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FRACTIONAL-ORDER CONTROLLERS FOR AN UNMANNED AERIAL VEHICLE A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Jesus Navarro 2017 SIGNATURE PAGE THESIS: FRACTIONAL-ORDER CONTROLLERS FOR AN UNMANNED AERIAL VEHICLE AUTHOR: Jesus Navarro DATE SUBMITTED: Summer 2017 Department of Mathematics and Statistics Dr. Hubertus von Bremen Thesis Committee Chair Mathematics & Statistics Dr. Subodh Bhandari Thesis Committee Co-Chair Aerospace Engineering Dr. Jennifer Switkes Mathematics & Statistics Dr. Robin Wilson Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis is dedicated to my family, friends, and mentors for all their support. Thank you to Megan Hamilton, for being a great friend and helping me in proof­ reading my thesis. Thank you Dr. Jung-Ha An and Dr. Michael Bice for being my REU advisors and showing me how much more there is to mathematics. Thank you Dr. John Rock, who always encouraged me to apply to grad school. Thank you Dr. Jennifer Switkes and Dr. Robin Wilson for being part of my thesis committee. Thank you Dr. Subodh Bhandari for being a co-advisor and allowing me to flex my mathematical knowledge into aerospace engineering. Thank you Dr. Hubertus von Bremen for being my thesis advisor and encouraging me to look into different STEM fields, so that I can apply what I know into a different field of study. Special thanks to my father and mother, who sacrificed so much to give me, my brothers, and sisters a future we could have never imagined. iii ABSTRACT Fractional calculus is a new field of study where the main focus is on differentiation and integration of non-integer order. We then expand upon this idea into linear fractional-order differential equations and apply those techniques into PIλDµ, PIλ , and PDµ controllers where λ and µ are arbitrary real numbers. The way we accom­ plish this is by implementing methods that have been proposed in Fractional-order Systems and Controls. There are two different models that we look into, the lon­ gitudinal model which deals with the pitch angle of the aircraft, and the lateral model which deals with the roll angle of the aircraft. In order to obtain an ideal controller, we optimize the performance index and obtain the best parameters for the system. Then, looking at the step information of the controller, we want to en­ sure that we get a reasonable overshoot, rise time, and settling time. In one of the examples, we notice a significant difference where the fractional-order controller’s rise time was 27.7 seconds faster, the settling time was 24.5 seconds faster, and our performance index gave us 4 more decimal places of accuracy compared to that of a regular controller. With these new techniques, we now have the opportunity to achieve better results than by using regular PID, PI, and PD controllers. iv Contents Signature Page ii Acknowledgements iii Abstract iv List of Tables viii List of Figures ix Chapter 1 Introduction 1 Chapter 2 Fractional Calculus 4 2.1 What is Fractional Calculus? . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Riemann-Liouville’s Fractional Derivative . . . . . . . . . . . . . . . 5 2.4 Caputo’s Fractional Derivative . . . . . . . . . . . . . . . . . . . . . 6 2.5 Gru¨nwald-Letnikov’s Fractional Derivative . . . . . . . . . . . . . . 7 2.6 Differences Between Riemann-Liouville and Caputo Derivatives . . 9 Chapter 3 Fractional-Order Differential Equations 13 v 3.1 Oustaloup Recursive Approximation . . . . . . . . . . . . . . . . . 16 3.2 Closed-Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 4 Controllers 18 4.1 PID-Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 PIλDµ-Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.1 Open-Loop System . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.2 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 5 Aircraft Dynamics 23 5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Longitudinal-Direction Dynamics . . . . . . . . . . . . . . . . . . . 26 5.3 Lateral-Directional Dynamics . . . . . . . . . . . . . . . . . . . . . 29 Chapter 6 Simulations 33 6.1 State-Space Equations to Transfer Function . . . . . . . . . . . . . 34 6.2 Optimal Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.3 Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3.1 Results for PID vs. PIλDµ Controllers . . . . . . . . . . . . 40 6.3.2 Results for PI vs. PIλ Controllers . . . . . . . . . . . . . . . 54 6.3.3 Results for PD vs. PDµ Controllers . . . . . . . . . . . . . . 62 6.4 Lateral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.4.1 Results for PID vs. PIλDµ Controllers . . . . . . . . . . . . 64 6.4.2 Results for PI vs. PIλ Controllers . . . . . . . . . . . . . . . 76 6.4.3 Results for PD vs. PDµ Controllers . . . . . . . . . . . . . . 84 vi Chapter 7 Summary 92 Bibiliography 102 Appendix 104 A MATLAB and Simulink Source Code 104 A.1 Optimal Sig Kadet State-Space Model - 1 . . . . . . . . . . . . . . 104 A.1.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . 108 A.2 Optimal Sig Kadet State-Space Model - 2 . . . . . . . . . . . . . . 108 A.3 Fractional-Derivative Block . . . . . . . . . . . . . . . . . . . . . . 113 A.4 Fractional-Order Controllers for Simulink . . . . . . . . . . . . . . . 115 vii List of Tables 6.1 PID vs. PIλDµ Controllers from Figure 6.13 . . . . . . . . . . . . . 51 6.2 PID vs. PIλDµ Controllers from Figure 6.15 . . . . . . . . . . . . . 54 6.3 PI vs. PIλ Controllers from Figure 6.21 . . . . . . . . . . . . . . . . 60 6.4 PID vs. PIλDµ Controllers from Figure 6.23 . . . . . . . . . . . . . 61 6.5 PID vs. PIλDµ Controllers from Figure 6.33 . . . . . . . . . . . . . 72 6.6 PID vs. PIλDµ Controllers from Figure 6.35 . . . . . . . . . . . . . 75 6.7 PI vs. PIλ Controllers from Figure 6.40 . . . . . . . . . . . . . . . . 81 6.8 PI vs. PIλ Controllers from Figure 6.42 . . . . . . . . . . . . . . . . 82 6.9 PD vs. PDµ Controllers from Figure 6.48 . . . . . . . . . . . . . . . 89 7.1 Optimal Tuning Results for Transfer Function 6.13 - Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Optimal Tuning Results for Transfer Function 6.14 - Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 Optimal Tuning Results for Transfer Function 6.17 - Lateral Model 97 7.4 Optimal Tuning Results for Transfer Function 6.18 - Lateral Model 99 viii List of Figures 5.1 Yaw, Pitch, and Roll rotations. Image obtained from the NASA website. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.1 Simulink model for fractional-order PID controller. . . . . . . . . . 35 6.2 PID vs. PI−0.25D0.25 - Longitudinal . . . . . . . . . . . . . . . . . . 41 6.3 PID vs. PI−0.5D0.5 - Longitudinal . . . . . . . . . . . . . . . . . . . 42 6.4 Comparison of various PIλDµ Controllers 1 - Longitudinal . . . . . 43 6.5 Comparison of various PIλDµ Controllers 2 - Longitudinal . . . . . 43 6.6 Comparison of various PIλDµ Controllers 3 - Longitudinal . . . . . 44 6.7 Contour plot of the index of performance. (Regular) - Longitudinal 45 6.8 Contour plot of the index of performance. (Regular with Logarithmic Scaling) - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.9 PID vs. PI−0.16D0.17 - Longitudinal . . . . . . . . . . . . . . . . . . 47 6.10 Contour plot of the index of performance. (Regular with Feedback) - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.11 PID vs. PI−0.14D0.99 - Longitudinal . . . . . . . . . . . . . . . . . . 48 6.12 Contour plot of the index of performance. (Tuned) - Longitudinal . 50 6.13 PID vs. PI−0.38045D0.56782 (Optimal) - Longitudinal . . . . . . . . . 51 6.14 Performance of the λ Values. (Tuned with Feedback) - Longitudinal 52 ix 6.15 PID vs. PI−0.6290D0.4820 (Optimal) - Longitudinal . . . . . . . . . . 53 6.16 Performance of the λ Values (Regular) - Longitudinal . . . . . . . . 55 6.17 PI vs. PI−0.19 - Longitudinal . . . . . . . . . . . . . . . . . . . . . . 56 6.18 Performance of the λ Values (Regular with Feedback) - Longitudinal 56 6.19 PI vs PI−0.1 - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . 57 6.20 Performance of the λ Values. (Tuned) - Longitudinal . . . . . . . . 58 6.21 PI vs. PI−3.415x10−7 (Optimal) - Longitudinal . . . . . . . . . . . . . 59 6.22 Performance of the λ Values. (Tuned with Feedback) - Longitudinal 60 6.23 PI vs. PI−0.000070268 (Optimal) - Longitudinal . . . . . . . . . . . . . 62 6.24 Index of Performance of the µ Values. (Regular) - Longitudinal . . 63 6.25 Contour plot of the index of performance. (Regular with Logarithmic Scaling) - Lateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.26 PID vs. PI0D0.8 - Lateral . . . . . . . . . . . . . . . . . . . . . . . . 66 6.27 PID vs. PI0D0.8 - Lateral . . . . . . . . . . . . . . . . . . . . . . . . 67 6.28 Contour plot of the index of performance. (Regular with Feedback) - Lateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.29 Modified PID vs. PI0D0.96 - Lateral . . . . . . . . . . . . . . . . . . 68 6.30 Modified PID vs. PI0D0.96 - Lateral . . . . . . . . . . . . . . . . . . 69 6.31 Index of Performance of the λ values. (Tuned) - Lateral . . . . . . . 70 6.32 Modified PID vs. PI−0.08D0 - Lateral . . . . . . . . . . . . . . . . . 70 6.33 PID vs. PI−0.7193D0.5236 (Optimal) - Lateral . . . . . . . . . . . . . 72 6.34 Contour plot of the index of performance. (Tuned with Feedback Logarithmic Scale) - Lateral . . . . . . . . . . . . . . . . . . . . . . 73 6.35 PID vs. PI−0.0090D0.7039 (Optimal) - Lateral . . . . . . . . . . . . . 75 6.36 Index of Performance of the λ Values. (Regular) - Lateral . . . . . . 76 x

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FOR AN UNMANNED AERIAL VEHICLE. AUTHOR: Jesus Navarro. DATE SUBMITTED: Summer 2017. Department of Mathematics and Statistics.
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FRACTIONAL-ORDER CONTROLLERS FOR AN UNMANNED AERIAL VEHICLE A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Jesus Navarro 2017 SIGNATURE PAGE THESIS: FRACTIONAL-ORDER CONTROLLERS FOR AN UNMANNED AERIAL VEHICLE AUTHOR: Jesus Navarro DATE SUBMITTED: Summer 2017 Department of Mathematics and Statistics Dr. Hubertus von Bremen Thesis Committee Chair Mathematics & Statistics Dr. Subodh Bhandari Thesis Committee Co-Chair Aerospace Engineering Dr. Jennifer Switkes Mathematics & Statistics Dr. Robin Wilson Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis is dedicated to my family, friends, and mentors for all their support. Thank you to Megan Hamilton, for being a great friend and helping me in proof­ reading my thesis. Thank you Dr. Jung-Ha An and Dr. Michael Bice for being my REU advisors and showing me how much more there is to mathematics. Thank you Dr. John Rock, who always encouraged me to apply to grad school. Thank you Dr. Jennifer Switkes and Dr. Robin Wilson for being part of my thesis committee. Thank you Dr. Subodh Bhandari for being a co-advisor and allowing me to flex my mathematical knowledge into aerospace engineering. Thank you Dr. Hubertus von Bremen for being my thesis advisor and encouraging me to look into different STEM fields, so that I can apply what I know into a different field of study. Special thanks to my father and mother, who sacrificed so much to give me, my brothers, and sisters a future we could have never imagined. iii ABSTRACT Fractional calculus is a new field of study where the main focus is on differentiation and integration of non-integer order. We then expand upon this idea into linear fractional-order differential equations and apply those techniques into PIλDµ, PIλ , and PDµ controllers where λ and µ are arbitrary real numbers. The way we accom­ plish this is by implementing methods that have been proposed in Fractional-order Systems and Controls. There are two different models that we look into, the lon­ gitudinal model which deals with the pitch angle of the aircraft, and the lateral model which deals with the roll angle of the aircraft. In order to obtain an ideal controller, we optimize the performance index and obtain the best parameters for the system. Then, looking at the step information of the controller, we want to en­ sure that we get a reasonable overshoot, rise time, and settling time. In one of the examples, we notice a significant difference where the fractional-order controller’s rise time was 27.7 seconds faster, the settling time was 24.5 seconds faster, and our performance index gave us 4 more decimal places of accuracy compared to that of a regular controller. With these new techniques, we now have the opportunity to achieve better results than by using regular PID, PI, and PD controllers. iv Contents Signature Page ii Acknowledgements iii Abstract iv List of Tables viii List of Figures ix Chapter 1 Introduction 1 Chapter 2 Fractional Calculus 4 2.1 What is Fractional Calculus? . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Riemann-Liouville’s Fractional Derivative . . . . . . . . . . . . . . . 5 2.4 Caputo’s Fractional Derivative . . . . . . . . . . . . . . . . . . . . . 6 2.5 Gru¨nwald-Letnikov’s Fractional Derivative . . . . . . . . . . . . . . 7 2.6 Differences Between Riemann-Liouville and Caputo Derivatives . . 9 Chapter 3 Fractional-Order Differential Equations 13 v 3.1 Oustaloup Recursive Approximation . . . . . . . . . . . . . . . . . 16 3.2 Closed-Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 4 Controllers 18 4.1 PID-Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 PIλDµ-Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.1 Open-Loop System . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.2 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 5 Aircraft Dynamics 23 5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Longitudinal-Direction Dynamics . . . . . . . . . . . . . . . . . . . 26 5.3 Lateral-Directional Dynamics . . . . . . . . . . . . . . . . . . . . . 29 Chapter 6 Simulations 33 6.1 State-Space Equations to Transfer Function . . . . . . . . . . . . . 34 6.2 Optimal Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.3 Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3.1 Results for PID vs. PIλDµ Controllers . . . . . . . . . . . . 40 6.3.2 Results for PI vs. PIλ Controllers . . . . . . . . . . . . . . . 54 6.3.3 Results for PD vs. PDµ Controllers . . . . . . . . . . . . . . 62 6.4 Lateral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.4.1 Results for PID vs. PIλDµ Controllers . . . . . . . . . . . . 64 6.4.2 Results for PI vs. PIλ Controllers . . . . . . . . . . . . . . . 76 6.4.3 Results for PD vs. PDµ Controllers . . . . . . . . . . . . . . 84 vi Chapter 7 Summary 92 Bibiliography 102 Appendix 104 A MATLAB and Simulink Source Code 104 A.1 Optimal Sig Kadet State-Space Model - 1 . . . . . . . . . . . . . . 104 A.1.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . 108 A.2 Optimal Sig Kadet State-Space Model - 2 . . . . . . . . . . . . . . 108 A.3 Fractional-Derivative Block . . . . . . . . . . . . . . . . . . . . . . 113 A.4 Fractional-Order Controllers for Simulink . . . . . . . . . . . . . . . 115 vii List of Tables 6.1 PID vs. PIλDµ Controllers from Figure 6.13 . . . . . . . . . . . . . 51 6.2 PID vs. PIλDµ Controllers from Figure 6.15 . . . . . . . . . . . . . 54 6.3 PI vs. PIλ Controllers from Figure 6.21 . . . . . . . . . . . . . . . . 60 6.4 PID vs. PIλDµ Controllers from Figure 6.23 . . . . . . . . . . . . . 61 6.5 PID vs. PIλDµ Controllers from Figure 6.33 . . . . . . . . . . . . . 72 6.6 PID vs. PIλDµ Controllers from Figure 6.35 . . . . . . . . . . . . . 75 6.7 PI vs. PIλ Controllers from Figure 6.40 . . . . . . . . . . . . . . . . 81 6.8 PI vs. PIλ Controllers from Figure 6.42 . . . . . . . . . . . . . . . . 82 6.9 PD vs. PDµ Controllers from Figure 6.48 . . . . . . . . . . . . . . . 89 7.1 Optimal Tuning Results for Transfer Function 6.13 - Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Optimal Tuning Results for Transfer Function 6.14 - Longitudinal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 Optimal Tuning Results for Transfer Function 6.17 - Lateral Model 97 7.4 Optimal Tuning Results for Transfer Function 6.18 - Lateral Model 99 viii List of Figures 5.1 Yaw, Pitch, and Roll rotations. Image obtained from the NASA website. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.1 Simulink model for fractional-order PID controller. . . . . . . . . . 35 6.2 PID vs. PI−0.25D0.25 - Longitudinal . . . . . . . . . . . . . . . . . . 41 6.3 PID vs. PI−0.5D0.5 - Longitudinal . . . . . . . . . . . . . . . . . . . 42 6.4 Comparison of various PIλDµ Controllers 1 - Longitudinal . . . . . 43 6.5 Comparison of various PIλDµ Controllers 2 - Longitudinal . . . . . 43 6.6 Comparison of various PIλDµ Controllers 3 - Longitudinal . . . . . 44 6.7 Contour plot of the index of performance. (Regular) - Longitudinal 45 6.8 Contour plot of the index of performance. (Regular with Logarithmic Scaling) - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.9 PID vs. PI−0.16D0.17 - Longitudinal . . . . . . . . . . . . . . . . . . 47 6.10 Contour plot of the index of performance. (Regular with Feedback) - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.11 PID vs. PI−0.14D0.99 - Longitudinal . . . . . . . . . . . . . . . . . . 48 6.12 Contour plot of the index of performance. (Tuned) - Longitudinal . 50 6.13 PID vs. PI−0.38045D0.56782 (Optimal) - Longitudinal . . . . . . . . . 51 6.14 Performance of the λ Values. (Tuned with Feedback) - Longitudinal 52 ix 6.15 PID vs. PI−0.6290D0.4820 (Optimal) - Longitudinal . . . . . . . . . . 53 6.16 Performance of the λ Values (Regular) - Longitudinal . . . . . . . . 55 6.17 PI vs. PI−0.19 - Longitudinal . . . . . . . . . . . . . . . . . . . . . . 56 6.18 Performance of the λ Values (Regular with Feedback) - Longitudinal 56 6.19 PI vs PI−0.1 - Longitudinal . . . . . . . . . . . . . . . . . . . . . . . 57 6.20 Performance of the λ Values. (Tuned) - Longitudinal . . . . . . . . 58 6.21 PI vs. PI−3.415x10−7 (Optimal) - Longitudinal . . . . . . . . . . . . . 59 6.22 Performance of the λ Values. (Tuned with Feedback) - Longitudinal 60 6.23 PI vs. PI−0.000070268 (Optimal) - Longitudinal . . . . . . . . . . . . . 62 6.24 Index of Performance of the µ Values. (Regular) - Longitudinal . . 63 6.25 Contour plot of the index of performance. (Regular with Logarithmic Scaling) - Lateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.26 PID vs. PI0D0.8 - Lateral . . . . . . . . . . . . . . . . . . . . . . . . 66 6.27 PID vs. PI0D0.8 - Lateral . . . . . . . . . . . . . . . . . . . . . . . . 67 6.28 Contour plot of the index of performance. (Regular with Feedback) - Lateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.29 Modified PID vs. PI0D0.96 - Lateral . . . . . . . . . . . . . . . . . . 68 6.30 Modified PID vs. PI0D0.96 - Lateral . . . . . . . . . . . . . . . . . . 69 6.31 Index of Performance of the λ values. (Tuned) - Lateral . . . . . . . 70 6.32 Modified PID vs. PI−0.08D0 - Lateral . . . . . . . . . . . . . . . . . 70 6.33 PID vs. PI−0.7193D0.5236 (Optimal) - Lateral . . . . . . . . . . . . . 72 6.34 Contour plot of the index of performance. (Tuned with Feedback Logarithmic Scale) - Lateral . . . . . . . . . . . . . . . . . . . . . . 73 6.35 PID vs. PI−0.0090D0.7039 (Optimal) - Lateral . . . . . . . . . . . . . 75 6.36 Index of Performance of the λ Values. (Regular) - Lateral . . . . . . 76 x

Description:
FOR AN UNMANNED AERIAL VEHICLE. AUTHOR: Jesus Navarro. DATE SUBMITTED: Summer 2017. Department of Mathematics and Statistics.
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