Adaptive BacksteppingFlight Control for ModernFighter AircraftISBN 978-90-8570-573-4Printed by W ohrmann Print Service, Zutphen, The Netherlands.Typeset by the author with the LATEX Documentation System.Cover design based on an F-16 image by James Dale.Copyright c 2010 by L. Sonneveldt. All rights reserved. No part of the materialprotected by this copyright notice may be reproduced or utilized in any form or by anymeans, electronic or mechanical, including photocopying, recording or by anyinformation storage and retrieval system, without the prior permission of the author.Adaptive BacksteppingFlight Control for ModernFighter AircraftPROEFSCHRIFTter verkrijging van de graad van doctoraan de Technische Universiteit Delft,op gezag van de Rector Magnicus Prof. ir. K.Ch.A.M. Luyben,voorzitter van het College voor Promoties,in het openbaar te verdedigen op woensdag 7 juli 2010 om 15.00 uurdoorLars SONNEVELDTingenieur luchtvaart en ruimtevaartgeboren te Rotterdam.Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. J.A. MulderCopromotor:Dr. Q.P. ChuSamenstelling promotiecommissie:Rector Magnicus voorzitterProf. dr. ir. J.A. Mulder Technische Universiteit Delft, promotorDr. Q.P. Chu Technische Universiteit Delft, copromotorProf. lt. gen. b.d. B.A.C. Droste Technische Universiteit DelftProf. dr. ir. M. Verhaegen Technische Universiteit DelftProf. dr. A. Zolghadri Universit e de BordeauxProf. Dr.-Ing. R. Luckner Technische Universit at BerlinIr. W.F.J.A. Rouwhorst Nationaal Lucht- en RuimtevaartlaboratoriumProf. dr. ir. Th. van Holten Technische Universiteit Delft, reservelidTo RianneSummaryOver the last few decades and pushed by developments in aerospace technology, the per-formance requirements of modern ghter aircraft became more and more challengingthroughout an ever increasing ight envelope. Extreme maneuverability is achieved bydesigning the aircraft with multiple redundant control actuators and allowing static in-stabilities in certain modes. A good example is the Lockheed Martin F-22 Raptor whichmakes use of thrust vectored control to increase maneuverability. Furthermore, the sur-vivability requirements in modern warfare are constantly evolving for both manned andunmanned combat aircraft. Taking into account all these requirements when designingthe control systems for modern ghter aircraft poses a huge challenge for ight controldesigners.Traditionally, aircraft control systems were designed using linearized aircraft models atmultiple trimmed ight conditions throughout the ight envelope. For each of theseoperating points a corresponding linear controller is derived using the well establishedlinear-based control design methods. One of the many gain scheduling methods can nextbe applied to derive a single ight control law for the entire ight envelope. However,a problem of this approach is that good performance and robustness properties cannotbe guaranteed for a highly nonlinear ghter aircraft. Nonlinear control methods havebeen developed to overcome the shortcomings of linear design approaches. The theo-retically established nonlinear dynamic inversion (NDI) approach is the best known andmost widely used of these methods.NDI is a control design method that can explicitly handle systems with known nonlin-earities. By using nonlinear feedback and exact state transformations rather than linearapproximations the nonlinear system is transformed into a constant linear system. Thislinear system can in principle be controlled by just a single linear controller. However,to perform perfect dynamic inversion all nonlinearities have to be precisely known. Thisis generally not the case with modern ghter aircraft, since it is very difcult to preciselyknow and model their complex nonlinear aerodynamic characteristics. Empirical datais usually obtained from wind tunnel experiments and ight tests, augmented by com-iii SUMMARYputational uid dynamics (CFD) results, and thus is not 100% accurate. The problemof model deciencies can be dealt with by closing the control loop with a linear, robustcontroller. However, even then desired performance cannot be expected in case of grosserrors, due to large, sudden changes in the aircraft dynamics that could result from struc-tural damage, control effector failures or adverse environmental conditions.A more sophisticated way of dealing with large model uncertainties is to introduce anadaptive control system with some form of online model identication. In recent years,the increase in available onboard computational power has made it possible to implementmore complex adaptive ight control designs. It is clear that a nonlinear adaptive ightcontrol system with onboard model identication can do more than just compensate forinaccuracies in the nominal aircraft model. It is also possible to identify any suddenchanges in the dynamic behavior of the aircraft. Such changes will in general lead to anincrease in pilot workload or can even result in a complete loss of control. If the post-failure aircraft dynamics can be identied correctly by the online model identication,the redundancy in control effectors and the y-by-wire system of modern ghter planescan be exploited to recongure the ight control system.There are several methods available to design an identier that updates the onboard modelof the NDI controller online, e.g. neural networks or least squares techniques. A disad-vantage of an adaptive design with separate identier is that the certainty equivalenceproperty does not hold for nonlinear systems, i.e. the identier is not fast enough to copewith potentially faster-than-linear growth of instabilities in nonlinear systems. To over-come this problem a controller with strong parametric robustness properties is needed.An alternative solution is to design the controller and identier as a single integratedsystem using the adaptive backstepping design method. By systematically constructinga Lyapunov function for the closed-loop system, adaptive backstepping offers the pos-sibility to synthesize a controller for a wide class of nonlinear systems with parametricuncertainties.The main goal of this thesis is to investigate the potential of the nonlinear adaptive back-stepping control technique in combination with online model identication for the designof a recongurable ight control (RFC) system for a modern ghter aircraft. The follow-ing features are aimed for: the RFC system uses a single nonlinear adaptive ight controller for the entiredomain of operation (ight envelope), which has provable theoretical performanceand stability properties. the RFC system enhances performance and survivability of the aircraft in the pres-ence of disturbances related to failures and structural damage. the algorithms, on which the RFC systemis based, possess excellent numerical sta-bility properties and their computational costs are low (real-time implementationis feasible).Adaptive backstepping is a recursive, Lyapunov-based, nonlinear design method, thatmakes use of dynamic parameter update laws to deal with parametric uncertainties. Theiiiidea of backstepping is to design a controller recursively by considering some of thestate variables as virtual controls and designing intermediate control laws for these.Backstepping achieves the goals of global asymptotic stabilization of the closed-loopstates and tracking. The proof of these properties is a direct consequence of the recur-sive procedure, since a Lyapunov function is constructed for the entire system includingthe parameter estimates. The tracking errors drive the adaptation process of the proce-dure. Furthermore, it is possible to take magnitude and rate constraints on the controlinputs and system states into account in such a way that the identication process is notcorrupted during periods of control effector saturation. A disadvantage of the integratedadaptive backstepping method is that it only yields pseudo-estimates of the uncertain sys-tem parameters. There is no guarantee that the real values of the parameters are found,since the adaptation only tries to satisfy a total system stability criterion, i.e. the Lya-punov function. Increasing the adaptation gain will not necessarily improve the responseof the closed-loop system, due to the strong coupling between the controller and the es-timator dynamics.The immersion and invariance (I&I) approach provides an alternative way of construct-ing a nonlinear estimator. This approach allows for prescribed stable dynamics to beassigned to the parameter estimation error. The resulting estimator is combined with abackstepping controller to form a modular adaptive control scheme. The I&I based esti-mator is fast enough to capture the potential faster-than-linear growth of nonlinear sys-tems. The resulting modular scheme is much easier to tune than the ones resulting fromthe standard adaptive backstepping approaches with tracking error driven adaptation pro-cess. In fact, the closed-loop system resulting from the application of the I&I basedadaptive backstepping controller can be seen as a cascaded interconnection between twostable systems with prescribed asymptotic properties. As a result, the performance of theclosed-loop system with adaptive controller can be improved signicantly.To make a real-time implementation of the adaptive controllers feasible the computa-tional complexity has to be kept at a minimum. As a solution, a ight envelope partition-ing method is proposed to capture the globally valid aerodynamic model into multiplelocally valid aerodynamic models. The estimator only has to update a few local modelsat each time step, thereby decreasing the computational load of the algorithm. An addi-tional advantage of using multiple, local models is that information of the models that arenot updated at a certain time step is retained, thereby giving the approximator memorycapabilities. B-spline networks are selected for their nice numerical properties to ensuresmooth transitions between the different regions.The adaptive backstepping ight controllers developed in this thesis have been evaluatedin numerical simulations on a high-delity F-16 dynamic model involving several controlproblems. The adaptive designs have been compared with the gain-scheduled baselineight control system and a non-adaptive NDI design. The performance has been com-pared in simulation scenarios at several ight conditions with the aircraft model sufferingfrom actuator failures, longitudinal center of gravity shifts and changes in aerodynamiccoefcients. All numerical simulations can be easily performed in real-time on an ordi-iv SUMMARYnary desktop computer. Results of the simulations demonstrate that the adaptive ightcontrollers provide a signicant performance improvement over the non-adaptive NDIdesign for the simulated failure cases.Of the evaluated adaptive ight controllers, the I&I based modular adaptive backstep-ping design has the overall best performance and is also easiest to tune, at the cost ofa small increase in computational load and design complexity when compared to inte-grated adaptive backstepping control designs. Moreover, the ight controllers designedwith the I&I based modular adaptive backstepping approach have even stronger provablestability and convergence properties than the integrated adaptive backstepping ight con-trollers, while at the same time achieving a modularity in the design of the controller andidentier. On the basis of the research performed in this thesis, it can be concluded that aRFC system based on the I&I based modular adaptive backstepping method shows a lotof potential, since it possesses all the features aimed at in the thesis goal.Further research that explores the performance of the RFC system based on the I&Ibased modular adaptive backstepping method in other simulation scenarios is suggested.The evaluation of the adaptive ight controllers in this thesis is limited to simulationscenarios with actuator failures, symmetric center of gravity shifts and uncertainties isindividual aerodynamic coefcients. The research would be more valuable if scenarioswith asymmetric failures such as partial surface loss are performed. Generating the nec-essary realistic aerodynamic data for the F-16 model would take a separate study in itself.Still an open issue is the development of an adaptive ight envelope protection systemthat can estimate the reduced ight envelope of an aircraft post-failure and that can feedthis information back to the controller, the pilot and the guidance system. Another im-portant research direction would be to perform a piloted evaluation and validation of theproposed RFC framework in a simulator. Post-failure workload and handling qualitiesshould be compared with those of the baseline ight control system. Simultaneously, astudy of the interactions between the pilots reactions to a failure and the actions taken bythe adaptive element in the ight control system can be performed.ContentsSummary i1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Recongurable Flight Control . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Recongurable Flight Control Approaches . . . . . . . . . . . 51.3.2 Recongurable Flight Control in Practice . . . . . . . . . . . . 91.4 Thesis Goal and Research Approach . . . . . . . . . . . . . . . . . . . 101.4.1 Nonlinear Adaptive Backstepping Control . . . . . . . . . . . . 111.4.2 Flight Envelope Partitioning . . . . . . . . . . . . . . . . . . . 111.4.3 The F-16 Dynamic Model . . . . . . . . . . . . . . . . . . . . 121.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Aircraft Modeling 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Aircraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Aircraft Variables . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Equations of Motion for a Rigid Body Aircraft . . . . . . . . . 202.2.4 Gathering the Equations of Motion . . . . . . . . . . . . . . . . 242.3 Control Variables and Engine Modeling . . . . . . . . . . . . . . . . . 262.4 Geometry and Aerodynamic Data . . . . . . . . . . . . . . . . . . . . 282.5 Baseline Flight Control System . . . . . . . . . . . . . . . . . . . . . . 312.5.1 Longitudinal Control . . . . . . . . . . . . . . . . . . . . . . . 312.5.2 Lateral Control . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.3 Directional Control . . . . . . . . . . . . . . . . . . . . . . . . 31vvi CONTENTS2.6 MATLAB/Simulink c Implementation . . . . . . . . . . . . . . . . . . 323 Backstepping 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Lyapunov Theory and Stability Concepts . . . . . . . . . . . . . . . . . 343.2.1 Lyapunov Stability Denitions . . . . . . . . . . . . . . . . . . 343.2.2 Lyapunovs Direct Method . . . . . . . . . . . . . . . . . . . . 363.2.3 Lyapunov Theory and Control Design . . . . . . . . . . . . . . 383.3 Backstepping Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . 413.3.2 Extension to Higher Order Systems . . . . . . . . . . . . . . . 443.3.3 Example: Longitudinal Missile Control . . . . . . . . . . . . . 474 Adaptive Backstepping 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Tuning Functions Adaptive Backstepping . . . . . . . . . . . . . . . . 544.2.1 Dynamic Feedback . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Extension to Higher Order Systems . . . . . . . . . . . . . . . 584.2.3 Robustness Considerations . . . . . . . . . . . . . . . . . . . . 634.2.4 Example: Adaptive Longitudinal Missile Control . . . . . . . . 664.3 Constrained Adaptive Backstepping . . . . . . . . . . . . . . . . . . . 684.3.1 Command Filtering Approach . . . . . . . . . . . . . . . . . . 694.3.2 Example: Constrained Adaptive Longitudinal Missile Control . 735 Inverse Optimal Adaptive Backstepping 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Nonlinear Control and Optimality . . . . . . . . . . . . . . . . . . . . 785.2.1 Direct Optimal Control . . . . . . . . . . . . . . . . . . . . . . 785.2.2 Inverse Optimal Control . . . . . . . . . . . . . . . . . . . . . 805.3 Adaptive Backstepping and Optimality . . . . . . . . . . . . . . . . . . 805.3.1 Inverse Optimal Design Procedure . . . . . . . . . . . . . . . . 815.3.2 Transient Performance Analysis . . . . . . . . . . . . . . . . . 855.3.3 Example: Inverse Optimal Adaptive Longitudinal Missile Control 865.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 Comparison of Integrated and Modular Adaptive Flight Control 936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Modular Adaptive Backstepping . . . . . . . . . . . . . . . . . . . . . 946.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 956.2.2 Input-to-state Stable Backstepping . . . . . . . . . . . . . . . . 976.2.3 Least-Squares Identier . . . . . . . . . . . . . . . . . . . . . 986.3 Aircraft Model Description . . . . . . . . . . . . . . . . . . . . . . . . 101CONTENTS vii6.4 Flight Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.4.1 Feedback Control Design . . . . . . . . . . . . . . . . . . . . . 1036.4.2 Integrated Model Identication . . . . . . . . . . . . . . . . . . 1056.4.3 Modular Model Identication . . . . . . . . . . . . . . . . . . 1066.5 Control Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.5.1 Weighted Pseudo-inverse . . . . . . . . . . . . . . . . . . . . . 1086.5.2 Quadratic Programming . . . . . . . . . . . . . . . . . . . . . 1086.6 Numerical Simulation Results . . . . . . . . . . . . . . . . . . . . . . 1106.6.1 Tuning the Controllers . . . . . . . . . . . . . . . . . . . . . . 1106.6.2 Simulation Scenarios . . . . . . . . . . . . . . . . . . . . . . . 1116.6.3 Controller Comparison . . . . . . . . . . . . . . . . . . . . . . 1126.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167 F-16 Trajectory Control Design 1197.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2 Flight Envelope Partitioning . . . . . . . . . . . . . . . . . . . . . . . 1207.2.1 Partitioning the F-16 Aerodynamic Model . . . . . . . . . . . . 1217.2.2 B-spline Networks . . . . . . . . . . . . . . . . . . . . . . . . 1247.2.3 Resulting Approximation Model . . . . . . . . . . . . . . . . . 1287.3 Trajectory Control Design . . . . . . . . . . . . . . . . . . . . . . . . 1287.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.3.2 Aircraft Model Description . . . . . . . . . . . . . . . . . . . . 1307.3.3 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . 1317.3.4 Model Identication . . . . . . . . . . . . . . . . . . . . . . . 1397.4 Numerical Simulation Results . . . . . . . . . . . . . . . . . . . . . . 1417.4.1 Controller Parameter Tuning . . . . . . . . . . . . . . . . . . . 1427.4.2 Maneuver 1: Upward Spiral . . . . . . . . . . . . . . . . . . . 1437.4.3 Maneuver 2: Reconnaissance . . . . . . . . . . . . . . . . . . 1457.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468 F-16 Stability and Control Augmentation Design 1498.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.2 Flight Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.2.1 Outer Loop Design . . . . . . . . . . . . . . . . . . . . . . . . 1518.2.2 Inner Loop Design . . . . . . . . . . . . . . . . . . . . . . . . 1528.2.3 Update Laws and Stability Properties . . . . . . . . . . . . . . 1538.3 Integrated Model Identication . . . . . . . . . . . . . . . . . . . . . . 1548.4 Modular Model Identication . . . . . . . . . . . . . . . . . . . . . . . 1558.5 Controller Tuning and Command Filter Design . . . . . . . . . . . . . 1578.6 Numerical Simulations and Results . . . . . . . . . . . . . . . . . . . . 1598.6.1 Simulation Scenarios . . . . . . . . . . . . . . . . . . . . . . . 1628.6.2 Simulation Results with Cmq = 0 . . . . . . . . . . . . . . . . 162viii CONTENTS8.6.3 Simulation Results with Longitudinal c.g. Shifts . . . . . . . . 1638.6.4 Simulation Results with Aileron Lock-ups . . . . . . . . . . . . 1648.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659 Immersion and Invariance Adaptive Backstepping 1679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.2 The Immersion and Invariance Concept . . . . . . . . . . . . . . . . . 1689.3 Extension to Higher Order Systems . . . . . . . . . . . . . . . . . . . 1739.3.1 Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . 1739.3.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.4 Dynamic Scaling and Filters . . . . . . . . . . . . . . . . . . . . . . . 1779.4.1 Estimator Design with Dynamic Scaling . . . . . . . . . . . . . 1779.4.2 Command Filtered Control Law Design . . . . . . . . . . . . . 1809.5 Adaptive Flight Control Example . . . . . . . . . . . . . . . . . . . . . 1829.5.1 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . 1839.5.2 Numerical Simulation Results . . . . . . . . . . . . . . . . . . 1859.6 F-16 Stability and Control Augmentation Design . . . . . . . . . . . . 1879.6.1 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . 1879.6.2 Numerical Simulation Results . . . . . . . . . . . . . . . . . . 1899.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19110 Conclusions and Recommendations 19310.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19310.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200A F-16 Model 203A.1 F-16 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203A.2 ISA Atmospheric Model . . . . . . . . . . . . . . . . . . . . . . . . . 204A.3 Flight Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . 205B System and Stability Concepts 207B.1 Lyapunov Stability and Convergence . . . . . . . . . . . . . . . . . . . 207B.2 Input-to-state Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 211B.3 Invariant Manifolds and System Immersion . . . . . . . . . . . . . . . 211C Command Filters 213D Additional Figures 215D.1 Simulation Results of Chapter 6 . . . . . . . . . . . . . . . . . . . . . 216D.2 Simulation Results of Chapter 7 . . . . . . . . . . . . . . . . . . . . . 221D.3 Simulation Results of Chapter 8 . . . . . . . . . . . . . . . . . . . . . 227D.4 Simulation Results of Chapter 9 . . . . . . . . . . . . . . . . . . . . . 234CONTENTS ixBibliography 239Samenvatting 263Acknowledgements 269Curriculum Vitae 271Chapter 1IntroductionThis chapter provides an introduction on modern high performance ghter aircraft andtheir ight control systems. It describes the current situation, the ongoing research andthe challenges for these systems. The position of the work performed in this thesis inrelation to existing research on control methods is explained. Furthermore, the solu-tion proposed in this thesis, as well as the research approach and scope are discussed.The thesis outline is claried in the nal part of the chapter, by means of a short topicdescription for each chapter and an explanation of the interconnections between the dif-ferent chapters.1.1 BackgroundAt the moment, most western countries, including the Netherlands, have started to re-place or are considering replacing their current eet of ghter aircraft with aircraft of thenew generation. Some of the better known examples of this new generation of ghteraircraft are the F-22 Raptor, the JAS-39 Gripen, the Euroghter and the F-35 LightningII (better known as Joint Strike Fighter). Pushed by Air Force requirements and by de-velopments in aerospace technology, the performance specications for modern ghteraircraft have become ever more challenging. Extreme maneuverability over a large ightenvelope is achieved by designing the aircraft unstable in certain modes and by usingmultiple redundant control effectors for control. Examples include the F-22 Raptor (Fig-ure 1.1(a)), which makes use of thrust vectored control to achieve extreme angles ofattack, and the highly unstable Su-47 prototype (Figure 1.1(b)) with its forward sweptwings and thrust vectoring.Human pilots are not able to control these highly complex nonlinear systems withoutsome kind of assistance for their various tasks. Modern ghter aircraft require digitalight control systems to ensure that aircraft possess the ying qualities pilots desire. Infact, ight control systems have been considered by inventors even before the rst ight12 INTRODUCTION 1.1(a) The F-22 Raptor (b) The Su-47 BerkhutFigure 1.1: Two examples of modern high performance ghter aircraft. The F-22 picture is bycourtesy of the USAF and the Su-47 picture is a photo by Andrey Zinchuk.of the Wright brothers in 1903 [139]. In 1893 Sir Hiram Maxim already made a work-ing model of a steam-powered gyroscope and servo cylinder to maintain the longitudinalattitude of an aircraft. The pitch controller weighted over 130 kg, which is still only afraction of the 3.5 ton total weight of his self-developed steam-powered ying machinedepicted in Figure 1.2. In 1913 Lawrence Sperry demonstrated hands-off ight when heand his co-pilot each stood on a wing of his biplane as it passed the exuberant crowd.Sperry used a lightweight version of his fathers gyroscope to control the pitch and rollmotion of his aircraft with compressed air.Both world wars not only stimulated the development of more advanced ight controlsystems, but also the fundamentals of classical control theory were laid.Figure 1.2: Sir Hiram Maxims heavier-than-air steam-powered aircraft.In the early 1950s it was found that constant-gain, linear feedback controllers had prob-lems to performwell over the whole ight regime of the new high-performance prototypeaircraft such as the X-15. After a considerable development effort it was found that gain-1.2 PROBLEM DEFINITION 3scheduling was a suitable technique to achieve good performance over a wide range ofoperating conditions [9]. Even today, modern ghter aircraft still make use of ightcontrol systems based on various types of linear control algorithms and gain-scheduling.The main benet of this strategy is that it is based on the well-developed classical linearcontrol theory. However, nonlinear effects, occurring in particular at high angles of in-cidence, and the cross-couplings between longitudinal and lateral motion are neglectedin the control design. Furthermore, it is difcult to guarantee stability and performanceof the gain-scheduled controller in between operating points for which a linear controllerhas been designed. This motivates the use of nonlinear control techniques for the ightcontrol system design of high performance aircraft.In the early 1990s a new nonlinear control methodology called feedback linearization(FBL) emerged [88, 187]. Nonlinear dynamic inversion (NDI) is a special form of FBLespecially suited for ight control applications; see e.g. [48, 123, 129]. The main ideabehind NDI is to use an accurate model of the system to cancel all system nonlinearitiesin such a way that a single linear system valid over the entire ight envelope remains. Aclassical linear controller can be used to close the outer loop of the system under NDIcontrol. The F-35 Lightning II will be the rst production ghter aircraft equipped withsuch a NDI based ight control system [20, 205]. The control law structure, presentedin Figure 1.3, permits a decoupling of ying qualities dependent portions of the designfrom that which is dependent on airframe and engine dynamics.Command Shaping(Desired Dynamics)Nonlinear Dynamic Inversion(Onboard Model )Control AllocationSensorProcessingPrecompensationStick InputFlying Qualities Dependent Airframe/Engine DependentFigure 1.3: The nonlinear dynamic inversion controller structure of the F-35: The onboard modelis used to cancel all system nonlinearities and a single linear controller which enforces the yingqualities closes the outer loop.1.2 Problem DenitionThe main weakness of the NDI technique is that an accurate model of the aircraft dynam-ics is required. A dynamic aircraft model is costly to obtain, since it takes a large number4 INTRODUCTION 1.3of (virtual) wind tunnel experiments and an intensive ight testing program. Small un-certainties can be dealt with by designing a robust, linear outer loop controller. However,especially for larger model uncertainties, robust control methods tend to yield ratherconservative control laws and, consequently, result in poor closed-loop performance[106, 202, 215]. A more sophisticated way of dealing with large model uncertaintiesis to introduce an adaptive control system with some form of online model identication.Adaptive control was originally studied in the 1950s as an alternative to gain-schedulingmethods for ight control and there has been a lot of theoretical development over thepast decades [8]. In recent years, the increase in available onboard computational powerhas made it possible to implement adaptive ight control designs.There are several methods available to design an identier that updates the onboard modelof the NDI controller online, e.g. neural networks or least squares techniques. A disad-vantage of a nonlinear adaptive design with separate identier is that the certainty equiv-alence property [106] does not hold for nonlinear systems, i.e. the identier is not fastenough to cope with potentially explosive instabilities of nonlinear systems. To overcomethis problem a controller with strong parametric robustness properties is needed [119].An alternative solution is to design the controller and identier as a single integrated sys-tem using the adaptive backstepping design method [101, 117, 118]. By systematicallyconstructing a Lyapunov function for the closed-loop system, adaptive backstepping of-fers the possibility to synthesize a controller for a wide class of nonlinear systems withparametric uncertainties.Obviously, a nonlinear adaptive (backstepping based) ight control system with onboardmodel identication has the potential to do more than just compensate for inaccuracies inthe nominal aircraft model. It is also possible to identify sudden changes in the dynamicbehavior of the aircraft that could result from structural damage, control effector failuresor adverse environmental conditions. Such changes will in general lead to an increase inpilot workload or can even result in a complete loss of control. If the post-failure aircraftdynamics can be identied correctly by the online model identication, the redundancyin control effectors and the y-by-wire system of modern ghter planes can be exploitedto recongure the ight control system.1.3 Recongurable Flight ControlThe idea of control reconguration can be traced back throughout the history of ight incases where pilots had to manually exploit the remaining control capability of a degradedaircraft. In 1971 an early theoretical basis for control reconguration appeared in [13],where the number of control effectors needed for the controllability of a linear system forfailure accommodation was considered. In fact, most of the studies in the 1970s werebased on the idea of introducing backup ight control effectors to compensate for thefailure of a primary control surface. Many of these studies are also relevant for controlreconguration. Two early studies that rst showed the value of control recongurationwere performed by the Grumman Aerospace Corporation for the United States Air Force(USAF) [23] and by the United States Navy [72]. The study done by Grumman demon-1.3 RECONFIGURABLE FLIGHT CONTROL 5strated the importance of considering reconguration during the initial design process.One of the aircraft studied at the time was the F-16, which would become a focus of laterUSAF studies as it appeared to be well suited for reconguration.Flight control reconguration became an important research subject in the 1980s and hasremained a major eld of study ever since. This section will try to provide an overviewof the many different recongurable ight control (RFC) approaches that have been pro-posed in literature over the past decades. Methods for accommodating sensor failures,software failures or for switching among redundant hardware will not be considered,although they are sometimes referred to as ight control reconguration. Here recon-gurable ight control is only used to refer to software algorithms designed specicallyto compensate for failures or damage to the ight control effectors or structure of theaircraft (e.g. lifting surfaces). This section is based on survey papers on recongurableight control by Huzmezan [84], Jones [92] and Steinberg [195]. Other relevant articlesare the more general fault-tolerant control surveys by Stengel [197] and Patton [162].1.3.1 Recongurable Flight Control ApproachesMost of the control conguration methods developed in the 1980s required a separatesystem for explicit failure detection, isolation and estimation (FDIE). An important earlyexample of this type of approach was developed by General Electric Aircraft Controls[55]. This design used a single extended Kalman estimator to perform all FDIE and apseudo-inverse approach based on a linearized model of the aircraft was used to deter-mine controller effector commands, so that the degraded aircraft would generate the sameaccelerations as the nominal aircraft. The single Kalman estimator approach turned outto be impractical, but the pseudo-inverse methods would become a major focus of re-search, even resulting in some limited ight testing at the end of the 1980s [140].By the beginning of the 1990s a set of ight tested techniques was available, whichcould be used to add limited recongurable control capability to otherwise conventionalight control laws for xed wing aircraft. FDIE was the main limiting factor and requiredcomplicated tuning based on known failure models, particularly for surface damage de-tection and isolation. Similarly, the control approaches could require quite a bit of designtuning and there was a lack of theoretical proofs of stability and robustness. However,these approaches were shown to be quite effective when optimized for a small number offailure cases [195].The increase of onboard computational power and advanced control development soft-ware packages in the 1990s led to an rapid increase in the number and types of ap-proaches applied to RFC problems. It became much easier and cheaper to experimentwith complex nonlinear design approaches. Furthermore, there had been considerabletheoretical advances in the areas of adaptive [9] and nonlinear control methods [187]throughout the 1980s. The late 1980s also saw a renewed interest in the use of emergingmachine intelligence techniques, such as neural networks and fuzzy logic [148]. Theseapproaches could potentially improve FDIE or support new control architectures that donot use explicit FDIE at all.6 INTRODUCTION 1.3An attempt is now made to organize the various RFC methods developed during the1990s and up until now. This has become increasingly more difcult, because a lot ofcombinations of different methods have been attempted over the years. In [92] the RFCmethods are subdivided in four categories. A short overview of each category will nowbe given. Note that this overview is by no means complete, but only serves to give an il-lustration of the advantages and disadvantages of the different methodologies. Also notethat many combinations of different methodologies have emerged over the years.Multiple Model ControlMultiple model control basically involves a controller lawexisting of several fault modelsand their corresponding controllers. Three types of multiple model control exist in liter-ature: multiple model switching and tuning (MMST), interacting multiple model (IMM)and propulsion controlled aircraft (PCA). In the rst two cases all expected failure sce-narios are collected during a failure modes and effects analysis, where fault models areconstructed that cover each situation. When a failure occurs MMST switches to a pre-computed control law corresponding to the current fault situation. Some examples ofMMST approaches can be found in [24, 25, 26, 71]. IMM removes the extensive faultmodeling limitation of MMST, by considering fault models which are a convex combi-nation of models in a predetermined model set. Again the control law can be based ona variety of methods. In [137, 138] a xed controller is used, while in [99, 100] a MPCscheme with minimization of the past tracking error is used. PCA is a special case ofMMST, where the only anticipated fault is total hydraulics failure and only the enginescan be used for control. There have been some successful ight tests with PCA on a F-15and an MD-11 in the beginning of the 1990s [32, 33].The advantage of multiple model methods is that they are fast and provable stable. Themain disadvantages are the lack of correct models when dealing with failures that werenot considered during the control design and the exponential increase of the number ofmodels required with the number of considered failures for large systems.Controller SynthesisController synthesis methods make use of a fault model provided by some form of FDIE.FDIE provides information about the onset, location and severity of any faults and hencethe reconguration problemis reduced to nding a proper FDIE. Many FDIE approachescan be found in literature, see e.g. [41, 169, 217, 216] and the references therein. Eigen-structure assignment (EA), the pseudo-inverse method (PIM) and model predictive con-trol (MPC) are three of the methodologies which can be used in this recongurable con-trol framework. The main idea of EA is to design a stabilizing output feedback law such that eigenstructure closed-loop system of the linear fault model provided by the FDIE unitis as close as possible to that of the original closed-loop system. The limitationswhen applying EA to recongurable ight control are obvious: only linear modelsare considered and actuator dynamics are not taken into account. Also, a perfect1.3 RECONFIGURABLE FLIGHT CONTROL 7fault model is assumed and the effect of eigenvectors in the failed system beingnot exactly equal to those in the nominal system is not well understood. Despitethese problems, some examples of EA and recongurable ight control exist inliterature, see e.g. [112, 214]. A method which closely resembles EA is the pseudo-inverse method. The idea ofthe PIM is to recover the closed-loop behavior by calculating an output feedbacklaw which minimizes the difference in closed-loop dynamics between the faultmodel and the nominal model. The PIM was popular in the 1980s and the early1990s, but has fallen out of grace due to difculties in ensuring stability. A surveywith several attempts to make this method stable can be found in [162]. MPC is an interesting method to use for RFC due to its ability to handle constraintsand changing model dynamics systematically when failure occurs. MPC also re-quires the use of a fault model since it relies on an internal model of the system.Several methods for changing the internal model have been proposed, such as themultiple model method in [99]. More examples of RFC using MPC can be foundin [94, 95]. In [74] a combination of MPC and a subspace predictor is suggestedand demonstrated in a recongurable ight control problem for a damaged Boeing747 model. A disadvantage of MPC is that the method requires a computation-ally intensive online optimization at each time step, which makes it difcult toimplement MPC as an aircraft controller. There is no guarantee that there exists asolution to the optimization problem at all time.Actuator OnlyActuator only methodologies are limited in the sense that they can only provide recon-gurable control in case of actuator failures. Sliding mode control (SMC) and controlallocation (CA) are two such methodologies: SMC is a nonlinear control method, which has become quite popular for RFCresearch [82, 83, 179, 180]. The advantages of SMC are its excellent provablerobustness properties by the use of a discontinuous term in the control law. Amajor disadvantage is that some assumptions have to be made which require thatthere is one control surface for each controlled variable, none of the controllersurfaces can ever be completely lost. This is not very realistic, as actuators areusually jammed completely when they fail. CA is mainly used in aircraft with redundant control surfaces, like high perfor-mance jet ghters [34, 35]. CA distributes the demanded forces and torque bythe controller over the actuators. CA handles actuator failures without the needto model the control law and has therefor received a lot of attention in literature,see [54] for a survey. A limitation of this approach is that the post-failure aircraftand actuator dynamics are not taken into account by the control law, so that thecontroller will still be attempting to achieve the original system performance while8 INTRODUCTION 1.3the actuators may not be capable of achieving this. Another problem is that thesystem will not necessarily be stable, even with a stabilizing control law, as the in-put seen by the system may not be equal to that intended by the controller. Severalextensions to the basic CA method have been proposed in literature, see e.g. [76]for an overview.Adaptive ControlAdaptive control approaches are by far the largest research area in RFC, especially inrecent decades. An adaptive controller is a controller with adjustable parameters and amechanism for adjusting these parameters. A good adaptive control law removes theneed for a FDIE system. However, it is often difcult or impossible to proof robust-ness and/or stability of such an algorithm. All the previously mentioned methods arealso somewhat adaptive, but all require FDIE or use pre-computed control laws and faultmodels. The bulk of the adaptive ight control approaches in literature can be roughly di-vided in two categories: model reference adaptive control (MRAC) and model inversionbased control, e.g. NDI or backstepping (BS), in combination with an online parameterestimation method, e.g. neural networks (NN), recursive least squares (RLS). MRAC is usually used as a nal stage in another algorithm. The goal of MRACis to force the output of the system to track a reference model with the desiredspecications. Adaptation is used to estimate the controller parameters needed totrack the model when failure occurs. There exists direct adaptation and indirectadaptation. Both these methods are compared in [18, 19]. Other publicationsabout RFC using MRAC are [73, 108, 107, 142]. In [85] a discrete version of thismethod is proposed. Several modications of standard MRAC have been proposedto provide stable adaptation in the presence of input constraints [91, 123]. NDI/BS in combination with NN/RLS basically uses nonlinear control for refer-ence tracking and NN/RLS to compensate for all modeling errors. In [30, 31, 37,38, 39] a controller using NDI in combination with NN is designed and (limited)ight tested on a tailless ghter aircraft under the USAF RESTORE program andon the unmanned X-36 (Figure 1.4(a)). NDI with NN was also used on the F-15ACTIVE (Figure 1.4(b)) under the intelligent ight control system program of theNASA [21, 22]. A Boeing 747 tted with a NDI controller combined with RLS forthe online model identication was successfully tested in a moving base simulator[131, 132, 133]. In recent years adaptive backstepping ight control in combina-tion with some formof neural networks has become a popular research subject, seee.g. [58, 125, 161, 176, 177, 196]. The main advantages of adaptive backsteppingover NDI are its strong stability and convergence properties. Some other approaches suggested in literature over the years include the adaptiveLQR methods in [2, 69]. In [62] a linear matrix inequalities framework for arobust, adaptive nonlinear ight control system is proposed. In [81, 185] a RFCfor the NASA F-18/HARV based on a QFT compensator and an adaptive lter is1.3 RECONFIGURABLE FLIGHT CONTROL 9used. Flight control based on reinforcement learning is the subject of [89, 90, 126].Indirect adaptive control using a moving window/batch estimation for partial lossof the horizontal tail surface is studied in [157].(a) The X-36 (b) The F-15 ACTIVEFigure 1.4: Two examples of aircraft used for recongurable ight control testing. Pictures bycourtesy of NASA.1.3.2 Recongurable Flight Control in PracticeIn 1998, an F-18E/F Super Hornet (Figure 1.5) was in the middle of a utter test ightwhen the right stabilizer actuator experienced a failure [53]. This failure would have trig-gered a reversion to a mechanical control mode in previous versions of the F-18, whichusually caused substantial transients and slightly degraded handling qualities. However,the E/F design included the replacement of the mechanical backup system with a recon-gurable control law. For this particular failure, the left stabilizer and rudder toe-in canbe used to restore some of the lost pitching moment and the aps, ailerons and rudderscan be used to compensate for the coupling in lateral/directional axis caused by asym-metric stabilizer deection. Although this control reconguration approach had beendemonstrated with simulated failures in ight tests, this was the rst successful demon-stration with an actual failure.In 1999 the F-18E/F was the rst production aircraft delivered with a recongurableight control law, which can only compensate for a single stabilizer actuator failuremode. Several more advanced RFC systems have been ight tested on the X-36 andthe F-15 ACTIVE, but manufacturers are cautious to implement them in production air-craft. One reason for this has been the difculty of certifying RFC approaches for safetyof ight. Therefore, part of the current research is focusing on the development of toolsfor the analysis of RFC laws and adaptive control algorithms that are easier to certify andimplement. For instance, in [27, 141] a retrot RFC law using a modied sequentialleast-squares algorithm for online model identication is proposed, which does not alter10 INTRODUCTION 1.4Figure 1.5: The F-18E/F Super Hornet with RFC Law. Picture by courtesy of Boeing.the baseline inner loop control and could be treated more like an autopilot for certica-tion purposes. A limited ight test programhas been performed by Boeing and the NavalAir Systems Command [158]. However, again only certain types of actuator failures areconsidered.1.4 Thesis Goal and Research ApproachThe main goal of this thesis is to investigate the potential of the nonlinear adaptive back-stepping control technique in combination with online model identication for the designof a recongurable ight control system for a modern ghter aircraft. The following fea-tures are aimed for: the RFC system uses a single nonlinear adaptive ight controller for the entiredomain of operation (ight envelope), which has provable theoretical performanceand stability properties. the RFC system enhances performance and survivability of the aircraft in the pres-ence of disturbances related to failures and structural damage. the algorithms, on which the RFC systemis based, possess excellent numerical sta-bility properties and their computational costs are low (real-time implementationis feasible).As a study model the Lockheed Martin F-16 is selected, since it is the current ghteraircraft of the Royal Netherlands Air Force and an accurate high-delity aerodynamicmodel has been obtained. The MATLAB/Simulink c software package will be used todesign, rene and evaluate the RFC system. A short discussion on the motivation of themethods and the aircraft model used in this thesis is now presented.1.4 THESIS GOAL AND RESEARCH APPROACH 111.4.1 Nonlinear Adaptive Backstepping ControlAdaptive backstepping is a recursive, Lyapunov-based, nonlinear design method, whichmakes use of dynamic parameter update laws to deal with parametric uncertainties. Theidea of backstepping is to design a controller recursively by considering some of the statevariables as virtual controls and designing intermediate control laws for these. Back-stepping achieves the goals of global asymptotic stabilization and tracking. The proofof these properties is a direct consequence of the recursive procedure, since a Lyapunovfunction is constructed for the entire system including the parameter estimates. Thetracking errors drive the adaptation process of the procedure.Furthermore, it is possible to take magnitude and rate constraints on the control inputsand system states into account such that the identication process is not corrupted duringperiods of control effector saturation [58, 61]. A disadvantage of the integrated adaptivebackstepping method is that it only yields pseudo-estimates of the uncertain system pa-rameters. There is no guarantee that the real values of the parameters are found, sincethe adaptation only tries to satisfy a total system stability criterion, i.e. the Lyapunovfunction. Furthermore, since the controller and identier are designed as one integratedsystem it is very difcult to tune the performance of one subsystem without inuencingthe performance of the other. In this thesis several possible improvements to the basicadaptive backstepping approach are introduced and evaluated.1.4.2 Flight Envelope PartitioningTo simplify the online approximation of a full nonlinear dynamic aircraft model andthereby reducing computational load, the ight envelope can be partitioned into mul-tiple connecting operating regions called hyperboxes or clusters [152, 153]. This canbe done manually using a priori knowledge of the nonlinearity of the system, automat-ically using nonlinear optimization algorithms that cluster the data into hyperplanar orhyperellipsoidal clusters [10] or a combination of both. In each hyperbox a locally validlinear-in-the-parameters nonlinear model is dened, which can be updated using the up-date laws of the Lyapunov-based adaptive backstepping control law.For an aircraft, the aerodynamic model can be partitioned using different state variables,the choice of which depends on the expected nonlinearities of the system. Fuzzy logicor some form of neural network can be used to interpolate between the local nonlinearmodels, ensuring smooth transitions. Because only a small number of local models isupdated at any given time step, the computational expense is relatively low. Anotheradvantage is that storing of the local models means retaining information of all ightconditions, because the local adaptation does not interfere with the models outside theclosed neighborhood. Hence, the estimator has memory capabilities and learns insteadof continuously adapting one global nonlinear model.12 INTRODUCTION 1.51.4.3 The F-16 Dynamic ModelThroughout this thesis the theoretical results are illustrated, where possible, by meansof numerical simulation examples. The most accurate dynamic aircraft model availablefor this research is that of the Lockheed Martin F-16 single-seat ghter aircraft. Thisaircraft model has been constructed using high-delity aerodynamic data obtained from[149] which is valid over the entire subsonic ight envelope of the F-16. Detailed engineand actuator models are also available, as well as a simplied version of the baselineight control system. However, structural failure models are not available, which poses alimitation on the recongurable ight control research in this thesis. In other words, thesimulation scenarios with the F-16 model are limited to actuator hard-overs or lock-ups,longitudinal center of gravity shifts and uncertainties in one or more aerodynamic coef-cients.Without any form of FDIE, these limited failure scenarios still pose a challenge, espe-cially the actuator failures, and can be used to evaluate the theoretical results in thisthesis work. Therefore, an FDIE system, such as sensor feedback of actuator positionsor actuator health monitoring systems, is not included in the investigated adaptive con-trol designs. In this way, the actuator failures are used as a substitute for more complex(a)symmetric structural failure scenarios.Note that the baseline ight control system of the F-16 model makes use of full statemeasurement and hence these measurements are also assumed to be available for thenonlinear adaptive control designs developed in this thesis.1.5 Thesis OutlineThe outline of the thesis is as follows:In Chapter 2 the high-delity dynamic model of the F-16 is constructed. The modelis implemented as a C S-function in MATLAB/Simulink c . The available aerodynamicdata is valid over a large, subsonic ight envelope. Furthermore, the characteristics of theclassical baseline ight control system of the F-16 are discussed. The dynamic aircraftmodel and baseline controller are needed to evaluate and compare the performance of thenonlinear adaptive control designs in later chapters.Chapter 3 starts with a discussion on stability concepts and the concept of Lyapunovfunctions. Lyapunovs direct method forms the basis for the recursive backstepping pro-cedure, which is highlighted in the second part of the chapter. Simple control examplesare used to clarify the design procedure.In Chapter 4 nonlinear systems with parametric uncertainty are introduced. The back-stepping method is extended with a dynamic feedback part, i.e. a parameter update law,that constantly updates the static control part. The parameter adaptation part is designedrecursively and simultaneously with the static feedback part using a single control Lya-punov function. This approach is referred to as tuning functions adaptive backstepping.Techniques to robustify the adaptive design against non-parametric uncertainties are also1.5 THESIS OUTLINE 13discussed. Finally, command lters are introduced in the design to simplify the tuningfunctions adaptive backstepping design and to make the parameter adaption more robustto actuator saturation. This approach is referred to as constrained adaptive backstepping.Chapter 5 explores the possibilities of combining (inverse) optimal control theory andadaptive backstepping. The standard adaptive backstepping designs are mainly focusedon achieving stability and convergence, the transient performance and optimality are nottaken explicitly into account. The inverse optimal adaptive backstepping technique re-sulting from combining the tuning functions approach and inverse optimal control theoryis validated with a simple ight control example.In Chapter 6 the constrained adaptive backstepping technique is applied to the designof a ight control system for a simplied, nonlinear over-actuated ghter aircraft modelvalid at two ight conditions. It is demonstrated that the extension of the method tomulti-input multi-output systems is straightforward. A comparison with a modular adap-tive controller that employs a least squares identier is made. Furthermore, the interac-tions between several control allocation algorithms and the online model identicationfor simulations with actuator failures are studied.Chapter 7 extends the results of Chapter 6 to nonlinear adaptive control for the F-16 dy-namic model of Chapter 2, which is valid for the entire subsonic ight envelope. A ightenvelope partitioning method to simplify the online model identication is introduced.The ight envelope is partitioned into multiple connecting operating regions and locallyvalid models are dened in each region. B-spline networks are used for smooth inter-polation between the models. As a study case a trajectory control autopilot is designed,after which it is evaluated in several maneuvers with actuator failures and uncertaintiesin the onboard aerodynamic model.Chapter 8 again considers constrained adaptive backstepping ight control for the high-delity F-16 model. A stability and control augmentation system is designed in such away that it has virtually the same handling qualities as the baseline F-16 ight controlsystem. A comparison is made between the performance of the baseline control system,a modular adaptive controller with least-squares identier and the constrained adaptivebackstepping controller in several realistic failure scenarios.Chapter 9 introduces the immersion and invariance method to construct a new type ofnonlinear adaptive estimator. The idea behind the immersion and invariance approach isto assign prescribed stable dynamics to the estimation error. The resulting estimator incombination with a backstepping controller is shown to improve transient performanceand to radically simplify the tuning process of the integrated adaptive backstepping de-signs of the earlier chapters.In Chapter 10 the concluding remarks and recommendations for further research arediscussed.Figure 1.6 depicts a ow chart of the thesis illustrating the connections between thedifferent chapters. Although this thesis is written as a monograph, Chapters 5 to 9 canbe viewed as a collection of edited versions of previously published papers. An (approx-imate) overview of the papers on which these chapters are based is given below.14 INTRODUCTION 1.5Chapter 5: L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Comparison of InverseOptimal and Tuning Functions Design for Adaptive Missile Control, Journal ofGuidance, Control and Dynamics, Vol. 31, No. 4, July-Aug 2008, pp. 1176-1182 L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Comparison of InverseOptimal and Tuning Function Designs for Adaptive Missile Control, Proc. of the2007 AIAA Guidance, Navigation, and Control Conference and Exhibit, HiltonHead, South Carolina, AIAA-2007-6675Chapter 6: E.R. van Oort, L. Sonneveldt, Q.P. Chu and J.A. Mulder, Full Envelope ModularAdaptive Control of a Fighter Aircraft using Orthogonal Least Squares, Journalof Guidance, Control and Dynamics, Accepted for publication E.R. van Oort, L. Sonneveldt, Q.P. Chu and J.A. Mulder, Modular Adaptive Input-to-State Stable Backstepping of a Nonlinear Missile Model, Proc. of the 2007AIAA Guidance, Navigation, and Control Conference and Exhibit, Hilton Head,South Carolina, AIAA-2007-6676 E.R. van Oort, L. Sonneveldt, Q.P. Chu and J.A. Mulder, A Comparison of Adap-tive Nonlinear Control Designs for an Over-Actuated Fighter Aircraft Model,Proc. of the 2008 AIAA Guidance, Navigation, and Control Conference and Ex-hibit, Honolulu, Hawaii, AIAA-2008-6786Chapter 7: L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Nonlinear AdaptiveBackstepping Trajectory Control, Journal of Guidance, Control and Dynamics,Vol. 32, No. 1, Jan-Feb 2009, pp. 25-39 L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Nonlinear AdaptiveTrajectory Control Applied to an F-16 Model, Proc. of the 2008 AIAA Guidance,Navigation, and Control Conference and Exhibit, Honolulu, Hawaii, AIAA-2008-6788Chapter 8: L. Sonneveldt, Q.P. Chu and J.A. Mulder, Nonlinear Flight Control Design UsingConstrained Adaptive Backstepping, Journal of Guidance, Control and Dynam-ics, Vol. 30, No. 2, Mar-Apr 2007, pp. 322-336 L. Sonneveldt, Q.P. Chu and J.A. Mulder, Constrained Nonlinear Adaptive Back-stepping Flight Control: Application to an F-16/MATV Model, Proc. of the 2006AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Col-orado, AIAA-2006-64131.5 THESIS OUTLINE 15 L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Nonlinear AdaptiveFlight Control Law Design and Handling Qualities Evaluation, Joint 48th IEEEConference on Decision and Control and 28th Chinese Control Conference, Shang-hai, 2009 L. Sonneveldt, et al., Lyapunov-based Fault Tolerant Flight Control Designs for aModern Fighter Aircraft Model, Proc. of the 2009 AIAA Guidance, Navigation,and Control Conference and Exhibit, Chicago, Illinois, AIAA-2009-6172Chapter 9: L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Immersion and Invari-ance Adaptive Backstepping Flight Control, Journal of Guidance, Control andDynamics, Under review L. Sonneveldt, E.R. van Oort, Q.P. Chu and J.A. Mulder, Immersion and Invari-ance Based Nonlinear Adaptive Flight Control, Proc. of the 2010 AIAA Guid-ance, Navigation, and Control Conference and Exhibit, To be presented16 INTRODUCTION 1.51. Introduction2. Aircraft Modeling3. Lyapunov Theory and Backstepping4. Adaptive Backstepping5. Inverse Optimal Adaptive Backstepping6. Comparison of Integrated and Modular Adaptive Flight Control7. F-16 Trajectory Control Design9. Immersion and Invariance Adaptive Backstepping10. Conclusions and Recommendations8. F-16 Stability and Control Augmentation System DesignFigure 1.6: Flow chart of the thesis chapters.Chapter 2Aircraft ModelingThis chapter utilizes basic ight dynamics theory to construct a nonlinear dynamicalmodel of the Lockheed Martin F-16, which is the main study model in this thesis work.The available geometric and aerodynamic aircraft data, as well as the assumptions madeare discussed in detail. Furthermore, a description of the baseline ight control systemof the F-16, which can be used for comparison purposes, is also included. The nal partof the chapter discusses the implementation of the model and the baseline control systemin the MATLAB/Simulink c software package.2.1 IntroductionIn this chapter a nonlinear dynamical model of the Lockheed-Martin F-16 is constructed.The F-16 is a single-seat, supersonic, multi-role tactical aircraft with a blended wing-fuselage that has been in production since 1976. Over 4,400 have been produced for 24countries, making it the most common ghter type in the world. A three-view of thesingle-engined F-16 aircraft is depicted in Figure 2.1.This chapter will start with a derivation of the equations of motion for a general rigidbody aircraft. After that, the available control variables and the engine model of the F-16 are discussed. The geometry and the aerodynamic data are given in Section 2.4. InSection 2.5 a simplied version of the baseline F-16 ight control system is discussed.The implementation in MATLAB/Simulink c of the complete F-16 dynamic model withight control system is detailed in the last part of the chapter.2.2 Aircraft DynamicsIn this section the equations of motion for the F-16 model are derived, this derivation isbased on [16, 45, 127]. A very thorough discussion on ight dynamics can be found in1718 AIRCRAFT MODELING 2.2Figure 2.1: Three-view of the Lockheed-Martin F-16.the course notes [143].2.2.1 Reference FramesBefore the equations of motion can be derived, some frames of reference are needed todescribe the motion in. The reference frames used in this thesis are the earth-xed reference frame FE, used as the inertial frame and the vehicle car-ried local earth reference frame FO with its origin xed in the center of gravity ofthe aircraft which is assumed to have the same orientation as FE; the wind-axes reference frame FW, obtained fromFO by three successive rotationsof ight path heading angle , ight path climb angle and aerodynamic bankangle ; the stability-axes reference frame FS, obtained from FW by a rotation of minussideslip angle ; and nally the body-xed reference frame FB, obtained from FS by a rotation ofangle of attack .The body-xed reference frame FB can also be obtained directly from FO by three suc-cessive rotations of yaw angle , pitch angle and roll angle . All reference frames areright-handed and orthogonal. In the earth-xed reference frame the zE-axis points to thecenter of the earth, the xE-axis points in some arbitrary direction, e.g. the north, and the2.2 AIRCRAFT DYNAMICS 19yE-axis is perpendicular to the xE-axis.The transformation matrices from FB to FS and from FB to FW are dened asTs/b =__cos 0 sin 0 1 0sin 0 cos __, Tw/b =__cos cos sin sin cos cos sin cos sin sin sin 0 cos __.2.2.2 Aircraft VariablesA number of assumptions has to be made, before proceeding with the derivation of theequations of motion:1. The aircraft is a rigid-body, which means that any two points on or within theairframe remain xed with respect to each other. This assumption is quite valid fora small ghter aircraft.2. The earth is at and non-rotating and regarded as an inertial reference. Thisassumption is valid when dealing with control design of aircraft, but not whenanalyzing inertial guidance systems.3. Wind gust effects are not taken into account, hence the undisturbed air is assumedto be at rest w.r.t. the surface of the earth. In other words, the kinematic velocity isequal to the aerodynamic velocity of the aircraft.4. The mass is constant during the time interval over which the motion is considered,the fuel consumption is neglected during this time-interval. This assumption isnecessary to apply Newtons motion laws.5. The mass distribution of the aircraft is symmetric relative to the XBOZB-plane,this implies that the products of inertia Iyz and Ixy are equal to zero. This assump-tion is valid for most aircraft.Note that the last assumption is no longer valid when the aircraft gets asymmetricallydamaged. However, the aerodynamic effects resulting from such damage will, in gen-eral, be much larger than the inuence of the center of gravity shift for a small ghteraircraft. A derivation of the equations of motion without this last assumption can befound in [11].Under the above assumptions the motion of the aircraft has six degrees of freedom (rota-tion and translation in three dimensions). The aircraft dynamics can be described by itsposition, orientation, velocity and angular velocity over time. pE = (xE, yE, zE)Tis theposition vector expressed in an earth-xed coordinate system. V is the velocity vectorgiven by V = (u, v, w)T, where u is the longitudinal velocity, v the lateral velocity andw the normal velocity. The orientation vector is given by = (, , )T, where isthe roll angle, the pitch angle and the yaw angle, and the angular velocity vector isgiven by = (p, q, r)T, where p, q and r are the roll, pitch and yaw angular velocities,respectively. Various components of the aircraft motions are illustrated in Figure 2.2.20 AIRCRAFT MODELING 2.2Figure 2.2: Aircraft orientation angles , and , aerodynamic angles and , and the angularrates p, q and r. The frame of reference is body-xed and all angles and rates are dened positivein the gure [178].The relation between the attitude vector and the angular velocity vector is given as =__1 sin tan cos tan 0 cos sin0 sin cos cos cos __ (2.1)Dening VT as the total velocity and using Figure 2.2, the following relations can bederived:VT =_u2+v2+w2 = arctan wu (2.2) = arcsin vVTFurthermore, when = = 0, the ight path angle can be dened as = (2.3)2.2.3 Equations of Motion for a Rigid Body AircraftThe equations of motion for the aircraft can be derived from Newtons Second Law ofmotion, which states that the summation of all external forces acting on a body must beequal to the time rate of change of its momentum, and the summation of the external2.2 AIRCRAFT DYNAMICS 21moments acting on a body must be equal to the time rate of change of its angular mo-mentum. In the inertial, earth-xed reference frame FE, Newtons Second Law can beexpressed by two vector equations [143]F = ddt(mV)_E(2.4)M = dHdt_E(2.5)where F represents the sum of all externally applied forces, m is the mass of the aircraft,M represents the sum of all applied torques and H is the angular momentum.Force EquationFirst, to further evaluate the force equation (2.4) it is necessary to obtain an expressionfor the time rate of change of the velocity vector with respect to earth. This process iscomplicated by the fact that the velocity vector may be rotating while it is changing inmagnitude. Using the equation of Coriolis in appendix A of [16] results inF = ddt(mV)_B+ mV, (2.6)where is the total angular velocity of the aircraft with respect to the earth (inertialreference frame). Expressing the vectors as the sum of their components with respect tothe body-xed reference frame FB givesV = iu + jv + kw (2.7) = ip + jq + kr (2.8)where i, j and k are unit vectors along the aircrafts xB, yB and zB axes, respectively.Expanding (2.6) using (2.7), (2.8) results inFx = m( u +qw rv)Fy = m( v +ru pw) (2.9)Fz = m( w +pv qu)where the external forces Fx, Fy and Fz depend on the weight vector W, the aerodynamicforce vector R and the thrust vector E. It is assumed the thrust produced by the engine,FT, acts parallel to the aircrafts xB-axis. Hence,Ex = FTEy = 0 (2.10)Ez = 0The components of W and R along the body-axes areWx = mg sinWy = mg sin cos (2.11)Wz = mg cos cos 22 AIRCRAFT MODELING 2.2andRx = XRy = Y (2.12)Rz = Zwhere g is the gravity constant. The size of the aerodynamic forces X, Y and Z isdetermined by the amount of air diverted by the aircraft in different directions. Theamount of air diverted by the aircraft mainly depends on the following factors: the total velocity VT (or Mach number M) and density of the airow , the geometry of the aircraft: wing area S, wing span b and mean aerodynamicchord c, the orientation of the aircraft relative to the airow: angle of attack and side slipangle , the control surface deections , the angular rates p, q, r,There are other variables such as the time derivatives of the aerodynamic angles that alsoplay a role, but these effects are less prominent, since it is assumed that the aircraft is arigid body. This motivates the standard way of modeling the aerodynamic force:X = qSCXT(, , p, q, r, , ...)Y = qSCYT(, , p, q, r, , ...) (2.13)Z = qSCZT(, , p, q, r, , ...)where q = 12V2T is the aerodynamic pressure. The air density is calculated accordingto the International Standard Atmosphere (ISA) as given in Appendix A.2. The coef-cients CXT, CYT and CZT are usually obtained from (virtual) wind tunnel data and ighttests. Combining equations (2.11) and (2.12) and the thrust components (2.10) with (2.9),results in the complete body-axes force equation:X +FT mg sin = m( u +qw rv)Y +mg sincos = m( v +ru pw) (2.14)Z +mg cos sin = m( w +pv qu)Moment EquationTo obtain the equations for angular motion, consider again Equation (2.5). The time rateof change of H is required and since H can change in magnitude and direction, (2.5) canbe written asM = dHdt_B+ H (2.15)2.2 AIRCRAFT DYNAMICS 23In the body-xed reference frame, under the rigid body and constant mass assumptions,the angular momentum H can be expressed asH = I (2.16)where, under the symmetrical aircraft assumption, the inertia matrix is dened asI =__Ix 0 Ixz0 Iy 0Ixz 0 Iz__ (2.17)Expanding (2.15) using (2.16) results inMx = pIx rIxz +qr(Iz Iy) pqIxzMy = qIy +pq(IxIz) + (p2r2)Ixz (2.18)Mz = rIz pIxz +pq(IyIx) +qrIxz.The external moments Mx, My and Mz are those due to aerodynamics and engine angu-lar momentum. As a result the aerodynamic moments areMx = LMy = M rHeng (2.19)Mz = N +qHengwhere L, M and N are the aerodynamic moments and Heng is the engine angular mo-mentum. Note that the engine angular momentum is assumed to act parallel to the bodyx-axis of the aircraft. The aerodynamic moments can be expressed in a similar way asthe aerodynamic forces in Equation (2.13):L = qSbClT(, , p, q, r, , ...) (2.20)M = qS cCmT(, , p, q, r, , ...)N = qSbCnT(, , p, q, r, , ...)Combining (2.18) and (2.19), the complete body-axis moment equation is formed asL = pIx rIxz +qr(Iz Iy) pqIxzM rHeng = qIy +pq(IxIz) + (p2r2)Ixz (2.21)N +qHeng = rIz pIxz +pq(IyIx) +qrIxz.24 AIRCRAFT MODELING 2.22.2.4 Gathering the Equations of MotionEuler AnglesThe equations of motion derived in the previous sections are now collected and writtenas a system of twelve scalar rst order differential equations. u = rv qw g sin + 1m( X +FT) (2.22) v = pw ru +g sincos + 1mY (2.23) w = qu pv +g cos cos + 1mZ (2.24) p = (c1r +c2p)q +c3L +c4( N +qHeng) (2.25) q = c5pr c6(p2r2) +c7( M rHeng) (2.26) r = (c8p c2r)q +c4L +c9( N +qHeng) (2.27) = p + tan (q sin +r cos ) (2.28) = q cos r sin (2.29) = q sin +r cos cos (2.30) xE = u cos cos +v(cos sin sin sin cos )+ w(cos sin cos + sin sin ) (2.31) yE = u sin cos +v(sin sin sin + cos cos )+ w(sin sin cos cos sin ) (2.32) zE = u sin +v cos sin +wcos cos (2.33)wherec1 = (IyIz)IzI2xz c4 = Ixz c7 = 1Iyc2 = (Ix Iy +Iz)Ixz c5 = IzIxIyc8 = Ix(IxIy) +I2xzc3 = Iz c6 = IxzIyc9 = Ixwith = IxIzI2xz.QuaternionsThe above equations of motion make use of Euler angle approach for the orientationmodel. The disadvantage of the Euler angle method is that the differential equations for2.2 AIRCRAFT DYNAMICS 25 p and r become singular when pitch angle passes through 2. To avoid these singular-ities quaternions are used for the aircraft orientation presentation. A detailed explanationabout quaternions and their properties can be found in [127]. With the quaternions pre-sentation the aircraft system representation consists of 13 scalar rst order differentialequations: u = rv qw + 1m_ X +FT_+ 2(q1q3q0q2)g (2.34) v = pw ru + 1mY + 2(q2q3 +q0q1)g (2.35) w = qu pv + 1mZ + (q20 q21 q22 +q23)g (2.36) p = (c1r +c2p)q +c3L +c4( N +qHeng) (2.37) q = c5pr c6(p2r2) +c7( M rHeng) (2.38) r = (c8p c2r)q +c4L +c9( N +qHeng) (2.39) q =__ q0 q1 q2 q3__= 12__0 p q rp 0 r qq r 0 pr q p 0____q0q1q2q3__(2.40)__ xE yE zE__ =__q20 +q21q22 q23 2(q1q2q0q3) 2(q1q3 +q0q2)2(q1q2 +q0q3) q20 q21 +q22 q23 2(q2q3q0q1)2(q1q3q0q2) 2(q2q3 +q0q1) q20 q21 q22 +q23____uvw__ (2.41)where__q0q1q2q3__= __cos /2 cos /2 cos /2 + sin /2 sin/2 sin/2sin /2 cos /2 cos /2 cos /2 sin/2 sin/2cos /2 sin/2 cos /2 + sin/2 cos /2 sin/2cos /2 cos /2 sin/2 sin/2 sin /2 cos /2__.Using (2.40) to describe the attitude dynamics means that the four differential equa-tions are integrated as if all quaternion components were independent. Therefore, thenormalization condition |q| =_q20 +q21 +q22 +q23 = 1 and the derivative constraintq0 q0 +q1 q1 +q2 q2 +q3 q3 = 0 may not be satised after performing an integration stepdue to numerical round-off errors. After each integration step the constraint may be re-established by subtracting the discrepancy fromthe quaternion derivatives. The correctedquaternion dynamics are [170] q= q q, (2.42)where = q0 q0 +q1 q1 +q2 q2 +q3 q3.26 AIRCRAFT MODELING 2.3Wind-axes Force EquationsFor control design it is more convenient to transform the force equations (2.34)-(2.36) tothe wind-axes reference frame. Taking the derivative of (2.2) results in [127]VT = 1m (D +FT cos cos +mg1) (2.43) = q (p cos +r sin ) tan 1mVT cos (L +FT sin mg3) (2.44) = p sin r cos + 1mVT(Y FT cos sin +mg2) (2.45)where the drag force D, the side force Y and the lift force L are dened asD = X cos cos Y sin Z sin cos Y = X cos sin + Y cos Z sin sin L = X sin Z cos and the gravity components asg1 = g (cos cos sin + sin sin cos + sin cos cos cos )g2 = g (cos sin sin + cos sin cos sin sin cos cos )g3 = g (sin sin + cos cos cos ) .2.3 Control Variables and Engine ModelingThe F-16 model allows control over thrust, elevator, ailerons and rudder. The thrust ismeasured in Newtons. All deections are dened positive in the conventional way, i.e.positive thrust causes an increase in acceleration along the xB-axis, a positive elevatordeection results in a decrease in pitch rate, a positive aileron deection gives a decreasein roll rate and a positive rudder deection decreases the yaw rate. The F-16 also has aleading edge ap, which helps to y the aircraft at high angles of attack. The deectionof the leading edge ap lef is not controlled directly by the pilot, but is governed bythe following transfer function dependent on angle of attack and static and dynamicpressures:lef = 1.382s + 7.25s + 7.25 9.05 qpstat+ 1.45. (2.46)The differential elevator deection, trailing edge ap, landing gear and speed brakes arenot included in the model, since no data is publicly available. The control surfaces ofthe F-16 are driven by servo-controlled actuators to produce the deections commandedby the ight control system. The actuators of the control surfaces are modeled as a rst-order low-pass lters with certain gain and saturation limits in range and deection rate.These limits can be found in Table 2.1. The gains of the actuators are 1/0.136 for theleading edge ap and 1/0.0495 for the other control surfaces. The maximum values and2.3 CONTROL VARIABLES AND ENGINE MODELING 27Table 2.1: The control input units and maximum valuesControl units MIN. MAX. rate limitElevator deg -25 25 60 deg/sAilerons deg -21.5 21.5 80 deg/sRudder deg -30 30 120 deg/sLeading edge ap deg 0 25 25 deg/sunits for all control variables are given in Table 2.1.The Lockheed Martin F-16 is powered by an after-burning turbofan jet engine, whichis modeled taking into account throttle gearing and engine power level lag. The thrustresponse is modeled with a rst order lag, where the lag time constant is a function of thecurrent engine power level and the commanded power. The commanded power level tothe throttle position is a linear relationship apart froma change in slope when the militarypower level is reached at 0.77 throttle setting [149]:Pc (th) =_ 64.94th if th 0.77217.38th117.38 if th > 0.77 . (2.47)Note that the throttle position is limited to the range 0 th 1. The derivative of theactual power level Pa is given by [149]Pa = 1eng(PcPa) , (2.48)wherePc =___Pc if Pc 50 and Pa 5060 if Pc 50 and Pa < 5040 if Pc < 50 and Pa 50Pc if Pc < 50 and Pa < 501eng=___5.0 if Pc 50 and Pa 501engif Pc 50 and Pa < 505.0 if Pc < 50 and Pa 501engif Pc < 50 and Pa < 501eng=___1.0 if (PcPa) 250.1 if (PcPa) 501.9 0.036 (PcPa) if 25 < (PcPa) < 50.28 AIRCRAFT MODELING 2.4The engine thrust data is available in a tabular formas a function of actual power, altitudeand Mach number over the ranges 0 h 15240 m and 0 M 1 for idle, militaryand maximum power settings [149]. The thrust is computed asFT =_ Tidle + (TmilTidle) Pa50 if Pa < 50Tmil + (TmaxTmil) Pa5050 if Pa 50 . (2.49)The engine angular momentumis assumed to be acting along the xB-axis with a constantvalue of 216.9 kg.m2/s.2.4 Geometry and Aerodynamic DataThe relevant geometry data of the F-16 can be found in Table A.1 of Appendix A. Theaerodynamic data of the F-16 model have been derived from low-speed static and dy-namic (force oscillation) wind-tunnel tests conducted with sub-scale models in wind-tunnel facilities at the NASA Ames and Langley Research Centers [149]. The aerody-namic data in [149] are given in tabular form and are valid for the following subsonicight envelope: 20 90 degrees; 30 30 degrees.Two examples of the aerodynamic data for the F-16 model can be found in Figure 2.3.The pitch moment coefcient Cm and the CZ both depend on three variables: angle ofattack, sideslip angle and elevator deection.20020406080200200.60.50.40.30.20.100.10.2beta (deg)alpha (deg)Cm ()(a) Cm for e = 02002040608020020321012beta (deg)alpha (deg)CZ ()(b) CZ for e = 0Figure 2.3: Two examples of the aerodynamic coefcient data for the F-16 obtained from wind-tunnel tests.The various aerodynamic contributions to a given force or moment coefcient as given2.4 GEOMETRY AND AERODYNAMIC DATA 29in [149] are summed as follows.For the X-axis force coefcient CXT:CXT = CX(, , e) +CXlef_1 lef25_+ q c2VT_CXq() +CXqlef()_1 lef25__ (2.50)whereCXlef = CXlef(, ) CX(, , e = 0o).For the Y-axis force coefcient CYT:CYT = CY (, ) +CYlef_1 lef25_+_CYa +CYalef_1 lef25___a20_+ CYr_r30_+ rb2VT_CYr() +CYrlef()_1 lef25__+ pb2VT_CYp() +CYplef()_1 lef25__ (2.51)whereCYlef = CYlef(, ) CY (, )CYa = CYa(, ) CY (, )CYalef= CYalef(, ) CYlef(, ) CYaCYr = CYr(, ) CY (, ).For the Z-axis force coefcient CZT:CZT = CZ(, , e) +CZlef_1 lef25_+ q c2VT_CZq() +CZqlef()_1 lef25__ (2.52)whereCZlef = CZlef(, ) CZ(, , e = 0o).30 AIRCRAFT MODELING 2.5For the rolling-moment coefcient ClT:ClT = Cl(, , e) +Cllef_1 lef25_+_Cla +Clalef_1 lef25___a20_+ Clr_r30_+ rb2VT_Clr() +Clrlef()_1 lef25__+ pb2VT_Clp() +Clplef()_1 lef25__+Cl() (2.53)whereCllef = Cllef(, ) Cl(, , e = 0o)Cla = Cla(, ) Cl(, , e = 0o)Clalef= Clalef(, ) Cllef(, ) ClaClr = Clr(, ) Cl(, , e = 0o).For the pitching-moment coefcient CmT:CmT = Cm(, , e) +CZT_xcgr xcg_+Cmlef_1 lef25_ (2.54)+ q c2VT_Cmq() +Cmqlef()__1 lef25_+Cm() +Cmds(, e)whereCmlef = Cmlef(, ) Cm(, , e = 0o).For the yawing-moment coefcient CnT:CnT = Cn(, , e) +Cnlef_1 lef25_CYT_xcgr xcg_ cb+_Cna +Cnalef_1 lef25___a20_+ Cnr_r30_+ rb2VT_Cnr() +Cnrlef()_1 lef25__+ pb2VT_Cnp() +Cnplef()_1 lef25__+Cn() (2.55)whereCnlef = Cnlef(, ) Cn(, , e = 0o)Cna = Cna(, ) Cn(, , e = 0o)Cnalef= Cnalef(, ) Cnlef(, ) CnaCnr = Cnr(, ) Cn(, , e = 0o).2.5 BASELINE FLIGHT CONTROL SYSTEM 312.5 Baseline Flight Control SystemThe NASA technical report [149] also contains a description of a stability and controlaugmentation system for the F-16 model. This ight control system is a simplied ver-sion of the actual baseline F-16 ight controller, which retains its main characteristics. Adescription of the different control loops of the system is given in this section, for moredetails see [149].2.5.1 Longitudinal ControlA diagram of the longitudinal ight control system can be found in Figure A.2 of Ap-pendix A.3. It is a command augmentation system where the pilot commands normalacceleration with a longitudinal stick input. Washed-out pitch rate and ltered normalacceleration are fed back to achieve the desired response. A forward-loop integration isincluded to make the steady-state acceleration response match the commanded acceler-ation. At low Mach numbers the F-16 model has a minor negative static longitudinalstability; therefore angle of attack feedback is used to provide articial static stability.The pitch control system incorporates an angle of attack limiting system, where againangle of attack feedback is used to modify the pilot-commanded normal acceleration.The resulting angle of attack limit is about 25 deg in 1g ight. Finally, the system alsomakes sure that the pitch control is deected in the proper direction to oppose the nose-upcoupling moment generated by rapid rolling at high angles of attack.2.5.2 Lateral ControlThe lateral ight control system is depicted in the block diagramgiven in Figure A.3. Thepilot can command roll rates up to 308 deg/s through the late