Document
(NASA-TP-3302) N93-19108 LONGITUDINAL-CONTROL DESIGN ,, APPROACH FOR HIGH-ANGLE-OF-ATTACK AIRCRAFT (NASA) 29 p Unc| as , ..... HI/08 0148129
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NASA
Technical
Paper
Longitudinal-Control
Design Approach for
High-Angle-of-Attack
Aircraft
Aaron J. Ostroff Langley Research Center Hampton, Virginia Melissa S. Proffitt Lockheed Engineering & Sciences Company Hampton, Virginia National Aeronautics and Space Administration Office of Management Scientific and Technical Information Program Contents °°, _i_,. _L_tl[,_Of.,i_kI._ ,r.,._:'_ _,_ PRECEDING PAGE BLANK NOT FILMED Summary This paper describes a control synthesis methodology that emphasizes a variable-gain output feedback technique that is applied to the longitudinal channel of a high-angle-of-attack aircraft.
The aircraft is a modified F/A-18 aircraft with thrust-vectored controls. The flight regime covers a range up to a Mach number of 0.7; an altitude range from 15000 to 35000 ft; and an angle-of-attack (c_) range up to 70 °, which is deep into the poststall region. A brief overview is given of tile variable-gain mathematical formulation as well as a description of the discrete control structure used for the feedback controller. This paper also presents an approximate design procedure with relationships for the optimal weights for the selected feedback control structure. These weights are selected to meet control design guidelines for high-a flight controls.
Those guidelines that apply to the longitudinal-control design are also summarized. A unique approach is presented for the feed-forward command generator to obtain smooth transitions between load factor and ct commands. Finally, representative linear analysis results and nonlinear batch simulation results are provided.
Results from linear single-loop stability and multiloop /_ analyses show a high degree of robustness. A sensitivity analysis of four stability derivatives shows that the minimum singular values for 38 out of 39 design eases are above 1, which indicates excellent robustness. Nonlinear batch simulations show good agility for both pitch-up and pitch-down maneuvers and good c_ regulation.
Introduction In recent years, researchers have investigated the feasibility of flight at a high angle of attack (c_) in the poststall regime. Operation in this flight regime enables the aircraft, to decrease speed rapidly and execute quick turns within a small turning radius so that the pilot can position the aircraft for the first shot at a target. High-a flight can be accomplished with thrust-vectored controls to augment the more classical aerodynamic control surfaces that lose effectiveness in the stall region. Control methodologies must now accommodate this highly nonlinear flight regime in addition to traditional ones.
The traditional approach for gain scheduling has been to develop individual, constant-gain feedback control laws at many operating points over the flight regime. These feedback gains are combined with a curve-fit technique (interpolation, straight line approxinmtion, or a least squares fit) to create a gain schedule for the final control gains. Several schedules are often combined when more than one independent variable is involved. In a modern control design where a matrix of feedback gains is generated, traditional gain scheduling may cause the loss of performance characteristics and possibly stability in sensitive high-order plants (mathematical representation of aircraft). These control characteristic changes can occur if the actual gains are significantly different from the design gains.
Recently, the variable-gain output feedback technique was developed (refs. 1 and 2). In this approach, the gain schedule is optimally generated internal to the design algorithm. Variable gain is an integrated design approach in which all design operating conditions are handled simultaneously, thereby creating a more efficient design process. All operating points that are considered in the integrated design are guaranteed to be stable. The developed controller is nonlinear; however, linear design and analysis techniques are used. Thus, the designer can rely upon the wealth of previously developed techniques. This paper describes the application of variable-gain methodology to the high-c_ aircraft. Results of the design validation simulations are also presented.
Variable-gain output feedback wasoriginally appliedto reconfigurable aircraft flight control technology (ref. 3). In that successful application,feedback gainswerecalculated as a function of control effectorfailures. A second applicationinvolveda high-a, high-performance aircraft in which feedback gainsvariedwith flight conditions(ref. 4). That work serves as a feasibility precursor to the controldesigndescribed here.
The variable-gain approach is appliedto a proportionalintegralfilter (PIF) discretecontrol
structure (refs.5 and 6) with a command-generator tracker(CGT) feed-forward path (refs.6
and 7). The PIF control structure used here is a direct digital formulation that accommodates the computational time lag from rate to position commands. Control of the rate command can ensure that actuators are not overdriven. With the CGT feed-forward structure, the pilot's command changes go directly to the rate command signal, which results in faster transient response. The PIF-CGT is part of tile feedback controller, which derives its signal from a feed- forward command generator (FFCG). The FFCG generates commands that are interpreted by the feedback controller based upon pilot stick commands. The FFCG includes a unique approach for integration and smooth transition between two command modes -the load-factor command that applies at high speeds and the a command that applies at high c_ and low speeds.
This paper commences with an overview of the mathematical formulation for the variable- gain methodology and the PIF-CGT formulation. The overview is followed by a description of the appropriate control design guidelines and the high-a aircraft model. Subsections contain descriptions of the feedback controller and the FFCC as well as the PIF-CGT controller and the FFCG implementation. An approximate design procedure is included for selection of the initial optimal weights for tile feedback controller; also provided are the relationships necessary to change these weights. The last subsection contains the design procedure for the FFCG.
Results are included for the linear analysis and nonlinear batch simulation. Linear analysis results include both gain and phase margins and a frequency response for a combined model composed of rigid body and servoelastic data. Nonlinear batch simulation results show closed- loop agility for both pitch-up and pitch-down time responses and _ regulation during 360 ° rolls.
All time responses are compared with design guidelines.
Nomenclature Ap continuous plant state matrix feed-forward coefficient matrices All, A12, A21, A22 Bp continuous plant control matrix Bw process noise matrix C plant and controller state to output matrix Q steady-state normalized coefficients, i = 1, 2, or 3 Cp plant state to output matrix c1, c2, c3 steady-state normalized coefficients for (_, q, and nz Dp plant control to output matrix E expectation operator E1 command-generator tracker feed-forward gain scalar-weighting matrix for cost function f acceleration due to gravity, ft/sec 2 g matrix that relates plant states to integrator states Hzx matrix that relates measurements to integrator states nzy I identity matrix J global cost local cost J K feedback gain matrix variable-gain feedback matrix partitions Ki,K0 proportional feedback gain matrix for nz, deg/sec/g Ku proportional feedback gain matrix for q Kq control filter feedback gain matrix, sec -1 Ku proportional feedback gain matrix Ky Kz integrator feedback gain matrix proportional feedback gain matrix for a, sec -1 K_ M individual operating points N integer for series summation load factor, g nz load-factor command, g TLZ_C load-factor command trim, g l_ZO_C partitions of covariance matrix where ij represents all nine combina- Pij tions representing x, u, and z components static pressure, lb/ft 2 P gain-schedule parameter stability-axis roll rate, deg/sec ps discrete-state weighting matrix Q impact pressure, lb/ft 2 Qc continuous weighting for nz continuous weighting for q Qq continuous weighting for control filter Qu continuous weighting matrix for outputs Oy continuous weighting for integrator Qz continuous weighting for a Q.
pitch rate, deg/sec q pitch acceleration, deg/sec 2 qc pitch-rate command, deg/sec R continuous-control weighting matrix Uc controller input command vector plant control input vector Up tip time derivative of vector Up V total airspeed, ft/sec Vc rate command vector, deg/sec W plant and controller process noise vector plant process noise vector Wp x() arguments of variable X x plant and controller state vector plant state vector Xp xp first derivative of plant state vector Y plant and controller output vector Yc controller output vector, deg Ycmd command from feed-forward command generator plant output vector yp Yp,SS normalized vector of Ci coefficients Yu output vectors for controller position command state Yz output vectors for integrator state z integrator state vector or z-transform variable time derivative of integrator state vector z o_ angle of attack, deg o_ c angle-of-attack command, deg angle-of-attack command trim, deg fl sideslip angle, rad F discrete plant and controller control matrix Fp discrete plant control matrix Fw discrete plant and controller process noise matrix "Tw discrete plant process noise /_ nZ_ C perturbation in nz,c, g Aqc perturbation in qc, deg/sec AT sampling period, sec Ay error signal to integrator perturbation in C_c, deg AOL c stabilator 5s _sc stabilator command, deg pilot stick command, in.
5sp 5v pitch thrust-vectored control pitch thrust-vectored command, deg ¢ damping ratio discrete plant and controller measurement noise vector plant measurement noise vector _p pitch attitude, deg structured singular value P ly measured variables used to calculate p discrete plant and controller state transition matrix discrete plant-state transition matrix _p ¢ bank angle, rad crossover frequency, rad/sec oJ c oJ n natural frequency, rad/sec Subscripts; controller or command c i,j scries integers k coefficient for sampling sequence plant P Superscripts: T transpose -1 inverse Abbreviations: CGT command-generator tracker FFCG feed-forward command generator PI proportional integral PIF proportional integral filter Mathematical Formulation Overview The control synthesis approach for variable-gain optimal output feedback is applied to a PIF discrete control structure. In this section is a review of the formulation for the PIF design with a single model; then, based upon that explanation, we show how to apply the variable- gain synthesis technique. Finally, the CGT formulation is reviewed. Please note that in the formulation that follows, many of the symbols defined as vectors are later used as scalars in the example problem.
PIF Formulation With Variable-Gain Application The dynamic process for the plant is represented by ±p = Apxp + Bpup + Bwwp
(1)
f
yp = Cpxp + rip where Xp, yp, and Up are the state, output, and control vectors for the plant; Wp and rip are process and measurement noise vectors; and Ap, Bp, and Cp are the plant state, control, and output matrices and Bw is the process noise matrix. Each noise process is assumed to be white with zero mean; the processes are uncorrelated.
The PIF controller is a rate-command system composed of a proportional integral (PI) section and a filter section (see fig. 1). The filter is constructed by feeding the control position command Yc back to the rate command Vc through a gain matrix. If Yc is connected to Up, the equations for the open-loop PIF model are tip = v_ (2) = Hzyyp = Hzxxp (3) where Vc is the rate command vector for the controller (control feedback point), z is the vector for tile integrator state, and Hzy and Hzx are matrices that select the measurements and states to be integrated. The controller states in equations (2) and (3) are also outputs used in the design process.
I Proportional Filter ]yc= Integral IKI(P_ _ - i_r'--l_ Figure 1. PIF control structure.
The design approach is to augment the PIF equations to the plant equation and discretize to form 0 I Up =
Up + wk (4a)
(AT) I v_.k +
{xp}
:){xp /
o] [o
(AT) Hzx 0 Z k+l I z k Yu = I Up + 0 (4b)
° }
Yz k 0 z k 0 k where (I)p, Fp, and 7w are the discrete matrices corresponding to Ap, Bp, and Bw; Yu and Yz are output vectors for the control command and integrator, respectively; AT is the discrete sampling period; subscript k is an integer representing the present time; and the other subscripts are the same as previously defined. Note that Yu is equivalent to Yc- The control Vc, k is related to the outputs by the feedback gain matrix as Vc,k=-[Ky Ku Kz] Yu (5) Yz k Equations (4a) and (4b) represent the system at a single operating point; however, design for a comprehensive flight envelope requires many operating points. The variable-gain synthesis approach is effective when many design conditions are integrated and operated upon simulta- neously. The resulting feedback gains are functions of the a priori gain schedule parameters that are chosen. The discrete state and output equations (eqs. (4) and (5)) can be rewritten in general form and in terms of arguments that represent scalar parameters p and sampling time k as (p, k + 1) = • (p) x (p, k) + r (p) vc (p, k) + (p, k) (6) y (p, k) = C (p) x (p, k) + vt (p, k) (7) Vc (p, k) = -g (p) y (p, k) (8) Each operating point described in equations (6) and (7) has a cost function J[p, K(p)] that is quadratic in states and controls. First, the cost function is formulated in the continuous domain, then the function is transformed into an equivalent discrete cost as N J [p, U (p)] = lim [ J N-_c 2(N+ 1) EE x(p,k+ 1)TQ(p)x(p,k+ 1) +vc(p,k)Ta(p)vc(p,k) k=0 (9) where K(p) is the feedback gain matrix and Q(p) and R(p) are the discrete weighting matrices.
For simplicity, the cross term between the state and control vectors is not included in this paper, although the term is in the design algorithm. The main objective is to minimize a global cost J(K), expressed by M (lO) J(K)= EfjJj_v,K(p)] (fj >_0) j=l where the local costs are summed and weighted by fj to assign relative priorities to the M individual operating points.
The feedback gain matrix in equation (8) has a linear, functional relationship with p and contains both constant- and variable-gain parts that are implemented as q (11) K(p) = K0 + EPi(_'i) Ki i=1 where the variable v i represents some measured variable that the designer selects for the gain schedule parameter. The relationship between the Pi and v i may be either linear or nonlinear.
First, the feedback gain matrix K(p) is partitioned into proportional gains Ky(p), integral gains g z (p), and filter gains Ku (p) as K(p) = [Ky (p) Kz (p) Ku (p)] (12)
Next, the gains are incorporatedinto the PIF control structure, as shownschematically in
figure1. The sumof the PI feedbacks goesto a first-order,low-pass filter beforeYcis generated.
This sum of all feedback signalsis vc, which is the rate commandand the controlsignalfor
the PIF design(eqs. (4a) and (4b)). Control of the rate commandensures that actuators
are not overdriven.The transfer functionvc to Yc is an integrationand accommodates one
computational time stepduringthe designphase(fig. 1 andeqs.(2) and (4)).
CGT Formulation The CGT is based on the theory that the integrated plant outputs can track the linearized, command model outputs (refs. 5 to 7). If the plant does not have a transmission zero at zero frequency, then the first step is to invert the plant to form coefficient matrices Aij. Thus, -1 (13) A21 A22 J Hzz where a single plant model is used for simplicity. If the output of the command model equals the input, then A22 is the only coefficient of interest. The solution for the feed-forward gain E1 can be calculated as K-P_-zl P T (14) where Pzz and Puz are partitions from tile matrix solution P to the output feedback cost equation (eq. (32) in ref. 2), shown here as Pxx Pzu Pxz l P = /PxT_ P_u Puz] (15) kPzz PTz Pzz In equation (15), tile subscripts x, u, and z correspond to the plant states, control position states, and integrator states, respectively. Equations (13) to (15) are solved for E1 at each design condition.
Before the theory just presented can be applied, control design guidelines must be established.
The following section provides a review of the design guidelines. Also in this section is a description of the high-c_ aircraft.
Guidelines Several preliminary design guidelines were established to assist in tile design effort. Many of these guidelines were developed through extensive piloted simulations. Although they are still being reviewed, these guidelines include linear criteria for flying qualities and robustness, large- amplitude criteria for agility and nonlinear coupling, and pilot-in-loop criteria for task-dependent agility and handling qualities. Those guidelines that apply to the longitudinal control system and the variable-gain control design approach are addressed in this paper. Other guidelines, such as pilot-related criteria, are used in real-time simulation and are not discussed here. Most of the guidelines that relate to agility are discussed in reference 8; those guidelines that relate to stability criteria and servoelastic attenuation are in reference 9.
Typical stability guidelines (ref. 9) include single-loop gain margins of 6 dB and phase margins of 45 °. Structured, singular-value, nmltiloop margins should be evaluated; however, quantitative guidelines have not been established yet. Agility guidelines (ref. 8) include minimum pitch rate q and pitch acceleration//, which are criteria for pitch-up and pitch-down maneuvers at an altitude
of 25000ft. Forexample, the minimum//and q criteria for a full-aft pitch stick command are
96 deg/sec 2 and 55 deg/sec, respectively, starting from a lg trim at c_ = 5° and the throttle commanded to full afterburner. The maximum // should be obtained within 1 sec from the onset of the pitch stick command, and the maximum q should be achieved within 1.75 sec.
The tactically desirable nose-down guidelines for 0 and q are -14.5 deg/sec 2 and -24 deg/sec, respectively. When starting from a 60 ° trim, recovery to 10 ° should occur within 7 sec for safety considerations.
During roll coordination tasks with full lateral stick, the c_-regulation guideline (ref. 8) is 6 ° for a 90 ° roll about the velocity vector; the guideline is 10 ° for a full 360 ° roll. Load factor nz excursions should not exceed 0.5g in either case.
A final design guideline relates to structural frequency attenuation (ref. 9) of all servoelastic modes by at least 8 dB (gain of 0.25).
Aircraft Model The mathematical model is representative of an F/A-18 class of aircraft that has been modified for thrust-vectored control. For this study, the aircraft has a gross weight of approximately 35765 lb, a wingspan of 40 It, and a length of 56 ft. Controls include two afterburner engines and the following aerodynamic control surfaces: horizontal stabilators; full- span, leading-edge flaps; trailing-edge flaps; ailerons; and twin vertical stabilizers. In addition, pitch and yaw thrust-vectored controls have been added for both longitudinal and lateral- directional maneuvers.
A longitudinal controller design was used here. (See fig. 2 for the main components of the longitudinal aircraft model used in the design and linear analysis and for the number of states.) The comprehensive aircraft model contains a series of models representing actuator dynamics, airplane longitudinal dynamics, and sensor and filter dynamics. Four states are used for the longitudinal equations of motion: total airspeed V, a, pitch rate q, and pitch attitude 0.
The actuator dynamics portion is represented by unity gain with a fourth-order model for the stabilator and a second-order model for thrust-vectored control. The result is six states. The natural frequency and damping ratio combinations (COn,¢) are (36.4, 0.41) and (105.0, 0.59) for stabilator 5s and (75.0, 0.59) for pitch thrust-vectored control 5v. The unit for wn is radians per second.
Control , Output inputs Actuator longitudinal and filter _ _ Aircraft _p._ Sensor _ements dynamics dynamics dynamics , 6 states 4 states 4 states - design 7 states - analysis Figure 2. Longitudinal aircraft model.
Three measurements---a, q, and nz are used in the design. These measurements are modified by both sensor and antialiasing filter dynamics that are included in the block-labeled filters. Only the sensor dynamics for the a probe is relevant here because that probe is modeled by a first-order response with a bandwidth of 14 rad/sec. The (con,_) combinations for each of the three antialiasing filters for the a, q, and nz measurement order are (209.0, 0.74), (78.5, 0.89), and (200.0, 0.89). Six states represent the three antialiasing filters and one state represents the a probe; thus, seven states are used in the analysis. Because the antialiasing filter in the q measurement loop is the filter of interest, the design model includes two states for this filter.
Thesetwo states,addedto onestatefor the c_ probe and one for approximation of the nz filter, equate to four states.
Controller Design In the example design, some of the symbols that have been shown as boldface (vector or matrix) will now be shown in italicized form for a scalar quantity. The two main parts of the controller that are illustrated in figure 3 are the FFCG and tile feedback controller. The FFCG transforms the pilot stick command into an equivalent command Ycmd that can be interpreted by the feedback controller. The controller has two command modes (each with its own stick sensitivity): one for load factor nz, which generally applies at high-speed flight; the other for c_, which generally applies at low-speed, high-a flight. The FFCG must select one of the command modes and make a smooth transition from one to the other. The feedback controller must maneuver the aircraft agilely to orientations defined by Yemd and regulate outputs about these new set points. In addition, the feedback controller must be robust to changes in plant parameters and must attenuate disturbances to avoid undesirable responses in the control loop.
Feedback controller CGT FFCG Feed-forward gain 8v'-_sc Stabilator command Yp variable-gain 8vc feedback Wash-out PIF structure _ t Pitch thrust- Feedback filter vectored measurements command
o J
Leading-edge flap Flap schedule Trailing-edge flap Figure 3. Overall controller configuration.
The control command output from the feedback controller (PIF structure) consists of the stabilator command 5sc and the input to a limited wash-out filter. This filter maintains the pitch thrust-vectored control 5vc at a neutral position during most flight conditions to keep the thrust-vectored vanes from overheating. The 5vc control assists during transient maneuvers and becomes the main control when 5s saturates.
One other key component of the control system is the flap schedule controller. Both leading- and trailing-edge flaps are driven by a flap schedule that is mainly a function of c_ and is gain scheduled with other air data parameters. This gain schedule is the same as the one being used on the F/A-18 aircraft.
The feedback controller implementation used for the high-c_ design is shown in figure 4. In this incremental approach, Ky(p) multiplies the incremental change in yp, Kz(p) multiplies the difference between the sum of the measured feedbacks and Ycmd (in this example, Ycmd has only one value), and Ku(p) is incorporated into the discrete filter loop. One advantage of this Ycmd riable limit Feed forward " _z Ay
+ __ F_
+
yp Vc
° A T [ K u (P)] t"e__ 8sc .006211 (z+l) VC z - .9876 Figure 4. Feedback controller implementation.
incremental approach is that sensor biases are subtracted out in the proportional feedback loop.
In the integrator loop, the pilot can move the pitch stick slightly to compensate for biases.
Position limiters are incorporated to prevent windup in the rate-to-position integrator. The discrete dynamics in the 5vc actuator loop represents the Tustin transformation for a low-pass filter with a bandwidth of 1 rad/sec.
The feed-forward gain (El in eq. (14)) is calculated from the CGT design approach for a step input. This gain, which varies continuously over the flight envelope, multiplies the incremental change in Ycmd" Gain E1 is independent of the feed-forward gains in the FFCG, which is discussed later. The CGT feed-forward path allows changes in pilot commands to go directly to the rate command signal, which results in faster transient response.
Feedback Controller Thirty-nine design conditions (table I) are used for thc feedback controller: 14 conditions are at 15 000 ft, 13 conditions are at 25 000 ft, and 12 conditions are at 35 000 ft. Nineteen of the design conditions are at lg (Earth axis) flight. The other cases are at various non-lg conditions.
The parameters in table I are the design case, altitude, Mach number, _, nz, and open-loop short period. Because nz is along the z axis of the aircraft, the lg trim cases are lower by the cosine of the pitch attitude (not shown). Most of the non-lg trim cases are at higher loads; however, design cases such as 13 and 14 are at lower load factors.
Variable-gain parameters. The variable-gain feedback shown in equation (11) is repeated here as K (p) = go + _Pi (_i) Ki (16) i=1
TableI. DesignConditions
Design Mach a, nz, period,
Short
case number deg g rad/sec Altitude of 15 000 ft 1 0.70 2.52 1.00 2.70 .60 3.37 1.00 2.10 .49 5 1.00 1.40 .27 20 .94 .57 5 .21 .82 .54 6 .20 50 .80 .83 .22 65 .82 1.30 .70 20 6.30 1.70 .60 20 4.90 1.60 10 .60 35 6.90 1.70 11 .40 20 2.10 .97 .40 35 3.10 .97 13 .30 5 .37 .79 14 .10 45 .22 .35 Altitude of 25 000 ft 0.70 3.58 1.00 2.10 .59 5 1.00 1.50 .33 20 .94 .63 .26 35 .88 .56 19 .26 50 .90 .90 .28 65 .92 1.30 .70 20 4.20 1.50 .60 20 3.20 1.40 .60 35 4.50 1.40 .40 20 1.40 .82 .40 35 2.00 .83 .30 5 .24 .64 27 .10 45 .14 .27 Altitude of 35 000 ft 28 0.70 5.34 1.00 1.50 .60 7.24 .99 1.00 30 .41 .94 .70 31 .34 35 .94 .60 .34 50 .95 .94 .35 60 .95 1.60 .70 20 2.70 1.20 35 .60 2.00 1.20 36 .60 35 2.90 1.10 37 .40 5 .28 .66 38 .40 50 1.40 1.10 39 .20 45 .34 .53 where six gain schedule parameters pi(vi) are used. These parameters are functions of ct, impact pressure Qc, and static pressure Ps. The Pi(_i) and their limits are Pl = O.la (1.5 < a < 65) P2 = O.O1Qc (10 _ Qc <_ 470) p3 = O.O01Ps (498 __ Ps <__1200) Qc (0.008 < P4 < 0.4) p4 = --_- s _ _ (17) P5 = 0.1a - 3.5 (a > 35) = o _<35) :06 = o.01Qc - 2.5 (Qc > 250) = 0 _<250) The feedback gains change continuously with the measured variables and the function is smooth except at two points. The first four parameters cover the entire flight envelope; the last two parameters cover only portions of the envelope. Parameter P5 is used only when (_ is 35 ° or greater, and parameter P6 is used when Qc is 250 lb/ft 2 or greater. Both P5 and P6 have lower limits of zero and are not differentiable at the breakpoints. When any value of v i exceeds the design limit, the variable is limited to the value shown in equation (17).
Design procedure. This section presents a description of the relationships between the weighting matrices, feedback gains, and design performance. It also provides a description of an approximate procedure for designing a multi-input, single-output PIF controller. Figure 1 graphically portrays the transfer function from Uc to Yc as Yc(Z) = _;--X-TT_ _ Ky + z-1 ' uc(z) (18) wherein the argument p has been dropped for simplicity. The first term (in brackets) represents the transfer function for the filter; the second term (in parentheses) represents the PI transfer function. The variables Ky and Hzy are row vectors, Uc is a column vector, Ku and Kz are scalars, and z is the z-transform variable. The proportional gain matrix can be partitioned into individual scalar gains as Ky=[Kc_ Kq KT,] (19) where the gains correspond to the measurements for a, pitch rate q, and load factor nz, respectively. In this section, Ky is the proportional gain. The relationship between the individual gains is discussed later. The row vector Hzy is selected as Hzy=[1 1 1] (20) so that the three output measurements are summed, then integrated.
In the following discussion, weights are referenced to the continuous domain. Feedback gains (Ky, Ku, Kz) are adjusted by varying the corresponding output penalty weights Qy, Qu, Qz, and the control-rate penalty weight R in the quadratic cost function. Changes in gains are approximately related to changes in the square of the corresponding penalty weight. The output measurements are related to the states through the C matrix in equation (7). First, all weights are implemented in the continuous domain; then, they are discretized to determine corresponding weights Q and R for the discrete cost function in equation (9).
The bandwidth and steady-state gain of the tow-pass filter are related to the control feedback gain Ku. A decreased control-rate penalty R results in a higher value of Ku, a correspondingly higher bandwidth, and a lower filter gain. Additional filtering can be obtained if Ku is lowered, but with the disadvantage of increased phase lag. A trade-off must be made between bandwidth and phase lag. The control bandwidth also can bc adjusted if the penalty weight Qu is changed.
When an adjustment to R has little effect on bandwidth, a change in Qu seems to help (and vice versa).
Higher gains for Ky and Kz result from increases in the penalty Qy for the output measurements and Qz for the integrator. The higher gain for Ky results in a faster time response to external commands; the larger Kz results in smaller low-frequency errors. The relationship between Ky and Kz is important because these gains affect the phase lag of the PI section.
Therefore, a reduction of the ratio of Qz to Qy also will reduce the phase lag.
A Nyquist test of equation (18) can be performed by moving z around the unit circle in the complex plane to evaluate whether the structural attenuation guideline is achieved. The attenuation guideline will be met if the singular value of the norm of yc(Z)/Uc(Z) for any one input has a gain of less than 1 (0 dB) at all modal frequencies because all the modes will have peaks of -8 dB or lower. The crossover frequency in the pitch-rate control channel is the most critical and should occur at a frequency below 75 rad/sec.
The approximate design procedure that is summarized below applies to all design models.
The suggested weights are provided as a starting point and can be modified with the appropriate relationships. The results described later in this report indicate that this procedure apparently works well in the design process.
1. Set Qu = 1 and adjust all other weights relative to this weight.
2. Fix the ratio of weights for the proportional measurements Qc_, Qq, and Qn. A rough guideline for magnitude is to set the norm of these weights ]]QyN inversely proportional to the square of the percentage of deviation error allowed. For example, if the deviation error is within 7 percent, HQyl] should be approximately 200.
3. Select R to adjust the bandwidth of the filter loop. For example, select the bandwidth to be approximately three to five times the short-period frequency, then calculate R as the inverse of the square of the bandwidth.
4. Select the integrator weight Qz as a function of desired system crossover frequency Wc and adjust the ratio 3Qz Wc < -- (21) Qc_ +Qq +Qn i0 to reduce the phase lag of the PI loop and maintain good, low-frequency characteristics.
The factor of 3 in the numerator is shown because all three proportional measurements are summed before integration. This approach yields a rough approximation that must be adjusted if any one loop has too much phase lag or poor low-frequency characteristics.
The selected weights are discretized and incorporated into the synthesis program. Various analyses are then performed (i.e., closed-loop eigenvalues and damping ratios, frequency re- sponses, stability margin evaluations, and time responses); based upon the results, weights are corrected according to the relationships described in this section. Typical corrections might include adjustment of R or the ratio of weights in equation (21) to change phase margin, ad- justment of Qz to correct the low-frequency response, and adjustment of Qy to modify time response. An adjustment of Q,, (from step 1 in the procedure) should be made only as one of the last steps or if the adjustment of R has little effect on filter bandwidth. An increase in Qu allows a trade-off of less agility but improved gain and phase margins.
Feed-forward command generator. The FFCG converts the pilot's stick command into an equivalent command Ycmd that can be interpreted by the feedback controller. The FFCG selects either an nz (nz,c) or a command mode (ac) and makes a smooth transition between the two.
modes without additional work by the pilot. The basis for designing the FFCG is to start at the error signal Ay in the feedback controller (fig. 4) so that (22) Ay = Hzyyp - Ycmd and, because Hzy is a row vector of ones (eq. (20)), all three output measurements are considered.
Expand equation (22) into components and assume that Ycmd consists of contributions from three sources. Thus, Ay = ((_ - (_c) + (q - qc) + (nz - nz,c) (23) where the subscript c represents tile command. Each of the variables can be separated into a perturbation plus a trim. The perturbations in the commands are related by the coefficients C1, C2, and C3 (to be determined) as AC_c _ C1 Ac_c _ C1 Aqc _ C2 (24) Anz,c C3 Aqc C2 Anz,c C3 and are substituted into equation (23). Relate the command perturbations Ac_c and Anz,c (Aqc is not considered in this design) to Aycm d as 3 3
ECi Eci
AYcm d _ i=1 Anz,c- i=1 Aac (25) C3 Ci Thus, Ycmd is the sum of Aycm d plus trims aoc and nzo,c.
The pilot's stick command 6sp and the stick sensitivity function determine the perturbation commands Anz,c and Aac. The sensitivity equations include a perturbation plus a bias nz, = 1.35sp (26) ac = 106sp + 20 (27) with the units in inches for 5sp, g for nz, and degrees for a. Normally, the bias in equation (26) would be 1; however, because lg is subtracted internal to the nz sensor, the bias command nzo,c must also be at 0g. The bias for the a stick function is 20 °. When all terms have been combined, the final equations for Ycmd are
Eci
(28a)
(1.358p) +
Ycmd -- C3
Eci
(28b) i=l (105sp) + 20 Ycmd -- C1 where aoc must be estimated.
Implementation of the FFCG is illustrated in figure 5. Selection of the command Ycmd is based upon choosing the solution that has the lowest absolute value biased at -5. The negative bias is incorporated to ensure that negative load factors can be commanded for small negative 5._p deflections. For positive 5sp stick deflections, transition can occur only when the two solutions are equal. When tile stick deflection is negative, one positive solution and one negative solution can occur simultaneously. A jump condition is possible when the two solutions are of opposite sign. A lockout feature was added to the implementation to minimize this jump. When the FFCG is in the a mode, that mode cannot change if the impact pressure is less than 80 lb/ft 2 or if tile _sp is negative.
Pilot stick 8sp 01(a) P2 (Oc) 03 {Ps) P4 Functional relationships* A nz,c 1 Aac 3 3 Z C i Z c i Feed-forward i=1 i=1 C1 ,C2,C3 coefficients --z_nz, c + O_oc -- aa c + 20 C3 Cl Feed-forward a O_oc trim _Ylc _Y2c lYlc+ 5l<ly2c+51 *Obtained off line by lyes _no least-squares solution Ycmd = Ylc Ycmd = Y2c Figure 5. Block diagram of FFCG.
Coefficients C1, C2, and C3 are determined for each design condition with a steady-state analysis of the open-loop short-period plant that was approximated. With an input of 1, the steady-state output is given as (29) Yp,ss = -CpAplBp + Dp The solution for the where yp,s.s is a vector based upon one control and three measurements.
coefficients is (3O)
c2
C1 }_ yp,ss c_ Evaluation of these coefficients shows that the solution from equation (28a) is dominant at high speed and low c_, whereas the solution from equation (28b) is dominant at low speed and high (_.
Figure 6 contains a plot of these coefficients (solid line) calculated every 5 ° as a function of for flight at lg trim and an altitude of 25 000 ft. In practice, the goal is both to reduce the time required to calculate the Ci and to have smooth feed-forward commands. Because interpolation is time-consuming, a functional relationship was obtained from an off-line, least-squares solution.
The selected inputs are the first four Pi (eq. (17)) used in the variable-gain feedback controller.
Figure 6 contains the least-squares estimate for the Ci.
1.0 -- .8 Calculated ............. Least-squares
_.6
estimate "O ._. C 2 1-- .4 _'_\, .2 C3 ,.
, . I 0 7O 10 20 30 40 50 60 (z, deg Figure 6. Feed-forward coefficients at lg and 25 000 ft.
Equation (29) assumes a stable plant, which applies to most of the 39 design conditions.
A few design cases in the stall region (high a) have short-period eigenvalues that are slightly unstable. However, c_ is the dominant coefficient (near 1) and Ycmd (from eq. (28b)) is the dominant selection for the high-a cases; thus, the gain is essentially unchanged.
The next two sections present the linear analysis and nonlinear batch simulation results, respectively, for tile longitudinal controller described above.
Results of Linear Analysis This section presents results of the linear analyses. The main components of the aircraft model are described in the section "Aircraft Model" and are illustrated in figure 2. The design model has 14 states- 6 for actuator dynamics, 4 for aircraft dynamics, and 4 for sensor and output filter dynamics.
Depending upon the type of linear analysis, the controller thrust-vectored wash-out filter can be incorporated either into the plant (which allows the input control loop to be broken at a single point rather than at two points) or the wash-out filter can be incorporated into the controller. The former configuration is used for single-loop gain and phase margins. The latter configuration is used for servoelastic frequency response and sensitivity studies where separate controls must be maintained.
Gain and Phase Margins Although the design included only 39 operating conditions with a maximum Mach number of 0.7, the stability margins were analyzed at 133 conditions ranging to a Mach number of 0.9.
All test cases within the flight envelope met both a gain margin guideline of 6 dB and a phase margin guidelineof 45°. Except for several high-speed casesat 15000ft, all other test cases
outsidethe flight envelope alsomet the guidelines. The nondesign cases wereequallyasrobust
to changes in gainandphase asthe 39designcases. That similar robustness between the design
andthe nondesign cases indicatesthat the controlleris not tunedto specific designpoints. The
resultsweresimilar for both the plant input andeachoutput loop. Tile q-measurement loop is
the critical path; gain and phase margins for this plant output loop were similar to those at the plant input.
tt Analysis A # analysis (ref. 10) for a multiplicative error was performed at the plant output with all three output loops opened. Of the 39 design conditions, tile worst cases were at an a between 20 ° and 50°; within these cases, the lowest magnitude (inverse of the maximum singular value) occurred just below 0.5. This result indicates that a simultaneous complex change of at least 50 percent would be required to produce an unstable control system; this safety margin is considered satisfactory for flight conditions.
A _t analysis also was used to evaluate the sensitivity of the aircraft stability and control derivatives that affect the short-period mode (a and q time derivatives). In the analysis of the four stability derivatives for the 39 design cases, the minimum singular values (with one exception) were above 1 for the frequency range near the short period. The exception was at an a of 60 ° at 35000 It; the minimum singular value was 0.9. In this type of analysis the phugoid, which is not directly controlled, has the most sensitivity with minimums near 0.1.
Other preliminary calculations with real p analysis indicate that the phugoid may also have minimums above 1. All four control derivatives in the 39 design conditions registered minimums of at least 0.6. Although no concrete guidelines exist for this type of analysis, these minimums can be considered quite good.
Loop Transfer To conduct a singular-value, loop transfer analysis, the control loop was opened first at the single plant input, then at the plant output. Because the analysis has only one control, the Bode response and the singular-value response were identical at the plant input. At both input and output locations, the crossover frequencies for all 39 design conditions ranged from 2.4 rad/sec to 9 rad/see, with a slope of approximately 6 dB per octave. The highest crossover frequency occurred at the highest Mach number. The filter bandwidth in the PIF feedback controller was reduced (lower filter gain) at higher a to obtain lower crossover frequencies.
Servoelastic Frequency Response The servoelastic analysis was conducted as an extension of the rigid-body analysis; however, the plant input was opened at two controls. First, a servoelastic modal model was paralleled with a series combination of the rigid body and actuator dynamics, then the new model was connected in series with the output filters. Analysis of both light and heavy airplanes shows that the open-loop, servoelastic transfer function from control input to q output (units of sec -1) exhibits peaks of -20 dB at 75 rad/sec and -8 dB at 100 rad/sec. In addition, the transfer function from control input to nz output (units of g/deg) exhibits peaks of -45 dB at 37 rad/sec, -38 dB at 50 rad/sec, and -33 dB at 100 rad/sec. For the 39 design cases, singular-value loop transfer analysis at the plant input and output locations showed that the structural mode at 100 rad/sec was attenuated to -20 dB. This rate of attenuation is significantly lower than the guideline of -8 dB.
Results of Nonlinear Batch Simulation Several nonlinear batch simulations were conducted to evaluate pitch-up and pitch-down agility and a regulation during stability-axis rolls of 360 °. Nonlinear compensation qcomp was added to the pitch-rate measurement to compensate for nonlinear effects of gravity and the kinematic term composed of stability-axis roll rate Ps and sideslip /3. The derivation for the compensation equation is shown in the appendix. The final result is qcomp -- V (cosOcos(_cosol -t- sinOsina - 1) - ps/3 (31) where 0 and ¢ are the body-axis pitch attitude and bank angle, respectively. The feedback controller implementation (fig. 4), which includes the wash-out filter for the pitch thrust-vectored command, was incorporated into the batch simulation. A rigid-body dynamic model with six degrees of freedom was used for the aircraft equations of motion. The aerodynamic tables were generated from a wind tunnel-derived data base; the tables include a range of -10 ° to 90 ° for a and a range of +20 ° for sideslip. Flex-rigid ratios were used to incorporate flexibility effects.
Pitch-Up Agility The objective of this maneuver was to evaluate the pitch-up response (fig. 7) to a full-pitch stick input of 5 in. when trimmed at an altitude of 25000 ft and a Mach number of 0.6. In particular, the c_, q, and //responses were measured. The latter two responses were compared with design guidelines. To simulate tile maneuver, maximum throttle was commanded at time equal to 0.01 sec; 2 sec later, after thrust had built up, the pitch stick was ramped to maximum within 0.3 sec to simulate the approximate response time of the pilot. The top plot in figure 7 shows ct reaching 60 ° in less than 2 sec, then slowly climbing to 70 ° after a slight bobble. The sluggish response after the c_ reached 60 ° was caused by the saturated actuator commands.
The thrust-vectored command came out of saturation at approximately 5.6 sec as the ct slowly converged toward 70 °, although the stabilator remained saturated. The pitch rate peaked at approximately 51 deg/sec (slightly under the desired guideline of 55 deg/sec), whereas the pitch acceleration was greater than the guideline of 96 deg/sec 2. The FFCG started in the nz mode and made a smooth transition to the ct mode at approximately 3 sec. Based oi1 this smooth transition, the pilot is unlikely to detect the transition. The smooth time response also indicated good integration between the feedback and feed-forward controllers.
Pitch-Down Agility The objective of this maneuver was to evaluate the response (fig. 8) to a full-forward pitch stick input of -2.5 in. starting from an a trim of 60 ° and at an altitude of 25 000 ft. The a decreased to 10 ° in approximately 2 sec and crossed 0 ° shortly thereafter. This response fell well within the safety guideline that specifies a decrease to 10 ° within 7 sec; an equivalent tactical guideline has not been developed. Similarly, both q and // were significantly greater than the tactical guidelines of -24 deg/sec and -14.3 deg/sec 2, respectively. Based on these results, the response is well damped. The system remained in the a mode with the nz mode locked out because the input stick command remained constant at a negative value.
Angle-of-Attack Regulation The objective of this maneuver was to evaluate the _ regulation during full-lateral stick stability-axis rolls. Figure 9 illustrates four a trim cases: 5 °, 30 °, 45 °, and 60 °. The dashed vertical lines denote the time in seconds for wind-axis bank angles of 180 ° and 360 ° . In all cases, the a regulation was considerably better than the ±10 ° guideline for a 360 ° roll. At an _ trim 9O g 30 _ ! ! ! I ! ! ! ! [ ! A__L__ I -'_I _ ! ! ! ! I ! ! ! ] I , I I I I i l 1 1 I I i l i _ • i J ¢,0 3O t- t_ I ! I I ! t I .... I ....
-30 .... ' t- O rl -80 2 4 6 8 10 Time, sec Figure 7. Pitch-up response, full stick at 25 000 ft.
_ 3o d 0 -20 _ -,10 c- o.. -60 ......... ' I __ - 30 _ o o_ _ 60 -_ ............................ "-'-_ "--'-_-_ ..............
0 2 4 6 8 10 Time, sec Figure 8. Pitch-down response, full forward stick at 25 000 ft.
2O
of 45 °, the stabilator (not shown)wassaturatedtrailing edgedown at approximately 6.2 sec
and the pitch thrust-vectored command wasquitecloseto saturation.
0_trimmed 5° 15_ I 0=3604 I 72001 I 108001 ! j _D "0 5 11,, , .... I,_,,,,1,, -5 cctrimmed 30 ° 180°1 360ol O3 -o 30 ........................... L_,__ ........... [ ......
o_trimmed 45 ° °1 - 001 -o 45 ..... I , , , I., 35 .....
c_trimmed 60 ° -o 60 0,70 50 0 2 4 6 8 10 Time, sec Figure 9. Angle-of-attack regulation, full lateral stick roll at 25 000 ft.
Conclusions This paper presents the methodology used to design a longitudinal controller for a high- angle-of-attack aircraft that operates in a highly nonlinear flight regime deep into the poststall regime. The paper covers information such as theoretical development, control guidelines, design application, and results. Mathematical formulations for both variable-gain output feedback and the proportional integral filter control structure were described, followed by a summary of appropriate longitudinal control design guidelines for fight at a high angle of attack (a).
The design approach for the feedback controller includes an approximate design procedure for determination of the starting point and for adjustment of the optimal weights. A description of the design approach for the feed-forward controller was also included. Finally, linear analysis results and nonlinear batch simulation examples were described. Based upon the design and application results, the following conclusions can be made: 1. The use of established relationships to adjust the optimal weights allows the designer to make trade-offs between control power, errors in regulated variables, bandwidth of the control filter, and phase lag. Such adjustments work well in the design process.
2. The derivation of the feed-forward command generator illustrates a good blending procedure between the load-factor command mode and a command mode; the transition between these command modes is undetectable in the time responses. The derivation also shows good
integrationbetween the feedback andfeed-forward controllers, whichis demonstrated by the
smoothresponse shownin the nonlinearsimulationrcsults.
3. Of 133test conditions,all cases in the flight envelope meetthe guidelines for both the 6-dB
gain margin and the 45°-phase margin. In addition, exceptfor a few high-speed casesat
15000ft, all other test cases outsidethe flight envelope alsomeettheseguidelines.
4. The filter in the PIF control structure providesadditional attenuationto help meet the
servoelastic guidelines.The peakstructural fi'equency at 100rad/secis attenuatedto at
least-20 dB, whichis significantlybetter than the guidelineof -8 dB.
5. A # analysiswasperformed for multiplicativeerrorsat the plant output andfor sensitivity
evaluations of multiplicativeerrorsin stability andcontrolderivatives. This analysis indicates
a reasonably robust controlsystem.Analysisof four stability derivativesfor the 39 design
cases showsthe minimum singularvalues(with oneexception)above1 for the frequency
rangenearthe short period;resultsfor four controlderivatives showall minimums above 0.6.
Although no concreteguidelineshave been establishedfor this type of analysis,these
minimumsarequite good.
6. Nonlinearbatchsimulations demonstrate goodagility in both pitch-upand pitch-downfull-
stickmaneuvers. Both the pitch-rateandpitch-acceleration responses aremuchbetter than
the agility guidelines for the pitch-downmaneuver.For the pitch-up maneuver, the pitch
acceleration is slightly greaterthan the guideline,whereasthe pitch-ratepeak is slightly
belowthe agility guideline.In addition, resultsof four trim conditionsfrom low to high c_
show that c_ regulation is very good and exceeds the guideline for all cases.
7. The variable-gain methodology is practical for high-o_ applications. Incorporation of an inter- nally generated, optimal gain schedule, using a priori selected gain schedule measurements, allows for an integrated design that is accomplished in a single process. This design ap- proach is more efficient than classical methods that use a separate design for each operating condition.
8. The PIF control structure and its incremental implementation are shown to work well in a nonlinear environment. Thc controller is a direct digital formulation that accommodates the computational time lag from rate command to position command. Control of the rate command can ensure that actuators are not overdriven. The incremental implementation means that sensor biases are subtracted out of the proportional feedback loop. In the integrator loop, the pilot can move the pitch stick slightly to compensate for biases. The command-generator tracker feed-forward gain allows changes in pilot commands to go directly to the rate command signal, which results in faster transient response.
NASA Langley Research Center Hampton, VA 23681-0001 Deccmbcr 23, 1992
Appendix
Pitch-Rate Compensation
This appendixhastwo sections.The first sectionis a nomenclature list. The second section
contains a derivationfor pitch-ratecompensation that drawsfromthe definitionsbelow.
Nomenclature
acceleration of gravity, ft/sec 2 g K constant L aerodynamic lift force, lb Tn mass, slugs P body-axis roll rate, deg/sec stability-axis roll rate, deg/sec ps body-axis pitch rate, deg/sec q pitch-rate compensation, deg/sec qcomp wind-axis pitch rate, deg/sec qw T body-axis yaw rate, deg/sec propulsive force along the normal wind axis V total airspeed, ft/sec Ol angle of attack, rad & time derivative of angle of attack, rad/sec sideslip angle, rad body-axis pitch attitude angle, rad O_ wind-axis pitch attitude angle, rad body-axis bank attitude angle, rad wind-axis bank attitude angle, rad Derivation of Compensation Start with the normal force equation of motion (ref. 11) in the wind axis at the aircraft center of gravity and use a fiat Earth approximation to obtain (A1) -mVqw = Tzw - L + mg cos Ow cos Cw Use the definition of the wind-axis pitch rate qw (ref. 11) qw = (q - (_) cos/_ - (p cos a + r sin c_) sin/_ (A2) to obtain after substitution (A3) mV (& - q) cos ¢? + mVps sin _ = Tzw - L + mg cos Ow cos Cw
wherethe definitionfor stability-axisroll rate p_ is included as
p.s = p cos a + r sin a (A4) Redefine the wind-axis Euler angles in terms of body-axis Euler angles and angle of attack
as
cos 0w cos ¢w = sin ct sin 0 + cos a cos 0 cos ¢ (A5) Substitute into equation (A3) to obtain mV (d - q) cos/3 + mVps sin/3 = Tzw - L + m9 (cos 0 cos ¢ cos a + sin 0 sin a) (A6) Because we are solving for the nonlinear coupling terms that affect q, let the external forces (Tzw and L) be assumed at constant K and let & be assumed as zero. This last assumption is used because the objective is to maintain a at trim during a stability-axis roll. The revised equation is -Vq cos/3 + Vp_ sin/3 - 9 (cos 0 cos ¢ cos a + sin 0 sin a) = K (A7) Use small-angle assumptions for/3 and solve for q to yield g (cos 0 cos ¢ cos a + sin 0 sin a) - K (AS) q = P_/3 - V where the first two terms on the right side of equation (A8) should be compensated. The pitch-rate compensation qcomp is 9 (cos 0 cos 0 COS c_ + sin 0 sin c_) -- Ps/3 (A9) qcomp = Use a constant value of 32.1 for the acceleration due to gravity (based upon an altitude range from 15 000 ft to 35 000 ft) and multiply the gravity compensation term by 180/77 to obtain units in degrees per second. The yield is 1839 (cos 0 cos ¢ cos c_ + sin0sina - 1) - Ps/3 (A10) qcomp - V where the -1 term is included to obtain a zero bias at the neutral stick position when ¢ and/3 are zero and tile flight path angle is zero (0 = a).
References 1. Halyo, Nesim; Mocrder, Daniel D.; Broussard, John R.; and Taylor, Deborah B.: A Variable-Gain Output Feedback Control Design Methodology. NASA CR-4226, 1989.
2. Halyo, Nesim: A Variable-Gain Output Feedback Control Design Approach. A Collection of Technical Papers, Part 2--AIAA Guidance, Navigation and Control Conference, Aug. 1989, pp. 1238 1248. (Available as AIAA-89-3575-CP.)
3. Moerder, Daniel D.; Halyo, Nesim; Broussard, John R.; and Caglayan, Alper K.: Application of Precomputed Control Laws in a Reconfigurabte Aircraft Flight Control System. J. Guid., Control, FJ Dyn., vol. 12, May June 1989, pp. 325 333.
4. Ostroff, Aaron J.: High-Alpha Application of Variable-Gain Output Feedback Control. J. Guid., Control, Dyn., vot. 15, Mar. Apr. 1992, pp. 491 497.
5. Broussard, John R.: Design, Implementation and Flight Testin9 of PIF AutopiIots for General Aviation Aircraft.
NASA CR-3709, 1983.
6. Maybeck, Peter S.: Stochastic Models, Estimation, and Control, Volume 3. Academic Press, 1982.
7. Broussard, John R.; and O'Brien, Mike J.: Feedforward Control To Track the Output of a Forced Model. IEEE Trans. Autom. Control, vot. AC-25, no. 4, Aug. 1980, pp. 851 853.
8. Forster, John V.; Bundick, W. T.; and Pahle, Joseph W.: Controls for Agility Research in the NASA High-Alpha Technology Program. SAE Paper 912148, Sept. 1991.
9. Military Specification Flight Control System General Specification For. MIL-F-87242 (USAF), U.S. Air Force, Mar. 31, 1986.
10. Doyle, John: Analysis of Feedback Systems With Structured Uncertainties. IEE Proe., vol. 129, pt. D, no. 6, Nov. 1982, pp. 242 250.
11. Etkin, Bernard: Dynamics of Atmospheric Fli9ht. John Wiley & Sons, Inc., c.1972.
Form Approved REPORT DOCUMENTATION PAGE OMB No 0704-0]88 Public reporting burden for this collection of information is estimated to average I hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions For reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, [215 Jefferson Davis Highway, Suite 1204. Arlington, VA 222024302, and to the OFfice of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503 1. AGENCY USE ONLY(Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED February 1993 Technical Paper 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Longitudinal-Control Design Approach for High-Angle-of-Attack Aircraft WU 505-64-30-01 6. AUTHOR(S) Aaron J. Ostroff and Melissa S. Progitt 8. PERFORMING ORGANIZATION 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) REPORT NUMBER NASA Langley Research Center Hampton, VA 23681-0001 L-17123 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY REPORT NUMBER National Aeronautics and Space Administration NASA TP-3302 "Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES Ostroff: Langley Research Center, Hampton, VA; and Proffltt: Lockheed Engineering _z Sciences Co., Hampton, VA.
12b. DISTRIBUTION CODE 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified Unlimited Subject Category 08 13. ABSTRACT (Maximum 200 words) This paper describes a control synthesis methodology that emphasizes a variable-gain output feedback technique that is applied to the longitudinal channel of a high-angle-of-attack aircraft. The aircraft is a modified F/A-18 aircraft with thrust-vectored controls. The flight regime covers a range up to a Math number of 0.7; an altitude range from 15000 to 35000 ft; and an angle-of-attack (c,) range up to 70 °, which is deep into the posts]all region. A brief overview is given of the variable-gain mathematical formulation as well as a description of the discrete control structure used for the feedback controller. This paper also presents an approximate design procedure with relationships for the optimal weights for the selected feedback control structure. These weights are selected to meet control design guidelines for high-c_ flight controls. Those guidelines that apply to the longitudinal-control design are also summarized. A unique approach is presented for the feed-forward command generator to obtain smooth transitions between load factor and c_ commands. Finally, representative linear atmlysis results and nonlinear batch simulation results are provided.
14. SUBJECT TERMS 15. NUMBER OF PAGES Longitudinal control; High c_; Variable gain; Feedback control; Feed-forward control; 28 Airplane control; Aircraft control 16. PRICE CODE 19. SECURITY CLASSIFICATIOf_ 20. LIMITATION 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATIOI_ OF THIS PAGE OF ABSTRACT OF ABSTRACT OF REPORT Unclassified Unclassified _SN 7540-01-280-5500 Standard Form 2g8(Rev. 2-89) Prescribed by ANSI Std Z39-18 298-102 NASA-Langley, 1993