Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
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Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo
Christopher Fielding, Andras Varga, Samir Bennani, Michiel Selier (Eds.)
Advanced Techniques for Clearance of Flight Control Laws With 244 Figures
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Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis
Editors Christopher Fielding Msc. Aerodynamics (W427D) BAE Systems Warton, Preston PR4 1AX England, UK
Dr.-Ing. Andras Varga Deutsches Zentrum f¨ur Luft- und Raumfahrt German Aerospace Center DLR - Oberpfaffenhofen Institute of Robotics and Mechatronics 82234 Wessling, Germany
Dr. Samir Bennani Delft University of Technology Faculty of Aerospace Engineering Kluyverweg 1 2629 HS Delft, The Netherlands
Michiel Selier Msc. National Aerospace Laboratory (NLR) Flight Mechanics Department Anthony Fokkerweg 2 1059 CM Amsterdam, The Netherlands
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Advanced techniques for clearance of flight control laws / Christopher Fielding . . . (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; 283) (Engineering online library) ISBN 3-540-44054-2
ISBN 3-540-44054-2
Springer-Verlag Berlin Heidelberg New York
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Preface
In recent years, a major aim of flight control systems research has been to achieve a high level of performance and safety by improving the design methods. Researchers and academia have concentrated their activities on the synthesis aspects of flight control systems, in particular by demonstrating the applicability and strengths of novel, robust, multivariable synthesis tools. Significantly less research effort has been spent on the global assessment of the achieved designs, which represents a key activity for the certification of today´s aircraft, which are equipped with safety-critical, highly complex flight control systems. Currently, the aeronautical industry is faced with the formidable task of clearance of the flight control laws. Before an aircraft can be tested in flight, it has to be proven to the authorities that the flight control system is safe and reliable, and has the desired performance under all possible operational conditions, and in the presence of failures. This motivated the research presented in this book: an exploration of the benefits of new analysis techniques for the clearance of flight control laws. It is a first step towards a better and deeper understanding of the industrial flight clearance process, with the objective to provide recommendations on how analysis techniques should evolve in order to improve the efficiency and reliability of this process. The Group for Aeronautical Research and Technology in Europe (GARTEUR) provided an ideal framework to bring together research institutes, academia and industry and pursue such a relevant research objective. This book is a result of a research effort performed by GARTEUR Flight Mechanics Action Group 11 FM(AG11). It would not have been possible without all individuals and organisations that have contributed to this group. GARTEUR FM(AG11) is also very thankful to all people from outside the group that have contributed with their constructive comments in the form of reviews or industrial evaluations.
June 2002
The Editors
Table of Contents
Part I Industrial Clearance of Flight Control Laws 1 Introduction Michiel Selier, Udo Korte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Tasks and Needs of the Industrial Clearance Process Udo Korte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Part II Tutorial on Analysis Methods 3 The Structured Singular Value and µ-Analysis Declan G. Bates, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 The ν-Gap Metric and the Generalised Stability Margin John Steele, Glenn Vinnicombe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 A Polynomial-Based Clearance Method Leopoldo Verde, Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Bifurcation and Continuation Method Mark Lowenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7 Optimisation-Based Clearance Andras Varga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Part III The HIRM+RIDE Benchmark 8 The HIRM+ Flight Dynamics Model Dieter Moormann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9 The RIDE Controller David Bennett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10 Selected Clearance Criteria for HIRM+RIDE Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
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Part IV LFT Modelling of Uncertainty Models 11 An Overview of System Modelling in LFT Form Jean-Fran¸cois Magni, Samir Bennani, Jean-Paul Dijkgraaf . . . . . . . . . . 169 12 Physical Approach to LFT Modelling Jean-Paul Dijkgraaf, Samir Bennani, GertJan Looye, Jean-Francois Magni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 13 Uncertainty Bands Approach to LFT Modelling Thomas Mannchen, Klaus H. Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 14 Flatness Approach to LFT Modelling Franck Cazaurang, Lo¨ıc Lavigne, Benoˆıt Bergeon . . . . . . . . . . . . . . . . . . . . 221
Part V Analysis Results 15 Baseline Solution Tobias Wilmes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 16 µ-Analysis of Linear Stability Criteria Declan G. Bates, Ridwan Kureemun, Martin J. Hayes, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 17 µ-Analysis of Stability Margin Criteria Thomas Mannchen, Klaus H. Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 18 ν-Gap Analysis of Stability Margin Criteria John Steele, Glenn Vinnicombe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 19 Polynomial-Based Clearance of Eigenvalue Criteria Leopoldo Verde, Federico Corraro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 20 Bifurcation-Based Clearance of Linear Stability Criteria Mark Lowenberg, Thomas Richardson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 21 Optimisation-Based Clearance: The Linear Analysis Andras Varga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 22 Optimisation-Based Clearance: The Nonlinear Analysis Lars Forssell, Andreas Sandblom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
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Part VI Conclusions and Recommendations 23 Industrial Evaluation Fredrik Karlsson, Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 24 Considerations for Clearance of Civil Transport Aircraft Robert Luckner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 25 Concluding Remarks Michiel Selier, Rick Hyde, Chris Fielding . . . . . . . . . . . . . . . . . . . . . . . . . . 457
A Nomenclature and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
1 Introduction Michiel Selier1 and Udo Korte2 1
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National Aerospace Laboratory NLR Flight Mechanics Department, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands.
[email protected] EADS Deutschland GmbH, Military Aircraft, MT 62 Flight Dynamics, 81663 M¨ unchen, Germany.
[email protected] Summary. We describe the background and motivation for the research carried out on advanced techniques for the clearance of flight control laws within the GARTEUR Flight Mechanic Action Group 11. This project involved 19 European partners representing research establishments, industry and universities. The core activity of this project was the HIRM+ clearance benchmark, whose main results are presented in this book.
1.1 The Importance of Research on Flight Control Law Clearance 1.1.1 Project Background Aircraft manufacturers have reached a high level of expertise and experience in flight control law design. The current design and analysis techniques applied in industry enable flight control engineers to address virtually any realistic design challenge. However, the development of flight control laws from concept to validation is a very complex, multi-disciplinary task and the many problems that have to be solved make it a costly and lengthy process. Researchers in universities and research institutes have developed new, advanced mathematical methods for design and analysis that have the potential to improve the flight control law development process. In the past decade, the Group for Aeronautical Research and Technology in Europe (GARTEUR) has established action groups to investigate the potential benefits and drawbacks of several of these new synthesis and analysis methods. From 1994 until 1997, the GARTEUR Flight Mechanics Action Group 08, FM(AG08), performed successful research on ”Robust Flight Control”. A design challenge was carried out, in which a set of robust control design methods were applied both to a civil and a military aircraft model. The results produced by this group (reports, two benchmark models and a book ”Robust Flight Control” [1]) are widely appreciated in the aerospace control community. In 1999 the Flight Mechanics Action Group 11, FM(AG11), was C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 3-11, 2002. Springer-Verlag Berlin Heidelberg 2002
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established to address a new and complementary challenge. This resulted in GARTEUR reports, three benchmark models and this book. This new challenge focused on the clearance process of the flight control algorithms. The clearance (or assessment) of control laws can be seen as the last step of the flight control system design, taking place when a mature controller design is available and ready for flight tests. In the clearance process it has to be proven that the flight control laws have been designed such that the aircraft is safe to fly throughout the whole flight envelope, under all parameter variability and failure conditions. The background and motivation for this research are briefly highlighted below. 1.1.2 Flight Control Systems As described in [2], the Flight Control System (FCS) enables the pilot to control the aircraft along a desired trajectory and provides safe and economic operation. Pilot inputs are translated into deflections of the aircraft’s control surfaces, which in turn change the aerodynamic forces and moments acting on the aircraft. In the early days, the FCS was a purely mechanical system, which connected the control devices of the pilot directly to the control surfaces of the aircraft by a system of rods, levers, cables and pulleys. The aerodynamic forces on the control surfaces were limited by the physical capabilities of the pilot. The aircraft in those days usually possessed natural aerodynamic stability. The size and operating speed of aircraft increased as aviation evolved with time. The control force that was needed to overcome the aerodynamic forces on the control surfaces also increased, until a point was reached where the required efforts exceeded the pilots’ physical capabilities. To assist the pilots, the FCS was augmented with hydraulic actuators to provide the required control force. This was the first step that removed the direct connection between the pilot and the control surfaces and the mechanical linkages between the pilots’ inceptors and the actuators now transmitted (mechanical) displacements instead of transmitting force. Eventually, developments in aviation, especially in the area of automation of flight, led to the development of the fly-by-wire FCS, in which electrical signals are transmitted between the pilot and the actuators instead of mechanical signals. Today’s high performance aircraft can no longer be flown directly by the pilot. This is especially true for fighter aircraft, which are often designed to be naturally unstable to improve performance. Redundant electronic flight control systems with sophisticated control algorithms running on digital computers are needed to assure integrity and reliability, and to provide the required stability, performance and handling characteristics. The design process for a modern FCS is a complex, multi-disciplinary activity, which has to be transparent, correct and well-documented in order to allow certification of the aircraft. The design and validation of the flight
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control laws (FCLs) is an important part of the FCS design. The FCL design process can be divided into the following five phases: 1. In the off-line design phase the control system architecture and FCL structure are defined and the control law parameters are tuned to achieve the desired handling qualities and closed-loop performance specifications. The control law design is completed by adding appropriate blocks to the control configuration (e.g., gains, filters, nonlinear functions) to guarantee the correct functioning of control laws within the allowed operational ranges. The stability and performance of the resulting design are assessed mainly by employing linear system analysis techniques and nonlinear simulations. 2. Via pilot-in-the-loop simulation the handling qualities and many operational issues of the augmented aircraft are assessed. 3. In iron-bird tests it is verified that the FCLs operate correctly with the FCS hardware in the loop. 4. In the clearance it is formally proven to the authorities that the designed FCLs fulfil all requirements for safe operation of the aircraft throughout the whole flight envelope, and under all foreseeable parameter variability and failure conditions. 5. Finally, flight tests are executed in which the FCS design is validated with respect to the aircraft specification derived from customer and airworthiness requirements. The FCL design process has a strong iterative nature, especially in the offline design phase. However, deficiencies in control laws found in later stages require iterations as well, since the design engineers have to go back to the first phase to improve the controller design. The cost for such modifications increases significantly with each phase. Once a controller design is considered to be sufficiently mature, the clearance task is started. Although FCL analysis and design clearance takes place during all phases of the design process, a formal clearance (phase 4) is required before flight testing (phase 5) can take place. The clearance process is described in more detail in the next section. 1.1.3 The Clearance Process and Potential Improvements As the safety of the aircraft operation is primarily dependent on the designed flight control laws, it must be proven to the clearance authorities that the flight controller is functioning correctly throughout the whole flight envelope in all normal and various failure conditions, and in the presence of all possible parameter variations. The role of clearance is to demonstrate, via exhaustive analyses, that a catalogue of selected criteria expressing stability and handling requirements is fulfilled. Typically, criteria covering both linear and non-linear stability, as well as various handling and performance requirements are employed for the purpose of clearance.
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The clearance of FCLs is a lengthy and expensive activity, especially for fighter aircraft, where many different store configurations have to be investigated involving large variations of mass, inertia, and centre of gravity location, as well as uncertain structural modes and aerodynamic data (often expressed through highly nonlinear dependencies). In addition, the error tolerances in the aerodynamic data and on air data signals used for control law scheduling, have to be taken into account. The complex aircraft models used for clearance purposes describe the actual aircraft dynamics, but only within given uncertainty bounds. One reason for this is the limited accuracy of the aerodynamic data set determined from theoretical calculations and wind tunnel tests. These parameters can even differ between two aircraft of the same type, due to production tolerances. Especially at high angles of attack, local flow separation effects can be different due to these tolerances. Furthermore, the employed sensor, actuator and hydraulic models are usually only linear approximations, where nonlinear effects are not fully modelled because they are either not known or it would make the model unacceptably complex. To perform the clearance, for each point of the flight envelope, for all possible configurations and for all combinations of parameter variations and uncertainties, violations of clearance criteria and the worst-case result for each criterion have to be found. Based on the clearance results, flight restrictions are derived when necessary. Since flying the aircraft in the presence of failures might involve the use of alternative control laws (e.g., by switching to a backup control law after the loss of a certain sensor or an engine failure), the number of additional cases that has to be investigated can be significant. The huge amount of assessment work, typically on systems of very high order, requires fast, efficient and numerically reliable methods and routines for the calculation and visualisation of results. A major improvement can be expected by increased automation of the tools used for model-based analysis of the aircraft’s behaviour. The objective should not be the faster production of analysis data, because a high degree of automation already exists. New techniques are needed for the faster detection of combinations of parameter values and manoeuvre cases for which flight clearance restrictions are necessary. Such ”worst cases” may be caused by rather obscure combinations of events and flight conditions, which makes it particularly difficult to detect them. Over the past two decades, several mathematical techniques have been developed for the analysis of linear and nonlinear systems with uncertain parameters. Each of these technique has its known strengths and weaknesses. However, at this moment it is still difficult for the aeronautical industry to assess whether their application would improve the efficiency of the FCL clearance process. The main objective of the research activity described in this book was thus to explore the potential benefits of using advanced
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analysis methods for the clearance of flight control laws, by demonstrating some of the most promising techniques on realistic flight clearance problems. Analysis of a complex system produces complex results. Good visualisation is essential to gain a deeper understanding of the clearance results on which important decisions about the airworthiness of an aircraft are based. For this reason, a secondary objective of the research activity was to explore, based on a ”wish list” from industry, new tools that would improve the visualisation of clearance results. It is important to keep in mind that the question addressed here is not a purely technical one, since industry is already technically able to successfully clear flight control laws. The main industrial benefits of new methods should be related to reducing the involved effort and cost, while getting sufficiently reliable results, or increasing the reliability of the analysis results within a reasonable amount of effort.
1.2 Description of GARTEUR FM(AG11) In 1999, GARTEUR FM(AG11) ”New Analysis Techniques for clearance of flight control laws - NEAT” was established to address the research objective described in the previous section. 1.2.1 Project Organisation In this group, 19 organisations from European research establishments, industry and universities participated: Research Establishments 1. Centro Italiano Ricerche Aerospaziali (CIRA, Italy, Capua) 2. Deutsches Zentrum fu ¨r Luft- und Raumfahrt e.V. – DLR-Braunschweig, Germany, Braunschweig – DLR-Oberpfaffenhofen, Germany, Oberpfaffenhofen 3. Totalf¨orsvarets Forskningsinstitut - The Swedish Defence Research Agency (FOI, Sweden, Stockholm) 4. Instituto Nacional de T`ecnica Aeroespacial (INTA, Spain, Madrid) 5. National Aerospace Laboratory (NLR, The Netherlands, Amsterdam) ´ 6. Office National d’Etudes et de Recherches A´erospatiale – CERT-ONERA, France, Toulouse – ONERA-Salon, France, Salon de Provence
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Industrial Members 7. 8. 9. 10. 11. 12. 13.
BAE SYSTEMS (BAE, United Kingdom, Warton) Dassault Aviation (DAv, France, Paris) Airbus Deutschland GmbH (Airbus, Germany, Hamburg) EADS Military Aircraft (EADS-M, Germany, Munich) QinetiQ Group (QG, United Kingdom, Bedford) Saab AB (SAAB, Sweden, Link¨oping) The MathWorks Ltd. (TMW, United Kingdom, Cambridge)
Universities 14. 15. 16. 17. 18. 19.
L’Universit´e Bordeaux (UBOR, France, Bordeaux) University of Bristol (BU, United Kingdom, Bristol) University of Cambridge (UCAM, United Kingdom, Cambridge) Delft University of Technology (DUT, The Netherlands, Delft) University of Leicester (ULES, United Kingdom, Leicester) Universit¨at Stuttgart (UST, Germany, Stuttgart)
The Action Group was chaired by CIRA (Dr. Stefano Scala) from April 1999 until May 2000 and by NLR (Mr. Michiel Selier) from May 2000 until the end of the activity in September 2002. Two workshops, intended to present the results obtained within this Action Group, have been organised: the first by INTA in Madrid (2000) and the second by CIRA in Capua (September 2002). 1.2.2 The HIRM+ Analysis Challenge This book describes the results of an analysis challenge in which seven analysis teams have applied five methods to the same problem. The aim of this design challenge was to describe how these advanced methods can be applied to the clearance process and to demonstrate this on the basis of a benchmark model. Initially, all analysis teams needed to get acquainted with the industrial clearance task. For this purpose a description of the current industrial clearance process of flight control laws was provided by industry, which proved to be of great informational value. In parallel, a clearance benchmark problem was defined. It was decided to use the High Incidence Research Model (HIRM), a generic fighter model with a canard, wing, horizontal tail and vertical tail. This model was available in a mature state from the previous GARTEUR FM(AG08) action group. Within FM(AG11), the flight envelope of HIRM was expanded to suit the needs of the group and parametric uncertainties representing the main variabilities in the model have been defined and included in the model. This updated model, called HIRM+, has been used as the basic aircraft model for nonlinear simulations, trimming and linearisations. The basic aircraft model was
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augmented with control laws from the previous GARTEUR FM(AG08). The controller, called RIDE, is based on a robust inverse dynamics estimation. Finally, for the clearance of closed-loop HIRM+RIDE configuration, analysis criteria have been defined that were representative of industrial practice. In a first stage, the current industrial practice has been demonstrated within a so-called Baseline Solution, which was intended to serve as a basis for comparisons with more advanced techniques. In a second stage of the project, several advanced analysis techniques have been applied to the HIRM+ benchmark problem: – – – – –
µ-analysis ν-gap analysis a polynomial-based analysis method bifurcation analysis optimisation-based worst case search.
The basis for the µ-analysis based approaches is the so-called ”Linear Fractional Transformation” (LFT) based parametric uncertainty model. An LFTmodel represents an approximation of a continuum of linear models, where a special (LFT) representation of parametric dependence is used to account for parametric model uncertainties. A complementary activity to the µ-analysis based approach was the generation and validation of LFT-models. Obtaining good quality LFT-models is time consuming. The order of LFT-models depends on the complexity of the parametric dependencies, the number of parameters, and the employed method for the LFT-model generation. Several approaches have been employed to illustrate the generation of LFT-models for the longitudinal dynamics of HIRM+ with five uncertain parameters. The results obtained by the analysis teams have been described in detailed reports, showing what steps were necessary to apply the method, presenting complete analysis results, and discussing advantages of methods and encountered difficulties. These reports served as basis for the industrial partners to identify and to evaluate, from their point of view, the benefits and limitations of each method. 1.2.3 Other Project Activities Developing visualisation tools for clearance. As already indicated, a very important aspect of the clearance is the presentation of the results of the analysis to the control design engineers, pilots and clearance authorities in a straightforward way. In FM(AG11) specifications for visualisation tools were identified based on input from current industrial practice and conceptual ideas generated by the analysis teams in the group. Setting up more realistic benchmark models. The HIRM+ was used mainly because it was readily available at the beginning of the project in a
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mature state. However, HIRM+ is a generic model and is not based on an actual aircraft. The industrial partners indicated their desire for demonstration of the analysis methods on more realistic and more complex models that are closer to the large and complex models used in industry. Thus two additional models were developed: ADMIRE and HWEM. ADMIRE (Aero Data Model In Research Environment) is a realistic fighter model, with a configuration comparable to a Gripen delta-canard configuration. This model is especially of interest for the clearance of the transonic region. HWEM (Harrier Wide Envelope Model) is a realistic model of an actual Harrier aircraft. An interesting aspect of this model is its use in the clearance of the control laws in the transition phase between normal wing-borne flight and hovering flight. The adjustment of these models to meet the needs of the group took a substantial amount of time and effort. Once the new benchmarks were defined, the analysis teams could choose one of the two models for application of their analysis techniques, in addition to HIRM+. The results of this additional work are not discussed in this book because of publication time constraints, but have been presented at the final workshop, which was held at CIRA in September 2002. A public web site on the FM (AG11) project is available at: http://www.nlr.nl/public/hosted-sites/garteur/rfc.html.
1.3 Objectives and Structure of This Book Due to the large amount of work performed by the members of the group, it was decided to gather the main results into a single book. This book focuses on the HIRM+ analysis activities of the group, since this model has been addressed by all analysis teams. The main objectives of this book are: – to describe the current clearance process for flight control laws for fighter aircraft and the typical flight clearance criteria, and to specify which improvements would increase efficiency; – to demonstrate advanced analysis techniques for the clearance of fighter aircraft flight control laws; – to indicate the advantages and limitations of each method and give directions for further improvements and research. This book consists of six parts. Part one describes the clearance process as it is currently applied in the military aircraft industry. In part two, tutorial sections provide the reader with a brief explanation of the theory behind each method, and references are given for more elaborate descriptions of the analysis techniques. Part three introduces the HIRM+ benchmark problem that the analysis teams have addressed for demonstration of their techniques.
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The basic aircraft model, the controller and a representative set of analysis criteria are described. Part four discusses the generation of LFTs, which is an essential task of µ-analysis. In part five the analysis techniques are applied to the HIRM+ benchmark and the results are described. As a reference, a baseline solution is provided, which used the classical clearance methods. Finally, the industrial view on the project’s achievements is given in part six, together with recommendations for future improvements and research.
References 1. J.F. Magni, S. Bennani, J. Terlouw (eds). Robust Flight Control - A Design Challenge. Springer-Verlag London Limited, United Kingdom, 1997. 2. C. Fielding and R. Luckner. Industrial Considerations for Flight Control. In Flight Control Systems. Co-published by IEE Control Engineering Series, London, UK, 2000 and American Institute of Aeronautics and Astronautics (AIAA), USA, 2000.
2 Tasks and Needs of the Industrial Clearance Process Udo Korte EADS Deutschland GmbH, Military Aircraft MT 62 Flight Dynamics 81663 Mu ¨nchen Germany
[email protected] Summary. The process of clearance of the flight control laws of a fighter aircraft is described. In order to better understand the industrial task and needs in this field, the different steps of the clearance assessment, the methods, procedures and criteria currently applied in industry to derive a clearance are explained. Finally, requirements on evaluation and visualisation tools are addressed.
2.1 Introduction In the introduction to this book it was pointed out that clearance of the flight control laws of a fighter aircraft is a tremendous task because of the many different store configurations, the large number of parameter variations and uncertainties and the large flight envelope for which a clearance has to be provided. This book chapter describes the process, methods and procedures which are currently used in industry to carry out this task. It will thus enable a better understanding of the industrial needs and stimulate research for improvements. The chapter starts with the requirements for the generation of an analysis model and addresses the need for familiarisation with the basic aircraft and the controller. It then demonstrates the effect of important variabilities and uncertainties and describes the methods and criteria applied in linear and nonlinear analysis and simulation to generate a clearance. Finally it addresses the needs for data processing and software tools and how flight limitations – derived from the worst case parameter combinations – can be visualised to give precise information on where the aircraft is allowed to fly.
2.2 Steps of the Analysis Process The clearance of the flight control laws provides information about the flight envelope, angle of attack/load factor limits and the manoeuvres which are C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 13-33, 2002. Springer-Verlag Berlin Heidelberg 2002
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allowed to be flown. In order to derive this information, a number of consecutive steps has to be performed. For the current industrial process these steps can be summarised as follows: Step 1: Generation of an analysis model This involves establishing a full-size nonlinear model which includes all parametric uncertainties. From this model, linear (small perturbation) models are derived (from trimming and linearising). These linear models have to be validated against the nonlinear model. Step 2: Familiarisation with aircraft and controller To provide the designer with a good appreciation of the uncontrolled aircraft, studies are carried out to produce plots of aerodynamic stability and control derivatives (unaugmented aircraft), plots of scheduled feedback gains and the open-loop eigenvalues. Step 3: Trend studies on the effect of uncertainties This involves studying the effects of the various uncertain parameters on the stability and handling of the given aircraft. Step 4: Linear stability analysis This step involves the calculation of open-loop stability margins (Nichols plots) for a narrow grid of flight envelope points and for different uncertainties. It also includes the calculation of closed-loop eigenvalues for all flight envelope grid points and for different uncertainties. It covers the identification of worst case results for all grid points and the visualisation of all results. Step 5: Linear handling analysis This involves the evaluation of appropriate frequency and time domain criteria, the identification of worst cases and the visualisation of the results via plots and tables. Step 6: Nonlinear analysis by off-line and manned simulation This step involves assessing the general flying characteristics with and without uncertainties, the identification of handling and control problems, and the derivation of manoeuvre limitations. Step 7: Clearance report This involves the derivation and visualisation of manoeuvre and flight envelope limitations based on linear and nonlinear analysis results, and the provision of flight test recommendations. Step 8: Improvement of the clearance This is to address some non-compliances and involves special investigations of reduced stability margins, based on reduced tolerances from flight test results. The details of the main steps are described in the following paragraphs.
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2.3 Aircraft Analysis Model The first task of the clearance process is the generation of a representative model of the aircraft and its flight control system (FCS) because this is the basis of all clearance work. 2.3.1 Nonlinear Model For design purposes a reduced model is often used but the clearance analysis work requires a full-size model. Such a model includes the nonlinear, elastified aerodynamic data-set, the nonlinear equations of motion with configuration dependent data, the nonlinear control laws and the hardware model of the actuators, sensors, filters, pilot inceptors, hydraulics, hinge moments, engine, computing delays and data processing. Models for atmospheric turbulence and gusts also have to be included. The nonlinear model will be used extensively both in mathematical simulation (non-realtime) and manned simulation to test stability, handling and control of the aircraft under all conditions, as realistically as possible, in the air and on the ground. Rig tests with real hardware in the loop (i.e. flight control computers with control law software and redundancy management, sensors, actuators and hydraulics) will complement these simulations and will be used to check the transient behaviour when system failures are occurring. 2.3.2 Linear Model The linear, small perturbation model is separated into longitudinal and lateral/directional motion and is derived from the nonlinear model by trimming and linearising the model at a large number of grid points over the entire flight envelope which are dependent on Mach number (M), angle of attack (AoA) and altitude (or dynamic pressure). It is important to choose the grid points narrow enough in order not to miss the critical points of the envelope where large changes of the nonlinear aerodynamics occur within a small AoA or Mach range. If, for instance, the normal gridding is in steps of 10 kft for altitude, 0.2 in Mach number and 2 ◦ in AoA it might be necessary to choose Mach number steps of 0.05 and AoA steps of 1 ◦ for the transonic area. Indications for a suitable choice will become evident from familiarisation with the model. For a complete assessment, thousands of grid points have to be trimmed, and hence fast trim routines are an important factor for time-efficient analysis. In the linear model the hardware (actuators, sensors, time delays etc.) is represented by linear transfer functions which approximate the nonlinear models in the frequency range of interest (up to 5 to 7 Hz). The linear model will be used for calculation of stability margins and eigenvalues and to check that the customer-agreed linear handling criteria are fulfilled. As the linear model may easily reach a dynamical order of 70 or more, numerically stable and efficient algorithms are required for analysis.
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2.3.3 Familiarisation with the Model Before clearance assessment work starts, the engineer should make himself familiar with the aerodynamic characteristics of the unaugmented aircraft and the controller logic and dynamics. Essential stability and control power derivatives should be plotted as a function of Mach number, AoA and control surface position to get an indication of potential problem areas such as reduced control surface effectiveness or sign reversal. For naturally unstable aircraft it is very useful to plot the unstable (positive) real eigenvalue as a function of M and AoA. An example is given in Fig. 2.1 for the pitch axis, where sigma denotes the magnitude of the aircraft’s unstable pole (originating from the short-period mode) and is plotted against AoA for a fixed Mach number.
i n
a
l
4
i t u
d
3
n
g
2
L
o
1
S
i g
m
a
0 - 1 0
2
4
A
6
o
A
8
[ d
e
g
1
0
1
2
1
4
]
Fig. 2.1. Plot of unstable short period eigenvalue
The plot shows a strong peak in instability (positive sigma) followed by a change to stability over a narrow range of AoA. In this area, stability margin problems might be expected due to the high instability level and the wide variation in stability. If the controller were to be scheduled with AoA then tolerances in the airdata system might lead to misadaptation of the controller gains and thus induce over- or undergearing. 2.3.4 Inclusion of Uncertainties in the Model For clearance, it must be demonstrated that the aircraft is safe under all conditions and variations. This means that the assessment must be performed not only for the nominal model but also for all possible deviations, operating conditions and store configurations. Therefore, the model has to be extended to enable the inclusion of such variabilities/uncertainties. Many of the variations are known to a large extent whereas others are uncertain and known only within certain confidence levels. The boundaries of c.g. position, mass
2 Tasks and Needs of the Industrial Clearance Process
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A
/
C
M
a
s
s
[
K
g
]
and inertia, for instance, are known for a given store configuration. For unusual fuel demand, fuel sloshing, fuel system failure or missile firing, the c.g. position will deviate from nominal but not exceed a specified most forward or most aft limit, as indicated in Fig.2.2.
L
o
n
g
i
t
u
d
i
n
a
l
C
G
P
o
s
i
t
i
o
n
[
%
m
.
a
.
c
.
]
Fig. 2.2. Centre of gravity diagram
The aerodynamic data-set is another example. The data-set values which are used in the initial analysis model are derived from wind tunnel measurements and theoretical calculations. They are therefore known only within given tolerance bands. The tolerance values will be largest at the beginning of the aircraft’s development. In-flight parameter identification during flight test will later improve the knowledge of the aerodynamics and allow a reduction in the tolerance levels and thus to delete possible clearance restrictions. For stability and handling investigations the change in pitching, yawing and rolling moments due to changes in AoA, M, control surface position, angular rates etc. are needed. Therefore, the aerodynamic tolerances are defined as deviations of the stability and control derivatives and not of the moment coefficients themselves. In paragraph 2.3.5 the four groups of variabilites / uncertainties are defined and how they can be implemented in the analysis model is described in chapter 8 of this book. In industry it is current practice not to derive separate clearances for each store configuration but to provide block clearances. For this purpose, a grouping of configurations which are aerodynamically equal (within tolerances) is made. For each group, mass/c.g. boundaries as shown in Fig. 2.2 are defined which cover the extreme values of all included configurations. For this analysis, several representative mass/c.g. points on the forward and aft boundary curves are chosen (i.e. maximum mass, minimum mass and medium mass). At each of these operating points (fixed values of mass, c.g., inertia, throttle setting and air data uncertainty) the aircraft is trimmed for all grid points and then linearised. After linearisation the aerodynamic uncertainties with their extreme values (e.g. Cmα tolerance = 0.1) are added to the nominal
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derivatives. This has the advantage that no new trimming is required when type or sign of the tolerance is changed, only the corresponding elements of the state-space matrices have to be re-calculated. This approach saves a significant amount of time and effort. 2.3.5 Type and Effect of Uncertainties Variabilities and uncertainties can be divided into four groups: Configuration dependent variabilities such as mass c.g. and inertia, which differ with stores and fuel Aerodynamic uncertainties on stability derivatives, control power and damping derivatives Hardware dependent variabilities such as changes of the actuator or sensor dynamics or computing delays Air data system dependent tolerances such as measurement errors in signals like AoA, M or dynamic pressure which are used for scheduling of the control laws or control surfaces (to optimise performance) The effect of these uncertainties on stability, handling and performance differs with aircraft type, store configuration, control laws and flight condition. Before actual analysis work is started, the relative importance of the different uncertainties on the clearance results of the aircraft to be assessed should be investigated in trend studies. Parameters with minor effect can thus be excluded from further assessment in order to reduce the amount of necessary calculations. For the most important tolerances the effects are pretty clear - at least for linear analysis. The trends are more straightforward for the longitudinal axis than for the lateral/directional axes, where coupling effects between roll and yaw can lead to different results at different flight conditions. Examples for the effect of variabilities / uncertainties are given below. Configuration dependent variabilities There are several configuration dependent variabilities which are now described. Shift of longitudinal c.g. position The c.g. position is a dominant parameter for the longitudinal characteristics of an aircraft (much less for the lateral characteristics) because it directly influences the stability. When the c.g. is moved aft, the longitudinal stability of the aircraft is reduced and when it is aft of the neutral point the aircraft becomes unstable. This means that the feedback gains of a controller designed for a given c.g. position will be higher than needed for forward c.g. (over-gearing) and less than needed for a more aft c.g. (under-gearing).
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In a Nichols diagram (see explanation in subsection 2.4.1), a more forward situated c.g. will shift the plot of the open-loop frequency response upwards and - in the low frequency range - to the right, because the aircraft becomes more stable. This means that more forward c.g. will decrease the upper gain margin (UGM) and increase the lower gain margin (LGM). Note that a LGM only exists for basically unstable aircraft.
NICHOLS PLOT 12
Pitch Loop Cut
f in [hz]
0.050 0.089
9
0.282
0.158 0.089
6
Gain [db]
3
0.282
0.501 0.891 0.501 0.891
0 -3
0.891 1.585 1.585
-6
1.585 2.818
-9
2.818 2.818
-12 -15 -220
0.501
0.282 0.158
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
nominal c.g. 2% forward c.g. 2% aft
Fig. 2.3. Pitch actuator loop cut: effect of 2% c.g. shift
A more aft c.g. will accordingly shift the frequency plot downwards (increased UGM) and - for low frequency - to the left (decreased LGM). An example is given in Fig. 2.3. An effect of an aft c.g. shift to be seen in the time domain would be a quickened response in a pull-up manoeuvre with the danger of overshooting the allowed limits of AoA or normal load factor. For the c.g. more forward than nominal, the pull-up response would be slowed down (creeping response). Shift of lateral c.g. position The effect of a lateral c.g. shift on linear stability is usually neglected. Investigations will concentrate on the handling aspect and are done by mathematical and manned simulations. A noticeable effect might be seen for strongly asymmetric configurations (i.e. jettison of an underwing
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tank). The roll stick deflection needed for compensation must not be too large (< 70%) and must leave some margin for manoeuvring. The induced sideslip due to lateral g must not lead to departure at high AoA. Shift of vertical c.g. position This effect is of minor importance for a fighter aircraft and can usually be neglected. Changes in inertia A reduction of the pitch inertia (Iy ) would mainly reduce the UGM by
NICHOLS PLOT 12
Pitch Loop Cut
f in [hz] 0.089
9
0.158 0.158 0.282 0.282
6
0.501 0.501
Gain [db]
3
0.891
0
0.891
-3
1.585 1.585
-6 2.818
-9
2.818
-12 -15 -220
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
nominal Iy 20% reduced
Fig. 2.4. Pitch actuator loop cut: effect of 20% reduction in Iy
shifting the high frequency part of the pitch Nichols plot upwards (Fig. 2.4). The effect of the yaw inertia Iz on the rudder loop and the roll inertia Ix on the aileron loop is similar. Changes in mass Mass changes are usually of much less importance than inertia or c.g. changes. This might be different with controllers where the gains are a function of mass.
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Aerodynamic uncertainties There are several aerodynamic uncertainties which are now described. Changes of stability derivative Cmα This derivative defines the static stability around the pitch axis of the
NICHOLS PLOT 12
Pitch Loop Cut
f in [hz]
0.252 0.449
0.142
9
0.252
6
0.449 0.449
3
0.798
Gain [db]
0.798 0.798
0
1.418 1.418 1.418
-3 -6
2.522 2.522 2.522
-9 -12 -15 -220
4.486
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
Nominal +0.1 Cm_alfa Tolerance -0.1 Cm_alfa Tolerance
Fig. 2.5. Pitch actuator loop cut : effect of Cmα tolerance (± 0.1)
aircraft. For positive Cmα the aircraft is usually unstable and a controller is needed to provide artificial stability. A positive Cmα tolerance will increase the basic instability and affect the low frequency part of the Nichols plot: the upper part is shifted left and downwards, thus reducing the LGM. The high frequency range (lower part of the plot) will not be affected. The effect is demonstrated in Fig. 2.5. Changes in weathercock stability Cnβ and dihedral effect Clβ For weathercock stability, Cnβ must be positive. Whether or not a negative value of Cnβtotal (nominal + tolerance value) will produce lateral instability can only be determined by a full lateral analysis. The influence of a Cnβ tolerance on the Nichols plot of the rudder loop cut is comparable to that of a Cmα tolerance on the pitch actuator loop cut: a reduction of the static stability term (negative Cnβ tolerance) will turn the upper (lower frequency) part of the plot to the left and downwards with the high frequency part remaining unaffected. An example is given in Fig.
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2.6, where the tolerance value is large in relation to the nominal Cnβ and therefore, a big effect is observed. Furthermore, the unaugmented aircraft is directionally unstable. The effect of a Clβ tolerance on the plot of the rudder loop cut is similar to that of a Cnβ tolerance. The effect on the aileron loop is small and negligible.
NICHOLS PLOT 12 9
Rudder Loop Cut 0.004
f in [hz]
0.014 0.008 0.025 0.045 0.080 0.142
6
Gain [db]
3
0.008 0.0040.014 0.025 0.045 0.080
0.252
0.003
0
0.142
0.449 0.252 0.449 0.798
-3
0.798
-6
1.418 1.418
-9 -12 -15 -220
2.522 2.522
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
nominal -0.04 Cn_beta Tolerance
Fig. 2.6. Rudder loop cut: effect of Cnβ tolerance (-0.04)
Changes in the pitch control effectiveness CmδT S , CmδCS Tolerances on the effectiveness of symmetric flap or canard have an effect similar to a change of the pitch feedback gains. An increase of the effectiveness (negative CmδT S or positive CmδCS tolerance) will shift the pitch actuator Nichols plot upwards and thus reduce the UGM. A decrease will have the opposite effect. This is shown in Fig. 2.7. Changes in the roll control effectiveness ClδT D , ClδCD An increase in the roll control effectiveness shifts the Nichols plot of the roll loop upwards and thus decreases UGM and possibly the phase margin (PM). Changes in the rudder control effectiveness CnδR Comparable to the pitch loop, an increase in rudder effectiveness decreases the UGM of the rudder loop and possibly the PM. Changes in coupling terms CnδT D and ClδR The effect of changes in these coupling terms is not as predictable as for the other tolerances. It will depend more on flight condition/AoA.
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Changes in damping derivatives Cmq , Clp , Cnr The effect of changes in these coupling terms is not as predictable as for the other tolerances. It will depend more on flight condition/AoA. The effects are usually small and negligible in linear analysis.
NICHOLS PLOT 12
Pitch Loop Cut
f in [hz]
0.089 0.089 0.158
9
0.158
6
0.282 0.282
Gain [db]
3
0.501 0.501 0.891
0
0.891
-3
1.585 1.585
-6 2.818
-9
2.818
-12 -15 -220
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
Nominal 0.04 Cm_dts Tolerance
Fig. 2.7. Pitch actuator loop cut: effect of CmδT S tolerance (0.04)
Hardware tolerances The effect of hardware tolerances is straightforward. The hardware dynamics (actuators, sensors, delays) are represented by transfer functions (filters) whose frequency response will change when the filter time constants change due to tolerances. This will introduce additional phase lag/lead and gain decrease/increase into the feedback loop as a function of frequency and thus influence the stability margins. In the example plot of Fig. 2.8, for simplicity, a single-input system is shown with an actuator transfer function of 1 for the nominal case and a transfer function of 1/(1 + 0.1s) for the tolerance case. Here, the difference in phase and gain between the two transfer functions will directly add to the Nichols plot (at 1.585Hz, we get for example, an additional phase shift of −44.9◦ and a gain decrease of -3dB from the actuator). Additional time delays from hardware tolerances will have a detrimental effect on handling qualities. With increasing delay it becomes more and more difficult for the pilot to predict the response of the aircraft. Too large a time delay will lead to pilot-involved oscillations (PIO). For good handling (Level 1) the ”equivalent time delay” between pilot input and aircraft response should be smaller than 100ms.
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Pitch Loop Cut
f in [hz] 0.089 0.089
9
0.158 0.158 0.2820.282
6
0.501
Gain [db]
3
0.501
0
0.891 0.891
-3 1.585
-6 1.585
-9
2.818
-12 -15 -220
5.012
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
Nominal Nominal plus Actuator
Fig. 2.8. Effect of actuator change on Nichols plot
Air data system tolerances Measurement errors in AoA, M, altitude or dynamic pressure could have a considerable effect on stability, handling and performance because they could lead to incorrect scheduling of the controller gains or incorrect positions of air data scheduled control surfaces. As already pointed out in chapter 2.3.3, clearance problems might then be found in areas of fast stability changes (Cmα ) over a small range of AoA or in the transonic area where the aerodynamics change considerably with small Mach number changes. An example for the transonic area is given in Fig. 2.9, where a pressure measurement error leads to a Mach error which in turn leads to wrong gain scheduling of the controller. The effect is a violation of the stability margin requirement as defined by the trapeziodal boundary.
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NICHOLS PLOT 12
Pitch Loop Cut
f in [hz] 0.142 0.252
9
0.252
6
0.449
0.449
3 0.798
Gain [db]
0
0.798 1.418
-3
1.418 2.522
-6
2.522
-9
4.486 4.486
-12 -15 -220
-200
-180
-160 -140 Phase [deg]
-120
-100
-80
-60
Nominal 0.085 Mach Tolerance
Fig. 2.9. Pitch actuator loop in the transonic area: effect of Mach tolerance (+0.085)
2.4 Clearance Requirements, Criteria, and Tasks 2.4.1 Stability Analysis The basic aim of all clearance work is to prove that the aircraft is stable over the entire flight envelope with sufficient margin against instability for a given set of uncertainties - just to prove stability (boundedness) is not enough. The first step in demonstrating sufficient stability is the calculation of linear stability margins for the open-loop frequency responses in pitch, roll and yaw at all points of the flight envelope within the required AoA or load factor limits. The open-loop frequency responses are obtained by breaking the loop at the input of each actuator or sensor. The results are plotted in Nichols diagrams where the required gain and phase margins are shown as exclusion zones which must not be violated by the plot. An example is given in Fig. 2.10, where it can be seen that with increasing AoA the UGM decreases and at 16 ◦ and 18◦ the stability margin requirements are no longer fulfilled. Therefore, a flight limitation for an AoA above 14◦ would be necessary unless the problem could be solved by a modification of the controller or a reduction of the tolerance. Such a deterioration of the stability margins would be reflected in decreased damping characteristics of the lateral response of the aircraft. The boundary values of the Nichols exclusion zone are based on experience. The outer boundaries are valid for the nominal case whereas the inner boundaries have to be observed when uncertainties are applied. The stability margin requirements [1] are valid for frequencies between 0.06Hz and the first
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rudder - loop cut
NICHOLS PLOT
12 9
0.014 0.025 0.008 0.045 0.004 0.080 0.003
6
0
0.252
0.142
0.025 0.014 0.045 0.008 0.080 0.004 0.142 0.0030.008 0.025 0.014 0.045 0.080 0.004 0.142 0.003
3
Gain [db]
f in [hz]
0.142
0.252 0.449 0.252 0.252
0.449 0.449 0.798 0.449 0.798
-3
0.798 0.798 1.418 1.418 1.418 1.418
-6 -9
2.522 2.522 2.522 2.522
-12 -15 -220
-200
ident 683 684 685 686
-180
alt.[ft] 40000. 40000. 40000. 40000.
-160 -140 Phase [deg]
ma/kcas[-/kts] 1.50/494. 1.50/494. 1.50/494. 1.50/494.
-120
-100
alpha[deg] 12.00 14.00 16.00 18.00
-80
-60
nz[g]
Fig. 2.10. Rudder loop with Cnβ -tolerance: margin degradation with AoA
elastic mode of the aircraft. The aim of the analysis is to find all violation points in the flight envelope and for each of these points, the worst case, i.e., that uncertainty combination which yields the biggest violation. If no violation is found then the aircraft is cleared without limitations. In addition to the open-loop stability margins, the worst case unstable eigenvalues (those with positive real part) of the augmented closed-loop system must be identified. More details about the stability margin and eigenvalue requirements can be found in chapter 10 of this book. The single-loop analysis is usually supported by a limited multi-loop analysis to check for the effects of multiple loop perturbations. Finally, it must be shown by simulation that the stability/controllability is not destroyed by non-linear effects such as rate and position saturation, inertia coupling etc. 2.4.2 Handling Analysis Apart from the stability requirements, the aircraft must fulfil the requirement of good handling. The clearance assessment must show that the pilot can control the aircraft precisely and easily to accomplish the mission. During the last two decades a number of linear and nonlinear criteria have been developed whose fulfilment will give a high degree of confidence that the aircraft will
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exhibit good handling qualities without pilot-in-the-loop oscillations and that flight testing can be started without large risk. The American military specification F-8785C [2] defines 3 levels of flying qualities: – Level 1: Handling clearly adequate (satisfactory) for the mission flight phase – Level 2: Handling adequate but some increase in pilot workload and/or degradation in mission effectiveness exists – Level 3: Aircraft can still be controlled but pilot workload is excessive and/or mission effectiveness is inadequate For combat aircraft Level 1 handling is required within the operational flight envelope and Level 2 handling within the service flight envelope [2] . Linear Handling Analysis Most of the linear handling criteria which are presently in use are based on frequency domain calculations like: – – – – – – – –
Pitch/bank attitude frequency response Pitch/bank average phase rate (to assess resistance to PIO) Absolute amplitude (to assess resistance to PIO) Frequency and damping of short period mode, Dutch roll and FCS modes [1], complemented by the low-order equivalent system approach for highorder systems Closed-loop pitch axis bandwidth (Neal Smith) Open-loop pitch axis bandwith (Hoh) Phase and gain margin criterion (R¨oger) Ride discomfort index [1]
Time domain criteria [3] in use are, among others: – Pitch rate overshoot/dropback, pitch rate peak time, pitch acceleration peak time, flight path angle time delay (Gibson) – Equivalent CAP (control anticipation parameter) – (Effective) Roll mode time constant, time to bank. 2.4.3 Nonlinear Analysis Linear analysis is complemented by nonlinear, non-realtime simulation and manned simulation which are used for detailed investigation of problem areas found in the linear evaluation, to check for the effect of nonlinearities, such as rate and position saturation, dead zones, inertia coupling etc., on aircraft stability, handling and control, and to finally decide whether the aircraft is fit for purpose.
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Off-line simulation Defined control inputs in pitch, roll and yaw are used to test whether the aircraft response is fast and precise without overshoots of AoA, AoS and load factor limits. Pull/push Modern flight control systems are usually of the manoeuvre demand type. This means that pitch stick deflection is proportional to AoA- or nz demand etc. By using full stick rapid pulls/pushes and ramp inputs of defined duration it is checked whether the aircraft shows the required response and stays within the limits for the nominal case. With uncertainties added, the deviations from the nominal trajectory should not exceed a given limit - the overshoot of the AoA-/nz -limit for instance should be less than 2◦ /0.4g and not lead to departure. Rapid rolling Important features to be checked for the nominal and the uncertainty cases are maximum roll rates/overshoots, maximum sideslip generated during roll, roll angle overshoot when trying to stop the roll, variation of normal load factor during full stick rapid roll and available pitch down control power at high AoA and high roll rate (absence of departure due to gyroscopic effects). Pedal response Pedal inputs will generate sideslip. It must be demonstrated for all uncertainties that the maximum sideslip does not exceed safe limits, defined by loading and aerodynamic considerations (values different for different aircraft). Roll due to pedal which is generated as a side effect should always be in the direction of the input to help with turn co-ordination. The pedal inputs will be step inputs and 3-2-1-1 inputs (consecutive steps of alternating sign and a duration of for instance 3s, 2s, 1s, 1s). The latter input is well suited to show whether dynamic inputs will lead to limit overshoots. Rate/position saturation If saturation is encountered it must not lead to control problems or PIO. Limit cycles The aircraft should be free of limit cycles, i.e. sustained oscillations. Integrator windup A problem connected with position saturation is potential integrator windup in control systems with integrator feedback. If for instance, in a pull-up the flaps and canards reach their limits before the commanded AoA has been reached, then the integrator value - if not stopped - would continue to increase. The aircraft could then not follow a nose down command until the integrator has run down below some threshold value. This could be disastrous.
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Fading/switching The correct functioning of fading and switching in the control laws or between different control modes should be demonstrated. Response after failures Depending on the type of failure, aircraft behaviour might change. Actuator failures for instance, might introduce additional phase lag and thus lead to a deterioration in stability and handling. In aircraft with two hydraulic systems, loss of one system could inhibit full use of the control surfaces due to high hinge moments produced by the aerodynamic forces at high dynamic pressure. It must be demonstrated that in such cases the aircraft is still safe - for all combinations of variabilities/uncertainties. Hinge moment effects are mainly seen in the transonic/supersonic area. Manned simulation/rig tests Real-time manned simulation is used to check for nominal and tolerance cases, whether handling and control are adequate in general and for special tasks such as air-to-air refuelling. The pilot will give the final answer whether the aircraft can be flown safely and allows fulfilment of the mission. Tasks to be addressed in manned simulation are listed below. They partly overlap with non-realtime simulation. Control sensitivity, overcontrol, PIO Checks should be made for tasks requiring tight control such as approach and landing, tracking, air-to-air refuelling. Control of AoA/normal load factor It must be demonstrated that the pilot can precisely control the aircraft and stay within allowed AoA/nz -limits. Local instabilities (pitch-ups) must not lead to exceedance of these limits. Bank angle control Precise control of bank angle must be possible. Rapid Rolling Manned simulation complements the non-realtime simulation. Tracking and gross manoeuvring Pilot opinion is required whether aircraft response is adequate. Steady heading sideslip It shall be possible to maintain constant heading without difficulty when applying up to full pedal. Application should be made in a progressive manner. Take-off and landing It should be possible to take-off and land the aircraft safely and without undue pilot workload for all combinations of uncertainties. This must be demonstrated for dry and wet runways, with and without crosswinds (up to 30kts dry) for calm and turbulent weather (moderate and severe turbulence). How turbulence can be included in the model is described in [2].
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Stick authority with asymmetric configurations After missile firing, underwing tank jettison failure (hang-up) or fuel failures, considerable asymmetries can be produced. It must be possible to compensate for the asymmetries with less than 70% roll stick deflection. Acceptable handling must be demonstrated. Carefree handling For those types of aircraft which have been designed to be carefree meaning that the FCS provides automatic protection of certain limits (i.e. AoA-, AoS-, nz -, roll rate limit) - the correct functionality must be demonstrated for nominal cases and under uncertainty. Crossed controls combined with throttle inputs should be used in such a sequence that would most likely generate departure. Handling after failures Depending on the failure, agile manoeuvring might no longer be required and the envelope to be cleared might be reduced. The criteria to be applied remain the same, but the required handling level however might be reduced by one. The transient behaviour during occurrence of a system component failure is checked in Rig (iron bird) tests with real hardware and redundancy management (multi-channel system) included.
2.5 Requirements on Clearance Methods and Tools 2.5.1 Methods Generally all analysis techniques - current ones and new ones-must be able to find out where the aircraft is safe to fly with the given control system and uncertainties. Therefore, they must give answers to the following questions: – Are there any stability margin violations in the required flight envelope? What is the strongest violation at each point and which uncertainty combination has caused it? – Are there any unstable eigenvalues which are outside the requirement limits? For which uncertainty combination are they largest? – Are there any limit cycles? Where and for what combination? – Is there any rate or position saturation? Which combination gives the longest duration? Does it lead to control problems? – Is there any handling criterion which indicates handling worse than Level 2? Where and for which uncertainty combination? – Are there any limit cycles? Where and for what combination? – Is there any rate or position saturation? Which combination gives the longest duration? Does it lead to control problems? – Are there any exceedances of +ve/-ve limits of AoA, AoS, load factor or roll rate? Which uncertainty combination yields the worst case and for which manoeuvre?
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– Which uncertainty combination yields the largest deviations from nominal manoeuvre response? – Is the aircraft still flyable with sufficient precision and ease in specified turbulence, crosswind and gusts for the given set of uncertainties? Current methods do not allow inclusion of all variabilities and uncertainties simultaneously in one calculation run. For each c.g. position or mass for instance, a new calculation is required, and this increases the assessment effort. The procedure to put a grid on the flight envelope and do all calculations at the grid points does not give an answer about the behaviour between these points. The hope is that with a very narrow grid, all critical points will be found - but there is no guarantee and more grid points lead to more calculations. In current practice in industry, only the maximum/minimum values of the uncertainties are considered because otherwise, the necessary analysis effort would increase tremendously and might become unaffordable. A guarantee that the extreme values will deliver the worst case does not exist. There is a need for methods which are not conservative and free from the above limitations, and they should reduce the computational effort - or at least they should not increase it above the level required by current methods. 2.5.2 Software Tools and Visualisation The control laws clearance assessment of a fighter with so many different configurations and parameter uncertainties requires a huge amount of data processing. Therefore, fast and efficient software tools are needed, for instance, tools for fast trim calculations or calculation of higher order frequency responses. Commercial-off-the-shelf software tools are often not well suited for this purpose. Besides the calculation of results, a difficult and time consuming part of the work is to mentally put together all pieces of information obtained from linear and nonlinear analysis in order to derive necessary clearance limitations. For this task database structures for easy interactive information retrieval in combination with good Visualisation and plotting tools are of high importance. Industry is presently using commercial software packages like MATLAB or MATRIXX combined with their in-house developed facilities. The commercial products provide many useful graphical capabilities but they are not tailored to the specific needs of the clearance task. Adaptation of existing tools in this respect is needed and tailored plotting packages for 2D- and 3Dplots are needed for: Familiarisation with the aircraft and the controller Plots of important aerodynamic parameters or controller gains against AoA, Mach number, dynamic pressure etc. will help to develop a feeling about where the weak points of the system with respect to clearance might be.
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Display of clearance assessment results Nichols plots, plots of minimum Gain and Phase Margin (see Fig.2.11), plots or tables of unstable Eigenvalues, plots of Handling Qualities metrics
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Visualisation of clearance limitations It is extremely important that clear and unmistakable information about necessary clearance restrictions is provided to the operators of the aircraft. The pilot must know exactly where he is not allowed to fly (prohibited areas) and in which area of the flight envelope restrictions in AoA or load factor must be observed. Plots of the flight envelope - altitude versus Mach number - are used to display the prohibited or restricted areas, as indicated in Fig. 2.12.
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A general requirement for all plotting information is that it contains a legend which allows clear identification of the presented results and the basis from
2 Tasks and Needs of the Industrial Clearance Process
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which they have been be derived (model standard and issue of the aerodynamic data set, hardware, configuration, control laws, flight condition, failure states etc.). This means the documentation must allow exact reproduction at a later time or by other users, if necessary.
2.6 Conclusions The different tasks in the process of clearance of flight control laws have been described and it has been made obvious that this process is time consuming and expensive due to the problem complexity and the huge amount of effort required. It has been pointed out that new methods and tools are needed which could help to reduce the computational effort and/or give a guarantee for the detection of the worst case results with respect to variabilities and uncertainties.
References 1. Military Specification of Flight Control Systems - Design, Installation and Test of Piloted Aircraft MIL-F-9490D (USAF), 6 June 1975 2. Military Specification of Flying Qualities of Piloted Airplanes MIL-F-8785C Department of Defense, USA, November 1980 3. Military Handbook of Flying Qualities of Piloted Airplanes HDBK-1797 Department of Defense, USA, 19 December 1997
3 The Structured Singular Value and µ-Analysis Declan G. Bates and Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, University Road, Leicester, LE1 7RH, UK.
[email protected],
[email protected] Summary. We introduce the structured singular value µ and discuss its use as an analysis tool for flight control applications. To apply µ-analysis tools to flight control law clearance problems, linear fractional transformation (LFT) based uncertainty models must first be generated to capture the effect of uncertain aircraft parameters on the closed-loop system dynamics. The clearance criteria which are most easily addressed with µ-analysis are those which are defined in the frequency domain such as the stability margin criterion or the unstable eigenvalue criterion. µ-analysis tools which have been developed to address these specific clearance criteria are discussed.
3.1 Introduction It is generally possible to arrange any linear time invariant (LTI) closedloop system which is subject to some unstructured and/or structured type of norm-bounded uncertainty in the form shown in Fig. 3.1, where P , K1 and K2 denote the plant, pre-filter and feedback controller respectively. With respect to this figure, unstructured uncertainty means that the uncertainty matrix ∆ is fully populated, while structured uncertainty means that it has some (typically diagonal or block diagonal) structure. In the context of a flight control clearance problem, unstructured uncertainty could correspond, for example, to unmodelled high frequency aircraft dynamics, while structured uncertainty is used to represent particular aircraft parameters such as stability derivatives, inertias, etc, which are subject to change, or known only to within a certain tolerance. Techniques for converting standard aircraft models into the form shown in Fig. 3.1 are discussed in Part IV of this book. Given a model in this form, it is then straightforward to rearrange the system into the form shown in Fig. 3.2, where M represents the known part of the system (aircraft model and controller) and ∆ represents the uncertainty present in the system. Partitioning M compatibly with the ∆ matrix, the relationship between the input and output signals of the closed-loop system shown in Fig. 3.2 is then given by the upper LFT: C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 37-55, 2002. Springer-Verlag Berlin Heidelberg 2002
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∆
w r -
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- h u6
z y -
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K2
Fig. 3.1. Interconnection structure of a general uncertain closed-loop system
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y = Fu (M, ∆) r = (M22 + M21 ∆(I − M11 ∆)−1 M12 ) r
(3.1)
Now, assuming that the nominal closed-loop system M (s) in Fig. 3.2 is asymptotically stable and that ∆ is a complex unstructured uncertainty block, the small gain theorem (SGT), [3], gives the following result: The closed-loop system in Fig. 3.2 is stable if and only if σ(∆(jω))
0 guarantees that the closed-loop system is stable, and higher values of ²P,C correspond to a greater degree of relative stability (and also some degree of performance). In the single-input singleoutput case ²P,C > 0.3, for example, ensures that the the Nichols diagram of P C avoids an elliptical region centred on −1 guaranteeing a phase margin of 35◦ and gain margin of 5.4dB. In the multi-loop case, closed loop stability is guaranteed in the face of simultaneous and independent gain/phase offsets at each input and output. The second fundamental object is the ν-gap distance δν (P, P∆ ) between a nominal plant P and a perturbed plant P∆ , which measures the importance of any difference between the open-loop systems from a closed-loop perspective. These tools will be used in an effort to exploit the structure of the feedback interconnection , and the inherent robustness of feedback systems, to reduce the complexity of the analysis.
4.2 The ²-Margin, a Generalised Stability Margin Consider the standard linear feedback configuration illustrated in Fig. 4.1 and referred to as [P, C]. The transfer function from the noise signals [ vy vu ]T to [ y u ]T is given by the 2 × 2 block transfer function matrix · ¸ £ ¤ P (I − CP )−1 −C I T [ vy ] → [ y ] = I vu u C. Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 57-75, 2002. Springer-Verlag Berlin Heidelberg 2002
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(i.e. Tvu →y = P (I − CP )−1 etc). We define a stability margin in terms of the H∞ norm of this transfer function matrix, the H∞ norm being defined as the worst case gain over frequency. 1 Definition 1 (Generalised stability margin ²P,C ). Given a closed-loop system [P, C], define °" # ° ° h i°−1 ° ° P −1 (I − CP ) if [P, C] is (internally) stable −C I ° ° ²P,C := ° I ° ∞ 0 otherwise. (4.1) This measure of stability is large when then the norm of the closed-loop system transfer function T[ vy ]→[ y ] is small (implying small amplification vu
u
of noise). It was shown, in [1], that ²P,C is a number never less than zero or greater than one. In that reference, ² was also first referred to as a stability margin, since it equals the size of the smallest perturbation to the normalised coprime factors of P for which the perturbed closed-loop system is destabilised. The stability margin ²P,C is well-known in the context of H∞ loop-shaping controller design [2, 3]. In this design process, the frequency response of the open-loop plant is shaped using weighting transfer functions to reflect the desired loop shape (e.g. large low-frequency gain for disturbance rejection and sufficiently fast high-frequency roll-off for noise immunity). The loop-shaping weights implicitly define the desired crossover frequency of the system. These weights are usually selected to be diagonal since they then have the direct interpretation as frequency-wise scalings which reflect the relative importance of each signal to closed-loop behaviour. wy
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We can also define a frequency-wise version of the stability measure ²P,C . 1
That is, if T (s) is the transfer function of some stable system, then kT k∞ is defined as maxω σ(T (jω)), where σ(T (jω)) is the maximum singular value of T (jω), defined in turn as the square root of the maximum eigenvalue of the matrix T (jω)∗ T (jw) (and T (jw)∗ is the complex conjugate transpose of T (jω)).
4 The ν-Gap Metric and the Generalised Stability Margin
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Definition 2 (Frequency-wise generalised stability margin ²P,C ). ¡ ¢ ρ P (jω), C(jω) :=
1 ¶ ¸ µ· £ ¤ P (jω) −1 −C(jω) I (I − C(jω)P (jω)) σ I
(4.2)
Comparing (4.1) and (4.2), it should be clear that, provided [P, C] is stable, ¡ ¢ (4.3) ²P,C = min ρ P (jω), C(jω) . ω
This frequency-wise computation of stability margin will be revisited in more detail in Section 4.4.3.
4.3 Using the ²-Margin for Analysis 4.3.1 The Effect of Weighting the ²-Margin In order to assess the ²-margin of a specified plant and controller, we will need to apply weights which scale the closed-loop disturbance signals in a similar manner to those used in H∞ loop-shaping design. Since the plant and controller are fixed, the loop shape must not be altered by our choice of weights. Consequently, we consider applying input and output weights to the plant along with the inverses of these weights to the controller as shown in Fig. 4.2. uˆ vˆu
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Fig. 4.2. Standard feedback configuration for H∞ loop-shaping analysis
It may be shown that ²Wo P Wi ,W −1 CWo−1 i
°· · −1 ¸ ¸°−1 ° Wo 0 Wo 0 ° ° v T =° y y ° 0 W −1 [ vu ]→[ u ] 0 Wi °∞ i
(4.4)
When faced with the problem of selecting weights for analysis, the user must consider the control system design specifications and capture the essential ideas with the analysis weights. If the controller was designed using H∞ loop-shaping, the same weights can be used for the analysis. When the given
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controller does not come with such weights, the analysis must begin with a selection of appropriate weights. If these weights have diagonal constant gains, then they can be interpreted as a change of engineering units to reflect the relative importance of each signal. However, we shall go further and use frequency-dependent diagonal weights to reflect frequency-wise the relative importance of each signal. 4.3.2 Selecting Weights for Gain/Phase Stability Analysis For a certain class of input and output disturbances perturbations, we shall see that robust stability is equivalent to considering the best case ²-margin over all diagonal frequency-dependent weights. The results in this section summarise those in [4]. That paper considers closed-loop robustness to simultaneous and independent multiplicative complex perturbations to each plant input and output as shown in Fig. 4.3. This concept of robustness is a generalisation of gain and phase margins used in single-loop analysis, and it will later be used to assess the stability margins of the HIRM+ model.
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Fig. 4.3. Input/output perturbations for multi-loop generalised gain/phase margins (|δ| < β)
Proposition 1 (Gain and phase offset robust stability [4]). Let ∆1 and ∆2 be complex diagonal matrices which perturb a nominal plant P to
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−1
P∆ = (I + ∆1 ) P (I − ∆2 ) . If ²Wo P Wi ,W −1 CWo−1 ≥ β for any diagonal ini put and output analysis weights, Wi and Wo , then [P∆ , C] is stable for any perturbations satisfying k∆2 k∞ < β and k∆1 k∞ < β. For a given β ∈ [0, 1), the perturbation structure P∆ −1 (I + ∆1 ) P (I − ∆2 ) for diagonal ∆1 and ∆2 can be rewritten as
=
p 1 −1 1 − β2 (I + ∆1 ) P (I − ∆2 ) P∆ = p 2 1−β ½ ¾ ½√ ¾ 1−β 2 Note that the sets √1+δ1 2 : |δ1 | < β and : |δ | < β are iden2 1−δ2 1−β
tical; therefore, the result in Proposition 1 is equivalent to saying that each input and output can be independently and simultaneously multiplica¾ ½ tively perturbed by a term δio ∈
√1+δ
1−β 2
: |δ| < β . This set of complex
gain/phase offsets describes a region of allowable input/output multiplicative perturbations δio satisfying: Ã
1
Re (δio ) − p 1 − β2
!2
¡ ¢2 + Im (δio )
0, then each flight condition in the hyperbox around F Ci is cleared. Otherwise, a locally finer grid can be considered if necessary and the clearance can be repeated on this finer grid. The main advantage of this approach is the complete and continuous coverage of both the flight envelope and the parameter space, and thus a higher confidence in the clearance results. Another advantage is the potentially lower total costs, by using a reduced set of only K ¿ N flight conditions. The mathematical optimisation problems to be solved is a NLP with only simple bounds on variables and linear constraints (see Section 7.3). It is important to note that robustness analysis problems are essentially global optimisation problems. When qualifying flight conditions as cleared or not cleared on basis of a local search, only the not cleared points are guaranteed. For a rigourous analysis, only the computationally very expensive global search approaches with guaranteed convergence are able to assess cleared points. Often, a restricted preliminary sensitivity analysis with only a few parameters can indicate the probable lack of multiple local minima. In such cases, the cheaper and more efficient local search methods can be used for solving clearance problems practically without any loss of reliability of the results.
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7.2 Description of the Analysis Cycle Let c(p, F C) be a given clearance criterion depending on the uncertain parameters grouped in a parameter vector p and the flight condition vector F C. The analysis cycle used for the clearance of a control configuration for the given clearance criterion c(p, F C) is illustrated by the flow diagram in Figure 7.1. Here we assume that a continuous search is performed only in the parameter space P for a finite set of flight conditions F Ci , i = 1, . . . , N .
þ
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Fig. 7.1. Optimisation-based clearance analysis cycle
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According to the flow diagram in Fig. 7.1, the following main steps have to be performed in an optimization-based clearance procedure: Step 0 is the initialisation step for the optimisation-based clearance procedure and usually involves choosing the flight conditions F Ci , i = 1, . . . , N where the worst-case parameter combinations are to be determined, the definition of vector p (e.g., most relevant or full set, longitudinal or lateral), setting of appropriate options for trimming, setting the values of criterion specific variables (e.g., frequency-grid, time-grid), or choice of optimisation method (see Section 7.3) and corresponding options (e.g., stopping tolerances, maximum number of iterations etc.) Step 1 is necessary to eliminate from the analysis those points where the clearance requirements are not fulfilled for the nominal values of parameters. Furthermore, here we can also check if the normal acceleration nz is within an allowed range of values (e.g., −3 g ≤ nz ≤ 7 g for HIRM+) or the control surface deflection saturation limits for δT S , δT D , and δR are reached. Points where such violations occur are not cleared and are automatically eliminated from the analysis. The neat effect of this check is a reduction of the overall computational effort. Step 2 is the basic optimisation step performed for each selected flight condition F Ci . The results of this step are the worst-case parameter combination pworst , F Ci ). The and the corresponding criterion value c(pworst i i performed number of function evaluations is an indication of the efficiency of the optimisation-based search in comparison with the classical grid-based approach. Step 3 is similar to Step 1 and the performed check is necessary because the worst-case parameter combination can lead to the same possible violations of some conditions as those occurring in the nominal case (e.g., violation of condition −3 g ≤ nz ≤ 7 g or of the deflection saturation limits). Note however, that such points are found only incidentally by the optimiser, and may exist in a particular flight condition F Ci even in the case when the determined worst-case parameter combination does not violate the above conditions. Step 4 is the outputting of computed data to a database. For each flight condition F Ci , the stored information contains typically the computed worst-case parameter combination pworst , the corresponding minimum i distance d(piworst , F Ci ), the number of performed function evaluations, cleared/not cleared status information, etc. Step 5 performs the graphical evaluation of obtained results by producing plots necessary for assessing and documenting the clearance results.
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7.3 Optimisation Algorithms Suitable for Clearance Each clearance analysis problem defined in Chapter 10 can be formulated as a standard nonlinear programming problem (NLP) of the form min f (x) subject to cj (x) ≥ 0, j = 1, . . . , m li ≤ xi ≤ ui , i = 1, . . . , n
(7.2)
to be solved for x ∈ IRn . Here the components of x includes, in general, variables defining the flight condition (e.g., M , h, and/or α) and components of the vector p representing the uncertain parameters of the model. Each component xi of x must lie between the corresponding lower bound li and upper bound ui . The lower and upper bounds are defined by restricting the flight conditions to lie within the admissible region defined by the flight envelope, while the bounds on uncertain parameters are defined on basis of their physical significance. The scalar constraints cj (x) may correspond, for example, to restricting the search to a typical polygonal region, whose boundary is defined by several line segments. Thus, in the most general case, the NLP (7.2) corresponding to a particular clearance problem is still a particular NLP subject only to simple bounds on variables and linear constraints. If the flight condition (i.e., M , h and α) is not part of x, then the clearance problem can be formulated as an even simpler NLP with only simple bounds on variables. The NLPs arising in clearance problems have several particular features: Low order. Since the optimisation variables are the uncertain parameters and possibly some components of the flight condition vector, the dimension of the optimisation problem is relatively small, satisfying n ≤ 25. Multiple local minima. The functions expressing clearance criteria exhibit very complex dependencies of parameters. It follows, that we can always expect that these functions have several local minima. Expensive function evaluation. The evaluation of criteria based on linearised models, involves trimming, linearisation and frequency response or eigenvalue computation of relatively high order systems (up to 60 state vector components). The evaluation of criteria based on nonlinear models usually involves simulations, preceded by trimming. Thus typically, the evaluation of clearance criteria is very time consuming. Fast and reliable trimming (e.g. via inverse models) is a prerequisite to increase the efficiency of function evaluations. Model reduction techniques can be efficiently used to reduce the order of linear models used to evaluate frequency-response based criteria. Discontinuous derivatives. Discontinuities in derivatives of functions arise from several sources. Naive implementation of criteria by defining distance functions to regions with polygonal boundaries will certainly lead to functions with discontinuous derivatives. By approximating boundaries
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by polynomials (e.g., spline functions) we can get rid of such discontinuities, but the clearance problem can be falsified by such an approximation. Other sources of discontinuities can lie in the model itself, if table-driven linear interpolations are present. Finally, failures to accurately evaluate the function (e.g., because of inaccurate trimming) lead to discontinuities even in the functions themselves. Handling trim failures, for example by setting the function values to a very large number, can rise severe problems for certain solvers. Noisy function. Noise in function values originates from various truncation errors made in intermediary computations such as trimming, linearisation, order reduction, numerical evaluation of gradients, simulation, as well as from the round-off errors associated with difficult numerical computations like eigenvalue computation. To handle such functions, the usage of more robust, derivative-free optimisation methods could be necessary (e.g., pattern search) or enhancements of gradient-search techniques are necessary (e.g., usage of central difference approximation of gradients, usage of gradually increased accuracy in gradient computations, etc.). For additional aspects of optimisation with noisy function see [1]. In the following paragraphs we present brief information on several optimisation algorithms which are suitable for solving the NLPs appearing in the clearance problems. For most algorithms software implementations are freely available on the Internet [2]. 7.3.1 Gradient-Based Local Search Methods Gradient-based minimisation methods use local information on the function through its gradient to achieve fast convergence rates. This is why, when applicable, many gradient-based search methods allow highest computational performance in solving general or particular NLPs. For the usage of most gradient-based techniques a basic requirement is the continuity of gradient with respect to the optimisation variables. Furthermore, for a satisfactory performance, the availability of an analytic expression of gradient is highly desirable. However, for complex functions like those typically arising in clearance problems, usually no analytic gradients are available. Therefore, numerical approximations of gradients have to be computed resultsing in a slower and less reliable execution, especially when function evaluations are noisy. The sequential quadratic programming (SQP) method to solve the general NLP with equality and inequality constraints can be used to solve the particular NLP of the form (7.2) which arises in clearance problems. The SQP method can be seen as a generalisation of Newton’s method for unconstrained optimisation in that it finds a step away from the current point by minimising a quadratic approximation of the problem function f (x). Under mild conditions this method has a fast, so-called superlinear convergence [3]. An alternative approach for problems with only simple bounds constraints on the vari-
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ables is the limited memory BFGS with bound constraints (L-BFGSB) described in [4] together with accompanying Fortran 77 software. This approach extends the standard Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method to handle NLPs with simple bounds, by using a gradient projection approach. The algorithm has a superlinear convergence and no violations of bound constraints on variables occur during optimisation. For a more detailed discussion of both approaches see [3]. 7.3.2 Gradient-Free Local Search Methods Derivative-free methods using only function evaluations are a real alternative to gradient-based methods, especially when function evaluations are noisy and/or discontinuities in the gradient are present. Two classes of derivativefree methods are known: direct-search methods which include the popular simplex and pattern search methods, and trust-region methods relying on linear or quadratic interpolation models. Derivative-free methods are useful when the function f (x) is not smooth (e.g., ”noisy” function) or when accurate derivatives are difficult to determine numerically. For more details on derivative-free methods see [5] and for performance comparisons see [6]. Pattern search (PS) algorithms are a class of direct search methods initially proposed for unconstrained minimisation which has a rigourous global convergence theory. The PS techniques has been recently extended to solve NLPs with simple bounds [7]. PS methods use a simple decrease criterion to accept a step as opposed to the sufficient decrease criterion used by gradientbased search. This is why, PS methods usually have a slower convergence rate than a gradient-based search. On the other hand, PS methods are often numerically more robust than gradient-based methods in avoiding local minima as well as tackling with noisy functions. PS methods may require a relatively large number of function evaluations, hence they tend to be effective primarily for problems of relatively small dimensions and low accuracy situations. Model-based trust region methods exploit the smoothness of the objective function and attempt to preserve the convergence properties of their gradient-based counterparts. The constrained optimisation by linear approximations (COBYLA) approach employs linear approximations to the objective and constraint functions [8]. The approximations are formed by linear interpolation at n + 1 points in the space of the variables (regarded as vertices of a simplex) and the size of the simplex is reduced as the optimisation advances. The main advantage of COBYLA over many of its competitors, is that it treats each constraint individually when calculating a change to the variables, instead of lumping the constraints together into a single penalty function. Therefore, COBYLA usually has better convergence than the pattern search method. One disadvantage of the COBYLA software, is that it does not address simple bounds explicitly and these must be transformed to 2n general constraints in the NLP (7.2) of the form cj = xj − uj ,
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cj+n = lj −xj , for i = 1, . . . , n. Unfortunately, this leads to frequent violations of bound constraints during the computation. The derivative-free optimisation (DFO) trust-region method uses a quadratic approximation of the objective function (see [6] and references therein). The quadratic model approximates the function well within a certain ”trust”-region of a given radius and serves to determine new points by minimising the current approximation instead of the function itself. The new points generated by the algorithm are used both to advance the optimisation and to update the approximation. Since the DFO algorithm needs only a relatively few function evaluations, this method is well-suited to minimise expensive functions which depend on few (some hundred at most) variables. 7.3.3 Global Search Methods For functions with many minima, the use of global optimisation techniques is the only alternative for successful computations. In this section we discuss three global optimisation approaches which can be employed for solving optimisation-based clearance problems with simple bounds on the parameters. Typically, these methods require a very large number of function evaluations and therefore they are primarily intended either to determine good starting points for local search based methods, or to address difficult clearance problems with many local minima. The simulated annealing (SA) algorithm is essentially an iterative random search procedure with adaptive moves along the coordinate directions [9]. It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered. It has been proved that the sequence of points sampled by the SA algorithm form a Boltzmann distribution and converges to a global minimum with a probability of one as the annealing ”temperature” goes to zero. The genetic algorithm (GA) is a global optimisation approach based on evolution strategies which guarantee the survival of the fittest individual in each population [10]. The GA can easily handle problems with simple bounds on the variables, and even general constraints by using penalty function techniques. There are several selection schemes which can be combined with a shuffling technique for choosing random pairs for mating. The GAs based on binary coding, use mutations (e.g., jump or creep mutations), crossover (single-point, uniform, etc.), niching and various other strategies to produce successive populations. The use of GA for function optimisation is quite costly in terms of the required number of function evaluations, but usually its cost can be predicted in advance by choosing the population size and the number of successive generations. To find the global extremum with high accuracy, this method typically requires a very large number of function evaluations. The global optimisation using multilevel coordinate search (MCS) attempts to find the global minimiser of the bound constrained optimisation problem using function values only, based on a multilevel coordinate search
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A. Varga
that balances global and local search [11]. The local search is done via SQP. The search is not exhaustive, so occasionally the global minimum may be missed. However, a comparison to other global optimisation algorithms shows excellent performance of the MCS method in many cases, especially in low dimensions.
7.4 Conclusions The main benefits of the optimisation-based search are the lower costs in the case of many simultaneous parameters and an increased reliability of the results because of the continuous exploration of parameter space. Prerequisites for the applicability of this approach are appropriate parameterised models, fast and reliable trimming and linearisation procedures (necessary for efficient function evaluation) and robust optimisation software capable of addressing the challenge of solving NLP problems with possibly non-smooth, expensive to evaluate and noisy functions. Taking into account all these aspects, the best suited approach appears to be the trust-region DFO method. For functions with only a few variables, DFO typically requires relatively few evaluations of the problem function. For more difficult problems with many local minima, the MCS method combining local and global search appears to be a viable alternative to more expensive GA and SA methods. The acceptance of the optimisation-based clearance approach by the industry depends on several aspects. Since the optimisation-based clearance can be seen just as a straightforward (more powerful) extension of the classical approach, the effort to learn this method is almost negligible. In fact, the classical gridding-based approach can always be used as a standard option even in an optimisation-based clearance methodology. This is why, the first time setting up of the method is not much different than for the grid based approach. However, the usage of sophisticated optimisation tools requires special care when defining suitable smooth distance functions on basis of standard clearance criteria. Further, the implementation of fast and reliable procedures to evaluate these functions is of crucial importance for the success of the optimisation-based worst-case search. The reusability of software to cope with new aircraft models and control laws can be enforced by performing the optimisation-based clearance within a dedicated software environment which supports the interchange of different models and criteria. Within such an environment, the effort for a new analysis setup with different models and control laws is expected to be easily affordable. For maximum flexibility, such an environment has to provide additional facilities for experimenting with various optimisation techniques, different parameter sets, different criteria, different optimisation options etc. A clearance software environment satisfying all above requirements will be implemented as an add-on to the optimization based design environment MOPS of DLR (Multi-Objective Parameter Synthesis) [12].
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References 1. C. T. Kelley. Iterative Methods for Optimisation. SIAM, Philadelphia, 1999. 2. H.D. Mittelmann and P. Spellucci. Decision tree for optimization software. World Wide Web, http://plato.la.asu.edu/guide.html, 2001. 3. J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer-Verlag, New York, 1999. 4. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal. Algorithm 778. L-BFGS-B: Fortran subroutines for Large-Scale bound constrained optimization. ACM Transactions on Mathematical Software, 23:550–560, 1997. 5. M.J.D. Powell. Direct search algorithms for optimization calculations. In A. Iserles, editor, Acta Numerica, Vol. 7, pages 287–336. Cambridge University Press, 1998. 6. A. R. Conn, K. Scheinberg, and Ph. L. Toint. A derivative free optimization algorithm in practice. In Proc. of AIAA St Louis Conference, 1998. (http://www.fundp.ac.be/∼phtoint/pht/publications.html). 7. R. M. Lewis and V. Torczon. Pattern search methods for bound constrained minimization. SIAM Journal on Optimization, 9:1082–1099, 1999. 8. M.J.D. Powell. A direct search optimization method that models the objective and constraint functions by linear interpolation. In S. Gomez and J.P. Hennart, editor, Advances in optimization and numerical analysis, pages 51–677. Kluwer Academic Publishers, 1994. 9. P. Laarhoven and E. Aarts. Simulated Annealing: Theory and Applications. D. Reidel Publishing, 1987. 10. D. E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA, 1989. 11. W. Huyer and A. Neumaier. Global optimization by multilevel coordinate search. J. Global Optimizationn, 14:331–355, 1999. 12. H.-D. Joos, J. Bals, G. Looye, K. Schnepper, and A. Varga. A multi-objective optimisation-based software environment for control systems design. In Proc. IEEE CADCS Symposium, Glasgow, UK, 2002.
8 The HIRM+ Flight Dynamics Model Dieter Moormann EADS Deutschland GmbH?? , Military Aircraft MT62 Flight Dynamcis, 81663 M¨ unchen, Germany
[email protected] Summary. The major objective of the GARTEUR Action Group on Analysis Techniques for Clearance of Flight Control Laws FM(AG-11) is the improvement of the flight clearance process by increased automation of the tools used for modelbased analysis of the aircraft’s dynamical behaviour. What is finally needed are techniques for faster detection of the worst case combination of parameter values and manoeuvre cases, from which the flight clearance restrictions are be derived. The basis for such an analysis are accurate mathematical models of the controlled aircraft. In this chapter the HIRM+ flight dynamics model is described as one of the benchmark military aircraft models used within FM(AG-11). HIRM+ originates from the HIRM (High Incidence Research Model) developed within the GARTEUR Action Group on Robust Flight Control FM(AG-08). In building the HIRM+, additional emphasis has been put on realistic modelling of parametric uncertainties.
8.1 Introduction The HIRM+ has been developed from the HIRM, a mathematical model of a generic fighter aircraft originally developed by the Defence and Evaluation Research Agency (DERA, Bedford). The HIRM is based on aerodynamic data obtained from wind tunnel tests and flight testing of an unpowered, scaled drop model. The model was set up to investigate flights at high angles of attack (-50◦ ≤ α ≤ 120◦ ) and over a wide sideslip range (- 50 ◦ ≤ β ≤ +50◦ ), but does not include compressibility effects resulting from high subsonic speeds. The origin of the model explains the unconventional configuration with both canard and tailplane, plus an elongated nose (see Fig. 8.1). The aircraft is basically stable. However, there are combinations of angle of attack and control surface deflections, which cause the aircraft to become unstable longitudinally and/or laterally. Engine, actuator and sensor dynamics models have been added within FM-AG-08 to create a representative, nonlinear simulation model of a twin-engined, modern fighter aircraft. The model building was done by using the object-oriented equation-based modelling environment Dymola [4]. ??
The work on this project was conducted while the author was employed at DLR, Institute of Robotics and Mechatronics, Oberpfaffenhofen, 82234 Wessling, Germany
C . Fielding et al. (Eds.): Advanced Techniques for Clearance of Flight Control Laws, LNCIS 283, pp. 121-139, 2002. Springer-Verlag Berlin Heidelberg 2002
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Fig. 8.1. High Incidence Research Model [1]
In building the HIRM+, the emphasis has been put on realistic modelling of parametric uncertainties. Parameters have been defined to specify uncertainties in mass, inertial data, position of the centre of gravity, aerodynamic control power derivatives, stability derivatives, and some coefficients in the actuator and engine dynamics. In spite of these changes, the nominal models of HIRM+ and HIRM (i.e., with all uncertain parameters set to zero) are essentially the same. Although variations of the uncertainty parameters affect the trim values of states and control surface deflections, due to the HIRM’s fairly linear aerodynamic derivatives over the specified flight envelope, the stability properties remain essentially unchanged Another aspect arising from the current industrial clearance practice is to allow the use of expected tolerance ranges of typical uncertain parameters (e.g., stability and control power derivatives) to be directly accessible in the nonlinear model. This allows the HIRM+ to mimic the industrial clearance approach, which is heavily based on both linear and nonlinear aircraft models. Usually, individual entries of the state-space matrices, with known physical meaning, are considered as uncertain and varied within the expected ranges.
8 The HIRM+ Flight Dynamics Model
123
automatic code generation
aerodynamic controls & engine controls & gust inputs measurements & evaluation outputs
Fig. 8.2. The HIRM+ aircraft dynamics model
8.2 The HIRM+ Object Model The HIRM+ aircraft dynamics model in the upper part of Figure 8.2 consists of four basic blocks denoted as: actuator dynamics, engine dynamics, flight dynamics and sensor dynamics. Zooming into the flight dynamics model displays its internal structure, as given in the lower half of Fig. 8.2: The flight dynamics block incorporates the mass properties including equations of motion and the models of aerodynamics, variations in thrust, gravity, atmosphere, and gust disturbances [2, 3].
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In contrast to the (input-output) block-oriented description of the aircraft dynamics model, the flight dynamics model itself is specified using an acausal model formulation [4]. Due to the acausal approach, interconnections between components are not limited to signal flows but represent physical system interactions, like energy flows, or kinematic constraints. Automatic code generation is used to import the flight dynamics into the overall model. The outputs of the flight dynamics model, which are used as measurements for control and evaluation outputs are specified in Table 8.1. Table 8.1. Measurements and evaluation outputs of HIRM+ Name
Description
Unit
measurements p
y(1)
Body-axis roll rate
rad/s
q
y(2)
Body-axis pitch rate
rad/s rad/s
r
y(3)
Body-axis yaw rate
θ
y(4)
Pitch angle
rad
φ
y(5)
Bank angle
rad
ψ
y(6)
Heading angle
ax
y(7)
Body-axis x-acceleration
m/s2
ay
y(8)
Body-axis y-acceleration
m/s2
az
y(9)
Body-axis z-acceleration
m/s2 m/s
VA
y(10)
Airspeed
M
y(11)
Mach number
rad
-
h
y(12)
Altitude
α
y(13)
Angle of attack
rad
m
β
y(14)
Angle of sideslip
rad
Flight path angle
rad m/s
evaluation γ
y(15)
VG
y(16)
Ground speed (magnitude)
x
y(17)
Earth-axes x-position (north)
m
y
y(18)
Earth-axes y-position (east)
m
Fp1
y(19)
Thrust of engine 1 (left engine)
N
Fp2
y(19)
Thrust of engine 2 (right engine)
N
The inputs of the aircraft dynamics model (aerodynamic controls, engine controls, and gust inputs) are specified in Table 8.2.
8 The HIRM+ Flight Dynamics Model
125
Table 8.2. Controls and gust inputs of HIRM+ Name δT S
u(1)
Description
Unit
Symmetric tailplane deflection
rad
δT D
u(2)
Differential tailplane deflection
rad
δCS
u(3)
Symmetric canard deflection
rad
δCD
u(4)
Differential canard deflection
rad rad
δR
u(5)
Rudder deflection
suction
u(6)
Nose suction
-
δT H1
u(7)
Throttle of engine 1 (left engine)
-
δT H2
u(8)
Throttle of engine 2 (right engine)
-
WXB
u(9)
Body-axes head wind
m/s
W YB
u(10)
Body-axes cross wind
m/s
WZB
u(11)
Body-axes vertical wind
m/s
The uncertain parameters of the HIRM+, their formulation, nominal values, upper and lower bounds, units and descriptions are given in sections 8.2.1 to 8.2.5. 8.2.1 Mass Characteristics and Geometric Data The body-object of Fig. 8.2 specifies the mass characteristics and the rigid body differential equations of motion with 6 degrees of freedom. For a derivation of these equations a reference such as [5] should be consulted. The HIRM+ mass characteristics are specified in Table 8.3 Variations in mass and moment of inertia are given by the following equations. For convenience, the uncertain parameters of the HIRM+ are denoted with an asterisk and parameters without, as their nominal values. The uncertainty itself is expressed by the subscript U nc: m∗ = (mU nc + 1) m
0 −Ixz (1 + Ixz U nc ) Ix (1 + IxU nc ) 0 Iy (1 + IyU nc ) 0 I∗ = 0 Iz (1 + IzU nc ) Ixz (1 + IxzU nc )
The centre of gravity varies with respect to its nominal value which is defined as body geometric reference BGR, see Fig. 8.2): ∗ = Xcg + XcgU nc Xcg ∗ Ycg = Ycg + YcgU nc ∗ Zcg = Zcg + ZcgU nc
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D. Moormann Table 8.3. Inertial parameters Name
Nominal value
m
15296.0
Unit kg
Description Aircraft total mass
2
Ix
24549.0
kg m
Iy
163280.0
kg m2
y body axis moment of inertia
x body moment of inertia
Iz
183110.0
kg m2
z body moment of inertia
2
Ixz
-3124.0
Xcg
0
m
Centre of gravity location along x-axis
Ycg
0
m
Centre of gravity location along y-axis
Zcg
0
m
Centre of gravity location along z-axis
kg m
x-z body axis product of inertia w.r.t. body geometric reference BGR w.r.t. body geometric reference BGR w.r.t. body geometric reference BGR
Table 8.4. Inertial uncertain parameters Name
Nominal
[min; max]
Unit
Description
value mU nc
0
[-0.2; 0.2]
-
Uncertainty level of aircraft mass
XcgU nc
0
[-0.15; 0.15]
m
Centre of Gravity offset along x-axis from nominal Xcg , positive toward nose
YcgU nc
0
[-0.10; 0.10]
m
Centre of Gravity offset along y-axis from nominal Ycg , positive toward starboard
ZcgU nc
0
[-0.04; 0.04]
m
Centre of Gravity offset along z-axis from nominal Zcg , positive down
IxU nc
0
[-0.2; 0.2]
-
Uncertainty level of Ix
IyU nc
0
[-0.05; 0.05]
-
Uncertainty level of Iy
IzU nc
0
[-0.08; 0.08]
-
Uncertainty level of Iz
IxzU nc
0
[-0.2; 0.2]
-
Uncertainty level of Ixz
The parametric uncertainties in the HIRM+ mass characteristics are defined using the parameters given in Table 8.4 in terms of their nominal values (see Table 8.3) and their set of uncertain parameters.
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127
In some cases (e.g. mass) physical units are not shown, because the uncertainties are expressed in terms of percentages (±20% for mass) of the nominal value. 8.2.2 Aerodynamics The aerodyn-object of Fig. 8.2 describes the aerodynamic forces and moments. The aerodynamic force and moment coefficients for HIRM+ are given by the summation of several components [1]. Most components have the form Cab (c, d). The derivative for a force or a moment a with respect to b is determined by linearly interpolating between the values given in a look-up table as a function of the variables c and d. The basic aerodynamic parameters are specified in Table 8.5 Table 8.5. Aerodynamic parameters Name
Nominal value
Unit
Description
c¯
3.511
m
Mean aerodynamic chord
S
37.16
m2
Wing planform area
b
11.4
m
Wingspan
To allow a physically meaningful interpretation of parametric variations with a direct influence on the stability and control power derivatives, the uncertain parameters in the HIRM+ have been defined such that they can be directly recovered in the linearised models. This has the undesired effect that trim values are explicitly used in the definition of uncertain parameters, which means, that the nonlinear simulations are now trim point dependent through initial state components (e.g.,αtrim ) and initial control surfaces (e.g., δCStrim ). Thus, strictly speaking, even for the nonlinear model this approach permits only small manoeuvres close to the trim point. This approach is convenient, in that it allows model upgrades to be made at the level of the nonlinear model, prior to linearisation. In what follows the expressions of the uncertain aerodynamic moment coefficients are given, where trim values of various parameters are specified with the subscript trim (e.g., αtrim ). Uncertain pitching moment coefficient: ∗ = Cm + Cm0 U nc + CmδCS U nc (δCS − δCStrim ) Cm + CmδT S U nc (δT S − δT Strim ) + Cmα U nc (α − αtrim ) c¯ + Cmq U nc (q − qtrim ) VA 2
with Cm as the nominal pitching moment coefficient of HIRM, depending on δT S , δCS , etc.
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Uncertain rolling moment coefficient: Cl∗ = Cl + Cl0 U nc + ClδCD U nc (δCD − δCDtrim ) + ClδT D U nc (δT D − δT Dtrim ) + ClδR U nc (δR − δRtrim ) b b + Clp U nc p + Clβ U nc (β − βtrim ) + Clr U nc r 2 VA 2 VA with Cl as the nominal rolling moment coefficient of HIRM. Uncertain yawing moment coefficient: Cn∗ = Cn + Cn0 U nc + CnδCD U nc (δCD − δCDtrim ) + CnδT D U nc (δT D − δT Dtrim ) + CnδR U nc (δR − δRtrim ) b b + Cnp U nc p + Cnβ U nc (β − βtrim ) + Cnr U nc r 2 VA 2 VA with Cn as the nominal yawing moment coefficient of HIRM. Table 8.6. Uncertain parameters of aerodynamic stability derivatives Name
Nom.
[min; max]
Unit
[0 ; 0]
-
Description
value Cl0 U nc
0
Uncertainty in rolling moment
Cm0 U nc
0
[0 ; 0]
-
Uncertainty in pitching moment
Cn0 U nc
0
[0 ; 0]
-
Uncertainty in yawing moment
Cmα U nc
0
[-0.1; 0.1]
1/rad
Uncertainty in Cmα stability derivative
Clβ U nc
0
[-0.04; 0.04]
1/rad
Uncertainty in Clβ stability derivative, where: k = 1 for α < 12◦ , k = 2 for α > 20◦ , and k is linearly interpolated for 12◦ ≤ α ≤ 20◦ between 1 and 2.
Cnβ U nc
0
[-0.04; 0.04]
1/rad
Uncertainty in Cnβ stability derivative
Cmq U nc
0
[-0.1; 0.1]
-
Uncertainty in pitching moment derivative due to normalised pitch rate
Clp U nc
0
[-0.1; 0.1]
-
Uncertainty in rolling moment derivative due to normalised roll rate
Clr U nc
0
[-0.03; 0.03]
-
Uncertainty in rolling moment derivative due to normalised yaw rate
Cnp U nc
0
[-0.1; 0.1]
-
Uncertainty in yawing moment derivative due to normalised roll rate
Cnr U nc
0
[-0.05; 0.05]
-
Uncertainty in yawing moment derivative due to normalised yaw rate
In Tables 8.6 and 8.7 the ranges of the uncertain aerodynamic stability derivatives and control power derivatives are given. For some parameters, no
8 The HIRM+ Flight Dynamics Model
129
value of uncertainty has been defined. These terms have been included to allow for future applications of the model. Table 8.7. Uncertain parameters of aerodynamic control power derivatives Name
Nom.
[min; max]
Unit
Description
CmδT S U nc 0
[-0.04; 0.04]
1/rad
Uncertainty in pitching moment derivative due to symmetrical tailplane deflection
CmδCS U nc 0
[-0.02; 0.02]
1/rad
Unc. in pitching moment derivative due to symmetrical canard deflection
ClδT D U nc
0
[-0.04; 0.04]
1/rad
Unc. in rolling moment derivative due to differential tailplane deflection
ClδCD U nc
0
[-0.02; 0.02]
1/rad
Unc. in rolling moment derivative due to differential canard deflection
ClδR U nc
0
[-0.006; 0.006]
1/rad
Uncertainty in rolling moment derivative due to rudder deflection
CnδT D U nc 0
[-0.02; 0.02]
1/rad
Unc. in yawing moment derivative due to differential tailplane deflection
CnδCD U nc 0
[-0.01; 0.01]
1/rad
Unc. in yawing moment derivative due to differential canard deflection
CnδR U nc
[-0.02; 0.02]
1/rad
Uncertainty in yawing moment derivative due to rudder deflection
value
0
8.2.3 Engine Dynamics Each engine-object of Fig. 8.2 is modelled as shown in Fig. 8.3. A throttle
Fig. 8.3. Engine dynamics model
demand of 0 selects idle which is 10 kN of thrust at sea level. A throttle demand of 1 corresponds to maximum dry thrust of 47 kN. Full reheat is selected when the throttle demand equals 2 and corresponds to a thrust
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D. Moormann
of 72 kN. The rate at which the thrust changes depends on whether the engine is in dry thrust or reheat. For dry thrust, the maximum rate of change is 12 kN/s whereas in reheat it is 25 kN/s. The sea level engine thrust is scaled with the relation of local density ρ to sea level density ρ0 . The engine setting angles are zero and so the thrust acts parallel to aircraft x-body axis. The engine positions with respect to the body geometric reference BGR are given in Table 8.8. Table 8.8. Engine parameters Name
Nom.
Unit
Description
value XT AP
-6.0
m
Body-axes x-position of thrust application point
YT AP
± 0.56
m
Body-axes y-position of thrust application point
0.35
m
Body-axes z-position of thrust application point
thridle
10 000
N
Idle thrust (at sea level)
thrdrymax
47 000
N
Maximum dry thrust (at sea level)
thrreheatmax
72 000
N
Maximum reheat thrust (at sea level)
ZT AP
thrdryrL
±12 000
N/s
Ratelimit at dry thrust
thrreheatrL
±25 000
N/s
Ratelimit at reheat thrust
Variations due to parametric uncertainties in engine rate limiters for dry thrust and reheat thrust are given by the following equations: thrdryrL ∗ = thrdryrL (1 + EngrLU nc ) thrreheatrL ∗ = thrreheatrL (1 + EngrLU nc ) The uncertainty level of the engine rate limits is given in Table 8.9. Table 8.9. Engine uncertain parameter Name
Nominal value
EngrLU nc
0
[min;
Unit
Description
max] [0 ; 0]
-
Uncertainty level of engine rate limits
8.2.4 Actuator Dynamics The actuator dynamics block of Fig. 8.2 is specified by the following transfer functions:
8 The HIRM+ Flight Dynamics Model
131
Taileron actuator transfer function: T ∗ (s) =
1 (1 + 0.026 (1 + δTbw U nc ) s) (1 + 0.007692 s + 0.00005917 s2 )
with an uncertain rate limit defined as ± 80 (1 + δTrL U nc )◦ /s Canard actuator transfer function: T ∗ (s) =
1 (1 + 0.0157333 (1 + δCbw U nc )s + 0.00017778s2 )
with an uncertain rate limit defined as ±80 (1 + δCrL U nc )◦ /s Rudder actuator transfer function: T ∗ (s) =
1 (1 + 0.0191401 (1 + δRbw U nc ) s + 0.000192367s2 )
with an uncertain rate limit defined as ± 80 (1 + δRrL U nc )◦ /s For the actuator dynamics block, currently no values of uncertainty has been defined. These terms have been included for future applications of the model. Table 8.10. Actuation uncertain parameters Name
Nom. value
δTrL U nc
0
[min;
Unit
Description
max] [0 ; 0]
-
Uncertainty level of tailplane rate limit
δTbw U nc
0
[0 ; 0]
-
Uncertainty level of tailplane bandwidth
δCrL U nc
0
[0 ; 0]
-
Uncertainty level of canard rate limit
δCbw U nc
0
[0 ; 0]
-
Uncertainty level of canard bandwidth
δRrL U nc
0
[0 ; 0]
-
Uncertainty level of rudder rate limit
δRbw U nc
0
[0 ; 0]
-
Uncertainty level of rudder bandwidth
8.2.5 Sensor Dynamics To reduce the complexity of the overall model, and thus the computation times required by simulations, the sensor dynamics for HIRM are replaced by lower order approximated sensor models for the HIRM+, described by transfer functions. The HIRM+ sensor dynamics for body axis angular rates [p, q, r] and body axis accelerations [ax , ay , az ]: T ∗ (s) =
1 − 0.005346 s + 0.0001903 s2 1 + 0.03082 s + 0.0004942 s2
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D. Moormann
The HIRM+ sensor dynamics for airspeed, Mach-number, altitude, angle of attack and angle of sideslip [VA , M a, h, α, β]: T ∗ (s) =
1 1 + 0.02 s
The HIRM+ sensor dynamics for body axis attitudes and heading angle [ϕ, θ, ψ]: 1 T ∗ (s) = 1 + 0.0323 s + 0.00104 s2 A measurement error signal is added to the signal of α and β. These errors are assumed to be constant during the period of simulation: α∗ = α + αU nc β∗ = β + βU nc For the HIRM+ sensor dynamics block, currently no value of uncertainty for α- and β-measurement errors have been defined. These terms have been included for compatibility with the HIRM, in which these uncertainties had been used. Table 8.11. Sensor uncertain parameters Name
nom
[min; max]
Unit
Description
αU nc
0
[0 ; 0]
[rad]
Uncertainty in sensed angle of attack (added to the α-measurement signal)
βU nc
0
[0 ; 0]
[rad]
Uncertainty in sensed sideslip angle (added to the β-measurement signal)
8.3 Automated Model Generation for Parametric Time Simulations and Trim Computations The object model in Fig. 8.2 is graphically specified using components from the Flight Dynamics Library [3], that are instantiated with HIRM+ specific system model parameters. From this object model, simulation and analysis models of the aircraft system dynamics and documentation can be generated automatically (see Fig. 8.4). In the mathematical model building process, the equation handler of Dymola solves the equations according to the inputs and outputs of the complete HIRM+ model. Equations that are formulated in an object, but that
8 The HIRM+ Flight Dynamics Model physical model composition
133
modelling
physical system model
3.511 15296.0
systemparameter
component libraries
y
dts dtd dr
u
V alpha beta
V alpha beta q gamma
y
8
u
8
specification of model inputs and outputs
dts dtd dr throttle1 throttle2
mathematical model building
(interactive)
(automatic) sorted & solved equations for simulation
sorted & solved equations for trim calculation
codegeneration
(automatic) simulation model e.g. Matlab/Simulink(cmex-) S-function
trim code e.g. Matlab/Simulink(cmex-) S-function
Fig. 8.4. Model building process
are superfluous for capturing the behaviour of the particular model, are automatically removed. The result is a nonlinear symbolic state-space description with a minimum number of equations for this task x˙ = f (x, u, p) y = h(x, u, p) From the symbolic description, numerical simulation code for different simulation environments is generated automatically. In this way, it is possible to generate, for example, a Matlab-Simulink m-file or cmex-code, or CCode according to the neutral DSblock standard [7], which can be used in any simulation environment, being capable of importing C-Code. From the sorted and solved equations for simulation, symbolic analysis code can be generated, describing a parameterised state-space model. This
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D. Moormann
code can be used to extract in an automated procedure the so-called Linear Fractional Transformation (LFT) standard form, that may serve as the basis for µ robustness analysis. A detailed description of the generation of an LFT representation from an object model as depicted in Fig. 8.4 can be found in [8]. One of the key aspects of the successful usage of an optimisation-based clearance methodology is an efficient trimming approach. The trimming of HIRM+ is a very challenging computational task, involving the numerical solution of a system of 60 nonlinear equations for the stationary values of state and control variables appearing in the HIRM+ state model. The difficulties mainly arise because of the lack of differentiability of the functions due to the presence of various look-up tables used for linear interpolations. Severe nonlinearities in the engine model and in the aerodynamics, as well as the presence of control surface deflection limiters make the numerical solution of this high order system of equations very difficult. To manage the trimming problem, an highly accurate and efficient approach has been employed in [2]. The facilities of an equation based modelling environment as Dymola [4] allows the generation of C-code for an inverse model to serve for trimming. Such a model has as inputs the desired trim conditions (such as Va , α, . . . ) and as outputs the corresponding equilibrium values of trimmed state (x) and control vectors (such as δT S , δCS ,. . . ). Dymola generates essentially explicit equations for the inverse model by trying to solve the 60th order nonlinear equation symbolically. Even if a symbolic solution cannot be determined, Dymola is still able to reduce the burden of solving numerically a 60th order system of nonlinear equations to the solution of a small core system of 13 nonlinear equations which ultimately must be solved numerically. Thus, the trimming procedure based on such an inverse model is very fast and very accurate.
8.4 Flight Conditions and Envelope Limits The analysis of HIRM+ is restricted to the flight conditions defined in Table 8.12. Depending on the clearance problem, the equilibrium conditions in these Table 8.12. Set of flight conditions for clearance analysis FC No.
F C1
F C2
F C3
F C4
F C5
F C6
F C7
F C8
M
0.2
0.3
0.5
0.5
0.6
0.7
0.8
0.8
h [ft]
5,000
25,000
40,000
15,000
30,0000
20,000
5,000
40,000
points are defined by the trimming conditions for straight and level flight for given γ, M and h or pull-up manoeuvres for given α, M and h. For the
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variation of the α the interval [−15◦ , 35◦ ] has been chosen, and for gridding a step size ∆α = 2◦ has been suggested. When several aerodynamic uncertainties are simultaneously used in the analysis, reduction factors must be applied on their absolute values as specified in Tables 8.6 and 8.7. The values of reduction factors for different numbers of aerodynamic uncertainties are given in Table 8.13. Table 8.13. Reduction factors for simultaneous aerodynamic uncertainties Number of aerodynamic uncertainties Reduction factor
2
3
4
≥5
0.62
0.46
0.37
0.31
Due to load factor limitations (section 8.4.1) and control surface deflection limits (section 8.4.2) it is not possible to trim all flight conditions of Table 8.12 for all angles of attack between −15◦ and 35◦ . This is already true for the nominal model, for which all uncertainty parameters are set to zero. The number of not trimmable points in the flight envelope increases with more uncertainty parameters being used. This fact must be accounted for during the assessment. 8.4.1 Load Factor Limits The clearance task is restricted to a ”true” flight envelope, where additional restrictions on variables must be satisfied. The first condition is to restrict the load factor to meet −3 [g] ≤ nz ≤ 7 [g] All flight conditions, where this condition is violated can not be cleared. A preliminary check involving only nominal cases has been performed. In Fig. 8.5 the values of load factors versus α for the eight flight conditions are presented. It can be seen that, because of violation of load-factor limit, F C6 is defined only for α ∈ [−9◦ , 29◦ ] and F C7 is defined only for α ∈ [−2◦ , 12◦ ]. It is helpful to have the dependence of nz on various parameters in mind. In general, nz can be expressed as nz = −
ρ VA 2 SCZ 2mg
and thus depends on the Z-force aerodynamic coefficient CZ , altitude (via air density ρ), airspeed VA , and mass of the aircraft m. For HIRM+, CZ is given by [1] CZ = CZδT S (α, δT S ) + CZδCS (α, δT S )δCS + 1.7555 CZq (α, δCS )
q VA
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Load factors for nominal cases FC 1 FC2 FC3 FC4 FC5 FC6 FC 7 FC 8
n
z
7 5
0 −3 −5
−10 −15
−10
−5
0
5
10 AoA [deg]
15
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Fig. 8.5. Nominal load factors for HIRM+
Because δCS = 0◦ (the canards are not used) and the term 1.7555 CZq (α, δCS ) VqA being much smaller than CZδT S , CZ can be approximated by the single term CZ ≈ CZδT S (α, δT S ), where the dependence on δT S , being not significant, can be dropped. Thus, if we neglect the pitching motion, nz for straight and level flight can be expressed as nz ≈ −
ρ VA 2 SCZδT S (α) 2mg
and depends finally only on α, altitude (influence on air density), the airspeed, and the mass of the aircraft. The uncertain parameters, with exception of the mass, do not have any influence on the values of nz . A remarkable property of HIRM+ is that, independently of any values of uncertain model parameters, nz ≈ 0 for α close to 2◦ , because CZδT S (2◦ ) ≈ 0. This particular feature of HIRM+ can be observed in Fig. 8.5. 8.4.2 Control Surface Deflection Limits A second set of conditions originate from the deflection limits on taileron and rudder actuators:
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−40◦ ≤ δT S + δT D ≤ 10◦ −40◦ ≤ δT S − δT D ≤ 10◦ −30◦ ≤ δR ≤ 30◦
Nominal actuator deflections for δTS+δTD
20
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−5
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Fig. 8.6. Summation of symmetrical and differential tailplane deflection
Nominal actuator deflections for δTS−δTD
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Fig. 8.7. Difference between symmetrical and differential tailplane deflection
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All flight conditions, where the above conditions are violated, lead to saturation of control surfaces, and thus are automatically not cleared. For the nominal cases, the variations of δT S + δT D and δT S − δT D for the rigid body equations of HIRM+ can be seen in Figures 8.6 and 8.7 1 . The values computed in these figures have been determined with the inverse trim routine where these limits are not present, and therefore the trimming is always possible. This is intentionally done, in order to make trimming numerically easier and to be able to study points also slightly outside of the limits for the control surface actuators. It follows that the trimming results are valid only if the above bounds are fulfilled. As a practical consequence, the above conditions must be checked after each trim computation. Ignoring these conditions leads to strange (but expected) effects, as for example, zero columns in the input matrix B of the linearised HIRM+ in F C1 for α ∈ [−15◦ , −10◦ ] because of saturation of inputs. This further leads to identically zero transfer function, when breaking the symmetric taileron loop. According to these plots, for the nominal parameters, F C1 is defined only for α ∈ [−9◦ , 35◦ ] because of violation for α ∈ [−15◦ , −10◦ ] of the conditions δT S ± δT D ≤ 10◦ . The variation of δR is within the allowed limits and is not shown here. Based on nominal case analysis results, the ”true” set of flight conditions to serve for analysis purposes must be restricted.
References 1. Ewan Muir. The HIRM design challenge problem description. In J. F. Magni, S. Bennani and J. Terlouw, editors, Robust Flight Control, A Design Challenge, Lecture Notes in Control and Information Sciences, vol. 224, pp. 419–443, Springer Verlag, Berlin, 1997. 2. D. Moormann. Automatisierte Modellbildung der Flugsystemdynamik (Automated Modeling of Flight-System Dynamics). Dissertation, RWTH Aachen. VDI Fortschrittsberichte, Mess-, Steuerungs- und Regelungstechnik, Reihe 8, Nr. 931, ISBN: 3-18-393108-7, 2002. 3. D. Moormann and G. Looye. The Modelica Flight Dynamics Library. Modelica 2002, Proceedings of the 2nd International Modelica Conference. Oberpfaffenhofen, Germany, March 18-19, 2002. 4. H. Elmqvist. Object-Oriented Modeling and Automatic Formula Manipulation in Dymola. In Scandinavian Simulation Society SIMS’93, Kongsberg, Norway, June 1993. 5. R. Brockhaus. Flugregelung. Springer Verlag, Berlin, 1994. 6. J. F. Magni, S. Bennani and J. Terlouw. Robust Flight Control, A Design Challenge. Lecture Notes in Control and Information Sciences, vol. 224, Springer Verlag, Berlin, 1997. 7. M. Otter and H. Elmqvist. The DSblock Model Interface for Exchanging Model Components. Simulation, 71:7–22, 1998. 1
Figures 8.6 and 8.7 are the same for α ≤ 20◦ , because δT D is zero for a trimmed straight-and-level-flight within this α-limit. δT D becomes different from zero due to a lateral asymmetry in the aerodynamic model for α > 20◦ .
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8. A. Varga, G. Looye, D. Moormann, and G. Gr¨ ubel. Automated generation of LFT-based parametric uncertainty descriptions from generic aircraft models. Mathematical and Computer Modelling of Dynamical Systems, 4:249–274, 1998.
9 The RIDE Controller David Bennett BAE SYSTEMS Aerodynamics (W427D) Warton, UK
[email protected] Summary. The Robust Inverse Dynamic Estimation (RIDE) control laws and HIRM+ aircraft model provide a suitable basis for engineering research within the Matlab / Simulink design environment. With the implementation of a dynamic pressure scheduler, the control laws provide consistent handling qualities across an extended flight envelope over which flight control analysis techniques can be applied. The model offers reliable trimming, linearisation, simulation and analysis capabilities through the functionality of the model and Matlab.
9.1 Introduction The following section describes the integration of the Robust Inverse Dynamic Estimation (RIDE) control laws with the HIRM+ aircraft model . The RIDE control laws and the HIRM+ were developed separately, and therefore the task was to integrate the two and ensure that the combined model was suitable for robustness analysis research by the GARTEUR Action Group, AG11. The RIDE control laws were developed for the HIRM model by DERA Bedford, for the GARTEUR Group FM-AG-08. A full description of the design process and implementation of RIDE is provided in [1] and therefore, it will not be repeated in this chapter. Replacing the HIRM model with that of the HIRM+ did not require any modifications to the controller design for the nominal model, i.e. with all the uncertainties set to zero. The only significant difference between the HIRM and HIRM+ that may have influenced the performance of the control system, is the reduced complexity of the sensor models. However, the modified sensor models are an approximation of the original models, i.e. there is no significant difference in low frequency (< 5 Hertz) dynamics, and therefore, no significant difference in control system performance. Modifications have been made to the control laws, to improve their performance, and to extend their Matlab/Simulink functionality for release to the GARTEUR Action Group. To summarise, the changes include: – – – –
improved trimming, implementation of a dynamic pressure gain scheduler, an improved process for controller initialisation, and improved linearisation.
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9.2 Augmenting HIRM+ with the RIDE Control Laws The complete HIRM+RIDE model was constructed by integrating the HIRM+ model described in the previous chapter and the RIDE control laws, extracted from the existing RIDE/HIRM model, which was provided by DERA. As both were readily available in Simulink, only minor modifications were required to integrate them. The resulting Simulink model, hirmpride.mdl, is shown in Fig. 9.1, which provides consistent notation between its individual components. In addition to the aircraft model and control laws, the model shown in Fig. 9.1 provides functionality for: – specifying pilot demands: roll rate, pitch rate, sideslip and airspeed, – specifying wind inputs: x, y and z axis components, and – forcing the actuator inputs directly. The block trim inputs provide the actuator inputs with the correct trim settings, calculated by the Matlab file trimhirmplus.m.
Fig. 9.1. The augmented HIRM+RIDE Simulink model, hirmpride.mdl
9.3 Trimming the RIDE Control Laws The HIRM+RIDE results presented in [1] showed slight mis-trims at the start of time responses. To remove the mis-trims a two-part trimming strategy was developed and implemented. Firstly, the sensor outputs are initialized in order to set the schedulers to obtain the correct trimmed state, and secondly, the model is initialised to force the control surfaces to their correct positions. The second part is achieved by summing the trimmed control surface deflections
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(block: trim inputs) with the controller outputs. This requires the controller outputs to be zero for a perfect trim. The RIDE control laws contain an inverse dynamics loop that outputs non-zero values for all trimmed flight conditions. As this inner-loop is summed with the integrators to form the output of the control laws, the integrators are initialised so that the total control law output is zero on all paths, as shown in Fig. 9.2.
Fig. 9.2. Initialisation of the integrators
To demonstrate the trimming process, the model was trimmed at Mach=0.4, h=10000 ft and a small amplitude, 1 deg/sec, pitch rate demand was applied. The results for the pitch rate response, before and after the modifications to the trimming process, can be seen in Figs. 9.3 and 9.4. It can be seen that the pitch rate trimming error has been eliminated.
Fig. 9.3. Pitch rate demand showing mis-trimmed response
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Fig. 9.4. Pitch rate demand showing correct trim
9.4 Dynamic Pressure Scheduler Implementation The recommended design envelope for the HIRM control laws as defined in [2] is Mach 0.15 to 0.5 and height from 100 to 20000 ft, with an AoA range of -10 degs to +30 degs, and a sideslip range of 10 degs. To accomplish this, a dynamic pressure scheduler was implemented to provide consistent handling qualities across the required flight envelope. The dynamic pressure scheduler is implemented using one equation and one look-up table, designed for the flight condition Mach=0.5, h=15000 ft. The method is best described with the aid of the following equation, which defines dynamic pressure: 1 2 (9.1) ρv 2 Fig. 9.5 shows the characteristics of the dynamic pressure gain scheduler. The gain scheduler will traverse a single curve as Mach number varies, and will change to a different curve as the altitude varies, thereby providing the gain value that produces the desired handling qualities for the current flight condition. The dynamic pressure scheduler is implemented using the equation: ¶ µ q5 (9.2) Gain = g5 qbar dynamic pressure =
Where:
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– g5 is the constant gain value from the original RIDE control laws (at Mach=0.5, h=15000 ft), – q5 is the dynamic pressure (at Mach=0.5, h=15000 ft, which remains constant), and – qbar is the dynamic pressure at the current flight condition.
Fig. 9.5. Dynamic pressure scheduler
The method works by calculating the ratio of dynamic pressure at Mach=0.5, h=15000 ft, to that at the current flight condition, and then modifying the gain at Mach=0.5, h=15000 ft by this ratio. This approach works well where air compressibility effects are not significant (or have not been modelled) and will give reliable results at low Mach numbers. The benefits of the modified scheduler are demonstrated by comparing Figs. 9.6 and 9.7, which show a comparison of the aircraft pitch responses over the flight envelope, before and after the addition of the dynamic pressure scheduler. Both Figures show the flight envelopes and pitch rate responses to pilot step demands at the envelope corner points. A pitch rate step demand of 1 deg/sec was commanded at 1 second, and removed at 4 seconds. It can be clearly seen that by designing a controller scheduler that takes into account both Mach number and height (in comparison to one that uses Mach number only), significantly improved and consistent responses, and therefore consistent handling qualities are obtained. It can also be seen that the flight envelope has been significantly increased.
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Fig. 9.6. Flight envelope with the Mach number scheduler
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Fig. 9.7. Flight envelope with the dynamic pressure scheduler
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9.5 Linearisation of the HIRM+RIDE The following presents an automated procedure for linearising the HIRM+ and RIDE control laws, that have been shown to produce excellent robustness. The procedure is contained completely within the trimming program trimhirmplus.m, which is executed when trimming the HIRM+. The linearisation process is performed in three stages. Initially a linear model is obtained for the HIRM+ using the results from the trimming routine and Matlab linearisation commands. Secondly, the RIDE control laws are linearised, and thirdly, the two linear models are integrated to form a state-space representation of the HIRM+RIDE model as described below, where the input and output vectors are shown in Fig. 9.8, and are defined as: ym yr ye uref uc ud ref mes
– – – – – – – –
sensor output (fed back to the controller) controller-generated actuator demands evaluation outputs (not fed back to the control laws) pilot demands actuator (control) inputs, gust (disturbance) inputs, controller reference input (trim actuator setting) controller measurement input
Fig. 9.8. Connectivity of the HIRM+ and RIDE models
The linearised HIRM+ is represented by the following state and output equations ¸ · uc (t) (9.3) x(t) ˙ = Ax(t) + [ B1 B2 ] ud (t) · ¸ · ¸ ¸ · ¸· ym (t) C1 0 D1 uc (t) = (9.4) x(t) + C2 0 D2 ye (t) ud (t)
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Similarly, for the linearised RIDE control laws,
149
¸ ref (t) (9.5) mes(t) ¸ · ref (t) (9.6) yr (t) = Cride xride (t) + [Dride,1 Dride,2 ] mes(t) The two sets of state-space equations are then combined using the following equations: ·
x˙ ride (t) = Aride xride (t) + [Bride,1 Bride,2 ]
mes = ym uc = uref + yr
(9.7) (9.8)
Combining equations (9.3) and (9.4), and (9.5) and (9.6) by using equations (9.7) and (9.8), gives the following state and output equations for the linearised HIRM+ and RIDE. ¸ · ¸· ¸ · A + B1 Dride2 C1 B1 Cride x(t) x(t) ˙ = + Bride2 C1 Aride x (t) x˙ ride (t) ride ¸ ref (t) · (9.9) B1 Dride1 B1 Dride2 D1 B1 ud (t) Bride1 Bride2 D1 0 uref (t) · ¸ · ¸ · ¸· ¸ ref (t) ym (t) C1 0 x(t) 0 D1 0 ud (t) = + (9.10) ye (t) C2 0 0 D2 0 xride (t) uref (t) Equations (9.9) and (9.10) are coded into the file trimhirmplus.m, resulting in the linearisation being performed automatically when the HIRM+ is trimmed.
9.6 Verification of the Linearised Model Fig. 9.9 shows the results of the linearisation process. The results from small amplitude non-linear and linear responses are overplotted to compare the differences in the two representations of the model for the flight condition Mach=0.4, h=10,000 feet. The figure shows the results of a 1 deg/sec, step demand in pitch rate, initiated after 1 second and held for a duration of 3 seconds. The results show a very accurate match in pitch rate, but the Mach number plot shows a slight difference. Engine rate limiting due to the speed hold function will contribute to this this difference, since rate limiting effects will not be captured by the linearisation routine, and hence the non-linear Mach number response is expected to reduce at a slightly lower rate. These results and those obtained for the other flight conditions were considered to be satisfactory in terms of the verification of the linearisation process and the resulting models.
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Fig. 9.9. Response to a small amplitude pitch rate demand
References 1. E.A. Muir. HIRM Design Challenge Presentation Document: The Robust Inverse Dynamics Estimation Approach. Technical Publication TP088-28, Group for Aeronautical Research and Technology in Europe, Technical Report, GARTEUR-FM(AG-08), 1997. 2. E.A. Muir. Robust Flight Control Design Challenge, Problem Formulation and Manual: The High Incidence Research Model (HIRM). Technical Publication TP088-4, Group for Aeronautical Research and Technology in Europe, Technical Report, GARTEUR-FM(AG-08), 1997.
10 Selected Clearance Criteria for HIRM+RIDE Federico Corraro? Centro Italiano Ricerche Aerospaziali Flight System Department Via Maiorise, 81043, Capua (CE) Italy
[email protected] Summary. In this chapter the main requirements for the HIRM+RIDE clearance problem are given. The HIRM+ flight envelope is firstly introduced and the points where clearance criteria have to be checked are defined. The uncertainty parameters to be considered for the analysis are listed and categorised. Finally, each clearance criterion selected for the HIRM+ is described and mathematically defined.
10.1 Introduction The industrial flight control law clearance is an extensive verification process based on several test criteria which can be grouped into four classes: I. II. III. IV.
Linear stability criteria Aircraft handling/PIO criteria Non-linear stability criteria Non-linear response criteria
In order to define a clear HIRM+RIDE [1] benchmark problem for verifying the capabilities of analysis techniques, a selection of criteria from the above set has been performed and their definition has been mathematically formulated in order to avoid interpretation ambiguities. The selected criteria are: 1. 2. 3. 4.
Stability margin criterion (class I) Unstable eigenvalues criterion (class I) Average phase rate and absolute amplitude criteria (class II) Angle of attack/normal load factor limit exceedance criterion (class IV)
The benchmark definition is based on the description of the unaugmented HIRM model [2] and on the description of the original RIDE (Robust Inverse ?
The content of this chapter is, to a large extent, based on the GARTEUR report [4] TP-119-2-A1v2 edited by S. Scala, F. Karlsson (SAAB) and U. Korte (EADSM). The author whishes to thank the above mentioned people for their support in revising the report [4] for the scope of this book.
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Dynamics Estimation) flight control laws for the HIRM [3]. Some degree of freedom has been still left on how to define worst cases and robustness measures.
10.2 Flight Envelope and Model Uncertainties Taking into account the limited scope of this benchmark problem, the analysis has been restricted to the discrete set of flight conditions defined in Table 10.1 and shown in Fig. 10.1. For each flight condition in Table 10.1, several flight cases must be considered. These flight cases are equilibrium conditions, in straight and level flight and in pull-up manoeuvres, characterised by different values of AoA and load factors, up to the maximum AoA or load factor nz (which respectively range [-15˚,35˚] and [-3g,7g], as reported in [1]).
Fig. 10.1. Flight envelope of the HIRM+ with RIDE control laws
It should be noted that in reality, industry needs to clear the flight control laws across the whole flight envelope, and that reducing the analysis to a limited number of selected flight conditions is only done for the HIRM+ benchmark problem to. As an example, for an aircraft with a flight envelope similar to the HIRM, a typical clearance analysis would be performed on a grid in the flight envelope (M, h), with typical steps of ∆M =0.2, ∆h=5,000ft.
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Table 10.1. Set of points in the flight envelope for the HIRM benchmark problem
FC No. FC1 M 0.2 h [ft] 5,000
FC2 0.3 25,000
FC3 0.5 40,000
FC4 0.5 15,000
FC5 0.6 30,000
FC6 0.7 20,000
FC7 0.8 5,000
FC8 0.8 40,000
In areas of the flight envelope where problems are expected, a more dense grid is used. Methods which can potentially perform ”global” analysis are desired because they allow clearance of whole sets of flight conditions (the entire envelope in the limit) in one shot, instead of single points one at a time. If a model linearisation is performed for the analysis, the linearised models should be those obtained in straight and level flight conditions and in pull-up manoeuvres 1 (with a given AoA grid spacing), for the selected points in the flight envelope. The HIRM+ model contains several uncertain parameters which are defined in Chapter 8 and [1], together with their nominal values and their uncertainty ranges. For the scope of this benchmark problem, the variation of the aircraft dynamics due to the model uncertainties listed in Table 10.2, has to be considered. Furthermore, separate analyses for longitudinal and lateral-directional uncertainties should be performed. In Table 10.2, a rating of the importance of the uncertain parameters of the HIRM+ model has also been given, both for longitudinal and for lateraldirectional axes, see Chapter 2. The intention is to reduce the number of parameters to be taken into account during a preliminary analysis of limited time length. Parameters rated as category 1 are the most relevant for clearance, and therefore they must be taken into account in the analysis from the beginning. Parameters rated as category 2 are less relevant and their variation can be ignored during a preliminary analysis. It should also be noted that the reduction factors in Table 10.3 have to be applied to the ranges of aerodynamic uncertainties only when several of them are applied simultaneously for the clearance analysis. This assumption is made to avoid unduly pessimistic assumptions from being made and is based on a probability argument. Regarding the figures in Table 10.2 and the classification of the uncertainties into two categories of decreasing relevance to clearance, the following should be noted: a problem in itself is to establish which uncertainties should be in which category for an aircraft. Aircraft designers usually have a good idea which parameters matter, firstly from their experience with earlier projects, and secondly, from their understanding of the aircraft’s linearised equations of motion and associated transfer functions. It is common practice to support this knowledge with a preliminary analysis with only one uncertainty at a time, in order to confirm the degree of importance of every uncertainty, and then to repeat the analysis with different sets of simultane1
Usually, trimmed banked turns for different aircraft load factors also considered.
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Table 10.2. Uncertain parameters for the HIRM+RIDE benchmark problem Longitudinal axis Lateral axis Category 1 Category 2 Category 1 Category 2 (most relevant) (less relevant) (most relevant) (less relevant) Xcg [-0.15;0.15] m [-0.2;0.2] Ycg [-0.1;0.1] ClδT D [-0.04;0.04] Iy [-0.05;0.05] Zcg [-0.04;0.04] Ix [-0.2;0.2] ClδCD [-0.02;0.02] Cm α [-0.1;0.1] Ixz [-0.2;0.2] Iz [-0.08;0.08] ClδR [-0.006;0.006] CmδT S [-0.04;0.04] CmδCS [-0.02;0.02] Clβ K [-0.04;0.04]∗ CnδT D [-0.02;0.02] Cm q [-0.1;0.1] Cnβ [-0.04;0.04] CnδCD [-0.01;0.01] Cnr [-0.05;0.05] CnδR [-0.02;0.02] C lp [-0.01;0.01] C lr [-0.03;0.03] Cn p [-0.1;0.1] ∗ K = 1 for α < 12◦ , K = 2 for α > 20◦ and K is linearly interpolated between 1 and 2 for 12◦ ≤ α ≤ 20◦ . Table 10.3. Weights on simultaneous aerodynamic tolerances Simultaneous number of aerodynamic uncertainties 2 3 4 ≥5
w = Reduction factor on uncertainty range 0.62 0.46 0.37 0.31
ous uncertainties, to which the reduction factors of Table 10.3 are applied. Within this analysis it is reasonable to assume that no more than five uncertainties at a time should be considered. Indeed, increasing the number of simultaneous uncertainties would lead to smaller values of the reduction factor in Table 10.3, such that the uncertainty set around the nominal point would become very small, and therefore, the analysis would give results not significantly different from those obtained for the nominal condition. In order to clearly define the selected clearance criteria for this benchmark problem, the following generic definition of the uncertainties is needed. Let k be the number of uncertain parameters to be taken into account in some criterion. For the i-th uncertain parameter, define Π i as the interval in which the parameter can vary according to Table 10.2 and the definition in Chapter 8 and [1]. The actual uncertainty range to be used in the analysis for the i-th parameter is defined as ½ Πi if the i-th uncertainty is not an aerodynamic parameter Πi,w = wΠi if the i-th uncertainty is an aerodynamic parameter where w is the reduction factor given in Table 10.3, when several aerodynamic uncertainty parameter are employed simultaneously in the analysis. For ex-
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ample, if the i-th uncertain parameter is cmαU nc and three aerodynamic uncertain parameters are considered simultaneously, then Πi = [−0.1, 0.1], w = 0.46 and Πi,w = wΠi = [−0.046, 0.046]. The complete uncertainty set is Π = Π 1,w ×. . . ×Π k,w ⊂Rk , i.e. the hyper-rectangle in which the vector of k uncertain parameters varies. In what follows, we denote by π ∈Π a particular value of the vector of uncertain parameters in the uncertainty set Π.
10.3 Stability Margin Criterion (Class I) This criterion requires identification of all flight cases (in terms of M , altitude and AoA) where the Nichols plot stability margin boundaries of Figs. 10.2 and 10.3 (see also [2]) are violated. It is also necessary to identify, for each flight condition, which uncertainty parameter values lead to the biggest violation - i.e. to define the worst-case tolerance combination. 6
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4
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−140
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Fig. 10.2. Nichols plot exclusion zones for single-loop analysis with uncertainties
Note that for clearance purposes, only violations of the boundaries are of interest, but for analysis and understanding purposes, the worst-case combination is of interest, even if the boundaries are not violated. This knowledge becomes important when considering possible future developments of an aircraft, as it gives an indication of what changes might be possible.
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3
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2 1
1
0
0
30
−1
−1
−2 −3
−3
−4 −30
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0
10 20 Phase offset [°]
30
40
50
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Fig. 10.3. Gain and phase offsets for multi-loop analysis with uncertainties
This criterion requires two analyses: a single-loop analysis and a multiloop one. Note that at least the single-loop analysis must be performed for this benchmark problem. In the following, details on both analyses are given. a) Single-loop analysis. The open-loop Nichols plots of the frequency response is obtained by breaking the loop at the input of each actuator, one at a time, while leaving the other loops closed. The frequency response should avoid the region shown in Fig. 10.2. This test is carried out mainly to assess the sensitivity of the system to changes in the dynamics of each actuation system, and to ensure that the system maintains adequate stability margins. It also gives a good indication of the sensitivity of the system to changes in aerodynamic control power. Note that when performing the frequency response, a gain of -1 needs to be included on the input or the output, to obtain the correct phase response. This requirement should be satisfied for each control loop. Note also that the RIDE control laws of the HIRM+ only use symmetrical thrust, symmetrical and differential tailplane, and rudder for control, while symmetrical and differential canards, nose suction and differential thrust are not used. b) Multi-loop analysis. The closed-loop system should be able to withstand the application of simultaneous and independent gain and phase offsets at the input of each of the actuators, without becoming unstable. This test is mainly carried out to check the system sensitivity to simulta-
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neous changes in the dynamics of the actuation systems and is aimed at identifying problem cases that might be missed by the single-loop tests. Simulataneous changes in actuation dynamics can occur, for example, due to a reduction in hydraulic pressure. The corresponding perturbation matrix P will be of the form: P =diag(K1 e−jϕ1 ,. . . ,K 4 e−jϕ4 ) Where Ki and ϕi are gain and phase offsets respectively, taking values in the region shown in Fig. 10.3. Note that P should be placed in series in front of the actuators, giving an uncertain gain scaling in the range [0.7, 1.4] and a maximum phase lag of 30deg on the input to the actuators. The matrix P is of dimension 4 by 4, since four controls are used in the RIDE control laws: symmetrical thrust, symmetrical tailplanes, differential tailplanes and rudder. For the scope of this benchmark, the minimum set of points to be tested can be the four corners of the region in Fig. 10.3. This criterion is not only a pass/fail test. It is further required to give the combination of uncertain parameters that lead to the worst case stability margin. Therefore, a suitable definition of the stability degree, say ρ, is required to identify the worst case.
6
ρ=1
4
Gain [dB]
2
O
0
B
A
D
C
−2 ρ = 0.57
−4
−6 −210
−200
−190
−180
−170 −160 Phase [°]
−150
−140
−130
−120
Fig. 10.4. Definition of a possible stability degree by scaling the exclusion region
158
F. Corraro
Several different possibilities for the definition of ρ exist [4]. For example, one could be to assign a normalised stability degree of unity to the regions shown in Figs. 10.2 and 10.3. and to scale the region of perturbation by preserving its aspect ratio. Smaller regions will have a stability degree less than one and greater regions will have a stability degree greater than one. The stability degree attained will be defined by the boundary of the greatest region around the critical point in the Nichols diagram, that is not crossed by the set of Nichols plots of the open-loop system (for the single-loop test) or the boundary of the greatest region for which all the internal perturbations do not destabilise the system (for the multi-loop test). An example is presented in Fig. 10.4 for the single-loop test. An hypothetical transfer function (continuous line) is plotted in the Nichols diagram against the stability margin boundary defined in the criterion, which is drawn with a bold line. The example transfer function crosses the boundary and therefore violates the requirements. A second region, of the same shape as the criterion boundary, is plotted in the figure. This is the greatest region having that shape that is not crossed by the transfer function line. The stability degree at the criterion region (with bold borders) is taken as unity. Thus, following the definition above, the stability degree at the inner region is defined by the ratio: ρS =
OB ∼ 20◦ ∼ = ◦ = 0.57 35 OA
10.4 Unstable Eigenvalues Criterion (Class I) For this criterion it is required to identify the flight cases (in terms of M , altitude and AoA) where unstable closed-loop eigenvalues (i.e. those with positive real part) occur, and for what tolerance combination these eigenvalues have the largest value of their real part. This test is to determine the most severe cases of divergent modes in the closed-loop system, to allow an assessment of their acceptability in terms of their influence on aircraft handling. A minimal requirement is to consider category 1 uncertainties, as described in section 10.2. In what follows, we define the precise bound on the real part of the unstable eigenvalues. Let λ = σ + j ω be an eigenvalue of the state matrix of the closed-loop linearised state space model. The real part σ must satisfy (see also Fig. 10.5) the following bounding condition: for ω ∈ Ω1 = {ω : |ω| ≥ 0.15 rad/s} σ1 = 0, σ < σ2 = (ln 2)/20, for ω ∈ Ω2 = {ω : 0 < |ω| < 0.15 rad/s} σ3 = (ln 2)/7, for ω ∈ Ω3 = {0}
(10.1)
The eigenvalues of the closed-loop system depend generally of the uncertain parameters. Let λ(π) = σ(π) + jω(π) be such an eigenvalue depending on
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parameter vector π ∈ Π. For each domain Ω i , i = 1, 2, 3 of the imaginary parts, we can define the set Λ+ i (π) = {λ (π) : σ (π) ≥ σi , ω (π) ∈ Ωi , } representing the eigenvalues whose real parts violates the condition (10.1). Let σi,max the maximum real part of eigenvalues in Λ+ i (π). Then, for each domain Ω i , the maximum unstable eigenvalue criterion can be defined as σi,wc = max σi,max (π) Π
and the corresponding worst-case parameter combination is πi,wc = arg max σi,max (π) Π
The criterion is defined for all flight conditions in Table 10.1, for which Λ+ i (π) is not empty, i.e. for which at least one unstable eigenvalue exists. Note that for real eigenvalues, the definition of worst case as the maximum real part among the positive eigenvalues is quite straightforward. For complex eigenvalues, different definitions of worst cases could have been chosen, such as the magnitude of the complex eigenvalue. Here, the maximum positive real part has been suggested because it can be directly linked to the existing handling qualities requirement on the minimum time to double amplitude of unstable modes. 1
ω
For ω ≥ 0.15 rad/s σ