TWISTING ALGORITHM FOR CONTROLLING F-16 AIRCRAFT WITH PERFORMANCE MARGINS IDENTIFICATION by AKSHAY KULKARNI A THESIS Submitted in partial fulfillment of the requirements For the degree of Master of Science in Engineering In The Department of Electrical and Computer Engineering To The School of Graduate Studies Of The University of Alabama in Huntsville HUNTSVILLE, ALABAMA 2014 In presenting this thesis in partial fulfillment of the requirements for a master's degree from The University of Alabama in Huntsville, I agree that the Library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by my advisor or, in his absence, by the Chair of the Department or the Dean of the School of Graduate Studies. It is also understood that due recognition shall be given to me and to The University of Alabama in Huntsville in any scholarly use which may be made of any material in this thesis. ____________________________ ___________ (Student signature) (Date) ii THESIS APPROVAL FORM Submitted by Akshay Kulkarni in partial fulfillment of the requirements for the degree of Master of Science in Engineering and accepted on behalf of the Faculty of the School of Graduate Studies by the thesis committee. We, the undersigned members of the Graduate Faculty of The University of Alabama in Huntsville, certify that we have advised and/or supervised the candidate on the work described in this thesis. We further certify that we have reviewed the thesis manuscript and approve it in partial fulfillment of the requirements for the degree of Master of Science in Engineering. _________________________________________ Committee Chair _________________________________________ _________________________________________ _________________________________________ Department Chair _________________________________________ College Dean _________________________________________ Graduate Dean iii ABSTRACT The School of Graduate Studies The University of Alabama in Huntsville Degree Master of Science in Engineering College/Department Engineering / Electrical and Computer Engineering. Name of Candidate Akshay Kulkarni. Title Twisting Algorithm for Controlling F-16 Aircraft with Performance Margins Identification A Twisting algorithm based second order sliding mode controller is designed for tracking of command angles for nonlinear flight dynamics of an F-16 aircraft, before and after the battle damages. The high frequency switching controller is designed in terms of control derivative so that the actual control function is continuous. The practical robustness performance margins are identified in terms of maximal additional gain (Practical Gain Margin) and phase lag (Practical Phase Margin) added to the frequency characteristic of the linear part of the open loop system that yield acceptable loss of command angle tracking performance in terms of admissible parameters of self sustained oscillations. The effects of parasitic dynamics, in terms of linear first order actuator dynamics, on the controller’s performance are analyzed. The performance margins in terms of parameters of parasitic dynamics are obtained. The proposed approach is verified via simulations. Abstract Approval: Committee Chair _______________________________________ Department Chair _______________________________________ Graduate Dean _______________________________________ iv ACKNOWLEDGMENTS The work described in this thesis would not have been possible without the assistance of a number of people who deserve special mention. First, I would like to thank Dr. Yuri Shtessel for his suggestion of the research topic and for his guidance throughout all the stages of the work. Second, the other members of my committee have been very helpful with comments and suggestions. I would like to thank my family and friends who encouraged me to begin work on this degree. v TABLE OF CONTENTS Page List of Figures…………………………………………………………………................ix List of Tables……………………………………………………………………………..xi List of Symbols…………………………………………………………………………..xii Chapter I. INTRODUCTION………………………………………………………………………1 II. MATHEMATICAL MODEL AND PROBLEM FORMULATION…………………..4 2.1 Mathematical model of F-16 Aircraft……………………………………………...4 2.2 Problem Statement…………………………………………………………………6 III. FUNDAMENTALS OF SECOND ORDER SLIDING MODE CONTROL……..…..9 3.1 Conventional Sliding Modes………………………………………………….….11 3.2 Second Order Sliding Mode Control…………………………………………….15 3.2.1 Twisting Controller…………………………………………………………19 3.2.2 Chattering Attenuation with 2-SMC ……………………………………….20 IV.CONCEPTS OF PERFORMANCE MARGINS IN SYSTEMS WITH SLIDING MODE CONTROL……………………………………………………………..……21 4.1 Performance Margins……………………………………………………….……23 4.1.1 Ideal Performance Margins…………………………………………………26 4.1.2 Practical Performance Margins ……………………………………….…....27 4.1.3 Performance Margins in terms of parameters of Parasitic Dynamics…...….28 4.2 Computation of Performance Margins………………………………………...…29 4.3 Describing Function of Twisting Controller…......................................................31 vi 4.4 Identification of Practical Performance Margins………………………………...35 4.4.1 Practical Phase Margin Identification ….......................................................36 4.4.2 Practical Gain Margin Identification …........................................................37 4.4.3 Identification of Practical Margins in terms of parasitic dynamics………...39 V. THE TWISTING CONTROLLER DESIGN FOR ATTITUDE CONTROL OF F-16 JET FIGHTER AND SIMULATIONS………………………………….….….40 5.1 Flight Control Design Approach…........................................................................41 5.2 SMC Design (Twisting Algorithm)…...................................................................45 5.3 Simulations……. ………………………………………………………………..51 5.3.1 Output tracking for unperturbed system …..................................................52 5.3.2 Effects of Parasitic Dynamics …………………………………………......54 VI. PERFORMANCE MARGINS IDENTIFICATION FOR ATTITUDE CONTROL OF F-16 JET FIGHTER AND SIMULATIONS...............................................................60 6.1 Transfer Functions for Simplified Linear Dynamics…………………………….61 6.2 Describing Functions of Twisting Controllers and the HB equations ………......62 6.3 Identification of Practical Performance Margins…...............................................63 6.3.1 Practical Phase Margin…………………………..…………………………65 6.3.2 Practical Gain Margin………………………………………………….…...69 6.3.3 Margins in Terms of Parameters of Actuator Dynamics …………………..71 6.4 Simulations…......................................................................................................78 6.4.1 Verification of Practical Phase Margin (PPM)…………………...…79 6.4.2 Verification of Performance Margins in terms of actuator dynamics…........84 6.4.3 Accuracy of the DF analysis…......................................................................89 VII. CONCLUSIONS…………………………………….……………………………...91 APPENDIX: DESCRIBING FUNCTION ANALYSIS…………………..…………….93 vii REFERENCES…..............................................................................................................97 viii LIST OF FIGURES Figure Page 4.1: DF analysis applied to linear systems with conventional SM controller ............................... 25 4.2: Block Diagram of Closed loop system with SMC controller ................................................ 29 4.3: Graphical solution of Harmonic Balance equation ................................................................ 35 4.4: Practical Phase Margin Identification .................................................................................... 37 4.5: Practical Gain Margin Identification ..................................................................................... 38 5.1: Reference profiles ................................................................................................................. 51 5.2: Command angle tracking ....................................................................................................... 52 5.3: Deflections of aerodynamic surfaces (Control functions) ..................................................... 53 5.4: Sliding Variables.................................................................................................................... 53 5.5: Effects of Actuator dynamics on command angle tracking performance with ................................................................................................................................ 55 5.6: Effects of Actuator dynamics on sliding variables with ............... 55 5.7: Effects of Actuator dynamics on command angle tracking performance with ................................................................................................................................ 56 5.8: Effects of Actuator dynamics on sliding variables with ................ 56 5.9: Effects of Actuator dynamics on command angle tracking performance with ................................................................................................................................ 57 5.10: Effects of Actuator dynamics on sliding variables with .......... …57 6.1: PPM for sliding variable dynamics ............................................................................ 66 6.2: PPM for sliding variable dynamics ............................................................................ 67 6.3: PPM for sliding variable dynamics ...........................................................................