Design, implementation and flight testing of PIF autopilots for general aviation aircraft

APPENDIX A AIRCRAFT EQUATIONS OF MOTION A. OVERVIEW This appendix reviews the nonlinear and linear perturbation equations of motion for a general aviation aircraft. These derivations make extensive use of vector-matrix differential equation ("state-space") notation, and the resulting equations are written in a form suitable for analysis using concepts of modern control theory.

The notation is fairly standard but a review of the List of Symbols may be desirable.

B. NONLINEAR DYNAMIC EQUATIONS The nonlinear rigid-body equations are reviewed in this section.

The equations are developed using "flat-earth" assumptions, i.e., the effects of earth curvature and rotation are assumed negligible. This means that earth- fixed and inertial reference frames are equivalent.

The origin of the inertial reference frame used here is located on the surface of the earth at the runway touchdown point. The x-axis points along the runway center line towards the approaching aircraft. The z-axis points down and the y-axis points to the right completing a right-hand coordinate system.

Detailed vehicle state equations can best be expressed in body-fixed axes.

These are the axes in which the pilot, the sensors, and the control surface locations are defined. Body axes are the only axes in which the moment-of- inertia matrix is constant. Dynamic data collected from flight tests usually are expressed in body axes.

The body-fixed axis system used in this report has its origin located at the body center of mass, and is fixed in orientation with respect to the vehicle. The body x-axis extends forward out the vehicle's nose, the y-axis extends out the right wing, and the z-axis extends out the bottom of the vehicle. The x-z plane is usually a plane of geometric symmetry, if the vehicle has one. For any nominal flight condition, another commonly used body-fixed axes system can be chosen so that the x-axis is aligned with the velocity vector, and the z-axis is in the body-axis plane.

This set of body-fixed axes is referred to as the stability-axis system and is not employed in this report.

A transformation from inertial to body axes is composed of a right-handed yaw through an angle q; then a right-handed pitch through an angle 8, and then a right-handed roll through an angle 4.

The inertial-to-body axis transform- ation proceeds as follows: Two aerodynamic angles used in this report are the angle of attack, ~1, Angle of attack represents the aircraft body pitch and angle of sideslip, (3.

angle above the velocity vector, and sideslip is the angle that the aircraft nose is yawed left of the velocity vector.

The nonlinear equations of motion of the aircraft are determined by con- structing kinematic relationships between aircraft translational and angular position states and by applying Newton's Second Law to the dynamics of the vehicle. The kinematic relationships are .

= HB" 1B (73) Ir, = Lgl EB (74) XB and the results from applying Newton's Second Law to translational and rotat- ional motions produces (Ref. 34) .

= (FB + TB> /m + Hi 2E - GBxB (75) !B .

-' (MB + GB) - I;'$ IB gB (76) WB = 'B is the position of the vehicle relative to the earth fixed co- The vector 3, ordinate System 2 = [xE, yEI ZE]. The inertial-body Euler angle vector is The Euler angle derivatives occur in three different T = rf$, 8 I qJI.

?&I XB references frames and are related to the body-axis vehicle rates as shown in Eq. 74 where -sine LB = 0 co.s$ sin$cose (77) 0 -sin@ c0s+20se The applied specific forces consist of gravitional forces and contact forces as shown in Eq. 75. The specific contact force is broken into two components, one of which is due to aerodynamic forces, FB, and one of which is due to thrust, The gravity force is 2; = [O 0 g]. tiB in Eq. 75 iS the cross product TB.

equivalent matrix for s and is given by (78) The inertia matrix, IB, in Eq. 76 contains all products and moments of inertia; (79) while the contact moments consist of aerodynamic components, EB, and thrust components, GJ. Equations 73 to 76 when combined fall into the general state equation form G=f(x u) (80) - - -9 - by defining the state vector (81) and noting that the aerodynamic forces and moments are functions of states, con- trols, and to some extent, the state history.

C. LINEAR DYNAMIC EQUATIONS The linearization procedure is performed by construction of a Taylor series expansion representing the nonlinear equations about some nominal trajectory: x = G + Al; + Higher Order Terms - - af A +% X Au + Higher Order Terms (82 1 = f(rZg'Q + z - au - x=llo x=llo the subscript "0" indicates the nominal value and the prefix “A” denotes Here, a small perturbation. All except first-order terms are then neglected by arguing that the higher-order terms are small compared to linear terms. The results of this procedure are separated into a nonlinear equation describing the nominal trajectory (Eq. 83) and a linear equation defining the dynamics of the perturbations about the nominal trajectory (Eq. 84): A; = AAx + BALJ (84) The formal linearization of the aerodynamic forces and moments and non- linear dyanmic kinematic equations is a lengthy but straight forward process which can be found in a number of references (Refs. 18, and 34). The linear- ization of each nonlinear dynamic equation in Eq. 78 results in the following: (85) -' AgB - L;' = LB L; AB (86)

A2B

0 0 = (AFB + ATBIb + ( (87)

AkB

(% + AG,> - Ii1 AkB = 1;' (Ix) 'EB (88) The perturbation equations contain as yet only general terms for the pertur- bation aerodynamic effects. The only remaining undefined term is in Eq. 86, a cLeB) (89) av -- B = ?Bo io taneo $ se& 0 0 0 -q L- = -Ij, 0 c0se 0 0 0 (90) BO 0 ho se& 0 Go taneo 0

I

The perturbation aerodynamic effects are assumed to be linear functions of the perturbation states, of the time derivatives of the perturbation states and of the perturbation values of the vehicle control variables.

This viewpoint leads to the conventional name of aerodynamic stability derivatives for the coefficients that relate the pertrubation states, state rates and controls to the perturbation forces and moments. Assuming insignificant altitude and orient- ation effects on contact forces and moments, and assuming insignificant state derivative and angular rate effects on thrust forces and moments, the pertur- bation aerodynamic forces and moments are as follows: (91) (92) (93) (94) By using the definition of non-dimensional stability derivatives given in Ref. 24, it is possible to define the dimensional stability derivative matrices in Eqs. 91 to 94 in terms of the usual body axis non-dimensional This is done in Eqs. 95 to 102.

stability derivatives.

wocx +‘/,v c + $Y vocx + ‘/2vocx vocxo uocxo V 0 O xw U (95) vocy +1/,v,c, wocy +1/,v,c, + 4 vocy

= poA

uOcYo W U 0 V 0 wocz +l/,v c + gzvocz vocz +$v,c, uoczo V 0 O zw U 0

I

CC X bCX

bCX

r q P CC (96) bCY Y bCY r P CC Z bCZ bCz r P q C c C % x6 xb x6T e a r C (97) C ‘SP~V;A cy6 % y6 Y6T e a r CZ6 cz6 i cz6 .cz6 T e a r CC CC bCX . x.

U ir xG aFB PO CC CC (98) s= TA bCY yil ir y;J [I I 1 CC CC zli bCZ ir zti uobC~o+~VObCR vObCQO+L2VObCQ wobCRo++VObCR U V W

aEB

--=

u cc + % voccm v cc + * v,cc (99) + % vo"cm

av pOp

wOccmo m

[ 1 -B

O m. U O m.

W V uob( +%VObCn +$ v bC wObCn +%VObCn

,bC*

vc

O no 0 u V 0 W b2CQ b:CR b2CR P r pO (100) = 4 VOA b:C C2C b:C m r mP mri b2C b:C b2C n

’ I r

nP % bC

bCJT

bCQ bCR 6T % 6

r

e a (101)

CC CC CC CC

m6

mbT m6 m6 r a e bC bC bC bC n6 n6 n6T % a r e bk, b:Ce b2CQ il ir i E2Crn. bk,, G2Crn.

(1021 V W U bk,.

b:C,. b2C n.

U V W 98 and 102 is included because of the affect of & on &M.

The A$, term in Eqs.

The subroutine FGAERO in the program PIFGCT, Ref. 5, accepts the non- dimensional stability derivatives and trim conditions shown in Eqs. 95 to 102 and uses the information and the derivation in this section to construct A and B in Eq. 84. Further details of the linear model construction are given in documentation of FGAERO.

Many elements in the stability matrices are known to be zero or can be neglected in all cases of interest. Considerable simplification of the equa- tions of motion for a specific vehicle can be performed and this is done for the NAVION aircraft in Appendix B.

APPENDIX B AIRCRAFT AERODYNAMIC MODEL -___- The aerodynamic and mass data and simplified equations of motion used in Comparisons of the model the NAVION model are discussed in this appendix.

response to the actual aircraft response for a pilot induced pulse in the air- craft control surface is also shown.

A. AERODYNAMIC AND MASS DATA The aerodynamic data for the NAVION aircraft is taken from Ref. 30.

Reference 30 uses flight data rather than wind tunnel data to identify the aerodynamic coefficients and therefore captures rate effects due to q, r, p, and &. A comparison of flight test results with linear model simulations caused some of the aerodynamic coefficients from Ref. 30 to be adjusted to better represent the aircraft dynamics. The original mismatch between the model and the aircraft was sufficient to cause the lateral-axis PIF control law to be unstable. The aircraft wheels in current flight tests are in the takeoff and landing configuration (increased drag) while the wheels in Ref. 31 are believed to be retracted. The NAVION airplane parameters are shown in Table 14.

The gross mass value represents the best estimate of the current fully instrumented NAVION. The mass value (WC) is 204.8 kg (452 lbs) heavier due to instruments than the value given in Ref. 30.

The instruments are near the c.g. The values for Ix, Iy, and IS remain unchanged from those given in Ref. 30.

The matrix differential equations for the nonlinear aircraft model in Appendix A can be simplified to the following series of scalar nonlinear equations for the NAVION aircraft.

6 = rv - qw - sin(B)g + % (103) pV A 4 C WT X ir = pw - ru - cos(e)sin(@)g + % oV2A & Cy (104) 6 = qu - pv + cos(e)cOs(~)g + % pV2A ;-$ cz (105) = -$- ((I-.-I,>qr + $ (106) pV2A b C,> (107 (108

e = cos($)q - sin(+)r

(109) $ = p + (q sin(+) + r cos($)) tan(B) (110) 4 = (sin($)q + c0d$9r>hde> (111) + cx (cl-ao) + cx (112) ( 6,-(JTo) cx = cxo a 6T cy = cyo + cy (B-B,) + Cy & (r-r01 + Cy j$ (p-p,) r B P (113) (pro) + cY r cz = czo + c z @-ao) + cz (6e-6eo) + cz.. 5 (q-4,) (114) a 6 e + CL (B-Be) + CL (6a-6ao) + CL (pro) cL = cLo B 6 'r a + CL ,(P-P,) + CL (r-ro) (115) r P

CM= CM0 + cMac cmo> + cM8 (6e-6eo) + CM,g

(q-qo)

e

CN= CNo+ CN (8-B > + c

r-ro) + CN g (P-P,) e 0 N, P

+c

(117)

(6a-6ao) + CN (6r-6ro)

N8

a r 30 to extract aerodynamic parameters.

Two flight conditions are used in Ref.

The wheels down fully weighted NAVION used in They are shown in Table 15.

flight tests cannot fly as fast as condition I. Corrections based on flight tests data have only been performed for condition II. The longitudinal aero- dynamic parameters at the two flight conditions are shown in Table 16. The lateral aerodynamic parameters at the two flight conditions are shown in Table The two parameters which have been adjusted after flight tests are Cydr 17.

(from -0.68 to -0.143) and CL&, (from -0.007 to -0.023).

Reference 35 was obtained after the completion of the PIF design and flight testing. The analog match from Ref. 35,and shown in Tables 16 and 17, is for the second NAVION, a variable stability aircraft equipped with side force pannels on the wings. CY6, and CL6, from Ref. 35 are in general agreement with the modifications used in the PIF design. The analog match in Ref. 35.did not use CSq and CM& as identifiable parameters and for this reason, the other longitudinal parameters in Table 16 do not agree well with Ref. 30.

B. SIGN CONVENTION In implementing a control law it is crucial that the sign conventions used during the control law design be explicitly detailed. For the following presentation of sign conventions, the aircraft is assumed to be near the glidepath approaching the runway. Left and right is referenced to the pilot perspective while setting in the cockpit. Vertical height and normal acceler- ation plotting from flight test are opposite in sign to the convention used in this section. The barometric altimeter complimentary filter also has the opposite sign convention.

The earth fixed (inertial) coordinate system and body axis coordinate system sign conventions are discussed in Appendix A. The following are sign conventions for positive step changes in control variables. The sign con- vention is chosen to match the sign convention onboard the NAVION aircraft Positive Rudder - Positive Yaw Positive Yaw - Aircraft Nose Rotates Right Positive Rudder - Rudder Trailing Edge Rotates Towards Right Wing Positive Aileron - Positive Roll Positive Roll - Right Wing Down Positive Aileron - Left Aileron Trailing Edge Rotates Down, Right Aileron Trailing Edge Rotates Up Positive Elevator - Positive Pitch Positive Pitch - Nose Up Positive Elevator - Elevator Trailing Edge Rotates Up As the aircraft progresses down the glideslope the flight path angle, y is given by y = sin-l (-2/V) (118) V is the aircraft total earth relative velocity and i is the aircraft's earth If the aircraft is descending the flight path angle relative vertical velocity.

is negative. When the aircraft has no lateral error, the distance in the verti- cal plane from the aircraft c.g. in a perpendicular line to the glidepath is d, and satisfies the equation

A = ,V sin(hGS + h)

[I191 Figure 45 shows the aircraft below the glideslope. The variable &S is the glideslope angle and is defined to be positive for typical glideslopes. Hence for a desired glideslope angle of 3 deg, the aircraft's flight path angle must If the be -3 deg for the aircraft to be tracking the glideslope, i.e., d = 0.

aircraft is above the glidepath, d is positive. The angular error, r, relative to the glidepath is I' = sin-'(d/g) (120) where R is the range to the aircraft. If the aircraft is above the glidepath, r is positive. In an MLS environment, d can be computed from x and z position using the equation d = x sin($S) - z cos($S) (121) The localizer beam centerline intercept and hold geometry is shown in Fig.

46. Projecting the velocity vector onto y produces $ = V cosy sin(E-<ref) (122) where 5 is the velocity heading angle measured relative to a reference value The lateral distance y satisfies the equation ref.

\ sin(E) Y= The localizer deviation angle, !, and the distance y shown in Fig. 46 are negative. The distance Rh is the projection of R onto the horizontal plane.

If the assumptions are made that the flight path angle and sideslip angle are small, (and the aircraft is in the approach mode) then the above equations can be simplified to i N V sin($-Qref) (124) y = R sin(E) can be estimated using Eqs. 124 and 125 The lateral position, y, and rate, $, and sensor information available onboard the aircraft. Projecting the velocity vector onto R and using the same simplifying assumptions produces R = V CO&/J-$ref - E'> [1X1 C. SIMULATION AND FLIGHT TEST COMPARISON During early flight tests, the safety pilot purposely made separate small pulse inputs to each aircraft surface. The resulting NAVION aircraft response from the sensors was recorded for comparison with the linear open-loop model simulation response for similiar pulse inputs. The results of the comparison for an elevator input and a rudder input are shown in Figs. 4 and 3, respec- tively.

The normal.accelerometer sampled output (10 samples per set) and pitch rate gyro sampled output for an elevator input agree well with the linear simu- lation as shown in Fig. 4. The normal accelerometer output sign convention is positive up and biased by -1.0 g to give zero output in level flight.

The aircraft response to a rudder pulse originally had poor agreement with the model response. After adjusting two aerodynamic coefficients as discussed in Section B, the new comparison in Fig. 3 is considered to be adequate. The initial response for the first few seconds match well then begin to diverge as The cross-axis aircraft response of roll rate for a the simulation continues.

rudder change has less agreement with the simulation than sideslip and the lateral accelerometer output. The aircraft's Dutch Roll mode appears to be better damped than the model response. The small rudder movements made by the pilot during the pulse command may have contributed towards the simulation and aircraft disagreements. The linear model simulation disagreements did not ad- versely affect the PIF control law performance in flight. On the other hand, better identification of all aerodynamic coefficients should further improve PIF control law performance.

D. SENSOR MEASUREMENT PERTURBATION EXPRESSIONS The PIF control law employs, as much as possible, sampled sensor outputs for direct feedback. The output of some sensors, particularly the accelero- meters, the pitot tube, the sideslip vane, and the angle-of-attack vane, are nonlinear functions of the aircraft body-axis states used in PIF feedback.

The PIF design procedure at a trim flight condition requires an expression for the perturbation sensor output in the form AZ = H Ax + D As where H and D are constant aircraft state control observation matrices.

The purpose of this section is to derive perturbation expressions in the form of Eq. 127 for the previously mentioned sensors. Sensor inaccuracies and noise effects are neglected.

The output of an orthogonal traid of body-mounted accelerometers aligned with body coordinates is expressed as (Ref. 3).

(128) A flat, non-rotating earth surface is assumed. The gravity vector, B, the Euler angle transformation matrix HE, the cross product notation i&, and the body-axis velocity vector, 3 are defined in Appendix A. kx is the position of the accelerometers with respect to the center of gravity in body axis.

Ax S AzB = Ay (129) S AZ S

[I

Perturbation expressions from individual componets in Eq. 127 are derived in Ref. 3 except for the third term which follows, (130) The perturbation expression for the accelerometers becomes (13.1) where LB is defined in Eq. 77. The variables A& and Ag are part of the perturbation aircraft dynamics, A% = A Ax + B AI-J (132) discussed in Appendix A.

Partitioning l&, A and B, it follows that AkB = AwAxB + AvuAg; + AmAxB + B Au (133) vu - A&; = AuvAxB -I- AwA$ + AwvAB + BwUAg (134) Substituting Eqs. 133 and 134 into Eq. 131 and combining terms results in the perturbation measurement equation for the body mounted accelerometers 135 is used as the measurement equation in constructing the PIF gain If Eq.

matrices then a sampled accelerometer output can be used directly as a feedback state in the control law without adding in gravity or correcting .for off center of gravity locations. There are certain advantages which can be obtained using off c-g. located accelerometers as discussed in Ref. 29. Although it was not feasible, the Y6, problem discussed in Chapter IV, Section C could have been alleviated by moving the lateral accelerometer forward along the body x-axis.

Corrections for gravity and off c.g. locations must be performed when the ac- celerometer outputs are used in complementary filtering as discussed in Chapter IV, Section A.

The pitot tube measurement of velocity magnitude, V, the sideslip vane measurement, B, and the angle of attach vane measurement, CX, are related to the body-axis velocities by (136) Perturbations of the above equation results in r1371 J, is a diagonal matrix which has elements 1.0, V,, and V. cos fi,.

Hz is the body to wind axis transformation and is the transpose of (138) Equation 137 can be used to form H and D in Eq. 127 when the air data sensor information is needed for feedback.

APPENDIX C AUTOPILOT MODE MODELS AND CONTROL SYSTEMS Each autopilot mode is composed of a dynamic model and a dynamic model When an autopilot mode is control system which are described in this appendix.

engaged, the internal states in the command model are initialized to the air- After initialization, the command model is propagated to generate craft states.

Some variables in the command state trajectories the PIF control law follows.

models (airspeed, range, pitch angle) are continually updated to match aircraft lateral position) are internal conditions. Other variables (yaw, roll, height, to the command model.

When the ALT SEL, HDG SEL, APR GS and APR LOC command models receive a a nonlinear feedback control system maneuvers the command model pilot command, to the new conditions. The feedback control system allows the autopilot command models to operate over a wide range of initial (boundary) conditions in real time. The nonlinear feedback control systems are designed solely to provide good ride quality. The simple autopilots BETA HOLD, ROLL SEL and PITCH SEL operate open loop. The PIF design process requires that linear representations of the command model be used, hence, both linear and nonlinear models, where applicable, are given in each of the following sections. The basic linear com- mand model representation is shown in Eqs. 6 and 7.

A- Pm, 4i-n COMMAND SYSTEM - BETA HOLD The sideslip, B,, roll angle, Qm, command system is used to compute the roll command to rudder command crossfeed gain used in ROLL SEL. The crossfeed is computed as an element in the feedforward matrix AZ2 in Eq. 9.

gain, akt The linear and nonlinear command models are identical for this command system, = 0 Sk+ 0 (139) %,k+l , k =osk+I 11401 G,k , k BETA HOLD is not intended to be an inflight command system.

COMMAND SYSTEM - ROLL SEL B* Qml 6, The roll angle, a,, rudder position, 6rm, command system performs the The linear and non- plus yaw damper autopilot feature.

standard wing leveler are identical for this command system, linear command models (1411 = 0 Sk %n,k+l , k [1421 is scheduled as a function of airspeed as shown in Eq.

The crossfeed gain, ak, The PIF autopilot with the ROLL SEL command model is designed to accept a 70.

roll command, $m, (usually 0.0) and a rudder command, 6,, (usually rudder pedal The autopilot will also capture and hold trim position) and hold these values.

a nonzero roll command input from the pilot.

COMMAND SYSTEM - HDG SEL

c. 4m + $,r drn

The Heading Select and Hold command system performs the bank-to-turn auto- pilot feature. When the autopilot is turned on, it holds the current aircraft When a new heading of the aircraft until a different heading is requested.

the autopilot banks the aircraft into a steady coordinated heading is requested, then establishes and holds the new heading.

turn, The bank to steady turn feature of the autopilot is accomplished using the basic nonlinear kinematics = $, k + At % tan Grn k [1431

4)

m,k+l , , vk 11441 + * f At @m k

G

m,k+l = m,k , The nonlinear relationship between $m and rjrn is derived in Ref. 34 and attempts to produce a coordinated turn where the areodynamic force lies in the body-axis x2 plane. The difference between actual and command heading is used to form a roll angle command I , The command error resembles a typical feature of a nested autopilot design where the heading error is used to compute a roll command in the ROLL SEL mode to pro- vide a heading select capability (Ref. 18).

The linear perturbation command used in the PIF design model is obtained The roll dynamics in Eq. 144, by linearizing Eq. 143 and regrouping Eq. 145.

are not represented in the linear model as discussed in Chapter 3, Section A, is used as the linear models con- because poor transient behavior occurs if orn The linear model is trol input, urn.

(g + v. VJ, tan @,I (146) "rn k+l = "m k + f , v.

(147) - kQ Ad', (148) Ayk = A$, + A6r k , .

uses $m,k as a model control variable The HDG SEL nonlinear command model -~ to cause qm,k to smoothly transfer from one pilot requesting heading command, system used to follow the $,, to another. The nonlinear command model control pilot command inputs is as follows: IF FIRST PASS THEN qrn o = $,I +m o = $,I 6m o = 6r k , I I I ;compute command model yaw error (149) 9, = 'c,k - 'm,k ;compute command model feedback (150) UC = HSKl*$E-HSK2*~m k I control command ;initialize roll rate command = UC (151) 'm,k ;set roll rate to zero if in IF ($m,kl>$max THEN $m,k=O.O steady turn ;reset roll rate command if IF b'E~~d'close THEN irn k=UC I approaching desired heading (1.52) = 0.0 +max2 ARE OPPOSITE SIGN IF '/', AND i, k THIN 4’max2 = i, k , I ;apply full rate command if (153) MAXIMUM (imax I (imax 'max = opposite sign occurs otherwise hold roll rate command below maxi IF 1orn k 1>$max THEN 6m k=SIGN (6, k) *bm, I I I = tan ;check to see if low velocity 0 -' 6;/RT*g) (154) maxl is decreasing turn radius ;keep roll angle below emax (155) @ = MINIMUM($maxlr $-nax2) max .

indicates it unless emax should be lower The control system feeds back the model yaw error and model roll angle using the gains HSKl and HSK2. If the model roll angle exceeds a specified value, the con- is reset to zero. During this phase the model dynamics are in a trol, @m,kr steady turn. If the roll rate control command causes the yaw angle to accelerate in the direction of the yaw angle command, qc, the roll rate control command is limited to the value emaxl. If the yaw angle error is below a threshold, @CLOSE, full yaw angle deceleration through the roll rate control command is used. The last feature causes the model and the aircraft to anticipate the roll back to zero just before completing the turn so that the yaw angle overshoot is small.

A minimum radius of turn, RT, is ysed'to reduce the maximum allowed back angle, if the filtered airspeed, V, drops to low values. The computed value of $yxr after Eq. 153, is used in Eq. 144 to generate the $m trajectory.

The re.&lting +m trajectory is used in Eq. 143 to generate the $m trajectory in flight.

D. zm COMMAND SYSTEM - ALT SEL The Altitude Select and Hold command system allows the pilot to hold the current altitude or select a new altitude the aircraft should establish.

When the pilot selects a new altitude, the autopilot slowly accelerates (or deceler- ates) the aircraft into a steady climb (or steady descent) until the new alti- tude is reached then captures and holds the aircraft at the new selected altitude.

The altitude select feature of the autopilot is accomplished using the basic command model [1561

[~~~~~] = [::Io::] [t::] +[“;:I ‘m,k

r1571 Xk - %,k = Zk - 'm,k is used as a control variable to transfer zm The altitude acceleration, zm,k, The linear command model used in from one desired altitude value to another.

and has the following simple form the PIF design does not use Em, (158) Azm k+l = Azm k + At i, k , , , (159) = Azk 'k (160) '&,k = "m,k The altitude select command model control system is as follows where zc is the requested altitude made by the pilot.

IF FIRST PASS THEN z = 2 - k' zm,O = o-0 m,o 161) ;compute command model height ZE = Zc,k - 'm,k error .

;compute command model feedback 162) UC = ASK~*ZR-ASK~*Z~,~ control command . .

;initialize acceleration command (163) Z = UC m,k THEN 2 m,k=O. 0 ;set acceleration to zero if in IF Jim,k/2i max steady descent (or ascent) THEN 6 ;reset acceleration command if =uc IF JZE"zCLOSE m,k approaching desired height ..

Z = 0.0 (164) max2 ARE OPPOSITE SIGN THEN';. =I; ) IF zE AND Em k max2 I m,k . .

. .

Z = MAXIMUM (i' z ;apply full acceleration if (165) I max2' maxl max opposite sign occurs otherwise hold acceleration command below . .

Zmaxl The control system feeds back the model height error and model vertical velocity using the gains ASK1 and ASK2. If the model vertical velocity exceeds a spe- cific value, the control, Zm,k, is reset to zero. During this phase the model If the vertical acceleration dyanmics are in a steady descent (or ascent).

command causes the model height to accelerate in the direction of the vertical height command, the vertical acceleration control command is limited to the . .

value Zmaxl. If the height error is below a threshold, zCLOSR, full deceler- ation through the vertical acceleration control command is used. The computed value of 'im,k after Eq. 165 is used in Eq. 156 to generate the Zm,k trajectory in flight. The altitude select and heading select model control systems are coincidently structurally identical.

grn COMMAND SYSTEM - PITCH SEL E.

The pitch angle command systems holds the current pitch angle or commands a new one using the simple model.

= 0.0 rr, k + 0.0 em k (166) %,k+l , , &,k = 0.0 s,k + 1.0 em,k (167) GLIDESLOPE CAPTURE AND HOLD - APR GS F.

The Approach Glideslope autopilot mode captures and tracks the glideslope.

The switch from ALT SEL to the APR GS mode occurs automatically when the air- craft is approaching the runway. The ALT SEL mode must be engaged and the air- craft must be holding a desired altitude prior to capturing the glideslope.

Glideslope capture can be performed either above or below the glideslope. The localizer capture and track should occur prior to glideslope capture.

These requirements are standard in most autopilots.

Glideslope capture and track is accomplished using the nonlinear model I I d d + At Vk * sin(XC5 + Xm k) (168) m,k+l = m,k , d^ (169) Yk - Y,,k = \ sin rk - dm,k = k - dm,k 1 I The distance, d,,k is the perpendicular distance from the aircraft to the glide- slope, XGS is the glideslope angle, X, k is the commanded aircraft flight path In the PIF control law, Eq. 168, is used to angle and V is the ground speed.

define a path that the measured value d tracks to zero. The values used for V, and &, do not have to be accurate in flight since inaccuracies only imply XGI In flight, a perturbation of the trajectory d, traces to zero. V is obtained is initialized by the pilot and A, is a control using measured airspeed, XGS feedforward variable. A measured aircraft flight path angle is not required for feedback. The linear perturbation command model used in the PIF design is obtained by linearizing Eq. 168, Adm k+l = Adm k + At Vk AXm k (170) , , , (171) = Adk (172) AYk The APR GS nonlinear command model uses Xm,k as a control variable to cause dm,k to smoothly transfer to zero from the measured value of d that occurs when the glideslope engage logic activates the APR GS command model controller.

The command model control system used to construct the path, which includes the range estimation equations, is as follows ;initialize APR GS flag DATA GSENGAE/FALSE/ ;initialize US to 0.1 DATA UC/O-l/ ;initial outer marker flag DATA OMKRCK/FALSE/ IF OMKRSOLID THEN OMKRCK=TRUE ;change flag when aircraft passes over outer marker IF NOT OMKRCK AND NOT GSENGAF OUTMARK ;initialize distance estimate THEN i$= (173) to outer marker distance ELSE 3EGIN ;initialize range estimate RK = 0.0 (174) filter gain to zero IF LOCENGE THEN fi, = ;jc-l-At*vk*cos($k-:k) ;propagate distance extimate (175) if 2 is a valid signal FWEAS = Z^k(t)/hGs ;estimate R using height (176) estimate and glideslope angle IF GSENGAE THEN RK=r *P +r2 ;increase RK as aircraft (177) lk descends on glideslope THEN RK=O.O IF RK < 0.0 IFRK>RKMAX THEN RK=RKMAX +RK* (E?MEAS- Isupdate distance estimate (178) i) END THEN IF ;I, < Gmin ;keep distance estimate above R min THEN IF IrkI < rmax BEGIN a sin(l?,) ;calculate d estimate when (1791 k=% glideslope is in range =a^ > 0.0 THEN d IF UC ;reset dm to d until the APR GS m,k k model command control system causes a pitch down command IF UC < 0.0 THEN GSENGAE=TRUE d = (0.0 - dm k) * 3.281 iconvert to feet (180) .E GSK = ;compute APR GS command model (181) i ldmk +g2 gl I control system gain IF GSK > g3 THEN GSK=g3 IF GSK < g4 THEN GSK=g4 UC = GSK * d * 0.01745 iconvert to radians (182) E UC = -A + UC (183) GS IF UC 2 ?, THEN UC=x should be 0.0 ;A max max max IF UC -< x THEN UC=Xmin ;keep flight path angle command min above x min x, = (UC - Am k-l) (184) I IF Ih,/ > x THEN i, = SIGN(Xm) * i ;keep flight path angle rate of (185) max change below Amax = x x ;propagate command model (186) m,k-1 + 'rn m,k = d d m k + At Gk sin(XGS + Xm k) (187) m,k I I :ND At the beginning of the control law computation, theglideslope angle is initial- ized by the pilot. The distance, R, is initialized to a typical outer marker distance. If the aircraft has intercepted or passed the outer marker and the the distance R is propagated. When the localizer deviation angle is in range, aircraft is descending on the glideslope an alternative estimate of R is made using the barometric altimeter 2 estimate and the geometry involved. The pro- pagated R and estimated R are combined in a complementary'filter fashion.

The filter gain, RK, is increased as the aircraft descends. When the estimated R falls below a prespecified value, R is fixed to the value.

Equation 180 forms the command model control error. When -GSK * dm,k is less than A,,, the flight path angle command, A, begins to command the aircraft to pitch down even when the aircraft is below the glideslope. The aircraft follows a d trajectory which intercepts the glideslope with no overshoot. The gain GSK is programmed to increase as the aircraft closes in to the glideslope. The computations after Eq. 183 check to determine if the commanded flight path angle is within spec- ified limits.

Setting hmax to zero ensures the command only executes pitch down glideslope captures. The rate of change of the flight path angle is also checked to keep the vertical accelerations small during capture.

LOCALISER CAPTURE AND TRACK - APR LOCI, APR LOCR, APR LOCP G.

The localizer capture and track autopilot mode is the most complicated command model design in the PIF control entourage. A number of options are available to specify a command model which captures and tracks the localizer beam centerline. Each option has advantages and disadvantages which are de- scribed in the flight test discussion, Chapter V. Three APR LOC command modes are presented (APR LOCI, APR LOCR, APR LOCP). PIF control law designs for all three modes are discussed in Chapters IV and V.

The APR LOC autopilot captures and tracks the inbound localizer beam centerline. All three approach localizer autopilot designs use the same basic command model. The autopilots differ in the construction of the command error signal which the PIF control law must regulate to zero.

The APR LOC mode requires that the aircraft be in HDG SEL mode prior to engage. The engage logic for transferring from HDG SEL to APR LOC is performed when the localizer signal is in range.

The internal intercept angle in the command model control system during capture is fixed. The APR LOC command system has considerable room for improvement as discussed in the recommendations section of Chapter 6.

Localizer capture and track is accomplished using the nonlinear dynamics shown in Eqs. 124 and 143 where, for simplicity, the reference value is assumed to be 0.0, + At Vk * sin $m k [1881 'm,k+l = 'm,k , + At F tan em k l4J [1891 m,k+l = %,k , k The three APR LOC modes have the following command errors APR LOCI (190) APR LOCR (192) -I The roll angle command, orn, is the control variable for the nonlinear command model control system.

The APR LOCI mode forms the command errors shown in Eq. 190 using roll angle and y position. The aircraft's y position is estimated from E measurements using Eq. 123.

The APR LOCR mode shown in Eq. 191 is simply the HDG SEL mode with an addi- tional modification. The qm,k command in Eq. 145 is replaced with ljm,k-ky(yk-xm,k).

An error in y position causes a $ angle error which in turn causes a coordinated roll command to bank the aircraft in the direction to null the y position error.

The APR LOCR mode is similiar in principle to a localizer autopilot mode dis- cussed in Ref. 18.

The APR LOCP mode is similiar to the APR LOCR mode except that the $ error is removed and only y position error commands the bank angle.

The $m state re- mains in the model dynamics and is used in the PIF control law to form the x-x* -- error (see Eq. 57) for $. The APR LOCP mode is able to perform a coordinated intercept the localizer beam centerline and track the beam centerline turn, crabbed, if necessary.

The linear perturbation command models used in the PIF designs are obtained by linearizing Eqs. 188 to 192, ,..,_.__........_, . ..- __..--.-. ..- -__- _.___ --- ..-..__..--.

__-- (1.0/57.3) cos qJ, * 1.0 At v.

1.0 1.0 0.0 0.0 Mm k , (193) A6 *t(g+VolClotado) .

rm,k

I[

V APR LOCI

0.0 0.0

(194) *%,k = 1.0 0.0

[ 1

(195) *Yk = k APR LOCR (196) -kdJkY *&n,k = -akk@ky

+ k+ky *yk

A@, - k+A’!‘, (197)

*, =

k k *Yk - ak 'I) *$, + ak $ y I A6r k , APR LOCP 'm (198)

[ 1

rmk - ky *Yk (199) *Yk = k - ak ky *Yk , The APR LOC autopilots use em as a control variable to cause ym to smoothly transfer to zero from the measured value of y that occurs when the localizer engage logic activates the APR LOC command model controller. The command model control system used to construct the path is as follows: DATA LOCENGE/FALSE/ ;initiaze APR LOC flag + RLOC) sin (Ek) ;estimate the y position using (200) 3, = the range estimate IF jEkj<: THEN LOCENGE=TRUE max IF NOT LOCENGE THEN ym k=$k ;reset the internal command model states I E is in range until 4 =ok m,k =; VJ m,k k 6 =6 rm,k rk THEN IF LOCENGE 3EGIN 1 LOC1=~l*Iym k*3.28081+12 ;compute APR LOC command model (201) I control system gain IF LOCl>13 THEN LOCl=e3 IF LOC1<14 THEN LOC1=14 LK1=LOCl(ck*3.28) ;try to keep closed-loop system (202 invariant by varying control gains with velocity LK2=LOC2/(g/ck) (203) ;compute qrn command $,=LK1*(O.O-ym k) (204) I THEN IF I~cI>~cm, should be less than the +c=sign WC) *+,,,, ;uJc cmax intercept angle dJ UC=LK2 * '$,-vJm ;compute -$m command (205) k) I THEN UC=sign(UC)*$max

IF Iucl>~max

;keep roll and roll rate below im=uc-+m k-1 (206) I maximum values THEN $m=sign(im)*Gmax IF IimI~imax ;the computed roll command used (207) 'm,k='m,k-l"rn in Eqs. 188 and 189 END The localizer model control system will begin to intercept the localizer beam centerline if the pilot has activated the approach mode (APR = TRUE) and a valid localizer signal is available. The pilot must set up an intercept angle.

The APR LOC mode performs best if the pilot orients the aircraft at a -60 deg intercept just before the outer marker. The command model values for ym, Qrn, and $+,, are continually updated to the corresponding measured aircraft's values as long as the APR LOC control system has not been activated. Equation 204 forms the command models heading command whose upper value is limited to the intercept angle set in the control system. As the aircraft nears the beam centerline the heading command gradually changes to the reference 0.0 deg runway heading. The measured heading is referenced to the runway heading. After the heading command is computed, the rest of the control system resembles the HDG SEL command model system discussed in Section C.

APPENDIX D TRIM, STEADY STATE AND THE STAR TRAJECTORY This appendix discusses trim for nonlinear systems, steady state for Steady state for linear linear systems and the relationship between the two.

systems is generalized to include the theory of feedforward control and model Feedforward control theory is used in the PIF derivation in following.

Chapter IV.

For a constant command,

= h(lfo, ~1 (208)

y-0

& = 0.0 (209) such that Eqs. 208 and 209 are an aircraft is in static trim if there is a u sh%n in Eq. 80, satisfy true and the nonlinear aircraft dynamics, 0 = f(x (210) --’ XJ The outer-loop states x, y, z, 9 are not included in Eq. 210. For the pertur- bation linear system, static trim or steady state is similiarily defined 0 = A AZ* -I- B AU* (211) - = H Ax* + D Au* = A (212) AY* - An Ai* = 0.0 (213) Grouping the linear equations, static trim for linear systems can be easily computed by solving the linear matrix equation (214) for Ax* and Au*. Ax* and Au* are defined to be the "star trajectories" of the piant forconst%t A%.- A solution to Eq. 214 exists if the plant has no transmission zeroes at zero, i.e., the quad partition matrix in Eq. 214 has full rank. Using the definition of perturbation systems, and the Taylor series expansion, different nonlinear trim points caused by perturbation represented as Ati, are related through the following equations, changes in yo, (215

Y. + A& = Y. + AY* = Y, new

, s + Ax* 2 x (216 - -0,new go + AU* g u (217) - -0,new Steady-state for discrete systems is similiarly derived.

The state trans- ition equation for a linear system is, Ref. 36, Ax(t) = eA(t-to)Az(to) + [ eAct-') BAG dT (218) - Starting from time t , Eq.

218 describes how Ax(t) evolves if AU(T) is known.

If AU(T) is held con&ant over the constant interval At (a zero-order hold implementation) then Eq. 218 can be represented at each time instant tk, where = tk + At, by tk+l Azk+l = @Ax+ + rA, (219) which is an exact representation.

The discrete PIF control law uses Eq.

219 as the discrete plant model.

In steady state, Eq. 219 can be combined with Eq. 212 to produce a result similiar to Eq. 214, (220) If the discrete plant has no transmission zeroes at 1.0 then As* and Ag* can be computed by c22i j [I:]=[:;; :;jLJ where The steady-state star trajectory computed for the discrete-time plant model is identical to the values computed using Eq. 214.

Tracking a constant command can be generalized to command model output tracking. The discrete-time plant representation for the command model shown in Eq. 7 and 8 is determined using Eq. 218, where As k is constant between I and has the following form sampling intervals, = @m k&,k + rm '%,k (223) '%I k+l , The solution for the control, Ak, which causes A& to track Ati,, for the discrete-time command model is accomplished by generalizing Eq. 220. The generalization primarily occurs by interpreting % and A$ as ideal traj- * ectories (the star trajectories) that occur when Ax = Ay in' To‘derive the equation for the star trajectories, the star trajector&es are assumed to be related to the discrete model through constant matriues (224) The matrices A.. in Eq. 224 are the feedforward matrices and remain to be determined. l' The derivation for a solvable expression for the feedforward matrices proceeds by forming two different expressions for then - A31 A:+1 r One expression is obtained using the plant dynamics, equating them.

(.225) The other expression is obtained using Eq. 224, 'x*k+l - 'x*k Allrm ci96 I AY*k Dm To proceed with the derivation, As,k+l - Agm,k in Eq. 226 is required to be zero, i.e., Agrn,k is assumed to be constant. Equating Eq. 225 and Eq.

226 results in the solvable feedforward matrix equation [2271 227 exists if the plant has A solution for the feedforward matrices in Eq.

no transmission zeroes at 1.0 and no transmission zero equal to an eigenvalue A numerical solution to Eq. 227 is given in Ref. 5 (the subroutine of Qrn' CGTPIF). The solution is unique if the number of plant controls and plant outputs are equal.

Assuming asalutionwith unknown coefficients then determining the values of the unknown coefficients by substituting the solution back into the equa- as is done for the feedforward control solution, is a technique also tions, used in differential equations to determine the homogeneous and particular solutions. If the forcing function to a differential equation can be repre- sented as a model, Eq. 223, with no control input, than the feedforward control solution in Eq. 229 can be shown to be the particular solution for that forcing function. Assuming the proper form for the solution has been simple for only the single-rate discrete-time case. The feedforward control solution for the multi-rate case has yet to be determined.

The feedforward control matrices can be used to relate incremental changes in nonlinear trim to incremental changes in the command model similiar to results in Eqs. 215 to 217. For a constant command model input, the relationship iS (228) %,k-+,k-1 = All@mo k-&o k-l) + A12@rao,k-%o,k-l) , , (229) = A21(&, k-s, k-l) + A22@rno,k-%o,k-l) %,k-%,k-1 , , the same purpose as small changes in & did in Small changes in -.I serves Eq. 215 and the PIF derivation in Ref. 3, (230) %lO 7no,k + *%I k '$EW=U 9 + *%-I-I k C23l)l %I0 -mo,k

,$(EW =x

*

The PIF control law is designed in this report to accommodate small changes in Au -In'

APPENDIX E CONTROLDATA HOLDS, COMPUTATIONDELAY AND TRIM ACCOMMODATION This appendix shows how different control data holds and modeled control law computation delay affect discrete-time plant model representations. Trim The concepts in this appendix aid in accommodation procedures are reviewed.

understanding the PIF derivation in Chapter IV and command model construction in Appendix C.

A. CONTROLDATA HOLD A control data hold is a procedure for constructing in continuous-time A zero-order hold, for example the control that commands the actuator surface.

has the following continuous-time representation for tk+l > t > tk

Au(t) = Au+ (232)

A triangular data hold has the following representation for tk+l > t 2 t, (233)

Ax(t) = A~I+ + (t-t,) (hk+,-A&)/At

Similiar expressionB can be determined for the first-order hold, slewer data If the intermediate variable . hold and others.

(234) is defined, then the triangular data hold becomes (235)

Au(t) = AE~ + (t-tk)Axk

(2361 = Afik + At Axk A2k+l The discrete plant representation at the sample points for the triangular data hold is obtained by substituting Eq. 235 into Eq. 218, --.-_- --_-. ~___~ -7.-- ;.,,, ., ---.---F-‘T; ‘:r.’ _._-,- -7- ._. ;.. 7,--- =., .-. ._--_.-- _.___ -.. .---A... .-- -- --. ----~- -1. __ -. . A _ -L- --A-- (237) &(t,,> = eFAt &(t,> +LAteFT d-rGAgk + eFAtLAtTemFT d'rGAxk The equation can be rewritten as [2381 where r = /At-a eFs ds G (239) = Jo eFs ds G lo r2 0 Equation 238 occurs exactly as shown as a partition when the sampled data regulator PIF discrete plant model is formed. The result is altered to obtain the desired autopilot zero-order hold mechanization.

COMPUTATIONDELAY B.

Digital control laws do not necessarily output the control command to the actuators at the sample points. There is a delay caused by computation and other factors that cause the control command to be released to actuator channels between sample points. The control command with computation delay can be represented in continuous-time as tk+l 't'tk+o (240) tk+o > t 2 t k where cf is the computational delay. Substituting Eq. 240 into 218 and simplifying produces (241)

+ rl ‘gk + r2 ‘x-1

‘?$+I = 0%

where At /T eFs rl = (I iAt-ceBFTdT G = L dsdT G (242) A discrete contra law can accommodate the computation delay, (that is the design evaluation and simulation exactly corresponds to the implementation response), by using Eq. 241 as the discrete model representation of the plant.

the PIF autopilot construction Rather than guess at (5 to perform a design, simply assumes the control is released to the actuators delayed one full cycle.

Equation 241 with a full cycle delay can be represented as follows 12431 where Au+ is the control applied at time tk+l. Equation 243 (with the uk index moved up one sample, the implementation is identical) occurs when the discrete PIF plant model is simplified.

C. TRIM ACCOMMODATION In typical autopilot operation, the autopilot feeds back an error signal to the actuators to drive the error signal to zero. The error signal is form- ed by subtracting the sensed signal from a command or trim value. Four popular ways to generate trim information in flight are 1) prestore.trim values with flight condition, 2) load the measured aircraft states at control initiation as the trim values, 3) estimate the trim value of a signal by low-pass filter- ing the signal (the error signal is constructed using a "wash-out" filter) and 4) estimate trim using integral control. The PIF control law uses the second, third and fourth mechanizations. The second mechanization is part of the control and command model initialization procedure. The third mechaniia- tion is part of the integral control implementation where the rudder command error is integrated and fed back. The following simple example demonstrates the effect of control error integral feedback.

Consider a scalar plant .

x = ax + bu A control law using integral control is formed which feeds back the integral of control error from some command value u=u c + kl x + k2 / (u - uc) .

(245) An implementable form for Eq.

245 is obtained by taking the derivative and simplifying.

If the variable -._.

-; ___...__-_ -___.-._-^---- ~.

.- .~.-.-- z=u-u (246) C is substituted into Eq. 245, the Laplace transform of the control law becomes u=z+u (247) C X (248) Equation 248 is recognized as a wash-out filter.

Digitally integrating the rudder control command error in a manner similiar to Eq. 245 is used in most of the lateral-directional PIF autopilots. The effect is similiar to a yaw damper except that all the lateral states fed back are washed out simulta- The value of uc in Eq.

neously. 247 for the PIF autopilots is the control position at engage.

_ _-_ .- -_--- __-_.- __- - _-__- __ ~._----__- ~--~----- .

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---- ------‘--~-.-.~~-.-- -. ._- -, --.,_- _..- -- ~~ _---.

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LIST OF SYMBOLS In general, matrices are represented by capital letters and vectors are underscored; exceptions to these rules are only made when they are contra- dicted by standard aerodynamic notation.

DESCRIPTION VARIABLE Discrete time feedforward matrix A Fundamental matrix (continuous-time system) Wing reference area ASK Gain in ALT SEL command model control system a Acceleration Roll to rudder crossfeed gain Component of the earth-relative acceleration in the an aircraft x-z plane normal to the velocity vector Component of the earth-relative acceleration normal to the aircraft x-z plane B Control effect matrix (.continuous-time system) Wing span b Bias estimate Control law feedback gains C Partial derivative of the nondimensional coefficient of cl2 force or moment 1 with respect to the nondimensional variable 2 (scalar) C Mean aerodynamic chord D Control observation matrix Drag d Perpendicular position from the glideslope to the air- craft F Aerodynamic contact force vector - f Vector-valued nonlinear function - GSK Gain in APR GS command model control system DESCRIPTION VARIABLE Magnitude of gravitational acceleration vector g Scalar gain value Command observation matrix H HSK Gain in HDG SEL command model control system Euler angle transformation form Frame 1 axes to Frame 2 axes Vector-valued nonlinear observation function

!L

Barometric altimeter output hB I Identity matrix Moment of inertia i Index integer Cost functional matrix J j Gain value K Index integer k LK Gain in APR LOC command model control system LOC Gain in APR LOC command model control system L Aerodynamic moment about the x-axis (scalar) Scalar gain value R Number of commands M Aerodynamic moment about the y-axis (scalar) Cross weighting matrix between states and controls m Mass of the vehicle Number of controls Meters Aerodynamic moment about the z-axis (scalar) N Newtons (kg-m/secm2) n Number of states DESCRIPTION VARIABLE Riccati matrix in the optimal regulator problem P Rotational rate about the body x-axis P State weighting matrix Q Integrator state weighting matrix QZ Rotational rate about the body y-axis q Free stream dynamic pressure (= $3Vz) Control weighting matrix R RD Control rate weighting matrix Range position from localizer transmitter to touchdown RLOC point Estimate value of range using vertical position and RMEAS glideslope angle r Rotational rate about the body z-axis Feedforward matrix for sensor measurements S S Laplace transform variable Thrust T t Time Command model control variable UC U Body x-axis velocity component U Control vector - V Velocity magnitude V Body y-axis velocity component Control difference Body z-axis velocity component Aerodynamic force along the x-axis (scalar) Position along the x-axis DESCRIPTION VARIABLE State vector X - Y Aerodynamic force along the y-axis (scalar) position along the y-axis Y Aerodynamic force along the z-axis (scalar) Position along the z-axis z VARIABLE DESCRIPTION (GREEK) Wind-body pitch Euler Angle (angle of attack) a Negative of wind-body yaw Euler angle (sideslip angle) l3 Discrete time control effect matrix

r

Glideslope deviation angle Y Inertial-velocity axis pitch Euler angle (flight-path) Aileron deflection Elevator deflection Rudder deflection Damping ratio 8 Inertial-body pitch Euler angle

x Eigenvalue

V Euler angle position vector - -2 Localizer deviation angle Eigenvector Integrator state Air density Summation .d Real part of an eigenvalue in radians/set Time delay VARIABLE (GREEK) DESCRIPTION Time constant cp Discrete-time system matrix Inertial-body axis roll Euler angle Inertial-body axis yaw Euler angle Matrix in quad partition matrix inverse w Frequency in radians/set Imaginary part of an eigenvalue Body angular rate vector DESCRIPTION SUBSCRIPTS a Aileron B Body axis b Bias C Command value E Earth-relative axis e Error quantity GS Glideslope H Horizontal i Element index for vectors and matrices Element index for vectors and matrices j Sampling instant index Aerodynamic moment about the x-axis Aerodynamic moment about the y-axis model variable n Aerodynamic moment about the z-axis 0 Nominal value SUBSCRIPTS DESCRIPTION Static pressure S Accelerometer offset from the cg Wind Horizontal perpendicular to y and z Horizontal perpendicular to x and z Vertical perpendicular to x and y Aerodynamic force along the x-axis Aerodynamic force along the y-axis Aerodynamic force along the z-axis SUPERSCRIPTS DESCRIPTION E Earth (inertial) axis T Transpose of matrix W Wind axis -1 Inverse of matrix * Steady state Trim PUNCTUATION DESCRIPTION Derivative of quantity with respect to time ('1 Vector quantity

(-1

Partial derivative of one variable with respect to a( )/a( I another Perturbation variable ( 1” Star trajectory (3 Estimated quantity Discrete cost function weighting matrix Infinity DESCRIPTION PUNCTUATION / Integral [a 1 Important Equation Equation P 1 Matrix equivalent to vector cross product, (‘-1 Specifically, _ if x is the three-dimensional vector

--z Y

x= then 2 = z 0 -x

- -yx 0

[ 1

and the cross product of x and f is equal to the product of the matrix % and the-vector f, xxf=%f - ACRONYM CORRESPONDING PHRASE ARA Avionics Research Aircraft Center of gravity cg DME Distance Measuring Equipment GA General Aviation GATOR General Aviation Terminal Area Operation Research ILS Instrument Landing System kt Knot MAX Maximum MLS ticrowave Landing System NASA National Aeronautics and Space Administration PI Proportional-Integral PIF Proportional-Integral-Filter rad Radians Seconds ACRONYM CORRESPONDING PHRASE VALT VTOL Approach and Landing Technology VTOL Vertical Takeoff and Landing TABLE 1 NAVION ACTUATOR CHARACTERISTICS CONTROL DISPLACEMENT RATE LIMIT, TIME CONSTANT, MAXIMUM SPECIFIC LIMIT, deg deglsec MOMENT (IAS=53.8m/s T, (-cl (105kt)) _ AILERON 70.0 0.17 9.2 rad/sec2 roll acceleration ELEVATOR 70.0 0.17 9.9 rad/sec2 pitch acceleration RUDDER 70.0 0.17 4.2 rad/sec2 TABLE 2 SENSORNOISE AND BIAS CHARACTERISTICS FOR THE NAVION RESEARCHAIRCRAFT SENSOR NOISE STANDARDDEVIATION BIAS STANDARDDEVIATION - Lateral Accelerometer 0.6 m/s2 0.3 m/s2 Vertical Accelerometer 0.6 m/s2 0.6 m/s Rate Gyro 0.2 deg/sec 0.2 deg/sec Attitude Gyro 0.1 deg 0.2 deg Heading Gyro 0.1 deg 0.5 deg Indicated Airspeed 0.15 m/s (0.5 fps) 0.05 m/s (0.15 fps) Barometric Altimeter 3.0 m (10 ft) 15.0 m (50 ft) Localizer/Glideslope Angular Deviation 0.02 deg 0.012 deg Outer Marker --- Indication 500.0 m (1640 ft) L TABLE 3 ROLM 1666 CHARACTERISTICS DESCRIPTION - General purpose 16 - bit minicomputer designed to MIL-E-5400 specifications - 65,536 words of 1 Vsec ferrite core MEMORY register to register EXECUTION TIME - (Time in I.lsec, operations) FLOATING POINT INSTRUCTION (32 bit) Add 1.0 1.8 - 4.8 Multiply 5.2 - 5.4 3.6 - 4.8 Divide 9.2 - 9.6 8.0 - 8.8 Load, Store 2.0 4.8 -- _ - _.- _.I.

TABLE 4 SAMPLE SUMMARY OF COMMERCIALLYAVAILABLE AUTOPILOTS JrI2l-l 0 0 3 0 0 0 0 0 l 0 0 @ l 0 0 0 SPZ 500 EDO-AIRE 0 0 0 MITCHEL 3 l l 0 l l 0 0 0 l CENTURY41 KING 0 l l 3aeao~o~e~@o~~ KFC 300 COLLINS 0 0 0 0 l 0 0 0 0 3o@eooo APS-80 0 l

I Em 131+1+l+l lW@I Pl*\~l~l@l

l l l 0 0 l 0 0 l TABLE 5 STATES AND CONTROLSFOR AUTOPILOT MODE DESIGNS BETA HOLD MODEL m I

= mm Gml

%n T

= mm $,I

%I

z= [13

m 'ml XT = LB

- = P $1

UT

= va srl

-

yT =

[B $1

ROLL SEL MODEL T

= mm 6,l

%ll

T

= Mm “ml

%tl

(a

orn+ 6m>l

k XT = [a.y r

P $1

- UT = wa 8= - = [G 6=1 YT HDG SEL NONLINEAR MODEL T

1 = rim 6*

%l T

= urn Gml

%I

2 = [ QJ--Icm)

(a,($-$m) + 6m) 1

xT = [a, = PG $1 -

UT = wa 6=1

-

yT =

I@ srl

If.

TABLE 5 (CONTINUED) STATES ANJl CONTROLSFOR AUTOPILOT MODE DESIGNS HDG SEL LINEAR MODEL T Au

m = Mm A6&

T Ax

m = DNJ~I

AC = [-AI/J -ak Qrn + A6ml m AxT = [Aay

A= AP WI

-

T Au = [A&,

A6=1

-

= [AC&AI)

-ak A+ + 6=1

AYT ALT SEL MODEL

= Ii’,]

%l

T =

[z iml

%I

ym = r.11

XT = [V an q 8 z]

- U

= WeI

= [zl

Y LINEAR MODEL Au

= W,l

Ax

= [Az,l

Ar, = b,l

kT=[AV Aa Ag A0 AZ] n Au

= MeI

= [AZ] AY TABLE 5 (CONTINUED) STATES AND CONTROLSFOR AUTOPILOT MODE DESIGNS -____----...----. -_ PITCH SEL u

= Iem1

m X = remi m = fern1 'rn XT = Iv an q ei - U = vel Y = vi ARP GS NONLINEAR MODEL U

= Pm1

m X = [d,l m

= Ed,1

ym XT = [V an q 8 d] - U

= vel

= [dl

Y LINEAR MODEL Au

= Wml

m Axm = [Adm]

AY = Mm1

m T Ax = [AV Aan Aq A@ Ad] - Au

= WeI

= [Ad] AY -__-?___-- TABLE 5 (CONTINUED) STATES m CONTROLS FOR AUTOPILOT MODEDESIGNS APR LOC NONLINEARMODEL T

= NJ, “ml

%n T

= [Ym vJrnl

%I xT = [a = P @ $ ~1 - Y T U

= Ma arl

-

APR LOCI

2 = [@

m 'ml = I$ Yl I APR LOCR 2 = [k,+$-ky(y-y 1) $,J m

= [(a 6=1

YT APR LOCP

IrlT,= [ky(w,) $J

yT =

[@ 6=1 LINEAR MODEL APR LOC AuT

- = Wm A6J

T Ax

- = MY, Wml

T Ax = [Aay A= AP A@ A9 AYI - T Au = [A6, AcYr] - APR LOCI AZ = Mm Ayrnl AyT = [A@ AYI APR LOCR AZ = a k k Ay + A6,n] 1kqkyAYm k$y m AyT = DW-k&W + kqkyAy -akkQA@+akk$kyAy + A6,l APR LOCP AZ = [-kyAym -a k Ay + A6m] ky m AyT = [A@+ AY -Y~~-ilkky?~f.t6~1 TABLE 6 AUTOPILOT MODEI: DESIGN PARAMETERS ___ ---- --_.- -- _ AUTOPILOT EQUATION PARAMETERS VALUE UNITS _- -~_-- .-. . . - _-.. -~.~- --- HDG SEL 150 1.0 --- 2.0 155 0.262 radians --- 0.157 radians 153 0.0873 radians --- 10.0 154 914.4 m --- ALT SEL 162 ASK1 0.8 --- ASK 2.0 --- z 2 6.0 Eclose m2 0.2286 m/s .maxl Z 167 2.53 m/s max2 a 0.24 radians max APR GS OUTMARIZ 9879.0 m --- RMIN 100.0 m --- I? 0.0436 radians max 182 -0.0004 ft-1 --- 0.07 --- --- 0.07 --- --- 0.04 184 0.0523 radians --- 0.0 radians max --- x -0.08726 radians -min --- A 0.000524 radians max --- -0.00037 m-l =1 --- --- 0.112776 --- --- &AX 0.08 APR LOC 200 RLOC 2205.0 m s m-m 0.0122 radians -max 201 -0.0002 ft-1 % --- 201 0.2 R2 --- --- 0.2 &3 --- --- 0.11 R4 9.81 g m/s --- 203 1.2 B-w 0.523 radians --a 0.175 radians --- 0.00873 radians --- APR LOCR 3.0 --- i91 0.017 Y --- APR LOCP k 192 0.04 Y TABLE 7 SQUAREROOT OF Q AND R DIAGONAL WEIGHTS IN THE PIF COST FUNCTIONS AUTOPILOT MODE STATE* OR ALT SEL ALT SEL PITCH SEL APR GS CONTROL (DESIGN) (LOWER 8 WEIGHT) A6e 0.0 0.0 0.0 0.0 A6a --- --- --- --- '% --- --- --- --- Au 0.0 0.0 0.0 0.0 Aw 0.0 0.0 0.0 0.0 Aq 0.0 0.0 0.0 0.0 AB 11.0 5.0 8.0 12.5 AZ 0.5 0.5 0.0 0.0 B-w Ad --- --- 0.5 Av --- --- --- --- Ar --- --- --- --- AP --- --- --- --- B-w A@ --- --- --- 4J --- m-w --- --- AY --- w-w --- --- ASlong 0.25 0.25 3.0 0.25 Atlatl --- --- --- --- 'slat2 --- m-w --- --- Ade 7.0 7.0 7.0 7.0 A$a --- --- --- --- A6r --- --- --- --- I (Ai or Ai> 1.0 1.0 0.0 1.0 * units are degrees and meters TABLE 7 (CONTINUED) SQUAREROOT OF Q AND R DIAGONAL WEIGHTS IN THE PIF COST FUNCTIONS _ -- --~-- AUTOPILOT MODE

-I

STATE* APR LOCI OR A.PR LOCR APR LOCP -m_.CoJTgoL -a- -mm A6 A6e 0.1 0.1 0.1 Aba 0.1 0.1 0.1 --- Au= -w- Aw -em Aq --- ae --- AZ --- Ad Av 0.0 0.0 0.0 Ar 11.0 11.0 11.0 0.0 0.0 0.0 AP 10.0 10.0 10.0 A@ 14.0 11.0 14.0 WJ 0.0 0.0 0.0 AY --- 2.5 1.0 1.0 0.15 2.2 2.2 --a -a- 7.0 7.0 7.0 7.0 7.0 7.0 * units are degrees and meters TABLE 7 (CONTINUED) SQUAREROOT OF Q AND R DIAGONAL WEIGHTS IN THE PIF COST FUNCTIONS AUTOPILOT MODE STATE* OR ROLL HOLD HDG SEL CONTROL A6 --- --- A6e 3.5 0.1 A6a 1.8 0.1 Al? --- --- mm- --- Aw Aq --- --- A8 --- --- AZ --- --- mm- --- Ad Av 0.0 0.0 Ar 8.0 11.0 AP 0.0 0.0 &J 6.0 10.0 4J --- 10.0 &ng --- --- --- --- 3.0 3.0 JAY

JAY latl 2.5 2.5

A61at2 --- --- A&" 3.5 7.0 AAa 4.0 7.0 -em --- Air * units are degrees and meters TABLE 8 OPEN- AND CLOSED-LOOP s-DOMAIN EIGENVALUES OPEN LOOP ALT SEL (DESIGN) ALT SEL (LdWJXR 0 WEIGHT DYNAMIC

un9 5, T, w*s 5, f, 5. =*

MODE wn'

rad/sec - set rad/sec - set rad/sec - set SHORTPERIOD 3.0 0.78 --- 3.07 0.69 --- 3.07 0.72 --- --- -- -- -- mm- -- PWGOID 0.25 0.025 --- -- -- -* --- - -- m-s e-w d- ii -8 1.58 0.99 --- 1.12 0.81 --- 0.36 0.80 --- &z 0.25 0.77 --- v --- --- 24.2 -- B-B 24.3 PITCH SEL APR GS DYNAMIC MODE 5. 'c, %' 5, 'c, uns rad/sec - set cad/set - set SHORTPERIOD 3.04 0.74 --- 3.21 0.67 --- --- --- --- --w --- PWGOID --- --- 5 -0 1.35 0.91 -- --- w-e w-e &d 0.24 0.77 --- --- --- 8.3 --- --- 24.2 --- --- -we :s --- --.- 2.7 OPEN LOOP ROLL HOLD HDG SEL DYNAMIC MODE un, . 5, T, 5. f,-- 5, 'c¶ wns wns cad/set - set cad/see - set rad/sec - set --a -em -me D-m B-w -em XJTCHROLL 2.07 0,22 --- --- --- --- e-w <OLL --e -em 0.16 0.16 0.16 -a- --- D-m -a- --- --a SPIRAL --- --- -31.0 mm- -me ,a --- --- -se -A- --- --- WING )UTCk ROLL-Gr 3.06 0.53 --- 2.84 0.46 --- rcs -v 0.55 0.74 z --- 0.24 0.79 --- 1.83 0.72 --- 1.77 0.67 --- $A '$-J, or /(I --- --- 2.0 1.42 0.82 --- APR LOCI APR LOCR APR LOCP DYNAMIC MODE 5, T, 5, f, 5, ‘c.

w*s wn’ %’ eadjsec - set rad/.sec - set rad/sec - set NJTCHROLL-&r 2.84 0.48 --- 2.9 0.47 --- 2.9 0.47 --- LOLL --s --- 0.16 --- --- 0.16 -- e-s 0.16 ry-v 0.29 0.58 --- 0.36 0.56 --- 0.35 0.61 --- 1.72 0.69 --- 1.73 0.69 --- 1.77 0.69 --- ii-4 I --- -- 1.05 --- a-- 1.07 (0:60 0.99)s ?

--- --- 2.00 m-w, mm- 3.08 m-m -me D-e m-w 6.04 m-m --- 5.9 5.6 e mode split into two real roots.

t 3 0 The roots combined to form a complex pair.

TABLE 9 ELEVATOR CONTROLGAINS FOR THE LONGITUDINAL AUTOPILOTS I AUTOPILOT MODES STATE* ALT SEL ALT SEL PITCH SEL APR GS (DESIGN) (LOWER 8 WEIGHT) 0.719 0.79 0.753 0.704 ve --- -a- --- --- V a --- --- -em --- vr -0.027 -0.025 -0.018 -0.00928 2 -0.021 -0.015 -0.011 -0.022 -0.428 -0.28 -0.294 -0.466 qz 8 -2.2 -1.48 -1.42 -2.34 Z 0.0 0.0139 0.012 0.0 zy --- --- --- --- --- --- --- --- F? --- -e- --- --- @ --- -em --- --- ti --- --- --- ---

--- --- ---

h -0.0144

--- --- --- --- Y 0.000176 0.000183 -0.038 -0.000175 long --- --- --- --- Jatl --- --- --- --- '1at2 u -0.0491 -0.029 1.08 2.49 mlong --- --- --- --- Umlatl --- -em --- --- Umlat2 I * units are radians and meters TABLE 10 CONTROLGAINS FOR THE ROLL SEL AND HDG SEL AUTOPILOTS AUTOPILOT MODES AILERON RUDDER STATE* ROLL SEL HDG SEL ROLL SEL. HDG SEL --- e-w --- B-w V e 0.021 0.061 V 0.65 0.69 a V 0.020 0.063 0.76 0.715 r D-w -a- --- D-w -aa --- -a- B-w VaT Z : -Me --- --- --- --- B-m --a -mm B-m --- mm- --- Z zy -0.053 -0.126 0.012 0.25 -0.74 0.178 -2.0 0.537 -0.26 -1.51 -0.53 0.011 -0.44 0.087 ; -0.31 -2.16 VJ B-m 0.50 Be- -5.9 s-m --- --- --- d em- Y --- --- --- -w- -a- --- 'long -~-~68 . -0.014 -0.022 0.032 'latl 'lat2 0.019 0.028 -0.512 0.013 --- --- Umlong --- --- 1.15 0.67 1.28 Umlatl 1.5 Umlat2 -0.35 -1.19 0.81 -0.59 * units are radians and meters TABLE 11 AILERON CONTROLGAINS FOR THE APPROACHAUTOPILOTS AUTOPILOT MODE STATE* AJ?R LOCI APR LOCP APR LOCR V --- --- --- e v 0.69 0.69 0.69 .a V 0.032 0.048 0.074 Vr --- --- --- aT --- --- --- Z e" --- --- --- --- --- --- Z --- --- --- zy -0.26 0.18 -0.356 0.278 -0.38 0.39 ; -0.25 -1.60 -0.245 -1.55 -0.25 -1.54 + -2.15 -2.95 -4.19 d --- --- --- Y -0.019 -0.014 -0.026 --- --- long -i-i21 . 0.000798 0.0075 ; 5 latl -0.00026 0.0269 0.0012 lat2 Yang I-i7 --- --- Ulatl 1.74 -1.25 Umlat2 0:o -1.53 2.05 9~ units are radians and meters - TABLE 12 RUDDER CONTROLGAINS FOR THE APPROACHAUTOPILOTS AUTOPILOT MODE STATE* APR LOCI APR LOCR APR LOCP --- --- V e V 0.000277 -0.000674 -0.00068 a V 0.827 0.821 0.824 r --- --- vT --- --- --- a Z --- --- --- --- --- --- --- --- Z --- 0.73 0.706 0.67 aY r -1.22 -1.24 -1.18 -0.027 -0.021 -0.027 -0.61 -0.66 -0.65 -5.54 -5.54 -5.45 --- --- --- -0.0224 -0.031 -0.028 Y --- --- 'long 0.0227 0.0123 0.00967 'latl -0.000237 -0.00171 -0.000947 Ylat2 --- _-- --- U mlong 1.41 1.59 1.16 Umlatl 0.0 0.310 1.72 Umlat2 * units are radians and meters - TABLE 13 Sll MATRIX ELEMENTS FOR THE PIF AUTOPILOT MODES APR GS ROLL SEL

[jj Lo !I

.

HDG SEL APR LOCI APR LOCR APR LOCP 0 0

0 00

0 0 0 00

0 0 00

0 0 0 1.0 0 1.0

[I

1.0 0.31

[ ~[,8, 1x0 1

L 1.0 0 0 0 0 0 1.0 0 0 0 0 0 1

TABLE 14 NAVION AIRPLANE PARAMETERS I 7 Wing area, A 17.112 m2 (184 ft2) Mean Aerodynamic Chord, c 1.74 m (5.7 ft) Wing Span, b 10.17 m (33.38 ft> Gross Mass, WT. - 1540.6 kg (3400 lb) - 1742.33 kg-m2 (1284.08 slug-ft2) IX Iy - 3762.4 kg-m2 (2772.86 slug-ft2) - 4389.1 kg-m2 (3234.72 slug-ft2) Iz 0.0 Ixz Control Surface Deflection, deg Flaps 0 to -43 Elevator -19 - to 29 Aileron - -18 to 19 Rudder - -20 to 25 TABLE 15 FLIGHT CONDITIONS FOR AERODYNAMICDATA --.- _ __ _ -_I _=_.

Ref. 31, Ref. 31, Ref. 38, Condition I Condition II --- Altitude 1524.0 m 1524.0 m 1980.0 m (5000 ft) (5000 ft) (6500 ft) Velocity 73.2 m/set 44.0 m/set 54.0 m/set (true airspeed) (2400 ft/sec) (144.0 ft/sec) (176.0 ft/sec) Flaps 0 deg 20 deg 10 deg TABLE 16 LONGITUDINAL AERODYNAMIC PARAMETERS Parameter Condition I, Condition II, Ref. 31 Ref. 31 Ref. 38 Design Values Analog Match % 0.262 -1.37 --- %l -4.33 -4.86 -4.6 C=I -15.9 -27.13 0.0 %s, 0.511 0.52 0.29 %i -0.77 -0.84 0.0 Cm -6.5 -6.0 -0.64 % -18.1 -16.4 -14.4 %s, 1.42 1.55 0.96 0 0.0055 rad 0.105 rad 0.036 rad q0 0.0 rad 0.0 rad 0.0 rad w 0 9.4 mjsec(l.3 ft/sec) 4.6 mfsec(15.2 ft/sec) 1.1 m/see (3.7 ft/sec u 0 73 m/sec(240.0 ft/sec) 44 m/sec(144.1 ftjsec) 55 m/set (185 ft/sec) aO 0.0055 rad 0.105 rad 0.02 rad C 0.0015 0.0015 --- x0 C -me 20 -0.27 -0.75 %o 0.0 0.0 -- %T 0.0061 0.0061 -- 9 9.8 m/se=' 9.8 m/sect 9.8 m/se=' .O.00912 kg/m3 0.00912 kg/m3 P 0.00912 kg/m3 (0.00205 slugs/ft3) (0.00205 slugs/ft3) (0.00205 slugs/ft3) TABLE 17 LATERAL AERODYNAMICPARAMETERS Parameter Condition I, Condition II, Ref. 31 Ref. 31 Ref. 38 Design Values Analog Match 0.0 0.0 0.0 FP 0.0 0.0 0.0 Er -0.6 -0.74 -0.92 SyB -0-33 -0.143 -0.21 q&r -0.026 -0.023 -0.025 Y6, -0.07 -0.053 -0.086 CM C -0.49 -0.53 -0.52 LP C 0.11 0.114 0.093 Lr 0.154 0.16 0.16 cL6a 0.0 0.0 -0.025 'LSr 0.073 0.080 0.103 %3 C -0.04 -0.147 -0.049 NP -0.09 -0.12 -0.11 CNr -0.004 -0.0015 -0.0039 cNSa 0.063 0.075 0.11 cN6r V 0.0 m/set 0.0 m/see 0.0 m/set r 0.0 rad 0.0 rad 0.0 rad 0.0 rad 0.0 rad 0.0 rad PO 0.0 rad 0.0 rad 0.0 rad 0.0 rad 0.0 rad 0.0 rad $0 0.0 0.0 0.0 0.0 0.0 0.0 cLo C 0.0 0.0 0.0 No * different .from Ref.. 31 value, FIGURE 1 WALLOPS FLIGHT TEST FACILITIES t PILOTPUSH COMMAND COMMAND MODEL OUTPUT I CONTROL FILTERING b , T AIRCRAFT & . CONTROL >.ACTLJATOR ' SENSOR - DYNAMICS I '\ ' /\ SAMPLE& FEEDBACK STATE FILTERING L OUTPUT FIGURE 2 BASIC PIF CONTROLLAW 1.96 - f T 4% - z A- -

-065 -

&

-i .3s;

0 5 TIME17 15

SEC)

9.00 -

c2

%

3.33 -

Gi

E

- -2.33 -

-8.00 _

TIME (SEC1

15 20

0 5

TIMEI: SEC)

FIGURE 3 COMPARISON OF OPEN-LOOP LINEAR MODEL RESPONSE VITH ATRCRAFT RESPONSE FOR A COMMANDIN RUDDER .

8--

,

3-

-2 1

TIME?

SEC)

FIGURE 3 (CONTINUED) Comparison of Open-Loop Linear Model Response with Aircraft Response for a Command in Rudder

v----- B

---

v .Y

11.0

16.5

22.0

TIME [ SEC1

3.92

r

FIGURE 4 COMPARISON OF OPEN-LOOP LINEAR MODEL RESPONSE WITH AIRCRAFT RESPONSE FOR A COMMAND -IN ELEVATOR ARA CONFiG’JRATION -----.-------- AIRCRAFTSYSTEM : 1 ,p&la-- SAFETY PliOT I MOTION

I

J

-I CONTROLSYSTEMS :

I

I

I

I I PiLOl

I J

EXPEF~IMENTAL c ISPlAYS I I ---- I FIGURE 5 ARA CONFIGURATION AVIONICS RESEARCH AIRCRAFT CONTROL/DISPLAY PANEL EVALUATION PILOT SAFETY PILOT STATION STATION FIGURE 6 AVIONICS RESEARCHAIRCRAFT CONTROL/DISPLAY PANEL LO8 %,k k rm(.)

I .

FIGURE 7 DETAILED PIF CONTROLLAW BLOCK DIAGRAM ENTRY ONE TIME TASKS x FILTER INITIAL- IZATION EXECUTE FILTERS 4.

FIGURE 8 FLOW DIAGRAM FOR THE MAIN EXECUTIVE APCMO= .FALSE.

INITIALIZE LPIF P .TRUE.

NO FIGURE 9 FLOW DIAGRAM FOR THE SUBROUTINE CONTROL

I

I !

i ; I I I 1.

YES f CALL HDG SEL ; i :I ROLL SEL 1 : 'I I CALL FPIF2 L-J RETURN FLOW DIAGRAM FOR THE SUBROUTINE COKTROL FIGURE 9 (CONTINUED) ENTRY UPDATE Vk-l*Vk

H

.COMPUTE ------------ COMPUTE ' %C-S-1 I i Sk 'S-1 YAWCOR.

I ,+-1 - %,k-1; I 1 %,k+l - %,k f

is>

APR GS

I>

i I f I ALT SEL !

h i I - 1 I I ALT SEL !

I> ' I i I I I f ' PITCH SEL. : I> I i I f I I> APR LOC ' t , I I I YES I I I ROLL SEL/$-++ . 9 !- I I i ‘I 1’ UPDATE %l=k F2 FIGURE 10 FLOW DIAGRAM FOR THE SUBROUTINE FPI Eq. 58 UPDATE Eq. 56 CONTROLS CONTROL LIMITS FIGURE 11 FLOW DIAGRAM FOR THE SUBROUTINE FPIFl CHANGE FILTER NOISE DESIGN I- CHAkGE COMMAND

‘I

I GAINS & MODEL LIMITS RESIGN I

!

I

i

I

I

I

.i PIF CGT

PROGRAM I CHANGE PIF QUADRATIC i DESIGN

/

I

I

I

I

1 1 I I I I / I I I Imm FLIGHT TESTS FIGURE 12 PIF DESIGN SEQUENCE

TIME (SEC1

%

Y

QJ 320 t 305 I

20 LiO TIME': 80

SEC)

350 r-

80 100

TIME? SEC1

FIGURE 13 A COMPARISONOF THE BAROMETRICALTIMETER OUTPUT, HBARM, THE COMPLEMENTARY FILTER HEIGHT ESTIMATE, HEIGHT EST, AND THE RADAR COMPLEMENTARY FILTER HEIGHT ESTIMATE, ZF

2!ko

50 100 150 200

TIME IS-EC)

0 50

100 150 200

TIME [SEC)

50 100 150

TIME ISEC)

FIGURE 14 SPIRAL DESCENTS USING THE HDG SEL AND ALT SEL PIF AUTOPILOTS 3.17

11 33

TIME2; SEC)

E E

-.5

r

; -1 no

a

-1.5

11 22 33

TIME (SEC1

-60

0 11 TIME2: 33 w

SEC1

FIGURE 15 A 45 DEG HEADING CHANGEUSING THE HDG SEL AUTOPILOT AND AN ay PREFILTER WITH A 0.01 SEC TIME CONSTANT SAFETY PILOT PULSED RUDDERPEDAL

0 11 33

T I ME2: SEC1

-1.5I -1..

.I I----l

.--.

0 11 ME2? 33 w

T I SEC)

0 11 33

T I ME2: SEC)

FIGURE 16 A 45 DEG HEADING CHANGEUSING THE HDG SEL AUTOPILOT AND AN ay PREFILTER WITH A 0.10 SEC TIME CONSTANT

G

-

ke

SENSOR BIAS ?

-

E -2

M

-6

0 11

T I ME2: SEC)

TIME I SEC)

0 11 33 Y!k

T I ME2; SEC1

FIGURE 16 (CONTINUED) A 45 DEG HEADING CHANGEUSING THE HDG SEL AUTOPILOT AND AN ay PREFILTER WITH A 0.10 SEC TIME CONSTANT SAFETY PILOT PULSED RUDDER PEDAL

-3 1 ~~ __~ -122

25 36 ME’: 58 63

T I SEC1

-.

>

a

-1 .l

25 36 58 63

T I ME’; SEC)

-60

-_A-~-

L

36 58

25 63

T I ME9: SEC)

FIGURE 17 A 45 DEG HEADING CHANGEUSING THE HDG SEL AUTOPILOT AND AN a, PREFILTER WITH A 0.22 SEC TIME CONSTANT 7 ?7 , IL.l

' TIME (SEC)

0 30 90

TIME? SEC1

-5

-7

-3

w-lll’---l--J

0 30 30

TIME6: SEC1

FIGURE 18 A 30.5 m (100 FT) ALTITUDE DESCENT USING THE ALT SEL AUTOPILOT, NO THROTTLE ADJUSTMENTAND THE LOW PITCH WEIGHT GAIN SET

F 655 -

0’ 30 90 120

TIME? SEC1

E5

CL

6 0 11-v I

0 30 ME? 90 120

T I SEC1

E -5

4:

- -7

c5

I-

a -3

d-11

0 30 30 120

T I ME6: SEC1

FIGURE 19 A 30.5 m (100 FT) ALTITUDE DESCENT USING THE ALT SEL AUTOPILOT, NO THROTTLE ADJUSTMENT AND THE DESIGN GAIN SET \----A PITCH (DEG) B-w------- PITCH* f DEGl I .-. - I -00 ~-El -es 0 7 TIPIE‘: SEC1 OB ( DEG/SEC) t a-l - ----m----m OBr f DEG/SECl a I -.-II -2_ m et 0 7 TIllE? SEC1 FIGURE 20 ALT SEL LINEAR AIRCRAFT MODEL RESPONSE SIMULATION I I II II VT l II/SEC) t a -6 -__------- VTr t fl/SECl ?i a -16 et es TI IlE? SEC) ELEVATOR ( DEG) d oz-I- --_------a ELEVATORr 1 DEGI s -0 er es 0 7 TlilE’: SEC) FIGURE 20 (CONTINUED) ALT SEL LINEAR AIRCRAFT MODEL RESPONSESIMULATION PSI ( DEGI PSI* (DEG) TME (SEC) ROLL t DEG) t g-20 - m--m------ ROLLm ( DEG) Y ;; -00, 21 20 0 7 TIHE? SEC1 .2- RY t n/s-21 ---------- RYr ( tl/S=m2) E -.Y I 7 e1 26 TIllE1lf SEC) FIGURE 21 HDG sEL LINEAR AIRCRAFT MODEL REsp0~sE sIwhYrIoN RB ( DEG/SECl -a-------- RBm ( DEG/SECl 21 2s TI”E’lf SEC) I- RILERON l DEG) ---------- AILERON= l DEGI -2 2a 7 TIHE%ECl FIGURE 21 (CONTINUED) .JJDG SEL LINEAR AIRCRAFT MODEL RESPONSESIMULATION ROLL (DE61 ---------- ROLLS ( DE61 0 Y TIllE ‘t SEC1 PB t DEG/SECl ---------- PBm l DEG/SECl al- I .I -1 12 16 0 % TI”E*l SEC) RY ( tl/S-21 t a-.06 -----e-e-- RYm I ll/Sr=2) iI Ei a -.12. I .--.-.-- A 0 % 12 16 TIIIE ‘l SEC) FIGURE 22 ROLL SEL LINEAR AIRCRAFT MODEL RESPPNSE SIMUIATION

r

RB I DEWSEC) ---------- RBn I DEWSECI -1 0 'L ‘t 12 10 TI”E SEC) e‘ ii % I- RI LERON ( DES) ---------- AILERON= ( DE61 Y I- k o- ----- -me Ir.L z z -I ie 0 ‘L TItlE ‘t SEC1 e

r

RUDDER f DE61 ---------- RUDDER= t DE61 -1 I 0 4 le 16 ‘l TItlE SEC) FIGURE 22 (CONTINUED) ROLL SEL LINEAR AIRCRAFT MODEL RESPONSESIMULATION e7 TItlEs: SEC1 PSI t DEGI t a-22- ---------- PSfr (DEGI E a -%I II 3 56 TZtlE’: SEC1

2or-

t ROLL (DEGI ROLL (DEGI a-10 --a-----a- --a-----a- ROLLa L DEGl ROLLa L DEGl E a -25 -25 0 0 3 3 TIllEs: 27 27 36 36 TIllEs: SEC) SEC) FIGURE 23 APR LOCI LINEAR AIRCRAFT MODEL RESPONSESIMULATION RY I tl/Saa121 t a-l -----a---- RYr ( tVSu112l E E .L.

t -2 -.__.- 0 3 27 36 TItlE1? SEC1 t RI LERON ( DEGI a-5 -___------ AILERON* f DEGI 0” a ii 8 -3 kk ii RUDDER f DEG) d 02 -3- I- - WJDDER~ 1 DEG) z FIGURIZ 23 (CONTINUED) APR LOCI LINEAR AIRCRAFT MODEL RESPONSESIMULATION Y (Ill ki z 100 - B------m-- Ym (Ill i?

i n z 60- E E 0 I 0 3 TIllE1: 27 96 SEC1 PSI i DEGI ---------- PSI* (DEG) -33 0 3 TItlE? 21 96 SECj ROLL (DEGI --m-m----- ROLLS ( DEG) z -25 I 0 3 et 36 TIflE1: SEC) FIGURE 24 A.PR LOCR LINEAR AIRCRAFT MODEL RESPONSESIMULATION RY f DEGI E-1 i5 t... ---------- RYr 1 DEGI a -2 I I -~ ~ 0 3 TIflE1: e7 36 SEC) - t AILERON ( DEGI a-5 ---------- RILERONr I DEG) & ;; -3 t I I 0 3 TIME%ECl 27 36 ---e--e ii "0 -3 zl ii RUDDER ( DEGI RUDDER ( DEGI E: lx -J- -----_---- c --e--_---w RUDDERS L DE61 RUDDERS L DE61 E u -15 I I --I --I 0 0 3 3 27 27 36 36 TlllE%ECI TlllE%ECI FIGURE 24 (CONTINUED) APR LOCR LINEAR AIRCRAFT MODEL RESPONSE SIMULATION .'

Y ftll : -w-------m em trill =lOO- z B 0 3 TIHE? 27 SEC) PSI (DEG) t a-22- ---------- PSIu f DEGl ai E -33 J 0 3 TlfiE? 27 SEC) ROLL (DEG) ---------- ROLLr ( DEGI a -25 0 3 TlHE? 21 36 SEC1 FIGURE 25 APR LOCP LINEAR AIRCRAFT MODEL RESPONSESIMULATION

r

RY (DEGI E-l- ---------- RY= (DEG) & z -2 0 3 27 s6 TlllE’: SEC1 AILERON ( DEG) E g-s- ---------- RI LERONH 1 DEGI z a -3 0 3 THE’: 27 36 SEC1 RUDDER 1 DEGI

d

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0 3 PIE’! 27 s6 TI SEC1 FIGURE 25 (CONTINUED) APR LOCP LINEAR AIRCRAFT MODEL RESPONSE SIMULATION 135,.

D fM1 ---------A 0’ ttl) L.I_J 7 21 26 TIME1: SEC) PITCH ( DEGI _------0-- PITCHm (DEG)

r

----------e---- -15, I IS 0 7 TIME’; 21 28 SEC1

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QB l DEG/SECl t 2-t ---------- OBr ( DEG/SECl t 2 FIGURE 26 APR GS LINEAR AIRCRAFT MODEL RESPONSE SIMULATION -2 21 26 0 7 TIME’: SEC) ,---- ____ -_----- 3 0 l- 0 0 -- VT 1 H/SEC) t Q -7- ---------- VT* ( PI/SEC1 E tz -16 TIIIE’: 21 26 0 7 SEC1 ELEVRTOR f DEGI d W-Y- __-------- ELEVRTOR= ( DEGI I3 -6, TI$: 21 26 0 7 SEC1 ’ I FIGURE 26 (CONTINUED) APR GS LINEAR AIRCRAFT MODEL RESPONSE SIMULATION

50 100 150 200

TIME [SEC1

E

E

10 -

OI

t- 5

n-

0 v 50 100 150 200

TIME (SEC)

0 50 100 150

TIME (SEC)

FIGURE 27 A 152.4 m (500 ft) ALITUTDE ASCENT USING THE ALT SEL AUTOPILOT, THROTTLE ADJUSTMENTS, LOW PITCH WEIGHT GAIN SET AND HEAVY TURBULENCECONDITION

2 720

LL

N 660

65.0 87.5

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TIME [SEC1

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FIGURE 28 A 152.4 m (500 ft) ALTITUDE ASCENT USING THE ALT SEL AUTOPILOT, THROTTLE ADJUSTMENTS, LOW PITCH WEIGHT GAIN SET AND MEDIUM TURBULENCECONDITIONS

-

k

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0 25

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FIGURE 29 A 152.4 m (500 ft) ALTITUDE ASCENT USING THE ALT SEL AUTOPILOT, THROTTLE ADJUSTMENTS, DESIGN GAIN SET AND LIGHT TURBULENCE

G

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FIGURE 29 (CONTINUED) A 152.4 m (5,0,0 ft) ALTITUDE ASCENT USING THE ALT SEL AUTOPILOT, THROTTLE ADJUSTMENTS, DESIGN GAIN SET AND LIGHT TURBULENCE

i=-56Ob - /---

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75 100

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75 100

T I ME’: SEC)

TIME I SEC1

FIGURE 30 A 152.4 m (500 ft) ALTITUDE ASCENT USING THE ALT SEL AUTOPILOT, NO THROTTLE ADJUSTMENTS, DESIGN GAIN SET AND LIGHT TURBULENCE

25 75 100

T I ME5: SEC>

25 75

T I ME57 SEC1

0 25 75

T I ME5: SEC1

FIGURE 30 (CONTlkJED) A 152.4 m (500 ft) ALTITUDE ASCENT USING THE ALT SEL AUTOPILOT, NO THROTTLE ADJUSTMENTS, DESIGN GAIN SET AND LIGHT TURBULENCE LL N

0 30 90 120

l TI ME? SEC)

0 30 90

T I ME? SEC)

0 30 90 120

T I PIE? SEC)

FIGURE 31 A 152.4 m (500 ft) ALTITUDE DESCENT USING THE ALT SEL AUTOPILOT, THROTTLE ADJUSTMENTS, DESIGN GAIN SET AND LIGHT TURBULENCE - LL

N 210 -

I

.

-

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0 30

T I ME? SEC1

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TIME (SEC1

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FIGURE 32 A 152.4 m (500 ft) MANUAL ALTITUDE DESCENT WITHOUT USING THROTTLE DURING LIGHT TURBULENCE

cs

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FIGURE 33 A 45 deg HEADING CHANGEUSING THE HDG SEL AUTOPILOT IN HEAVY TURBULENCE

se 440

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FIGURE 34 A 5 deg PITCH CHANGEUSING THE PITCH SEL AUTOPILOT

2 -160

TIME ISECI

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FIGURE 35 A 5 deg ROLL CHANGEUSING THE ROLL SEL AUTOPILOT

r

TIME (SEC1

-

(-!I -20 I

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100 130

160 190 220

TIME (SEC)

_ ‘.

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FIGURE 36 APR GS GLIDESLOPE CAPTURE AND TRACK WITH OPEN-LO,OPRANGE ESTIMATION AND THE AIRCMFT CROSSING THE OUTER MARK&R ..

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FIGURE 36 (CONTINLJED) DR GS GLIDESLOPE CAPTURE AND TRACK WITH OPEN-LOOP RANGE ESTIMATION AND THE AIRCRAFT CROSSING THE OUTER MARmR

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LL

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TIME (SEC)

60 -

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z r

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FIGURE 37 APR GS GLIDESLOPE CAPTURE AND TRACK WITH CLOSED-LOOP RANGE ESTIMATION AND THE AIRCRAFT CROSSING THE OUTER MARKER

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FIGURE 38 APR GS GLIDESLOPE CAPTURE AND TRACK WITH CLOSED-LOOP RANGE ESTIMATION AND THE AIRCRAFT MISSING THE OUTER MARKER

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FIGURE 38 (CONTINUED) m GS GLIDESLOPE CAPTURE AND TRACK wmi CLOSED-LOOP RANGE ESTIMATION ANTI THE AIRCRAFT MISSING THE OUTER MARKER

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FIGURE 39 APR LOCR LOCALIZER CAPTURE AND TRACK FOR A 45 deg INTERCEPT ANGLE i__,-,;,;,

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FIGURE 40 APR LOCP LOCALIZER CAPTURE AND TRACK FOR A 45 deg INTERCEPT ANGLE

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FIGURE 41 APR LOCI LOCALIZER CAPTURE AND TRACK FOR A 45 deg INTERCEPT ANGLE

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FIGURE 42 MANUAL LOCALIZER TRACKING

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c3

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FIGURE 43 APR LOCR LOCALIZER CAPTURE AND TRACK FOR A 60 deg INTERCEPT ANGLE

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z 20-

-

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FIGURE 43 (CONTINUED) APR LOCR LOCALIZER CAPTURE AND TRACK FOR A 60 deg INTERCEPT ANGLE

-65

-15

0 LlO

120 160

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E-2

m

-6

120 160

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FIGURE 44 APR LOCR LOCALIZER CAPTURE AND TRACK FOR A 45 deg INTERCEPT ANGLE AND THE AIRCRAFT MISSING THE OUTER MARKER -7.5 -5.0 -2 -5 0 x103 -10.0

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FIGURE 44 (CONTINUED) APR LOCR LOCALIZER CAPTURE AND TRACK FOR A 45 deg INTERCEPT ANGLE AND THE AIRCRAFT MISSING THE OUTER MARKER - ---- ircraft Velocity FIGURE 45 GLIDESLOPE GEOMETRY X - Axis Y - Axis Aircraft Velocity -------A----- Y L Aircraft cg FIGURE 46 LOCALIZER GEOMETRY - 1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No.

NASA CR-3709 -- 4. Title and Subtitle 5. Report Oate July 1983 DESIGN, IMPLEMENTATION AND FLIGHT TESTING OF PIF 6. Performing Or*niration Code AUTOPILOTS FOR GENERAL AVIATION AIRCRAFT 7. Author(s) 8. Performing Organization Report No.

TR-681102 JOHN R. BROUSSARD . 10. Work Unit No.

9. Performing Organization Name and Address 11. Contract or Grant No.

INFORMATION & CONTROL SYSTEMS, INCORPORATED NASl-16303 28 RESEARCH DRIVE HAMPTON, VA 23666 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address CONTRACTOR REPORT NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 14. Sponsoring Agency Code 20546 WASHINGTON, DC

I

- 15. Supplementary Notes WAYNE BRYANT NASA LANGIJZY TECHNICAL MONITOR: 6. Abstract This report presents the designs of Proportional-Integrated-Filter (PIF) auto- pilots for a General Aviation (NAVION) aircraft. The PIF autopilot uses the sampled- data regulator and command generator tracking to determine roll select, pitch select, altitude select and localizer/glideslope capture and hold autopilot heading select, The PIF control law uses typical General Aviation sensors for state feedback, modes.

command error integration for command tracking, digital complementary filtering and a control filter for computation analog prefiltering for sensor noise suppressiori, delay accommodation and the incremental form to eliminate trim values in implemen- Theoretical developments described in detail, were needed to combine the tation.

sampled-data regulator with command generator tracking for use as a digital flight control system.

The digital PIF autopilots are evaluated using closed-loop eigenvalues and linear simulations. The implementation of the PIF autopilots in a digital flight The successful computer using a high order language (FORTRAN) is briefly described.

flight test results for each PIF autopilot mode is presented.

7. Key Words kuggested by Author(s)) 18. Dktribution Statement Optimal Control-Theory Optimal Control-Application Unclassified-Unlimited Autopilot Digital Flight Control System Subject Category 08 General Aviation Flight Control System Model FollowinP I g. kurity Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pqes 22. Price Unclassified Unclassified A09 For Sale by lhe Nallonal Techcal lnfotmatlon Serwce. Sprinelleld. Vlrglnla 22161 NASA-Langley, 1983