Advances in Industrial Control

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Other titles published in this Series:

Energy Efficient Train Control P.G. Howlett and P.J. Pudney

Hierarchical Power Systems Control: Its Value in a Changing Industry Marija D. Ilic and Shell Liu

System Identification and Robust Control Steen T0ffner-Clausen

Genetic Algorithms for Control and Signal Processing K.F. Man, K.S. Tang, S. Kwong and W.A. Halang

Advanced Control of Solar Plants E.F. Camacho, M. Berenguel and F.R. Rubio

Control of Modern Integrated Power Systems E. Mariani .and S.S. Murthy

Advanced Load Dispatch for Power Systems: Principles, Practices and Economies E. Mariani and S.S. Murthy

Supervision and Control for Industrial Processes Bjorn Sohlberg

Modelling and Simulation of Human Behaviour in System Control Pietro Carlo Cacciabue

Modelling and Identification in Robotics KrzysztofKozlowski

Spacecraft Navigation and Guidance Maxwell Noton

Robust Estimation and Failure Detection Rami Mangoubi

Adaptive Internal Model Control Aniruddha Datta

Price-Based Commitment Decisions in the Electricity Market Eric Allen and Marija Ilie

Compressor Surge and Rotating Stall Jan Tommy Gravdahl and Olav Egeland

Radiotherapy Treatment Planning Oliver Haas

Feedback Control Theory For Dynamic Traffic Assignment Pushkin Kachroo and Kaan 6zbay

Control Instrumentation for Wastewater Treatment Plants Reza Katebi, Michael A. Johnson and Jacqueline Wilkie

Rick Lind and Marty Brenner

Robust Aeroservoelastic Stability Analysis Flight Test Applications

With 70 Figures

Springer

Rick Lind, PhD, MS, BS, BS

Marty Brenner, MS, MS, BS Dryden Flight Research Center MS4840D/RS CA93523 USA

ISBN-13:978-1-4471-121S-0 Springer-Verlag London Berlin Heidelberg

British Library Cataloguing in Publication Data Lind. Rick

Robust aeroservoelastic stability analysis: flight test applications. - (Advances in industrial control) l.Flight control2.Aeroelasticity - Stability I.Title II.Brenner. Marty 629.1'326 ISBN-13:978-1-4471-1215-0

Library of Congress Cataloging-in-Publication Data Lind. Rick, 1968-

Robust aeroservoelastic stability analysis: flight test applications / Rick Lind and Marty Brenner.

p. em. -- (Advances in industrial control) Includes bibliographical references (p. ). ISBN-13: 978-1-4471-1215-0 e-ISBN-13: 978-1-4471-0849-8 DOl: 10.1007/978-1-4471-0849-8

I. Stability of airplanes--Mathematical models. 2. Aeroelasticity. I. Brenner. Marty. 1955- . II. Title. III. Series. TL574.S7L47 1999 629.132'362'OI5118--dc21 98-52346

CIP Apart from any fair dealing for the purposes of research or private study. or criticism or review. as permitted under the Copyright. Designs and Patents Act 1988. this publication may only be reproduced, stored or transmitted, in any form or by any means. with the prior permission in writing of the publishers. or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

© Springer-Verlag London Limited 1999 Softcover reprint of the hardcover 1st edition 1999

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69/3830-543210 Printed on acid-free paper

Advances in Industrial Control

Series Editors

Professor Michael J. Grimble, Professor ofIndustrial Systems and Director Professor Michael A. Johnson, Professor of Control Systems and Deputy Director

Industrial Control Centre Department of Electronic and Electrical Engineering University of Strathclyde Graham Hills Building 50 George Street Glasgow G1 1QE United Kingdom

Series Advisory Board

Professor Dr-Ing J. Ackermann DLR Institut fur Robotik und Systemdynamik Postfach 1116 D82230 WeBling Germany

Professor I.D. Landau Laboratoire d'Automatique de Grenoble ENSIEG, BP 46 38402 Saint Martin d'Heres France

Dr D.C. McFarlane Department of Engineering University of Cambridge Cambridge CB2 1 QJ United Kingdom

Professor B. Wittenmark Department of Automatic Control Lund Institute of Technology PO Box 118 S-221 00 Lund Sweden

Professor D.W. Clarke Department of Engineering Science University of Oxford Parks Road Oxford OX1 3PJ United Kingdom

Professor Dr -Ing M. Thoma Institut fur Regelungstechnik Universitat Hannover Appelstr. 11 30167 Hannover Germany

Professor H. Kimura Department of Mathematical Engineering and Information Physics Faculty of Engineering The University of Tokyo 7-3-1 Hongo Bunkyo Ku Tokyo 113 Japan

Professor A.J. Laub College of Engineering - Dean's Office University of California One Shields Avenue Davis California 95616-5294 United States of America

Professor J.B. Moore Department of Systems Engineering The Australian National University Research School of Physical Sciences GPO Box 4 Canberra ACT 2601 Australia

Dr M.K. Masten Texas Instruments 2309 Northcrest Plano TX 75075 United States of America

Professor Ton Backx AspenTech Europe B.V. De Waal32 NL-5684 PH Best The Netherlands

SERIES EDITORS' FOREWORD

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology impacts all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, .... , new challenges. Much of this deVelopment work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination.

The high performance control systems applications in aerospace and astronautics almost have a tradition of exploiting the most advanced control theoretical developments first. The optimal control and ffitering paradigm associated with the names of Kalman, Bucy, Anderson and Moore found application in the astronautics of the 1960'S and 1970'S.

At the beginning of the 1980'S, control theory moved on to robustness, singular values and mu-analysis. This new work was associated with the names of Zames, Doyle, Glover, Balas among others. The Advances in Industrial Control monograph series have published several volumes over the years which have archived the applications experience garnered from applying robust control to the aerospace sector problems. Rick Lind and Marty Brenner add to this set with their volume on robust aeroservoelastic stability. This volume reports the application of the structured singular value to aeroelastic and aeroservoelastic aerospace problems. A complete and systematic presentation is made of this area so that the monograph is essentially self-contained. A strong motivation for the work was the cost saving obtained from reducing the number of test flights needed to probe and determine safe flight envelopes.

The monograph is recommended for all specialists in the aerospace sector and also for the general reader seeking an insight into the practical application of the ~u-analysis method.

M.J. Grimble and M.A. Johnson Industrial Control Centre

Glasgow, Scotland, UK

ACKNOWLEDGEMENTS

The authors would like to thank the Structural Dynamics group at NASA Dryden Flight Research Center for their invaluable assistance in developing the material for this text. Leonard Voelker was instrumental in formulating the f..£ method by continuously providing lessons about the physics. of flut­ter and how to interpret f..£ in terms of this phenomenon. Larry Freudinger shared his vast knowledge of data analysis and helped to develop the con­cepts of uncertainty for aeroservoelastic models. Dave Voracek gave in-depth information about the workings of an F / A-18 and demonstrated methods of flight flutter testing that could be improved by applying the f..£ method. Roger 'Iruax and Tim Doyle generated the finite element model of the F / A-18 and computed nominal stability margins using classical methods. Mike Kehoe suggested many research avenues to address practical concerns of the aeroelasticity community and strengthen the applicability of the f..£ method to flight test programs. Also, Kajal Gupta deserves significant recognition for his efforts at developing the high fidelity tool for aeroservoelastic modeling and analysis that we relied upon to investigate stability of the F / A-18.

The authors would also like to ackn9wledge several other people that have contributed to the making of this text. Gary Balas has taught us many lessons about robust stability and the concept of f..£ in addition to providing insight into uncertainty modeling. Kari Appa was a dedicated and inspiring mentor in the area of aero elasticity and flight dynamics. Tom Strganac and Andy Kurdila developed an aeroservoelastic testbed and generously provided models as an example with which to compute robust stability margins. Ad­ditionally, we would like to thank Rudy Yurkovich, Ken Griffin, Mordechay Karpel and Eli Livne for their helpful comments and support with regard to using f..£ for flight test programs.

CONTENTS

1. Introduction........................ . . . . . . . . . . . . . . . . . . . . . . 1

2. Robust Stability ..... .................................. '" 7 2.1 Signals................................................ 7 2.2 Systems............................................... 9 2.3 Small Gain Theorem ................................. '... 11 2.4 Robust Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12

3. Structured Singular Value : J.L. • • . • • • . . • . • • . . • • • • • • • • • • • • •. 15 3.i Linear Fractional Transformations. . . . . .. . . .. . . . . . . .. . . . .. 15 3.2 Structured Uncertainty. . . .. . . . . . . .. . . . . .. . . . . . . . . . . . . . .. 18 3.3 Structured Singular Value: J.L • • • • • • • • • • • • • • • • • • • • • • • • • • •• 21 A.3 Upper Bound for J.L •••• •••••••• •• •• •••••• ••••••••••••••• 23

4. Aeroservoelasticity....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29 4.1 Lagrangian Derivation of Equations of Motion ............. 29 4.2 Aerodynamic Derivative Representation . . . . . . . . . . . . . . . . . .. 34 4.3 Servoelastic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 4.4 Aeroelastic and Aeroservoelastic Stability. . . . . . . . . . . . . . . . .. 46 A.4 Inertial to Body-fixed Coordinate Transformation .......... 50

5. Aeroelastic and Aeroservoelastic Models.. . ........ ....... 55 5.1 Aeroelastic Equation of Motion .......................... 55 5.2 Nominal Aeroelastic Model. . . . . . . .. . . . . .. . . . . . . . . . . . . . .. 57 5.3 Robust Aeroelastic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60 5.4 Robust Aeroservoelastic Model. . . . . . . . . . . . . . . . . . . . . . . . . .. 61 A.5 Models with Specific Aerodynamic Elements . . . . . . . . . . . . . .. 64

6. Uncertainty Descriptions.. .... ................ .. .. .. . . ... 67 6.1 Parametric Uncertainty in Structural Models. . . . . . . . . . . . . .. 67 6.2 Parametric Uncertainty in Aerodynamic Models. . . . . . . . . . .. 70 6.3 Parametric Uncertainty in Modal Models. . . .. . . . . . . . . . . . .. 74 6.4 Dynamic Uncertainty. . .. . . . . . . . . . . . . . . . . .. . . . . . . .. . . . .. 80 6.5 Uncertainty Associated with Nonlinearities ................ 84 6.6 Uncertainty Associated with Flight Data. . . . . . . . . . . . . . . . .. 89

7. Incorporating Flight Data....... .. .. ...... ............... 91 7.1 Model Validation. . . . . . .. .. . . . . . . . . .. . . . . .. . . . . . . . . . . . .. 91 7.2 Determining Levels of Uncertainty. . . . . . . . . . . . . . . . . . . . . . .. 93

XII Contents

8. Stability Margins.... .. .... .... ...... .... . . .... .. .. .... ... 99 8.1 Robust Aeroelastic Stability Margins. . . . . . . . . . . . . . . . . . . . .. 99 8.2 Robust Aeroservoelastic Stability Margins ................. 105 8.3 Properties of Robust Stability Margins . . . . . . . . . . . . . . . . . . .. 108

9. f..£ Method ................................................ 111 9.1 Model Updating ........................................ 111 9.2 Approaches to Utilize Flight Data ........................ 113

10. Robust Stability Margins of a Pitch-Plunge System ....... 117 10.1 Equations of Motion .................................... 117 10.2 Nominal Aeroelastic Model .............................. 120 10.3 Robust Aeroelastic Model ............................... 124 10.4 Robust Aeroservoelastic Model ........................... 132 10.5 Aeroelastic Stability Margins ............................ 138 1O.~ Aeroservoelastic Stability Margins ........................ 140 A.lO Computer Code ........................................ 143

11. Robust Flutter Margins of the F / A-18 SRA .............. 153 11.1 Flight Flutter Test ..................................... 153 11.2 Flight Data Analysis .................................... 155 11.3 Analytical Model ....................................... 159 11.4 Uncertainty Description ................................. 161 11.5 Nominal and Robust Flutter Pressures .................... 164 11.6 Nominal and Robust Flutter Margins ..................... 167 11. 7 Computational Evaluation ...... ; . . . . . . . . . . . . . . . . . . . . . . .. 170

12. Robust Aeroservoelastic Stability of the F / A-18 HARV .. 173 12.1 Sensing and Control Elements ............................ 173 12.2 Analytical Model ....................................... 174 12.3 Uncertainty Description ................................. 176 12.4 Stability Margins ....................................... 179

13. On-Line Analysis during a Flight Test .................... 183 13.1 Flutterometer............. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183 13.2 Aircraft Model ......................................... '185 13.3 Simulated Flight Test ................................... 187 13.4 On-Line Robust Flutter Margins ......................... 191

14. Conclusion ............................................... 195

References .................................................... 197

Index ......................................................... 205

NOMENCLATURE

9 9 k kT ko,bo,.mo k1,b1,ml n ns ni

no nQ nw nz

p,q,r q qo

qnom

qrob r s

u

body axis perturbation accelerations wing span total actuator damping half mean aerodynamic chord mean aerodynamic chord aeroelastic damping gravitational acceleration reduced frequency total control surface stiffness aero stiffness, aero damping, and control surface mass modal stiffness, damping, and mass number of actuator hydraulic systems number of states number of inputs number of outputs number of states in unsteady aerodynamic force matrix number of feedback signals from uncertainty to plant number of feedback signals from plant to uncertainty perturbation roll, pitch, and yaw rate dynamic pressure nominal value of dynamic pressure dynamic pressure associated with nominal stability analysis dynamic pressure associated with robust stability analysis position vector Laplace variable actuator controller state time input signal output signal feedback signal from uncertainty to plant state vector derivative of x with respect to time actuator ram moment arm displacement of piston relative to main ram body displacement of main control valve displacement of control surface mass displacement of structural modal mass feedback signal from plant to uncertainty discrete variable

aerodynamic influence coefficient matrix ith term in Roger's form of unsteady aerodynamic forces

XIV Nomenclature

Ap Ax, Ay,Az AQ Be BQ Cc

C1,Cm,Cn Cp

Cv

CD, CL , Cy CQ DQ C D V Fe Fu (" .)

Fl(·, .) G 9 K In M P P Pj

Pij

P1 ,Ql,R1

Q Q s S T U1 , VI, WI W Wq-

DEI KEAS LFT LMI LTI LTV SRA

main actuator ram piston area body axis accelerations state matrix of unsteady aerodynamic force matrix effective stiffness correction matrix input matrix of unsteady aerodynamic force matrix cross piston leak flow coefficient coefficients of roll, pitch, and yaw moment actuator fluid flow pressure constant main valve flow coefficient coefficients of drag, lift, and side force output matrix of unsteady aerodynamic force matrix feed through matrix of unsteady aerodynamic force matrix damping matrix scaling matrix set of scaling matrices externally applied actuator test force upper loop linear fractional transformation in Definition 3.1.1 lower loop linear fractional transformation in Definition 3.1.2 scaling matrix set of scaling matrices stiffness matrix identity matrix of dimension n mass matrix transfer function operator set of system operators pressure difference across actuator ram piston transfer function element of P reference (trim) roll, pitch, and yaw rate unsteady aerodynamic force matrix Q without the rigid air loads state-space system operator reference area of wing sample time for digital controller reference (trim) velocities in body-fixed frame of reference matrix scaling uncertainty operators matrix scaling feedback perturbation to dynamic pressure

Dynamic Engineering Incorporated knots of equivalent airspeed linear fractional transformation linear matrix inequality linear time-invariant linear time-varying Systems Research Aircraft

a al

a(w) (3 (3i o Od Jq ~

..!l

'TJ (),¢,'ljJ 8 1 , Pl,!]il

r ri ]

A,A,Ai,Ai /L II p (J

(j

T

cnxm

R nxm

1-£2 1-£00 n1-£oo £00 £2 £2 (-00,00) £2[0, (0) S SJw

Ilx(t)112 Ilf(]w)112 11P1100

angle of attack trim angle of attack equivalent downwash angle of sideslip aerodynamic lag terms uncertainty scalar actuator command

Nomenclature XV

uncertainty perturbation on dynamic pressure uncertainty matrix set of uncertainty matrix operators generalized coordinate perturbation pitch, roll, and yaw angles reference (trim) pitch, roll, and yaw angles stability margin expressed as difference in flight condition inertial to body-fixed coordinate transformation matrices imaginary unit of A inertial to body-fixed coordinate transformation matrices structured singular value stability margin expressed as percentage of flight condition spectral radius singular value maximum singular value system time delay actuator filter time constants rigid, elastic, and control mode shape matrices frequency in units of radians per second set of frequency points

space of complex valued matrices of dimension n by m space of real valued matrices of dimension n by m Hardy space in Definition 2.1.8 stable rational transfer functions in Definition 2.2.3 stable rational transfer functions in Definition 2.2.4 Lesbegue space of transfer functions in Definition 2.2.2 frequency domain Lebesgue space in Definition 2.1.7 time-domain Lebesgue space in Definition 2.1.3 time-domain Lebesgue space in Definition 2.1.4 set of signals in Definition 2.1.1 set of frequency domain signals in Definition 2.1.5

2-norm of time domain signal in Definition 2.1.2 2-norm of frequency domain signal in Definition 2.1.6 1-£oo-norm induced by £2 signals in Definition 2.2.1