the surface deflection rate. Numerical difficulties were encountered in the full-state feedback gain projection to output feedback when actuator dynamics were included in the augmented state vector. These difficulties were primarily a function of the limited number of outpui parameters that were selected for feedback, rather than a fun- damental limitation of the technique. The output parameters chosen were those typically used as feedback signals in classical control systems, especially those parameters related to the lateral-directional axis. Signal reliability and sensor redundancy were also factors in the selection. The output vector was identical for the model and the plant.

The output error y, is given below as Pe roll rate error, rad/sec qe i pitch rate error, tad/see re [ yaw rate error, tad/sex Y" = q5, roll angle error, tad (19) a,,e I normal acceleration error, g . a_, ] lateral acceleration error, g

withthe error between the three angular rates of the plant and model

roll rate error, rad/see (20) pitch rate error, rad/see YZe = qe yaw rate error, rad/see _e used as the input to the integral error feedback Yte. The angular rates were chosen as integral error feedbacks to improve the steady-state model following and add additional damping to cross-axis perturbations.

RESULTS The design flight condition was M = 0.8, altitude = 20,000 ft, and wing skew = 45 °. This flight condition was chosen because it was contained in the proposed initial flight test envelope, and the cross-axis coupling was V near a worst case for the entire flight envelope. Additionally, the total aircraft drag was near a minimum at the 45 ° wing-skew condition (fig. 4). Tables 1 and 2 show the system matrices for the plant and model respectively.

Many design iterations were used to determine a set of weighting matrices that resulted in good system perfor- mance with low control surface activity. The final matrices used were the following: Q+ = DIAG ( [ 10,20, 10,500, 10,100]), Q Ie = DIAG ( [ 500,500, 100]), and R = DIAG ( [ 200 I5; 10 7/'5 ]). The large values in R are applied to ura to eliminate any feedback loops to the model. The output weighting matrix W was constructed to eliminate all but the desired feedback gains. In this case, W = DIAG([Ig; 10-sI6]). The resulting output feedback gain matrices K_,, Kte, Km are shown in table 4. For this design case, it is interesting to note that the number of unique outputs is less than the number of states (Rank (H) = 9 ,Rank (A) = 15), yet the eigenvalues for the output feedback system remain close to the eigenvalues of the full-state system (table 3).

The responses of the linearized open-loop aircraft model to a l*-ramp in elevator and aileron are shown in figures 5-6. The ramp begins at 1 sec and ends at 3 sec for each input. The open-loop aircraft responses arc lightly damped (note or, q, and the accelerations) and the roll angle (_) due to pitch command is larger in magnitude than the roll angle due to roll command. The roll response in the pitch axis was found to be a strong function of angle of attack as well as nonlinear at this flight condition. For this reason, only small amplitude inputs were used in the design evaluations, since the linear models were not valid for large amplitude maneuvers. Time histories for the closed-loop system were run initially with linear aerodynamics and nonlinear actuator models with rate and surface limits included.

After some design iterations, the control laws were implemented and tested in the non, ear six-degree-of- freedom simulation. Figures 7-14 show the ideal linear model and closed-loop aircraft responses (both linear and nonlinear) to a 4°-roll ramp, while figures 15-22 show the responses to a 5°-pitch ramp. The closed-loop system is mucfi_re damped, and __ue'to pitch c0mm_d h-_ b_n al| but e_ted, Gotd-m0del-_iitw_mg is :evi dent, but with some significant time delay evident for both linear and nonlinear cases; note _ and lateral acceleration for the roll input, and q, or, for the pitch input. Oscillations can be seen in the nonlinear responses, especially in the cross-axis components, but the amplitudes are quite small.

Ames Research Center Dryden Flight Research Facility National Aeronautics and Space Administration Edwards, California, October 6, 1989 + Table 1. Open-loop system matrices flight condition; M = 0.8, altitude = 20,000 ft, wing skew = 45 °.

Ap -0.7826 0.0958 0.00(30 0.0030 0.9926 -0.0003 -0.0592 -0.2908 0.0387 0.0259 0.0001 -0.9920 0.0000 0.0000 0.0000 1.0000 0.0000 0.0275 33.1431 -53.6928 0.0000 -3.1250 2.0552 1.7210 -8.6816 0.7975 0.0000 0.1679 -1.0352 0.1810 - 1.0092 10.7521 0.0000 -0.0213 0.0080 -0.7129 Bp -0.0974 -0.0974 -0.0198 -0.0302 0.0000 -0.0166 0.0166 0.0008 -0.0005 0.0647 0.0000 0.0000 0.0000 0.0000 0.0000 12.9304 -22.2654 15.8467 - 11.8422 13.2774 -9.4073 - 10.8655 - 1.2311 0.8797 0.5694 1.9854 -2.2579 0.5262 -0.3276 -6.2499 H, 1.0000 0.0(_ 0.0000 0.0000 0.0000 0.0000 0.00130 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00(30 1.0000 0.13000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0302 0.1895 0.0097 20.1511 -2.4782 0.00130 -1.5285 -7.4982 0.0000 -0.0426 0.0036 0.1972 0.0000 0.0000 0.0000 0.00130 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0_0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.7802 -0.0033 2.5097 2.5097 0.5103 -0.4291 0.4291 0.0216 -0.0120 1.6674 Table 2. Desired model systcm matrices.

Atlrl -1.1568 0.0000 0.0000 0.0000 0.9566 0.0000 0.0000 -0.2852 0.0387 -0.0148 0.0000 -0.9600 0.0000 0.0000 0.00(30 1.0000 0.0000 0.0000 0.0000 -41.7362 0.0000 -8.0000 0.0000 10.0000 -16.8252 0.0000 0.0000 0.0000 -4.0271 0.0000 0.0003 10.9181 0.0000 -0.2726 0.0000 -3.3632 mtl'l -0.0848 -0.0848 -0.0494 -0.0494 0.0000 -0.0166 0.0166 0.0000 0.0(_ 0.0647 0.0000 0.0000 0.0000 0.0000 0.0000 11.7086 -11.7086 29.2528 -29.2528 8.1349 -7.8228 -7.8228 0.0000 0.0000 0.0000 0.0000 0.0000 0.00(30 0.0000 -6.4014 Hylt$ 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.00_ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 1.0000 0.0000 0.2342 0.0000 28.5286 0.0000 0.0000 0.0000 -7.3538 0.0000 -0.0394 0.0000 0.2075 0.00(30 0.0000 0.0000 0.0000 0.0000 0.0000 0.00_ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.1840 2.1840 1.2743 1.2743 0.0000 -0.4291 0.4291 0.0000 0.0000 1.6674 Table 3. Closed-loop cigcnvalues for full-state and output feedback.

Full-state feedback Output feedback -7.4002 4- 4.1541 i -7.1095 4- 4.8533i -6.1927 4- 3.7359i -5.3189 4- 4.1278i -3.5624 + 2.3180/ -3.3258 4- 2.4188i -0.2347 -0.2271 -0.0341 -0.0317 -0.0011 -0.0011 Model eigenvalues -2.5920 4- 3.7464i Shoapefiod -3,5624 ± 2.3180i Du_h-roD -0.0120 Spiral -7.9137 RoU Table 4. Model feedforward, error feedback, and integral error feedback gain matrices flight condition; M = 0.8, altitude = 20,000 ft, wing skew = 45 °.

Kilt -0.4449 0.1051 0.0006 -0.0409 0.0485 0.0261 1.1031 -0.3790 0.0017 0.0613 0.1025 0.0051 -0.5542 0.2179 -0.0001 -0.0359 -0.0234 0.0340 0.5025 -0.1434 -0.0021 0.0258 0.0164 -0.0275 -0.7614 -0.5524 0.0012 -0.0286 -0.1194 0.1726 -0.1694 0.4585 -0.1858 -0.6725 0.0055 -0.1032 0.2076 0.3803 0.1436 0.7338 0.0717 0.0521 -0.1346 0.0891 -0.0243 -0.4903 -0.0200 -0.0760 0.0992 -0.0900 0.0101 0.3344 -0.0179 0.0476 -0.1042 0.0085 0.5972 -0.1658 -0.0255 -0.0145 K'_e -0.4697 0.7455 -0.1821 0.5597 0.6492 0.1385 -0.3838 0.1587 -0.0805 0.3108 -0.3873 0.0219 -0.4845 0.2165 0.3367 APPENDIX The following informal proof shows that the projection from state space to output space preserves the augmented system closed-loop ¢igenvalues if the number of unique outputs is greater than or equal to the number of states.

Given the system equations = Ax + Bu (21) y = Hx + Fu (22) Where (A, B) is stabilizable, (H, A) is observable, H is full rank, and Rank (H) > Rank (A).

Given the optimal control in the form u = -Kffix. Then the closed-loop system equation becomes the following (23) = (A - BK._)x -1 HT to get To determine the output form of the control, premultiply the output equation by ( H T H) (HT H)-IHTy = x + (HT H)-IHT Fu (24) This inverse exists since H is full rank (by assumption). Substituting into the optimal control equation yields u = -Kz(HTH)-IHTy + Kx(HTH)-IHTFu (25) which can be rewritten in the final output form = = u = - [I - Kz(HTH)-IHTF] -1Kz(HTH)-IH T y (26) K_ To examine the closed-loop eigenvalues of the output feedback system, substitute equation (26) into the system equations to yield (27) = Ax - BKz, y y = Hx - FK_y (28) Rewriting this last equation y = [/+ FK_]-lHx (29) Tiros the closed-loop output feedback system equation becomes (30) f_ = (A- BKv[I + FKv]-IH)x To show that the closed-loop eigenvalues of the state feedback (eq. 23) and output feedback (eq. 30) systems are the same, it is sufficient to show that the closed-lo0p dynamic equations of the two systems are identical. Comparing the two equations, it is necessary to show that If_ = Kv[ I + FK_]-IH.

From the definition of K v (eq. 26), (31) [I - Kx( HT H) -I HT F] K v = K,( HT H) -I H v (32) Kv = Kz(H "r H)-I H T [ I + FKv] Collecting K v terms and post multiplying by H results in, (33) K, = Ks,[ I + FK_] -I H REFERENCES Alag, G.S., R.W. Kempel, J.W. Pahle, J.J. Bresina, and E Bartoli, Model Following Control for an Oblique-Wing Aircraft, NASA TM-88269, 1986.

Bryson, A.E., and Y.C. Ho, Applied Optimal Control, Revised Edition, Hemisphere Publishing Corporation, 1975.

Duke, E.L., B.E Patterson, and R.E Antoniewicz, User's Manual for LINEAR, a FORTRAN Program to Derive Linear Aircraft Models, NASA TP-2768, 1987.

Graham, A., R.T. Jones, J. Summers, Wind Tunnel Test of an F-8 Airplane Model Equipped with an Oblique Wing, NASA TM X-62273, 1973.

Gregory, T., "Oblique Wing Ready for Research Aircraft," Aerospace America, June 1985, pp. 78-81.

McMurtry, T.C., A.G. Sire, and W.H. Andrews, "AD- 1 Oblique Wing Aircraft Program," AIAA-81-2354, Nov. 1981.

Nelms, W.E, Jr., "Appfications of Oblique Wing Technology - An Overview," AIAA-76-943, Sept. 1976.

Smith, R., R.T. Jones, and J. Summers, Transonic Wind Tunnel Tests of an F-8 Airplane Model Equipped with 12 and 14-percent Thick Oblique Wings, NASA TM X-62478, 1975.

Smith, R., R.T. Jones, and J. Summers, Transonic Longitudinal and Lateral Control Characteristics of an F-8 Airplane Model Equipped with an Oblique Wing, NASA TM X-73103, 1976.

Tyler, J.S., Jr., "The Characteristics of Model Following Systems as Synthesized by Optimal Control," IEEE Trans- actions on Automatic Control, Oct. 1964.

Vincent, J.H., "Direct Incorporation of Flying Qualifies Criteria Into Multivariate Flight Control Design," AIAA- 84-1830-CP, Aug. 1984.

White, S., et al., A Feasibility Design Study for an F-8 Oblique Wing Research Demonstrator, Final Report, NASA CR NAS-11409, 1984.

Wiler, C.D., and S.N. White, "Projected Advantage of an Oblique Wing Design on a Fleet Air Defense Mission," J. Aircraft, Vol. 22, No. 10, Oct. 1985, pp. 896-900.

v 8,311 Figure1.Three viewof proposed OWRAconfiguration.

From alrcraft sensors Ym

I

Fm Pilot Proportlonal I Model commends Kle J

Kplr

error -_ I _ Integral _,j err= 4- _.+ Up Km _,_ _ To acluetors Figure 2. Closed-loop control system block diagram of explicit model following.

14-- ¢, s deg 6 -2.7 -2.4 -2.1 -1.8 -1.5 -1.2 -.9 -.6 -.3 0 Sideslip, deg No7 Figure 3. Trim sideslip as a function of roll angle flight condition; M = 0.8, altitude = 20,000 ft, wing skew = 65 °.

4000 -- B 300O 0 o 000000000 0 Drag, Ib 2000 IOO0 I t I I I I 1 I I I 1 I i 5 1015 20 25 3035 4045 SO 556065 Wing sweep, (leg 9eo6 Figure 4. Total aircraft drag as a function of wing skew flight condition; M = 0.8, altitude = 20,000 It, sideslip trim.

Elevator 0 deflection, A I I | I i !

deg -1 deg 0 , , , _ , , I i i deg 0 _ - | | • ! i i A n

J

_, 30_ deg I I _ I I I I I I I I I I I deg sec _ . l - - ! , n I I i a a deg 0 • , I, , ! i i deg r, se_'_ 0 g 0 v ay, g ".1 0 1 2 3 4 5 6 7 8 9 10 Time, sec Figure 5. Open-loop aircraft response to l°-pitch ramp.

Aileron 1 deflection,

oF

i I I I I I I dog

dog 1 E

I I I . I I ....... I _ I I I I I 0, dog 1 ,,I ., I i i I !

0 ....... I P!

deg i I I I I I | I I I | O n'] : .....

dog q' "$I_ _v ,-t- I O F_ L -_-- J ..... , , :I I I I I -,3 rt dog I, I I 0 i I i I I I I OOO "n, .OS _ g 0 F __, I I I I i v -.05 ' i I I I I . |, , ! I I 8 9 10 0 1 2 3 4 S 6 7 Time, leo Figure 6. Open-loop aircraft response to l°-roU ramp.

2.0 m 1.9 1.8 w 1.7 - .._'_ "%. .......... l Llnesr 1.6 - 1.5 - \ _-.-., deg 1.4 m "--'-----___,--;; 1.3 1.2 m 1.1

1 I I I I I I ! I 1

1.0 2 3 4 5 6 7 8 9 10 0 1 TiMe, lie© 9614 Figure 7. Closed-loop aircraft response to roll ramp angle of attack.

.10 /-- Nonllnnr •t ¢ q9 ._,_/_, _.__ ...........

(leg imc ..2 L..., --o6 -.4 f _o8

I I,I I I I I I I I

-.10 1 2 3 4 S 0 7 8 9 10 Time, seo _ls Figure 8. Closed-loop aircraft response to roll ramp pitch rate.

.8 .6 A _| |near ._ _ Nonlinear /-- Model _°6 ,d. r -.4 I

-.lo I I ! .1 I I I I I I

0 1 2 3 4 5 6 7 $ 9 10 "nine, _c _16 Figure 9, Closed-loop aircraft response to roll ramp sideslip angle.

2°1 D 1.8_ 1.5 1.2 .9 r, .6 ,--Unelr deg .3 -.3 f-Model -.9

I I I f I I I I I I

-1°2 0 1 2 3 4 5 6 7 8 9 10 Time, sec 9617 Figure 10. Closed-loop aircraft responseto roll ramp yaw rate.

2O m deg 12 D

a

i I I I I I I I

I

-3 3 4 5 6 7 8 9 10 0 1 "[11110, IleC 9618 Figurc 11. Closed-loop aircraft response to roll ramp roll angle.

27 _ Model 2421 _t_ _ Nonlinear 1815 H ; /---Unelr _ o d_P'g12 /l_

4 1 I I"l-I I I I-] I

0 1 2 3 4 5 6 7 8 9 10 Time, sec 9619 Figure 12. Closed-loop aircraft response to roll ramp roll rate.

1.5 m 1.4 1.3 1.2 m /--- Linear /- Model 1.1 an, . ,,,,_,- _.L_ l.

g .9 YHonllneer .5 ,5 B

I I I I I I I I I I

.5 0 1 2 3 4 5 6 7 8 9 10 Time, sec Rgurc 13. Closed-loop aircraft response to roll ramp normal acceleration.

.01 -.01 -- ,,, /--- Nonlinear -.02 -- / \ ./ ...........

° i

-.03 -- _,; =----'-"" .......

i ny, -.04 -- I _L]n_ar :: !: -.05 m g -.07 -.o0 -- l_lf_ Uodet: -.09

-.10 I 1 Vl 1 I 1 1 1 I I

0 1 2 3 4 5 6 7 8 9 10 Time, eec 9621 Figure 14. Closed-loop aircraft response to roll ramp lateral acceleration.

1.5 -- --- Model 1.2

'!

G, 0 deg -.8 -- ,4.L_IL._ _nUnear _'AFm_ drU-.,L,__ _ ---.-,..% -1.2 -- L -1.5--

I It'_ t I I I I

I

-1.8 I 2 3 4 5 6 7 8 9 0 1 Time, IleC 9s22 Figure 15. Closed-loop aircraft response to pitch ramp angle of attack.

6 -- _. Linear fl 5 -- j- Model 4-- q, 2 deg s---_ 1 M 0 al ff -- -- -1 -- _.,(_,- Nonlinear -2

I I !

__ I I I I I

8 9 10 0 1 2 3 4 5 6 7 Time, lec ee_ Figure 16. Closed-loop aircraft response to pitch Tamp pitch rate.

I, 2 , , --11 _" \._., -.4 I _.-.5 _ el Nonlinear deg --,7-'8 _._1 -.8 I --.9

-1.o 1 1 I I I I I I I I

0 1 2 3 4 $ 6 7 8 9 10 Time, sec 962,1 Figure 17. Closed-loop aircraft response to pitch ramp sideslip angle.

1.0

::P

r, . _, /- Model

i,r

--.4

_\-C,..r

--°6 seo -.2 I --.8

I 1 l l ! l I I

-1.o I I

0 1 2 3 4 5 6 7 8 9 10 Time, sec Figure 18. Closed-loop aircraft response to pitch ramp yaw rate.

•32 nllnoar

-'r X'I'._ -°''_ _"_"

,,,

:r

-"E

-:,,,,,,,,,,A

0 1 2 $ 4 S 0 7 8 O 10 Time, IleO 9e26 Figure 19. Closed-loop aircraft response to pitch ramp roll angle.

2.0 m 1.5-- -Nonlinear 1.0 ,5 Pl d_ n NO ._/-, ..%_._., _,5 -1.0 -1,5

I I I ! I I I I I 1

-2.O 0 1 2- 3 4 5 6 7 8 9 10 Time, IleC Figure 20. Closed-loop aircraft response to pitch ramp roll rate.

2.2 2.0 1.8

- A _I/__ _:, '

an , 1.6 g 1.4 1.2 1.0

i i I_i i'i i I-T--I

.8 0 1 2 3 4 5 6 7 8 9 10 Time, uc g628 Figure 21. Closed-loop aircraft response to pitch ramp normal acceleration.

0 _ _ u,, -.03 --.04 ly, -.05 _ Unear O tl -.06 -.07 k _ Nonllnur --.0_ --.09 -.10 I_ I I I ] 1 1 I I I 012345678910 Time, sec Figure 22. Closed-loop aircraft response to pitch ramp lateral acceleration.

Report Documentation Page Nalo-_/_a_cs and _ _lct, ?,Ono_ralo_ 3. Recipient's Catalog No.

2. Government Accession No.

1. Report No.

NASA TM- 100454 5. Report Date 4. Title and Subtitle April 1990 Output Model-Following Control Synthesis for an 6. Performing Organization Code Oblique-Wing Aircraft 8. Performing Organization Report No.

7. Author(s} H-1522 Joseph W. Pahle 10. Work Unit No.

RTOP 533-06-01 9. Performing Organization Name and Address 11. Contract or Grant No, NASA Ames Research Center Dryden Hight Research Facility P.O. Box 273, Edwards, CA 93523-0273 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Memorandum 14. Sponsoring Agency Code National Aeronautics and Space Administration Washington, DC 20546 15. Supplementary Notes 16. Abstract Recent interest in oblique-wing aircraft has focused on the potential aerodynamic performance advantage of a variable-skew oblique wing over a conventional or symmetric sweep wing. Unfortunately, the resuk- ing asymmetric configuration has significant aerodynamic and inertial cross-coupling between the aircraft longitudinal and lateral-directional axes. This paper presents a decoupling control law synthesis technique that integrates stability augmentation, decoupling, and the direct incorporation of desired handling qualifies into an output feedback controller. The proposed design technique uses linear quadratic regulator concepts in the framework of explicit model following. The output feedback strategy used is a suboptimal projection from the state space to the output space. Dynamics are then introduced into the controller to improve steady- state performance and increase system robustness. Closed-loop performance is shown by application of the control laws to the linearized equations of motion and nonlinear simulation of an oblique-wing aircraft.

18. Distribution Statement 17. Key Words (Suggested by Author(s)} Unclassified m Unlimited Automatic control Control law synthesis Model-following Subject category 0_ Oblique wing 22. Price 21. No. of pages 20. Security Clauif. (of this page) 19. Security Classif. (of this report) 29 A03 Unclassified Unclassified NASA FOaM 1626 OCT 86 *For sale by the National Technical Information Service, Springfield, VA 22161-2171.