"N AD-776 297

APPLICATION OF PRACTICAL OPTIMAL CONTROL THEORY TO THE C-5A LOAD IMPROVEMENT CONTROL SYSTEM (LICS)

Albert J. Van Dierendonck, et al

Honeywell, Incorporated

Prepared for:

Air Force Flight Dynamics Laboratory

October 1973

DISTRIBUTED BY:

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APPLICATION OP PRACTICAL OPTIMAL CONTROL THEORY TO THE C-5A LOAD IMPROVEMENT CONTROL SYSTEM (LICS)

Final Technical Report.15 August 1972 ' 15 September 1972

Albert J. VanOierendonck Charles R. Stone

October 1973

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Air Force Flight Dynamics Laboratory Air Force Systems Command Wright-Patterson Air Force Baae, Ohio

II illfliCf - Practicalizing quadratic optimal control algorithms were used to design load relief

systems for the C-5A, a large flexible aircraft. The predicted rms stresses at the wing root were reduced by more than 40 percent. Handling qualities or stability were not compromised. The control is realised with a gyro and three acceler- ometerti affecting ailerons and elevator - two accelerometers more than an existing stability augmentation system. The quadratic performance index is defined to enforce good handling qualities and to limit the control »ystem bandwidth. ;

Reproduced by

NATIONAL TECHNICAL INFORMATION SERVICE U S Department of Commtrt«

Springfield VA 22151

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Lockheed C-5A Optimal Control Maneuver Load Control Wing Root Stress Reductions Practical Optimal Control

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APPLICATION OP PRACTICAL OPTIMAL CONTROL THIORY TO THI C-SA LOAD IMPROVIMINT

CONTROL SYSTIM (LICf)

A. J. VANDIERENDONCK C. R. STONE M. D. WARD

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FOREWORD

The technical work reported was conducted by the Research Department of the Systems and Research Center of Honeywell Inc., Minneapolis, Minnesota. The principal investigator was Dr. A. J. VanDierendonck. He was assisted by Mr. C.R. Stone and Mr. M.D. Ward. Dr. E. E. Yore was Program Manager at Honeywell. The Project Engineers were Major B. Kujawski (AFFDL/FGB) and Mr. Charles Stockdale. Their help is grate- fully acknowledged.

This work was performed under Project No. 487T0105, Contract No. F33615-72-C-2008, sponsored by the Air Force Flight Dynamics Laboratory. The period of performance was from 15 August 1972 to 15 September 1972.

This work has benefited considerably from the past efforts of the author's colleagues at Honeywell. They are too numerous to be individually mentioned.

The manuscript was released by the authors in September 1973. The number assigned to this report by Honeywell Systems and Research Center is F0161-FR, Volume III.

This report has been reviewed and is approved.

Flight Control Division

It

ABSTRACT

Practical!zing quadratic optimal control algorithm! were used to design load relief systems for the C-5A, a large flexible aircraft. The predicted rms stresses at the wing root were reduced by more than 40 percent. Handling qualities or stability were not compromised. The control is realized with a gyro and three accelerometers affecting ailerons and elevator -• two accel- erometers more than an existing stability augmentation system. The quadratic performance index is defined to enforce good handling qualities and to limit the control system bandwidth.

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TABLE OF CONTENTS

SECTION I

SECTION II

SECTION in

SECTION IV

SECTION V

SECTION VI

APPENDIX

REFERENCES

INTRODUCTION AND SUMMARY

PROBLEM FORMULATION Design Procedure Control Law Performance Index The Problem Aircraft Model Response Selection

OPTIMAL STATE FEEDBACK CONTROL

SIMPLIFIED FEEDBACK CONTROL

LICS FUNCTIONAL BLOCK DIAGRAMS

CONCLUSIONS AND RECOMMENDATIONS

SYSTEM MATRICES FOR FLIGHT CONDITION 37

Page

1

3

3 4 4 4 S

11

13

18

26

30

31

37

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▼

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LIST OP ILLUSTRATIONS

Page

1 The Lockheed C-5A 7

2 C-5A LICS Sensor Locations 9

3 Chronological Weighting of Responses for Optimal 14 State Feedback Control Design

4 Chronological Stress Responses for Optimal State 14 Feedback Control Design

5 Chronological Model-Following Responses for Optimal IS State Feedback Control Design

6 Aileron and Elevator Responses for Optimal State IS Feedback Control Design

7 Incremental Gradient for C-SA Design 23

8 Simplified Control Block Diagram (Controller 14C) 27

9 System I LICS (Without Spoilers) Block Diagram 28

10 System H LICS (With Spoilers) Block Diagram 29

LIST OP TABLES

I Wing Root Stress Relief Summary 1

n Comparison of RMS Responses Due to 1-FPS RMS Gust 10

HI Chronological Frequencies, Damping Ratios, and Actuator 16 Roots for Optimal State Feedback Control Design

IV Practical Design Results Summary 19

V Design Results Summary with Six-Mode Model 20

VI Spoiler Effectiveness for Gust Relief 20

VII Candidate Measurement for Practical Designs 21

vm Maneuver Load Control 25

vi

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LIST OF SYMBOLS

VECTORS

X State vector u Control vector

y Measurement vector r Response vector r\ Disturbance vector F. . F« Row vectors of stability matrix of model-following model

MATRICES

F Gl G2 H D M K* K

Q

System stability matrix Control input matrix

Disturbance input matrix

Response state output matrix Response control output matrix Measurement output matrix Measurement gains matrix State gains matrix Quadratic weighting matrix

VARIABLES

J w 0/n2

Performance index Vertical velocity Normalized pitch rate ■

vli

" T.:

LIST OP SYMBOLS -- CONCLUDED

VARIABLES (continued)

8

*L aL Wg P u

C Ä/n2 a E Tr

Bending-mode coordinate Elevator displacement Aileron diiplacexuent

Spoiler displacement Stress Lagged gyro output Lagged accelerometer output Gust vertical velocity Wind model state Frequency Damping ratio Model-following error Acceleration Expectation operator Trace operator Gradient incrementing parameter

SUBSCRIPTS AND SUPERSCRIPTS

(•) Time derivative ()m

Model parameter -

()T Transpose of a vector or matrix (,.p 1

Short period Inner surface

■

o Outer surface ■•

I Left .

T Right

vüi

- r ,

SECTION I INTRODUCTION AND SUMMARY

Quadratic methodology was applied to the design of a pitch-axis load- relief control system for the C-5A aircraft. Predicted rms stresses due to wind gusts at the wing root were reduced by 50 percent, while stress rates were reduced by 31 percent. This was done without serious degradation of the handling quantities or the stability of the aircraft. Similar reductions in peak and steady-state stresses due to maneuvers were realized. Symmetric ailerons and the inboard elevator are driven by control signals from acceler- ometers and a gyro. One accelerometer and the gyro already exist in the stability augmentation system (SAS). The additional load improvement con- trol system (LICS) acceleiometers would be placed on outer wing panels.

The effectiveness of active control for effecting load improvement is summarized in Table I. Results are given for two systems. System I uses ailerons to reduce the loads, whereas System II uses spoilers in addition to

Table I. Wing Root Stress Relief Summary

ParMitlar

Ratio of ControUod Airert» to Fr»« Aircraft

Syittm I: AUarona, On« Extra Senaor Sat

Syatam II: Ailarona ♦ Speilara, Two Extra Sanaor S«tac

Wing root •treat* Wing root stroH rmt«'

PMk mantmroriiif

StMdy-stat«

O.SO o.ae O.ST

O.ST

0.33 0.91 0.38

0.31

"For Loekhaad fllfM condition 3T (w > S03, IM lb: M • 0.933; h ■ 10,000 ft) uaing a als-flaxurw-mod« rapraaantation.

"Vor *l, 9 Incramantal g uaing 89 dag of up aileron. The allerona would hit the down atope of 19 deg at -0.8 ineramental load factor.

cFor 10 deg of up apoller at +1.S incremental g.

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the ailerons. Table I shows that rms stress and stress rate can be reduced by 50 percent and 31 percent, respectively, using System 1. Peak and steady- state maneuvering stresses can be reduced by 43 percent of their nominally attained values using System I. System I requires one additional set of sen- sors to achieve the gust relief ir-provements cited. In the study conducted, virtually no improvement was obtained using additional sensors.

By using spoilers, performance could be improved as indicated for System II in Table I. For System n, use of a pilot-ope rated switch is recommended to activate the spoiler portion of the system. In normal use the spoilers would not be deflected; the performance would be that of System I. In rough air, the spoilers would be activated to achieve the results noted for System n. The spoilers would be biased at 10 degrees in this mode of operation.

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SECTION n PROBLEM FORMULATION

DESIGN PROCEDURE

The design approach used is based on quadratic optimal control theory -- optimal with respect to u quadratic performance index subject to practical con- straints. The background for this methodology is given in References 1 and 2.

The design is accomplished by mlnlroizing a quadratic performance Index which weights mean square stresses, stress rates, model-following errors, control surface rates and control surface deflections. Simple compensation filters were included in the measurement constraints to improved bendlng- mode damping and handling qualities. The model-following errors were weighted to enforce good handling qualities.

The quadratic optimal design technique requires that the aircraft be modeled as a linear time-invariant plant representing a single flight condition:

x

r

y

Px + GjU + GgTi

Hx + Du

Mx

(1)

(2)

(8)

where ■

State vector (including rigid-body states, actuator and servo states, flexure-mode states, sensor states, model-following states, and wind states) ■,,

u > Control input vector

r\ ' Unit-variance white noise vector

r ■ Reeponee vector

y > Measurement vector

CONTROL LAW

The control law is constrained to be of the feedback gain form

u ■ K*y (4)

where the asterisk is used to differentiate this matrix K* from the optimal full-state feedback form

Kx

PERFORMANCE INDEX

(5)

The performance index is defined to be

J • E{TrCQ r rT] )

THE PROBLEM

(6)

Using Equations (I) through (6), the problem reduces to minimising the performance index J with respect to the gains matrix K* subject to the con- straint of system stability and Equations (1), (8), and (3).

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There are two parts lu this problem. This first part Involves determining the full-state feedback [Equation (5)] . The second part involves constraining the feedback [Equation (4)] and is also known as the fixed-form optimal con- trol problem. The first part of the problem is discussed in Section HI. The simplified control law is discussed in Section IV.

AIRCRAFT MODEL

The design approach was applied to the pitch axis of the C-SA aircraft at one flight condition (FC-37) which is a low-altitude cruise condition with the following parameters:

• Gross weight - 593.154 lb (50* fuel. 50* cargo)

• Mach number - 0.533

• Altitude - 10.000 ft

• Dynamic pressure - 292 psf

• True airspeed - 577 fps

• Center-of-gravity location - 31< MAC

The general arrangement of the C-5A is shown in Figure 1. The spoilers being considered are the outboard spoilers (which consist of three panels). Figure 2 shows the locations of the sensors that will be considered in the con- trol synthesis.

The procedure described in Reference 3 was used to reduce a C-5A model to the form of Equation (1). The procedure included the quasi-elastic effects of bending modes neglected, as only the three most prominent modes were included in the design. The final designs were checked using a six-mode model.

5

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The Lockheed date (listed in Ref. 4) for FC-37 were processed by the computer programs developed for s com '-configured vehicle (CCV) (Ref. 3). Satisfactory agreement with load allevla n and mode stabilization (LAMS) results (Ref. 5) was not achieved. Two errors in the processing program (Table 11-13 of Vol. II of Ref. 3) were found. After making these corrections, the results shown in Table n were obtained. Agreement is now considered to be sufficient for the intended purposes.

The notation used for column headings in Table n (e.g., LAMS-6) refers to the data source [LAMS (Ref. 5) or this contract] and the number of bending modes used. The number ru (shown in row 2) is used to convert 0 from rad/ sec to in/sec. It is equal to 0.6066 x 10~ .

The state vector, x, for optimal state feedback designs was (17 states):

XT - (W. e/Uj, ^ IIQ, llg, 1\V Tig, Tig. öa. öe , Pj. P2. Pg,

p4. p5. p6, wg)

where w is the rigid-body vertical velocity (in. /sec), ö/n, U- a normalized rigid-body pitch rate, rij is the first bending-mode coordinate. Tig is the sixth bending-mode coordinate, rig is the third-mode coordinate, 0 is symmetric aileron displacement, 0 is the inboard elevator displacement, p. through p- are wind distribution and lift growth states, and w is the vertical gust velocity.

The control input vector, u, contains the inputs to the actuator models (two controls):

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WING PANEL 23 WING STATION 1200

WING PANEL 15 WING STATION 768

PITCH RATE GYRO

ACCELEROMETER

VERTICAL ACCELEROMETERS

Figure 2. C-5A LICS Sensor Locations

Pricriiiipw

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Table n. Comparison of RMS Reeponfles Due to 1-FPS RMS Gust

P«r»mtttr LAMS-« LAMS-15 LICS-3 LICS-«

w (in/MC) 10.41 11.08 11.03

«/nj (In/itc) 1.4« ... 1.12 1.133

n, (In.) 0.421 ... 0.549 0.550

r\l (ln/i«e) 1.81 ... 1.933 1.94

n, (in.) 0.0332 ... 0.0331 0.0339

ti, (in/ate) o.sn ... 0.314 0.339

n9 (to.) 0.0126« ... 0.009« 0.00968

ii6 (ln/Mc) 0.1483 >■>«■ 0.0411 0.0456

•, (pal) 138.8 22» 4 178.0 179.0

i, (pai/atc) 533.4 1040.0 571.0 564.0

■2 (pal) 143. S 343.1 183.8 181.0

a, (pal/aae) 853. T 1578.0 841.0 870.0

No. of modaa 6 15 3 6 Mode« uaad 1.3,4,6,

7,11 An 1.3,8 1,8,3,4,5,8

Ouat panatrailon No Yaa Yaa Yaa 'Vagntr dynamici Conatant

(unity) Sacoad- ortar

Conatant (unity)

Conatant (unity)

Mod« approsl- Tnincatad Complatad Staady- atat« ratainad

Staad-atat« ratainad

Seala langth (ft) 1000 1000 1750 1750

The complete transfer function for the elevator and aileron Is third- order and was used for evaluation only:

a 4

it - (9)

i

10

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RESPONSE SELECTION

The objective of the LICS design was to significantly reduce wing root stress and stress rate using active control without degrading the existing handling qualities. This reduction was to consider both maneuvering relief as well as gust-induced stresses. Therefore, the response vector, r. mini- mized was

rT - {BV s2. ßv i2, öa. öe. öa, Öe. B/t^) (10)

where s. is the stress at the wing root. Sg is the stress at a mid-wing panel, and d/ng is a model-following error. The model-following error represents the difference between the pitching moment equation of the aircraft and a rigid model with desired handling qualities. That is.

»/«a • (F2 - F2m) x (11)

where F2 Is the second row al the stability matrix F of the aircraft and F- is the second row of the stability matrix of the model. This row is given by

2m (F21m' F22m' 0 0' ^.ll.-" P2,17) (12)

Here. F21m and F22 are selected so that the desired short-period frequency and damping ratio are computed from

u 2 ■P

F F - F F ll,22m ,21mr12

2 C w * wsp wsp ■Fll ' F22m

(IS)

(14)

where F. ^ and F12 are elements of the first row of F. This is because the z force (row 1 of the model) is taken to be

Flm " (F11'F12'0,*,'0'F1,11.,,"F1.17) (15)

11

im '

• _ ■.

The wind coefficients of the model are taken to be the same as the air- craft itself. The wind coefficients of the model in Equation (12) should probably be adjusted to agree *rith an adjustment in F.. the coefficient of

w. However, in this design. F«, and ^22m Bre the 0ame a8 F21 ^ F22,

Thus, the model-following error tends only to decouple the bending modes from the rigid body.

The matrices of Equation (1), (2), and (3) corresponding to this flight condition are tabulated in the Appendix.

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SECTION III OPTIMAL STATE FEEDBACK CONTROL

Optimal full-state feedback designs were obtained for various quadratic weighting selections on the responses of Equation (10). Not all responses were weighted} some were only monitored to ensure that they were not com- promised. This was especially true with 82 and 82, the mid-wing stress and stress rate.

The quadratic weights selected versus chronological designs are shown in Figure 3. Zero weights are not shown. The weights on 6 and e/n« are not shown, since they were held constant at 20 and 10' , respectively. These weights, as well as the initial weights (Iteration 1), were determined in previous designs with erroneous stress equations. The weights on 6. and e/n« were iteratively selected to constrain the rigid-body frequency and damping ratio. This was done with relatively little effect on stress control. Figures 4, 5, und 6 present the root mean square (rms) responses and rigid- body frequencies and damping ratios corresponding to the weights of Figure 3. The dashed lines represent responses which are not weighted. Table III lists the corresponding frequencies, damping ratios, and actuator root locations of 14 optimal (full-state feedback) control designs. The rationale used in the designs will now be discussed.

On iteration 1, the aileron actuator root was large (caused by a signifi- cant negative feedback of 6 to UQ ). Thus« the aileron rate weight, Qß , was increased for iteration 2. There was a corresponding increase in stress s.. In an attempt to simultaneously decrease 0 feedback and increase aileron effectiveness, the weight on aileron displacement, Qg , was removed for iteration 3. This had very little effect. Thus, for iteration 4, this weight was reinstated and the aileron rate weight, Qg . was increased further. The aileron feedback became positive and the stress increased} so, on the fifth iteration, the QA weight was cut in half. a

18

_ -

Q, (xlO')

ITERATION

Figure 3. Chronological Weighting of Responses for Optimal State Feedback Control Design

NOTE: ITERATION ZERO REFERS TO FREE AIRCRAFT

10,

5j & 8 UJ ♦- $ (A 6 M UJ

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Figure 4. Chronological Stress Responses for Op Imal State Feedback Control Design

14 •

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•**•

NOTE: ITERATION ZERO REFERS TO FREE AIRCRAFT

(17,42)

s s

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Figure 5. Chronological Model-Following Responses for Optimal State Feedback Control Design

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MM ill • > *» A. it*4

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2-

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i I I i l i i i i i i i i t 2 4 6 8 10 12 14

ITERATION

Figure 6. Aileron and Elevator Responses for Optimal State Feedback Control Design

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The Iteration 3 controller was quite acceptable, if the 0 feedback could be successfully eliminated later- during practicalization designs. Stress was reduced 67 percent on that iteration, and stress rate SO. 5 percent. The iteration 5 controller is a compromise, with a 49. S-percent reduction in stress and a 42. 3-percent reduction in stress rate.

In an attempt to reduce stress rate even more, the stress rate weight, Qg , was increased and Q^ was removed for iteration 6. Both stress and stress rate decreased, with a significant increase in bending-mode damping, although the model-following error, short-period damping ratio, and 0 feedback to UQ increased somewhat. This controller was also considered a candidate for practical design; however, from past experience, it was expected to lead to difficulty. Generally, practical measurements cannot produce this much bending-mude damping without excessive filtering. This is especially true when slow actuators are used.

Iterations 7 through 11 were further attempts to re'iuce stress. Finally, by deemphasizing reductions in stress rate, stress couid be reduced much more within the constraints of the actuator bandwidths. The stress rate weight, Qg , was removed by Iteration 12. However, the 0 feedback to UQ

was still high. Iterations 13 and 14 brought this feedback to within reason for practical design. The iteration 14 controller was also a candidate for practical design. With this controller, stress was reduced 67.8 percent and stress rate 36 percent, with small öa to u« feedback. wa

i.

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SECTION IV SIMPLIFIED FEEDBACK CONTROL

This section summarizes the LICS control design effort. The first set of controllers used ailerons and elevator and was based on a three-mode model. Both full-state and measurement feedback were evaluated (Table IV). Three of the 14 full-state feedback controllers were successfully simplified.

The second set of controllers also used aileron and elevator but was based on a six-mode model. .Both full-state and measurement feedback were evaluated (Table V).

The last set of controllers used spoilers and elevator with a three-mode model. Only full-state control was considered (Table VI).

With the iteration 3, 5, 6, and 14 controllers as baselines« practical designs were attempted using their respective quadratic weights and their optimal gains as starting points. The practical designs were attempted with different measurement complements. The candidate measurements are listed in Table VII.

T For most of the practical designs, the state vector, x , included 18 states:

• • • • |w, e/nj, ti1# TI6, TI3, r\v Tig, Tij, öa, öe , eL,

PV p3' p3* p4# p5* pe* wg) (16)

18

.

j.

■

■

fr cd 8

w

■

c»

1

I

u

V •* 4 44*4444 m •»' d**4d*4d*4 • • • • •

9 •* 4 44 444444 44 44 44 44 4 A 4 \ •• • •

• • •

pl

* o M' e *.' o e o e- • S, 4J444md*J4 \ •

• 9

m n •

2 O - o « rf o- o o «• - o- • ef •' o- • o «* r-- ! t 3I .1 8 nisüSiS !6flai9lcti .

4*4444444 ^44444 44^44j • •

»4

• •

4 liiSiSiil mSsIslKsi . 444444444 J44444 44^J41

•

J1 • •

m

• « •

n 4~4444444 4 4 4 4 4 4 44441 — * II 3I 3|

s 444444444 J444J4444J4 ! * 4

m m •

» SiHiSÜi liilsltiasi, 444444444 ^44444 44 444 ! — 5 1 M | 1

• •

m* • • • —

s 5H5 = iiii5 S^SsscSssl i 444444444 J44444444J4\ * - T ^ . •

«4

9 • •

s ssaiSÜM leiiiSftiiftti 444444444 44444444J^4'

" " iii

• •

P«

• • M «f * ■«1

m IHSsiSSis BSSKsSsiRI i . 4J4J44444 J4444444~4\ \ mm II A d

1 ■ 11 ■ - ? 5

«.

19

1

•

Table V. Design (flrwlnd

Results Summary with Six-Mode Model »5.2 fps)

Parameter Fraa Aircraft

Controllar

1 3 3 4 5

., (J03p»l> 0.030 0.411 0.311 0.470 0.581 0.487

«, dO9 (Ml) 0.040 0.580 0.943 0.873 0.810 0.887

• , (10s pat/Me) 2. 938 1.888 1.SS7 3.301 1.181 1.318

•2(103p*i/Me) 1.587 a. 417 3.879 1.117 1.918 3.010

•W.1 (*•, 43.1$ 49.10 43,00 38.43 38.78 30.83

«/nj (in/MC» s.as ».10 8.81 4.98 8.15 4.38

«, (rad/Mc) 0 0.0411 0.0784 0.0171 0.0430 0.0008

«, (rad/Me) 0 0.0180 0.0183 0.0184 0.0108 0.0184

Sa (rad) 0 0.0118 0.0311 0.1355 0.01031 0.0108

«, (rad) 0 0.0048 0.0088 0.0048 0.0033 0.0047

Actuator modal ... 1« I' 3* 3«» L sb

Foodbocka Nona All atatoa

All atataa

Sam« a« SCof Tabla IV

Sam* a« BCof TabU IV

Sama aa SCof Tabla IV

'Flrat-ordar modal ClUfor to Equation (0) j. bThlrd-ordar modal [Rafar to Equation (0)].

Table VI. Spoiler Effectiveness for Gust Relief ^wind-5-2fP8)a

Paramatar Fraa Aircraft

State Control 1

State Control 8

a, (10s pal) 0.M5 0.037 0.004

a, (10S pal) 0.981 0.331 0.811

i, (10s pai/aac) 1.973 0.880 0.009

i, (10S pai/aac) 4.100 0.944 1.407

"total "»^ 87.84 43.94 48.87

»/nj (in/aae) 8.81 7.39 0.08

», (rad/aac) 0 0.0080 0.084

«, (rad/aac) 0 0.0188 0.0140

«a(rad) 0 0.0084 0.0100

».(rad) 0 0.0080 0.0088

«s (rad) 0 0.038 0.0840

it (rad/aae) 0 0.004 0.0000

*Thrae-mode rapraaantatioo with flrat-ordar actuatora, daftacted apollar ayatam, atata control.

20

,

f

•

'

Table VII. Candidate Measurement for Practical Designs

Mtaturtmant Number

t

2

Oaicrlptlon

AcceleroiTi«t«r located at fuatlaf« panel 4 (forward)

Difference between accelrrometer located at fuaelaM panel 4 and averafe of rcceleromctera located at wing panels 2 i 'tip)

Difference between ac ■-aerometer located at fuaelaee panel 4 and average of accelarametera located at wing panel! IS (mid-wing)

Combination of lagged and highpaeeed rate gyro located at fuaelage panel 4 (forward) with frequency cutoff at 0. 88 rad/eec (-Zw)

Rate gyro located at fuaelage panel 24 (aft)

Lagged meaaurement 2 with frequency cutoff at 2 rad/eec

which is the same as Equation (7) with the addition of 9L, the lagged fuselage panel 4 rate gyro output used to construct measurement 4 of Table VII. In the designs where the lagged acceleration measurement was used, the state vector included 19 states:

(17) x - (w, e/^. nj. rig. TI3, Tir rig. TI3. öa. öe. eL.

aL' pl' p2' p3' p4' p5' p8' V

where aL is the lagged acceleration measurement.

The design procedure described in References 1 and 2 was used to realize the practical designs. Using this procedure, the measurement gains are written as a function of a sealer parameter. \. such that

K*(\) - K1 (JO + \K2 0< X< 1 (18)

21

"

The incrementing parameter, \, is equal to 1 for the optimal state feedback controller, and X. is equal to zero for the optimal measurement feedback controller.

The starting point (\ > 1) is found by using the optimal state feedback gains and the measurement matrix (augmented with direct measurements of states not necessarily measurable so M' exists):

K*(l) ' KM" (19)

The measurement constraints are applied gradually by stepping \ to zero. The matrix K (0) is the fixed-form solution and has the gain structure desired.

This procedure of "backing off" from the state feedback controller is illustrated in Figure 7 for the design of controller 14C (defined below). The same quadratic performance Index was minimized while the measurement constraints were gradually applied.

The cost J is minimized at each increment of \, with respect to the gains on the measurements. The gains are predicted for each increment. The minimization of J is the correction. The differences between the gradient norms before and after the corrections are also shown in Figure 7.

The practical designs are summarized in Table IV, except for the practicalization of the iteration 6 controller. In an attempt to practlcalize this controller using measurements 1, 2, 3, and 4, the rigid-body and first bending-mode frequencies and damping ratios became unacceptable, as their roots appeared to merge with actuator roots. The successful practical designs will now be discussed.

■

22

■

•Uivjll 0.48-,

s1

10001

900 -

800 •

700 •

600

500 ■

400..

300 -

200 •

100 ■ 0 •

U2

10 r

5<-

2"

I"

0 .

$1 FREE VC J FREEA/C

i I i i IX OX 0.6 04 0.2

Figure 7. Incremental Gradient for C-5A Design

Succeeatal design« were achieved using iteration 3,5, and 14 controllers as baselines. Controller 8A had one more accelerometer measurement than SB. This accelerometer gained very little stress and stress rate reduction at the expense of overcontrolling ihe rigid body. It slightly improved mode 3 damping, with less damping on modes 1 and 6. On controller 5C, aileron feedbacks were permitted. This defined a prefllter for the actuator. There was an improvement over 5B in stress rate and stress Sj (not weighted) at the expense of bending-mode damping. The aft rate gyro was included in controller 5D, with some improvement over 5B in stress, stress rate, and mode damping.

i

23

'■■ ■'»"

Controller? 3A and 3B are based on the iteration 3 controller. On con- troller 3A. aileron feedbacks were allowed, with significant improvements in stress and stress rate at the expense of bending-mode damping.

Controller 14 practical controllers 14A, 14B, and 14C produced lower stress s j levels than the others. Stress rate levels were higher. On con- troller 14C, measurement 6 (lagged acceleration) increased bending-mode damping and lowered the short-period frequency. The short-period damping ratio was somewhat higher, however.

Controllers 5D, 3B, and 14C are the most desirable controllers from the bending- mode damping viewpoint, which could be Important for stability margins. Controller 3B is the least complex of the three; however, 14 C is only different by a lag network. Controller SD uses an extra sensor.

Table V summarizes the performance of two state-feedback and three practical controllers for a more complex model containing six bending modes. The results are not markedly different using a more complete model.

Table VI shows the effectiveness of spoiler controls in providing gust load relief. These are for state controls. Since full-state feedback was used, conservative depreciation was used from these answers to provide the numbers of Table I representing the results of simplified control.

Table VIII presents results for maneuvering load contrcl (MLC). The maneuver is a step column input that attains a steady-state value of 1.5 Incremental g.

The free aircraft does not use symmetric ailerons in the steady state. At +1.5 Incremental g, the wing root perturbation stress is -17,400 psi. The peak perturbation stress achieved during the transient for the step column input is -18,700 psi (row 1 of Table VIII).

24

■

■

■

Controller! 3A and SB are based on the Iteration 3 controller. On con- troller 3A. aftaron feedbacks were allowed, with significant improvements in stress and »tress rate at the expense of bending-mode damping.

Controller 14 practical controllers 14A, 14B, and 14C produced lower stress Sj levels than the others. Stress rate levels were higher. On con- troller 14C, measurement 6 (lagged acceleration) Increased bending-mode damping and lowered the short-period frequency. The short-period damping ratio was somewhat higher, however.

Controllers 5D, SB, and 14C are the most desirable controllers from the bending-mode damping viewpoint, which could be Important for stability margins. Controller SB is the least complex of the three; however, 14 C is only different by a lag network. Controller 5D uses an extra sensor.

Table V summarizes the performance of two state-feedback and three practical controllers for a more complex model containing six bending modes. The results are not markedly different using a more complete model.

Table VI shows the effectiveness of spoiler controls in providing gust load relief« These are for state controls. Since full-state feedback was used, conservative depreciation was used from these answers to provide the numbers of Table I representing the results of simplified control.

Table VIII presents results for maneuvering load control (MLC). The maneuver is a step column input that attains a steady-state value of 1. S Incremental g.

■■ •

The free aircraft does not use symmetric ailerons in the steady state. At +1,5 incremental g, the wing root perturbation stress is -17,400 psi. The peak perturbation stress achieved during the transient for the step column input is -18,700 psi (row 1 of Table Vin).

24

'"

*

-4^

Table Vm. Maneuver Load Control8

•l StMdy- Stett

(lO'pti) (lO'pM)

A. Stvady- sutt (d«()

Steady- State (de(>

Remarfct

-17.4 - 0.11 - 6.27

- 9.81

-18.7

-H.l - 8.68 -10.6

0 -as -38

-as

0 0

-10 0

Free aircraft Free aircraft Free aircraft

With feedback

Vor ♦!• 8 ineremeatal-f command. Relief ia linear with control. Hence, reaulta may be uaed to determine effectiveneaa with different aurface deflecliona.

The effects of connecting the input of the aileron actuator to the control column are shown in row 2 of Table VIII. Steady-state values at 1.5

perturbation g of s. and Ö are -9810 psi and 25 deg. Thus, the steady-state MLC relief is 1 - i.?Vnn = 0.4362. The ratio of peak stresses without and with MLC for the free aircraft is 1 • flf ygj - 0. 3529.

Table VIII (row 3) shows that adding the spoiler to the free aircraft provides further reductions in both steady and peak wing root stresses.

Table VIII (row 4) shows that feedback provides attenuation of peak maneuvering stresses; but, of course, it can do no better than the free air* craft (with MLC ailerons) in the steady state at -»-1.5 incremental g.

_.

SECTION V LICS FUNCTIONAL BLOCK DIAGRAMS

The block diagram corresponding to simplified controller 14C (Table IV) Is shown in Figure 8. The MLC feedforward gain was computed to enforce the appropriate steady-state aileron surface deflection per g. At a single flight condition this was satisfactory. However« flight throughout the enve- lope would have required excessively complex scheduling because of the large variation in control column deflection per g with center-of-gravity variations. The system was then revised to provide maneuvering relief with a feedback control (Figure 9) using the Inputs to the aileron and elevator servos. The only feedforwards remaining are the existing mechanical links in the aircraft. Gains and time constants were not determined for this con- figuration.

Similarly, Figure 10 illustrates a possible functional block diagram of a system using the spoilers in addition to the aileron and elevator. The various gains shown were not determined.

System I (Figure 9) requires one additional pair of accelerometers in the wings as input sensors, and it makes use of the autopilot normal accel- erometers and the pitch augmentation rate sensors already on board the air- craft. System II (Figure 10) requires two additional pairs of dual accelerom- eters in each wing, and also requires the addition of dual-hydraulic servos to control the spoilers. A norm-rough weather switch is also required in the cockpit.

. ,.. .——.- " ' * ■ —■. ,■■ i ii .- ii ■ « ' I i-ir Lin ■ II.M».I—

_ .._

'

27

■

B

I

CO Ü

I Q

a>

28

__.

■ ■

29

b..

i

i

1 ■■'

SECTION VI CONCLUSIONS AND RECOMMENDATIONS

Results of the LICS study showed that rms wing root stress and stress rate can be reduced by factors of 50 percent and 31 percent, respectively, by using symmetric aileron and elevator control. However, two deficiencies become apparent;

• Handling qualities are degraded somewhat (this is noticeable when observing transients from step commands).

e No means for enforcing steady-state load relief are included.

At the time this report was being written, additional work had been done on the C-5A under contract from the Air Force.

Results from this study (Ref. 6) indicate that the handling qualities can be maintained with some loss in rms stress performance. The second deficiency can be corrected by at least two techniques:

e Integral control can be used to enforce steady-state load relief proportional to normal acceleration.

e Direct accelerometer-to-aileron feedback can be used to give steady-state load relief. A highpass network is used on the aileron input to wash out all other steady-state inputs.

In future studies it is recommended that wing torsion be considered.

■

■

• • • •

■

APPENDIX SYSTEM MATRICES FOR FLIGHT CONDITION 37

For the usual notation the model has the form

x ■ Fx + GjU + G2il

r » Hx + Du

Mx

where

• • x ■ (w, ö/iij, Tij, iig, Tjj, r\y Tig, r\y 6^, 6^ , pj, p2, Pg,

.'

p4. P5. P6. w^

r « (Sj, 82, Sj, ij, ö^, ö^, Öft, fl^, tf/aj)

and y is defined in Table VII.

The six matrices (F, Gj, Gg. H, D, and M) corresponding to flight condition 37 follow.

.<■

.

. ■

31

■

■

r •MMii

*M 1 -.MI«K«fl» .«M««C*9| • «»«*<sr-n| •.SMSU'Ol -.•OOOTI-OI -.leion •01 •.SMIW*01 •• M«NC*ei -.HTMf .11 •.iiso>r*i*3 o . ». -.Tomt'oo •.«oii«e • 01 -.•«««»(«00 0. 1. .», «

•W» 1 •.**llte*no •.iN««i«ei *»»?>r-rti .II»»3C »fl •.1MT7C*0« •.)t3*re • 00 .i7ii*e*o> •. IHNC*tf •.•«•««r*») -.Jll»»f0» « • . .3*T«ie*oi -.utm • 01 -.7OIMC*01 0. I« ••• 9

*M s -•lftM|*«l -.UlTtC-«! - i*ri«r*Ai -.jjftsoe »01 -.TMIlfOO -.3I1SK< •0> •.MII3C*02 •• O«T«C*0> •.MflMC*A« • IM4«r*«* 0 •. .M*««c*oe '.tuft >9l .3*T4oe*ei o. •• >•. •

•M 4 .«MITC« •«• .♦tii»e»oo • 3T»«Jr-el -.IMMC • 01 ,7MNC*ti .»MtK • 00 -.3T500f03 , »IMtC*ll

•.*MNC< »«J .ium*o* o 0. .M6»Tt»00 .3**3je • 01 •miot.oi o. •. ••. •

•9« « .TWT4C« '<)• •.)i««sr*«o .iimrteo .«iriic-ii •.ISIMC*«! .»3T5*f • 01 .l*37Te*0> •< iiistr*03

-.II1W.«4 •.t*l«M*«* 0 •* •.IMITC*«! .»•Tk4C •ot •.*I«IK*01 0 •• •. •

•9M * «. lomr»nl 0. 9. ( 0. 0, i. • • . • . « 0. 0, •. •

•OH* T 0. 0 .I0000t< • 01 0. < t. 0. «. 0 •* 0. t 0. 0 •• o

*M* • 0. 0 •« .ioeooe*oi ( 0. 0. 1. 0 •• i. ( 0. 0 1. 0

•OW* • 0. 0, 0. i. < 0. 0<

•UtNM« •1 0. • •< 0. 1 0. 1. t. 0

«9M* If 1. • o. •. i 0. • ■•MtOtC*ll 0 •• •. i 0. o. • • 0

•ON* 11 •tNMr-M • •TI0«r-«* .»*Wdt-«3 -.WOOC-03 ( 0. 0

•9«* It

». 0. •

.MlMr*e« •. 0. * 0. 0

.1•»••£< 01 .«M^rc'O« IMMffll .I19>M**I .l*MU*OI .<M*M • 01 itMNf*! INSSC'OS

.«•7MC*** .7tMM*t9 •. -.tMOKMl •.fum*ti .IM>*e*0t .ia««w*ii o. •. t. •

•9M 11 •• t. 0. •. ( u •. • •« 1. 0. -.imtc.ot « 1. •• o •• 1. 3T7**r^-'t

•OH U • . 1. 0. •. ( 1. •• • •. ». 0. Pg •• •.im«c**< •* • .itflMr*Af #. •.

•9M 11 •. ( •t t, •.ACinc*ii ( •• !io«ooe*oi

•0«* 1*

:: o. * •. •

•OW* IT

.|30I*C*03 ( -.Mtne**« • !iifMe*ot

•• t. 1 •« (

0. t •• 1. 1

'.in*Wn* t. l<»»iT»»nf •aw i*

t. • . 0 •• I« ( |« e •• 1. • o. • . ( 0, 1 t. -.MMVM • IO«T|r.f»0

•OM 1« 0. 1. •.

:: • . 1. k

•t «. 0, • . ( 0. 0 • . .i»«o«r*ei o, ,

■

... .

Gl MATRIX

ROW 0.

ROW' 0.

ROW* 0.

ROW' 0.

ROW* 0.

ROW* 0.

ROW* 0.

ROW* 0.

ROW* •60000E >0l 0.

ROW 0. •60000E*01

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

ROW 0. 0.

33

r,2 MATRIX

ROW 1 0.

ROW 2 0.

ROW 3 0.

ROW 4 0.

ROW 5 0.

ROW 6 0.

ROW 7 0.

ROW 8 0.

ROW 9 0.

ROW 10 0.

ROW 11 0.

ROW 12 0.

ROW 13 0.

ROW 14 0.

ROW 15 0.

ROW 16 0.

ROW 17 0.

ROW It -.24279E*0l

ROW 19 ,517171*01

. . ■

H «*r»M

•3M 1 -.«*I«*C »m -.loioor »01 • »W»r •ii .»UNI • 00 -•«Mt« • 00 .2377IC*03 -.«♦*•« • 03 -.Il047«r*03 •.SM«7e ••»J -.«siue »0« A. o. .US07C »00 -.3*S7ie*02 -.MJ«K »01 •• 0. • M '.

*9M 2 -.T»i??r »Al -.TMMf >4A • 11633' •Ai .MtlU • 01 .?MSSE •01 .3M6K*I3 .I7«*7e »M .I3507C.04 .IMftO* •A* -.«MIIIT >0? o. 0. .»•3TIE •01 -.SIMX*<2 -.34tSK ••• ••

0. 0. «. *3N 1

.mrfr •A| -.|*|*X »0? .?33«7r • ol -.M96ie •03 •.no«*6e • 03 •.7033002 -.2»733C • •3 .I3|4*f03 -.?H1*C »A« .2MMC • 05 0. •. .MS1K •0? .3M««e*t3 .t«M«C »M -.*«3«2e*0l -.3n*«r »AJ 0. . Ml«r • A?

•M * .^AMSC >AI -.uo«oe •e? • >*702r •03 .i7«ne • 0« .13*»H •0« •.2«S«4C*M -.11M1C ••♦ -.*l23Se*03

-.«Mire .»< .«717K • 04 o. •. ••ItMll •03 .MMK*«3 .10MM ••2 •.342Ste>H -.5*65V »Al 0. . m«? •03

*M « 0. -.60*9or

•• • Al 1. 0. •«

9. •. • • «.

«. a.

•. 0. o. II9M A

«. «. o. 0. • . 0. t. a. 0. -.«OOAOt >0l 0. 0. •. f. «. a. t. • . o.

*M 7 •• 0. 0. •. •. •. 1. a. .i«eeer*Ai e. o. 0. •. • . •. a.

•« 0* 0. •M •

0. a. •. •. •• a. *lMHt« •1 1. •. : •• «. a.

1. ■9M* •

•. •. «. • . •. a. •. •. •. t. 0. a. «.

•M* 1* 0. 0. 0. •. •• a. •• 0. 0. 1. •• •• •«

»M* 11 .622?2r< Al .iuue< 01 -••Mm« •• -.303»7e»00 .I71I«C*M -.IMNCM»

•I*«M*f< •3 -IJII7W« M 0. •• t« 0. «. a. • .

•«** l> .1»M7E.«I .2t«m< «1 ,*????'■ Al .ll*63t« •1 -.S3*77t. •• -.3A347f00 .I711M« •2 •.imar*02

•.•««*«r< Bi -,3ii7?e. 0* • . t. ••• t. 0. a. •. •• t. *9M 13

.IMMf. 11 1. o. o. •• 0. •. a. «. «. 0. • . t. .1MMC*M 9. a. •• •. A.

!

1 ■■"■"

■

■

0 MATRIX

»a« i o. 0.

ROM 2 0. 0.

ROW 3 -•3S7S8E< »04 -•16M0E*0S

ROM 4 .lf596E< »ns -.59191E*03

ROM 5 •60000E«01 0.

ROM 6 0. •60000E«01

ROM 7 0. 0.

ROM A 0. 0.

ROM 9 •IDOOOC« •01 0.

ROM 10 0. •lOOOOE^Ol

ROM 11 0. 0.

ROM 12 0. 0.

ROM 13 0. 0.

35

M ««»TJI«

0. 1

0. 0, 0 .iooooe*ci 0. 0. 0, 0. 0. ft. 0 0 0. 0. 0. A. n. A.

•w ? -. imir ► no .j^Mir • 00 -.*10W-«> - 33»or».Aa - .»««..lE-Ol .mm >0I .MOArC'02 -. 3«26«P»02 •. »?«3*ir >n> .s**i»r >01 ,|*«Ur-ft3 0 .'7li<if0l -.««»•toe »01 .i*i6oe»oj o. 0. 0. 0.

»r* 1 • lOfOAf »A| •MMM no .156"»r»ol ^^^Hr,o^ .l*»«.IE»01 .29K60C »02 .27«0«e*03 , l29S3C*e3 . HT^Or »c» .Tf?«ir »oj 0. 0 .>ur«c*oi .iz*2*e »02 .iao«He*oi o

e. A. 0. •3t« *

«. A. .loooo«-.m o 0. 0. 0 0. 1. A. 0 0. 0. 0, 0. 0. A,

II3M % , 1 J?3f >"! .6A»1K 00 .*l?SAr.nO -, *»*»ie«oo - .2S5»AC«00 .•♦1*H »01 •.9S26SC*02 -. 97«22r»02 § MT04r .')•« .•lAT*e 01 A. 0. .»90**f0l .U*3*C »02 .20T07t«01 0.

0. 0. 0. • 3W A

.«MT«? on .*Tftm*oe .IIWOrtAO •. »I0JAE.00 - .^6»iac«oo .2«72ie »01 -.17802C«02 -, »nutfoi -. IISMf .«1 .7II10K 03 0. 0. .MVMC*!! .•soioe »01 .i«ii**e*»i o.

«. A. MK* T

0. , 10000J»01 • 0. 0. 0.

• DK* It 0. 0 .IA0eer*Al 0

0. •OK* 4

e. 0 loesor >-)! 0. 1

o. ■ 3«* 10

9. 0 Iioooot •01 0. 0.

0. •a»' u

0. 0. o. o,

.1A0A0»»A| ■9*' i?

0. 0, «. •. !ieoeee*oi •.

•3W* n «. 0. «. o liooooe »01 0.

•9K* i« A. 0, 1. •. Iieooee*oi o 1.

•9««' 14 0. 0. 0. •. lOOOOf.Ol 0.

• 3t«* 1* o. o 0. 0,

ie«m*Ai •. •M* >T

0. •, o. a.

I moor »01 •. • DU* 14

.<S0V.AF-03 -.Tftllftf-A» «l»*2r-03 - .1043AE-03 A. •.IOO0O»*AI 0. A. 0.

•3«** 1« 0. 0.

0. 0 ioeooe*oi o :: !

0. A.'

36

■ ■

REFERENCES

1. VanDierendonck, A. J., "Practical Optimal Flight Control for Aircraft with Large Flight Envelopes, " AIAA Paper No. 73-169, AIAA 11th Aerospace Sciences Meeting, January 1973.

2. VanDierendonck, A. J., "Practical Quadratic Optimal Control for Systems with Large Parameter Variations," Conference Record. Sixth Asilomar Conference on Circuits ar.^ Systems, pp. 391-396, November 1972.

3. Stone, C.R., Ward, M. D., Karvey, C.A., Ebsen, M.E., McBridle, E. E., and Hollenbeck, W. W., "Studies on the Compatibility of Relaxed Static Stability and Maneuver Load Control to C-SA-Type Aircraft," Volumes I and II, Technical Report AFFDL-TR-72-38, Wright- Patterson Air Force Base, Ohio, June 1972.

4. Edinger, L. and Lahn, T., "C-5A Data Base for Load Alleviation and Mode Stabilization Program." Report 20564-DB1, Honeywell Inc., Government and Aeronautical Products Division, Minneapolis, Minnesota, 1 April 1968.

5. Anonymous, "Aircraft Load Alleviation and Mode Stabilization (LAMS): C-5A System Analysis and Synthesis," Technical Report AFFDL-TR- 68-162, Wright-Patterson Air Force Base, Ohio, November 1969.

6. Barrett, M.F. and McLane, R.C., "Procedure and Results for ALDCS Task IU, Subtask IB (Part 1) (Old LICS Continuation at FC-37). " Honeywell Inc. Memorandum MR-12223. 4 September 1973.

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