Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle: Roadmap to Improve Tracking-Trajectory Performance in the Presence ... in Systems, Decision and Control, 384) 9783030810139, 9783030810146

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Studies in Systems, Decision and Control 384

Moussa Labbadi Yassine Boukal Mohamed Cherkaoui

Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle Roadmap to Improve Tracking-Trajectory Performance in the Presence of External Disturbances

Studies in Systems, Decision and Control Volume 384

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

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Moussa Labbadi · Yassine Boukal · Mohamed Cherkaoui

Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle Roadmap to Improve Tracking-Trajectory Performance in the Presence of External Disturbances

Moussa Labbadi Engineering for Smart and Sustainable Systems Research Center, Mohammadia School of Engineers Mohammed V University in Rabat Rabat, Morocco

Yassine Boukal Department of Aeronautics, Space and Defense Capgemini Engineering Toulouse, France

Mohamed Cherkaoui Engineering for Smart and Sustainable Systems Research Center, Mohammadia School of Engineers Mohammed V University in Rabat Rabat, Morocco

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-81013-9 ISBN 978-3-030-81014-6 (eBook) https://doi.org/10.1007/978-3-030-81014-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This work is dedicated to My brother Mohamed LABBADI, deceased in the prime of his life. May Allah, the Almighty, have him in his holy mercy! —Moussa Labbadi My father Larbi BOUKAL, who passed away too early, and who always pushed and motivated me in my studies. May God, the Almighty, have him in his holy mercy! —Yassine Boukal

Preface

The present book aims to develop and design some robust nonlinear flight control strategies for the quadrotor UAV (QUAV) system in the presence of external disturbances, and the system uncertainties. This book presents new control approaches to synthesize the nonlinear control of the trajectory of the QUAV. Despite a rich literature, the problem has not yet been adequately addressed and it is difficult to provide high-precision flight path following in the presence of these disturbances. This work proposed robust control techniques to achieve adequate set-point tracking of different complex trajectories of QUAV and unknown disturbance rejection with smoother control action. Based on sliding mode control (SMC) theory, backstepping technique, fractional-order calculus, and adaptive laws, tracking methods are presented for the QUAV. The main motivations of this book are given by the following points: 1.

2.

3.

Designing a control strategy for QUAV to meet certain requirements (precision, minimum energy consumption, and simple structure) poses an additional challenge. Also, the design of flight controllers for these multi-rotor drones presents three important challenges: (i) the vehicle dynamics are multi-input, multi-output (MIMO) and strongly nonlinear coupled; (ii) the dynamics of the quadrotor involve various sources of uncertainties, including parametric uncertainties, unmodeled uncertainties, and external disturbances; (iii) there are multiple states varying over time and entry delays into control systems. So, this book proposes robust flight control schemes against these perturbations. In the presence of external disturbances, the trajectory following performance of the QUAV may be degraded, and its dynamic systems may also lead to instability. The coupling between the position and attitude of the QUAV increases the stability problem with satisfied performance. Hence, disturbances that affected the QUAV system should be considered in the design of the flight controller to enhance the QUAV control performance. The high-order SMC is usually applied to nonlinear complex systems under disturbances. The super-twisting and its version controls are successfully adapted to nonlinear systems. It is generally known that the SMC technique applied for QUAV is robust control, but does not guarantee finite time convergence. Also, the system uncertainties can

vii

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4.

5.

Preface

degrade the control performance. As a result, the nonlinear sliding mode variables are designed and parametric uncertainties are rejected to enhanced the performance of the closed-loop system. In addition, some control laws are developed to guarantee the sliding mode of the QUAV states in the initial time. Therefore, the development of hybrid finite-time control and global SMC is significant for the control of QUAV systems with perturbations. Design adaptive laws with finite-time control of the errors to zero of the QUAV in the presence of disturbances and uncertainties are a challenging problem for this type system. Therefore, considering these control problems and guaranteeing the finite-time stability as well as specific criteria such as perturbation rejection and robustness to the parametric uncertainties of the system are very significant for the QUAV. Also, the upper bound of these perturbations is assumed to be known in advance: in reality, this limit is unknown and difficult to determine. Moreover, it is a challenging problem to develop the control laws for addressing these perturbations and guarantee the finite-time stability. Therefore, time-varying SMC, terminal SMC, and nonsingular terminal SMC schemes with adaptive laws need to be further studied for QUAV systems in the presence of external wind disturbances and parametric uncertainties. The design of robust nonlinear controls with the fractional-order dynamics such as high-order SMC, TSMC, nonsingular TSMC, backstepping techniques, the path following of the QUAV is realized and better robustness can be obtained against parametric uncertainties and perturbations. These flight control methods are important when the QUAV is affected by unknown complex disturbances during flight missions. The control performance of these fractional-order controllers may be better than integer controllers. Therefore, it is important to study these control schemes for the QUAV. The control performance including the transient and steady states of the tracking errors should be enhanced for the QUAV in the presence of unknown disturbances to minimize the consummation error energy. The study uses robust nonlinear controllers including fractional dynamics of these actions. Using these, the above performance can be improved under the effect of disturbances.

The main contributions of the research work carried out in this book are shown as follows: Firstly, on the basic of the result research on the high-order SMC for the QUAV, a combination of the integral SMC with the super-twisting algorithm to eliminate the reticence phenomenon is presented. This control method is proposed to stabilize the QUAV to follow a path in the presence of perturbations. Vehicle state variables converge to their desired values in a short time at a specified time. In order to increase the robustness of the control system, a new optimization is used to adjust the proposed controller parameters. In addition, a novel modified super-twisting is combined with the nonlinear sliding mode controller to improve the stabilization of this vehicle under time-varying disturbances. Using nonlinear sliding mode variables, fast convergence of position/attitude outputs is established. This control technique offers some performance such as rejection of disturbances and reduction of chattering phenomenon.

Preface

ix

The proposed control methods utilize the Lyapunov theory to prove the stability of the system. To demonstrate the effectiveness of these control approaches, numerical simulations are given. Secondly, for the path following of the QUAV under unknown disturbances, a hybrid finite-time control technique is proposed, which combines by three control algorithms including integral terminal sliding mode control for attitude subsystem, adaptive backstepping for altitude channel, and backstepping for horizontal position. To eliminate the reaching phase and ensure sliding mode of the state variables of the QUAV, a robust controller is designed which tacking account the initial time. Also, the upper bound of these perturbations is addressed using the proposed control approach. The obtained results in the presence of disturbances demonstrate the robustness of the proposed controllers. Also, these obtained results are compared with nonlinear controllers. Thirdly, a robust approach to controlling the QUAV has been proposed. The control technique is designed on the basis of on-line estimators of the dynamic parameters. These adaptation methods make it possible to improve the control performance of this system, and to compensate for the parametric errors due to the coupling of the position with the orientation of the QUAV. The suggested control method is based on the backstepping fast terminal SMC and a new version of the adaptive backstepping. The on-line rules are presented to estimate exactly some unknown parameters caused by wind gust and other factors. To assess/emphasize the efficacy of the proposed control methods for the QUAV, various simulations under various scenarios in terms of external perturbations and parametric uncertainties are performed. A comparison analysis clearly demonstrates that the proposed control schemes outperform the competition. Fourthly, it should be noted that all research works have focused on adaptive upper bound uncertainty estimation using terminal SMC and fast terminal SMC techniques. In this view, a second control approach is proposed using a new adaptive nonsingular fast terminal SMC controller (ANFTSMC) for the QUAV under complex perturbations. The main advantage ANFTSM control is the avoidance of singularity, the rapidity when states are far from the origin, and the high robustness against system uncertainty and external disturbances. Also, this controller ensures fast convergence, avoids singularities, resolves the reticence effect, and offers robustness against unknown external perturbations and uncertainties. In addition, the upper limit of uncertainty and unknown external perturbations of the system are covered by the proposed control approach. On-line estimation of these upper limits is introduced only by velocity and position measurements. Various simulations under different scenarios in terms of external perturbations and parametric uncertainties are performed to assess/emphasize the effectiveness of the proposed control methods for the QUAV. A comparative study clearly shows the outperformance of the proposed control schemes. Fifthly, the aerodynamics disturbances, parametric uncertainties, and noise measurements are considered for the design of adaptive global time-varying SMC (RAGTVSMC) for the trajectory tracking of the QUAV under the random disturbances/uncertainties. The problem of initial control effort and reaching phase is

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addressed by the proposed controller. Designing time-varying sliding manifolds for the QUAV system meet the impact time and given the convergence in a specific time of the state variables in the presence of disturbances. In addition, the upper bound of these perturbations is addressed by designing the adaptive laws using only velocity and position of the tracking errors. To demonstrate the effeteness of the RAGTVSMC method, simulations are conducted. Sixthly, an improved FO control is proposed for the QUAV to solve the problems in the control of this vehicle including disturbances, uncertainties, and the variations of the drag coefficients. The FO control method uses super-twisting based on FO integral sliding mode variables to improve the tracking performance. Also, a generic switching law with FO dynamic is designed for the QUAV to address the disturbances. The corresponding controllers with fractional calculus ensure high precision, fast convergence, and good robustness against perturbations. Good scenarios under different situations in the flight are proposed to evaluate the efficiency of the FO control method, and its numerical results show better performance compared with other control strategies. Seventhly, motivated by the advantage of the finite-time stability in the control theory and fractional calculus, a FO global SMC approach is proposed for the control problem of the uncertain QUAV under complex disturbances. This presented method uses the specific sliding manifolds to ensure faster convergence speed and weakening error. The global stabilization with a specific time of the QUAV under disturbances is guaranteed based on Lyapunov theory. Finally, simulations included a good comparison with robust controllers are performed to check the correctness of theoretical analysis. The nonlinear SMC such nonsingular fast TSMC with FO operators enhances the tracking performance of the QUAV against random disturbances/uncertainties and other perturbations. Also, adaptive control law is developed to augment the robustness of the proposed FO control method in the presence of disturbances. The designed control technique ensures high accuracy and fast speed for the QUAV. Using FO nonsingular fast terminal sliding manifolds, a finite-time convergence of the tracking errors is achieved. The present book intends to provide the readers an excellent understanding on how to obtain the robust controller path following approaches for the QUAV system in the presence of known or unknown uncertainties, disturbances, and other problems of the control. The designed control methods in this book can be applied for nonlinear systems under complex disturbances in various fields in engineering. The book can be used as a reference for the academic research on the control theory, drones, and terminal sliding mode control, and related to this or used in Ph.D. study of control theory and their application in field engineering. Rabat, Morocco Toulouse, France Rabat, Morocco May 2021

Moussa Labbadi Yassine Boukal Mohamed Cherkaoui

Acknowledgements

This book is based on a series of articles and conference proceedings that we have been published during my Ph.D. study. Our great thanks go to Profs. Abdellah Benzaouia, Mohamed Djemai, and Maarouf Saad who worked with us on this subject of advanced controllers for Quadrotor system and Fractional order control. I would also like to thank all my colleagues from the E.3S-EREE&C research center, Department of Electrical Engineering, Mohammadia School of Engineers, E.3S-EREE&C research center that I had the pleasure to work with during my thesis. I address all my gratitude to my friends Mr. Chakib Chatri, Mr. Kamal Elyaalaoui, and Mr. Ali Agga, for his indirect help while wishing them a very good courage. A special thought goes to Mrs. Karima Boudaraia, and Miss. Imane Hammou Ou Ali for their indirect help. Finally, my special thanks go to my mother Fatima and my father M’hamed, my brothers Abdalghani, Redouan, Yassine, Mehdi, and Abderahmane for their moral, encouragement, and sacrifices. Moussa Labbadi

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Significance and Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Guidance, Navigation, and Control . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Guidance System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Navigation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Flight Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Advanced Flight Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Flight Control Methods Based on FO Control . . . . . . . . . . . . . . . . 1.6 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 4 5 6 10 11 14

2

QUAV Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Review of Multirotors Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Preliminaries and Frame Representation . . . . . . . . . . . . . . . . . . . . . 2.4 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Flight Modeling of the Quadrotor with Newton-Euler’s Formalism . . . . . . . . . . . . . . . . . . . . 2.4.2 Aerodynamic Forces and Moments Applied to the Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Full Multi-rotor Dynamic Model . . . . . . . . . . . . . . . . . . . . 2.4.4 Rotor Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Model of the Vehicle Flying in a Gust of Wind . . . . . . . . . . . . . . . 2.5.1 Modeling of Wind Gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Additional Applied Forces and Moments . . . . . . . . . . . . . 2.5.3 Dynamic Model of Multi-rotor Under the Effect of the Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Technical Simplifications for Implementation Purposes . . . . . . . . 2.6.1 Simplified Simulation Model . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Control Oriented Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Control Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 19 21 24 24 25 31 32 33 33 35 37 37 37 39 40 xiii

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2.6.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation of the Integral and Derivative Operators of Non-integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Useful Functions in Fractional Calculation . . . . . . . . . . . 2.7.2 Integration of Non-integer Order . . . . . . . . . . . . . . . . . . . . 2.7.3 Differentiation of Non-integer Order . . . . . . . . . . . . . . . . . 2.7.4 Approximation of Non-integer Order Systems . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stabilization of QUAV Under External Disturbances Using Modified Novel ST Based on Finite-Time SMC . . . . . . . . . . . . . . . . . . . 3.1 Premolars of the Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Synthesis of the Sliding Surface . . . . . . . . . . . . . . . . . . . . . 3.1.4 Design of the Control Law . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Quadrotor Control by First Order Integral Sliding Mode . . . . . . . 3.3 Quadrotor Control by Higher Order PID Sliding Modes . . . . . . . . 3.3.1 Principle of the Second Order Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Super-Twisting Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Application of the Higher Order SM-PID Control to Quadrotor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Simulation Results with Controller Gains Optimization . . . . . . . . 3.4.1 Scenario 1: Robustness Analysis (constant Disturbances) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Scenario 2: Robustness Analysis (Time-Varying Disturbances) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Novel Terminal Sliding Mode Control for the Position and Attitude of a Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Modified Super-Twisting NSMC for the Quadrotor System . . . . . 3.8 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control of the QUAV by a Hybrid Finite-Time Tracking Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hybrid Finite-Time Trajectory Tracking Technique for Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Control of the Altitude Subsystem by Adaptive Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Control by Backstepping Technique for the Horizontal Position Subsystem . . . . . . . . . . . . . . .

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Attitude Control Using the Full Terminal Sliding Mode Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Analysis of the Simulation Results by the Hybrid Control . . . . . . 85 4.4 Adaptive Global Nonlinear SMC for a Quadrotor [1] . . . . . . . . . . 88 4.4.1 Stability Analysis of the Proposed Controller . . . . . . . . . 93 4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5

6

Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Finite-Time Adaptive Flight Control of a Quadrotor . . . . . . . . . . . 5.2.1 Quadrotor Position Control by a New Adaptive Backstepping Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Attitude Control of a Quadrotor Using the Adaptive Fast Terminal Sliding Mode Technique with the Backstepping Approach . . . . . . . . . . 5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Scenario 1: Without Disturbances . . . . . . . . . . . . . . . . . . . 5.3.2 Scenario 2: In the Presence of Parametric Uncertainty and Disturbances . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robust Nonsingular Fast Terminal SMC for Unceratin QUAV Subjected to External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Design Methodology of a New Controller for the Quadrotor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Design of a Trajectory Tracking Controller for the Quadrotor Position Based on the NFTSMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Design of the Trajectory Tracking Controller for the Quadrotor Position Based on the RANFTSMC Method . . . . . . . . . . . . . . . . . . 6.2.3 Design of a Trajectory Tracking Controller for Quadrotor Attitude Based on the RANFTSMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Scenario 1: Nominal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Scenario 2: Constant Disturbance . . . . . . . . . . . . . . . . . . . 6.3.3 Scenario 3: Time Variation of the Wind Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Scenario 4: Noise from Sensors . . . . . . . . . . . . . . . . . . . . . 6.3.5 Scenario 5: Parametric Uncertainties . . . . . . . . . . . . . . . . .

103 103 104 104

107 112 112 113 121 121 123 123 124

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131 133 134 138 140 141 143

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6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7

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Robust Adaptive Global Time-Varying SMC for QUAV Subjected to Gaussian Random Uncertainties/Disturbances . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Controller Design Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Design Controller Based on a Global Time-Varying SMC for the QUAV . . . . . . . . . . . . . . . . . . 7.2.2 Design Controller Based on an Adaptive Time-Varying SMC for the QUAV System . . . . . . . . . . . . 7.3 Simulation Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Order Fractional Controller Based on PID-SMC for the QUAV Under Uncertainties and Disturbance . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Position Controller Design Based on FO-ST-PID-SMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Attitude Control Method Design Based on FO-ST-PID-SMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Simulation Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Comparisons Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Fractional Controller Based on SMC for the QUAV Under Uncertainties and Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Design of Fractional Sliding Manifold for Quadrotor System Without Disturbances and Modeling Uncertainties . . . . . 9.3 Design of Fractional Sliding Manifold for the QUAV System in the Presence of Modeling Uncertainties and Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 150 150 154 158 161 164 165 165 166 166 170 173 174 177 179 187 187 189 191 191 191

196 198 198 201 203 206 210

Contents

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV Under Random Gaussian Disturbances . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Translational Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Stability Analysis for the Translational Loop . . . . . . . . . . 10.2.3 Rotational Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 With Drag Coefficients Uncertainties and Stochastic Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 With Drag Coefficients Uncertainties and Random Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 With Random Uncertainties (Random Uncertainty 30% Added in Mass and Rotary Inertia) and External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

213 213 213 214 215 217 219 222 225

226 234 236

11 Summary and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 11.1 Summary of Full Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 11.2 Future Research Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Appendix A: Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Appendix B: Simulations of Non-integer Systems and Stability in the Lyapunov Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Acronyms

tw 0 λ μ ρ σ ui (t), (t), (t) ω r

Blade torsion angle (deg) Angle of attack of the blade root profile (rad) Entry rate (−) Advance ratio (−) Density of the air (kg/m3 ) Rotor strength ratio (−) Rotor torque (N.m) Angles of the quadrotor (rad) Angular speed of the rotor (rad/s) Angular speed of the propeller (rad/s)

Ensembles R, C R+ Rn , Rn Rn×m , Cn×m

The set of real (resp. complex) numbers The set of non-negative real numbers R+ = [0, ∞) Real (resp. complex) Euclidean space of dimension n Set of real (resp. complex) matrices of dimension n × m

u, v, w

Wind speeds in relation to the earth and within the body (m/s) Control inputs (N; N.m) Position of the quadrotor in the earth frame (m) Trajectory tracking error (−) Sliding Mode Monifold (−) Calculated disturbance for the quadrotor in the body frame (−) Computed disturbance for the quadrotor around the attitude (−)

Vx , Vy , Vz , u , u , u x(t), y(t), z(t) e(t) s(t) dx (t), dy (t), dz (t) d (t), d (t), d (t)

xix

xx

Acronyms

g I irot Ixx , Iyy , Izz l Jrot m p(t), q(t), r(t) R R Rrot u(t), v(t), w(t) uj , vj , wj

The acceleration of gravity (m/s2 ) Matrix of inertia of the quadrotor (kg.m2 ) Current in the electrical circuit of the rotor (A) Inertia of the quadrotor (kg.m2 ) Quadrotor arm length (m) Moment of inertia of the rotor (kg.m2 ) Mass of the quadrotor (kg) Angular speeds of the quadrotor (rad/s) Rotation matrix (−) Rotor radius (m) Internal resistance of the rotor (Ohm) Linear speeds of the quadrotor in the body frame (m/s) Linear speeds of each rotor in the body frame (m/s)

AB AB-ABFTSMC AHRS ANFTSMC

Adaptive Backstepping Adaptive Backstepping Fast Terminal Sliding Mode control Attitude and Heading Reference System Adaptive Nonsingular Fast Terminal Sliding Mode Controller Backstepping-Integral Terminal Sliding Mode Controller East-North-Up Fractional Adaptive Nonsingular Fast Terminal Sliding Mode Controller Feedback Linearization Fractional-order Fractional Order Backstepping Sliding Mode Control Fractional Order Global Sliding Mode Global Nonlinear Sliding Mode Control Global Positioning System Horizontal Take-Off and Landing Integral Absolute Error Inertial Measurement Unit Inertial Navigation System Integral Squared Error Integral Terminal Sliding Mode Controller Linear-Quadratic Regulator Micro or Miniature Air Vehicles Nano Air Vehicles North-East-Down Nonlinear Internal Model Control Proportional–Integral–Derivative Quadrotor Unmanned Aerial Vehicle Riemann-Liouville, Caputo Sliding Mode Control

B-ITSMC ENU FANFTSMC FL FO FO-BSMC FOGSM GNSMC GPS HTOL IAE IMU INS ISE ITSMC LQR MAVs NAVs NED NLIMC PID QUAV RL, C SMC

Acronyms

STA STC TSMC UAV VTOL

xxi

Super-Twisting Algorithm Super-Twisting Control Terminal Sliding Mode Control Unmanned Aerial Vehicle Vertical Take-Off and Landing

Chapter 1

Introduction

1.1 Introduction This chapter illustrates some general knowledge about the control of multi-rotor systems, in particular the quadrotor. The research work has been carried out over three diverse and complementary studies: the design of a set of sliding mode control laws based on the super-twisting algorithm and a finite time hybrid control with respect to perturbations, the design of non-integer order sliding mode controllers in the presence of parametric uncertainty and exogenous perturbations, and the design of adaptive finite time control laws for the perturbed uncertain quadrotor, based on on-line estimators of the dynamic parameters. This research work is at the intersection of the robustness themes of the multi-rotor control system in a complex environment, and the design of robust controllers; more precisely, it deals with improved sliding mode control, improved fractional order sliding mode, and adaptive non-singular fast terminal sliding mode for the multi-rotor system while taking into account the constraints generated by endogenous and exogenous perturbations.

1.2 Significance and Purpose Beyond our motivations, the object of this book is the design of control systems for autonomous aerial vehicles, in particular multirotors (mainly quadrotors) flying in disturbed areological conditions. Consequently, these problems treated in this book are complex problems touching several domains. The other difficulty is that these problems are treated in different ways by the many researchers who are interested in this topic according to their disciplines and the community to which they belong. Therefore, this research work is part of a multidisciplinary framework combining aerodynamics, control theory and estimation theory.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_1

1

2

1 Introduction

The problems presented in the previous section are poorly addressed by existing techniques to satisfy a good compromise between the level of robustness and control performance. In this context, the present work provides controllers to the QUAV system in order to compensate the effect of external perturbations, based on the modified super-twisting algorithm and a hybrid control. Most of the control approaches proposed in the literature for these vehicles are whole-order control methods, the present work proposes fractional-order controllers to deal with perturbations and variation of the QUAV parameters. However, we are interested in the design of adaptive controls whose robustness properties to parametric uncertainties and perturbations have been demonstrated. In addition, the objective is to guarantee the stability as well as specific criteria such as the rejection of perturbations and the robustness to the parametric uncertainties of the system. The upper limit of these perturbations is supposed to be known in advance, in reality this limit is unknown and difficult to determine. In order to circumvent this constraint, we propose an adaptive estimator that allows to estimate the upper limit of these perturbations in finite time using only the tracking error and its speed as measures. In addition, a low-cost, real-time QUAV test bench is proposed to verify the 4 DOF flight controller of the multi-rotors.

1.3 Guidance, Navigation, and Control During the last 20 years, several research works have been carried out on the guidance, navigation and control (GNC) of the QUAV, resulting in various techniques. Some researchers have tried to examine different CNG systems and subsets of CNG. In addition, the authors of [1] have recently completed a comprehensive and organized survey report on the wide variety of CNG methods. They gave an overview of CNG systems to increase the autonomous capabilities of drones. More details about GNC systems can be found in [1]. Subsequently, the different parts of the GNC system are presented. CNG consists essentially of three parts: the guidance module, the navigation module and the control module [2]. The rotors receive the appropriate control signals from the control module in which the QUAV variables and their reference trajectories are taken into account to generate the appropriate control signals. The reference trajectories are managed by the guidance system where the system states are provided by the navigation system. The outputs, generated by these last two modules, are based on the behavior of the multi-rotor, which can be expressed by sensor measurements. The Fig. 1.1 shows the main architecture of the multi-rotor vehicle.

1.3 Guidance, Navigation, and Control

3

Fig. 1.1 Global CNG architecture

1.3.1 Guidance System The guidance system can be defined as the “driver” of a QUAV that performs planning and decision-making functions to achieve missions or objectives. The role of a guidance system for QUAV is to replace the cognitive processes of a human pilot and operator. It uses signals from the navigation system as inputs. Then, it takes the appropriate decisions, selects the appropriate maneuvers and manages the corresponding reference trajectories such as angles, positions, speeds, etc. desired. These trajectories are directed to the flight control system that allows the multi-rotor to reach the desired configuration. It includes various functions that promote autonomy, including trajectory management, trajectory planning, mission planning, high-level reasoning and decision making. The main tasks performed by the guidance system are: • Mission planning: Mission planning involves the process of managing flight paths, coordinating itinerary and tactical objectives, synchronizing the vehicle, in advance or in real time, by a human pilot or by the on-board software system centrally or distributed. It also has to manage the flight modes (flight based on vision, flight based on GPS, target tracking mode, landing, hovering, tracking, etc.). These different modes are either selected by the human pilot via the GCS interface, or by the integrated security system which contains additional functionalities for security purposes such as: emergency landing in the event of an unintentional emergency, return home if the onboard energy is not sufficient, etc.

4

1 Introduction

Each flight mode has predefined conditions to be respected to minimize errors and risks, otherwise the task will be rejected. • Path planning: Path planning is a process of using the accumulated navigation data and a priori information to allow the vehicle to find the best and safest way to reach a specific position/configuration goal or task. Dynamic trajectory planning refers to onboard trajectory planning in real time. Thus, by respecting these thresholds, a basic level of performance is guaranteed in terms of accuracy and without excessive delays. • Generation of trajectories: The role of trajectory generation is to compute and provide time-trended parametric reference trajectories for the flight control system, taking into account the vehicle dynamics, physical constraints (maximum speed and acceleration, etc.) the flight path, the flight mode and other specifications such as: smooth trajectories, optimality, etc. The reference trajectories can be ready loaded, programmed or managed in real time.

1.3.2 Navigation System The design of an autonomous QUAV with higher levels of autonomy requires an appropriate navigation system for detection, state estimation, environmental perception and situational awareness. It can be defined as the process of data acquisition, data analysis and information extraction about the vehicle’s state and its environment. This information is then exploited by the flight control system for motion control and by the guidance system for mission accomplishment. The navigation system can be divided into conventional navigation systems as shown in Fig. 1.2. • Inertial Measurement Unit (IMU): By definition, an IMU is mainly composed of three accelerometers and three gyroscopes to measure the specific acceleration of a vehicle as well as its rotational speed. • Attitude and Heading Reference System (AHRS): An AHRS designed to provide attitude (roll and pitch) and heading estimates in addition to raw acceleration and angular velocity information. It contains gyroscopes, accelerometers and magnetometers. • AHRS assisted by GPS, GPS/Inertial Navigation System (INS): INS is an integration (AHRS and GPS, GPS and INS) that corrects the accumulated error of a pure system (AHRS or INS), and provides the complete navigation solution (position, speed, attitude and heading). In addition, advanced navigation systems such as optical flow, visual odometer, etc., using additional sensors, which are required for specific missions (mapping, localization, etc.) or to bring some autonomy to the system.

1.3 Guidance, Navigation, and Control

5

Fig. 1.2 Categories of inertial systems

1.3.3 Flight Control System The flight control system generates the moments and forces necessary to stabilize the vehicle, compensate for internal and external disturbances, follow the desired trajectory or perform a given navigation task. Due to the remarkable properties of multirotor dynamics (multi-variable, highly coupled, hierarchical and under-actuated nonlinear system), an enormous amount of traditional and advanced control strategies, such as PID control, sliding mode control, backstepping, fuzzy logic, etc., have been proposed. In fact, the flight control domain is well developed and can offer many solutions. Many efforts have been made to control multi-rotors and several strategies have been developed to solve the trajectory tracking problem for this type of system. There is a rich literature describing different control techniques. Recent investigations by [3–5] compare some control techniques applied to VTOL vehicles and discuss the advantages of each approach. Thereafter, different flight modes will be detailed in the following section.

Flight Modes The multirotor has six DOF (three translations χ (t) = (x(t), y(t), z(t)) and three rotations η(t) = ((t), (t), (t)) and four entres which are the global thrust u m and the torques (u  , u  , u  ). Indeed, only four outputs are selected among the six degrees of freedom. According to the principle of multi-rotor flight, the pairs (x(t), (t)) and (y(t), (t)) cannot be considered as independent exits. In other words, the (x(t), (t)) =⇒ (xd (t), d (t)) or (y(t), (t)) =⇒ (yd (t), d (t)) control types are prohibited. Depending on the set of states, we can distinguish the following possible combinations: (x(t), z(t), (t), (t)): Longitudinal movement (y(t), z(t), (t), (t)): Lateral movement (z(t), (t), (t), (t)): Teoperated mode or radio control (requires human assistance) (x(t), y(t), z(t), (t)): Autonomous UAV. The dfi is then to deal with the last case where the vehicle can rotate around the three axes and can reach a given configuration in space (x(t), y(t), z(t), (t)) in order to accomplish the mission. We present below some basic control flight modes [2].

6

1 Introduction

• Control the hovering flight: This is the most basic maneuver whose objective is to keep the vehicle stationary at the desired altitude by maintaining a given yaw angle or yaw rate. Most dynamic models are derived by assuming hovering conditions or slow speeds to approximate hovering flight. • Yaw or heading control: The yaw dynamics are totally decoupled during hovering flight, whereas in translational flight a change of heading directly affects the laterallongitudinal dynamics. • Attitude or orientation control: This is the stabilization of the multi-rotor orientation sub-system (roll, pitch and yaw). • Altitude control: During the flight, altitude control is used to ensure that the vehicle reaches the desired height. This is achieved by simultaneously increasing the speed of the rotors. In general, the altitude is coupled with attitude dynamics. • Position control: The goal is to bring the multirotor to a particular point in space. To follow a desired trajectory, one can use a combination of all the controls and teeth. This control can be realized by using two loops. The outer loop of the control loop determines the necessary orientation, taking into account the desired position, while the inner loop of the control loop determines the necessary control inputs. • Speed control: Speed control is taken into account when the multi-rotor has to ensure a given speed. This control is used in parallel with the heading control to mimic fixed-wing vehicles.

1.4 Advanced Flight Control Methods This section presents a remarkable literature on non-linear control techniques with the context of multi-rotor flight control design. Non-linear control methods that have been studied and applied to the QUAV autopilot system include feedback linearization, sliding mode and predictive model control. In addition to conventional non-linear approaches, augmented studies include adaptation and observationbased approaches that are mainly presented. Coupled nonlinear dynamics, parametric uncertainties, input time delay problems, convergence rates, optimal operating regime, stability problems, etc., are addressed. Generally, the most common structures for tracking multi-rotors are cascade control strategies, which use an internal control loop for the rotation subsystem, or in some cases for activated degrees of freedom, combined with an external loop to control the translation movements. Reference [3] provides a brief overview of control architectures for this vehicle. Control systems using this strategy can be found in [6]. Although several control strategies have been tested on multirotors, most of them do not take into account external perturbations over the six degrees of freedom, unmodelled dynamics and parametric uncertainty over the whole model. For example, in [7, 8]. However, in recent years, researchers have begun to take these effects into account at the design stage of the control law for the QUAV. When designing flight controls for these types of air vehicles, there are three important challenges [9, 10]: First,

1.4 Advanced Flight Control Methods

7

the vehicle dynamics is multiple input multiple output (MIMO) and highly nonlinear and coupled; second, the dynamics of the QUAV involves various sources of uncertainties, including parametric uncertainties, unmodelled uncertainties and external perturbations; and third, there are multiple state delays and time-varying input delays in the control systems. Numerous studies have been performed over the years on the problem of controlling aerial vehicles for systems involving non-linearity and coupling dynamics. In order to compensate for external disturbances and improve the trajectory following performance of the QUAV, many robust nonlinear control approaches have been designed for the aforementioned problems, such as controller based on the assignment-passivity of damping and an interconnection strategy [11], backstepping controller with sliding mode [12, 13], disturbance observer [14], control strategies for continuous sliding modes [15], adaptive monitoring of the sliding mode with disturbance algorithms [16], robust adaptive control with input saturation [17], robust observer with linear variation of parameters [18], adaptive sliding mode [19], model predictive control [20], adaptive nonlinear estimation techniques [21], robust nonlinear PID combined with H∞ [22], sliding hierarchical adaptive robust nonlinear mode [23], control methods LQR [24], and non-linear PID type controller [25]. In [26], a new adaptive sliding controller is proposed to control the QUAV. This controller uses Lyapunov analysis to ensure system stability. Fuzzy logic is used to determine the best coefficients for this controller. In [27], a nonlinear control strategy is developed by combining integral backstepping with sliding mode control to stabilize the attitude of the quadrotor and follow the desired trajectory. Likewise, in [28], a robust controller is designed to stabilize a quadrotor attitude. This controller consists of a nominal controller and a robust signal-based compensator. The proposed control method parameters have been tuned to increase system performance. In [29], a robust structured control system design for attitude and position tracking of the quadrotor is developed. In [30], an integral action predictive control strategy is developed to follow the position path and the H∞ control is designed to stabilize the orientation of the quadrotor. The work developed in [31] has dealt with the problem of trajectory tracking under aerodynamic moments and forces using the integral and nonlinear backstepping H∞ . In the case of the variation of the payload of the quadrotor, a hierarchical nonlinear control, based on the thrust allocation algorithm and the Lyapunov technique, is developed in [32]. In [33], the authors propose a robust control method to improve the position following performance of the quadrotor, which is based on a second order sliding mode approach. In [34], the quadcopter drone trajectory following problem was studied by designing a combination between hybrid model predictive controllers and fuzzy logic controllers. The external disturbances and the unknown states of the quadrotor are estimated using an observer in [35]. In order to estimate disturbances and control the quadrotor, an active disturbance rejection control based on virtual variables is designed in the Ref. [36]. In [37], robust nonlinear controllers are proposed to control the quadrotor subjected to uncertainties, noise and disturbances. In [38], a geometric control theory addressed the problem of designing a controller for the quadrotor system with actu-

8

1 Introduction

ation constraints. In [39], three control methods are designed for MAV quadcopter based on linear and nonlinear optimal control theory. The reference [40] presents an observer of the pitch/roll angles of the quadrotor subjected to complex disturbances, which is based on algorithms in sliding mode. The reference [41] proposes a controller based on the passivity damping for a quadrotor system. In [42], a visual mechanism control based on artificial neural networks to control the attitude and position of the quadrotor is designed. A finite-time hybrid control technique is presented in [43], which is based on adaptive integral sliding mode, backstepping, and non-singular terminal MG approaches to solve quadrotor path following problems with dynamics and disturbances. In [44], a cascade control is proposed using a PID control and Lyapunov analysis to stabilize the quadcopter. The reference [45] proposes a robust hierarchical control technique based on neural networks and SMC to deal with quadcopter control under parametric uncertainty and disturbances. In [46], a robust nonlinear control strategy based on a modified active perturbation rejection technique is suggested for the dynamic model of quadrotor position and attitude. This control technique consists of three parts: a nonlinear proportional derivative, a tracking deferential and an extended state observer/predictor. In order to compensate for external disturbances and parameter uncertainties, a robust adaptive control approach to monitoring quadcopter AVMs is presented in Ref. [47]. In [48], a new controller based on continuous control in sliding mode is proposed to solve the problem of the pursuit trajectory of a group of quadcopters in the presence of disturbances. In [49], a model predictive controller for the problem of following the trajectory of both translational and attitude movements of the quadrotor, a time-varying subject of disturbances is presented. The works presented by the authors in [50] propose an adaptive super-twisting sliding mode algorithm to control a biotechnological process under uncertainties and disturbances. In [51], a new adaptive control strategy for robust backstepping is presented. This technique makes it possible to control the dynamic model in the presence of unknown payloads and wind disturbances which vary with time. The authors of [52] introduce a nonlinear controller approach to follow the uncertain quadcopter position trajectory with perturbations. A disturbance observer is proposed to estimate the state variables of this vehicle. The authors of [53] propose an adaptive super-twisting sliding mode control approach for the quadcopter. In [54], an internal model control approach was proposed to stabilize the quadrotor. The wind gust disturbances affecting the vehicle were overcome in the presence of the sensor fault and uncertainties. In [55], a new control procedure for fourth order systems is developed based on adaptive super torsion and terminal sliding mode control approaches. These control techniques reduce the reluctance problem, establish finite-time convergence of the system, and provide a law of parameter adjustment to eliminate external disturbances. In [56], a new fast specification finite-time non-singular terminal slip mode control scheme is proposed for robotic airship trajectory tracking. In order to achieve finitetime convergence and to guarantee the stability of third-order nonlinear systems, a new range law based on the control of the terminal sliding mode has been designed [57]. In [58], backstepping with finite-time convergence techniques is used to generate a control law to stabilize a mini-rotorcraft. In [59], an adaptation mechanism

1.4 Advanced Flight Control Methods

9

and the Nussbaum gain technique are used to control the attitude and position of the rotorcraft. These techniques attenuate immeasurable disturbances, compensate for the parametric uncertainties of the system and compensate for actuation faults. In [60], a hybrid controller is designed for full system path following and closed loop stability is provided using Lyapunov analysis. In [61], a sliding mode terminal controller is developed to control second order nonlinear systems in the presence of perturbations. In order to face the perturbations, non-linearities and uncertainties of nonlinear systems, a new global adaptive approach to terminal sliding mode control is proposed in [62]. In [63], a nano quadrotor is used for the development of autonomous flight controls in environments without a global positioning system. A nonlinear flight controller is designed to keep the quadrotor in the desired flight path position and to ensure attitude stability. In [12], a mathematical model of a quadrotor is presented and a robust nonlinear controller, which combines the sliding mode control technique and the backstepping control technique. A sliding mode controller is designed for the attitude subsystem and the rollback technique is applied to the position loop. An adaptive observer is considered for the take-off mode, this observer is based on a fault estimate. In [64], an adaptive controller is presented to provide increased robustness to parametric uncertainties and to effectively mitigate the effects of a loss of thrust anomaly. In [65], an omnidirectional multirotor vehicle was designed, modeled and tested. This controller makes it possible to simultaneously follow the desired position and attitude of the trajectory of the vehicle. The proposed control is based on multiple cascade control loops, where internal control loops can arbitrarily follow input commands and each control loop is designed through feedback linearization. In [66], a control algorithm is proposed for the visual target tracking system which consists of a fixed-wing drone. Seven fuzzy controllers are used to stabilize the QUAV and to compensate for external disturbances, the information obtained from the images are used to generate the roll command. These proposed algorithms are capable of accomplishing a moving target. Based on the backward sliding mode control approach, the authors of [67] propose an adaptive fuzzy control technique for stratospheric satellites subject to uncertainty and input constraints. In [68], a nonlinear controller combines a non-singular modified super-twisting controller with a high-order sliding mode observer to allow a quadrotor to follow a desired trajectory in the presence of unmodeled dynamics and external disturbances. In addition, two points deserve to be pointed out. On the one hand, most control applications assume that calculated control actions will never reach actuator saturation limits, although in practice this is possible. For example, when the drone is far from its destination, the generated control signals are normally higher than the allowable values. In addition, vehicles are made up of mechanical and electrical parts, also subject to physical constraints. In this book, the problem of following the trajectory of a quadrotor will be addressed, where the main objective is to improve the robustness of control strategies when the vehicle is flying in the presence of external disturbances, non-moving dynamics and uncertainties. parametric, and unknown inputs.

10

1 Introduction

1.5 Flight Control Methods Based on FO Control Fractional-order (FO) control techniques are recognized as an effective tool to improve the structure for designing control schemes for nonlinear systems, in recent years. These controllers can be applied in various complex systems. A FO-PID controller is used [69], FO combined with sliding mode control in the Ref. [70], in [71, 72], an adaptive FO nonsingular fast terminal SMC is proposed. The authors of [73] proposed an adaptive FO-ST nonsingular terminal SMC, and FO-SMC is presented in [74, 75]. Combined the SMC and the FO theory to control the QUAV in the presence of external disturbances, drag coefficients, moment of inertia, and the timevarying load. A nonlinear FO P I λ D α is proposed for controlling attitude/position of the QUAV in [76]. To select their FO parameters, the Black-Nichols method is used. In order to guarantee an exponential tracking of attitude dynamics, the authors of [77] combine the SMC and FO calculus. In this view, the authors of [78] presented an adaptive FOSMC strategy to address the varying load. Further works have been conducted for controlling QUAV in the presence of uncertainties and disturbances that uses continuous-time FO and the proportional-derivative (PD) in [79]. However, in most of the existing publications on the quadrotor drone, research efforts on control techniques have mainly focused on whole order controllers and not on fractional order controllers. In order to perform attitude control and position tracking of a quadrotor, fractional sliding mode and fractional fast terminal sliding mode control techniques have been investigated in [80, 81]. Other studies consider the FO operators in the design of the flight controllers. For example, in [82], an original FO controller, which is based on backstepping SMC method for QUAV. Similarly, in [88], a robust FO-SMC method is proposed for the QUAV. On the other hand, the work in [83] presented a FO-PID controller based on a neural network method. Therefore, the FO controllers can be used to compensate the effect of the uncertainties/disturbances affecting the rotational and translational subsystems. The SMC is one on best suitable robust controller for the QUAV. The tracking performance can be increased combining the SMC and FO theory [84]. Design the sliding mode variables with FO operators, two degrees of freedom are added to allow the structure of the adopted control more flexible. The work developed in [85] proposed a FO-SMC based on backstepping to address the problem of wind disturbance and effects of variations in load momentums of inertia. Some studies including [86] proposed another type of FO-SMC for the tracking problem of uncertain QUAV under time-varying state constraints. In [88], FO attitude-reactive controller is suggested to control the QUAV. A continuous high order SMC and FO control law are proposed by the authors of [89] to address the tracking problem of the QUAV under complex disturbances. Also, they can further benefit implementing the FO-SMC based on backstepping to attenuate wind disturbance and effects of variations in load momentums of inertia in [85]. The authors of [87] presented a robust FO for position/yaw tracking of QUAV. Although the above studies have addressed the design techniques and applications of FO-SM controllers, flight control schemes based on FO adaptive NFTSM controllers have not been fully investigated

1.5 Flight Control Methods Based on FO Control

11

so far. Therefore, for the QUAV system with unknown disturbances, FO finite-time techniques based on TSMC control methods can be further investigated.

1.6 Structure of the Book In this book, the QUAV dynamics model, the high order sliding mode control, terminal SMC, global SMC, backstepping technique, time-varying SMC, backstepping fast TSMC, NFTSMC, FO-super twisting PID SMC, FO-global SMC, FONFTSMC, external disturbances, parametric uncertainties, random noise, random disturbances/uncertainties will be studied. The relations among chapters are shown in Fig. 1.3.

Fig. 1.3 A block diagram of this book

12

1 Introduction

The rest of this book is organized as follows: To validate the efficiency of the different control laws in order to stabilize the QUAV in a wind field, an accurate modeling is important taking into account the wind speed and the external disturbances acting on the vehicle. In Chap. 2, we present a generic dynamic model for any wind-driven multi-rotor vehicle. In addition, general notions of certain mathematical formulae, such as the definitions of fractional order differentiation, approaches to approximation and simulation of a fractional drifter, stability conditions in the sense of Lyapunov are presented. In Chap. 3, two control strategies to solve the tracking problem of the QUAV subjected to finite energy perturbations, are proposed. These proposed control laws must minimize the effect of these perturbations on the tracking error. Next, the controllers proposed in this chapter must ensure the convergence of the tracking errors towards zero. Subsequently, each controller was compared with existing controllers in the literature to show the effectiveness of these approaches. According to suppertwisting algorithm (STA), a combination of PID-SMC and modified STA with an optimization method are designed. The design of this controller is based on Lyapunov theory, which ensures the stability of the system. Based on the research in the first part, a modified STA and nonsingular terminal SMC are proposed for the QUAV under external disturbances in the second part of this chapter. Finally, the results are presented to demonstrate the effectiveness of the high order SMC schemes. In the Chap. 4, a hybrid finite-time control scheme is proposed for both attitude and position subsystems. The attitude subsystem is commanded by integral-type terminal SMC to obtain finite stability of the attitude tracking errors under the influence of disturbances. an adaptive backstepping method is designed for the altitude channel, which gives an adaptive law to estimate exactly the disturbances. The horizontal position is commanded by a simple backstepping to generate the desired tilting angles. Secondly, adaptive global SMC is designed for the QUAV under perturbations. The main objective of this control method is the elimination of the reaching phase by forcing the state variables of the QUAV in the sliding mode in the initial time. the stability of these control schemes proposed in this chapter is guaranteed according the Lyapunov theory. Also, in order to show the effectiveness of the proposed control methods, various simulations with comparison are given. The Chap.5 offers several controllers with estimators of dynamic system parameters. Three adaptive control schemes for estimating time-varying parameters are presented, compared and finally merged. In this context, the interest of these algorithms is important. First of all, the need for an accurate estimation of the upper bound of these perturbations is essential to improve the flight safety of the QUAV in a perturbing environment. In addition, these control algorithms have a strong robustness against time-varying uncertainties, non-linearities and external perturbations, and have features such as simplicity and continuous control signals. Each approach proposed in this chapter is compared with robust controllers to show the superiority of these proposed techniques. Finally, an on-line estimation is performed for the upper limits of uncertainties and perturbations of the QUAV. Finally, simulation results show the validity of adaptive nonlinear SMC control scheme.

1.6 Structure of the Book

13

In Chap. 6, the problem of the elimination of the reaching phase and reduction of the control effort in the initial time are addressed. In addition, under the aerodynamics perturbations, noise measurements, and uncertainties, the design of the control method becomes more difficult for QUAV to satisfied these performances. In this view, global time-varying SMC with an online estimator are designed for the QUAV. Using the proposed sliding manifolds, the initial control efforts of the QUAV inputs are eliminated. Also, the stability of the tracking errors is obtained with specific time. Finally, the effectiveness of the adaptive global time varying SMC is demonstrated by simulation results. According to the research in Chap. 3, a robust control scheme is proposed for the QUAV. The Chap. 7 presents a combination of the STA, FO switching law, and FOPID-SMC techniques to solve the path following problem in the presence of complex disturbances. The problems of time-varying loading and changes in drag coefficients are addressed in this chapter. The FO switching control law is designed against the disturbances, STA is given to compensate the unknown complex disturbances caused by wing gust. Various numerical simulations are conducted to validate the tracking performance of this FO control scheme. In Chap. 8, FO global SMC (FOGSMC) control scheme is suggested, a designed FOGSMC and the prescribed performance control for the QUAV with system uncertainties and external disturbance. Firstly, the global sliding mode manifolds are designed for the position and attitude of the QUAV. The use of these sliding mode manifolds fast tracking performance and high accuracy of attitude and position tracking are provided. A generic FO switching control law is proposed to compensate the negative influence of parametric uncertainties and disturbance on the QUAV control performance. The Lyapunov theory is used to prove the stability of the proposed control method. Finally, under the complex flight trajectories, multiple simulation results are given to validate the theorical performance. Based on the research in Chaps. 5, 7, and 8, the Chap. 9 proposes an adaptive FO nonsingular fast terminal sliding mode (AFONFTSM) control scheme for the QUAV under Random disturbances/parametric uncertainties. The designed sliding mode variables are based on FO operators to achieve high accuracy and fast speed of the QUAV under the influence of the perturbations. The tracking errors of the translational and rotational subsystems are converged in the finite-time using the AFONFTSM method. In addition, based on velocity and position the tracking errors, an adaptive law is designed to reject the upper bound of disturbances/uncertainties. Finally, the effectiveness of the AFONFTSMC is shown under different situations by simulation results. The Chap. 10 proposes the design and construction of a test bench to set up and check the 4 DOF flight controller. This test bench is used to evaluate altitude and attitude stability (roll, pitch and yaw angles). It is based on a dual-core F28379D DSP microcontroller via serial communication with a host computer. Some experimental results are given in this chapter. In Chap. 11, several future research directions are predicated.

14

1 Introduction

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Chapter 2

QUAV Modeling

2.1 Introduction To validate and test the efficiency of the different control laws in order to stabilize the quadrotor in a wind field, accurate modeling is important, taking into account the wind speed and the external disturbances acting on the vehicle. For this reason, this chapter deals with the modeling of the elements constituting the dynamics of the quadrotor. In addition, a reminder of some mathematical formulas, such as the definitions of non-integer order differentiation, approaches to approximation and simulation of a fractional drifter, stability conditions using the Lyapunov theory. It is organized as follows: In the Sect. 2.2, a brief review is presented showing the progress of the modeling. Some important preliminaries are presented in the Sect. 2.3. The forces and moments applied to multi-rotor systems are detailed in Sect. 2.4. Wind generation and its interaction with the vehicle cell are modeled and presented in Sect. 2.5. Some technical simplifications for implementation are presented in Sect. 2.6. The formulation of the integral and non-integer derivative operators is presented in Sect. 2.7. Finally, some concluding remarks are made in Sect. 2.8.

2.2 Review of Multirotors Modeling The quadrotors are highly non-linear and coupled systems. In addition, uncertainties, caused by the environment and induced by aerodynamic phenomena and disturbances, make the modeling task more difficult. However, simplified models can be used to approximate the vehicle dynamics requiring effective control strategies. In fact, many models have been proposed in the literature, with varying levels of complexity and completeness. Based on different works in the literature, different hypotheses are considered in the modeling of a quadrotor, [1–3]. Drone modeling based on the Lagrange-Euler and Newton-Euler formalisms is presented in [4]. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_2

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basic concepts of helicopter aerodynamics are illustrated in [4]. The authors of [5], treat the aerodynamic coefficients of the rotating blades in hovering flight, the thrust and drag are assumed to be proportional to the square of the rotation speed of the propellers in [5]. Due to the low translation and angular velocities, friction forces and moments are neglected in [6]. A model considering the aerodynamic coefficients in forward and vertical flight is presented in [7] at higher speeds. The majority of the works assume that the origin of the coordinate system coincides with the center of mass of the vehicle as [8]. The authors of [9] have derived a dynamic model for a small quadrotor called Mesicopter. This model was developed as one of the first proposed models used for hovering. The model was created assuming that the dynamics are decoupled along the two planes of symmetry. It was called a 2-D dynamic model. However, the authors explicitly gave the expression of some aerodynamic forces (mainly thrust and hub). In [10], a dynamic model for the quadrotor in X configuration has been proposed. It was a simplified model where the attitude dynamics is modeled by dual integrators. The representation of the Euler angles was chosen to describe the rotations by considering the East-North-Up (ENU) frame standard. The reference [11] proposes a compact model based on Newton’s equations for the motion of a rigid object where several aerodynamic forces and moments are neglected as the drag force. The authors considered the North-East-Down (NED) convention. The reference [12] presents the dynamic model of a crossed quadrotor, including the blade flapping effect in addition to most of those mentioned above, which used EulerLagrange’s formalism as well as Newton-Euler’s approach to derive the dynamic model and which is closest to the real vehicle. However, the various effects are not well defined in the appropriate frameworks. In addition, the authors considered only the roll and pitch flapping moment assuming that the thrust forces are collinear with the bz axis of the vehicle. A rich literature shows considerable progress in modeling dealing with other aspects such as the non-rigidity of the propellers [13], the ground effect [14], etc. For a more in-depth knowledge, the reader can refer to [15] for example and the references it contains. Despite the interest shown by the community for modeling, few articles describe the interaction of wind with a quadrotor even though many wind models exist. It is generally considered as an unknown limited disturbance or a constant component, which is handled by a robust controller. Then, a dynamic model, describing the quadrotor flying under the effect of the wind, is proposed. This model is a generic representation for all existing multirotor typologies [16]. It is selected in order to better represent the dynamic behavior of quadrotors in the presence of wind disturbances. The modeling given in this book is based on different research work [16, 17]. Moreover, the modeling part is very important for quadrotors. However, the same established models are valid for other forms of multirotors. For external flight, this modeling is important to validate the control algorithms. In the design of these controllers, some works use simplified models in order to obtain simple control laws. To approach the reality, a quadrotor model in the presence of perturbations and parametric uncertainties will be presented in this chapter.

2.3 Preliminaries and Frame Representation

21

2.3 Preliminaries and Frame Representation Modeling the quadrotor system requires knowledge of the aerodynamics, mechanical characteristics and dynamics of the actuators. Then, the modeling step contains different levels, including vehicle motion, propeller aerodynamics and actuator dynamics. In the first part, the equations of motion of an air vehicle are presented, and can be divided into the following two sets: • Kinematic equations: give the position and orientation of the vehicle in relation to the frame fixed to the ground. • Dynamic equations: connect  the external forces F ext affecting the vehicle to the translational accelerationa ( Fext = ma) and the external moments M ext at rotational accelerations ( mathcal Mext = I dot ) where m is the mass and I is the inertia of the vehicle. The drone used in this work is a nano quadrotor shown in Fig. 2.1. This machine is a rigid body equipped with four rotors. The movement of the quadrotor is controlled by adjusting the speed of the rotors. The yawing motion is obtained by creating a difference in speed between two rotors in the opposite direction. In fact, between the rotors (2, 4) and (1, 3). The vertical movement of the quadrotor is obtained by decreasing or increasing the total speeds of the rotors. The forward movement is obtained by modifying the speed of rotation of the propeller (1 and 3). Lateral movement is obtained by modifying the speed of rotation of the propeller (2 and 4). This vehicle operates in two coordinate frames: the inertial reference frame linked to the earth E(Oe , ex , e y , ez ) and the frame linked to the quadrotor B(Ob , bx , b y , bz ) (as shown in Fig. 2.1). E is considered to be an inertial frame of reference, in which Newton’s and Lagrange’s laws are valid (the rotation speed of the earth is neglected.) and B is fixed to the vehicle and constrained to move with it when the vehicle is considered to be a rigid body. Consequently, the body is intrinsically unstable; its

Fig. 2.1 Quadcopter configuration with earth and body frames

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mechanical system is strongly coupled and under-actuated. Moreover, in this thesis, the dynamic model of the quadrotor is obtained using the following assumptions: Assumption 2.1 The structure and the propellers are rigid and perfectly symmetrical. Assumption 2.2 The dynamics of the rotors is relatively fast and will be neglected. Assumption 2.3 The influence of the soil is neglected. Assumption 2.4 Torque (u  , u (t) , u  ) and thrust u m , produced by the rotor speeds, are proportional to the square of the rotor speeds. The bx axis is directed towards the front of the quadrotor, the z axes are chosen to be directed upwards, the y axes are directed to the left (see Fig. 2.1). Thus, the East-North-Up standard is adopted. The origin of the O B frame attached to the body is an invariant point and belongs to the vehicle structure. To simplify, it is chosen to be the center of gravity (CoG) of the vehicle. Note that Tait-Bryan angles (also called gimbal angles or nautical angles.) are adopted. Thus, roll is chosen to be around the bx axis, pitch is chosen to be around b y and finally yaw is chosen to be around the bz axis. The roll, pitch and yaw angles ((t), (t), (t)) are commonly called Euler angles. Since the different coordinate frames are defined, the relationship between them, i.e. the coordinate transformation, must be found. Starting from the fixed frame of the body to the fixed frame of the earth, the sequence of rotations chosen is: the sequence x(t) −→ y(t) −→ z(t), which means that the attitude is obtained first by the roll angle “(t) , then by the pitch angle “(t) and finally by the yaw “  . Therefore, the rotation matrix R((t), (t), (t)) is given by R((t), (t), (t)) = R(z(t), (t))R(y(t), (t))R(x(t), (t)) (2.1a) ⎤⎡ ⎤ ⎡ ⎤⎡ 1 0 0 C(t) −S(t) 0 C(t) 0 S(t) (2.1b) = ⎣ S(t) C(t) 0⎦ ⎣ 0 1 0 ⎦ ⎣0 C(t) −S(t) ⎦ 0 S(t) C(t) 0 0 1 −S(t) 0 C(t) ⎡ ⎤ C(t) C(t) S(t) S(t) C(t) − C(t) S(t) C(t) S(t) C(t) + S(t) S(t) = ⎣ C(t) S(t) S(t) S(t) S(t) + C(t) S(t) C(t) S(t) S(t) − S(t) C(t) ⎦ −S(t) S(t) C(t) C(t) C(t) (2.1c) R is an orthonormal matrix, so that R ∈ S O(3) = [R ∈ R3×3 |RT R=I3×3 , det (R) = 1]. The index 3 × 3 means that the dimension of the matrix is 3 × 3 and I3×3 is the identity matrix of dimension 3. s(.) and c(.) are the abbreviations of sin(.) and cos(.) respectively. Conversely, to go in the opposite direction, i.e. from the land frame to the body frame, you have to follow the sequence yaw, pitch and roll. All orientations of the vehicle can be achieved using the three elementary rotations (x yx, x zx, yzy, etc.). We note that there is a very useful mathematical tool to represent this rotation, it has great advantages over the representation of Euler angles more commonly used because of, for example, the absence of singularities and the mathematical simplicity

2.3 Preliminaries and Frame Representation

23

[16]. Based on the previous description, the transformation between the earth bound and the vehicle body bound benchmark can be expressed explicitly using the rotation matrix R. Thus, the velocity vector VB (t) = (u(t), v(t), w(t))T ∈ R3 of the vehicle, expressed in a fixed frame of the body, can be rotated in the fixed frame of the earth as follows χ˙ (t) = R((t), (t), (t))VB

(2.2)

where χ˙ (t) = (x(t), ˙ y˙ (t), z˙ (t))T ∈ R3 is the velocity vector of the quadrotor in E. A rotation sequence x(t) −→ y(t) −→ z(t) is selected for the Euler angles. This means that the frame attached to the body is first rotated around the x(t) axis. Therefore, the ˙ angular velocity (t) is the same as the angular velocity p(t) of the vehicle around ˙ on the y(t) axis is influenced by the first step of rotation bx . The angular velocity (t) around the x(t) axis. Therefore, this speed must be multiplied by the rotation matrix ˙ R.(x, ). Finally, the rotation (t) around the z axis must be multiplied by the two rotation matrices R(x(t), (t)) and R(y(t), (t)). Therefore, the angular velocities ( p(t), q(t), r (t)) in the fixed frame of the body can be written according to Euler’s law of angular velocities as ⎤ ⎡ ⎤ ˙ p(t) (t) −1 ⎣ q(t) ⎦ = W ⎣(t) ˙ ⎦ ˙ r (t) (t) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ˙ 0 0 (t) ˙ ⎦ + RT (x(t), (t))RT (y(t), (t)) ⎣ 0 ⎦ = ⎣ 0 ⎦ + RT (x(t), (t)) ⎣(t) ˙ 0 0 (t) ⎤ ⎡ ⎤⎡ ˙ (t) 1 0 −S(t) ˙ ⎦ ⎣0 C(t) S(t) C(t) ⎦ = ⎣(t) ˙ 0 −S(t) C(t) C(t) (t) ⎡

(2.3a) (2.3b) (2.3c)

The relation between the angular velocities  = ( p(t), q(t), r (t))T ∈ R3 of the T ˙ ˙ ˙ quadrotor in the fixed frame and the angular velocities η(t) ˙ = ((t), (t), (t)) ∈ R3 in the frame fixed to earth is presented as follows: ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ˙ (t) 1 S(t) tan(t) C(t) tan(t) p p(t) ⎣(t) ˙ ⎦ = W ⎣ q ⎦ = ⎣0 C(t) −S(t) ⎦ ⎣ q(t) ⎦ ˙ r r (t) 0 S(t) sec(t) C(t) sec(t) (t) ⎡

(2.4)

Obviously, the representation of Tait Bryan’s angles suffers from certain peculiarities: (t) = ±π and (t) = ±π . In practice, this limitation does not affect the quadrotor in normal flight mode. The Eqs. (2.2) and 2.3 represent the kinematic model of the quadrotor.

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2.4 Mathematical Modeling In this section, the differential equations, which link the output of the system (position and orientation) to its input (forces and torques), are derived. The dynamics of the quadrotor can be formulated by the Newton-Euler and Lagrange-Euler methods. Based on the work developed in Refs. [9, 16, 18, 19], the dynamics of the quadrotor is obtained. In the first part, the Newton-Euler equations for a rigid body system are established. Then, the different forces and moments applied to the vehicle are presented.

2.4.1 Flight Modeling of the Quadrotor with Newton-Euler’s Formalism The rigid body has six degrees of freedom, it has a mass m and an inertia I ∈ R3×3 around the center of gravity. Let VB (t) = (u(t), v(t), w(t))T ∈ R 3 represent the linear velocity of the center of gravity and as defined in the previous section (t) = ( p(t), q(t), r (t)) ∈ R 3 is its angular velocity expressed in the fixed frame of the body, η(t) = ((t), (t), (t))T ∈ R 3 its orientation (Roll, Pitch, Yaw) and let χ (t) = (x(t), y(t), z(t))T ∈ R 3 its absolute position with respect to E. The relation between velocities and external forces F B = (FxB , FyB , FzB )T ∈ R3 and the moments M B = (MxB , M yB , MzB )T ∈ R3 , applied to CoG, expressed in the frame B, can be written using the Newton-Euler formalism as:         × mVB m I3×3 O3×3 V˙ B FB ˙ +  × I  = MB O3×3 I 

(2.5)

where O3×3 is a zero matrix of dimension 3 × 3 and × designates the cross product. On the basis of the symmetry property claimed by assumption 1 and with the appropriate choice of the fixed frame of the body as represented in Fig. 2.1, the inertia matrix becomes the diagonal I = diag(Ix x , I yy , Izz ). Explicitly, in terms of translational and angular acceleration, the system (2.5) is written for a multi-rotor vehicle as: ⎡ ⎤ ⎡ ⎤ ⎡ B⎤ Fx u(t) ˙ r (t)v(t) − q(t)v(t) ⎣ v˙ (t) ⎦ = 1 ⎣ p(t)w(t) − r (t)u(t)⎦ + 1 ⎣F yB ⎦ (2.6) m q(t)u(t) − p(t)v(t) m w(t) ˙ FzB ⎤ ⎡ ⎤ ⎡ B⎤ Mx Ix x p(t) (I yy − Izz )q(t)r (t) ˙ ⎣ I yy q(t) ˙ ⎦ = ⎣ (Izz − Ix x ) p(t)r (t) ⎦ + ⎣M yB ⎦ Izz r˙ (t) (Ix x − I yy ) p(t)q(t) MzB ⎡

(2.7)

2.4 Mathematical Modeling

25

The next step is to determine the forces F and moments M.

2.4.2 Aerodynamic Forces and Moments Applied to the Quadrotor The movement of multirotors is subject to various forces and moments from different sources. Here we provide a global and generic model for a large class of multirotors. All the recurrent parameters used in this sub-section are given in Table 2.1 where the explicit formulations of the aerodynamic coefficients Ci |i=T,H,Q,B are given in [18]. In the previous part, the Newton-Euler equations (2.5) for a rigid body system were presented, which are suitable to describe any system with several rotors. The main difference lies essentially in the various aerodynamic forces and moments which are defined as a function of the rotor speeds by the shape of the vehicle and its number of rotors. Before presenting the main external forces and moments, an overall description of the multi-rotor vehicles considered was given. Multirotor vehicles have practically the same principle of flight. They differ by the location and size of the arms and the number of rotors. Let  O = {r oi , i = 1, . . . , 2Nr } the set of rotors where Nr ∈ N is the number of pairs of rotors (propellers) with 2Nr 4 i.e. the multirotors considered are those with an even number of rotors. Each rotor r oi |i=1,...,2Nr located at oi |i=1,...,2Nr in the same plane as the CoG, is supported by an arm of length li |i=,...,2Nr and rotates around an axis, which is parallel to the axis bz (the rotors are not tilted). Thus, a rotor r oi is defined by the polar coordinates oi (li , α) ∈ R × [0, 2π ] where αi designates the angle between its arm and the axis bx , the speed of rotation i and the direction of rotation where Nr rotors rotate clockwise (S p = 1) while the others rotate counterclockwise (S p = −1) (see the representation of a rotor on Fig. 2.2. For reasons of symmetry, each pair of rotors is placed on two opposite sides of the cell αi+Nr = αi + π |i=1,...,Nr ) with the same arm length (li = Nr + i|i=1,...,Nr ) . This description is valid for a large class of multi-rotor vehicles. It provides a global expression of the aerodynamic forces and moments as a function of the rotors speeds.

Table 2.1 Modeling of recurring parameters Parameter Description A

Parameter

Description

Rr

Propeller radius

ρ

Effective area of the propeller disc Air Density

Ci |i=T,H,Q,B

Jr

Rotor inertia



Aerodynamic coefficients Rotor rotation speed

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Fig. 2.2 Description of the rotor

Table 2.2 External forces applied on a multi-rotor Force − Thrust Hub forces

Ti Hi

Aerodynamic forces

Fd

Gravity

g

Direction According to the axis bz In the plan bx , b y , According to the direction of the linear speed In the space bx , b y , bz , Depending on the direction of the linear speeds According to the axis ez

Forces Using the theory of blade elements, the expressions for the aerodynamic forces are deduced. Also, neglecting the ground effects, four main forces acting on the quadrotor are considered. These expressions are used in modeling. The three forces mentioned in this subsection and their directions are shown in the Table 2.2. All forces are expressed as part of the body. The first is the force of gravity: G = −mg

(2.8)

It is along the ez axis in the negative direction where m is the mass of a quadrotor, g is the gravity coefficient. In addition, each rotor r oi produces thrust and hub forces, which depend on its angular velocity . The thrust force Ti is along the axis bz in the positive direction while the hub force Hi is in the plane bx b y in the negative direction of the horizontal speed Vh (the projection of the forward speed V f in the plane bx b y ). Thus, the hub force can be broken down into two components Hxi on the bx axis and Hyi on the b y axis in the bx b y plane in the frame attached to the body.

2.4 Mathematical Modeling

27

Figure 2.2 presents a graphical representation of these forces, for a rotor. The analytical expressions of the thrust and hub forces for a rotor are given below [16, 18]. Ti = C T ρ A(i Rr )2 ,

Hi = C H ρ A(i Rr )2

(2.9)

ρ = 1.293 kg/m3 is the air density, A is the effective area of the propeller disc and Rr its radius. The coefficients of thrust and hub force are [9]: 1 1 tw 1 C T = σ a[( + μ)0 − (1 + μ2 ) − λr ] 6 4 8 4 1 ¯ 1 tw C H = σ a[ μCd + λr μ(0 − )] 4a 4 2

(2.10a) (2.10b)

where a is the lift slope, 0 and tw are the linear lift parameters, Cd is the drag coefficient, σ is the strength ratio that defines the ratio of the blade surface to the disk surface. The solidity is defined as [18]

σ =

N c¯ πR

(2.11)

where N is the number of blades, c¯ is the blade chord. More details on the parameters can be found in [9]. The total thrust and the hub are the sum of the forces generated by each rotor. As the multirotor has rotors of 2Nr , the forces acting on the multirotor are [16, 18]: T =

2Nr i=1

Ti ,

Hx = −

2Nr i=1

Hxi ,

Hy = −

2Nr

Hyi

(2.12)

i=1

The resultant of the forces produced along the axes (x, y, z) is given by the following relation: ⎡ ⎤ Kdx 0 0 Fd = ⎣ 0 K dy 0 ⎦ χ˙ (2.13) 0 0 K dz where K d x , K dy and K dz indicate the translational drag coefficients. Once the main forces have been examined, the net external force vector acting on the vehicle is expressed in the frame attached to the body, using the rotation matrix (2.1) as [16, 18].

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Table 2.3 The external forces applied on a multi-rotor Moment Notation Roll moment

M

Pitching moments

M(t)

Yaw moment

Qi

Aerodynamic moments

Md

Gyroscopic moment

mathcalG x , mathcalG y

Blade beat moment

Bx , mathcal B y

Direction On the bx axis in the frame of the body On the b y axis in the frame of the body On the axis bz in the frame of the body On the bx , b y , bz axes in the frame of the body On axes bx and b y in the frame of the body On the bx and b y axes in the frame of the body

⎡

FB

⎡ ⎤ ⎤ Hxi 0 = ⎣ Hyi ⎦ + RT ⎣ 0 ⎦ G T ⎤ ⎡ ⎤ ⎡ 2Nr ⎤ ⎡ − i=1 Ti 0 Kdx 0 0 2Nr 1 = ⎣− i=1 Hxi ⎦ + ⎣ 0 K dy 0 ⎦ + RT ⎣ 0 ⎦ 2Nr m −mg 0 0 K dz − i=1 Hyi

(2.14)

(2.15)

where mathcal R −1 = mathcal R T is the rotation matrix from the fixed frame of the earth to the fixed frame of the body. Moments In this subsection we present the main external moments that express themselves within the body and their directions. Using the blade element theory, Fay also explicitly deduced the expressions for the aerodynamic moments. These moments are used in the modeling. The external moments mentioned in this sub-section and their directions are given in the Table 2.3. All moments are expressed in the body frame. Roll and pitch moments The roll and pitch moments, M and M(t) , about the bx and b y axes respectively, are obtained by the difference in combined thrusts on opposite sides of the vehicle. They are among the most important moments because they contribute to the control of the vehicle. The direction of the moments is decided according to the right-hand rule. They are defined as [16, 18] M =

2Nr i=1

li sαi Ti ,

M(t) =

2Nr i=1

li cαi Ti

(2.16)

2.4 Mathematical Modeling

29

where s(.) and c(.) are abbreviations of sin(.) and cos(.) respectively and αi is the angle between the arm, length li , supporting the rotor r oi and the axis bx .

Yaw moment The aerodynamic forces acting on the blade elements cause a moment Q i around the rotor shaft. The sum of these moments, considering all the rotors, induces the rotation of the multirotor around the axis bz . The yaw moment Q i for each rotor is modeled as follows, Q i = C Q ρ A Rr (i Rr )2

(2.17)

The yaw moment coefficient is expressed as follows: C Q = σ a[

1 1 1 1 (1 + μ2 )C¯ d + λr ( 0 − tw − λr )] 8a 6 8 4

(2.18)

There are 2Nr rotors which are separated by different directions of rotation (clockwise and counterclockwise) inducing moments in the different direction. The net yaw moment is given by: Q=

2Nr

S pi Q i

(2.19)

i=1

where S pi = 1 if r oi turns clockwise and Spi = −1 if r oi turns counterclockwise. The moment of the center As discussed previously, the hub forces have two components along the bx and b y axes. These hub forces exert moments on the vehicle around the axis bz . These two components contribute to the movement according to their location oi |i=1,...,2Nr . Therefore, the resulting moment of the hub is given by [16]. H = Hx + H y =

2Nr i=1

li sαi Hx +

(2.20) 2Nr i=1

li cαi H y

(2.21)

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Fig. 2.3 Aerodynamic phenomena

Gyroscopic moment Rotations about two axes cause a third rotation about a third axis, which is perpendicular to the plane formed by the first two axes (see Fig. 2.3). This effect is called the gyroscopic effect. Therefore, when the multirotor rotates on the bx axis, a moment on the b y axis is created on each rotating rotor. When the multirotor rotates on the b y axis, each rotor also has a moment on the bx axis. The total gyroscopic components are: 2Nr 2Nr S pi i , G y = −Jr p S pi i (2.22) Gx = Jr q i=1

i=1

where Jr is the inertia of a rotor. Flapping moment of the blade During translational flight, there is a difference in lift between the advancing and retreating blades. This effect is called blade flapping. The difference in lift applies a moment to the rotor disk (see Fig. 2.3). The flapping moment Bi is about an axis perpendicular to the plane formed by the rotor shaft and the forward speed of the multirotor. It is given by: Bi = CB ρ A Rr (i Rr )2

(2.23)

For more convenience, the flapping moments of each rotor B can be separated into moments Bxi and B yi . The total flapping moment of the blades of a multi-rotor is:

Bx =

2Nr i=1

S pi Bxi ,

By =

2Nr

S pi B yi

(2.24)

i=1

The resultant torques due to gyroscopic effects and the resultant aerodynamic friction torque, the expressions are given by

2.4 Mathematical Modeling

31

⎤ K d 0 0 Md = ⎣ 0 K d 0 ⎦ 2 0 0 K d ⎡

(2.25)

where K d , K d , and K d are positive orientation drag coefficients. Once the different moments affecting the multirotor are presented, the total moment vector is presented in the body-related frame as follows [16, 18]. ⎡

⎤ M + Gx + Md + Bx M B = ⎣M(t) + G y + Md + B y ⎦ Md + Q + H 2Nr 2Nr ⎤ ⎡ 2Nr 2 i=1 li sαi Ti + Jr q i=1 S pi i − K d  + i=1 S pi Bxi 2N 2Nr 2Nr = ⎣ i=1r li cαi Ti − Jr p i=1 S pi i − K d 2 + i=1 S pi B yi ⎦ 2Nr 2Nr 2Nr 2 i=1 S pi Q i − K d  + i=1 li sαi Hxi − i=1 li cαi H yi

(2.26)

(2.27)

2.4.3 Full Multi-rotor Dynamic Model In the previous sections, the Newton-Euler equations for a rigid body have been deduced. Then, the important external forces and moments are analyzed. In this section, the complete model of a multi-rotor system will be proposed. In addition, the control problems are studied taking into account the motion of the vehicle around its CoG and the relative motion of the CoG with respect to the inertial reference E. Therefore, the travel speeds and accelerations are defined in the E frame fixed to the earth, while the angular velocities and accelerations are defined in the B frame fixed to the vehicle body. This simplifies the vehicle dynamics. From the Eqs. (2.2), (2.5)– (2.7), (2.14) and (2.26), et les travaux de recherche [16, 19], the complete dynamic model of a multirotor is written ⎤ ⎡ ⎤ ⎤⎡ C(t) C(t) S(t) S(t) C(t) − C(t) S(t) C(t) S(t) C(t) + S(t) S(t) x(t) ¨ u(t) ⎣ y¨ (t)⎦ = ⎣ C(t) S(t) S(t) S(t) S(t) + C(t) S(t) C(t) S(t) S(t) − S(t) C(t) ⎦ ⎣ v(t) ⎦ z¨ (t) w(t) −S(t) S(t) C(t) C(t) C(t) ⎡

(2.28a)

⎤ ⎡ C S S C − C(t) S(t) C(t) S(t) C(t) + S(t) S(t) C 1 ⎣ (t) (t) (t) (t) (t) = C(t) S(t) S(t) S(t) S(t) + C(t) S(t) C(t) S(t) S(t) − S(t) C(t) ⎦ m −S(t) S(t) C(t) C(t) C(t) ⎡ 2N ⎤ ⎤ ⎡ ⎤ ⎡ r H − 0 Kd x 0 0 xi 1 ⎢ i=1 ⎥ T 2Nr ˙ ⎣− i=1 H yi ⎦ + R ⎣ 0 ⎦ + ⎣ 0 K dy 0 ⎦ χ(t) m 2Nr −mg 0 0 K dz − i=1 Ti ⎡ ⎤⎡ ⎤ ⎤ ⎡ ˙ p(t) (t) 1 S(t) tan(t) C(t) tan(t) ⎣(t) ˙ ⎦ = ⎣0 C(t) −S(t) ⎦ ⎣ q(t) ⎦ ˙ r (t) 0 S(t) sec(t) C(t) sec(t) (t)

(2.28b) (2.28c)

(2.28d)

32

2 QUAV Modeling

Fig. 2.4 Rotor model

⎤ 2Nr i=1 li sαi Ti + Jr q(t) i=1 S pi i (t)  ⎥ ⎢ 2Nr −K d 2 (t) + i=1 S pi Bxi ⎥ ⎤ ⎡ ⎤ ⎢ ⎡ ⎥ ⎢ 2Nr 2Nr ˙ I x x p(t) (I yy − Izz )q(t)r (t) ⎥ ⎢ l c T − J p(t) S  r i αi i pi i i=1 i=1 ⎥ ⎣ I yy q(t) ˙ ⎦ = ⎣ (Izz − I x x ) p(t)r (t) ⎦ + ⎢  2Nr ⎥ ⎢ 2 −K d  (t) + i=1 S pi B yi ⎥ ⎢ Izz r˙ (t) (I x x − I yy ) p(t)q(t)  ⎥ ⎢2Nr 2N r 2 ⎣ i=1 S pi Q i − K d  (t) + i=1 li sαi Hxi ⎦ 2Nr − i=1 li cαi H yi ⎡

2Nr

(2.28e)

The translational dynamics and the rotational dynamics are presented respectively to the Eq. (2.47) which are expressed respectively in the fixed frame of the earth and the fixed frame of the body. Based on this model to test, analyze and compare different control strategies afterwards. This mathematical representation of the vehicle is necessary to understand their overall dynamics. This modeling part expresses the main effects acting on the multi-rotor for different flight conditions. Moreover, modeling can be a good test bench to test, analyze and identify physical phenomena. In order to test the proposed control algorithms in this work, some simplifications can be considered for various purposes.

2.4.4 Rotor Dynamics The rotors are driven by direct current motors which connect the electrical and mechanical quantities. As in [25], the rotors can be represented as shown in the Fig. 2.4, and described by the following dynamic equations L r ot

dir ot = u r ot − Rr ot ir ot − K e ω, dt

Jr ot

dω = τm − τd , dt

where u r ot is the input of the rotor, Rr ot is the internal resistance of the rotor, L r ot is the inductance of the rotor, K e is the constant EMF (Force Electro Magnetic) rear of the rotor, tau r is the torque of the rotor, ω is the angular speed of the rotor, τd is the load of the rotor, Jr ot is the moment of inertia of the rotor, ir ot is the current. Considering small motors with very low inductance, the given system can be simplified and approximated by linearization around an operating point ω0 , then the

2.4 Mathematical Modeling

33

system takes the form of ω˙ = Aω + Bu wor d + C, where A, B, C are the parameters of the rotors, and it can be described in more detail by a transfer function G(s) =

1 , bs + 1

(2.29)

where b is the time constant of the rotors to be identified. The value of parameter b is small for small rotors with a small time delay, and reciprocally.

2.5 Model of the Vehicle Flying in a Gust of Wind One of the objectives of this work is to provide robust controllers for a multi-rotor vehicle flying under an environment totally disturbed by a gust of wind. In this context, particular attention is paid to wind modeling. Subsequently, the motion of the vehicle in a gusty environment was described. In order to do so, we first present a model that can mimic as much as possible the real behavior of a wind gust. Then, the detail, of the expressions of forces and moments acting on the multirotor, will be given. Once the model of the system is presented, the application of control and estimation theories on the UAV system will be made.

2.5.1 Modeling of Wind Gusts In this part, the wind gust is represented by its speed, which is considered as the sum of two components: a deterministic component and a stochastic component. The first component represents the average wind speed, which is set according to the daily use situation. It can be initialized with the weather forecast and is assumed to vary on a time scale slightly larger than the vehicle’s flight time scale. The second component represents the wind gust turbulence which is superimposed on the mean wind when data is collected from statistical properties such as spectra, scale length, turbulence intensity [9]. VwB = [u w vw ww ]T ∈ R3 designates the wind speed vector expressed in the fixed frame B as: VwB = vwB + δVwB ⎡ ⎤ Uw + δu w = ⎣ Vw + δvw ⎦ Ww + δww

(2.30) (2.31)

where vwB = [Uw Vw Ww ]T represents the mean wind speed vector expressed in the fixed body frame (B). The term δV B = [δudδvdδwd]T represents the deviations

34

2 QUAV Modeling

from the mean wind speeds and represents gust turbulence. The wind speed the wind ¯ wB whereV ¯ wB ∈ R+ indicates the wind vector is supposed to be limited as VwB ≤ V speed limit for which the multirotor remains controllable [18]. It is usually specified by the manufacturer as a safety statement. u w = Gu Hu (s),

vw = Gv Hv (s),

ww = Gw Hw (s)

(2.32a)

where Gu , Gv and Gw are non-zero average white Gaussian noise signals that pass through the filters Hu (s), Hv (s) and Hw (s) responsively. The expression of these transfer functions is given by [4]:

Hu (w) = σu

1 Lu 2 , πU 1 + L u Us

√ L v 1 + 3L v Us Hv (w) = σv , πU (1 + L v Us )2 √

L w 1 + 3L w Us Hw (w) = σw πU (1 + L w Us )2

(2.33)

where s is the Laplace variable. The expressions of L u , for multirotor vehicles flying at low altitude ( 0).

2.7.2 Integration of Non-integer Order The calculation of derivatives and integrals of an analytical function is very important in the study and analysis of the behavior of dynamic systems. In this paragraph, we recall the definition of the non-integer order integral calculation. To introduce the calculation of integrals of non-integer order, we start by giving Cauchy’s formula for the repeated integration of an analytic function f (t) of a real variable t.

2.7 Formulation of the Integral and Derivative Operators of Non-integer Order

43

The Cauchy formula [20, 21] is defined by J f (t) :=

 t

n

a

a

τ1

 ... a

τn−1

1 f (τ ) dτ . . . dτ2 dτ1 = (n − 1)!



t

f (τ )(t − τ )n−1 dτ

a

(2.59) with n ∈ N, a, t ∈ R, a < t. If n is replaced by a real number (n − 1)! and the term (n − 1)! by its generalization which is the Gamma function, the Eq. (2.59) becomes a non-integer order integral. Definition 2.1 Suppose that α > 0, a < t, α, a, t ∈ R. The left-hand RiemannLiouville integral of a function f (t) for a non-integer order α is defined by  t 1 RL α I f (t) = f (τ )(t − τ )α−1 dτ , 0  t (2.60) a t (α) a

2.7.3 Differentiation of Non-integer Order After the introduction of the non-integer order integral calculus operator, it is reasonable to define also the non-integer order differentiation. There are different definitions in the literature, which in general do not coincide. We quote some of those existing in the literature, knowing that the two most used definitions are those of RiemannLiouville and of the operator defined by Caputo.

2.7.3.1

Definition of Riemann-Liouville

In this section we present the Riemann-Liouville definition for the calculation of the non-integer derivative. Definition 2.2 Suppose that α > 0, a < t, α, a, t ∈ R. The expression given by RL α a Dt

dn 1 f (t) = (n − α) dt n

 a

t

f (τ ) dτ , n − 1 < α < n ∈ N (t − τ )α+1−n

(2.61)

is the non-integer Riemann-Liouville derivative of order α of the function f (t).

2.7.3.2

Definition of Caputo

The definition of the non-integer order derivative introduced by Caputo [22] and [23], is given as follows: Definition 2.3 Suppose that α > 0, a < t, α, a, t ∈ R. The expression given by

44

2 QUAV Modeling C α a Dt

1 f (t) = (n − α)

 a

t

dn dt n

f (τ ) dτ , n − 1 < α < n ∈ N (t − τ )α+1−n

(2.62)

is the non-integer derivative of Caputo of order α of the function f (t). For some time, Caputo’s definition was considered advantageous for initial value problems.

2.7.3.3

Definition of Grünwald-Letnikov

The Grünwald-Letnikov derivative is an extension of the Euler method in non-integer calculus, it allows to take the derivative a non-integer number of times, where the coefficients are binomial coefficients of non-integer order, which are recursively defined and positive. Definition 2.4 Suppose that α > 0, a < t, α, a, t ∈ R. The expression given by 

t−a



  h 1 GL α j α (−1) f (t − j h) a Dt f (t) = lim α j h→0 h j=0

(2.63)

is the non-integer Grünwald-Letnikov derivative of order α of the function f (t). We note that the Grünwald-Letnikov definition is equivalent to the Riemann-Liouville definition.

2.7.4 Approximation of Non-integer Order Systems The main problem of the temporal simulation of non-integer systems is that the evaluation of the output requires the knowledge of the whole past of the system, which is however difficult to take into account.

2.7.4.1

Approximation Method for Non-integer Order Operators

The multitude of works on the approximation of non-integer order operators is due to the fact that the peculiarities of each fractional problem impose a different approach for its solution. The approximation by poles and recursive zeros “CRONE” The objective of this method is to approximate the non-integer integration or derivation operator by a frequency bounded integer model of finite dimension. This approx-

2.7 Formulation of the Integral and Derivative Operators of Non-integer Order

45

imation requires two steps, a step of frequency truncation of the non-integer integration or derivation operator, then a second step of approximation of the non-integer integration or derivation operator bounded in frequency by an integer model. The first step is to introduce the frequency bounded integration operator proposed in [24] by  α 1 + ωsh −α , −1 < α < 1 (2.64) s[ω A ,ω B ] = C 1 + ωsb where ωb < ωh and C is obtained in order to have a unitary gain in the center of the interval [ω A , ω B ]. ( jω)−α = C

1+ ωs

α

(2.65)

h

1+ ωs

b

√ where ωc = ωb ωh . The transitional pulses ωb and ωh are defined by ωA σ ωh = σ ω B ωb =

(2.66a) (2.66b)

where σ is generally set at 10. The widening of the frequency band [ω A , ω B ] allows to avoid the phenomenon of edge effects and to obtain a good approximation of s −α in [ω A , ω B ]. The second step is to approximate the irrational part of (2.64) by a recursive distribution of real zeros and poles [24]. α    Ns ! 1 + ωsk 1 + ωsh ≈ (2.67) 1 + ωsb 1 + ωs k=1 k

where Ns is the number of recursive poles and zeros. The recursiveness of the zeros and poles is characterized by the following relations ω ωk =  k = βη ωk+1 ωk+1 ωk =β ωk  ωk+1 =η ωk+1

(2.68a) (2.68b) (2.68c)

The recursive factors β and η are related to the derivation order α and defined by

46

2 QUAV Modeling

 β=  η=

ωh ωb ωh ωb

 Nα

s

(2.69a)  1−α N s

(2.69b)

The interest of this approximation method lies in its simplicity of implementation. If the order of derivation 1 < α (respectively order of integration α < −1) only the non-integer part is approximated by an integer order model. Remark 2.1 Unlike the approximations, the “CRONE” pole and re-cursive zero approximation can easily be extended to the case where the α order is complex. Such a derivation order is used only in very specific circumstances such as the robust “CRONE” order, in order to have more degrees of freedom.

2.8 Conclusion In this chapter, the dynamic modeling of multi-rotors has been provided. A six DOF rigid body model has been developed using the Newton-Euler formalism. Then, the main moments and forces acting on this vehicle were included. In addition, the effect of wind was discussed by presenting a model for wind gusts and its effect was incorporated into the main model of the multirotors. The resulting equations are nonlinear and therefore their direct application for the synthesis of control and estimation algorithms is complicated. To overcome this problem, some simplifications have been considered in order to develop relatively simple control laws for application purposes by adopting the hierarchical control architecture. The overall system of the quadrotor is rewritten as a state space to show the controls and wind influences on the quadrotor. Using this representation in state space, the control laws and the estimation of the algorithms are calculated in the following chapters. Finally, some definitions which seem to us essential to know in fractional computation have been presented. Afterwards, a short introduction of some applications has been proposed in a section. Then, a few approaches to approximate a non-integer derivative were recalled.

References 1. Norouzi Ghazbi, S., Aghli, Y., Alimohammadi, M., Akbari, A.A.: Quadrotor unmanned aerial vehicles: a review. Int. J. Smart Sens. Intell. Syst. 9(1), 309–333 (2016) 2. Xiaodong, Z., Xiaoli, L., Kang, W., Yanjun, L.: A survey of modelling and identification of quadrotor robot. Abs. Appl. Anal. 320526 (2014) 3. Quan, Q.: Introduction to Multicopter Design and Control. Google-Books-ID: 2017932434, isbn 978-981-10-3381-0 (eBook). Springer Nature Singapore Pte Ltd. (2017)

References

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4. Bouabdallah, S.: Design and control of quadrotors with application to autonomous flying. Ph.D. Thesis. EPFL (2007) 5. Bouabdallah, S., Murrieri, P., Siegwart, R.: Design and control of an indoor micro quadrotor. In: 2004 IEEE International Conference on Robotics and Automation. In: Proceedings. ICRA 04, vol. 5. Apr. 2004, pp. 4393–4398 (2004). https://doi.org/10.1109/ROBOT.2004.1302409 6. Mian, A., Wang, D.: Dynamic modeling and nonlinear control strategy for an underactuated quad rotor rotorcraft. J. Zhejiang Univ. Sci. A 9(4), 539–545 (2008) 7. Matko, O., Stjepan, B.: Influence of forward and descent flight on quadrotor dynamics. In: Recent Advances in Aircraft Technology, InTech (2012) 8. Mokhtari, A., Benallegue, A.: Dynamic feedback controller of Euler angles and wind parameters estimation for a quadrotor unmanned aerial vehicle. In: IEEE International Conference on Robotics and Automation, vol. 3, pp. 2359–2366 (2004) 9. Fay, G.: Derivation of the aerodynamic forces for the mesicopter simulation. https://www. scienceopen.com/document-vid=b3638dea-34c3-4033-b05b-ec69db015c1d (2001) 10. Altug, , E., Ostrowski, J.P., Mahony, R.: Control of a quadrotor helicopter using visual feed back. In: Proceedings 2002 IEEE International Conference on Robotics and Automation, vol. 1, pp. 72–77 (2002) 11. Hamel, T., Mahony, R., Chriette, A.: Visual servo trajectory tracking for a four rotor VTOL aerial vehicle. In: Proceedings 2002 IEEE International Conference on Robotics and Automation, vol. 3, pp. 2781–2786 (2002) 12. Castillo, P., Dzul, A., Lozano, R.: Real-time stabilization and tracking of a four-rotor mini rotorcraft. IEEE Trans. Control Syst. Technol. 12(4), 510–516 (2004) 13. Bristeau, P.J., et al.: The role of propeller aerodynamics in the model of a quadrotor UAV. In: 2009 European Control Conference (ECC), pp. 683–688 (2009) 14. Danjun, L., et al.: Autonomous landing of quadrotor based on ground effect modelling. In: 2015 34th Chinese Control Conference (CCC). pp. 5647–5652 (2015) 15. Bangura, M., Mahony, R.: Nonlinear dynamic modeling for high performance control of a quadrotor. In: Proceedings Australasian Conference on Robotics and Automation 2012. Australian Robotics and Automation Association (2012) 16. Bouzid, Y.: Guidance and control system for the navigation of autonomous air vehicles. Ph.D. Thesis. Paris-Saclay University (2018) 17. Bertrand, S.: Miniature rotary wing drone control. Ph.D. Thesis. Nice Sophia Antipolis University (2007) 18. Wang, J.: Quadrotor analysis and model free control with comparisons. PhD thesis. University of Paris Sud-Paris XI (2013). https://tel.archives-ouvertes.fr/tel-00952401/document 19. Perozzi, G.: Exploration sécurisée dun champ aérodynamique par un mini drone. Automatic Control Engineering. Ecole Centrale de Lille. English. NNT: 2018ECLI0007.tel-0210536 (2019) 20. Olsson, G., Newell, B.: Wastewater Treatment Systems: Modelling. Diagnosis and Control. IWA Publishing, London (1999) 21. Podlubny, I.: Fract. Differ. Equ. Acad. New York (1999) 22. Caputo, M.: Linear model of dissipation whose Q is almost frequency independent. Geophys. J. Roy. Astron. Soc. 13, 529–539 (1967) 23. Benzaouia, A., Hmamed, A., Mesquine, F., Benhayoun, M., Tadeo, F.: Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Trans. Autom. Control 59(8), 2203–2208 (2014) 24. Oustaloup, A.: La Dérivation Non Entière?: Synthése et Applications. Hermes, Paris (1995) 25. Bouabdallah, S. Becker, M., Siegwart, R.: Autonomous miniature flying robots: coming soon Research, Development, and Results. IEEE Robot. Autom. Mag. 14(3), 88–98 (2007). https:// doi.org/10.1109/MRA.2007.901323

Chapter 3

Stabilization of QUAV Under External Disturbances Using Modified Novel ST Based on Finite-Time SMC

The design of control laws for autonomous aerial vehicles is an important step. It is therefore essential to develop a suitable control algorithm for a quadrotor subjected to wind disturbances. During the sixties, a new type of disturbance-robust control laws under the name of sliding mode control, which is simple to compute, appeared. In this chapter, newly revisited control strategies have been studied and applied to the wind-influenced quadrotor, which do not seem, at least to our knowledge, to be given in the literature. They could be implemented without much difficulty and provide some improvements over existing control techniques. In the case of vehicle subjected to finite energy disturbances, these proposed control laws should minimize the effect of these disturbances on the tracking error. On the other hand, if the disturbances are considered as unknown inputs, the controllers proposed in this chapter must ensure the decoupling of the unknown inputs from the tracking error. Moreover, in the absence of the disturbances, these controllers must ensure the convergence to zero of the tracking errors. Then, each controller has been compared with existing controllers in the literature to show the efficiency of the approaches proposed in this chapter. The rest of this chapter is structured as follows. The basics of SMC was presented in Sect. 3.1. The design of the integral SMC (ISMC) is described in Sect. 3.2. Then, the combination of ISMC and a super-twisting (ST) algorithm is presented in Sect. 3.3. The simulation results with controller gains optimization are then presented in Sect. 3.4. A discussion is presented in Sect. 3.5. The modified super-twisting with NSMC for the quadrotor system is presented in Sect. 3.7. The simulation results of the modified ST-NSMC are exposed in Sect. 3.8. The conclusion of this chapter is given in the last Sect. 3.9.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_3

49

50

3 Stabilization of QUAV Under External Disturbances …

3.1 Premolars of the Sliding Mode Control 3.1.1 Sliding Mode Control This section is devoted to the presentation of the basic concepts of classical sliding mode control theory. Interested readers can refer to [1–4] for more details.

3.1.2 Basic Concepts The main objective of the sliding mode control is to force the system trajectories to reach in a finite time a defined sliding surface in the state space and then stay there. The sliding mode surface represents the desired dynamics of the system to be controlled. The synthesis of the sliding mode control law consists of: 1. determine a suitable sliding surface so that the system dynamics produce a desired behavior; 2. design a discontinuous control law that forces the system trajectories to reach the sliding surface and stay there despite uncertainties and disturbances.

3.1.3 Synthesis of the Sliding Surface Since the control objective is to ensure the tracking of a desired trajectory xd (t) by the vector x(t). The synthesis of the sliding surface is based on the definition of the sliding variable: T  s(t) = s1 (t), s2 (t), s3 (t), . . . , sn (t)

(3.1)

where si (t) are sufficiently differentiable functions considered as a fictitious output of the system such that their cancellation will satisfy the control objective. Definition 5 We say that there is an ideal sliding regime if there is a finite time Tc such that the solution of the system satisfies s(t) = 0 for all t  Tc . When the trajectories of the system evolve on the sliding surface, its dynamics becomes immersed in the state of a reduced system of dimension less than 1 to its relative degree. This reduced dynamic is determined by the choice of the sliding surface. The classical choice is a linear surface passing through the origin of the state space. The reduced system is therefore linear and converges exponentially to the origin of the phase plane (see Fig. 3.1) if it is stable (except if the reduced system is of order 0). A necessary condition for the establishment of a sliding regime is

3.1 Premolars of the Sliding Mode Control

51

.

e(t)

e(0)

Surface de glissement

Phase de convergence

Phase de glissement

e(t)

e(Tc)

Fig. 3.1 Phase plan of the sliding mode control

that the sliding surface has a relative degree equal to 1. The linear sliding surface is chosen as follows: s(t) =

d dt

+

r −1

e(t)

(3.2)

In our case, the relative degree r is equal to 2, the matrix  ∈ Rn × n is a diagonal matrix whose elements are chosen strictly positive so that s(t) is a Hurwitz polynomial [4, 5]. Thus, when the sliding surface is forced to zero, the tracking error e(t) converges asymptotically to zero with a desired dynamics imposed by the coefficients of the  matrix. Moreover, the relative degree condition is satisfied as long as the input u(t) appears in the expression of e(r ) (t) obtained directly after computing the first derivative with respect to time of the sliding surface s(t). Once the sliding surface is determined, the second step of designing the control law can be undertaken.

3.1.4 Design of the Control Law In order to achieve the objective of the control, the attractivity condition [3] ensuring the stability of s(t) = 0 must be verified. For this purpose, a class of quadratic Lyapunov functions are used: V =

1 s(t)T s(t) 2

(3.3)

this function is strictly positive ∀s(t) = 0. In order for the sliding surface to converge to zero, the derivative with respect to time of the Lyapunov function must be negative:

52

3 Stabilization of QUAV Under External Disturbances …

V˙ = s(t)T s˙ (t) < 0

(3.4)

However, this condition is not sufficient to ensure the convergence of the system trajectories to the sliding surface in a finite time. Therefore, a stronger condition must be verified. In the SMC case, a nonlinear attractivity condition called the K attractivity condition is given by: si (t)˙si (t) ≤ −K i |si (t)| ,

K i > 0|i=1,...,n

(3.5)

In other words, it comes down to choosing: s˙ (t) = −K sign(s(t))

(3.6)

where K = diag(K 1 , K 2 , . . . , K n ) is a diagonal matrix whose elements are chosen T  strictly positive, sign(s(t)) = sign(s1 (t)), . . . , sign(sn (t)) with ⎧ ⎨ 1, i f s(t) > 0 sign(s(t)) = 0, i f s(t) = 0 ⎩ −1, i f s(t) < 0

(3.7)

Using this law of convergence, si (t)v er ti=1,...,n reaches 0 in a finite time less than: Tc,i ≤

|si (0)| Ki

(3.8)

where si (0) is the initial value of the sliding surface. Furthermore, without considering uncertainties, s˙i (t) is calculated as follows: ∂s(t) ∂s(t) ∂ x(t) = x(t) ˙ ∂ x(t) ∂t ∂ x(t) ∂s(t) = [F (x(t)) + G(x(t))u(t)] ∂ x(t)

s˙ (t) =

(3.9)

Solving s˙ (t) = −sign(s(t)) using the previous equation, we obtain the sliding mode control law as follows: u(t) = −−1

∂s(t) F (x(t)) − −1 K sign(s(t)) ∂ x(t)

(3.10)

This command exists only if the matrix  = ∂∂s(t) G(x(t)) is invertible. Note that the x(t) control law u(t) in (3.10) is composed of two parts:

3.1 Premolars of the Sliding Mode Control

∂s(t) F (x(t)) ∂ x(t) u dis (t) = −−1 K sign(s(t)) u eq (t) = −−1

53

(3.11a) (3.11b)

The first part, u eq (t) is continuous and called the equivalent command. This command is responsible for the sliding phase which corresponds to the time interval t ∈ [Tc,i , ∞] during which the trajectories of the system are confined with the sliding surface. The behavior of the system during this phase depends only on the sliding surface. The second part u dis (t) is a discontinuous control. It is responsible for the convergence phase or the so-called reaching phase. It corresponds to the time interval t ∈ [0, Tc,i ] during which the trajectories of the system are not in the sliding surface but they converge there. Its duration depends strongly on the choice of the matrix K , the larger the elements of K are, the smaller the convergence time becomes and vice versa. In the literature, different structures of the discontinuous control are possible. We quote the most popular ones: Constant gain convergence law: s˙ (t) = −K sign(s(t))

(3.12)

where K is a diagonal matrix whose gains are strictly positive constants. The strong point of this law is the simplicity of the synthesis of the control law. However, if the gains chosen are too small, the convergence time will be very long. Moreover, the choice of large gains causes the phenomenon of reluctance. Convergence law with a linear state feedback: s˙ (t) = −K 1 s(t) − K 2 sign(s(t))

(3.13)

where K 1 and K 2 are diagonal matrices whose gains are strictly positive constants. This law allows a fast convergence when the value of the sliding surface is large. Law of exponential convergence: s˙ (t) = −K (s(t))sign(s(t))

(3.14)

where K (s(t))=diag N1Ks11(t) , NnKsnn(t) with K i > 0 = |i=1,...,n while the term Ni (si (t)) is defined in [6, 7] by : Ni (si (t)) = δ0i + (1 − δ0i )e − γi |si (t)| qi , 0 < δ0i )0i < 1, γi > 0, qi > 0 (3.15) The term Ni (si (t)) is defined as positive. This variation allows some adaptation. In other words, when the trajectories of the system converge towards the sliding surface, the value of |si (t)| becomes small which makes Ni (sKii(t)) −→ K i while when the value

54

3 Stabilization of QUAV Under External Disturbances …

of |si (t)| increases (the trajectories of the system are far from the sliding surface), the value of the term Ni (si (t)) decreases and Ni (sKii(t)) allowing a fast convergence.

3.2 Quadrotor Control by First Order Integral Sliding Mode The sliding mode control offers relatively many advantages over other robust nonlinear control methods [8] such as insensitivity to unmodeled dynamics/external disturbances, good robustness and the ability to handle nonlinearities, the controller structure is simple and easily adjustable. In order to eliminate the steady state error in the MSC design procedure, an integral tracking error action was added in the sliding surfaces. Recall that the objective of the control is to obtain the virtual control signals, these generate the desired attitude (d (t), d (t)) and the total thrust u m . The internal loop control ensures the stability of the quadrotor by generating the roll, pitch and yaw torques. In addition, the control method presented in this section takes into account and compensates for matched disturbances. The big issue for the considered problem is that the disturbance di (t) depends on the wind signals, the control itself and the system state, as discussed in the previous chapter. Therefore, it is necessary to slightly expand the ISMC approach. To this end, the sliding surfaces in this section are selected to the form of the classical PID control, in this way we will control the dynamics proportional to the position and velocity. Let’s introduce the tracking errors of the position and attitude subsystems as follows: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ˙ d (t) X2 −  e1 (t) X1 − d (t) e2 (t) ⎣e3 (t)⎦ = ⎣X3 − d (t)⎦ , ⎣e4 (t)⎦ = ⎣X4 −  ˙ d (t)⎦ , ˙ d (t) e5 (t) X5 − d (t) e6 (t) X6 −  ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ e7 (t) X7 − xd (t) e8 (t) X8 − x˙d (t) ⎣ e9 (t) ⎦ = ⎣ X9 − yd (t) ⎦ , ⎣e10 (t)⎦ = ⎣X10 − y˙d (t)⎦ e11 (t) X11 − yd (t) e12 (t) X12 − z˙ d (t) ⎡

(3.16)

(3.17)

The PID sliding surfaces are selected as follows:  ⎧ s7 (t) = K px e7 (t) + K i x e7 (t)dt ⎪ ⎪ ⎪ ⎧  +K d x e˙7 (t) ⎪  ⎨ ⎨ s1 (t) = K p e1 (t) + K i  e1 (t)dt + K d e˙1 (t) ⎪ s9 (t) = K py e9 (t) + K i y e9 (t)dt s3 (t) = K p e3 (t) + K i  e3 (t)dt + K d e˙3 (t) +K dy e˙9 (t) ⎪ ⎩  s5 (t) = K p e5 (t) + K i e5 (t)dt + K d e˙5 (t) ⎪ ⎪ ⎪ ⎪ s11 (t) = K pz e11 (t) + K i z e11 (t)dt ⎩ +K dz e˙11 (t)

(3.18) where K pj , K i j , K d j | j=(,,,x,y,z) are positive parameters. The time derivatives of the PID sliding surfaces are given by:

3.2 Quadrotor Control by First Order Integral Sliding Mode

55

⎧ ⎧ ⎨ s˙1 (t) = K p e˙1 (t) + K i e1 (t) + K d e¨1 (t) ⎨ s˙7 (t) = K px e˙7 (t) + K i x e7 (t) + K d x e¨7 (t) s˙3 (t) = K p e˙3 (t) + K i e3 (t) + K d e¨3 (t) s˙9 (t) = K py e˙9 (t) + K i y e9 (t) + K dy e¨9 (t) ⎩ ⎩ s˙5 (t) = K p e˙5 (t) + K i e5 (t) + K d e¨5 (t) s˙11 (t) = K pz e˙11 (t) + K i z e11 (t) + K dz e¨11 (t)

(3.19) The use of reach control laws such as: s˙ (t) = −K s sign(s(t))

(3.20)

where sign denotes the signum function and K s is the non-negative coefficient. The switching control laws, for the translation and rotation subsystems, are given by the following equations. ⎧ ⎧ ⎨ s˙1 (t) = −K 1 sign(s1 (t)) ⎨ s˙7 (t) = −K 7 sign(s7 (t)) s˙3 (t) = −K 3 sign(s3 (t)) s˙9 (t) = −K 9 sign(s9 (t)) ⎩ ⎩ s˙5 (t) = −K 5 sign(s5 (t)) s˙11 (t) = −K 11 sign(s11 (t))

(3.21)

Using (3.19) and (3.21) the virtual commands for the position loop are given in (3.22). ⎧ 1 ⎪ ⎨ Vx = −ρ X X8 + x¨d (t) − K d x (K px e˙7 (t) + K i x e7 (t) + K s7 sign(s7 (t))) Vy = −ρY X10 + y¨d (t) − K1dy (K py e˙9 + K i y e9 (t) + K s9 sign(s9 (t))) ⎪ ⎩ V = −ρ X + z¨ (t) − 1 (K e˙ (t) + K e (t) + K sign(s (t))) z Z 12 d pz 11 i z 11 s11 11 K dz (3.22) Similarity, we can easily deduce the control signals of the inner loop as follows: ⎧ ¨ d (t) u  = ρ11 (−ρ1 X4 X6 − ρ2 X4 − ρ3 X2 2 +  ⎪ ⎪ ⎪ 1 ⎪ + K d (K p e˙1 (t) + K i e1 (t) + K s1 sign(s1 (t)))) ⎪ ⎪ ⎪ ⎨u = 1 (−ρ1 X2 X6 − ρ2 X2 − ρ3 X4 2  ρ2 1 ¨ (t) + (K p e˙3 (t) + K i e3 (t) + K s3 sign(s3 ))) +  ⎪ d ⎪ K d ⎪ ⎪ 1 ¨ d (t) ⎪ = (−ρ u  1 X2 X4 − ρ2 X6 2 +  ⎪ ρ3 ⎪ ⎩ 1 + K d (K p e˙5 (t)K i e5 (t) + K s5 sign(s5 (t)))) where K si |i=(x,y,z,,,) is a positive parameter. Theorem 1 For the x-subsystem studied with the controller Vx , we can conclude that the x-subsystem is stable. Proof 1 The Lyapunov function of the x-subsystem is defined as follows: V7 =

1 2 s (t) 2 7

(3.23)

with V7 (0) = 0 and V7 (t) > 0 for s7 (t) = 0. The time derivative of V7 is given by:

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3 Stabilization of QUAV Under External Disturbances …

V˙7 = s7 (t)˙s7 (t)

(3.24)

The use of (3.50), so the equation (3.24) becomes V˙7 = s7 (t)(K px e˙7 (t) + K i x e7 (t) + K d x e¨7 (t))

(3.25)

The substitution of the double time derivative of the tracking error in (3.25) as    dx (t) ˙ − Vx V7 = s7 (t) K px e˙7 (t) + K i x e7 (t) + K d x x¨d (t) − ρ X X8 − m (3.26) By substituting the expression of the control law Vx given by (3.22), we obtain  dx (t) + ρ X X8 V˙7 = s7 (t) K px e˙7 + K i x e7 (t) + K d x (x¨d (t) − ρ X X8 − m  dx (t) 1 + (K px e˙7 + K i x e7 (t) + K s7 sign(s7 (t)))) − x¨d (t) − m Kdx

(3.27)

After a simple calculation, we have V˙7 = − s7 (t)K s7 sign(s7 (t)) = − K s7 |s7 (t)| ≤ 0

(3.28)

It may be clear that the Eq. (3.28) will be less than zero. However, despite the robustness of the ISM control to disturbances, the phenomenon of reluctance (high frequency oscillations of a discontinuous control signal in the steady state mode caused by the inaccuracy of the values of the model parameters, numerical and measurement noises) is very present which could lead to the deterioration of the system in the case where the force is acceptable by the rotors. This problem can increase the energy consumption and ruin the rotors, so below several solutions for the reduction of the reluctance phenomenon are compared. For this purpose, several modifications are introduced in the control algorithm for comparison. In the literature, this problem is a well-known topic and discussed in many articles. Saturation functions are popular solutions used for reluctance reduction in ISMC which leads to practical stability in the closed loop system, as in Fig. 3.2. ). The hyperbolic tangent function replaces the sign as: sign = tanh( s(t) ξ In practice, ξ should be chosen small enough to find a compromise between reducing the reluctance and a minimum acceptable error in r steady state. Another way to reduce the reticence problem is to use a higher order sliding mode control. The latter is the most relevant solution that we will introduce and explain in the following.

3.3 Quadrotor Control by Higher Order PID Sliding Modes

57

Fig. 3.2 Approximation of the sign using the linear saturation function

3.3 Quadrotor Control by Higher Order PID Sliding Modes The higher order sliding mode control was introduced in the late eighties by Emelyanov and Korovin and in the early nineties by Arie Levant (Levantovsky) [6, 9]. It allows, in addition to the good robustness properties of the classical sliding mode control, the reduction or the elimination of the reticence phenomenon, while preserving the system performances. The classical sliding regime acts on the sliding surface in order to cancel it, whereas the sliding regime of order r (noted r -slipping) acts on the surface and its (r − 1) first successive derivatives with respect to time [10–14]. The objective is to force the system to evolve not only on the surface but also on its (r − 1) first successive derivatives and to maintain the sliding set at zero: s(t) = s˙ (t) = s¨ (t) = · · · = s r −1 (t) = 0

(3.29)

where r denotes the relative degree of the system. However, the main problem in implementing this technique is the need for the often unavailable measurement of the sliding surface derivatives. For example, for the second-order sliding mode, s(t) and s˙ (t) should be available. As the subsystems of the quadrotor considered in this work have a relative degree equal to 2, we will be interested in the second order sliding mode control. To do so, we will introduce its theory as well as the super-twisting algorithm with an integral surface.

3.3.1 Principle of the Second Order Sliding Mode Control For second-order sliding mode control, the control action acts directly on the sign and magnitude of s(t) and appropriate switching logic, which can be based on s(t) and its derivative s(t) in order to ensure finite-time convergence of the system states

58

3 Stabilization of QUAV Under External Disturbances …

Fig. 3.3 Trajectory of second order sliding modes

to the sliding hyperplane s(t) = s˙ (t) = 0. The trajectory of the second order sliding modes is shown in Fig. 3.3. As mentioned earlier, for the second-order sliding mode, measurements of s(t) and its derivative s˙ (t) are required. The super-twisting algorithm requires only the measurement of s(t), although it is a second-order sliding mode. In this algorithm, the derivative of s(t) is not required to realize s(t) = s˙ (t).

3.3.2 Super-Twisting Control This control algorithm was first developed for systems of relative degree equal to one, then for systems of relative degree equal to two in order to avoid the reticence phenomenon. The trajectories on the phase plane are characterized by an oscillation around the origin as in the Fig. 3.4. Contrary to other second order sliding mode algorithms, the control law is directly designed for an input signal u(t) without taking into account its derivative u(t). ˙ The control law is computed by solving the following equation: 

s˙ (t) = −K 1 |s(t)|(1/2) sign(s(t)) +  ˙ = −K 2 sign(s(t))

(3.30)

(1/2) , K 1 et K 2 are diagonal matriwith |s(t)|(1/2) =diag |s(t)|(1/2) (t), . . . , |s(t)| (t) n 1 ces whose elements are strictly positive. The main advantage of the super-twisting algorithm is that it is a second order sliding mode and does not require the measurement of the derivative of the sliding surface. In this section, basic notions on the control of higher order sliding modes in particular the super-twisting algorithm have been presented and discussed. In order to control the attitude and position of

3.3 Quadrotor Control by Higher Order PID Sliding Modes

59

Fig. 3.4 Phase plan of the super-twisting algorithm

the quadrotor, a modified super-twisting algorithm with an integral surface is applied to the vehicle system. This controller is capable of generating a continuous control signal that eliminates reluctance and reduces control effort.

3.3.3 Application of the Higher Order SM-PID Control to Quadrotor System In order to increase the robustness of the quadrotor system and eliminate the reticence phenomenon in ISMC, a new strategy of ISMC super-twisting is proposed to subject to time-varying perturbations. In order to develop the super-twisting-ISMC, sliding mode integral surfaces. By combining the super-twisting algorithm and the PIDISMC controller, the new control law is presented as: u(t) = u I S MC (t) + u ST C (t)

(3.31)

where u I S MC (t) is the sliding mode integral control defined in Fig. 3.3 and u C ST represents the super-twisting control law, the latter is formulated by the following equation [15]. 1 u C ST (t) = −1 |s(t)| sign(s(t)) − 2 2

 sign(s(t))dτ

(3.32)

where 1 and 2 are positive constants. Using this controller, the dynamics of the closed-loop sliding surface is recast as:  s˙ (t) = − 1 K d |s(t)|sign(s(t)) − 2 K d − K S sign(s(t))

sign(s(t))dτ

(3.33)

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3 Stabilization of QUAV Under External Disturbances …

Fig. 3.5 Block diagram of the method proposed

We define 1 K d = K 1 and 2 K d = K 2 . The control scheme proposed in this part for the quadrotor vehicle is presented in Fig. 3.5. Thus, the new virtual control laws are given by the following equation. ⎧  Vx = −ρ X X8 + x¨d (t) + K 1x |s7 (t)| 21 sign(s7 (t)) − K 2x sign(s7 (t))dτ ⎪ ⎪ ⎪ ⎪ − K1 (K px e˙7 (t) + K i x e7 (t) + K s7 sign(s7 (t))) ⎪ ⎪ ⎪ ⎨ V = −ρ d xX + y¨ (t) − K |s (t)| 1 sign(s (t)) − K  sign(s (t))dτ y Y 10 d 1y 9 9 2y 9 2 1 (K e ˙ (t) + K e (t) + K sign(s (t))) − ⎪ py 9 i y 9 s9 9 K ⎪ dy ⎪ 1 ⎪ ⎪ ⎪ Vz = −ρ Z X12 + z¨ d (t) − K 1z |s11 (t)| 2 sign(s11 (t)) ⎪ ⎩ 1 −K 2z sign(s11 (t))dτ − K dz (K pz e˙11 (t) + K i z e11 (t) + K s11 sign(s11 (t))) (3.34) and the new attitude loop control signals are given as: ⎧ ¨ d (t) + K 1 |s1 (t)| 1 sign(s1 (t)) u  = − ρ11 (−ρ1 X4 X6 − ρ2 X4 − ρ3 X22 −  ⎪ 2 ⎪  ⎪ 1 ⎪ +K 2 sign(s1 (t))dτ − K d (K p e˙1 + K i e1 (t) + K s1 sign(s1 (t)))) ⎪ ⎪ ⎪ ⎨ u = − 1 (−ρ X X − ρ X − ρ X 2 −  ¨ d (t) + K 1 |s3 (t)| 1 sign(s3 (t))  2 2 3 4 ρ2  1 2 6 2 1 +K 2 sign(s3 (t))dτ − K d (K p e˙3 (t) + K i e3 (t) + K s3 sign(s3 (t)))) ⎪ ⎪ ⎪ ⎪ ¨ d (t) + K 1 |s5 (t)| 1 sign(s5 (t)) ⎪ u  = − ρ13 (−ρ1 X2 X4 − ρ2 X26 −  ⎪ 2 ⎪  ⎩ 1 +K 2 sign(s5 (t))dτ − K d (K p e˙5 + K i e5 (t) + K s5 sign(s5 (t)))) (3.35) where K 1i and K 2i for i = (x, y, z, , , ) are positive constants. Theorem 2 For the system studied with the Super Twisting control law, we can conclude that the system is stable and the tracking errors ei (t) and e(t) ˙ will asymptotically converge to zero. Proof 2 In order to verify the stability of the system, let us consider the standard super-twisting algorithm (STA) in [16, 17] as: z˙ i = − i |z i |1/2 sign(z i ) + z i+1 z˙ i+1 = − i+1 sign(z i )

(3.36)

3.3 Quadrotor Control by Higher Order PID Sliding Modes

61

where z i ∈ R and z i+1 ∈ R are the state variables, i and i+1 are the positive gains. The quadratic form of the Lyapunov function can be written as: V (z i ) = ζ T Pζ

(3.37)

with ζ = [|z i |1/2 sign(z i ), z i+1 ]and P = P T is a positive matrix. The latter is a solution of a Lyapunov algebraic equation (LAE) (3.38). A T P + A P = −Q

(3.38)

where Q = Q T > 0 et A is Hurwitz, defined in (3.39). 

− 21 i −i+1

1 2



0

(3.39)

In addition, the time derivative V˙ in [16, 17] satisfies the differential inequality. 1 V˙ ≤ −γ (Q)V (z) 2 with γ (Q) 

λmin (Q)λmin 21 (P) λmax (P)

(3.40)

(3.41)

is a scalar depending on Q. Therefore, the authors of [16, 17] provide the details of this technique (STA).

3.4 Simulation Results with Controller Gains Optimization In this section, the dynamic vehicle attitude and position model established in the modeling chapter is used to verify and show the effectiveness of the proposed control laws under constant and time-varying external disturbances. A group of numerical simulations of the modified super-twisting PID sliding mode control method is presented in this work under two scenarios. The simulation results are performed to prove the superior performance of the proposed ST-PID-ISM controller over the full-action sliding mode controller presented in [18, 19]. In order to obtain the optimal control parameters, the Matlab/Simulink software toolbox optimization method is used to solve this problem. This optimization technique uses the Check Step Response Characteristics block. The details of this optimization method are presented in [20]. Remark 2 The goal of the optimization is to minimize the tracking error and to obtain a better performance of the trajectory tracking. In addition, the optimization is performed using the Simulink “Design Optimization and Response Optimization” tools of Matlab/Simulink software. The quadrotor system consists of six subsystems.

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3 Stabilization of QUAV Under External Disturbances …

For this purpose, the “Check Against Reference” and “Check Step Response Characteristics” blocks are inserted in each output of the quadrotor control. By defining some performances in the block ‘Check Step Response Characteristics’, such as the rise time, the rise %, the stabilization time, the stabilization %, the overshoot % and the undershoot %. Then, we initialize the controller parameters, after some tests, the optimization provides the best values of the control parameters allowing to converge the tracking errors.

3.4.1 Scenario 1: Robustness Analysis (constant Disturbances) Several simulation tests are performed to evaluate the robustness and effectiveness of the proposed trajectory tracking technique. The simulation results show the performance of the ST-PID-ISMC quadrotor trajectory tracking technique. In the STPID-ISMC method, the use of integral action eliminates the steady-state error, while the use of the super-twisting algorithm eliminates the reluctance effect and the insensitivity to disturbances. In this scenario, we verify whether the performance of the ST-PID-ISMC strategy is preserved even in the presence of the constant external disturbances. The expressions of the latter are given as follows  dx (t) = 

0 m/s 2 t ∈ [0, 5) d y (t) = 1 m/s 2 [5, 40]

0 1  0 d (t) = 1

dz (t) =

m/s 2 t m/s 2 t rad/s 2 rad/s 2



0 m/s 2 t ∈ [0, 15) 1 m/s 2 t ∈ [15, 40]  ∈ [0, 25) 0 rad/s 2 t ∈ [0, 10) d (t) = ∈ [25, 40] 1 rad/s 2 t ∈ [10, 40]  t ∈ [0, 20) 0 rad/s 2 t ∈ [0, 30) d (t) = t ∈ [20, 40] 1 rad/s 2 t ∈ [30, 40]

(3.42) (3.43) (3.44)

The initial conditions of the vehicle are [0, 0, 0]m and [0, 0, 0.5] rad. The reference trajectory used in the simulations of this scenario is a set of several types of sections as follows: ⎧ 0.5 cos(0.5t) m ⎪ ⎪ ⎨ 0.5 m xd (t) = 0.25t − 4.5 m ⎪ ⎪ ⎩ 3m

t t t t

⎧ 0.5 sin(0.5t) m ⎪ ⎪ ∈ [0, 4π ) ⎪ ⎪ ⎨ 0.25t − 3.14 m ∈ [4π, 20) y (t) = 5 − π m ∈ [20, 30) d ⎪ ⎪ −0.2358t + 8.94 m ⎪ ⎪ ∈ [30, 80] ⎩ −0.5 m

t t t t t

∈ [0, 4π ) ∈ [4π, 20) ∈ [20, 30) ∈ [30, 40] ∈ [40, 80]

⎧ π t ∈ [0, 4π ) ⎨ 0.125t + 1 m rad t ∈ [0, 50) t ∈ [4π, 40) d (t) = 4 z d (t) = 0.5π + 1 m 0 rad t ∈ (50, 80] ⎩ exp(−0.2t + 8.944) m t ∈ [40, 80)

(3.45) (3.46)

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3.4 Simulation Results with Controller Gains Optimization 3

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The simulation results that are given to compare with the full sliding mode controller proposed in [18, 19], are shown in Figs. 3.7, 3.8, 3.9. These figures show that the control strategy exhibits a robust tracking trajectory and achieves fast tracking performance when sustained disturbances are applied to the quadrotor system and abrupt flight changes from the desired trajectory. The results of these figures indicate that the performance of ST-PID compares better with ISMC. Figure 3.6 shows the performance of the desired trajectory tracking system. we can observe that the ST-PID-ISMC method tracks the trajectory accurately and remains stable. The ISMC method follows the trajectory more slowly at the beginning (see Fig. 3.6). We can clearly indicate that the ST-PID-ISMC method provides a faster response and excellent tracking in the presence of perturbations. Moreover, Fig. 3.7 describes the attitude performance. We can see that the roll and pitch angles computed by the outer loop trajectory following ST-IMSC and the yaw angle follow the reference values in a finite time. Although the roll and pitch angles are not controlled directly, the proposed controller is able to stabilize the orientation of the quadrotor. Figure 3.8 shows the results of the trajectory tracking in 3D space. We can observe how, starting from an initial condition of the position away from the reference, the ST-PID-ISM control strategy is able to make the quadrotor follow the desired trajectory.

3 Stabilization of QUAV Under External Disturbances …

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3.4 Simulation Results with Controller Gains Optimization

65

Fig. 3.9 3D trajectory of the quadrotor in scenario 1

Finally, Fig. 3.9 shows the responses of the total thrust u m and the three rotation torques (u  , u  , u  ). The amplitudes of these four control inputs are physically feasible and show no reluctance, these control input signals have obviously smaller values and are smooth. The control input signals obtained by the ISMC method are similar to those obtained by the proposed ST-PID-ISMC method.

3.4.2 Scenario 2: Robustness Analysis (Time-Varying Disturbances) In the scenario 3.4.1, the external perturbations are considered as constant. In this part, we give a second collection of simulations taking into account the timevarying perturbations caused by external factors and wind gusts on the position and attitude of the quadrotor ((t), (t), (t), x(t), y(t), z(t)). The expressions of the perturbations applied to the six degrees of freedom are given as follows: di |(i=i,,,x,y,z) = 0.5 cos(t). The six degrees of freedom are initialized by the following conditions: [0.5, 0.5, 1]m and [0, 0, 0.5] rad. The reference trajectory used in the simulations of this scenario is defined by the following equations. xd (t) = cos(t) + 1 m, yd (t) = sin(t) + 1 m z d (t) = 0.5t + 1 m, d (t) = 0.5 sin(t) rad

(3.47)

The ST-PID-ISMC controller and the performance of ISMC are shown in Figs. 3.11, 3.12, 3.13. To improve the robustness of the ISMC controller, a robust controller is introduced, which reduces the effect of disturbances on the quadrotor and thus increases the robustness. Through the results presented in Figs. 3.11, 3.12, 3.13, the proposed controller performance is better than the ISMC presented by the authors of [18, 19]. Figure 3.10 describes the position control performance, we can see that, the proposed

66

3 Stabilization of QUAV Under External Disturbances …

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ST-PID-ISMC strategy can drive the quadrotor to the desired trajectory in a finite time. The outer loop generates the pitch and roll angle reference. Figure 3.11 shows the result of the attitude responses. We can observe that, the yaw angle follows the desired sine wave with precision, the Euler angles ((t), (t)) follow the original values even if the desired trajectory is modified. The four control inputs of the four-rotor system are shown in Fig. 3.12. These input control signals are smoother and of good quality compared to the ISMC method. Figure 3.12 indicates that the control signals obtained by ISMC have the problem of reluctance caused by discontinuous components. The flight path in 3D space is plotted on Fig. 3.13. We can observe that, the controller proposed in this chapter ensures a better follow-up of cyclic trajectories in the presence of external perturbations at all times. Remark 3 The first disturbed period for the tests of scenario 1 and 2, the system curves are able to clearly show the difference between the ISMC and the proposed controller. For example, Figs. 3.6 and 3.10 show that the proposed method has better tracking performance during the initial period of flight.

3.4 Simulation Results with Controller Gains Optimization

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(rad)

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U (N.m)

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Fig. 3.11 Quadrotor attitude ((t), (t), (t)) 10

STPIDSMCM ISMC

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0 -2

Fig. 3.12 Inputs of the quadrotor in scenario 2

time (s)

68

3 Stabilization of QUAV Under External Disturbances … Reference

ISMC

STPIDSMCM

z (m)

15 10 5 0 2 1

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x (m)

Fig. 3.13 3D trajectory of the quadrotor in scenario 2

3.5 Discussion The reluctance phenomenon caused by a wide choice of switching gains to cope with external disturbances poses a serious problem for ISM control in practice. One of the effective ways proposed to reduce and avoid this problem is the second order sliding mode control. In particular, the super-twisting algorithm which requires neither the measurement of the derivative of the sliding surface nor its sign. The Table 3.1 summarizes the characteristics of the integral sliding mode control and some algorithms of this control in order two. In this part, a second order integral sliding mode controller based on the super-twisting algorithm for the control of the inner and outer loop of a four-wheel scooter vehicle is designed. New methods that can guarantee a reduction in reluctance, excellent robustness to wind disturbances and simplicity of design will be developed in the next section.

Table 3.1 Evaluation of methods to reduce the reticence phenomenon Method Chattering Robustness ISMC SMC with saturation Twisting algorithm Sub-optimal

High Reduced Reduced Reduced

Excellent Low Excellent Excellent

Super-twisting

Reduced

Excellent

Design Simple Simple function Very complex Very complex algorithm Complex algorithm

3.6 Novel Terminal Sliding Mode Control for the Position and Attitude of a Quadrotor

69

3.6 Novel Terminal Sliding Mode Control for the Position and Attitude of a Quadrotor In this subsection, novel nonlinear sliding mode variables (NSMV) are suggested for both position/attitude to increase the control performance efficiency in terms of path following and disturbance rejections. Its expressions can be defined as: ⎧ ⎨ s1 (t) = K pφ e1 (t) + K dφ e˙1 (t) + γ1 e1 (t)μm s3 (t) = K pθ e3 (t) + K dθ e˙3 (t) + γ3 e3 (t)μ ⎩ s5 (t) = K pψ e5 (t) + K dψ e˙5 (t) + γ5 e5 (t)μ5 ⎧ ⎨ s7 (t) = K px e7 (t) + K d x e˙7 (t) + γ7 e7 (t)μ7 s9 (t) = K py e9 (t) + K dy e˙9 (t) + γ9 e9 (t)μ9 ⎩ s11 (t) = K pz e11 (t) + K dz e˙11 (t) + γ11 e11 (t)μ11

(3.48)

(3.49)

where γi , μi , K pk , K ik , and K dk for k = (φ, θ, ψ, x, y, z) are positive coefficients. The objective of the designed controller is to force the position tracking-errors (3.17) and attitude tracking-errors (3.17) to approach the designed sliding variables (3.48) and then move along the NSMV to the origin. The time-derivative of si (t) can be presented as: ⎧ ⎨ s˙1 (t) = K pφ e˙1 (t) + K dφ e¨1 (t) + γ1 μm e˙1 (t)e1 (t)μm −1 s˙3 (t) = K pθ e˙3 (t) + K dθ e¨3 (t) + γ1 μ e˙3 (t)e3 (t)μ −1 ⎩ s˙5 (t) = K pψ e˙5 (t) + K dψ e¨5 (t) + γ5 μ5 e˙5 (t)e5 (t)μ5 −1 ⎧ ⎨ s˙7 (t)(t) = K px e˙7 (t) + K d x e¨7 (t) + γ1 μ7 e˙7 (t)e7 (t)μ7 −1 s˙9 (t) = K py e˙9 (t) + K dy e¨9 (t) + γ1 μ9 e˙9 (t)e9 (t)μ9 −1 ⎩ s˙11 (t) = K pz e˙11 (t) + K dz e¨11 (t) + γ11 μ11 e˙11 (t)e11 (t)μ11 −1

(3.50)

(3.51)

Substituting the time-derivative of the tracking errors (3.17) in (3.50) and (3.51), we have

⎧ μm −1 )e˙ (t) s˙1 (t) = (K pφ + ⎪ 1 ⎪ γ1 μm e1 (t)

⎪ ⎪ 1 2 + ρ u ) + d (t) − φ¨ (t) ⎪ +K (−ρ X X − ρ X − ρ X ⎪ dφ 1 4 6 2 4 3 1  φ d 2 ρ1 ⎪ ⎪ ⎪ ⎨ s˙3 (t) = (K pθ + γ1 μ e3 (t)μ −1 )e˙3 (t)

(3.52) +K dθ ρ12 (−ρ1 X2 X6 − ρ2 X2 − ρ3 X24 + ρ2 u  ) + dθ (t) − θ¨d (t) ⎪ ⎪ ⎪ ⎪ ⎪ s˙5 (t) = (K pψ + γ5 μ5 e5 (t)μ5 −1 )e˙5 (t) ⎪ ⎪

⎪ ⎪ ⎩ +K dψ ρ13 (−ρ1 X2 X4 − ρ2 X26 + ρ3 u  ) + dψ (t) − ψ¨ d (t) ⎧

⎪ s˙7 (t)(t) = (K px + γ7 μ7 e˙7 (t)e7 (t)μ7 −1 )e˙7 (t) + K d x Vx − ρ X X8 + dxm(t) − x¨d (t) ⎪ ⎪ ⎨

d (t) s˙9 (t) = (K py + γ1 μ9 e˙9 (t)e9 (t)μ9 −1 )e˙9 (t) + K dy Vy − ρY X10 + ym − y¨d (t) ⎪

⎪ ⎪ ⎩ s˙11 (t) = (K pz + γ11 μ11 e˙11 (t)e11 (t)μ11 −1 )e˙11 (t) + K dz Vz − ρ Z X12 + dz (t) − z¨ d (t) m

(3.53) The equivalent control law can be obtained by forcing s˙i = 0 without disturbances e.g., di (t) = 0, then the equivalent control inputs can be defined as:

70

3 Stabilization of QUAV Under External Disturbances …

For the position-loop, ⎧ 1 μ7 −1 ⎪ )e˙7 (t) ⎨ V1eq = −ρ X X8 + x¨d (t) − K d x (K px + γ7 μ7 e˙7 (t)e7 (t) 1 μ9 −1 )e˙9 (t) V2eq = −ρY X10 + y¨d (t) − K dy (K py + γ9 μ9 e˙9 (t)e9 (t) ⎪ ⎩ V = −ρ X + z¨ (t) − 1 (K + γ μ e˙ (t)e (t)μ11 −1 )e˙ (t) 3eq Z 12 d pz 11 11 11 11 11 K dz

(3.54)

For the attitude-loop,  ⎧ U2eq = ρ11 − ρ1 X4 X6 − ρ2 X4 − ρ3 X22 + φ¨ d (t) ⎪  ⎪ ⎪ ⎪ − K1dφ (K pφ + γ1 μm e˙1 (t)e1 (t)μm −1 )e˙1 (t) ⎪ ⎪  ⎪ ⎨ U = 1 − ρ X X − ρ X − ρ X2 + θ¨ (t) 3eq 1 2 6 2 2 3 4 ρ2 d − K1dθ (K pθ + γ3 μ e˙3 (t)e3 (t)μ −1 )e˙3 (t) ⎪ ⎪  ⎪ ⎪ ⎪ U4eq = ρ1 − ρ1 X2 X4 − ρ2 X26 + ψ¨ d (t) ⎪ 3 ⎪  ⎩ − K1dψ (K pψ + γ5 μ5 e˙5 (t)e5 (t)μ5 −1 )e˙5 (t)

(3.55)

The reaching control law u s for position and attitude are added to the equivalent control laws in order to improve the control performance in the presence of external disturbances as: U = Ueq + Us

(3.56)

where Us = −K s sign(s(t)) is the switching law. Then, the total control laws can be designed as: ⎧ K dφ K s1 ⎪ ⎨ U = u 2eq − ρ1 s1 (t)sign(s1 (t)) U = u 3eq − K dθρ2K s3 s3 (t)sign(s3 (t)) ⎪ ⎩ U = u − K dψ K s5 s (t)sign(s (t))  4eq 5 5 ρ3 ⎧ ⎨ Vx = v1eq − K d x K 7s s7 (t)sign(s7 (t)) Vy = v2eq − K dy K 9s s9 (t)sign(s9 (t)) ⎩ Vz = v3eq − K dz K 11s s11 (t)sign(s11 (t))

(3.57)

(3.58)

where K ks is the reaching gain. Theorem 3 Consider the x-subsystem with the equivalent control u 2eq given in (3.54) and a hitting control term vs designed in (3.58), the sliding variable s7 (t) converge to origin value, then the tracking error e7 (t) is stable. Proof Consider the Lyapunov function candidate as follows: V7 =

1 2 s (t) 2 7

(3.59)

with V7 (t) > 0 and V7 (0) = 0 for s7 (t)(t) = 0. The derivative of V7 is given by:

3.6 Novel Terminal Sliding Mode Control for the Position and Attitude of a Quadrotor

V˙7 = s7 (t)˙s7 (t)

71

(3.60)

Substitute (3.51) into (3.61) produces  ˙ V7 = s7 (t) (K px + γ7 μ7 e˙7 (t)e7 (t)μ7 −1 )e˙7 (t)  dx (t) − x¨d (t)) + K d x (Vx − ρ X X8 + m

(3.61)

Using Vx the virtual law presented in (3.58), we get  s˙7 (t) = s7 (t) (K px + γ7 μ7 e˙7 (t)e7 (t)μ7 −1 )e˙7 (t) + K d x − ρ X X8 + x¨d (t) 1 (K px + γ7 μ7 e˙7 (t)e7 (t)μ7 −1 )e˙7 (t) − K 7s s7 (t)sign(s7 (t)) Kdx  dx (t) − x¨d (t)) ρ X X8 + m −

(3.62)

After a simple calculation, we have   dx (t) V˙7 =s7 (t) −K s7 sign(s7 (t)) + m

(3.63)

By assuming the | dxm(t) | ≤ K s7 , then the time- derivative of Lyapunov function presented in (3.63) will be less than zero. Theorem 4 The ultimate control laws presented in (3.54) and (3.55) applied to dynamics system and the STMFTSMC technique ensures the overall closed-loop system stability. Proof Let define the Lyapunov function for the quadrotor system as: V =

1 2 1 1 1 1 1 2 s1 (t) + s32 (t) + s52 (t) + s72 (t) + s92 (t) + s11 (t) 2 2 2 2 2 2

(3.64)

with V (t) > 0 and V (0) = 0 for si (t) = 0. The derivative of V is given by: V˙ = s1 (t)˙s1 (t) + s3 (t)˙s3 (t) + s5 (t)˙s5 (t) + s7 (t)˙s7 (t)(t) + s9 (t)˙s9 (t) + s11 (t)˙s11 (t) Using the same calculation procedure presented in Theorem 1, we have

(3.65)

72

3 Stabilization of QUAV Under External Disturbances …

    dφ (t) dθ (t) + s3 (t) −K s3 sign(s3 (t)) + V˙ = s1 (t) −K s1 sign(s1 (t)) + J1 J2     dψ (t) dx (t) + s7 (t) −K s7 sign(s7 (t)) + + s5 (t) −K s5 sign(s5 (t)) + J3 m     d y (t) dz (t) + s11 (t) −K s11 sign(s11 (t)) + + s9 (t) −K s9 sign(s9 (t)) + m m (3.66) d (t) Assuming the | dim(t) | ≤ K si and | jJ | ≤ K s j , then the time-derivative of Lyapunov function presented in (3.66) will be less than zero.

3.7 Modified Super-Twisting NSMC for the Quadrotor System A modified super-twisting is combined with the control method presented in in the previous part in order to increase the robustness and tracking control against disturbances. The proposed scheme can be defined as: U = Unsmc + Umst

(3.67)

where Unsmc is the new sliding-mode control given in the previous subsection and Umst denotes the proposed modified super-twisting control law, which can be given by [21]: Umst = −c1 |s(t)|0.5 sign(s(t)) − c2 s(t) + ε (3.68) ε˙ = −c3 sign(s(t)) − c4 s(t) where c1 , c2 , c3 and c4 are positive parameters. The closed-loop sliding-surface dynamics using the proposed controller, can be rewritten as:  s˙ (t) = − c1 K d |s(t)|

0.5

− c2 K d s(t) − K d

[c3 sign(s(t)) − c4 s(t)]dτ

(3.69)

We define C1 = c1 K d , C2 = c2 K d , C3 = c3 K d , and C4 = c4 K d . The block diagram of the proposed control scheme with the optimization tool are presented in 5.1. Thus, the novel virtual control laws are modified as:

3.7 Modified Super-Twisting NSMC for the Quadrotor System

⎧ 1 μ7 −1 ⎪ e˙7 (t) ⎪ Vx = −ρ X X8 + x¨d (t) − K d x (K px + γ7 μ7 e˙7 (t)e7 (t) ⎪ ⎪ 0.5 ⎪ s (t)sign(s (t)) − C |s (t)| −K 7s 7 1x 7 ⎪  7 ⎪ ⎪ ⎪ −C2x s7 (t) − [C3x sign(s7 (t)) − C4x s7 (t)]dτ ) ⎪ ⎪ ⎪ 1 μ −1 ⎪ ⎨ Vy = −ρY X10 + y¨d (t) − K dy (K py + γ9 μ9 e˙9 (t)e9 (t) 9 e˙9 (t) 0.5 −K 9s s9 (t)sign(s  9 (t)) − C1y |s9 (t)| ⎪ ⎪ ⎪ −C2y s9 (t) − [C3y sign(s9 (t)) − C4y s9 (t)]dτ ) ⎪ ⎪ ⎪ ⎪ Vz = −ρ Z X12 + z¨ d (t) − K1dz (K pz + γ11 μ11 e˙11 (t)e11 (t)μ11 −1 e˙11 (t) ⎪ ⎪ ⎪ 0.5 ⎪ −K ⎪ ⎪  11s s11 (t)sign(s11 (t)) − C1z |s11 (t)| − C2z s11 (t) ⎩ − [C3z sign(s11 (t)) − C4z s11 (t)]dτ )

73

(3.70)

and the novel controllers of the attitude-loop are presented as: ⎧ U = ρ11 (−ρ1 X4 X6 − ρ2 X4 − ρ3 X22 + φ¨ d (t) ⎪ ⎪ ⎪ ⎪ − K1dφ (K pφ + γ1 μm e˙1 (t)e1 (t)μm −1 )e˙1 (t) + K 1s s1 (t)sign(s1 )) ⎪ ⎪  ⎪ ⎪ ⎪ −C1φ |s1 (t)|0.5 − C2φ s1 (t) − [C3φ sign(s1 (t)) − C4φ s1 (t)]dτ ) ⎪ ⎪ ⎪ 1 ⎪ ⎨ U = ρ2 (−ρ1 X2 X6 − ρ2 X2 − ρ3 X24 + θ¨d (t) − K1dθ (K pθ + γ3 μ e˙3 (t)e3 (t)μ −1 )e˙3 (t) + K 3s s3 (t)sign(s3 (t)))  ⎪ ⎪ 0.5 ⎪ −C |s (t)| − C s (t) − [C3θ sign(s3 ) − C4θ s3 (t)]dτ ) ⎪ 1θ 3 2θ 3 ⎪ ⎪ 1 1 2 ¨ ⎪ ⎪ ⎪ U = ρ3 (−ρ1 X2 X4 − ρ2 X6 + ψd (t) − K dψ (K pψ ⎪ ⎪ μ −1 5 ⎪ +γ5 μ5 e˙5 (t)e5 (t) )e˙5 (t) +K 5s s5 (t)sign(s5 (t))) ⎪ ⎩ −C1φ |s5 (t)|0.5 − C2ψ s5 (t) − [C3ψ sign(s5 (t)) − C4ψ s5 (t)]dτ )

(3.71)

where C1 j , C2 j , C3 j and C4 j for j = (φ, θ, ψ, x, y, z) are positive constants. The closed-loop of the tracking-error dynamics s˙i (t) can be defined as:  s˙i (t) = −c1 K d |si (t)|0.5 − c2 K d si (t) − K d

[c3 sign(si (t)) − c4 si (t)]dτ + di (t) (3.72)

Defining new variables Z1 and Z2 as Z1 = si (t)  Z2 = −K d

[c3 sign(si (t)) − c4 si (t)]dτ + di (t)

We obtained the following dynamics system. ˙ 1 = −c1 K d |Z1 |0.5 sign(Z1 ) − K d c2 Z1 + Z2 Z ˙ 2 = −c3 K d sign(Z1 ) − K d c4 Z1 + d˙i (t) Z

(3.73) (3.74)

Theorem 5 Assume that the disturbances derivative the affecting the quadrotor subsystems is globally bounded, and according to [21], the positive parameters are selected. Then, the modified super-twisting with the NSMC yields finite-time

74

3 Stabilization of QUAV Under External Disturbances …

Fig. 3.14 Block diagram of the proposed control method

convergence of the si (t) = 0 and the tracking error ei and its derivative e˙i (t) will converge to zero [16, 17, 21–24]. The proof of the above theorem can be found in [16, 17, 21–24].

3.8 Simulation Results and Discussion To illustrate the effectiveness of the proposed method MSTFNSMC, numerical simulations will be presented in this part. In addition, the MSTFNSMC is compared with a traditional super-twisting SMC technique. Also, the disturbances are considered as: di (t)|x,y,z,φ,θ,ψ = 0.4 cos(t). The desired trajectory is chosen as: xd (t) = cos(t) m yd (t) = cos(t) + 2 m z d (t) = 0.5t + 5 m ψd (t) = 0.5 rad

(3.75)

The initial conditions are [x0 , y0 , z 0 , ψ0 ] = [0, −2, −2, 0]. Remark 4 All the gains of the controller can be computed based on the optimization toolbox in MATLAB software, the main objective of this optimization method is to minimize the position/attitude tracking errors. We define some tracking performance of the quadrotor in the block Check Step Response Characteristics such as rise time, % rise, settling time, % settling, % overshoot and % undershoot. Then, the controller parameters are initialized, after many tests, the optimization gives the best values of these parameters, which make the tracking errors converge.

3.8 Simulation Results and Discussion

75

0.6

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1.3

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Fig. 3.15 Quadrotor’s position (x(t), y(t), z(t))

The simulation results of the proposed approach and super-twisting SMC are plotted in Figs. 3.15, 3.16, 3.17, 3.18, 3.19, 3.20. The simulation shown the behavior of the proposed controller compared with the ST-SMC. From these we can see that proposed method is able to properly mitigate the effect of time-varying disturbances. The tracking performance of the attitude/position is depicted in Figs. 3.15 and 3.16. It’s clear that the MSTFTSMC is able to steer the attitude/position to the origin faster than ST-SMC. The total thrust and control torques are plotted in Fig. 3.17, these inputs are characterized by a chattering-free smooth and faster low-frequency responses. Besides, the path following in 3D is plotted in Fig. 3.18, it can be seen from this result that the proposed method is able to follow the reference of the trajectory under disturbances. The sliding mode variable responses of the translational and rotational are respectively presented in Figs. 3.19 and 3.20. Clearly, it can be seen that the nonlinear sliding variables converge to the

76

3 Stabilization of QUAV Under External Disturbances … Reference STSMC STMFTSMC

(rad)

0.2 0 -0.2 -0.4 0

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Fig. 3.16 Quadrotor’s attittude (φ, θ, ψ) Fig. 3.17 Quadrotor’s control inputs Um (N)

20 15 10

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U (N.m)

5 0

10

20

30

40

0

10

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30

40

0

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20

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40

0

10

30

40

0 -0.2 -0.4

0.6 0.4 0.2 0

0.6 0.4 0.2 0 20 time (s)

3.8 Simulation Results and Discussion

77

z (m)

Fig. 3.18 Quadrotor’s 3-D trajectory tracking

y (m )

Fig. 3.19 Position sliding variables

) x (m

s7

0 -1 -2 0

10

20

30

40

0

10

20

30

40

30

40

s9

0 -5

-10

s 11

0 -5

-10

0

10

20 time (s)

origin with a short finite-time. Noticeably, the results confirm the efficiency of the MSTFTSMC method. Some curves are not able to show the differences between the and the MSTFTSMC method and ST-SMC technique. Therefore, the integral absolute error (IAE) performance indexes are used and summarized in Table 3.2 to make a quantitative comparison of these control methods. It can be observed that in the presence of disturbances/uncertainties, the proposed control approach provides a more accurate tracking compared to ST-SMC method.

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3 Stabilization of QUAV Under External Disturbances …

s1

2 1 0 0

10

20

30

40

0

10

20

30

40

s3

0 -2 -4

s5

0 -1 -2 0

10

20

30

40

time (s)

Fig. 3.20 Attitude sliding variables Table 3.2 IAE performance indexes State ST-SMC x(t) y(t) z(t) (t) (t) (t)

0.94 4.41 11.09 0.35 0.60 0.04

STMFTSMC 0.90 4.42 9.58 0.31 0.53 0.35

3.9 Conclusion In this chapter, we have presented two flight control strategies: – A new way of designing the super-twisting controller was described. By making some changes, it operated in a closed loop. This reformulation increased the performance with respect to disturbances. Since the use of a discontinuous function in the ISMC leads to an undesirable reticence phenomenon, we introduced the super-twisting controller as a possible solution to overcome this drawback. This combination of ISMC and super-twisting leads to a good robustness/performance compromise. – A modified super-twisting and nonlinear sliding mode controller is combined for controlling the quadrotor system. From the results presented in the simulation section, we can show that the proposed control method improved the tracking-

3.9 Conclusion

79

performance against disturbances and reduce the chattering phenomenon. Simulation results of the proposed MSTFTSMC show a better performance tracking in comparison with the classical super-twisting SMC. Moreover, the proposed controllers can quickly and accurately track the quadrotor trajectory, the asymptotic stability of the closed-loop system is ensured. It has been shown that the described control strategies guarantee at least the asymptotic stability. Furthermore, robustness against disturbances by choosing appropriate control parameters has been guaranteed. The numerical simulations obtained are compared with some popular strategies to show the efficiency of the controllers proposed in this chapter.

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19. Labbadi, M., Cherkaoui, M., Houm, Y.E., Guisser, M.: Modeling and robust integral sliding mode control for a quadrotor unmanned aerial vehicle. In: 2018 6th International Renewable and Sustainable Energy Conference (IRSEC) (2018) 20. Freire, F.P., Martins, N.A., Splendor, F.: A simple optimization method for tuning the gains of PID controllers for the autopilot of cessna 182 aircraft using model-in-the-loop platform. J. Control Autom. Electr. Syst. 29, 441–450 (2018) 21. Muoz, F., Gonzlez-Hernndez, I., Salazar, S., Espinoza, E.S., Lozano, R.: Second order sliding mode controllers for altitude control of a quadrotor UAS: Real-time implementation in outdoor environments. Neurocomputing 233, 61–71 (2017). Apr 22. Jayakrishnan, H.J.: Position and Attitude control of a Quadrotor UAV using Super Twisting Sliding Mode. IFAC-PapersOnLine 49(1), 284–289 (2016) 23. Bouyahia, S., Semcheddine, S., Talbi, B., Boutalbi, O., Terchi, Y.: An adaptive super-twisting sliding mode algorithm for robust control of a biotechnological process. Int. J. Dyn. Control 8(2), 581–591 (2019). May 24. Babaei, A.-R., Malekzadeh, M., Madhkhan, D.: Adaptive super-twisting sliding mode control of 6-DOF nonlinear and uncertain air vehicle. Aerospace Sci. Technol. 84, 361–374 (2019). Jan

Chapter 4

Control of the QUAV by a Hybrid Finite-Time Tracking Technique

4.1 Introduction In this chapter two designs of nonlinear control schemes will be presented for a quadrotor system. A hybrid finite-time nonlinear controller is proposed for the quadrotor under disturbances. Also, an adaptive backstepping technique is suggested for altitude subsystem. In the other hand, an adaptive nonlinear global SMC is proposed for controlling the quadrotor system in the presence of uncertainties/disturbances. The chapter is structured as follows. The design of the hybrid control is presented in Sect. 4.2. The analysis of the simulation results by the hybrid control is given in Sect. 4.3. Then, the global sliding mode control of trajectory tracking of a quadrotor and the simulation results are presented in Sects. 4.4 and 4.5 respectively. The conclusion of this chapter is given in the last Sect. 4.6.

4.2 Hybrid Finite-Time Trajectory Tracking Technique for Quadrotor This part presents a robust control structure for a quadrotor under disturbances. The hybrid finite-time control method is divided into two attitude and position loops. For the position loop, an adaptive backstepping (AB) technique is designed for the altitude subsystem. The backstepping techniques proposed for the tracking of the desired altitude z d (t) and an accurate estimation of the unknown disturbance dz (t) acting from the altitude subsystem. The backstepping method is used to generate the virtual command (Vx , Vy ) for the horizontal position. For the attitude loop, a terminal ISMC (TISMC) is proposed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_4

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4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

4.2.1 Control of the Altitude Subsystem by Adaptive Backstepping The Lyapunov candidate function is considered as follows, Vz1 =

1 2 e (t) 2 11

(4.1)

By taking the time derivative of the equation V˙z1 = e11 (t)e˙11 (t) = e11 (t)(X12 − z˙ d (t))

(4.2)

The virtual control law is, αz = −cz1 e11 (t) + z˙ d (t)

(4.3)

with cz1 > 0 is a positive constant. Define the tracking error in step 2 as: ˙ 12 − αz ez2 (t) = X

(4.4)

The virtual control and adaptive law of the altitude subsystem are given by Eq. (4.5) and Eq. (4.6), respectively. Vz = −(e11 (t) + cz1 (ez2 (t) − cz1 e11 (t)) + cz2 ez2 (t) − ρ Z X12 + g + dˆz (t) d˙ˆz (t) = γz ez2 (t)

(4.5)

(4.6)

where cz2 , γz are the positive constants and dˆz (t) indicates the estimate of dz (t). Remark 4.1 The proposed control method requires that disturbances be limited. It should be noted that the above constraints on limiting disturbances are reasonable since all external effects on the quadcopter are in fact limited and bounded in practice. Theorem 4.1 Consider the dynamic model of the altitude subsystem. Then, the errors between the actual and desired outputs converge to zero and this system is located at the desired position at infinity according to the laws presented in (4.5) and (4.6). Proof The Lyapunov function is considered as follows: Vz2 =

1 2 1 2 1 ˜2 e (t) + ez2 (t) + d (t) 2 11 2 2γz z

The time derivative of Vz2 is given by

(4.7)

4.2 Hybrid Finite-Time Trajectory Tracking Technique for Quadrotor

83

1 V˙z2 = e11 (t)e˙z1 + ez2 (t)e˙z2 (t) + d˜z (t)d˙˜z (t) + e11 (t)(ez2 (t) − cz1 e11 (t)) γz 1 + ez2 (t)(X11 + cz1 e˙z1 (t) − z¨ d (t)) − d˜z (t)dˆ˙z (t) γz = −cz1 e11 (t)2 + ez2 (t)[ρ Z X12 − g + Vz + cz1 (ez2 (t) − cz1 e11 (t)) − z¨ d (t)] dˆ˙z (t) + ez2 (t)dˆz (t) + dz (t)(ez2 (t) − ) γz (4.8) Considering the control law equation (4.5) and the adaptive law equation (4.6), we obtain 2 2 (t) − cz2 ez2 (t) ≤ 0 V˙z2 = −cz1 e11

(4.9)

The condition of the stability of the altitude subsystem is ensured by the inequality in the Eq. (4.9). Remark 4.2 If one seeks to further improve this control, in addition to using an adaptive technique to counteract the complex unknowns, disturbance observers can likely be designed to estimate the disturbances. However, since the position of the quadrotor is an underactuated system, therefore it becomes difficult to estimate the perturbations and ensure convergence to the desired trajectory, simultaneously.

4.2.2 Control by Backstepping Technique for the Horizontal Position Subsystem The adaptive backstepping controller applied to the altitude subsystem can guarantee the fast convergence of the quadrotor to the desired altitude and estimate the disturbances acting on the vertical axis. However, for the position subsystem, there are 3 outputs (z(t), x(t), y(t)) that must be provided by a single control input u m . Therefore, it is impossible to design an AB controller to reach the output, we try to design a backstepping based controller for the horizontal subsystem. The Lyapunov candidate functions for the horizontal position are defined as Vx1 =

1 2 1 e7 (t), Vy1 = e92 (t) 2 2

(4.10)

The time derivative of the Eq. (4.10) is V˙ y1 = e9 (t)e˙9 (t) = e9 (t)(X9 − y˙d (t)) (4.11) According to the Eq. (4.11), the virtual control laws are, V˙x1 = e7 (t)e˙7 (t) = e7 (t)(X7 − x˙d (t)),

αx = −cx1 e7 (t) + x˙d (t)

α y = −c y1 e9 (t) + y˙d (t)

(4.12)

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4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

with cx1 , c y1 > 0 are the positive constants. Consider the tracking errors in step 2 as ex2 (t) = X8 − αx

e y2 (t) = X10 − α y

(4.13)

The virtual control laws of the horizontal subsystems are given by the Eq. (4.14). Vx = −(e7 (t) + cx1 (ex2 (t) − cx1 e7 (t)) + cx2 ex2 (t)) − ρ X X8 Vy = −(e9 (t) + c y1 (e y2 (t) − c y1 e9 (t)) + c y2 e y2 (t)) − ρ X X10

(4.14) (4.15)

Proof Using the same procedure, presented in the previous part demonstrates the stability of the subsystem of the horizontal position.

4.2.3 Attitude Control Using the Full Terminal Sliding Mode Technique As mentioned before, the synthesis of the sliding surface is a very important step. Their design may result in unacceptable performance. The selection of the sliding surface is a tedious and complicated task [2]. The objective of this section is to design a robust control for the vehicle orientation using TISMC to ensure the attainment of the sliding mode as well as the tracking control in a finite time for the quadrotor subsystem subjected to external disturbances. Consider the TISMC surface of the quadrotor attitude as follows: [3–5]. ⎧  ⎪ s1 (t) = e˙1 (t) + ⎪ ⎪ ⎨  s3 (t) = e˙3 (t) + ⎪ ⎪  ⎪ ⎩ s5 (t) = e˙5 (t) +

  q q α e˙1 (t) p + β e1 (t) 2 p −q dt   q q α e˙3 (t) p + β e3 (t) 2 p −q dt   q q α e˙5 (t) p + β e5 (t) 2 p −q dt

(4.16)

where αi et βi |(i=,,) are the positive parameters and pi are qi positive integers with pi > qi . The surface dynamics is given by ⎧ q q ⎪ ⎨ s˙1 (t) = e¨1 (t) + α e˙1 (t) qp + β e1 (t) 2 pq −q   s˙3 (t) = e¨3 (t) + α e˙3 (t) p + β e3 (t) 2 p −q ⎪ q q   ⎩ s˙5 (t) = e¨5 (t) + α e˙5 (t) p + β e5 (t) 2 p −q

(4.17)

Using the exponential laws of attainment, we obtain, ⎧ ⎨ s˙1 (t) = −λ s1 (t) − k sign(s1 (t)) s˙3 (t) = −λ s3 (t) − k sign(s3 (t)) ⎩ s˙5 (t) = −λ s5 (t) − k sign(s5 (t))

(4.18)

4.2 Hybrid Finite-Time Trajectory Tracking Technique for Quadrotor

85

According to the Eq. (4.17) and the Eq. (4.21), the control laws of the attitude subsystem are given as follows, ⎡ q ⎤  ¨ d (t) − α e˙1 (t) p 1 ⎣− ρ1 X4 X6 + ρ2 X4 + ρ3 X22 − λ s1 (t) − k sign(s1 (t)) +  ⎦ U = q ρ1 −β e1 (t) 2 p −q ⎡ q ⎤  ¨ d (t) − α e˙3 (t) p 1 ⎣− ρ1 X2 X6 + ρ2 X2 + ρ3 X24 − λ s3 (t) − k sign(s3 (t)) +  ⎦ U = q ρ2 −β e3 (t) 2 p −q ⎡ q ⎤  ¨ d (t) − α e˙5 (t) p 1 ⎣− ρ1 X2 X4 + ρ2 X26 − λ s5 (t) − k sign(s5 (t)) +  ⎦ (4.19) U = q ρ3 −β e5 (t) 2 p −q

Theorem 4.2 Consider the  subsystem and the ITSM surface defined in (4.16) then the origin (e1 (t), e˙1 (t)) is globally finite and stable in time by the law developed in the Eq. (4.19). Proof To demonstrate the stability of the roll subsystem, the Lyapunov candidate function is given as follows: V =

1 2 s (t) 2 1

(4.20)

The time derivative of V is given by: V˙ = s1 (t)˙s1 (t)   q q ˙ d (t) + α e˙1 (t) p + β e1 (t) 2 p −q = s1 (t) X2 −    q q ˙ d (t) + α e˙1 (t) p + β e1 (t) 2 p −q = s1 (t) ρ1 X4 X6 + ρ2 X4 + ρ3 X22 + ρ1 U + − 1 = s1 (t)(ρ1 X4 X6 + ρ2 X4 + ρ3 X22 + ρ1 q

 1 [− ρ1 X4 X6 + ρ2 X4 + ρ3 X22 − λ s1 (t) ρ1 q

q

q

¨ − α e˙1 (t) p − β e1 (t) 2 p −q ] −  ¨ d (t) + α e˙1 (t) p + β e1 (t) 2 p −q ) − k sign(s1 (t)) +  = s1 (t)(−λ − k sign(s1 (t))) = −λ s12 (t) − k |s1 (t)|

(4.21)

4.3 Analysis of the Simulation Results by the Hybrid Control To validate the performance of the proposed controllers, numerical simulations will be presented in this section. The initial attitude and position of the quadrotor are chosen as [0, 0, 0]rad and [0, 0, 0]m. The desired yaw angle path and position are shown in the Table 4.1.

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4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

Table 4.1 The reference trajectories of the position and the yaw angle Variable Value Time (s) [xd (m), yd (m), zd (m)]

[d (rad)]

[0.6, 0.6, 0.6] [0.3, 0.6, 0.6] [0.3, 0.3, 0.6] [0.6, 0.3, 0.6] [0.6, 0.6, 0.6] [0.6, 0.6, 0.0] [0.5] [0.0]

0 10 20 30 40 50 0 50

Remark 4.3 In order to achieve fast and smooth tracking performance, the design parameters of the proposed AB-RITSTM, B-SMC, and SMC techniques were tuned using a toolbox optimization method in MATLAB/Simulink (see for Example [6]). Furthermore, in order to highlight the superiority of the proposed control laws, comparisons with the sliding mode backstepping control and the first-order sliding mode control technique are performed. The simulation results are presented in Figs. 4.2, 4.3, 4.4, 4.5 and 4.6. The desired and actual tracking positions of variables x(t), y(t) and z(t) are shown in Fig. 4.1, where the proposed control strategy that causes the quadrotor to follow the desired flight path faster and more accurately than the conventional sliding mode control method and the sliding mode control with backstepping can be seen. Constant disturbances are added in the x(t) position subsystem at t = 5. It seems that the proposed control approach is managed to efficiently maintain the quadrotor position in finite time unlike MSC and B-SMC, the same behavior can be observed at t =15 for the y(t) position and at t = 25 for the z(t) position of the quadrotor system. Moreover, the attitude of the trajectory (t), (t) and (t) is shown in Fig. 4.2. We can see that the attitude of the quadrotor follows the desired angles in a short and finite time. The attitude slip variables (s1 (t), s3 (t), s5 (t)) are shown in Fig. 4.4, the convergence to zero in finite time of the slip surfaces can be observed. The trajectory tracking errors of the position are shown in Fig. 4.3. The estimated force acting in the z direction is shown in Fig. 4.6. To demonstrate the superiority of the proposed control strategy, the performance of the quadrotor trajectory in 3-D space is shown in Fig. 4.5. We can clearly see that the proposed strategy can accurately track the quadrotor trajectory in the presence of external disturbances. The proposed control approach obtains better performance compared to the classical SMC and B-SMC methods in terms of disturbance rejection and trajectory tracking. These results obtained in this section show that the proposed controls can reduce the control effort as well as guarantee a better trajectory tracking. However, the little reluctance present in the control signals may not be accepted by the controlled systems. In this section we

4.3 Analysis of the Simulation Results by the Hybrid Control

87

Fig. 4.1 Quadrotor position (x(t), y(t), z(t))

have presented a hybrid finite-time trajectory tracking control for the quadrotor by improving the following: 1. Precise tracking of a quadcopter with complex unknowns is solved. 2. The control and the observation at finite time are carried out simultaneously. 3. A compound control scheme with precise tracking is created. The reticence phenomenon caused, the discontinuous laws in the sliding mode control and the overshoot of the quadrotor time response generates physical oscillations when arriving at the waypoints. To address these undesirable behaviors, we suggest the use of a control strategy based on the super-twisting algorithm. The technique of tuning controller parameters increases tracking performance to decrease the energy consumed. Moreover, the hybrid representation of ISMC controller and adaptive backstepping is adapted in order to generate a controller able to reject the disturbances acting on the vehicle.

88

4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

Fig. 4.2 Quadrotor attitude ((t), (t), (t))

4.4 Adaptive Global Nonlinear SMC for a Quadrotor [1] The problem of the trajectory tracking of a QUAV under complex disturbances and tacking accounts the initial conditions of the state variables of this vehicle is addressed in this section. Then, an adaptive global nonlinear SMC (AGNSMC) technique for the position/attitude subsystems is proposed. The procedure of the proposed control method is given as: Firstly, tacking the sliding variables presented in Mobayen S, Tchier F and Ragoub L [7] for the quadrotor state variables. s1 (t) = Hφ (g1 (t) − g1 (0) exp(−β1 t)) s3 (t) = Hθ (g3 (t) − g3 (0) exp(−β3 t))

(4.22a) (4.22b)

s5 (t) = Hψ (g5 (t) − g5 (0) exp(−β5 t)) s7 (t) = H7 (g7 (t) − g7 (0) exp(−β7 t))

(4.22c) (4.22d)

s9 (t) = H9 (g9 (t) − g9 (0) exp(−β9 t)) s11 (t) = H11 (g11 (t) − g11 (0) exp(−β11 t))

(4.22e) (4.22f)

e7 (m)

4.4 Adaptive Global Nonlinear SMC for a Quadrotor [1]

89

B-ITSMC

0.2

SMC

B-SMC

0 -0.2 -0.4 -0.6

0

10

20

30

40

50

60

70

80

time (s) 0.5

e9 (m)

B-ITSMC

SMC

B-SMC

0

-0.5 0

10

20

30

40

50

60

70

80

time (s)

e11 (m)

0.5

B-ITSMC

SMC

B-SMC

0

-0.5 0

10

20

30

40

50

60

70

80

time (s)

Fig. 4.3 Tracking errors

where g1 (t) = e˙1 (t) + λ1 e1 (t)(t) g3 (t) = e˙3 (t) + λ3 e3 (t)(t)

(4.23a) (4.23b)

g5 (t) = e˙5 (t) + λ5 e5 (t)(t) g7 (t) = e˙7 (t) + λ7 e7 (t)(t) g9 (t) = e˙9 (t) + λ9 e9 (t)(t)

(4.23c) (4.23d) (4.23e)

g11 (t) = e˙11 (t) + λ11 e11 (t)(t)

(4.23f)

with Hi for i = φ, θ, ψ, x, y, z is a positive constant. λi and βi are the positive parameters. Using the GSMC, the tracking error is forced to approach the sliding variables from the initial time, then move in this sliding mode variables to the origin. Remark 4.4 Unlike the classical integral sliding surface, the global sliding surface forces the error to reach the surface from the initial time. Hence, the robust perfor-

90

4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique 1 B-ITSMC

s1

0.5

SMC

B-SMC

0

-0.5 0

10

20

30

40

50

60

70

80

time (s) 2 B-ITSMC

s3

1

SMC

B-SMC

0 -1 -2

0

10

20

30

40

50

60

70

80

time (s) 6 4 B-ITSMC

s5

2

SMC

B-SMC

0 -2

0

10

20

30

40

50

60

70

80

time (s)

Fig. 4.4 Sliding mode variables (s1 (t), s3 (t), s5 (t)) Fig. 4.5 Trajectory tracking in 3D space

Fig. 4.6 The estimation of the parameter dz (t)

1

^

^

dz

dz

dz

0.5 0 0

10

20

30

40

time (s)

50

60

70

80

4.4 Adaptive Global Nonlinear SMC for a Quadrotor [1]

91

mance of the quadrotor system in the presence of disturbances the disturbances is satisfied. The time-derivative of the sliding surface of the position variable x(t) is given by: s˙7 (t) = H7 (g˙ 7 (t) + β7 g7 (0) exp(−β7 t)) = H7 (¨e7 (t) + λ7 e˙7 (t) + β7 g7 (0) exp(−β7 t))

(4.24)

where e¨7 = ρ X X8 + Vx + dx (t) − x¨d (t). Substituting the double time derivative of tracking error into (4.24), we have, s˙7 (t) = H7 (ρ X X8 + Vx + dx (t) − x¨d (t) + λ7 e˙7 (t) + β7 g7 (0) exp(−β7 t))

(4.25)

The equivalent control law for x-subsystem can be found when the time derivative equal zero i.e. s7 (t) with dx (t). Then, the control Vxe is given as: Vxe = x¨d (t) − ρ X X8 − λ7 e˙7 (t) − β7 g7 (0) exp(−β7 t))

(4.26)

However, the equivalent control signal cannot ensure the tracking performance in the presence of uncertainties and disturbances. to enhance the control performance, a switching control signal is designed. In addition, we assume that the additive disturbance on the quadrotor system unknown bounded i.e. K 7 > |dx (t)|. Thus, the control law can be developed as: Vxs = −sign(s7 (t))[ 7 s7 (t) + Kˆ 7 sign(s7 (t)H7 )]

(4.27)

The term 7 represents a positive gain, the sign indicates the signum function, and Kˆ 7 represents the estimate of K 7 . The dynamics of Kˆ can be defined by the following equation. K˙ˆ 7 = ξ7 |s7 (t)H7 |

(4.28)

where ξ7 is a positive coefficient. Then, the global control law for x-subsystem can be written as: Vx = Vxe + Vxs = x¨d (t) − ρ X X8 − λ7 e˙7 (t) − β7 g7 (0) exp(−β7 t)) − sign(s7 (t))[ 7 s7 (t) + Kˆ 7 sign(H7 s7 (t))]

(4.29)

Similar to x(t)-subsystem, the controller design process of other subsystems can be defined. We use the same design presented for x(t)-subsystem, the control laws and adaptive laws for other subsystems are given as:

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4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

Position variable y(t) motion controller: ⎧ ⎪ ⎨ Vy = y¨d (t) − ρY X10 − λ9 e˙9 (t) − β9 g9 (0) exp(−β9 t)) −sign(s9 (t))[ 9 s9 (t) + Kˆ 9 sign(H9 s9 (t))] ⎪ ⎩ K˙ˆ = ξ |s (t)H | 9 9 9 9

(4.30)

Position variable z(t) motion controller: ⎧ ⎪ ⎨ Vz = z¨ d (t) − ρ Z X12 − λ11 e˙11 (t) − β11 g11 (0) exp(−β11 t)) −sign(s11 (t))[ 11 s11 (t) + Kˆ 11 sign(H11 s11 (t))] ⎪ ⎩ K˙ˆ = ξ |s (t)H | 11 11 11 11

Roll (t) motion controller: ⎧ u  = ρ11 {φ¨ d (t) − (ρ1 X4 X6 + ρ2 X4 + ρ3 X2 2) − λ1 e˙1 (t) ⎪ ⎪ ⎪ ⎨ −β1 g1 (0) exp(−β1 t)) ˆ −sign(s ⎪ 1 (t))[ 1 s1 (t) + K 1 sign(H1 s1 (t))]} ⎪ ⎪ ⎩ ˙ˆ K 1 = ξ1 |s1 (t)H1 | Pitch (t) motion controller: ⎧ U = ρ12 {φ¨ d (t) − (ρ1 X2 X6 + ρ2 X2 + ρ3 X24 ) − λ3 e˙5 (t) ⎪ ⎪ ⎪ ⎨ −β3 g3 (0) exp(−β3 t)) ˆ −sign(s ⎪ 3 (t))[ 3 s3 (t) + K 3 sign(H3 s3 (t))]} ⎪ ⎪ ⎩ ˙ˆ K 3 = ξ3 |s3 (t)H3 | Yaw (t) motion controller: ⎧ U = ρ13 {ψ¨ d (t) − (ρ1 X2 X4 + ρ2 X26 ) − λ5 e˙5 (t) ⎪ ⎪ ⎪ ⎨ −β5 g5 (0) exp(−β5 t)) ˆ −sign(s ⎪ 5 (t))[ 5 s5 (t) + K 5 sign(H5 s5 (t))]} ⎪ ⎪ ⎩ ˙ˆ K 5 = ξ5 |s5 (t)H5 |

(4.31)

(4.32)

(4.33)

(4.34)

The control signals u m , u  , u  , and u  are influenced by the following saturation function ⎧ ⎨ Ui L U¯ i < Ui L ¯ Ui = sat (Ui L , Ui , Ui L ) = (4.35) U¯ Ui L < Ui L < UiU ⎩ i UiU U¯ i > UiU where Ui L and UiU (i = 1, 2, 3, 4) are the lower/upper limit bounds of the four control inputs respectively. Once the saturation occurs, both tracking errors increase, which leads to an oscillation in the adaptive laws and decreased tracking performance. To reduce the influence of saturation nonlinearity, modifications and compensations in the design of the control laws are introduced as shown in the Ref. [8].

4.4 Adaptive Global Nonlinear SMC for a Quadrotor [1]

93

4.4.1 Stability Analysis of the Proposed Controller The result of the ANGSMC is listed in Theorem 4.1 and its proof. Theorem 4.1 Consider the x(t)-subsystem and the nonlinear sliding variable (4.22d), if the control Vx is designed as in (4.29) and the adaptive law is designed as in (4.28), then state variables of x(t)-subsystem are converged to s7 (t) = 0 in finite-time. Proof a Lyapunov function V7 is chosen as follows: V7 =

1 2 1 ˆ s7 (t) + ( K 7 − K 7 )2 2 2ξ7

(4.36)

The derivative of Vx with respect to time is, ˙ 7 = s7 (t)˙s7 (t) + 1 ( Kˆ 7 − K 7 ) K˙ˆ 7 V ξ7

(4.37)

Substituting the tracking error and (4.29) into Eq. (4.22d), the time-derivative of s7 (t) can be written as: s˙7 (t) = H7 {−sign(s7 (t))[ 7 s7 (t) + Kˆ 7 sign(s7 (t)H7 )] + dx (t)} = −H7 7 |s7 (t)| − H7 Kˆ 7 |s7 (t)H7 | + H7 dx (t)

Substituting Eqs. (4.38) and (4.28) into Eq. (4.37), we can get, ˙ 7 = s7 (t){−H7 7 |s7 (t)| − H7 Kˆ 7 |s7 (t)H7 | + H7 dx (t)} V 1 + ( Kˆ 7 − K 7 ) K˙ˆ 7 ξ7 = − H7 7 s7 (t)2 − Kˆ 7 (s7 (t)H7 )2 + H7 s7 (t)dx (t) 1 ˆ ( K 7 − K 7 ) K˙ˆ 7 ξ7 = − H7 7 s7 (t)2 − Kˆ 7 (s7 (t)H7 )2 + H7 s7 (t)dx (t) +

1 ˆ ( K 7 − K 7 )ξ7 |s7 (t)H7 | ξ7 = − H7 7 s7 (t)2 − Kˆ 7 [(s7 (t)H7 )2 − |s7 (t)H7 |] +

+ H7 s7 (t)dx (t) − K 7 |s7 (t)H7 |  − H7 7 s7 (t)2 − Kˆ 7 [(s7 (t)H7 )2 − |s7 (t)H7 |] + H7 |s7 (t)|.|dx (t)| − K 7 |s7 (t)H7 | = − H7 7 s7 (t)2 − Kˆ 7 [(s7 (t)H7 )2 − |s7 (t)H7 |] − H7 |s7 (t)|(K 7 − |dx (t)|)

(4.38)

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4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

 − H7 7 s7 (t)2 − Kˆ 7 [(s7 (t)H7 )2 − |s7 (t)H7 |] 0

(4.39)

From the above analysis, it can be observed that Theorem 4.1 is proved. In order to prove the stability of other subsystems, the same stability process as the position x(t)-subsystem can be used.

4.5 Results and Discussion In this section, simulation results are presented to show the performance tracking and to validate the efficiency of the AGNSMC proposed in this work. The suggested controller is compared with the results of the method of Labbadi and Cherkaoui presented in Ref. [9]. The initial conditions are selected as [0.05, 0.05, 0.01]m and = [0.01, 0.01, 0.01]rad. The desired trajectory is given by: 

0.6 m i f  10 or t > 30 0.3 m other wise

(4.40)

0.6 m i f  20 or t > 40 0.3 m other wise

(4.41)

0.6 m i f  50  (t) = 0.5 rad 0 m other wise d

(4.42)

xd (t) =  yd (t) =  z d (t) =

The disturbances used in the simulation are chosen as shown in Figs. 4.7 and 4.8, respectively. In order to test the performance of the proposed controller, the variation of the drag coefficients are considered as shown in Fig. 4.9.

Fig. 4.7 Disturbances applied on position subsystem

4.5 Results and Discussion

95

Fig. 4.8 Disturbances applied on attitude subsystem

Fig. 4.9 Variation effect of the drag coefficients

Remark 4.5 In order to avoid the chattering phenomena, the signum function can be modified by the tangent hyperbolic. The simulation results of the AGNSMC and the ST-PID are plotted in Figs. 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18 and 4.19. As shown in Figs. 4.10, 4.11, 4.12 and 4.13, the AGNSMC control smethod has successfully maintained the position/attitude in short finite-time. The attitude and position of the QUAV are affected by abruptly changed flight trajectory. Then, the ANGSM control method is capable of driving the quadrotor’s outputs back to the new reference angles and position. Moreover, the disturbances and the variation of the drag coefficients are taken into account in the AGNSMC design. Those show the robustness of the designed controller and the effectiveness of the AGNSMC control technique. Those show the efficiency of the developed controller and its robustness against the disturbances. The linear and angular velocities, depicted in Figs. 4.12 and 4.13, respectively, depict the same behavior as the angles and positions. It is shown in Fig. 4.14, the AGNSMC has archived a good tracking in 3D space under disturbances. The control signals are

96

4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

Fig. 4.10 Position tracking of the proposed control design

Fig. 4.11 Attitude tracking of the proposed control design

4.5 Results and Discussion

Fig. 4.12 Linear velocity tracking of the proposed control design

Fig. 4.13 Angular velocity tracking of the proposed control design

97

98

4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

z (m)

0.6 0.4

Reference STISMC AGNSMC

0.2 0 0.6 0.4 0.2

y (m)

0

0

0.2

0.4

0.6

x (m)

Fig. 4.14 Square trajectory with proposed control design

Fig. 4.15 Control inputs

depicted in Fig. 4.15. It is obvious from Fig. 4.15 these signals are continuous. The time responses of the sliding variables are displayed in Figs. 4.16 and 4.17, we can observe that all these variables converge to their origins exponentially. In the other hand, the responses of the adaptive parameters are depicted in Figs. 4.18 for the QUAV position and Fig. 4.19 for the QUAV attitude. These parameters converge to their desired values. Then, the method proposed in the part achieves an excellent control tracking against the external disturbances. As a result, the AGNSM controller is validated via these results.

4.5 Results and Discussion Fig. 4.16 Position nonlinear global sliding surfaces

Fig. 4.17 Atitude nonlinear global sliding surfaces

Fig. 4.18 Position parameter estimations

99

100

4 Control of the QUAV by a Hybrid Finite-Time Tracking Technique

Fig. 4.19 Attitude parameter estimations

4.6 Conclusion In this chapter, we have presented two flight control strategies: • A control strategy consisting of three different parts: the control of an altitude, a horizontal position and an attitude subsystem. First, the altitude subsystem is addressed on the basis of adaptive backstepping. In addition, the backstepping technique is designed to control the x(t) and y(t) position. • an adaptive global nonlinear SMC technique is designed for QUAV subjected to uncertainties/disturbances. The results of the AGNSM control method shown the robustness against disturbances, eliminates the reaching interval in the initial time of the flight and confirm its effectiveness. Also, the upper bounds of the disturbances are estimated based on adaptive law.

References 1. Labbadi, M. et al.: Robust adaptive global nonlinear sliding mode controller for a quadrotor under external disturbances and uncertainties. Adv. Mech. Eng. 12(11), 168781402097523 (2020). Available at: http://dx.doi.org/10.1177/1687814020975237 2. Kali, Y.: Control of Uncertain Non-Linear Systems by Sliding Modes with Estimation by Delay Application to a Redundant Manipulator Robot. Mohammed V University of Rabat (2018) 3. Riani, A.: Control and observation of exoskeletons for upper limb functional rehabilitation. Robotics [cs.RO]. Paris-Saclay University. NNT: 2018SACLV026.tel-01906453 (2018) 4. Zhang, R., Dong, L., Sun, C.: Adaptive nonsingular terminal sliding mode control design for near space hypersonic vehicles. IEEE/CAA J. Autom. Sin. 1, 155–161 (2014). https://doi.org/ 10.1109/JAS.2014.7004545 5. Riani, A., Madani, T., Benallegue, A., Djouani, K.: Adaptive integral terminal sliding mode control for upper-limb rehabilitation exoskeleton. Control Eng. Pract. 75, 108–117 (2018). https:// doi.org/10.1016/j.conengprac.2018.02.013

References

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6. Freire, F.P., Martins, N.A., Splendor, F.: A simple optimization method for tuning the gains of PID controllers for the autopilot of Cessna 182 aircraft using model-in-the-loop platform. J. Control. Autom. Electr. Syst. 29, 441–450 (2018). https://doi.org/10.1007/s40313-018-0391-x 7. Mobayen, S., Tchier, F., Ragoub, L.: Design of an adaptive tracker for n-link rigid robotic manipulators based on super-twisting global nonlinear sliding mode control. Int. J. Syst. Sci. 48, 1990–2002 (2017) 8. Li, S., Wang, Y., Tan, J.: Adaptive and robust control of quadrotor aircrafts with input saturation. Nonlinear Dyn. 89, 255–265 (2017) 9. Labbadi, M., Cherkaoui, M.: Novel robust super twisting integral sliding mode controller for a quadrotor under external disturbances. Int. J. Dyn. Control 8, 805–815 (2020). https://doi.org/ 10.1007/s40435-019-00599-6

Chapter 5

Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

5.1 Introduction In this chapter, the problem of controlling the orientation and position of the quadrotor is considered in the presence of parametric uncertainties and external disturbances. Previous works usually assume that the parameters of the flight controllers are constants. In reality, these parameters depend on the desired trajectory. Two approaches to control the quadrotor have been proposed. The control techniques are designed on the basis of on-line estimators of the dynamic parameters. These adaptation methods allow to improve the control performance of this system, and to compensate the parametric errors due to the coupling of the position with the orientation of the UAV. The first control is based on the robust nonlinear fast control, structured for the tracking of the position and attitude trajectory of the quadrotor. Two control loops have been designed, an outer loop (position loop) uses a new version of the adaptive backstepping (AB) based control while the inner loop (attitude loop) uses a new controller based on a combination of the backstepping technique and the fast terminal sliding mode control (AB-FTSMC) to control the yaw motion and the tilt angles [1]. In order to estimate the proposed parameters of the position controller and the upper bounds of uncertainties and attitude perturbations, adaptive online rules are proposed. Different simulations have been performed in MATLAB environment to show the effectiveness of the proposed controller. The supremacy, of the controller proposed in this chapter, is highlighted by comparing its performances with various approaches such as the classical sliding mode control, the integral backstepping and the second order sliding mode control. The rest of the chapter is presented as follows: the Sect. 5.2 expresses the finitetime adaptive flight control. The Sect. 5.3 presents the results and discussion of this control. The conclusion is given in Sect. 5.4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_5

103

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5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

5.2 Finite-Time Adaptive Flight Control of a Quadrotor This section is devoted to the design of the adaptive flight controller for the quadrotor system. The objective of this control law is to design the quadrotor position (lateral and vertical) and attitude control with an adaptive mechanism to estimate the upper bounds of external factors acting on the vehicle. Under the flight controller proposed in this part, the stability in finite time is ensured in closed loop and the trajectories follow their references. In what follows, a new controller is proposed by integrating a fast terminal slip mode controller and backstepping theories for attitude trajectory tracking control. The BFTSMC technique is designed for fast convergence of attitude state variables to stabilize pitch/roll angles and yaw angle trajectory tracking in finite time. The regular backstepping control technique is designed to ensure the tracking of the desired flight position trajectory.

5.2.1 Quadrotor Position Control by a New Adaptive Backstepping Method The adaptive backstepping technique has a good robustness for flight path tracking problems. In this section we propose an original method based on backstepping to deal with uncertainties and disturbances up to a certain level and to stabilize the system based on the Lyapunov approach. The first step of the design of this control is the same as the one presented in Chap. 3, which leads us to the direct application of the first Lyapunov function. Let us define the first Lyapunov function as follows: ⎤ ⎡1 2 ⎤ e (t) v7 2 7 ⎣ v9 ⎦ = ⎣ 1 e92 (t) ⎦ 2 1 2 v11 e (t) 2 11

(5.1)

⎤ ⎡ ⎤ ⎡ ⎤ e7 (t)e˙7 (t) v˙ 7 e7 (t)(X8 − x˙d (t)) ⎣ v˙ 9 ⎦ = ⎣ e9 (t)e˙9 (t) ⎦ = ⎣ e9 (t)(X10 − y˙d (t)) ⎦ v˙ 11 e11 (t)e˙11 (t) e11 (t)(X12 − z˙ d (t))

(5.2)

⎡

The time-derivative of (5.1) is ⎡

Virtual control inputs x8d , x10d and x12d are selected [6] as follows: ⎡

⎤ x8d = −cx1 e7 (t) + x˙d (t) ⎣ x10d = −c y1 e9 (t) + y˙d (t) ⎦ x12d = −z z1 e11 (t) + z˙ d (t) where cx1 , c y1 and cz1 are positive non-zero constants.

(5.3)

5.2 Finite-Time Adaptive Flight Control of a Quadrotor

105

By replacing (5.3) by (5.2) this gives the functions of Lyapunov derivatives. ⎡

⎤ ⎡ ⎤ v˙ 7 −cx1 e72 (t) ⎣ v˙ 9 ⎦ = ⎣ −c y1 e92 (t) ⎦ ≤ 0 2 v˙ 11 (t) −cz1 e11

(5.4)

Consequently, the Lyapunov theorem will ensure the stability of the quadrotor position. Step 2: The second tracking errors are given by: ⎡

⎤ ⎤ ⎡ e8 (t) X8 − x8d ⎣e10 (t)⎦ = ⎣X10 − x10d ⎦ X12 − x12d e12 (t)

(5.5)

In the same way, the Lyapunov candidate functions of step 2 are determined as ⎡

⎤ ⎤ ⎡ v7 + 21 e82 (t) v8 2 ⎣v10 ⎦ = ⎣ v9 + 1 e10 (t) ⎦ 2 1 2 v12 v11 + 2 e12 (t)

(5.6)

The time derivative of Lyapunov functions is ⎡

⎤ ⎡ ⎤ v˙ 8 v˙ 7 + e8 (t)e˙8 (t) ⎣v˙ 10 ⎦ = ⎣ v˙ 9 + e10 (t)e˙10 (t) ⎦ v˙ 12 v˙ 11 + e12 (t)e˙12 (t)

(5.7)

According to (5.3), (5.5) and (5.7) we have ⎡

⎤ ⎡ ⎤ v˙ 8 −cx1 e72 (t) + e7 (t)e8 (t) + e8 (t)(ρ X X8 + Vx − x¨d (t)) ⎣v˙ 10 ⎦ = ⎣ −c y1 e92 (t) + e9 (t)e10 (t) + e10 (t)(ρY X10 + Vy − y¨d (t)) ⎦ 2 v˙ 12 (t) + e11 (t)e12 (t) + e12 (t)(ρ Z X12 + Vz − z¨ d (t)) −cz1 e11

(5.8)

The corresponding control laws for the position of the quadrotor are designed as follows: Vx =(−e7 (t) − cˆx2 e8 (t) − cx1 (e8 (t) − cx1 e7 (t)) − ρ X X8 + x¨d (t)) Vy =(−e9 (t) − cˆ y2 e10 (t) − c y1 (e10 (t) − c y1 e9 (t)) − ρY X10 + y¨d (t))

(5.9)

Vz =(−e11 (t) − cˆz2 e12 (t) − cz1 (e12 (t) − cz1 e11 (t)) − ρ Z X12 + z¨ d (t)) where cˆx2 , cˆ y2 and cˆz2 are the estimates of cx2 , c y2 and cz2 , respectively. Theorem 5.1 If the control laws presented in (5.9) with the adaptation law in (5.10) are applied to the position system of the quadcopter, the asymptotic stability of the system is guaranteed.

106

5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

Thus the adaptation laws of the parameters are indicated as follows: ⎧˙ ⎨ cˆx2 = γ7 e82 (t) 2 c˙ˆ = γ9 e10 (t) ⎩ ˙y2 2 cˆz2 = γ11 e12 (t)

(5.10)

where γ7 , γ9 and γ11 are positive constants. Proof To prove the stability of the system and determine the parameters cˆx2 , cˆ y2 and cˆz2 , we use the Lyapunov approach. For cˆx2 : we introduce the Lyapunov candidate function of the positional subsystem. v78 = v8 +

1 2 c 2γ7 x2

(5.11)

where cx2 represents the estimation error. The time derivative of (5.11) is 1 c˙ x2 cx2 γ7 1 = −cx1 e72 (t) − (cx2 − cx2 )e82 (t) − cx2 c˙ˆx2 γ7 1 = −cx1 e72 (t) − cx2 e8 (t)2 + cx2 (e82 (t) − c˙ˆx2 ) γ7

v˙ 78 = −cx1 e72 (t) − cˆx2 e82 (t) +

In Eq. (5.12), the term cx2 (e82 (t) −

(5.12)

1 ˙ cˆ ) γ7 x2

is equal to 0. Considering cx2 as constant, the time derivative of widetildecx2 is written as c˙ x2 = 0 − c˙ˆx2 . Thus, the dynamics of the Lyapunov function defined in the Eq. (5.12) becomes the following: v˙ 78 = −cx1 e72 (t) − cx2 e82 (t) ≤ 0

(5.13)

Thus, the stability condition is satisfied by (5.13). To ensure the stability of the position system, the Lyapunov candidate function is chosen as: vsp =

1 2 1 2 2 (t) + 1 2 (t) + e2 (t) + 1 (e (t) + e82 (t) + c + e92 (t) + e10 c2 + e11 c2 ) 12 2 7 γ7 x2 γ9 y2 γ11 z2

(5.14)

The time derivative of the Lyapunov position is 2 2 2 (t) − cz1 e11 (t) − cz2 e12 (t)) ≤ 0 v˙ sp = (−cx1 e72 (t) − cx2 e82 (t) − c y1 e92 (t) − c y2 e10 (5.15) where cx1 , c y1 , cz1 , cx2 , c y2 and cz2 are the position parameters. Thus, the stability of the position system is guaranteed by (5.14) and (5.15), ensuring the ability to track the flight path.

5.2 Finite-Time Adaptive Flight Control of a Quadrotor

107

In this section, an adaptive backstepping law for the trajectory control of a quadrotor UAV is designed. The adaptive laws are used to determine the second parameters of the proposed control. The stability of the position system is proved by Lyapunov analysis. Based on the previous backstepping and SMC laws, a robust adaptive controller fast terminal slip mode with backstepping for attitude trajectory tracking control is presented in the following.

5.2.2 Attitude Control of a Quadrotor Using the Adaptive Fast Terminal Sliding Mode Technique with the Backstepping Approach The adaptive backstepping controller for the quadrotor position presented above is effective in terms of flight path tracking. It guarantees the stability of the position subsystem and provides variable (x(t), y(t), z(t)) tracking capability. However, the controller has some shortcomings in the realization of attitude control. For this, a combination of fast terminal slip mode control and backstepping control is proposed to control the attitude of the quadrotor. The FTSMC has many advantages: suppression of the reticence phenomenon, convergence of the state variables in finite time, the non-linear surfaces of the FTSM allow a faster convergence and compensate for parametric variations and external disturbances. This section presents an adaptive fast terminal backstepping mode controller for attitude trajectory tracking control. The main objectives of the proposed controller are to ensure the stability of the quadrotor and to converge the Euler angle trajectory ((t), (t), (t)) to the reference trajectories (d (t), d (t), d (t)). In order to ensure the stability of the attitude, let us present the following first functions of the Lyapunov candidate. ⎡ ⎤ ⎡1 2 ⎤ e (t) v1 2 1 ⎣v3 ⎦ = ⎣ 1 e32 (t)⎦ 2 1 2 v5 e (t) 2 5

(5.16)

⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ˙ d (t)) e1 (t)(X2 −  v˙ 1 e1 (t)e˙1 (t) ⎣v˙ 3 ⎦ = ⎣e3 (t)e˙3 (t)⎦ = ⎣e3 (t)(X4 −  ˙ d (t))⎦ ˙ d (t)) v˙ 5 e5 (t)e˙5 (t) e5 (t)(X6 − 

(5.17)

Let us derive (5.16) as follows:

The virtual input controls of the system are selected [6] as follows: ⎤ ˙ d (t) x2d = s1 (t) − c e1 (t) +  ⎣x4d = s3 (t) − c e3 (t) +  ˙ d (t)⎦ ˙ d (t) x6d = s5 (t) − c e5 (t) +  ⎡

(5.18)

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5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

c , c and c are positive constants. s1 (t), s3 (t) and s5 (t) are the sliding surfaces of the attitude. Let’s choose the attitude sliding surfaces: Let’s use the fast terminal sliding surfaces [3, 6] as follows: ⎡

⎤ s1 (t) = e˙1 (t) + ϒ e1 (t) +  e1 (t) p /q ⎣ s3 (t) = e˙3 (t) + ϒ e3 (t) +  e3 (t) p /q ⎦ s5 (t) = e˙5 (t) + ϒ e5 (t) +  e5 (t) p /q

(5.19)

where (ϒ , ϒ , ϒ ,  ,  ,  ) are positive non-zero constants and 0 < ( qp , qp , qp ) < 1. The time derivative of the terminal slip surfaces (5.19) is as follows: ⎡

s˙1 (t) = e¨1 (t) + ϒ e˙1 (t) + ⎢ s˙ (t) = e¨ (t) + ϒ e˙ (t) + 3  3 ⎣ 3 s˙5 (t) = e¨5 (t) + ϒ e˙5 (t) +

⎤

p  e (t)( p /q −1) e˙1 (t) q  1 p  e (t)( p /q −1) e˙3 (t) ⎥ ⎦ q  3 p  e5 (t)( p /q −1) e˙5 (t) q

(5.20)

Remark 5.1 Note that the term ei (t)( pi −qi )/qi approaches infinity if ei (t) = 0, i.e. the singularity. However, to avoid this problem, a sliding surface is modified as follows [10]: si (t) = e˙i (t) + ϒi ei (t) + i (ei (t))

(5.21)

The function (ei (t)) is defined as follows, (ei (t)) =

pi

ei (t) qi , i f si (t) = 0 or si = 0, |ei (t)| > μi ei (t), i f si (t) = 0, |ei (t)|  μi p /qi

where si = e˙i (t) + ϒi ei (t) + i ei i and i = (, , ).

(5.22)

(t), μi indicates a small threshold constant,

In order to obtain stabilizing control laws of the quadrotor attitude, let us define the Lyapunov candidate functions. ⎤ ⎡ ⎤ ⎡ v1 + 21 s12 (t) v2 ⎣v4 ⎦ = ⎣v3 + 1 s32 (t)⎦ 2 v6 v5 + 21 s52 (t)

(5.23)

The time derivative of (5.23) is ⎡ ⎤ ⎡ ⎤ v˙ 1 + s1 (t)˙s1 (t) v˙ 2 ⎣v˙ 4 ⎦ = ⎣v˙ 3 + s3 (t)˙s3 (t)⎦ v˙ 6 v˙ 5 + s5 (t)˙s5 (t)

(5.24)

5.2 Finite-Time Adaptive Flight Control of a Quadrotor

109

According to (5.20), the Lyapunov functions can be written as follows: ⎡ ⎤ ⎡ −c e2 (t) + e (t)s (t) + s (¨e (t) + ϒ e˙ (t) + p  e (t)( p /q −1) e˙ (t) ⎤  1 1 1  1  1 1 v˙ 2 q  1 ⎥ ⎢ ⎥ ⎢ p 2 (t) + e (t)s (t) + s (t)(¨ ( p /q −1) e˙ (t) ⎥ ⎢ −c e e (t) + ϒ e ˙ (t) +  e (t) v ˙ 3 3 3 3  3 3 ⎣ 4⎦ = ⎣  3 q  3 ⎦ v˙ 6

−c e52 (t) + e5 (t)s5 (t) + s5 (t)(¨e5 (t) + ϒ e˙5 (t) +

p q

 e5 (t)( p /q −1) e˙5 (t)

(5.25) According to (5.25), the equivalent control law of the quadrotor attitude is designed as follows: ⎧ Ueq = ⎪ ⎪ ⎪ ⎪ + qp  e1 (t)( p /q −1) e˙1 (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ueq = p ( p /q −1) e˙ (t))  e (t) + ⎪  3 3 ⎪ q ⎪ ⎪ ⎪ ⎪ ⎪ Ueq = ⎪ ⎪ ⎩ p + q  e5 (t)( p /q −1) e˙5 (t))

1 ρ1 (−(ρ1 X4 X6 1 ρ2

¨ d (t) − ϒ (s1 (t) − c e1 (t)) + ρ2 X4 + ρ3 X22 ) + 

¨ d (t) − ϒ (s3 (t) − c e3 (t)) (−(ρ1 X2 X6 + ρ2 X2 + ρ3 X24 ) + 

1 ρ3 (−(ρ1 X2 X4

¨ d (t) − ϒ (s5 (t) − c e5 (t)) + ρ2 X26 ) + 

(5.26) The control laws for the attitude switching of the quadrotor are chosen as follows: ⎧ Us = ⎪ ⎪ ⎨ Us = ⎪ ⎪ ⎩U = s

1 (−k sign(s1 (t))) ρ1 1 (−k sign(s3 (t))) ρ2 1 (−k sign(s5 (t))) ρ3

(5.27)

Therefore, the set of control inputs are given by: ⎧ ⎨ U = Ueq + Us U = Ueq + Us ⎩ U = Ueq + Us

(5.28)

Therefore, the attitude AB-FTSM controls are designed as follows: U = + U = + U = +

1 ¨ d (t) − ϒ (s1 (t) − c e1 (t)) (5.29) ((ρ1 X4 X6 + ρ2 X4 + ρ3 X22 ) +  ρ1 p  e1 (t)( p /q −1) e˙1 (t) − kˆ sign(s1 (t))) q 1 ¨ d (t) − ϒ (s3 (t) − c e3 (t)) (−(ρ1 X2 X6 + ρ2 X2 + ρ3 X24 ) +  ρ2 (5.30) p  e3 (t)( p /q −1) e˙3 (t) − kˆ sign(s3 (t))) q 1 ¨ d (t) − ϒ (s5 (t) − c e5 (t)) (−(ρ1 X2 X4 + ρ2 X26 ) +  (5.31) ρ3 p  e5 (t)( p /q −1) e˙5 (t) − kˆ sign(s5 (t))) (5.32) q

110

5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

Remark 5.2 In order to solve the reluctance problem, the discontinuous control component (sign(.) function) in (5.32) is replaced by the tanh(.) function. In addition, the parameters of the adaptive laws are as follows: ⎧˙ ⎪ ⎨ kˆ = γ |s1 (t)| k˙ˆ = γ |s3 (t)| ⎪ ⎩˙ kˆ = γ |s5 (t)|

(5.33)

where (kˆ , kˆ , kˆ ) indicate the line estimate of (k , k , k ). γ , γ and γ are positive non-zero constants. Proof To prove the stability of the subsystem and determine the adaptation parameters. We will rely on a candidate Lyapunov function, for example the Lyapunov function for the roll subsystem is given as follows: v12 = v2 +

1 2 k 2γ 

(5.34)

where k is the corresponding estimation error. The time derivative of (5.34): v˙ 12 = −c e12 (t) + s1 (t)e1 (t) + s1 (t)(ρ1 X4 X6 + ρ2 X4 + ρ3 X22 + ρ1 u  p 1 + ϒ (s1 (t) − c e1 (t)) +  e3 (t)( p /q −1) e˙1 (t)) + k k˙  q γ (5.35) Replacing (5.32) by (5.35), we obtain 1 k k˙  v˙ 12 = −c e12 (t) + s1 (t)e1 (t) + s1 (t)(−kˆ sign(s1 (t))) + γ 1 = −c e12 (t) − s1 (t)(k − k )sign(s1 (t)) − k k˙ˆ γ 1 ˙ˆ = −c e12 (t) − k |s1 (t)| + k (|s1 (t)| − k ) γ = −c e12 (t) − k |s1 (t)| ≤0

(5.36)

Let us define tr = [tr  , tr  , tr  ] as the time to reach, when si (i = , , ) reaches zero. The attitude states can reach the terminal sliding surfaces in finite time tr . Using (5.19), we obtain lim si (t) = lim si (t) = [e˙i + ϒi ei (t) + i ei (t) pi /qi ] = 0

t→t f

t→t f

(5.37)

5.2 Finite-Time Adaptive Flight Control of a Quadrotor

111

where t f = tr + ts , ts is the time interval during which the initial error e(0) = [e1 (t)(0), e3 (t)(0), e5 (t)(0)] = 0, which is expressed as follows [5].  ts =

q ϒ e(tr  )(q − p )/q +  ln , ϒ (q − p )  q ϒ e(tr  )(q − p )/q +  , ln ϒ (q − p )   ϒ e(tr  )(q − p )/q +  q ln ϒ (q − p ) 

(5.38)

By defining i , ϒi , pi , qi  0, we obtain lim ei (t)(i = , , ) = 0,

t→t f

lim e˙i (t)(i = , , ) = 0

t→t f

(5.39)

The synoptic diagram of the ABFTSMC for the attitude of the quadrotor is presented in the Fig. 5.1. Finally, the control laws are obtained from (5.9) and (5.32) via the adaptive backstepping and adaptive terminal fast sliding mode controller techniques. However, these control laws ensure the stability of the quadrotor in closed loop and guarantee the achievement of a better flight path tracking.

Fig. 5.1 Block diagram of the AB-ABFTSMC for the control of the quadrotor

112

5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

5.3 Results and Discussion In this section, simulation results are presented to verify the effectiveness and performance of the adaptive fast backstepping terminal slip mode controller for tracking the position and attitude of the quadrotor UAV. The initial state values of the quadrotor for the simulation tests are [0, 0, 0]rad and [0, 0, 0]m. Remark 5.3 The gains of the proposed AB-ABFTSMC, full backstepping and SMC-2 controllers were tuned for smooth and fast tracking performance using a toolbox optimization in MATLAB/Simulink (Check Step Response Characteristics block). The quadrotor is controlled to follow the 3D square trajectory in the presence of aerodynamic forces and moments. In order to demonstrate the robustness of the proposed flight controller, the nonlinear slip mode controller, the full backstepping method [6], and the second-order slip mode control technique [7] are considered for comparison. In addition, the simulations are performed with parametric uncertainties and external disturbances caused by wind gusts and other factors.

5.3.1 Scenario 1: Without Disturbances In this case, disturbances are neglected in the controller design. In order to highlight the superiority of the proposed AB-ABFTSMC method, comparisons with the backward sliding mode control technique are performed. The simulation results of the AB-ABFTSMC flight controller are presented in Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.9: these results demonstrate that the proposed controller successfully tracked the position and attitude of the quadrotor in finite time, as shown in Figs. 5.2 and 5.3. Even if the reference trajectories of the horizontal position and altitude of the quadrotor are changed, the proposed controller is able to maintain all state variables on the new trajectories (see Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.7). The 3D trajectory of the quadrotor position in Fig. 5.7 shows the best tracking in a few seconds, similarly in Figs. 5.2 and 5.3, the position trajectory (variables x(t), y(t) and z(t)) shows satisfactory performance. Figure 5.3 shows the evolution in time of the attitude variables ((t) and (t)), which also coincide for the operation of the quadrotor and the desired yaw angle. Figures 5.4 and 5.5 represent the linear and angular velocities, we can observe that these state variables converge to zero in finite time. Figure 5.6 represents the actual thrust and input torques for the quadrotor, which converge to their steady state values (4.768, 0, 0, 0), these results demonstrate the efficiency and robustness of the proposed flight controller. The reluctance phenomenon caused by the switching control action is resolved. Indeed, the system inputs have better performance with respect to this problem. In addition, the pulses shown in Figs. 5.2, 5.3, 5.4, 5.5, 5.6, 5.9 of the horizontal position, roll/pitch angles, and roll/pitch torques are caused by the tight coupling of

5.3 Results and Discussion

113

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the position and attitude state variables. The adaptive gains of the proposed controller are given in Figs. 5.8, 5.9, this estimate tracks the desired reference position and attitude with satisfactory accuracy. Finally, the robustness and supremacy of the proposed controller is demonstrated by comparing its performance with that of a conventional sliding mode controller. Furthermore, the AB-ABFTSM control scheme has better performance (setup time, rise time and overshoot) than the B-SMC. The AB-ABFTSMC strategy has good 3D trajectory tracking and quadrotor position tracking (see Figs. 5.2 and 5.7). In addition, the amplitudes of the roll, pitch and yaw torques are greatly reduced compared to the results presented in reference [8]. The latter means that the drone will be more stable.

5.3.2 Scenario 2: In the Presence of Parametric Uncertainty and Disturbances In this scenario, external disturbances and uncertainties in parameter values are taken into account. The terms di (t)(i = x, y, z) = 0.01 cos(0.1t) m/s 2 and d j (t)( j = , , ) = 0.5 cos(0.7t) rad/s 2 are added in the position and attitude accelera-

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(rad)

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Fig. 5.5 Scenario 1: quadrotor angular velocity performance

tions, respectively as perturbations. In addition, an uncertainty of ±30% in the values of mass m and inertias (Ix x , I yy , Izz ) is taken into consideration to show the robustness of the proposed controller. Comparative simulations with the second order sliding mode control method proposed in reference [7] and the full backstepping control approach in reference [6] are also given. The trajectory tracking (x(t), y(t), z(t), (t), (t), (t)), position tracking errors, and quadrotor input signals in the presence of parametric uncertainties and perturbations are shown in Figs. 5.10, 5.11, 5.12, 5.13. The simulation results indicate that both AB-ABFTSMC and SMC-2 methods can achieve robust tracking of the square trajectory. On the contrary, the trajectory tracking via full backstepping cannot be achieved in the presence of disturbances, but the proposed method can achieve better tracking performance than the others. Figures 5.10 and 5.11 show the position and attitude trajectory tracking respectively using three control approaches. We can see that the AB-ABFTSMC controller is able to reject disturbances and uncertainties,

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5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances

Fig. 5.6 Scenario 1: quadrotor inputs performance Fig. 5.7 Scenario 1: quadrotor trajectory tracking performance

5.3 Results and Discussion

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moreover the full backstepping follows the desired position trajectories with large oscillations. These results validated the effectiveness of the proposed control method when disturbances and other factors are considered. Figure 5.12 shows the position tracking error. We can observe that the proposed AB-ABFTSM control strategy achieves the better position tracking than SMC-2 and full backstepping. Finally, the input signals, shown in Fig. 5.13, are easy to implement in a real model and reach their stable values in a short time. Therefore, the AB-ABFTSMC method provides more accurate tracking and greater robustness to time-varying disturbances and parametric uncertainties. In this section we presented the design of a robust fast controller to track a desired trajectory of an attitude and position of the quadrotor. The attitude controller is based on the adaptive fast terminal sliding mode backstepping control (ABFTSMC).

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Fig. 5.9 Scenario 1: the adaptation of the parameters of the attitude flight controller

The position controller is based on adaptive backstepping (AB). This involves the following: the development of a nonlinear adaptive control that satisfies attitude stability, compensates for parametric uncertainties, external disturbances, and fast convergence for the quadrotor state variables. In addition, a procedure for online adaptation of the controller parameters has been presented. The following points list the contributions of this part of this chapter: • Compared with integral backstepping, and first and second order sliding mode controllers, the proposed AB-ABFTSM controller exhibits strong robustness against time-varying uncertainties, non-linearities and external disturbances, and has features such as simplicity and continuous control signals. • A new position and attitude controller is proposed for uncertain quadcopters with external disturbances. • Finite time control, precise tracking and fast convergence of a quadcopter can be achieved. • An on-line estimate is performed for the upper limits of uncertainties and attitude disturbances.

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5 Robust Nonlinear Backstepping SMC for QUAV Subjected to External Disturbances 1

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5.3 Results and Discussion

121

The AB-ABFTSMC proposed in this section is formulated assuming that the uncertainty and disturbance boundary of the system must generally be known in advance. This control law offers no guarantee of singularity. To avoid these problems, a new adaptive control law will be presented in the next section.

5.4 Conclusion We have successfully tested the application of two robust adaptive controllers for flight path tracking and stabilization of a quadrotor UAV in the presence of complex disturbances, parametric uncertainties. For the first part of the proposed work, we designed, two new robust controllers to control the quadrotor subjected to parametric uncertainties and disturbances, the first proposed controller is the adaptive backstepping (AB) to control the position loop, based on this design and the SMC approach, we combine the backstepping control and the fast terminal slip mode control to design a robust AB-ABFTSMC to control the attitude loop. However, the fast terminal slip surfaces are designed for finite time convergence of the attitude system. Adaptive laws are developed to estimate some parameters of the proposed controllers and the upper bound of the uncertainty of the system parameters. The results verify the effectiveness of the proposed controllers and show great 3D flight path tracking. Moreover, these results show that the AB-ABFTSMC control strategy, proposed in the first part of this chapter, has high performance and good robustness against disturbances and better than a classical SMC, SMC-2, and full backstepping techniques.

References 1. Labbadi, M., Cherkaoui, M.: Robust adaptive backstepping fast terminal sliding mode controller for uncertain quadrotor UAV. Aerosp. Sci. Technol. 93, 105306 (2019). Available at: http://dx. doi.org/10.1016/j.ast.2019.105306 2. Fazeli Asl, S.B., Moosapour, S.S.: Adaptive backstepping fast terminal sliding mode controller design for ducted fan engine of thrust-vectored aircraft. Aerosp. Sci. Technol. 71, 521–529 (2017). https://doi.org/10.1016/j.ast.2017.10.001 3. Zuo, Z.: Non-singular fixed-time terminal sliding mode control of non-linear systems. IET Control Theory Appl. 9, 545–552 (2015). https://doi.org/10.1049/iet-cta.2014.0202 4. Hua, C., Chen, J., Guan, X.: Adaptive prescribed performance control of QUAVs with unknown time-varying payload and wind gust disturbance. J. Franklin Inst. 355(14), 6323–6338 (2018) 5. Liu, J., Wang, X.: Advanced sliding mode control for mechanical systems: design, analysis and MATLAB simulation. Springer Science & Business Media, Berlin, pp. 155–158 (2012) 6. Bouabdallah, S., Siegwart, R.: Full control of a quadrotor. IEEE/RSJ Int. Conf. Intell. Robots Syst. 2007, 153–158 (2007). https://doi.org/10.1109/IROS.2007.4399042

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7. Zheng, E.-H., Xiong, J.-J., Luo, J.-L.: Second order sliding mode control for a quadrotor UAV. ISA Trans. 53, 1350–1356 (2014). https://doi.org/10.1016/j.isatra.2014.03.010 8. Rosales, C., Gandolfo, D., Scaglia, G., Jordan, M., Carelli, R.: Trajectory tracking of a mini fourrotor helicopter in dynamic environments a linear algebra approach. Robotica 33, 1628–1652 (2015). https://doi.org/10.1017/S0263574714000952 9. Freire, F.P., Martins, N.A., Splendor, F.: A simple optimization method for tuning the gains of PID controllers for the autopilot of Cessna 182 aircraft using model-in-the-loop platform. J. Control Autom. Electr. Syst. 29, 441–450 (2018) 10. Hua, B., Chen, J., Guan, X.: Fractional-order sliding mode control of uncertain QUAVs with time-varying state constraints. Nonlinear Dyn. (2018). https://doi.org/10.1007/s11071-0184632-0

Chapter 6

Robust Nonsingular Fast Terminal SMC for Unceratin QUAV Subjected to External Disturbances

6.1 Introduction It should be noted that all the research work has mainly focused on adaptive upper bound uncertainty estimation using terminal SM control and fast terminal SM control. In this chapter, a second control is proposed using a new non-singular adaptive fast terminal SMC (NAFTSMC). Note that the main advantage of using NAFTSMC control is the avoidance of singularity, the speed when states are far from the origin, and the strong robustness against system uncertainty and external disturbances. The ANFTSM control law: (i) ensures fast convergence, i.e., the outputs of the quadrotor reach the origin values in a short time; (ii) avoids singularity; (iii) solves the reluctance effect; (iv) provides robustness against unknown external disturbances and uncertainties. In addition, the upper bound of uncertainty and unknown external disturbances of the system are covered by the proposed control approach. The online estimation of these upper bounds is only introduced by velocity and position measurements. Moreover, the control law applies the Lyapunov theory and it guarantees the closed-loop stability of the quadrotor system. Various simulations under different scenarios in terms of external disturbances and parametric uncertainties are performed to evaluate/enhance the effectiveness of the NAFTSMC strategy proposed in this work [1]. In addition, a comparative study is performed at the end of this paper and clearly shows the outperformance of the proposed control scheme. The rest of the chapter is presented as follows: The NFTSMC proposed in this chapter, for the control of the quadrotor subject to disturbances and uncertainties, is given in Sect. 6.2. The simulation results of the proposed controller, compared to other techniques, are given in Sect. 6.3. The conclusion is given in Sect. 6.4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_6

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6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

6.2 Design Methodology of a New Controller for the Quadrotor System In this section, we will present a new controller for attitude and position subsystems in the presence of disturbances and parametric uncertainties. The objective of the controller is to track the position trajectory reference and stabilize the attitude in the presence of unknown wind gusts and additive disturbances. A novel robust control approach is adopted for both the outer and inner loops. The proposed controller is not only capable of achieving zero steady-state error tracking, handling complex disturbances, and establishing a faster convergence rate, but it is also capable of guessing the unknown upper bounds of uncertainties and disturbances. The proposed control improves the reference trajectory tracking performance and increases the robustness of the quadrotor control system to the external disturbances caused by the wind gust, compared to the sliding with backstepping, sliding mode with full backstepping and feedback linearization.

6.2.1 Design of a Trajectory Tracking Controller for the Quadrotor Position Based on the NFTSMC Method In this section, we will present the design procedure of the virtual control input V j ( j = x, y, z) for the position subsystem. This control ensures the convergence of the finite-time position tracking errors to zero asymptotically. The position and attitude subsystems of the quadrotor are a type of second-order nonlinear systems. However, the outer loop subsystem subject to uncertainties and disturbances can be expressed as follows ˙ i =Xi+1 (i = 7, 9, 11) X ˙ i+1 =F (P) + P(X) + d(t) + G(X)V X Yi =Xi

(6.1)

|Dk (X)| = |(P(X) + d)| ≤ δk

(6.2)

where δk is the upper limit of uncertainty and disturbance. The expression of δk can be given as [2–4]: δk = a0n + a1n |en | + a2n |en+1 |(n = 7, 9, 11)

(6.3)

where a0n , a1n , and a2n are non-zero positive numbers. Let us introduce the sliding surfaces of the position subsystem as [2, 5]:

6.2 Design Methodology of a New Controller for the Quadrotor System

125

s7 (t) =e7 (t) + b7 |e7 (t)|ϒ7 sign(e7 (t)) + b8 |e8 (t)|7 sign(e8 (t)) s9 (t) =e9 (t) + b9 |e9 (t)|ϒ9 sign(e9 (t)) + b10 |e10 (t)|9 sign(e10 (t)) s11 (t) =e11 (t) + b11 |e11 (t)|

ϒ11

sign(e11 (t)) + b12 |e12 (t)|

11

(6.4)

sign(e12 (t))

where bn (n = 7, 9, 11) and bn+1 are positive constants, 1 < n < 2 and ϒn > n . Remark 6.1 Using the NFSM surface presented in Eq. (6.4) and for any given initial condition ei (0) = e0 = 0, the state of the system converges very quickly to ei = 0 in finite time. However, when the state of the system is far from the steady state, The sub-term bi |ei (t)(t)|ϒi sign(ei (t)) dominates ei+1 (t)|i sign(ei+1 (t)) which guarantees a high convergence rate. Moreover, when the state of the system is close to the equilibrium state, the sub-term bi+1 |ei+1 (t)|i sign(ei+1 (t)) guarantees the convergence of the system in a finite time. The time derivative of the sliding surfaces can be given as: s˙7 (t) =e8 (t) + ϒ7 b7 |e7 (t)|ϒ7 −1 e8 (t) + 7 b8 |e8 (t)|7 −1 (ρ X X8 + Vx + dx (t) − x¨d (t)) s˙9 (t) =e10 (t) + ϒ9 b9 |e9 (t)|ϒ9 −1 e10 (t) + 9 b10 |e10 (t)|9 −1 (ρY X10 + Vy + d y (t) − y¨d (t)) s˙11 (t) =e12 (t) + ϒ11 b11 |e11 (t)|ϒ11 −1 e12 (t) + 11 b12 |e12 (t)|11 −1 (ρ Z X12 + VZ + dz (t) − z¨ d (t)) By defining s˙n (t) = 0, the equivalent control laws are given by:

(6.5)

1 |e8 (t)|2−7 (1 + ϒ7 b7 |e7 (t)|ϒ7 −1 )sign(e8 (t))) 7 b8 1 =(−(ρY X10 − y¨d )) − |e10 (t)|2−9 (1 + ϒ9 b9 |e9 (t)|ϒ9 −1 )sign(e10 (t))) 9 b10 1 =(−(ρ Z X12 − z¨ d (t)) − |e12 (t)|2−11 (1 + ϒ11 b11 |e11 (t)|ϒ11 −1 )sign(e12 (t))) 11 b12

Veq x =(−(ρ X X8 − x¨d (t)) − Veqy Veqz

(6.6)

In order to reject external factors and achieve robustness against their influences on the position subsystem, the switching control law u sw = −cs(t) − K sing(s(t)) is added to the equivalent control law, where c and K denote the switching gains. The appropriate value of K is chosen as [6]:K = δk + h, where h is a nonnegative parameter and δk represents the upper bound on disturbances and uncertainties. We assume that the control signal contains only velocity and position measurements, so the switching controller is modified as u sw = −cs(t) − (a0n + a1n |en | + a2n |en+1 | + h)sing(s(t)). Thus, the commutation control laws for the quadrotor position are given by:

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6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Vswx =(−c7 s7 (t) − K 7 sign(s7 (t))) =(−c7 s7 (t) − (δ7 + h 7 )sign(s7 (t))) Vswy

=(−c7 s7 (t) − (a07 + a17 |e7 (t)| + a27 |e8 (t)| + h 7 )sign(s7 (t))) =(−c9 s9 (t) − K 9 sign(s9 (t))) =(−c9 s9 (t) − (δ9 + h 9 )sign(s9 (t))) =(−c9 s9 (t) − (a09 + a19 |e9 (t)| + a29 |e10 (t)| + h 9 )sign(s9 (t)))

Vswz =(−c11 s11 (t) − K 11 sign(s11 (t))) =(−c11 s11 (t) − (δ11 + h 11 )sign(s11 (t))) =(−c11 s11 (t) − (a011 + a111 |e11 (t)| + a211 |e12 (t)| + h 11 )sign(s11 (t))) (6.7) where h n and cn are positive constants. Therefore, the virtual position control laws are designed as: Vx =Vswx + Veq x =(−c7 s7 (t) − (a07 + a17 |e7 (t)| + a27 |e8 (t)| + h 7 )sign(s7 (t)) − ρ X X8 (6.8) 1 2−7 ϒ7 −1 + x¨d (t) − |e8 | (1 + ϒ7 b7 |e7 (t)| )sign(e8 )) 7 b8 Vy =Vswy + Veqy =(−c9 s9 (t) − (a09 + a19 |e9 (t)| + a29 |e10 (t)| + h 9 )sign(s9 (t)) − ρY X10 (6.9) 1 2−9 ϒ9 −1 + y¨d (t) − |e10 (t)| (1 + ϒ9 b9 |e9 (t)| )sign(e10 (t))) 9 b10 Vz =Vswz + Veqz =(−c11 s11 (t) − (a011 + a111 |e11 (t)| + a211 |e12 (t)| + h 11 )sign(s11 (t)) − ρ Z X12 1 + z¨ d (t) − |e12 (t)|2−11 (1 + ϒ11 b11 |e11 (t)|ϒ11 −1 )sign(e12 (t))) 11 b12 (6.10) Theorem 6.1 Considering the quadrotor x-subsystem presented in the modeling chapter and the surface (5.19), if the control law is designed as (6.8), then the state variables of the x-subsystem converge to s7 (t) in finite time tr . Furthermore, the tracking error variables can converge to zero in finite time. Proof In order to prove the Theorem 6.1, the Lyapunov candidate function of x-subsystem is considered as: 1 (6.11) V7 = s72 (t) 2 The time derivative of V7 is, V˙7 = s˙7 (t)s7 (t) = s7 (t)(e8 (t) + ϒ7 b7 |e7 (t)|ϒ7 −1 e8 (t) + 7 b8 |e8 (t)|7 −1 e˙8 (t)) (6.12)

6.2 Design Methodology of a New Controller for the Quadrotor System

127

By substituting the position tracking error and (6.8) into (6.12), the dynamics of V7 can be written as: V˙7 =7 b8 |e8 (t)|7 −1 (D7 s7 (t) − c7 s7 (t)2 − (δ7 + h 7 )|s7 (t)|) ≤7 b8 |e8 (t)|7 −1 (|(D7 |.|s7 (t)| − c7 s7 (t)2 − (δ7 + h 7 )|s7 (t)|) 7 −1

=7 b8 |e8 (t)|

(6.13)

(|(D7 | − δ7 )|s7 (t)| − c7 s7 (t)2 − h 7 |s7 (t)|)

Using (6.2), we can obtain, V˙7 ≤7 b8 |e8 (t)|7 −1 (−c7 s72 (t) − h 7 |s7 (t)|) ≤ 0

(6.14)

The stability condition is guaranteed from the Eq. (6.13). The state variables of the system converge to s7 (t) = 0 asymptotically. In order to demonstrate this convergence, (6.14) can be rewritten as: d V7 ≤ −27 b8 c7 |e8 (t)|7 −1 V7 − V˙7 ≤ dt

√ 1/2 27 b8 h 7 |e8 (t)|7 −1 V7

(6.15)

√ By defining 1 = −27 b8 c7 |e8 (t)|7 −1 and 2 = − 27 b8 h 7 |e8 (t)|7 −1 , we get d V7 1/2 ≤ −1 V7 − 2 V7 dt

(6.16)

After some calculations, we obtain, dt ≤

1/2

−d V7 1/2

1 V7 + 2 V7

=

−d V7 1 V7 + 2

(6.17)

Now, integrating (6.17) from t0 to tr , we can obtain, tr ≤ t0 +

2 1 V7 (t0 )1/2 + 2 ln( ) 1 2

(6.18)

This completes the above proof. Remark 6.2 The first part of the robust control term, Veqi , is used to construct a NFTSSMC-like attainment law, which can ensure fast convergence in finite time as the system states are either far or near the sliding surface. While the second part, Vswi is adopted to make the system more robust against unmodeled uncertainties and external disturbances. Remark 6.3 The characteristics of the proposed RANFTSMC are (a) fast convergence in finite time, which can be easily adjusted; (b) in addition to avoiding singularity since the control law (6.8) does not include negative power; (c) strong robustness against uncertainties and disturbances due to the properties of RANFTSMC.

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6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Remark 6.4 It should be noted that the control law proposed above can achieve convergence of the tracking error to zero in finite time when the upper bounds on the system uncertainty and external disturbances are known. However, this assumption is difficult to satisfy due to the complexity of the uncertainty and disturbance structure. In order to overcome this problem, an adaptive tuning law is used to estimate the parameters of the upper bounds of the system uncertainty δ in the expression (6.3). This adaptive tuning method does not require prior knowledge of the upper bound of the system uncertainty.

6.2.2 Design of the Trajectory Tracking Controller for the Quadrotor Position Based on the RANFTSMC Method The RANFTSMC approach is developed to estimate the unknown upper bounds of the disturbances that affected the position subsystem. Therefore, the position control signals are modified as follows. Vx =Vswx + Veq x =(−c7 s7 (t) − (aˆ 07 + aˆ 17 |e7 (t)| + aˆ 27 |e8 (t)| + h 7 )sign(s7 (t)) − ρ X X8 (6.19) 1 2−7 ϒ7 −1 + x¨d (t) − |e8 | (1 + ϒ7 b7 |e7 (t)| )sign(e8 )) 7 b8 Vy =Vswy + Veqy =(−c9 s9 (t) − (aˆ 09 + aˆ 19 |e9 (t)| + aˆ 29 |e10 (t)| + h 9 )sign(s9 (t)) − ρY X10 1 + y¨d (t) − |e10 (t)|2−9 (1 + ϒ9 b9 |e9 (t)|ϒ9 −1 )sign(e10 (t))) 9 b10 (6.20) Vz =Vswz + Veqz =(−c11 s11 (t) − (aˆ 011 + aˆ 111 |e11 (t)| + aˆ 211 |e12 (t)| + h 11 )sign(s11 (t)) − ρ Z X12 1 + z¨ d (t) − |e12 (t)|2−11 (1 + ϒ11 b11 |e11 (t)|ϒ11 −1 )sign(e12 (t))) 11 b12 (6.21) where aˆ 0n , aˆ 1n , and aˆ 2n (n = 7, 9, 11) are the estimates of a0n , a1in and a2n , respectively. The parameters aˆ 0n , aˆ 1n , and aˆ 2n are modified by the adaptive laws. a˙ˆ 07 = μ07 |s7 (t)|.|e8 (t)|7 −1 , a˙ˆ 09 = μ09 |s9 (t)|.|e10 (t)|9 −1 , and a˙ˆ 011 = μ011 |s11 (t)|.|e12 (t)|11 −1 a˙ˆ 17 = μ17 |s7 (t)|.|e7 (t)|.|e8 (t)|7 −1 , aˆ˙ 19 = μ19 |s9 (t)|.|e9 (t)|.|e10 (t)|9 −1 , and a˙ˆ 111 = μ111 |s11 (t)|.|e11 (t)|.|e12 (t)|11 −1 a˙ˆ 27 = μ27 |s7 (t)|.|e8 (t)|7 , aˆ˙ 29 = μ29 |s9 (t)|.|e10 (t)|9 and a˙ˆ 211 = μ211 |s11 (t)|.|e12 (t)|11

where μ0i , μ1i , and μ2i (i = 7, 9, 11) are positive non-zero constants.

(6.22) (6.23) (6.24)

6.2 Design Methodology of a New Controller for the Quadrotor System

129

Theorem 6.2 If we consider the subsystem x with the designed controller (6.19) and the adaptive control laws (6.22)–(6.24), then the state variables of the subsystem x converge to the sliding surface (6.4) in finite time. Proof To determine the parameters (aˆ 07 , aˆ 17 , aˆ 27 ) and prove the stability of the system, the Lyapunov approach is used. The Lyapunov candidate function of the x-subsystem is considered as follows:  1 1 2 s7 (t) + 7 b8 (aˆ i7 − ai7 )2 2 2μ i7 i=0 2

V7 =

(6.25)

The time derivative of Eq. (6.25) is, V˙7 = s˙7 (t)s7 (t) + 7 b8

2  1 (aˆ i7 − ai7 )a˙ˆ i7 μi7

(6.26)

i=0

From Eq. (6.5), we can obtain, V˙7 = s7 (t)(e8 (t) + ϒ7 b7 |e7 (t)|ϒ7 −1 e8 (t) + 7 b8 |e8 (t)|7 −1 e˙8 ) + 7 b8

2  1 (aˆ i7 − ai7 )a˙ˆ i7 μi7 i=0

(6.27) Using the tracking error and Eq. (6.19), the dynamics of V7 is given by: V˙7 =7 b8 |e8 (t)|7 −1 (D7 s7 (t) − c7 s72 (t) − (aˆ 07 + aˆ 17 |e7 (t)| + aˆ 27 |e8 (t)| + h 7 )|s7 (t)|) + 7 b8

2  1 (aˆ i7 − ai7 )a˙ˆ i7 μi7 i=0

(6.28) Using (6.22)–(6.24), we can obtain, V˙7 =7 b8 |e8 (t)|7 −1 (D7 s7 (t) − h 7 |s7 (t)| − c7 s72 (t) − (aˆ 07 + aˆ 17 |e7 (t)| + aˆ 27 |e8 (t)|)|s7 (t)|)

(6.29)

From Eq. (6.2), we obtain, V˙7 ≤7 b8 |e8 (t)|7 −1 (|(D7 |.|s7 (t)| − h 7 |s7 (t)| − c7 s72 (t) − (a07 + a17 |e7 (t)|a27 |e8 |)|s7 (t)|) 7 −1

≤7 b8 |e8 (t)| ≤0

(−h 7 |s7 (t)| − c7 s72 (t))

(6.30) (6.31) (6.32)

The Lyapunov equation V˙7 ≤ 0 can guarantee that the position velocity errors will converge to zero in a finite time. Theorem 6.3 The controllers Vx , Vy , and Vz applied to the studied system ensure the asymptotic stability of the translation subsystem.

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6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Proof Consider the Lyapunov function for the translation subsystem as follows:  1  1 1 2 1 s7 (t) + 7 b8 (aˆ i7 − ai7 )2 + s92 (t) + 9 b10 (aˆ i9 − ai9 )2 2 2μi7 2 2μi9 2

VT =

2

i=0

i=0

 1 1 2 + s11 (t) + 11 b12 (aˆ i11 − ai11 )2 2 2μi11 2

(6.33)

i=0

The dynamics of VT is, V˙T = s˙7 (t)s7 (t) + 7 b8

2 2   1 1 (aˆ i7 − ai7 )a˙ˆ i7 + 9 b10 (aˆ i9 − ai9 )a˙ˆ i9 μi7 μi9 i=0

+ s˙9 (t)s9 (t) + s˙11 (t)s11 (t) + 11 b11

i=0

2  i=0

(6.34)

1 (aˆ i11 − ai11 )a˙ˆ i11 μi11

From Eqs. (6.2), (6.5), (6.19)–(6.22), (6.22) and (6.22), we can obtain V˙T ≤ 7 b8 |e8 (t)|7 −1 (−h 7 |s7 (t)| − c7 s72 (t)) + 9 b10 |e10 (t)|9 −1 (−h 9 |s9 (t)| − c9 s92 (t)) 2 (t)) + 11 b12 |e12 (t)|11 −1 (−h 11 |s11 (t)| − c11 s11

≤0

(6.35) From the above analysis, it is obvious that the stability condition of the outer loop is guaranteed. Remark 6.5 The NFTSM surface (6.4) was previously used to control many nonlinear systems, for example, the control of the nonlinear dynamical system in [7], the control of an electrically actuated micro-resonator in [8], the control of a mobile satellite communication antenna in [9], to solve the ground moving target tracking problem for fixed-wing unmanned aerial vehicles in [10], the trajectory tracking control of uncertain manipulator robots with actuation faults in [11]. However, none of the above work has developed an adaptive technique combined with the SMTRN surface (6.4) to cope with the unknown upper bound of uncertainties and external disturbances that cause the performance degradation of the proposed controller in real applications. Remark 6.6 Note that all the works mentioned in [7–11], were limited to the prior knowledge of the upper bounds of the system uncertainties. However, adaptive laws are designed to estimate the parameters of the system uncertainty bounds. Thus the controller proposed in this section still has acceptable performance, such as high accuracy, strong robustness, no singularity, less interference, and fast convergence in finite time. In this part, a robust RANFTSMC law for position control is provided. The stability of the outer loop is guaranteed by using Lyapunov analysis. Similarly, a robust RANFTSMC law for the attitude loop is presented in the next subsection.

6.2 Design Methodology of a New Controller for the Quadrotor System

131

6.2.3 Design of a Trajectory Tracking Controller for Quadrotor Attitude Based on the RANFTSMC Method In this section, we will present a RANFTSM controller for the perturbed attitude. The control law generated by this loop ensures the stability of the attitude in the closed loop. The controller proposed in this work for the quadrotor attitude allows the roll, pitch and yaw state variables (X1 , X3 , X5 ) to converge to the desired values (d (t), d (t), d (t)) in a short time. Therefore, the same control procedure presented in the previous section for the quadrotor position can be performed to design the quadrotor attitude. The corresponding RANFTSMC surfaces for the quadrotor attitude are chosen as: s1 (t) =e1 (t) + b1 |e1 (t)|ϒ1 sign(e1 (t)) + b2 |e2 (t)|1 sign(e2 (t)) s3 (t) =e3 (t) + b3 |e3 (t)|ϒ3 sign(e3 (t)) + b4 |e4 (t)|3 sign(e4 (t)) ϒ5

(6.36)

5

s5 (t) =e5 (t) + b5 |e5 (t)| sign(e5 (t)) + b6 |e6 (t)| sign(e6 (t)) where b j ( j = 1, 3, 5) and b j+1 are positive constants, 1 <  j < 2 and ϒ j >  j . The corresponding RANFTSMC controllers for the inner loop are designed as: u  =u sw + u eq =ρ1 [−c1 s1 (t) − (aˆ 01 + aˆ 11 |e1 (t)| + aˆ 21 |e2 (t)| + h 1 )sign(s1 (t)) − (ρ1 X6 X4 1 ¨ d (t) − + ρ2 X4 ) − ρ3 X22 +  |e2 (t)|2−1 (1 + ϒ1 b1 |e1 (t)|ϒ1 −1 )sign(e2 (t))] 1 b2 u  =u sw + u eq

(6.37)

=ρ2 [−c3 s3 (t) − (aˆ 03 + aˆ 13 |e3 (t)| + aˆ 23 |e4 (t)| + h 3 )sign(s3 (t)) − (ρ1 X6 X2 + ρ2 X2 ) 1 ¨ d (t) − |e4 (t)|2−3 (1 + ϒ3 b3 |e3 (t)|ϒ3 −1 )sign(e4 (t))] − ρ3 X24 +  3 b4 u =u sw + u eq

(6.38)

=ρ3 [−c5 s5 (t) − (aˆ 05 + aˆ 15 |e5 (t)| + aˆ 25 |e6 (t)| + h 5 )sign(s5 (t)) − (ρ1 X4 X2 + ρ2 X26 ) 1 ¨ d (t) − + |e6 (t)|2−5 (1 + ϒ5 b5 |e5 (t)|ϒ5 −1 )sign(e6 (t))] 5 b6

(6.39) Where aˆ 0 j , aˆ 1 j , and aˆ 2 j ( j = 1, 3, 5) are the estimates of a0 j , a1 j , and a2 j , respectively. c j ( j = 1, 3, 5) and h j are nonnegative parameters. The adaptive laws of the attitude subsystem are given by: a˙ˆ 01 = μ01 |s1 (t)|.|e2 (t)|1 −1 , aˆ˙ 03 = μ03 |s3 (t)|.|e4 (t)|3 −1 , and a˙ˆ 05 = μ05 |s5 (t)|.|e6 (t)|5 −1 a˙ˆ 11 = μ11 |s1 (t)|.|e1 (t)|.|e2 (t)|1 −1 , a˙ˆ 13 = μ13 |s3 (t)|.|e3 (t)|.|e4 (t)|3 −1 , and a˙ˆ 15 = μ15 |s5 (t)|.|(t)|.|e6 (t)|5 −1

(6.40) (6.41)

a˙ˆ 21 = μ21 |s1 (t)|.|e2 (t)|1 , a˙ˆ 23 = μ23 |s3 (t)|.|e4 (t)|3 , and a˙ˆ 25 = μ25 |s5 (t)|.|e6 (t)|5

(6.42)

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6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

where μ0 j , μ1 j and μ2 j ( j = 1, 3, 5) are positive non-zero constants. Theorem 6.4 The controllers u  , u  , and u applied to the studied system ensure the asymptotic stability of the rotation subsystem. Proof The Lyapunov candidate function for the rotation subsystem is considered as follows:  1  1 1 1 VR = s12 (t) + 1 b2 (aˆ i1 − ai3 )2 + s32 (t) + 3 b4 (aˆ i3 − ai3 )2 2 2μ 2 2μ i1 i3 i=0 i=0 2

2

 1 1 + s52 (t) + 5 b6 (aˆ i5 − ai5 )2 2 2μ i5 i=0 2

(6.43) The time derivative of VR is given by: V˙ R =˙s1 (t)s1 (t) + 1 b2

2 2   1 1 (aˆ i1 − ai1 )a˙ˆ i1 + 3 b4 (aˆ i3 − ai3 )a˙ˆ i3 μi1 μi3 i=0

i=0

2  1 + s˙3 (t)s3 (t) + s˙ (t)5 s5 (t) + 5 b6 (aˆ i5 − ai5 )a˙ˆ i5 μi5

(6.44)

i=0

According to Eqs. (6.2), (6.37)–(6.42), we can obtain, V˙ R ≤ 1 b2 |e2 (t)|1 −1 (−h 1 |s1 (t)| − c1 s1 (t)2) + 3 b4 |e4 (t)|3 −1 (−h 3 |s3 (t)| − c3 s3 (t)2) + 5 b6 |e6 (t)|5 −1 (−h 5 |s5 (t)| − c5 s5 2(t)) ≤0

(6.45) From the above analysis, it is obvious that the condition of achieving the stability of the attitude loop is guaranteed. Theorem 6.5 The ultimate control laws Eqs. (6.19)–(6.19) and (6.37)–(6.39) applied to the UAV system and the adaptive laws are obtained via the RANFTSMC technique which guarantees the global stability of the system in closed loop. Proof The Lyapunov function for the quadrotor system is chosen as follows: VS = VR + VT

(6.46)

The time derivative of VS is given by: V˙ S = V˙ R + V˙T From Eqs. (6.35) and (6.45), we have,

(6.47)

6.2 Design Methodology of a New Controller for the Quadrotor System

133

V˙ S ≤ 1 b2 |e2 (t)|1 −1 (−h 1 |s1 (t)| − c1 s12 (t)) + 3 b4 |e4 (t)|3 −1 (−h 3 |s3 (t)| − c3 s32 (t)) + 5 b6 |e6 (t)|5 −1 (−h 5 |s5 (t)| − c5 s52 (t)) + 7 b8 |e8 (t)|7 −1 (−h 7 |s7 (t)| − c7 s72 (t)) 2 + 9 b10 |e10 (t)|9 −1 (−h 9 |s9 (t)| − c9 s92 (t)) + 11 b12 |e12 (t)|11 −1 (−h 11 |s11 (t)| − c11 s11 (t))

≤0

(6.48) Remark 6.7 The global stability of the rotational and translational tracking errors is proved using the Lyapunov approach. Remark 6.8 Most adaptive works in the literature for UAV control, the update laws are derived using well-known properties, such as symmetric, antisymmetric, regression properties. These properties are not required in this work which makes the proposed control laws, applicable to any UAV system. Remark 6.9 From the control engineering point of view, the main difficulties that significantly limit the control performance in real applications are (i) the knowledge of the system model or upper bound of the dynamic system, (ii) the presence of system uncertainties and external disturbances, (iii) the feasibility of control inputs. It is obvious that neither the detailed dynamics of the system nor the prior knowledge of the upper bounds are required in our proposed control scheme. Moreover, good robustness against uncertainties and disturbances is ensured due to the efficiency of the RANFTSMC control. It should also be noted that for real implementations of the adaptive algorithm, the position and velocity states must be accessible for feedback. In summary, the proposed controller promises to be very suitable for practical applications.

6.3 Simulation Results In this section, the effectiveness of the proposed control strategy, based on the RANFTSMC method for the trajectory tracking problem, has been tested under different disturbances using numerical simulations. In order to show the improvement obtained by using the ANFTSMC approach, a comparative study with three other controllers is considered. The existing controllers presented in this work are the nonlinear sliding mode backstepping (BSMC) controller [12, 13], the feedback linearization (FL) technique [14], and the integral sliding mode backstepping (IBSMC) controller [15]. Remark 6.10 These parameters must be adjusted to achieve the performance requirement of the quadrotor system. For these reasons, the MATLAB software optimization toolbox is used to select the best values of the controller parameters in simulation 1 (without disturbances), and then we kept the same parameters for the rest of the proposed scenarios (see Ref. [16]). In addition, to evaluate the proposed control strategy, several scenarios are proposed.

134

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Remark 6.11 The simulations are performed on MATLAB software, which is equipped with a Lenovo computer that includes a 3.4 GHz processor with 8 GB of RAM and a 466 GB SSD.

6.3.1 Scenario 1: Nominal In this scenario, the quadrotor follows a square reference trajectory without external perturbations. Furthermore, the initial conditions of the vehicle in this case are zero. The simulation results are illustrated in Figs. 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 and 6.7 via comparisons of FL, BSMC, and the proposed RANFTSMC. Figures 6.1 and 6.2 display the performance tracking of the quadrotor attitude and position. We can observe that the yaw angle and position variables ( (t), x(t), y(t), z(t)) are returned to their references faster and more accurately. In addition, the roll and pitch angles converge to zero (see Fig. 6.2), indicating that the attitude subsystem is stable. Figure 6.3 displays the performance of the quadrotor control inputs (u m , u  , u  , u ), as we can see, the control signals are smooth and converge to their initial values (7.259, 0, 0, 0), which proves the efficiency of the RANFTSMC strategy. Indeed, the input system gives better performance with respect to this problem. The reluctance phenomenon is avoided by replacing the sign(.) function in the control laws with the tanh(.) function. In addition, the attitude tracking and position tracking errors are illustrated in Figs. 6.4 and 6.5. Good tracking performance is

Fig. 6.1 Scenario 1: Quadrotor position (x, y, z)

6.3 Simulation Results

Fig. 6.2 Scenario 1: Attitude du quadrotor (, , )

Fig. 6.3 Scenario 1: The control signals (u m , u  , u  , u  )

135

136

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Fig. 6.4 Scenario 1: Error tracking (e7 (t), e9 (t), e11 (t))

Fig. 6.5 Scenario 1: Error tracking (e1 (t), e3 (t), e5 (t))

6.3 Simulation Results

137

Fig. 6.6 Scenario 1: Estimation of the parameters of the position

obtained for the (x(t), y(t), z(t)) position using the ANFTSMC technique. The attitude tracking errors converge quickly to zero. The adaptive mechanism ensures good tracking performance. However, we can clearly see from the two Figs. 6.6 and 6.7 that the estimated parameters converge to constant values. Compared to the backstepping slip mode control and feedback linearization methods, these techniques cannot provide good accuracy for tracking the desired yaw angle and position (see Figs. 6.1 and 6.2). Therefore, the tracking error performance is not as good as the proposed RANFTSMC method. Figures 6.1 and 6.2 show that the FL and BSMC techniques that provide damped response with significant overshoot compared to the proposed controller.

138

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Fig. 6.7 Scenario 1: Estimation of attitude parameters

6.3.2 Scenario 2: Constant Disturbance The simulation is performed with a circular path defined as follows: xd (t) =

1 πt cos( m), 2 20

yd (t) =

1 1 πt πt sin( )m, z d (t) = 2 − cos( )m, d (t) = 0 rad 2 20 2 20

(6.49) The initial conditions of the vehicle are [0, 0, 0.5]rad and [0, 0, 0]m. In this simulation, the RANFTSMC proposed in this work is applied for the quadrotor subjected to the constant disturbances, their expressions are given by:  dx (t) =  dz (t) =

0 m/s2 t ∈ [0, 5) d y (t) = 1 m/s2 [5, 40] 0 m/s2 1 m/s2



0 m/s2 t ∈ [0, 15) 1 m/s2 t ∈ [15, 40]  t ∈ [0, 25) 0 rad/s2 t ∈ [0, 10) d (t) = t ∈ [25, 40] 1 rad/s2 t ∈ [10, 40]

(6.50) (6.51)

6.3 Simulation Results

139

Fig. 6.8 Scenario 2: Position of the quadrotor (x(t), y(t), z(t)) under constant external perturbations

Fig. 6.9 Scenario 2: Flight path tracking under the effect of constant external disturbances

 d (t) =

0 rad/s2 t ∈ [0, 20) d (t) = 1 rad/s2 t ∈ [20, 40]



0 rad/s2 t ∈ [0, 30) 1 rad/s2 t ∈ [30, 40]

(6.52)

The position tracking results are shown in Fig. 6.8. It can be seen that the proposed control strategy successfully tracks the position in the presence of disturbances. However, the B-SMC and LF methods cannot provide good accuracy in tracking the desired position under sustained disturbances. The 3D trajectory tracking results are

140

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

displayed in Fig. 6.9. The perturbation is introduced at t = 10 s for the x(t) position, we can observe that the RANFTSMC technique returns to the desired trajectory without perturbation, similarly for the y(t) position at t = 15 s. This means that RANFTSMC is able to reject perturbations. However, the FL and B-SMC achieve the desired values of the quadrotor position and 3D flight path just under undisturbed conditions (see Figs. 6.8 and 6.9).

6.3.3

Scenario 3: Time Variation of the Wind Disturbance

This part is devoted to the verification of the stability of the quadrotor under timevarying disturbances. The disturbances are added to the quadrotor model equations. These perturbations are accelerations caused by wind gusts when the quadrotor is flying outside. Therefore, the expression of the disturbances is as follows: dx (t) = − (0.8 sin(0.1013t − 3.0403) + 0.4 sin(0.4488t − 13.464) + 0.08 sin(1.5708t − 15π ) + 0.056 sin(0.2856t − 8.568) m/s2 t ∈ [10, 30] d y (t) =0.5 sin(0.4t) + 0.5 cos(0.7t) m/s2 t ∈ [10, 50] dz (t) = 0.5 cos(0.7t) m/s2 t ∈ [0, 80] d (t) =0.5 cos(0.4t) rad/s2 t ∈ [0, 80] d (t) = 0.5 sin(0.5t) rad/s2 t ∈ [0, 80] d (t) =0.5 sin(0.7t) rad/s2 t ∈ [0, 80]

(6.53)

The desired trajectory is given in 6.54. ⎧1 t ⎪ 2 sin( 2 ) m t ∈ [0, 4π ) ⎪ cos( 2t )m ⎪ ⎨ 0.25t − 3.14 m t ∈ [4π, 20) 0.5 m yd (t) = 5 − π m xd (t) = 0.25t − 4.5 m t ∈ [20, 30) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ −0.2358t + 8.94 m t ∈ [30, 80] 3m −0.5 m ⎧ t ∈ [0, 4π ) ⎨ 0.125t + 1 m t ∈ [4π, 40) d (t) = 0 rad z d (t) = 0.5π + 1 m ⎩ exp(−0.2t + 8.944) m t ∈ [40, 80] ⎧ ⎪ ⎪ ⎨

1 2

t t t t t

∈ [0, 4π ) ∈ [4π, 20) ∈ [20, 30) ∈ [30, 40) ∈ [40, 80]

(6.54)

The initial conditions of the vehicle are [0, 0, 0.5]rad and [0, 5, 0, 5, 0, 5]m. The simulation is performed under abrupt reference changes and time-varying disturbances. The performance of the proposed control strategy is shown in Figs. 6.10 and 6.11. A comparison with the FL technique and B-SMC proposed in Refs. [12, 13] are also given. The simulation results indicate that the robustness of the RANFTSMC approach against complex disturbances is better than the other methods. Figure 6.10 describes the performance of the position controller. We see that the position controllers can accurately track the desired values even if the references change rapidly.

6.3 Simulation Results

141

Fig. 6.10 Scenario 3: Position of the quadrotor (x, y, z) under time-varying external perturbations

4

z (m)

Fig. 6.11 Scenario 3: Flight path tracking under time-varying external disturbances

2

0 0.5 0.5

0

y (m)

0 -0.5

-0.5

x (m)

Figure 6.11 shows the desired 3D flight path. We observe from the above results that the RANFTSMC did track the desired trajectory well compared to the other methods.

6.3.4 Scenario 4: Noise from Sensors Low-cost sensors are typically used for UAVs where accuracy is rather low and the outputs are always noisy and biased. In addition, the implementation process involves additional drift and digital noise, especially with initially noisy data. Mechanical vibrations can also cause unwanted noise. For these reasons for further evaluation before the performance of the RANFTSMC approach, the effects of measurement noise are added to all state variables. This effect is displayed in Fig. 6.12.

142

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Fig. 6.12 Measurement noise Fig. 6.13 Scenario 4: Position of the (x, y, z) quadrotor in the case of measurement noise

The initial conditions of ((t), (t), (t), x(t), y(t), z(t)) are: [0, 0, 0.5]rad and [0.5, 0.5, 0.5]m. The results of the RANFTSMC simulation are shown in Fig. 6.13 comparing their performance to FL and B-SMC methods. Figure 6.13 shows the position trajectory. The performance results obtained by the proposed control show the worst profile compared to the other techniques (see Fig. 6.13).

6.3 Simulation Results

143

6.3.5 Scenario 5: Parametric Uncertainties In this scenario, the moments of inertia, the aerodynamic coefficients and the total mass are underestimated by 50% of the nominal values. In addition, the simulation is performed under time-varying disturbances. In addition, the RANFTSMC is compared to the full recoil sliding mode controller (IB-SMC) presented by the authors in Ref. [15]. The expression for the disturbances is given by: dx (t) = −(0.8 sin(0.1013t − 3.0403) + 0.4 sin(0.4488t − 13.464) + 0.08 sin(1.5708t − 15π ) + 0.056 sin(0.2856t − 8.568) m/s2 t ∈ [10, 30] d y (t) = 0.5 sin(0.4t) + 0.5 cos(0.7t) m/s2 t ∈ [10, 50] dz (t) = 0.5 cos(0.7t) + 0.7 sin(0.3t) m/s2 t ∈ [0, 120] d (t) = 0.5 cos(0.4t) + 1 rad/s2 t ∈ [0, 120] d (t) = 0.5 sin(0.5t) + 1 rad/s2 t ∈ [0, 120] d (t) = 0.5 sin(0.7t) + 1 rad/s2 t ∈ [0, 120]

(6.55)

The desired spatial trajectory is given as The desired spatial trajectory is given as:  xd (t) =

0 t ∈ [0, 55) 0.3 cos( π6t ) m t ∈ [55, 120]

 yd (t) =

⎧ ⎨ 0.5 m t ∈ [0, 42) z d (t) = 0.7 m t ∈ [42, 87) d (t) = 0 rad ⎩ 0.8 m t ∈ [87, 120]

0 t ∈ [0, 55) 0.3 sin( π6t ) m t ∈ [55, 120]

(6.56)

The initial conditions of attitude and position are [0, 0, 0.5]rad and [0.5, 0.5, [0.5]m. The tracking performance under time-varying perturbations and parametric uncertainties are plotted in Figs. 6.14 and 6.15, as we can see the results obtained have an excellent tracking of the proposed trajectory. The RANFTSMC technique allows the stabilization of the attitude loop under the effect of the perturbations and the parametric uncertainties (see Figs. 6.14 and 6.15). The quadrotor can maintain the desired x and y positions with good accuracy compared to the IBSMC approach presented in Ref. [15] during the first (0.55) period of flight. In addition, the altitude result of the IBSMC method shows strong oscillations and overshoots during the perturbed period of the desired altitude. Therefore, it is interesting to note that the proposed RANFTSMC is capable of rejecting disturbances. Moreover, it is clear from Fig. 6.14 that the proposed approach performs very well for the tracking trajectory with 50% uncertainty in the quadrotor and perturbation parameters. The result of the proposed internal controller, which allows the quadrotor to track the reference angles, is shown in Fig. 6.15. Here we can see that the quadrotor system is highly coupled. Each subsystem is affected by a different perturbation (see Eq. 6.55). Therefore, any change in the parameters of the quadrotor can lead to a change in the roll and pitch angle references. It is clear that we can observe that the proposed RANFTSMC can accurately track the desired yaw, roll and pitch angles.

144

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Fig. 6.14 Scenario 5: Position of the (x, y, z) quadrant under external perturbations and parameter uncertainties

A quantitative comparison between the proposed RANFTSMC, FL and B-SMC methods is considered based on the integral square error (ISE) performance indices defined in Ref. [17] as follows:  I SE =

tf

e2 dt

(6.57)

ti

where t f and ti denote respectively the final and initial instants, and e represents the tracking error. The results are presented in the Table 6.1. The RANFTSMC method is significantly superior to the FL and B-SMC control approaches, which can be observed in Table 6.1, the proposed RANFTSMC provided more accurate tracking even under sensor noise and time-varying disturbances. Remark 6.12 The ISE values shown in Table 6.1 are obtained from simulation 2.

6.3 Simulation Results

145

Fig. 6.15 Scenario 5: Attitude of the quadrotor (, , ) under external perturbations and parameter uncertainties Table 6.1 ISE performance indices State varibale FL x(t) y(t) z(t) (t) (t) (t)

0.2995 0.0255 0.2239 0.0064 0.0079 0.0572

B-SMC

RANFTSMC

0.2525 0.0175 0.0002 0.0034 0.0030 0.0438

0.1295 1.322e−05 5.782e−07 0.0019 0.0015 0.1485

Similarly, Table 6.2 presents a comparison based on ISE performance indices between the proposed RANFTSMC technique and the IB-SMC method. We can observe from the Table 6.2 that the proposed approach is still the best and has an excellent ISE compared to IB-SMC. Finally, the proposed controller presents an excellent flight path tracking and a strong robustness in all the proposed scenarios. The simulation results proved that RANFTSMC has better tracking performance than FL, BSM and IBSM control methods. Due to the adaptive mechanism capability provided by RANFTSMC and the strong robustness of the proposed RANTSTM, parametric uncertainties, external disturbances and measurement noise are well compensated with the RANFTSM controller.

146

6 Robust Nonsingular Fast Terminal SMC for Uncertain QUAV Subjected …

Table 6.2 ISE performance indices State variable RANFTSMC x(t) y(t) z(t) (t) (t) (t)

0.1551 0.1253 0.0193 0.0093 0.0090 0.1495

IB-SMC 0.1851 0.1665 0.00245 4.638 4.699 8.969

6.4 Conclusion In this work, a new RANFTSMC technique is proposed to solve the problem of trajectory control of an uncertainty quadrotor under disturbances. In addition, the fast convergence of all state variables has been achieved and the influence of the reluctance effect in SMC has been eliminated, while the online parameter estimation has been introduced and the singularity problem of TSMC has been avoided. The effectiveness of the proposed RANFTSMC approach was demonstrated in multiple test scenarios (constant external disturbances, parametric uncertainties, measurement noise, and time-varying external disturbances). The results presented in this section show that the RANFTSMC approach proposed in this work has good tracking of the desired trajectory, fast finite-time convergences of the slip surfaces with higher accuracy, a steady state null where the errors cancel out of tracking, and a high level of robustness against external disturbances compared to the feedback linearization, integral backstepping SM, and feedback SM control methods.

References 1. Labbadi, M., Cherkaoui, M.: Robust adaptive nonsingular fast terminal sliding-mode tracking control for an uncertain quadrotor UAV subjected to disturbances. ISA Trans. [Internet]. Elsevier BV 99, 290–304 (2020). Available from: http://dx.doi.org/10.1016/j.isatra.2019.10. 012 2. Boukattaya, M., et al.: Adaptive nonsingular fast terminal sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Trans. 77, 1–19 (2018) 3. Lin, P., Ma, J., Zheng, Z.: Robust adaptive sliding mode control for uncertain nonlinear MIMO system with guaranteed steady state tracking error bounds. J. Franklin Inst. 353, 303–321 (2016) 4. Zhihong, M., Yu, X.: Adaptive terminal sliding mode tracking control for rigid robotic manipulators with uncertain dynamics. JSME Int. J. Ser. C Mech. Syst. Mach. Elem. Manuf. 40, 493–502 (1997) 5. Yang, L., Yang, J.: Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems. Int. J. Robust Nonlinear Control 21, 1865–1879 (2016)

References

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6. Fazeli Asl, S.B., Moosapour, S.S.: Adaptive backstepping fast terminal sliding mode controller design for ducted fan engine of thrust-vectored aircraft. Aerosp. Sci. Technol. 71, 521–529. https://doi.org/10.1016/j.ast.2017.10.001 (2017) 7. Yang, L., Yang, L.: Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems. Int. J. Robust Nonlinear Control 21(16), 1865e71865e79 (2011) 8. Han, J., Zhang, Q., Wang, W., Jin, G., Qi, H., Li, Q.: Chaos suppression of an electrically actuated microresonator based on fractional-order nonsingular fast terminal sliding mode control. Math. Probl. Eng. e12 (2017) 9. Zhang, X., Zhao, Y., Guo, K.., Li, G., Deng, N.: An adaptive B-spline neural network and its application in terminal sliding mode control for a mobile satcom antenna inertially stabilized platform. Sensors 17(5), 1e21 (2017) 10. Wu, K., Cai, Z., Zhao, J., Wang, Y.: Target tracking based on a nonsingular fast terminal sliding mode guidance law by fixed-wing UAV. Appl. Sci. 7(4), 1e18 (2017) 11. Van, M., Ge, S., Ren, H.: Finite time fault tolerant control for robot manipulators using time delay estimation and continuous nonsingular fast terminal sliding mode control. IEEE Trans. Cybern. 47(7), 1e13 (2017) 12. Chen, F., Jiang, R., Zhang, K., Jiang, B., Member, S.: Robust Backstepping Sliding Mode Control and Observer-Based Fault Estimation for a Quadrotor UAV (2016). https://doi.org/10. 1109/TIE.2016.2552151 13. Basri, M.A.M.: Design and application of an adaptive backstepping sliding mode controller for a six-DOF quadrotor aerial robot. Robotica 36, 1701–1727 (2018). https://doi.org/10.1017/ S0263574718000668 14. Voos, H.: Nonlinear control of a quadrotor micro-UAV using feedback-linearization. IEEE Int. Conf. Mechatron. (2009) 15. Jia, Z., Yu, J., Mei, Y., Chen, Y., Shen, Y., Ai, X.: Integral backstepping sliding mode control for quadrotor helicopter under external uncertain disturbances. Aerosp. Sci. Technol. 68, 299–307 (2017). https://doi.org/10.1016/j.ast.2017.05.022 16. Freire, F.P., Martins, N.A., Splendor, F.: A simple optimization method for tuning the gains of PID controllers for the autopilot of Cessna 182 aircraft using model-in-the-loop platform. J. Control Autom. Electr. Syst. 29, 441–450 (2018) 17. Bouzid, Y., Siguerdidjane, H., Bestaoui, Y.: Nonlinear internal model control applied to VTOL multi-rotors UAV. Mechatronics 47, 49–66 (2017). https://doi.org/10.1016/j.mechatronics. 2017.08.002

Chapter 7

Robust Adaptive Global Time-Varying SMC for QUAV Subjected to Gaussian Random Uncertainties/Disturbances

7.1 Introduction In this chapter, the problem of designing an adaptive global time-varying slidingmode-controller (AGTVSMC), is investigated for the altitude and attitude of the QUAV under unknown external disturbances. The present chapter can be highlighted as: (1) Improvement of the convergence-time of the tracking errors based on AGTVSMsC method under external disturbances. (2) Using the AGTVSMC, a smaller amplitude of the input control signal and faster convergence are obtained, also the problem of the reaching phase is addressed. (3) The upper bound of the random uncertainties/disturbances is estimated using an adaptive control laws. The present chapter is organized as follows. The adaptive time-varying GSMC [1] is given in the Sect. 7.2. The results of the AGTVSMC carried out in the Sect. 7.3. Finally, the conclusion is presented in the Sect. 7.4. Problem 7.1 In this part we define the problem considered in this chapter. Consider the external perturbations applied on the model of the QUAV system. [dz (t) d (t) d (t) d (t)]. In order to control the QUAV system under these disturbances, the design of the controllers [u m u  u  u  ]T are considered as the problem of this chapter. Assumption 7.1 The disturbance d (t)( = , , , z) is assumed as  δ Q . with δ Q represents the upper bound of uncertainty/disturbance. The expression of δ Q can be written as [9, 10]: δ Q = a0k + a1k |ek (t)| + a2k |ek+1 (t)|(k = 1, 3, 5, 7)

(7.1)

with a0k , a1k , and a2k are positive constants. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_7

149

150

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

7.2 Controller Design Methodology Unlike ASMC [2], ATSMC [3], ANFTSMC [4], the present chapter proposes an AGTVSMC for the QUAV under random disturbances/parametric uncertainties. As known, the time-varying sliding-mode technique is composed by two phases: the first is the sliding phase with time-varying formulated to drive QUAV to origin in the pre-specified-time. The second phase is reaching phase which offers the switching law against the disturbances/time-varying of the QUAV parameters. This phase offers a finite-time convergence, smooth inputs. In the future subsection, we present the design of the AGTVSMC proposed in this work for the attitude and altitude in the presence of unknown disturbances.

7.2.1 Design Controller Based on a Global Time-Varying SMC for the QUAV The control objective is to track of the desired outputs with a finite-time from any initial condition. Let define the time-varying sliding variables as [5] na e1 (t) − ϒ (t) t − tfa nb e3 (t) − ϒ (t) s3 (t) = e2 (t) + t − tfb nc e5 (t) − ϒ (t) s5 (t) = e2 (t) + t − tfc nd e11 (t) − ϒz (t) s11 (t) = ez2 (t) + t − tfd s1 (t) = e2 (t) +

(7.2a) (7.2b) (7.2c) (7.2d)

whither t f j ( j = a, b, c, d) represents the prespecified time, the term e j2 (t) is the time-derivative of e j (t), and ϒi (t)(i = , , , z) is the function given by:  ϒi (t) =

Aj pj tfj

(t f j − t) pj f or 0  t  t f j 0

f or t > t f j

(7.3)

A j , n j and t f j are the positive coefficients. Setting s j (t)=0, we have e˙ j1 (t) + nj A e (t) − pjj (t f j − t) pj = 0, the analytical solution of eh (t) can be written as t−t f j j tfj

Aj (t f j − t) pj+1 and of pj t f j ( p f j −n f j +1) A ( p +1) t)n j−1 + pj j f j (t f j − t) pj , if t t f j ( p f j −n f j +1)

eh (t) = C j (t f j − t)n j −

e j2 (t) can be given by:

e j2 (t) = C j n j (t f j −

goes to t f j both tracking

errors converge to zero. Differencing (7.2) as:

7.2 Controller Design Methodology

na na e2 (t) + e1 (t) − ϒ˙  (t) t − tfa (t − t f a )2 nb nb e2 (t) + e3 (t) − ϒ˙  (t) s˙3 (t) = e˙ 2 (t) + t − tfb (t − t f b )2 nc nc e2 (t) + e5 (t) − ϒ˙  (t) s˙5 (t) = e˙ 2 (t) + t − tfc (t − t f c )2 na nd ez2 (t) + e11 (t) − ϒ˙ z (t) s˙11 (t) = e˙ z2 (t) + t − tfd (t − t f d )2 s˙1 (t) = e˙ 2 (t) +

151

(7.4a) (7.4b) (7.4c) (7.4d)

By forcing the time derivative of s(t) = equal zero e.g. s˙ (t) = 0 without any disturbances/uncertainties, e.g. d (t) the equivalent control laws can be obtained. ¨ d (t) ρ1 X4 X6 + ρ2 X4 + ρ3 X22 + ρ1 u  −  na na e2 (t) + e1 (t) − ϒ˙  (t) = 0 + t − tfa (t − t f a )2 ¨ d (t) ρ1 X2 X6 + ρ2 X2 + ρ3 X24 + ρ2 u  −  nb nb e2 (t) + e3 (t) − ϒ˙  (t) = 0 + t − tfb (t − t f b )2 ¨ d (t) ρ1 X2 X4 + ρ2 X26 + ρ3 u  −  nc nc e2 (t) + e5 (t) − ϒ˙  (t) = 0 + t − tfc (t − t f c )2 1 − g + ρ Z X12 − g + (cos X1 cos X3 )u m − z¨ d (t) m nd nd + ez2 (t) + e11 (t) − ϒ˙ z (t) = 0 t − tfd (t − t f d )2

(7.5a)

(7.5b)

(7.5c)

(7.5d)

Then, the equivalent control laws are calculated as follows: 1 ¨ d (t) (ρ X X + ρ2 X4 + ρ3 X22 ) −  ρ1 1 4 6 na na e (t) + e1 (t) − ϒ˙  (t)) + t − t f a 2 (t − t f a )2

u eq = −

1 ¨ d (t) (ρ X X + ρ2 X2 + ρ3 X24 ) −  ρ2 1 2 6 nb nb e (t) + e3 (t) − ϒ˙  (t)) + t − t f b 2 (t − t f b )2

(7.6a)

u eq = −

1 ¨ d (t) (ρ X X + ρ2 X26 −  ρ3 1 2 4 nc nc + e (t) + e5 (t) − ϒ˙  (t)) t − t f c 2 (t − t f c )2

(7.6b)

u eq = −

m nd (ρ X − g − z¨ d (t) + e (t) cos X1 cos X3 Z 12 t − t f d z2 nd + e11 (t) − ϒ˙ z (t)) (t − t f d )2

(7.6c)

u zeq = −

(7.6d)

In the presence of disturbances, another control law should be added to the equivalent law. Then, we define the switching control laws by the following equations.

152

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

1 (K  s1 (t) + (δ Q + ηa )sign(s1 (t)))) ρ1 1 = − (K  s3 (t) + (δ Q + ηb )sign(s3 (t)))) ρ2 1 = − (K  s5 (t) + (δ Q + ηc )sign(s5 (t))) ρ3 m =− (K z s11 (t) + (δ Qz + ηd )sign(s11 (t)))) cos X1 cos X3

u sl = −

(7.7a)

u sl

(7.7b)

u sl u zsl

(7.7c) (7.7d)

whither δ Q is the upper bound defined as: δ Q j ( j=,,,z) = a0k + a1k |ek (t)| + a2k |ek+1 (t)|(k = 1, 3, 5, 7)

(7.8)

whither K j and η j are the positive constants. The total control inputs are given by: 1 ¨ d (t) (ρ1 X4 X6 + ρ2 X4 + ρ3 X22 −  ρ1 na na e2 (t) + e1 (t) − ϒ˙  (t) + t − tfa (t − t f a )2

u = −

+ K  s1 (t) + (δ Q + ηa )sign(s1 (t)))) 1 ¨ d (t) u  = − (ρ1 X2 X6 + ρ2 X2 + ρ3 X24 −  ρ2 nb nb e2 (t) + e3 (t) − ϒ˙  (t) + t − tfb (t − t f b )2

(7.9a)

+ K  s3 (t) + (δ Q + ηb )sign(s3 (t)))) 1 nc ¨ d (t) + u  = − (ρ1 X2 X4 + ρ2 X26 −  e2 (t) ρ3 t − tfc nc e5 (t) − ϒ˙  (t) + (t − t f c )2

(7.9b)

+ K  s5 (t) + (δ Q + ηc )sign(s5 (t)))) m (ρ Z X12 + g − z¨ d (t) um = − cos X1 cos X3 nd nd + ez2 (t) + e11 (t) − ϒ˙ z (t) t − tfd (t − t f d )2

(7.9c)

+ K z s11 (t) + (δ Qz + ηd )sign(s11 (t))))

(7.9d)

Lemma 7.1 Assume that a positive function V satisfies the following inequality [6] V˙  −aV − bV 

(7.10)

7.2 Controller Design Methodology

153

whither a and b are two positive parameters, and  (0 <  < 1), Then, the functional V converges to zero in the finite-time. Theorem 7.1 Consider roll-subsection under the Assumption 1, using the timevarying sliding variable defined in (7.2) and the controller presented in (7.9a), then the tracking errors of the roll-subsection will both converge to zero at the prespecified time t f a . Proof In order to prove the stability of the roll-subsection, the Lyapunov function candidate for the roll be described by: V =

1 2 s (t) 2 1

(7.11)

By using (7.4a), the time derivative of V is, V˙ = s˙1 (t)s1 (t) = s1 (t)(˙e2 (t) + +

na e2 (t) t − tfa

na e1 (t) − ϒ˙  (t)) (t − t f a )2

(7.12)

By substituting the tracking error and Eq. (7.9a) into Eq. (7.12), the dynamic of V is given as: V˙ = s˙1 (t)s1 (t) = s1 (t)(d (t) − K  s1 (t) − (δ Q + ηa )sign(s1 (t))) = s1 (t)d (t) − K  s1 (t)2 − (δ Q + ηa ) |s1 (t)| It is obvious that V˙ = s1 (t)d (t) − K  s1 (t)2 − (δ Q + ηa ) |s1 (t)|  |s1 (t)| |d (t)| − K  s1 (t)2 − (δ Q + ηa ) |s1 (t)| = |s1 (t)| (|d (t)| − δ Q ) − K  s1 (t)2 − ηa ) |s1 (t)| = −K  s1 (t)2 − ηa ) |s1 (t)| 0

(7.13)

Then, the state trajectories of the roll-subsystem are convergent to the switching curve s1 (t) = 0 in the finite time t f a and stay on the curve thereafter. Remark 7.1 The proposed controller with a TVGS surface is designed so that the QUAV outputs converge towards the origin values in finite-time t f . For t > t f i , the control input will maintain this equilibrium state when z(t) = 0. According nd e11 (t). After some simple calculato (7.2), we can write s70 (t) = ez2 (t) + t−t fd tions, the corresponding analytic solutions are e11 (t) = C z (t f d − t)n d and ez2 (t) =

154

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

−C z n d (t f d − t)n d −1 . It is clear that e11 (t f d ) = e11 (t f d ) = 0 and s70 (t) would be zero at t = t f z . In order to address the unknown disturbances assumed known in this subsection, an adaptive law will be designed in the following subsection.

7.2.2 Design Controller Based on an Adaptive Time-Varying SMC for the QUAV System In this part, an adaptive global TVSMC method is developed to estimate exactly the upper bound of these disturbances. Thus, adaptive switching controllers are obtained by modifying Eq. (7.7) as: 1 (K  s1 (t) + (δˆ Q + ηa )sign(s1 (t)))) ρ1 1 = − (K  s1 (t) + (aˆ 01 + aˆ 11 |e1 | + aˆ 21 |e2 | ρ1 + ηa )sign(s1 (t)))) 1 = − (K  s3 (t) + (δˆ Q + ηb )sign(s3 (t)))) ρ2 1 = − (K  s3 (t) + (aˆ 03 + aˆ 13 |e3 | + aˆ 23 |e2 | ρ2 + ηb )sign(s3 (t)))) 1 = − (K  s5 (t) + (δˆ Q + ηc )sign(s5 (t))) ρ3 1 = − (K  s5 (t) + (aˆ 05 + aˆ 15 |e5 | + aˆ 25 |e2 | ρ3 + ηc )sign(s5 (t)))) m =− (K z s11 (t) + (δˆ Qz + ηd )sign(s11 (t)))) cos X1 cos X3 m =− (K z s11 (t) + (aˆ 011 + aˆ 111 |e11 | + aˆ 211 |e2 | cos X1 cos X3 + ηd )sign(s11 (t))))

u sl = −

u sl

u sl

u zsl

(7.14a)

(7.14b)

(7.14c)

(7.14d)

whither aˆ 0 j , aˆ 1 j , and aˆ 2 j ( j = 1, 3, 5, 11) are the estimates of a0 j , a1 j , and a2 j , respectively. Then, the ATVGSM controllers for the altitude and attitude are given by:

7.2 Controller Design Methodology

155

1 ¨ d (t) (ρ1 X4 X6 + ρ2 X4 + ρ3 X22 −  ρ1 na na e2 (t) + e1 (t) − ϒ˙  (t) + K  s1 (t) + t − tfa (t − t f a )2

u = −

+ (aˆ 01 + aˆ 11 |e1 | + aˆ 21 |e2 | + ηa )sign(s1 (t)))) 1 ¨ d (t) u  = − (ρ1 X2 X6 + ρ2 X2 + ρ3 X24 −  ρ2 nb nb e2 (t) + e3 (t) − ϒ˙  (t) + K  s3 (t) + t − tfb (t − t f b )2

(7.15a)

+ (aˆ 03 + aˆ 13 |e3 | + aˆ 23 |e2 | + ηb )sign(s3 (t)))) 1 ¨ d (t) u  = − (ρ1 X2 X4 + ρ2 X26 −  ρ3 nc nc e2 (t) + e5 (t) − ϒ˙  (t) + K  s5 (t) + t − tfc (t − t f c )2

(7.15b)

+ (aˆ 05 + aˆ 15 |e5 | + aˆ 25 |e2 | + ηc )sign(s5 (t)))) m k6 (g − z˙ − z¨ d (t) um = − cos X1 cos X3 m nd nd + ez2 (t) + e11 (t) − ϒ˙ z (t) + K z s11 (t) t − tfd (t − t f d )2

(7.15c)

+ ((aˆ 011 + aˆ 111 |e11 | + aˆ 211 |e2 | + ηd )sign(s11 (t))))

(7.15d)

Also, the adaptive control laws are given by: a˙ˆ 01 = μ01 |s1 (t)| , a˙ˆ 03 = μ03 |s3 (t)| , a˙ˆ 05 = μ05 |s5 (t)| , a˙ˆ 011 = μ011 |s11 (t)|

(7.16)

a˙ˆ 11 = μ11 |s1 (t)| |e1 | , aˆ˙ 13 = μ13 |s3 (t)| |e3 | , a˙ˆ 15 = μ15 |s5 (t)| |e5 | , a˙ˆ 111 = μ111 |s11 (t)| |e11 |

(7.17)

a˙ˆ 21 = μ11 |s1 (t)| |e2 | , a˙ˆ 23 = μ13 |s3 (t)| |e2 | , aˆ˙ 25 = μ15 |s5 (t)| |e2 | , a˙ˆ 211 = μ111 |s11 (t)| |ez2 |

(7.18)

whither μ0i , μ1i , and μ2i (i = 1, 3, 5, 11) are non-zero positive parameters. Figure 7.1 depicts the flowchart diagram of the ATVGSMC method. Theorem 7.2 Consider roll-subsystem under the effect of disturbances, where the time-varying sliding variable presented in (7.4a) and in which the adaptive control law is given in (7.15a) and the update signals as (7.16)–(7.18), then the both errors of the roll-subsystem will converge to zero in finite-time t f a .

156

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

Fig. 7.1 The flowchart diagram of the AGTVSMC

Proof To obtain the parameters (aˆ 01 , aˆ 11 , aˆ 21 ) and prove the system stability, the Lyapunov function candidate of the -subsystem is considered in the following equation: 2  1 1 2 (7.19) V = s1 (t) + (aˆ i1 − ai1 )2 2 2μ i1 i=0 Calculating the time-derivative of V as, V˙ = s˙1 (t)s1 (t) +

2  1 (aˆ i1 − ai1 )a˙ˆ i1 μ i1 i=0

(7.20)

Substituting (7.4a) in (7.20), We obtain: V˙ = s1 (t){˙e2 (t) + +

na na e2 (t) + e1 (t) − ϒ˙  (t)} t − tfa (t − t f a )2

2  1 (aˆ i1 − ai1 )a˙ˆ i1 μ i1 i=0

Substituting the tracking error in control law (7.15a), we get: V˙ = s1 (t)(d (t) − K  s1 (t) − (aˆ 01 + aˆ 11 |e 1| + aˆ 12 |e 2|) |s1 (t)| 1 1 1 (aˆ 01 − a01 )a˙ˆ 01 + (aˆ 11 − a11 )a˙ˆ 11 + (aˆ 21 − a21 )a˙ˆ 22 + μ01 μ11 μ21 Using the update laws (7.16)–(7.18), we have:

(7.21)

7.2 Controller Design Methodology

157

V˙ = d (t)s1 (t) − K  s1 (t)2 − (aˆ 01 + aˆ 11 |e 1| + aˆ 12 |e 2| + ηa ) |s1 (t)| + (aˆ 01 − a01 ) |s1 (t)| + (aˆ 11 − a11 ) |s1 (t)| |e1 (t)| + (aˆ 21 − a21 ) |s1 (t)| |e2 (t)| (7.22) = d (t)s1 (t) − K  s1 (t)2 − ηa |s1 (t)| − (aˆ 01 + aˆ 11 |e 1| + aˆ 12 |e 2|) |s1 (t)| It is obvious that V˙ = d (t)s1 (t) − K  s1 (t)2 − ηa |s1 (t)| − (aˆ 01 + aˆ 11 |e 1| + aˆ 12 |e 2|) |s1 (t)|  |s1 (t)| |d (t)| − K  s1 (t)2 − ηa |s1 (t)| − (aˆ 01 + aˆ 11 |e 1| + aˆ 12 |e 2|) |s1 (t)|

(7.23)

= −K  s1 (t)2 − ηa |s1 (t)| The Lyapunov function (7.23) reduces gradually and the switching curve methods to the origin in the finite-time t1 . Remark 7.2 When t = t1 , lead to a singularity in the controllers of the altitude and attitude presented in (7.15a)–(7.15d), to solve this problem the control laws are changed. For example, the controller of the roll-subsection can be rewritten as:  ¨ d (t) + na e2 (t)  = − ρ1 (ρ1 X4 X6 + ρ2 X4 + ρ3 X22 ) −  t−t f a 1     +C n a 1 − n a (t − t f a )na −2   n −1   2 (t−t f a ) a +(1 + A pa ) t pa ( p −n +1) + K  s1 (t) + (aˆ 01 + aˆ 11 |e1 |   a a ta     +aˆ 21 |e2 | + ηa )sign(s1 (t)))) f or 0  t  t f a  u   na 1 2  ¨  = − ρ1 (ρ1 X4 X6 + ρ2 X4 + ρ3 X2 ) − d (t) + t−t f a e2 (t)   n a −2 +C n a 1 − n a (t − t f a )   n a −1   (t−t f a )   |e | |e | + a ˆ + + K s (t) + ( a ˆ + a ˆ pa  1 01 11 1 21 2 tta ( pa −n a +1)     +η )sign(s (t)))) f or t > t a

1

(7.24)

fa

Hence, as long as n a > 2 and pa > 1, the singularities are avoided in the control laws. The advantages of the controller presented in this chapter the robustness against unknown disturbances, the convergence to the origin at the predefined-time, high precision, and fast convergence rate.

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7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

Fig. 7.2 The imitated drag coefficients

Fig. 7.3 Gaussian random disturbances

7.3 Simulation Results and Analysis The simulation results of the AGTV sliding mode controller will presented in this section, indeed two scenarios are proposed to verify the advantages of the AGTVSMC method. In these scenarios, the change, of the drag coefficient of the translational and rotational subsystems, is used with gaussian random form as depicted in Fig. 7.2. The effect of this change is more real flying environment for the QUAV flight. Figure presented in Fig. 7.2 demonstrates the variation of the drag coefficients. All numerical results have performed with Gaussian random disturbances acting on the QUAV dynamics. These perturbances are depicted in Fig. 7.3. The simulations compare the RATVGSM controller with RTVGSMC to show the improvement. Moreover, some analysis and discussion about the results will be given in this section. Therefore, the nominal parameters of the QUAV are: g = 9.81m/s 2 , m = 0.74 kg, Ix x = I yy = 0.004 kg.m2 , Izz = 0.0084 kg.m2 , K i = 5.5670e − 4 m−1 .s.N K j 120, η = 0.5. The parameters of the proposed RATVGSMC are chosen as p j = 5, μ0 j = 0.0118, μ2 j = 0.01, n j = 5, μ1 j = 6.2942e − 04. The simulations results of the RATVGSMC and with RTVSMC methods for this scenario are plotted in Figs. 7.4, 7.5, 7.6, 7.7, 7.8. The outputs trajectories tracking are shown in Fig. 7.4. We can see from these results the position motion in the z direction follows its reference trajectory in finite-time with fast transient response can be obtained by the suggested approach than other controller. This altitude result confirms the superiority of the RATVGSM controller in term of disturbances. Moreover, we can observe from Fig. 7.4, the RATVGSM controller

7.3 Simulation Results and Analysis

159

Fig. 7.4 Quadrotor output tracking

Fig. 7.5 Velocity of the quadrotor output tracking

allows the QUAV to track the reference of the angles (d (t), d (t), d (t)) under complex environment. It can be seen from Fig. 7.4, the roll trajectory follows the desired reference signal with a good performance. The velocities of the attitude and position are plotted in Fig. 7.5. The responses of the sliding mode variables are displayed in Fig. 7.7, in which it shown that by using the RATVGSMC method, the all sliding mode variables converge to zero. The results of input control signals (u m , u  , u  , u  ) are displayed in Fig. 7.6, in which it is evident that the RATVGSMC method inputs produce low overshoot and faster/better setting time. From these results, we can observe the inputs converge to original values. The trajectories of the adaptive parameters a00 , a01 , a02 , are plotted in Fig. 7.8. It can be seen the uncertain parameters are stabilized at constant values.

160

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

Fig. 7.6 Control inputs

Fig. 7.7 Sliding surfaces

For show further robustness of the control method proposed in this chapter, the effect of the Gaussian noise on the QUAV states is considered with a comparison to RTVGSMC. In this scenario, the RATVGSMC is compared with SMC and backstepping SMC (BSMC) proposed respectively in [7, 8]. The altitude and attitude are affected by the Gaussian noise as shown in Fig. 7.9. The results are plotted in Figs. 7.10, 7.11, 7.12, 7.13, 7.14. Figure 7.10 displays the tracking trajectories of the altitude motion, yaw, pitch, and roll angles under the measurement noise. As can be observed, the tracking outputs have slight chattering due to the measurement noise. From 7.10 we can observe the desired trajectories were tracked with more accuracy. Moreover, the angular and linear velocities are displayed in Fig. 7.11. It can be observed from the result obtained, the convergence of these states to the original values is achieved. The control signals plotted in Fig. 7.12

7.3

Simulation Results and Analysis

161

Fig. 7.8 Adaptive parameters

Fig. 7.9 The measurement noise

remain smooth in the admissible range limit, and without any chattering which clearly shows the effectiveness of the suggested controller. The responses of the time-varying sliding manifolds plotted in Fig. 7.13 converge to zero in the finite time. The adaptive parameter responses are illustrated in Fig. 7.14. It can be seen from this result that the proposed adaptive laws yield a better performance of the tracking trajectories with more accuracy against the random disturbances, and the effect of the noise monuments.

7.4 Conclusions In this research chapter, the problem of the concept, a finite-time control for the QUAV in the presence of the random disturbances/uncertainties has been investigated. The control technique presented in this chapter is based on robust adaptive TVGSMC method to drive the QUAV to the equilibrium in specific time of convergence. Two

162

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

Fig. 7.10 Tracking trajectories (with noises)

Fig. 7.11 Linear and angular tracking trajectories (with noises)

unknown coefficients are added to the sliding manifolds to avoid the reaching phase. To reject the upper bound of the disturbances, suitable adaptive laws are designed. The presented results confirmed the robustness of the control method proposed in this chapter.

7.4

Conclusions

Fig. 7.12 Control inputs (with noises)

Fig. 7.13 Sliding surfaces (with noises)

Fig. 7.14 Adaptive parameters (with noises)

163

164

7 Robust Adaptive Global Time-Varying SMC for QUAV Subjected …

References 1. Labbadi, M., Cherkaoui, M.: Robust adaptive global time-varying sliding-mode control for finitetime tracker design of quadrotor drone subjected to Gaussian random parametric uncertainties and disturbances. Int. J. Control. Autom. Syst. 19(6), 2213–2223 (2021). Available at: http://dx. doi.org/10.1007/s12555-020-0329-5 2. Mofid, O., Mobayen, S.: Adaptive sliding mode control for fi nite-time stability of quad-rotor UAVs with parametric uncertainties (2017). https://doi.org/10.1016/j.isatra.2017.11.010 3. Labbadi, M., Cherkaoui, M.: Novel robust super twisting integral sliding mode controller for a quadrotor under external disturbances. J. Dyn. Control Int. (2019). https://doi.org/10.1007/ s40435-019-00599-6 4. Labbadi, M., Cherkaoui, M.: Robust integral terminal sliding mode control for quadrotor UAV with external disturbances. Int. J. Aerosp. Eng. (2019). https://doi.org/10.1155/2019/2016416 5. Boukattaya, M., Gassara, H., Damak, T.: A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Trans. (2019). https://doi.org/10.1016/ j.isatra.2019.07.003 6. Mobayen, S., Baleanu, D.: Stability analysis and controller design for the performance improvement of disturbed nonlinear systems using adaptive global sliding mode control approach. Nonlinear Dyn. 83, 1557–1565 (2016). https://doi.org/10.1007/s11071-015-2430-5 7. Xiong, J.-J., Zheng, E.-H.: Position and attitude tracking control for a quadrotor UAV. ISA Trans. 53(3), 725–731 (2014) 8. Chen, F., et al.: Robust backstepping sliding mode control and observer- based fault estimation for a quadrotor UAV. IEEE Trans. Ind. Electron. 63, 5044–56 (2016) 9. Lin, P., Ma, J., Zheng, Z.: Robust adaptive sliding mode control for uncertain nonlinear MIMO system with guaranteed steady state tracking error bounds. J. Frankl. Inst. 353, 303–321 (2016) 10. Zhihong, M., Yu, X.: Adaptive terminal sliding mode tracking control for rigid robotic manipulators with uncertain dynamics. JSME Int. J. Ser. C Mech. Syst. Mach. Elements Manuf. 40, 493–502 (1997)

Chapter 8

High Order Fractional Controller Based on PID-SMC for the QUAV Under Uncertainties and Disturbance

8.1 Introduction Applications of fractional-order (FO) calculus in automation have grown considerably in recent years, especially in robust control. FO control is widely used to improve the performance of closed-loop systems by improving trajectory tracking, transient steady-state responses, and ensuring better control performance for both integer order (IO) and FO systems. In this sense, quadrotors are highly maneuverable unmanned aerial vehicles (UAVs), which are sensitive to uncertainties in parameters such as mass, drag coefficients, and moments of inertia. The nonlinearities, aerodynamic disturbances, and high coupling between the rotational and translational dynamics of these vehicles pose a problem that requires a robust control system. Recently, these individual small and other autonomous vehicles have been observed to transport and deploy payloads in several domains. This chapter deals with the problem of payload transfer by a quadrotor. The effects of a variable load on the trajectory of the quadrotor in the presence of disturbances such as wind are considered. The variable load causes an overweight on the quadrotor and a modification of its moments of inertia. The overweight reduces its height, especially in case of uncertain load. Asymmetry, density uncertainty, load vibration, and internal displacement of components during flight and variable load cause changes in the quadrotor’s center of gravity resulting in changes in its moments of inertia. In this chapter, OF controllers have been designed to obtain a better flexibility in the regulation of complex flight trajectories in the presence of stochastic disturbances. An improved super-twisting OF proportional integral derivative action sliding mode control is proposed for the quadrotor system. This control is based on fractional operators while taking into account the problem of random variation of the drag coefficients of the translational and rotational motions under wind condition. under the wind condition. This will be considered useful when analyzing the stability and robustness of the system.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_8

165

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8 High Order Fractional Controller Based on PID-SMC …

The remainder of this chapter is organized as follows. Section 8.2 provides a detailed description of the FO-ST-PID-SMC based robust tracking control strategy. Simulation results of the FO-ST-PID-SMC approach and comparisons are presented in Sect. 8.3. Finally, the chapter concludes in Sect. 8.4.

8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System In the present section, we propose a robust FO controller for the position/attitude of the QUAV under the effect of the disturbances and parameter uncertainties. Based on the papers developed in [1–3], a new fractional-order super twisting-PID sliding mode control(FO-ST-PID-SMC) technique for the QUAV is designed. The control proposed in this chapter will be designed to compensate the stochastic disturbances/uncertainties.

8.2.1 Position Controller Design Based on FO-ST-PID-SMC The proposed control method addresses the problems of stability, finite-time convergence, and robustness against the disturbances and uncertainties. To enhance the control tracking and response speed of the system, a PIλ Dα sliding mode variables are developed as follows: s7 (t) = k x0 e7 (t) + k x1 D −λx e7 (t) + k x2 D αx e7 (t) s9 (t) = k y0 e9 (t) + k y1 D

−λ y

αy

(8.1)

e9 (t) + k y2 D e9 (t)

(8.2)

s11 (t) = k z0 e11 (t) + k z1 D −λz e11 (t) + k z2 D αz e11 (t)

(8.3)

with k xi , k yi , k zi (i = 0, 1, 2), λ j and α j ( j = x, y, z) are adjustable control parameters to be designed, Eq. (8.1) can be rewritten as s7 (t) = k x0 e7 (t) + k x1 D −λx e7 (t) + k x2 D αx −1 (X8 − x˙d (t))

(8.4)

The time-derivative of s7 (t) is ˙ 8 − x¨d (t)) s˙7 (t) = k x0 e˙7 (t) + k x1 D 1−λx e7 (t) + k x2 D αx −1 (X = k x0 e˙7 (t) + k x1 D 1−λx e7 (t) + k x2 D αx −1 (Vx + ρ X X8 + dx (t) − x¨d (t))

(8.5)

In order to obtain the equivalent control law, the time -derivative of sliding variable should be to s˙7 (t) = 0 without considering any disturbances. Then, this law is given as

8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System

167

1 1−αx D (k x0 e˙7 (t) + k x1 D 1−λx e7 (t)) k x2

(8.6)

Vxeq = −ρ X X8 + x¨d (t) −

To increase the robustness of the system, the FO switching control law is added to Vxeq law. Vx1 = Vxeq + Vx f os 1 1−αx D (k x0 e˙7 (t) k x2 + k x1 D 1−λx e7 (t)) + Vx f os

= ρ X X8 + x¨d (t) −

(8.7)

The term Vx f os denotes the switching law its expression as [2]. Vx f os = −

εx2 εx1 qx s7 (t) − D sign(s7 (t)) k x2 k x2

(8.8)

where εx1 and εx2 are positive parameters. qx denotes the fractional-order operator satisfying 0  qx < 1. q

Lemma 8.1 [3] Consider the RL fractional derivative t0 Dt x ϕ(t) = ϕ(τ ) dτ, 0  qx < 1, and the sign function, we have (t − τ )qx  qx t0 Dt sign(s7 (t))

=

t 1 d dt 0 (1 − qx )

> 0, i f s7 (t) > 0, t > 0 < 0, i f s7 (t) < 0, t > 0

(8.9)

Using Lemma 8.1, the virtual control signalis given by: Vx1 = Vxeq + Vx f os εx2 1 1−αx = −ρ X X8 − s7 (t) + x¨d (t) − D (k x0 e˙7 (t) k x2 k x2 εx1 qx + k x1 D 1−λx e7 (t)) − D sign(s7 (t)) k x2

(8.10)

Theorem 8.1 For the position variable x with control law (8.10), we can deduce that the position subsystem is asymptotically stable. Proof Define the Lyapunov candidate function as: V =

1 2 s (t) 2 7

(8.11)

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8 High Order Fractional Controller Based on PID-SMC …

The time-derivative of Eq. (8.11) is V˙ = s7 (t)˙s7 (t) = s7 (t)[k x0 e˙7 (t) + k x1 D 1−λx e7 (t) + k x2 D αx −1 (Vx − ρ X X8 + dx (t) − x¨d (t))] = s7 (t)[k x0 e˙7 (t) + k x1 D 1−λx e7 (t) + k x2 D αx −1 {ρ X X8 εx2 1 1−αx − s7 (t) + x¨d (t) − D (k x0 e˙7 (t) + k x1 D 1−λx e7 (t)) k x2 k x2 εx1 qx − D sign(s7 (t)) + ρ X X8 + dx (t) − x¨d (t)}] k x2    εx1 qx εx2 D sign(s7 (t)) − s7 (t) + dx (t) = s7 (t) k x2 D αx −1 − k x2 k x2 k x2 dx (t)] = −D αx −1 εx2 s72 (t) − D αx −1 s7 (t)[εx1 D qx sign(s7 (t)) − m  −D αx −1 εx2 s72 (t) − D αx −1 s7 (t)[εx1 D qx sign(s7 (t)) − Dx ] = −D αx −1 εx2 s72 (t) − D αx −1 |s7 (t)|[εx1 − |Dx |]

(8.12)

with Dx > kmx2 dx (t). From Lemma 8.1 the first part of Eq. (8.12) is negative and Dx denotes the upper bound of the uncertainty /disturbances satisfying |Dx |  εx1 , thus V˙  0. To improve the control performance, the proposed method will be combined with super-twisting. Thus, the total virtual control law is given by Vx = Vx1 + Vxst εx2 1 1−αx s7 (t) + x¨d (t) − D (k x0 e˙7 (t) k x2 k x2 εx1 qx + k x1 D 1−λx e7 (t)) − D sign(s7 (t)) + Vxst k x2

= −ρ X X8 −

(8.13)

The term Vxst denotes the ST control defined as follows:

Vxst

bx1 bx2 1 =− |s7 (t)| 2 sign(s7 (t)) − k x2 k x2

t sign(s7 (t))dτ

(8.14)

0

where bx1 and bx2 are positive parameters. Using Eq. (8.14), we can then rewrite the closed loop sliding mode variable dynamics as: 1 2

t

s˙7 (t) = −bx1 |s7 (t)| sign(s7 (t)) − bx2

sign(s7 (t))dτ − εx2 s7 (t) 0

(8.15)

8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System

169

So, the FOSTPIDSMC for the position x can be written as: Vx = Vx1 + Vxst εx2 1 1−αx s7 (t) + x¨d (t) − D (k x0 e˙7 (t) k x2 k x2 εx1 qx + k x1 D 1−λx e7 (t)) − D sign(s7 (t)) k x2 t bx1 bx2 1 2 − |s7 (t)| sign(s7 (t)) − sign(s7 (t))dτ k x2 k x2

= −ρ X X8 −

(8.16)

0

In the same way, the virtual controllers for another two subsystems such as the position y and z are given in Eqs. (8.17) and (8.18), respectively. Vy = Vy1 + Vyst ε y2 1 1−α y s9 (t) + y¨d (t) − D (k y0 e˙9 (t) k y2 k y2 ε y1 q y + k y1 D 1−λ y e9 (t)) − D sign(s9 (t)) k y2 t b y2 b y1 1 |s9 (t)| 2 sign(s9 (t)) − sign(s9 (t))dτ − k y2 k y2

= −ρY X10 −

(8.17)

0

Vz = Vz1 + Vzst εz2 1 1−αz s11 (t) + z¨ d (t) − D (k z0 e˙11 (t) k z2 k z2 εz1 qz D sign(s11 (t)) + k z1 D 1−λz e11 (t)) − k z2 t bz2 bz1 1 |s11 (t)| 2 sign(s11 (t)) − sign(s11 (t))dτ − k z2 k z2

= −ρ X X12 −

(8.18)

0

In this part of the chapter the proposed controller has been applied on the positionloop in the presence of complex disturbances and the stability is proved based on the Lyapunov theory. In the will be focused on the attitude-loop control based on the proposed control method.

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8 High Order Fractional Controller Based on PID-SMC …

8.2.2 Attitude Control Method Design Based on FO-ST-PID-SMC The attitude of the QUAV is controlled by u , u , u  . Let introduce the sliding mode variables for the attitude as: s1 (t) = k 0 e1 (t) + k 1 D −λ e1 (t) + k 2 D α e1 (t) s3 (t) = k 0 e3 (t) + k 1 D

−λ

s5 (t) = k0 e5 (t) + k1 D

−λ

α

e3 (t) + k 2 D e3 (t) α

e5 (t) + k2 D e5 (t)

(8.19) (8.20) (8.21)

where k i , k i , ki (i = 0, 1, 2), λ j ( j = , , ) are the non-negative constants. In this part, we use the same design steps presented for position controller for designing the attitude controller. Theorem 8.2 Consider the attitude-subsystem and assume that the disturbances d (t), d (t), d (t) are bounded. To ensure the convergence of s1 (t) = s3 (t) = s5 (t) = 0 in the finite-time and the tracking errors [e1 (t), e3 (t), e5 (t)] and [e˙1 (t), e˙3 (t), e˙5 (t)] converge to zero, the control laws can be developed as: u 1 = u eq + u f os = −(ρ1 X4 X6 + ρ2 X4 + ρ3 X22 )  1 1−α ¨d − D (k 0 e˙1 (t) + k 1 D 1−λ e1 (t)) + Ix x k 2  ε 2 ε 1 q D sign(s1 (t)) − s1 (t) − k 2 k 2

(8.22)

u 1 = u eq + u f os = −(ρ1 X2 X6 + ρ2 X2 + ρ3 X24 )  1 1−α

¨ d (t) − D (k 0 e˙3 (t) + k 1 D 1−λ e3 (t)) + I yy

k 2  ε 2 ε 1 q

D sign(s3 (t)) − s3 (t) − k 2 k 2

(8.23)

8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System

171

u 1 = u eq + u  f os = −(ρ1 X2 X4 + ρ2 X26 )  1 1−α ¨ d (t) − D (k0 e˙5 (t) + k1 D 1−λ e5 (t)) + Izz  k2  ε2 ε1 q D sign(s5 (t)) − s5 (t) − k2 k2

(8.24)

Proof To prove the stability of the system, the Lyapunov function can be given as: VR =

1 2 (s (t) + s32 (t) + s52 (t)) 2 1

(8.25)

The time-derivative of VR is, V˙ R = s1 (t)˙s1 (t) + s3 (t)˙s3 (t) + s5 (t)˙s5 (t)

(8.26)

Using the time-derivative of the sliding surfaces (8.19)–(8.21) and substituting the control laws (8.22)–(8.24) in (8.26), one can obtain:   k 2 d (t) V˙ R = −D α −1 ε 2 s12 (t) − D α −1 s1 (t) ε 1 D q sign(s1 (t)) − Ix x   k

2 − D α −1 ε 2 s32 (t) − D α −1 s3 (t) ε 1 D q sign(s3 (t)) − + d (t) I yy   k2 d (t) − D α −1 ε2 s52 (t) − D α −1 s5 (t) ε1 D q sign(s5 (t)) − Izz  −D α −1 ε 2 s1 (t)2 − D α −1 |s1 (t)|[ε 1 − |D |] − D α −1 ε 2 s52 (t) − D α −1 |σ |[ε 1 − |D |] − D α−1 ε2 s52 (t) − D α−1 |s5 (t)|[ε1 − |D |]

(8.27)

with D > k 2 d (t), D > k 2 d , D > k2 d (t). From Lemma 1 the first part of Eq. (8.27) is negative and D , D and D denote the upper bound of the disturbances and uncertainty satisfying |D |  ε 1 , |D |  ε 1 and |D |  ε1 , thus V˙ R  0. The rolling, pitching, and yawing control signal laws can be defined based on FO-PID-SMC and STA as:

172

8 High Order Fractional Controller Based on PID-SMC …

u = u 1 + V st = −(ρ1 X4 X6 + ρ2 X4 + ρ3 X22 )  1 1−α ¨d − D (k 0 e˙1 (t) + k 1 D 1−λ e1 (t)) + Ix x k 2 ε 1 q ε 2 − D sign(s1 (t)) − s1 (t) k 2 k 2  t b 2 b 1 1 |s1 (t)| 2 sign(s1 (t)) − sign(s1 (t))dτ − k 2 k 2

(8.28)

0

u = u 1 + V st = −(ρ1 X2 X6 + ρ2 X2 + ρ3 X24 )  1 1−α

¨ d (t) − D (k 0 e˙3 (t) + k 1 D 1−λ e3 (t)) + Iy

k 2 ε 1 q

ε 2 − D sign(s3 (t)) − s3 (t) k 2 k 2  t b 2 b 1 1 2 |s3 (t)| sign(s3 (t)) − sign(s3 (t))dτ − k

k 2

(8.29)

0

u  = u 1 + Vst = −(ρ1 X2 X4 + ρ2 X26 )  1 1−α ¨d − D (k0 e˙5 (t) + k1 D 1−λ e5 (t)) + Iz  k2 ε1 q ε ˙ 2 K 6 ) − 2 s5 (t) − D sign(s5 (t)) −  k2 k 2  t b2 b1 1 |s5 (t)| 2 sign(s5 (t)) − sign(s5 (t))dτ − k k2

(8.30)

0

Using the controller u the time-derivative of s1 (t) can be rewritten as: 1 2

t

s˙1 (t) = −b 1 |s1 (t)| sign(s1 (t)) − ε 2 s1 (t) − b 2

sign(s1 (t))dτ 0

Defining new variables ϒ 1 and ϒ 2 as:

(8.31)

8.2 Fractional-Order Controllers Design and Stability Analysis for the QUAV System

173

ϒ 1 = s1 (t) t ϒ 2 = −b 2

sign(s1 (t))dτ 0

The s1 (t) close-loop error dynamics (8.31) can be obtained as: 1 ϒ˙ 1 = −b 1 |ϒ 1 | 2 sign(ϒ 1 ) − ε 2 ϒ 1 + ϒ 2 ϒ˙ 2 = −b 2 sign(ϒ 1 )

(8.32) (8.33)

Theorem 8.3 The statements presented in the following points are equivalent:   1 1 ε − 2 b 1 2 2 is Hurwitz. • Matrix A = −b 2 0 • The constant parameters are positive b 1 > 0, b 2 > 0, and ε 2 > 0. T T as the positive matrix, the algebraic Lyapunov equation A P + • Define Q =Q A P = −Q has a unique positive and symmetric dene solution P =P T . Then, the controller u yields the convergence of s1 (t) = 0 and the both tracking errors e and e˙ converge to zero [4, 5]. The proof of Theorem 3 can be found on [4, 5].

8.3 Simulation Results and Discussions The section is devoted to evaluate the effectiveness and performance of the FOST-PID-SMC approach proposed in this work. The simulations for algorithms are performed out on MATLAB software. The control laws of the quadrotor employed in the simulations are defined in Eqs. (8.16)–(8.18) and (8.28)–(8.30). Moreover, the proposed controller was tested under different scenarios, which such as the stochastic disturbances, time-varying load and time-varying moment of inertia and complex profiles of the drag coefficients, etc. Furthermore, several robust control techniques were implemented for comparison. First, the nonlinear internal model control (NLIMC) controller was tested presented by the authors of [6]. Subsequently, the FO backstepping sliding mode control system was implemented, which presented in [2]. Therefore, all control technique parameters such as the FOSTPIDSMC, FOBSMC, and NLMIC control approaches have been tuned by using the optimization toolbox in MATLAB, in which shown in Table A.4. This technique of optimization is presented by the authors in [7]. Finally, the chattering problem caused by the sign function in the control laws of the FOSTPISSMC was replaced by the hyperbolic tangent.

174

8 High Order Fractional Controller Based on PID-SMC …

Table 8.1 Parameter settings of the proposed controller Parameter Value Parameter ki0 ki1 ki2 εi1 εi2 αi λi qi bi1 bi2

0.7474 0.0144 0.1374 0.6617 1 0.9 0.9 0.1 0.6617 1

k j0 k j1 k j2 ε j1 ε j2 αi λj qj b j1 b j2

Table 8.2 The desired reference Variable Value [xd (t), yd (t), z d (t)] (m)

[d (t)] (rad)

Value

[0.6,0.6,0.6] [0.3,0.6,0.6] [0.3,0.3,0.6] [0.6,0.3,0.6] [0.6,0.6,0.6] [0.6,0.6,0.0] [0.5] [0.0]

7.6041 0.0375 0.2133 71.2189 1 0.9 0.9 0.1 0.11 0.1

Time (sec) 0 10 20 30 40 50 0 50

Remark 1 For fractional operators, original mathematical concepts developed were integrated in CRONE toolbox [8]. Thus, the fractional-order terms in the proposed controller are approximated using 10-order oustaloup modified filter with its frequency range from 0.01 to 100 rad/s.

8.3.1 Simulation 1 In this simulation, the FO controller has been applied on the dynamics of the QUAV with a square trajectory in 3D space. Specially, the desired path is presented in Table 8.2. Let’s take the starting path like [0, 0, 0] m and [0, 0, 0] rad. Besides, to highlight the superiority of the control approach (called as FO-ST-PID-SMC), the drag coefficients change like white noise, which is plotted in Fig. 8.1 and the sixdegrees of freedom are affected by disturbances. The disturbances are [6, 9].

8.3 Simulation Results and Discussions

175

Fig. 8.1 The drag coefficients

 dx (t) = − 0.8 sin(0.1t − 0.97) + 0.4 sin(0.45t − 4.3)) π  t − 15 + 0.056 sin t − 2.73 + 0.08 sin 2 11 π

s −2 .1 t ∈ [10, 30] d y (t) = 0.5 sin(0.4t) + 0.7 cos(0.7t) s −2 .1 t ∈ [10, 50] d y (t) = 0.5 cos(0.7t) + 0.5 sin(0.3t) s −2 .1 t ∈ [0, 80] d (t) = 0.5 cos(0.4t) + 1 rad/s2 t ∈ [0, 80] d (t) = 0.5 sin(0.5t) + 1 rad/s2 t ∈ [0, 80] d (t) = 0.5 sin(0.7t) + 1 rad/s2 t ∈ [0, 80]

(8.34)

The results of this simulation are shown in Figs. 8.2, 8.3, 8.4, 8.5, 8.6, and 8.7. Figure 8.2 depicts the flight trajectory in 3D space results. It can be seen that the path following is driven to their desired value with the FO control proposed in this chapter. Then, the NLIMC method cannot compensate the disturbances, in which Fig. 8.2 depicts higher oscillations by the NLIMC method. The FO-ST-PID-SM control method is able to make the QUAV tracks the reference trajectory in the presence of the variation of drag coefficients with high accuracy. The FO-BSMC technique can track trajectory tracking under these conditions less than the proposed controller. Due to strong robustness of FO-ST-PID-SMC, the influence of the disturbances and drag coefficient effects are well compensated. The tracking errors of the system are depicted in Figs. 8.2 and 8.3. It can be observed that the FO-ST-PID-SMC achieves a

176

8 High Order Fractional Controller Based on PID-SMC …

Fig. 8.2 Trajectory tracking in 3D space

x (m)

0.6 Reference NLIMC FOBSMC FOSTPIDSMC

0.4 0.2 0 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

y (m)

0.6 0.4 0.2 0 0.8

z (m)

0.6 0.4 0.2 0

time (s)

Fig. 8.3 Responses of position

8.3 Simulation Results and Discussions

177

2 Reference Real

ϕ (rad)

1 0 -1 -2 10

20

30

40

50

10

20

30

40

50

60

70

80

60

70

80

70

80

2

θ (rad)

1 0 -1 -2

Ψ (rad)

1.8 1.6

0 -3 ×10

NLIMC FOBSMC FOSTPIDSMC

1.4 1.2 10 0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

60

time (s)

Fig. 8.4 Responses of attitude angles

good tracking performance under disturbances. Moreover, the FO-ST-PID-SMC has advantages including the convergence rate of the all-states variables can be achieved with high tracking accuracy. The attitude/position results are presented in Figs. 8.2 and 8.3. We can see that the QUAV outputs converge to their original values in finite-time by the proposed controller. Finally, the control inputs are plotted in Fig. 8.7. So, the quadrotor with FOSTPIDSMC shows more performance level than NLIMC and FOBSMC.

8.3.2 Simulation 2 In this simulation, we consider a 45% uncertainty of both moment of inertia mass in the presence of external disturbances. A comparative study is conducted with the existing FO-BSMC and NLIMC and controllers. Let’s taking the start path as [0.5, 0.5, 0.5]m and [0, 0, 0.5] rad. Define the desired circle reference as:

178

8 High Order Fractional Controller Based on PID-SMC …

NLIMC FOBSMC FOSTPIDSMC

0.4

e7 (m)

0.2 0

-0.2 -0.4 -0.6 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

e9 (m)

0.5

0

-0.5

e11 (m)

0.5

0

-0.5 0

time (s)

Fig. 8.5 Tracking errors of position

xd (t) = cos(t) m, yd (t) = sin(t) m, z d (t) = 0.5t m, d = sin(0.5t) rad.

(8.35)

The disturbances are [9, 10], dx (t), d y (t), d y (t) = 0.2 sin(1.5t + 5) s−2 .1 i f t ∈ [0, 10] dx (t), d y (t), d y (t) = 1 s−2 · 1 i f t ∈ [10, 20] d (t), d (t), d (t) = 0.2 sin(2t + 5) s−2 · rad i f t ∈ [0, 10] d (t), d (t), d (t) = 1 s−2 · rad i f t ∈ [10, 20]

(8.36)

The position response presented in Fig. 8.10 depicts satisfactory performance in terms of tracking error, error steady-state, and convergence rapidity in the presence of 45% uncertainly of the QUAV parameters. Figure 8.11 depicts the attitude result. It can be seen that the actual angles track the desired yaw, roll, and pitch and response

8.3 Simulation Results and Discussions

179

e1 (rad)

2 NLIMC FOBSMC FOSTPIDSMC

1 0 -1 -2 0

10

20

30

40

50

60

70

80

0 ×10-3

10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

2

e3 (rad)

1 0 -1 -2 -3

e5 (rad)

1

0

-1

-2 0

time (s)

Fig. 8.6 Tracking errors of attitude

references with high accuracy. Moreover, it can be seen from both Figs. 8.11 and 8.12 that the position and yaw tracking errors converge to zero in a short seconds. Compared with FO-BSMC and NLIMC the proposed FO control method can be achieved an good path following. The position variables (x(t), y(t), z(t)) result obtained using the BSMC technique shows a high oscillation when introducing disturbance due to wind gust. Figures 8.8 and 8.9 depict that the 3D and 2D space trajectories. It can be observed that the proposed FO-ST-PID-SMC method achieves the best trajectory performances among the both NLIMC and FO-BSMC control techniques.

8.3.3 Simulation 3 In this simulation, the problem of varying load effect on trajectory tracking is addressed in the presence of complex disturbances. The variations of the drag coefficients are also taken account in this simulation. In addition, the change of the mass of the QUAV is considered in the form shown in Fig. 8.13. The varying load can

180

8 High Order Fractional Controller Based on PID-SMC … 20

Um (N)

15 10 5 0 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

U (N.m)

5

0

-5

U (N.m)

5

0

-5 0

×10-3

U (N.m)

0

-5

-10

-15 0

time (s)

Fig. 8.7 Control inputs

have two effects on the path following, first the altitude of the QUAV is directly influenced by this load. Secondly, the moment of inertia is affected. To simulate the uncertainties of the moment of inertia created by the load variations, sine waves at different frequencies are added to nominal values moment of inertia according to Fig. 8.14. The stochastic disturbances applied on the dynamics system are given as [9, 10]: ⎧ dx (t) = 0.2 sin(2t + 5) + 12 exp(−0.5((t − 15)/(0.1))2 ) ⎪ ⎪ ⎪ ⎪ s−2 · 1 t ∈ [5, 80] ⎪ ⎪ ⎪ ⎪ ⎨ d y (t) = 0.2 sin(2t + 5) + 12 exp(−0.5((t − 20)/(0.1))2 ) s−2 · 1 t ∈ [0, 70] (8.37) ⎪ 2 ⎪ (t) = 0.2 sin(2t + 5) + 12 exp(−0.5((t − 41)/(0.1)) ) d ⎪ y ⎪ ⎪ ⎪ s−2 · 1 t ∈ [0, 60] ⎪ ⎪ ⎩ d (t), d (t), d (t) = 0.5 sin(1.5t + 5) s−2 · rad

8.3 Simulation Results and Discussions

181

Fig. 8.8 Trajectory tracking in 3D space 1.5

y (m)

1 Reference NLIMC FOBSMC FOSTPIDSMC -Parameter +45 % FOSTPIDSMC-Nominal Parameter

0.5 0

-0.5 -1 -1.5 -1.5

-1

-0.5

0

0.5

x (m)

1

1.5

Fig. 8.9 Horizontal position

To prove the efficiency of the FO-ST-PID-SMC, using a complex form of reference trajectory as: ⎧1   cos 21 t m ⎪ ⎪ ⎨ 21 m xd (t) = 2 0.25t − 4.5 m ⎪ ⎪ ⎩ 3m

t t t t

∈ [0, 4π ) ∈ [4π, 20) ∈ [20, 30) ∈ [30, 80]

(8.38)

182

8 High Order Fractional Controller Based on PID-SMC …

1

x (m)

0.5 0

-0.5 -1 0

2

4

6

8

10

12

14

16

18

20

2

4

6

8

10

12

14

16

18

20

y (m)

1 0.5 0

-0.5 -1 0 10

z (m)

8 6 Reference NLIMC FOBSMC FOSTPIDSMC -Parameter +45 % FOSTPIDSMC-Nominal Parameter

4 2 0

0

2

4

6

8

10

12

14

16

18

20

time (s)

Fig. 8.10 Responses of position

⎧1   sin 21 t m ⎪ 2 ⎪ ⎪ ⎪ ⎨ 0.25t − 3.14 m yd (t) = 5 − π m ⎪ ⎪ −0.2358t + 8.94 m ⎪ ⎪ ⎩ 1 −2 m

∈ [0, 4π ) ∈ [4π, 20) ∈ [20, 30) ∈ [30, 40] ∈ [40, 80]

(8.39)

⎧ t ∈ [0, 4π ) ⎨ 0.125t + 1 m t ∈ [4π, 40) z d (t) = 21 π + 1 m ⎩ exp(8.944 − 0.2t) m t ∈ [40, 80)

(8.40)

d (t) =

t t t t t

π

rad t ∈ [0, 50) 4 0 rad t ∈ (50, 80]

(8.41)

Let’s use the start points trajectory as [0.5, 0.5, 0.5]m and [0, 0, 0.5] rad. The results in this simulation are shown in Figs. 8.15, 8.16, 8.17, 8.18, and 8.19 with the results of the control method presented in [6] and FOBSMC proposed in [2]. In order to demonstrate the performance of the FOI control proposed in this chapter, the simu-

8.3 Simulation Results and Discussions

183

1 Reference Real

ϕ (rad)

0.5 0

-0.5 -1 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

2

θ (rad)

1 0 -1 -2 1.5 Reference NLIMC FOBSMC FOSTPIDSMC

1

ψ (rad)

0.5 0

-0.5 -1 0

2

4

6

8

10

time (s)

Fig. 8.11 Responses of attitude

lation is carried under complex perturbations and abrupt reference changes. Based on the optimization toolbox proposed in [7], the parameters of the three control methods have been tuned in the same condition. The results displayed in Figs. 8.15, 8.16, 8.17, 8.18, and 8.19. It can be seen that the QUAV follows the desired trajectory after a few seconds. The results obtained by the three control methods show no significant changes in the path trajectory when adding the load. The upper bounds of the perturbations acting in the x(t), y(t), and z(t)-axis are introduced after respectively 15 s, 20 s and 41 s. However, the proposed control method resolves the above problems and obtains a high tracking performance compared to two other control techniques. Based on those results presented in this simulation, the applied FO nonlinear control provides a good tracking of the path trajectory. The control tracking obtained by the position and attitude are displayed in both Figs. 8.17 and 8.18. Therefore, Fig. 8.19 demonstrate the tracking errors of the attitude and position of the system. Based on the results depicted in Figs. 8.15 and 8.16 for respectively 3D and 2D, the proposed control method is able to achieve a stable flight.

184

8 High Order Fractional Controller Based on PID-SMC …

e7 (m)

0.5

0 NLIMC FOBSMC FOSTPIDSMC

-0.5 0

2

4

6

8

10

12

14

16

18

20

18

20

e9 (m)

0.5

0

-0.5 0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

e11 (m)

0.5

0

-0.5

18

20

e5 (rad)

0.5

0

-0.5 18

20

time (s)

Fig. 8.12 Tracking errors

m+load (Kg)

1.8 1.6 1.4 1.2 1 0.8 0.6 0

10

20

30

40

time (s)

Fig. 8.13 The mass and load variation

50

60

70

80

8.3 Simulation Results and Discussions 10

185

×10-3

Ixx(N.m/rad/s)

8 6 4 2 10

0 ×10-3

10

20

30

40

50

60

70

80

0 ×10-3

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

Iyy(N.m/rad/s)

8 6 4 2

Izz(N.m/rad/s)

15

10

5

time (s)

Fig. 8.14 Moment of inertia variations arising from the load shakes

3

z (m)

2

Reference NLIMC FOBSMC FOSTPIDSMC

1

0 3 2 1 0

y (m)

-1

Fig. 8.15 Trajectory tracking in 3D space

-1

0

1

x (m)

2

3

4

186

8 High Order Fractional Controller Based on PID-SMC …

3

y (m)

2 Reference NLIMC FOBSMC FOSTPIDSMC

1 0

-1 -1

-0.5

0

0.5

1

x (m)

1.5

2

2.5

3

3.5

Fig. 8.16 Horizontal position

x (m)

3 Reference NLIMC FOBSMC FOSTPIDSMC

2 1 0 -1 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

y (m)

2 1 0 -1 3

z (m)

2

1

0

time (s)

Fig. 8.17 Responses of position

8.3 Simulation Results and Discussions

187 Reference Real

ϕ (rad)

1 0 -1 -2 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

2

θ (rad)

1 0 -1 -2 0.5

NLIMC FOBSMC FOSTPIDSMC

ψ (rad)

0.4 0.3 0.2 0.1 0 -0.1 0

10

20

30

40

50

60

70

80

time (s)

Fig. 8.18 Responses of attitude

8.3.4 Comparisons Analysis Some curves presented in the simulations specially in the simulation 2 are impotent to show clearly the differences between the control strategies. Therefore, a comparative analysis is proposed to show the differences between these controllers based on the performance of the integral absolute error IAE. The comparative results are summarized in Table 8.3. It can be seen that the FO-ST-PID-SMC has superior performance in terms of accuracy and has larger IAE index performances in all scenarios.

8.4 Conclusion The development of robust fractional order controllers for the quadrotor control system allows for improved flight path tracking under complex conditions. In this chapter, we have proposed a fractional order control approach based on sliding mode control. The use of a PID sliding surface leads to a fast convergence of the state variables of the system thus ensuring a stability of the QUAV system. The proposed controller differs from existing controllers by providing optimal trajectory tracking.

188

8 High Order Fractional Controller Based on PID-SMC … 0.3 NLIMC FOBSMC FOSTPIDSMC

e7(m)

0.2 0.1 0 -0.1 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

e9(m)

0.5

0

-0.5

e11(m)

0.5

0

-0.5

e5 (rad)

0.5

0

-0.5

time (s)

Fig. 8.19 Tracking errors

Moreover, the problems of load variation and time variation of drag coefficients are solved by this controller in the presence of stochastic disturbances. Finally, the simulation results show that the proposed controller is capable of performing trajectory tracking tasks under conditions of model uncertainty and wind disturbance with accuracy and speed, compared to existing robust approaches. In this chapter, fractional order control laws have been designed to control the UAVs specifically the quadrotor in the presence of complex aerodynamic disturbances and time-varying parametric uncertainties of the system.

References

189

Table 8.3 IAE performance indexes Variable

FOSTPIDSMC

FOBSMC

NLMIM

x(t) y(t) z(t) (t)

(t) (t)

0.6388 0.6433 0.6286 0.1802 0.1155 0.0027

1.109 1.121 1.407 0.1043 0.1068 0.0428

1.116 1.116 1.7338 1.092 1.5 0.0088

x(t) y(t) z(t) (t)

(t) (t)

0.2911 0.1777 0.3302 0.0335 0.0325 0.0233

2.4332 2.258 1.892 0.1024 0.1166 0.0315

0.2507 0.1827 0.5599 0.4421 0.6105 0.0235

x(t) y(t) z(t) (t)

(t) (t)

0.290 0.2826 0.3156 0.8718 0.1644 0.0244

0.7364 1.101 1.271 0.2129 0.1995 0.0357

1.1685 1.4654 0.5387 1.376 1.013 0.0631

Scenario 1

Scenario 2

Scenario 3

References 1. Fayazi, A., Pariz, N., Karimpour, A., Hosseinnia, S.H.: Robust position-based impedance control of lightweight single-link flexible robots interacting with the unknown environment via a fractional-order sliding mode controller. Robotica 36, 1920–1942 (2018). https://doi.org/10. 1017/S0263574718000802 2. Shi, X., Cheng, Y., Yin, et al.: Design of fractional-order backstepping sliding mode control for quadrotor UAV. Asian J. Control 21, 156–171 (2019). https://doi.org/10.1002/asjc.1946 3. Yin, C., Chen, Y., Zhong, S.M.: Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50(12), 3173–3181 (2014) 4. Alejandro, D., Moreno, J.A., Fridman, L.: Reaching time estimation for super-twisting based on Lyapunov function. IEEE Conf. Decis. Control 0-6 (2009) 5. Derafa, L., Benallegue, A., Fridman, L.: Super twisting control algorithm for the attitude tracking of a four rotors UAV. J. Franklin Inst. 349, 685–699 (2012). https://doi.org/10.1016/ j.jfranklin.2011.10.011 6. Bouzid, Y., Siguerdidjane, H., Bestaoui, Y.: Nonlinear internal model control applied to VTOL multi-rotors UAV. Mechatronics 47, 49–66 (2017) 7. Freire, F.P., Martins, N.A., Splendor, F.: Simple optimization method for tuning the gains of PID controllers for the autopilot of Cessna 182 aircraft using model-in-the-loop platform. J. Control Autom. Electr. Syst. 29, 441–450 (2018). https://doi.org/10.1007/s40313-018-0391x

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8. Tepljakov, A., Petlenkov, E., Belikov, J.: FOMCON: fractional-order modeling and control toolbox for MATLAB. In: Proceedings of the 18th International Conference Mixed Des. Integrated Circuits Systems, pp. 684–689 (2011) 9. Labbadi, M., Cherkaoui, M.: Robust adaptive nonsingular fast terminal sliding-mode tracking control for an uncertain quadrotor UAV subjected to disturbances. ISA Trans. 99, 290–304 (2020) 10. Li, Z., Ma, X., Li, Y.: Robust tracking control strategy for a quadrotor using RPD-SMC and RISE. Neurocomputing 331, 312–322 (2018)

Chapter 9

Global Fractional Controller Based on SMC for the QUAV Under Uncertainties and Disturbances

9.1 Introduction This chapter investigates to design fractional-order (FO) global sliding mode control scheme for path following of the QUAV under external disturbances. The designed controller ensures finite-time convergence of both tracking errors with high accuracy by introducing FO sliding mode manifolds. A global stabilization is obtained using the suggested control law. The Lyapunov stability is used to guarantee the global stabilization of the QUAV system. in order to validate the performance of the proposed controller, a comparative analysis is done. The chapter is organized as follows. In the Sect. 9.2, the FO controller is presented with the stability analysis without disturbances. Section 9.3 provides the design of he FO global SMC in the presence of disturbances. Simulation results is presented in Sect. 9.4. Section 9.5 concludes the chapter. Throughout the chapter, Z+ and R+ are the set of integer numbers and positive real, respectively; while C and R represent separately the set of complex real numbers. Euclidean space. For a vector ι = [ι1 , ι2 , . . . , ιn ]T ∈ Rn represents the n-dimensional  n 2 Rn , let us use ι2 = i=1 |ιi | to denote the 2-norm of vector ι, while ι denotes an arbitrary norm of vector ι.

9.2 Design of Fractional Sliding Manifold for Quadrotor System Without Disturbances and Modeling Uncertainties In this section, a fractional-order global SMC is developed for the QUAV to enhance the trajectory tracking. The FOGSMC design procedure is presented as [1]:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_9

191

192

9 Global Fractional Controller Based on SMC for the QUAV …

Let rewrite the tracking error as: ei (t) = X7 − xd (t), e˙i (t) = X8 − x˙d (t)

(9.1)

In this part will be designed a sliding manifold that ensures e(t) = x(t) − xd (t)(t) −→ 0 when t −→ ∞. Let us introduce the RL integral of tracking error ei (t) as [2] to [5]:

I γ ei (t) =

ei (0) γ 1 t + (γ + 1) (γ )



t 0

ei (τ ) − ei (0) dτ (t − τ )(1−γ )

(9.2)

It is necessary that ei (0) = 0 in order to ensure that the tracking error e(t) is differentiable at t = 0. Then, we define the initial values of the tracking error and their time-derivative as [2]. According to [2–5], consider the bended tracking error as Z i (t) = ei (t) − ϒi (t)

(9.3)

for ϒi (t) a smooth function, such that, {Z i (0), Z˙ i (0)} = {0, 0} and {ϒi (ts ), ϒ˙ i (ts )} = {0, 0}, where ϒi (t) can be defined as [2]:  ϒi (t) =

b0i + b1i t + b2i t 2 + b3i t 3 f or 0 ≤ t ≤ ts 0 f or t ≥ ts

(9.4)

where the constants {b0i , b1i , b2i , b3i } can be defined as: b0i = ei (0) b1i = e˙i (0) e˙i (0) ei (0) b2i = −2 −3 2 ts ts e˙i (0) ei (0) b3i = 2 + 2 3 ts ts

(9.5)

Remark 9.1 The bended tracking error Z i (t) provides additional terms, in contrast to classical error ei (t) and enhances the tracking performance of the control law. The sliding mode manifolds of the position and attitude are defined by the following equations, as [6] s7 (t) = β1 I γ1 Z 7 (t) + γ1 D 1−γ1 Z 7 (t) + D 1−γ1 Z 8 (t)

(9.6a)

s9 (t) = β3 I γ3 Z 9 (t) + γ3 D 1−γ3 Z 9 (t) + D 1−γ3 Z 10 (t)

(9.6b)

9.2 Design of Fractional Sliding Manifold for Quadrotor System …

193

s11 (t) = β5 I γ5 Z 11 (t) + γ5 D 1−γ5 Z 11 (t) + D 1−γ5 Z 12 (t)

(9.6c)

s1 (t) = β7 I γ7 Z 1 (t) + γ7 D 1−γ7 Z 1 (t) + D 1−γ7 Z 2 (t)

(9.7a)

and,

γ9

s3 (t) = β9 I Z 3 (t) + γ9 D s5 (t) = β11 I

γ11

1−γ9

Z 5 (t) + γ11 D

Z 3 (t) + D

1−γ11

1−γ9

Z 5 (t) + D

Z 4 (t)

1−γ11

Z 6 (t)

(9.7b) (9.7c)

where γi ∈ (0, 1), βi and γi are positive parameters for i = 1, 3, . . . , 11. According Property 1 and Property 2 and tacking type Riemann-Liouville fractional derivative for the sliding surfaces (9.6) and (9.7) as D γi si (t) = βi Z i (t) + γi Z i+1 (t) + Z˙ i+1 (t) i = 1, 3, . . . , 11

(9.8)

Substituting the tracking error into (9.8), one has D γ1 s7 (t) = β1 Z 7 (t) + γ1 Z 8 (t) + ρ X X8 + Vx − x¨d (t) D γ3 s9 (t) = β3 Z 9 (t) + γ3 Z 10 (t) + ρ X X10 + Vy − y¨d (t) D γ5 s11 (t) = β5 Z 1 1(t) + γ5 Z 12 (t) + ρ Z X12 + Vz − g − z¨ d (t)

(9.9a) (9.9b) (9.9c)

and, D γ7 s1 (t) = β7 Z 1 (t) + γ7 Z 2 (t) + ρ1 X4 X6 + ρ2 X4 + ρ3 X22 + ρ1 u (9.10a) D γ9 s3 (t) = β9 Z 3 (t) + γ9 Z 4 (t) + ρ1 X2 X6 + ρ2 X2 + ρ3 X24 + ρ2 u (9.10b) D γ11 s5 (t) = β11 Z 5 (t) + γ11 Z 6 (t)ρ1 X2 X4 + ρ2 X26 + ρ3 u 

(9.10c)

The equivalent control laws can be obtained by canceling the terms on the right-hand side of (9.9) and (9.10), then the equivalent control laws of the position and attitude can be expressed as u xeq = −[β1 Z 7 (t) + γ1 Z 8 (t) + ρ X X8 − x¨d (t)]

(9.11a)

u yeq = −[β3 Z 9 (t) + γ3 Z 10 (t) + ρY X10 − y¨d (t)] (9.11b) u zeq = −[β5 Z 11 (t) + γ5 Z 12 (t) + ρ Z X12 − g − z¨ d (t)] (9.11c)   1 ¨ d (t) u eq = − β7 Z 1 (t) + γ7 Z 8 (t) + ρ1 X4 X6 + ρ2 X4 + ρ3 X22 − ρ1 (9.11d)   1 ¨ d (t) u eq = − β9 Z 9 (t) + γ9 Z 10 (t) + ρ1 X2 X6 + ρ2 X2 + ρ3 X24 −

ρ2 (9.11e)  1  ¨ d (t) u eq = − β11 Z 11 (t) + γ11 Z 12 (t) + ρ1 X2 X4 + ρ2 X26 −  (9.11f) ρ3

194

9 Global Fractional Controller Based on SMC for the QUAV …

The switching control signal laws can be defined as: Vx = Vxeq − [ε1 I 1−γ1 sign q1 s7 (t) + σ1 I 1−γ1 s7 (t)]

(9.12a)

Vy = Vyeq − [ε3 I

(9.12b)

1−γ3

sign s9 (t) + σ3 I q3

1−γ3

s9 (t)]

Vz = Vzeq − [ε5 I sign s11 (t) + σ5 I s11 (t)] 1 u = u eq − [ε7 I 1−γ7 sign q7 s1 (t) + σ7 I 1−γ7 s1 (t)] ρ1 1 u = u eq − [ε9 I 1−γ9 sign q9 s3 (t) + σ9 I 1−γ9 s3 (t)] ρ2 1 u  = u eq − [ε11 I 1−γ11 sign q11 s5 (t) + σ11 I 1−γ11 s5 (t)] ρ3 1−γ5

q5

1−γ5

(9.12c) (9.12d) (9.12e) (9.12f)

where εi and σi are positive coefficients, sign qi si (t)  sgn(s(t)i ) |s(t)i |qi , qi ∈ (0, 1), for i = 1, 3, . . . , 11. Where sgn(∗) = 1 if ∗ > 0 or sgn(∗) = −1 if ∗ < 0. Remark 9.2 To solve the chattering problem in (9.12a)–(9.12f), the discontinuous functions have been replaced tanh( μ∗ ), with μ is a small positive parameter. Theorem 9.1 The FOGSMC (9.12a) for the x-subsystem without disturbances in path following can assure the system finite-time stability and its state vari1 ln(1 + ables converge to the sliding manifold s1 (t) with the finite-time Tr ≤ σ1 (1−q 1) σ1 ε1

1 s1 (0)1−q ). 2

Proof Choose the Lyapunov function candidate for the x-subsystem as V1 =

1 2 s (t) 2 7

(9.13)

The time derivative of V1 can be given by V˙1 = s7 (t)˙s7 (t) = s7 (t)D 1−γ1 (D γ1 s7 (t))

(9.14)

Substituting (9.10a) and (9.12a) into (9.14), we have   s7 (t) q1 1−γ1 1−γ1 1−γ1 ˙ |s7 (t)| − σ1 I −ε1 I tanh s7 (t) V1 ≤ s7 (t)D μ1   s7 (t) |s7 (t)|q1 − σ1 s7 (t) = s7 (t) −ε1 tanh μ1   s7 (t) q1 |s7 (t)| − σ1 s72 (t) = sgn(s7 (t)) |s7 (t)| −ε1 tanh μ1  s7 (t) |s7 (t)|q1 +1 − σ1 s72 (t) = −ε1 sgn(s7 (t)) tanh μ1



s7 (t)



. |s7 (t)|q1 +1 − σ1 s72 (t) = −ε1 tanh μ1

(9.15) (9.16) (9.17) (9.18) (9.19)

9.2 Design of Fractional Sliding Manifold for Quadrotor System …

195

≤ −ε1 |s7 (t)|1+q1 − σ1 s72 (t)

(9.20)

Hence, one obtains V˙1 ≤ −ε1 |s7 (t)|1+q1

(9.21)

The above ineqaulity (9.21) can be rewritten as 1+q V˙1 ≤ −ε1 s7 (t)2 1

(9.22)

Hence, from analysis above, it can be concluded that sate variables will asymptotically converge to s7 (t) = 0. Using Eq. (9.15), we have V˙1 ≤ −ε1 |s7 (t)|1+q1 − σ1 s72 (t) = −ε1 (2V1 )

1+q1 2

− σ1 (2V1 )

(9.23)

Hence, after simple calculation we can get dt ≤ −

(2V1 )− 2 d(2V1 ) 1 =− q 2 ε1 (2V1 ) 21 + σ1 (2V1 ) 21 + σ1 (2V1 ) 1

d V1 ε1 (2V1 )

1+q1 2

1

≤− ≤

d(2V1 ) 2 ε1 (2V1 )

q1 2

+ σ1 (2V1 ) −q1 s7 (t)2 ds7 (t)2 = 1−q ε1 + σ1 s7 (t)2 1

1 2

=−

ds7 (t)2 q ε1 s7 (t)21 + σ1 s7 (t)2 1−q

−

d(σ1 s7 (t)2 1 1 1 σ1 (1 − q1 ) ε1 + σ1 s7 (t)1−q 2

(9.24)

By integraling Eq. (9.24) from 0 to tr and s1 (tr ) = 0, one gets 1 tr − 0 ≤ − σ1 (1 − q1 )

tr

1−q1

d(σ1 s7 (t)2

1−q1

0

ε1 + σ1 s7 (t)2

1 1−q ln(ε1 + σ1 s7 (t)2 1 )|t0r σ1 (1 − q1 ) 1 1−q [ln(ε1 ) − ln(ε1 + σ1 s1 (0)2 1 )] ≤− σ1 (1 − q1 )  1 σ1 1−q ≤ ln 1 + s1 (0)2 1 σ1 (1 − q1 ) ε1 ≤−

(9.25)

Let Tr be the finite time, which satisfies  σ1 1 1−q1 Tr = tr − 0 ≤ ln 1 + s1 (0)2 σ1 (1 − q1 ) ε1 This completes the proof.

(9.26)

196

9 Global Fractional Controller Based on SMC for the QUAV …

Fig. 9.1 Relationship between the reaching time tr and parameters σ , and ε with tacking s(0)2 = 1.1 and q = 35

Figure 9.1 shows the variations effect of the parameters on the reaching time. From this result, we can see that the reaching time tr is more sensitive to variation in the parameter σ than to variations in the parameter ε.

9.3 Design of Fractional Sliding Manifold for the QUAV System in the Presence of Modeling Uncertainties and Disturbances The FO global SMC is designed for the QUAV system in the previous section without uncertainties and disturbances. in the present section will be included the disturbances in the control design. We conduce the same design process given in the previous section to design FOGSM control law for the QUAV in the presence of disturbances and model uncertainties. Assumption 9.6 Let dx,y,y, , , (t) = dx,y,y, , , (t) +  f x,y,y, , , be the additive disturbance and uncertainty on the QUAV dynamics, a positive  there exists  function δdi for i = x, . . . ,  such as di  ≤ I 1−γi δdi and  D 1−γi di  ≤ δdi . Consequently, the FOGSMC laws for the QUAV system in the presence of disturbances and uncertainties are modified as: Vx = − β1 Z 7 (t) + γ1 Z 8 (t) + ρ X X8 − x¨d (t) + I 1−γ1 δd x 

s7 (t) q1 1−γ1 1−γ1 |s7 (t)| + σ1 I + ε1 I tanh s7 (t) (9.27a) μ1 Vy = − β3 Z 9 (t) + γ3 Z 10 (t) + ρY X10 − y¨d (t) + I 1−γ3 δdy 

s9 (t) q3 1−γ3 1−γ3 |s9 (t)| + σ3 I + ε3 I tanh s9 (t) (9.27b) μ3

9.3 Design of Fractional Sliding Manifold for the QUAV System …

197

Vz = − β5 Z 11 (t) + γ5 Z 12 (t) + ρ Z X12 − g − z¨ d (t) + I 1−γ5 δdz 

s11 (t) |s11 (t)|q5 + σ5 I 1−γ5 s11 (t) + ε5 I 1−γ5 tanh (9.27c) μ5 1 ¨ d (t) β7 Z 1 (t) + γ7 Z 2 (t) + ρ1 X4 X6 + ρ2 X4 + ρ3 X22 − u = − ρ1 

s1 (t) |s1 (t)|q7 + σ7 I 1−γ7 s1 (t) (9.27d) + I 1−γ7 δd + ε7 I 1−γ7 tanh μ7 1 ¨ d (t) β10 Z 3 (t) + γ9 Z 4 (t) + ρ1 X2 X6 + ρ2 X2 + ρ3 X24 −

u = − ρ2 

s3 (t) |s3 (t)|q9 + σ3 I 1−γ9 s3 (t) (9.27e) + I 1−γ9 δd + ε9 I 1−γ9 tanh μ9 1 ¨ d (t) β11 Z 5 (t) + γ11 Z 6 (t) + ρ1 X2 X4 + ρ2 X26 −  u = − ρ3 

s5 (t) |s5 (t)|q11 + σ11 I 1−γ11 s5 (t) (9.27f) + I 1−γ11 δd + ε11 I 1−γ11 tanh μ11 Theorem 9.2 Consider the x-subsystem dynamics with dx (t) = 0 controlling by u x (9.27a), then the FOGSM surfaces s7 (t) be reached in finite time tr . Moreover, the tracking error variables can converge fast to zero in short finite-time. Proof Consider the positive Lyapunov function as: V1d =

1 2 s (t) 2 7

(9.28)

Calculating the time derivative of V1d , we get V˙1d = s˙7 (t)s7 (t) = s7 (t)D 1−γ1 (D γ1 s7 (t))

(9.29)

Substituting (9.10a) and (9.27a) into (9.29), we have   s7 (t) |s7 (t)|q1 − σ1 I 1−γ1 s7 (t) V˙1d = s7 (t)D 1−γ1 dx (t) − I 1−γ1 δ H x − ε1 I 1−γ1 tanh μ1   (t) s 7 |s7 (t)|q1 − σ1 s7 (t) = s7 (t) D 1−γ1 dx (t) − δ H x − ε1 tanh μ1  s7 (t) |s7 (t)|q1 − σ1 s7 (t)s7 (t)) = D 1−γ1 s7 (t)dx (t) − s7 (t)δ H x − ε1 s7 (t) tanh μ1



s7 (t)

|s7 (t)|q1 +1 − σ1 s7 (t)s7 (t)) = D 1−γ1 s7 (t)dx (t) − s7 (t)δ H x − ε1

tanh μ1



s7 (t)

|s7 (t)|q1 +1 − σ1 s7 (t)s7 (t)) ≤ −ε1

tanh μ1

198

9 Global Fractional Controller Based on SMC for the QUAV … ≤ −ε1 |s7 (t)|q1 +1 − σ1 s72 (t) ≤0

(9.30)

Thus, it can be concluded that the sate variables of the QUAV system will converge in short finite-time. The remain of proof is similar to the end of Theorem 9.1 proof.

9.4 Results and Discussion In this part of the present chapter, three simulation results will be presented. To show the robustness and superiority of the FOGSMC scheme, a comparative study is given including the results of fractional order backstepping sliding mode controller (FOBSMC) [9] and integral backstepping sliding mode control (IBSMC) method [8], and backstepping sliding mode control (BSMC) technique [10, 11]. to obtain good tracking performance, the parameters of all controllers are tuned carefully. Remark 9.3 The optimization toolbox in Simulink is applied to chose the optimal parameters (see Ref. [12]). Remark 9.4 In this chapter, to approximate FO dynamic used in the designed controllers, the CRONE toolbox is used [13, 14] designed since the nineties by the CRONE team [15–17]. In the simulations, we approximated FO operators by using 2-order Oustaloup modified filter with its frequency range from 0.02 to 50 rad/s.

9.4.1 Simulation 1 In this simulation, we consider the time varying of the drag coefficients as illustrated in Fig. 9.2. In addition, the simulation is conducted under disturbances. In the same situations, the FOGSM controller is compared with other control methods. Taking the initial conditions [x0 , y0 , z 0 , 0 , 0 , 0 ]T = [0 m, 4.6 m, 0 m, 0 rad, 0 rad, 0.1 rad]. The disturbances used in this simulation are given by the following equation: 

0 m · s−2 t ∈ [0, 10] 1 m · s−2 t ∈ [0, 155] ⎧2 ⎨ 0.4 m · s−2 t ∈ [0, 20] d y (t) = −0.2 m · s−2 t ∈ [20, 50] ⎩ 0.2 m · s−2 t ∈ [50, 155] ⎧ ⎨ 0 m · s−2 t ∈ [0, 10] dz (t) = −0.2 m · s−2 t ∈ [10, 40] ⎩ 0.3 m · s−2 t ∈ [40, 155]  1 1 d (t) = sin t rad · s−2 2 2 dx (t) =

9.4 Results and Discussion

199 Drag Coefficients for transitional Motions

0.015

kx, ky, kz

0.01 0.005 0 -0.005 -0.01 -0.015 0

10

20

30

40

time (sec)

50

60

70

80

60

70

80

Drag Coefficients for transitional Motions

0.015

kϕ, kθ, kψ

0.01 0.005 0 -0.005 -0.01 -0.015 0

10

20

30

40

time (sec)

50

Fig. 9.2 The drag coefficients

1 cos(t) rad · s−2 2 1 d (t) = sin(0.7t) rad · s−2 2 d (t) =

(9.31)

The results of this simulation are presented in Figs. 9.3, 9.4, 9.5, 9.6, 9.7, and 9.8. The QUAV outputs x(t), y(t), z(t), (t), (t), and (t) under the controllers (9.27) are separately depicted in Figs. 9.3 and 9.4. The FO global SM control scheme exhibits the best tracking performance in terms of stability and tracking accuracy in comparison with fractional order backstepping sliding mode and integral sliding mode controllers (see Fig. 9.3). The linear and angular velocities are depicted in Figs. 9.5 and 9.6. It can be observed that outputs are driven to their references within seconds. The signal inputs are plotted in Fig. 9.7 and the tracking errors Fig. 9.8. From Fig. 9.7, we observe that the controllers applied to the dynamics of the QUAV under disturbances are smooth, practical without having oscillations, and capable of overcoming the disturbance. Also, good performance of tracking errors is obtained as shown Fig. 9.8. The tracking performance in 2D and 3D spaces are depicted in Figs. 9.9 and 9.10. The control method proposed in this chapter achieved better tracking performance than other control methods.

200

Fig. 9.3 Position response

Fig. 9.4 Attitude response

9 Global Fractional Controller Based on SMC for the QUAV …

9.4 Results and Discussion

201

Fig. 9.5 Angular velocity

9.4.2 Simulation 2 In order to further verify the control approach proposed in this chapter, an uncertainty %30 is added in all parameters of the QUAV system. Complex form of the desired trajectory and disturbances are also considered in this simulation. On the other hand, the FO-BSMC and IBSMC techniques are simulated under the same condition of the FOGSMC in order to show the better control performance of this one. The disturbances are given as: dx (t) = sin(t) m · s−2 d y (t) = cos(t) m · s−2 dz (t) = sin(t) cos(t) m · s−2 d (t) = 0.5 cos(0.5t) rad · s−2 d (t) = 0.5 sin(t) rad · s−2 d (t) = 0.5 sin(0.7t) cos(0.7t) rad · s−2

(9.32)

202

9 Global Fractional Controller Based on SMC for the QUAV …

Fig. 9.6 Linear velocity

Tacking the desired trajectory as  xd (t) =  yd (t) =

0 m t ∈ [0, 7] )] m t ∈ [7, 155] 10[1 − cos( 2π(t−7) 23 0 m t ∈ [0, 7] ) m t ∈ [7, 155] 5 sin( 4π(t−7) 23

z d (t) = 7(1 − exp(−0.3t) m d (t) = 0.5 sin(t) rad

(9.33)

The initial conditions are [x0 , y0 , z 0 , 0 , 0 , 0 ]T = [0 m, 0 m, 0 m, 0 rad, 0 rad, 0.1 rad]. The numerical simulation curves are plotted in Figs. 9.11, 9.12, 9.13, 9.14, 9.15. The comparisons between x(t), y(t), z(t), and (t) under suggested controller, FOSMC, and IBSMC are shown in Fig. 9.11. Figure 9.12 shows the performance tracking of roll and pitch angles using the proposed controller. The 2D and 3D trajectories are depicted in Figs. 9.9 and 9.10. The results of the tracking errors are depicted in Figs. 9.15. They show that the results under the suggested FOGSMC law own a faster response due the adjusted fractional parameters, higher tracking performance, and lower chattering problem.

9.4 Results and Discussion

203

Fig. 9.7 Signal inputs

9.4.3 Simulation 3 In this simulation, the proposed control law is compared with another nonlinear BSM controller to show the superiority of the proposed controller. Different perturbations are used in this case to make the QUAV system more real. The disturbances are given by the following equation. dx (t) = 0.15 sin(2.5t) + 0.1 cos(4t) m · s−2 d y (t) = 0.15 sin(t) + 0.1 cos(2.5t) m · s−2 dz (t) = 0.5 sin(1.5t) + 0.3 cos(2t) m · s−2 d (t) = 0.8 sin(2t) + 1.5 cos(3t) rad · s−2 d (t) = 2 cos(2.5t) + 0.9 sin(1.5t) rad · s−2 d (t) = 1.3 cos(2t) + 2 sin(3t) rad · s−2

(9.34)

204

Fig. 9.8 Tracking errors Fig. 9.9 Three dimensional tracking results

Fig. 9.10 Two dimensional tracking results

9 Global Fractional Controller Based on SMC for the QUAV …

9.4 Results and Discussion

Fig. 9.11 Position response

Fig. 9.12 Attitude response

205

206

9 Global Fractional Controller Based on SMC for the QUAV …

Fig. 9.13 Three dimensional tracking results Fig. 9.14 Two dimensional tracking results

The simulation results in this case are shown in Figs. 9.16, 9.17, 9.18. The black curves are the desired trajectories, the green curves are the BSM control performance under disturbances, maroon curves are the FOBSM control performance under disturbances, the blue curves are the IBSM control performance under disturbances, and the red curves are the proposed controller performance under disturbances. The position and yaw angle responses of the control methods are depicted in Figs. 9.16 and 9.17, which show that the suggested controller is enable to track the desired trajectories with a accuracy and higher performances compared to other controllers. The 3D position results are displayed in Fig. 9.18, which shows that the FO global SMC law enable to force the QUAV to follow the desired trajectory under disturbances.Three simulations results show that the FO global SM control technique has been quite effective for control the quadrotor under exogenous disturbances and parametric uncertainties, using global sliding mode controller with fractional operators.

9.5 Conclusions In this chapter, the problem of designing nonlinear FO control scheme is addressed and applied to the uncertain QUAV subjected to disturbances. The robust FO global SMC is presented for the QUAV system in order to stabilize it in the presence of

9.5 Conclusions

207

Fig. 9.15 Tracking errors

complex disturbances. The control scheme developed in this chapter reduced the chattering effect and provided a global stabilization of the QUAV system. Simulation results demonstrate that the control proposed in this chapter is able to track the trajectory tracking in the different scenarios in terms of disturbances and model uncertainties.

208

Fig. 9.16 Position response

9 Global Fractional Controller Based on SMC for the QUAV …

9.5 Conclusions

Fig. 9.17 Attitude response

Fig. 9.18 Three dimensional tracking results

209

210

9 Global Fractional Controller Based on SMC for the QUAV …

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16. Boukal, Y., Zasadzinski, M., Darouach, M., Radhy, N. E.: Stability and stabilizability analysis of fractional-order time-varying delay systems via diffusive representation. In: 2016 5th International Conference on Systems and Control (ICSC), Marrakesh, pp. 262–266 (2016). https:// doi.org/10.1109/ICoSC.2016.7507077 17. Boukal, Y., Zasadzinski, M., Darouach, M., Radhy, N. E.: Robust H ∞ observer-based stabilization of disturbed uncertain fractional-order systems using a two-step procedure. In: Domek, S., Dworak, P. (eds) Theoretical Developments and Applications of Non-Integer Order Systems. Lecture Notes in Electrical Engineering, vol 357. Springer, Cham (2016). https://doi.org/ 10.1007/978-3-319-23039-9

Chapter 10

Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV Under Random Gaussian Disturbances

10.1 Introduction One of the suitable control schemes for the QUAV system is nonsingular fast terminal SMC (NFTSMC). Combined NFTSMC and FO, robustness against the disturbances, fast speed, singularity avoidance, fast convergence due to the existence of the fractional integral and derivative in the sliding manifolds, can be obtained. In addition, the use of the adaptive law with FONFTSMC, the upper bounds of the disturbances are coped. In this context, the FO based on SMC has been developed in various applications such as adaptive fuzzy fractional-order sliding mode control for permanent magnet linear synchronous motors [2], adaptive fractional-order nonsingular fast terminal SMC for robot manipulators [3], fractional-order fuzzy SMC for the deployment of tethered satellite system [4], adaptive FO super-twisting NFTSMC for cable-driven manipulators [5]. Therefore, for the QUAV system in the presence of disturbances, the FO with ANFTSMC methods need to be studies. According to the above discussion and the results developed in the Chap. 8, an adaptive FONFTSMC is proposed for the QUAV system under random disturbances. The remaining parts of this chapter are organized as: The proposed adaptive FO control scheme including the stability proof is presented in Sect. 10.2. Three simulation results compared with recent flight control methods are given in Sect. 10.3. Section 10.4 concludes this chapter.

10.2 Controller Design In this part, a new adaptive FONFTSMC is proposed for the QUAV under unknown random disturbances and uncertainties [1]. The design process in presented in the following subsections for attitude and position.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_10

213

214

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

10.2.1 Translational Subsystem The present work is inspired by [6, 7] and motivated by the problems of unknow disturbances including drag coefficients, moment inertial, and total mass. This part is dedicated to design the virtual control laws. Let’s introduce the FO nonsingular FTSM manifolds as for the translational subsystem as: s7 (t) = e8 (t) + D γx1 −1 cx1 e7 (t) px /qx + I γx2 cx2 e7 (t)ϕx sgn(e7 (t)) s9 (t) = e10 (t) + D s11 (t) = e12 (t) + D

γ y1 −1

γz1 −1

c y1 e9 (t)

cz1 e11 (t)

p y /q y

pz /qz

γ y2

+I

+I

γz2

ϕy

c y2 e9 (t) sgn(e9 (t)) ϕz

cz2 e11 (t) sgn(e11 (t))

(10.1) (10.2) (10.3)

where ci1 , ci2 , and ϕi for i = x, y, z are positive parameters. The parameters pi and qi satisfied that 1 < pi /qi < 2. The fractional orders γi1 and γi2 are 0 < γi1 , γi2 < 1. where sgn(x) equals 1, x ≤ 0 or 0, x < 0. Remark 10.1 In order to solve the singularity problem, the terminal sliding manifold can be rewritten as: si (t) = ei+1 (t) + D γi1 −1 ci1 ei (t) pi /qi + I γi2 ci2 ei (t)ϕi sgn(ei (t))

(10.4)

if 0.5 < pi /qi < 1. The singularity problem may occur in the TSMS (10.4) in this case. The following condition can be applied, which is known as the nonsingular TSMC. If so that if 1 < pi /qi < 2, then when si (t) −→ 0, u i is bounded. The dynamic of FONFTSM manifolds can be written as: s˙7 (t) = e˙8 (t) + D γx1 cx1 e7 (t) px /qx + I γx2 cx2 ϕx |e7 (t)|ϕx −1 e8 (t) c y1 e9 (t)

p y /q y

s˙11 (t) = e˙12 (t) + D cz1 e11 (t)

pz /qz

s˙9 (t) = e˙10 (t) + D

γ y1

γz1

+I

γ y2

+I

γz2

c y2 ϕ y |e9 (t)|

(10.5)

ϕ y −1

e10 (t)

(10.6)

ϕz −1

e12 (t)

(10.7)

cz2 ϕz |e11 (t)|

Thus, we shall give the form of the AFONFTSM controller for the position subsystem based on the fractional integral and derivative. By using the system dynamics, tracking error, and (10.5)–(10.7) and letting s˙7 (t) = s˙9 (t) = s˙11 (t) = 0; dx (t) = d y (t) = dz (t) = 0 in (10.5)–(10.7), the equivalent control inputs for the position loop can be obtained as: Vx0 = −ρ X X8 + x¨d (t) − D γx1 cx1 e7 (t) px /qx − I γx2 cx2 ϕx |e7 (t)|ϕx −1 e8 (t) Vy0 = −ρY X10 + y¨d (t) − D γ y1 c y1 e9 (t) p y /q y − I γ y2 c y2 ϕ y |e9 (t)|ϕ y −1 e10 (t) Vz0 = −ρ Z X12 + z¨ d (t) − D γz1 cz1 e11 (t) pz /qz − I γz2 cz2 ϕz |e11 (t)|ϕz −1 e12 (t) (10.8) d (t)

Assumption 10.1 The external disturbances/uncertainties pm and and assumed to be bounded respectively by D p (t) and Dηϕ (t), i.e.

dηϕ (t) are unknown I d p (t) ≤ D p (t) ≤ m

10.2 Controller Design

215

  ≤ Dηϕ (t) ≤  Dηϕ (t). We assume that the upper  bounds D p (t) and  D p  = b0k + b1k ei (t) + Dηϕ (t) contain only tracking errors and their derivatives,  b2k ei+1 (t) and  Dηϕ = b0 + b1 e j (t) + b2 e j + 1(t , for i = x, y, z, j = φ, θ, ψ, and k = 1, 3, 5, 7, 9, 11.

   D p  and

dηϕ (t) I

The reaching control inputs are configured as follows to ensure good tracking accuracy in the presence of unknown random disturbance/uncertainty and to ensure quick convergence of posttranscriptional states: Vx1 = −x1 sgn(s7 (t)) − x2 s7 (t) − (bˆ07 + bˆ17 e7 (t) + bˆ27 e8 (t) + μ)sgn(s7 (t)) Vy1 = − y1 sgn(s9 (t)) −  y2 s9 (t) − (bˆ09 + bˆ19 e9 (t) + bˆ29 e10 (t) + μ)sgn(s9 (t)) Vz1 = −z1 sgn(s11 (t)) − z2 s11 (t) − (bˆ011 + bˆ111 e11 (t) + bˆ211 e12 (t) + μ)sgn(s11 (t))

(10.9)

where i1 and i2 are the control parameters, and μ is a smaller parameter; and bˆ0k , bˆ1k , and bˆ2k for k = 7, 9, 11 are revised by adaptive rules, which are expressed as: b˙ˆ07 = ηx0 |s7 (t)| , b˙ˆ17 = ηx1 e7 (t) |s7 (t)| , b˙ˆ27 = ηx2 e8 (t) |s7 (t)| bˆ˙09 = η y0 |s9 (t)| , b˙ˆ19 = η y1 e9 (t) |s9 (t)| , b˙ˆ29 = η y2 e10 (t) |s9 (t)| b˙ˆ011 = ηz0 |s11 (t)| , b˙ˆ111 = ηz1 e11 (t) |s11 (t)| , b˙ˆ211 = ηz2 e12 (t) |s11 (t)|

(10.10) (10.11) (10.12)

where ηi0 , ηi1 , and ηi2 for i = x, y, z to be designed. As a result, the total virtual controllers obtained using the AFONFTSMC are: Vx = Vx0 + Vx1 , Vy = Vy0 + Vy1 , Vz = Vz0 + Vz1

(10.13)

with Vx0 , Vx1 , Vy0 , Vy1 , Vz0 , and Vz1 given by (10.8) and (10.9).

10.2.2 Stability Analysis for the Translational Loop Theorems 10.1 and 10.2 summarize and describe the ANFTSMC’s output for the position-loop problem. In addition, the translational subsystem’s stability is examined. Theorem 10.1 With Assumption 2.3, consider the x-subsystem control problem of the QUAV under disturbances. Design the AFONFTSMC method in (10.9) and their adaptation laws in (10.10), and the tracking errors will converge to zero in finite time. Proof To begin, the adaptive estimation errors are described as follows:  b07 = b17 = bˆ17 − b17 , and  b27 = bˆ27 − b27 . Assuming the time-derivative of bˆ07 − b07 ,  estimation errors as:  b˙ 07 = bˆ˙07 ,  b˙ 17 = bˆ˙17 , and  b˙ 27 = bˆ˙27 . p The Lyapunov function Vx is then defined as:

216

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

 1 1 2 2  , i = 1, 2, 3 bi7 s7 (t) + 2 2η xi i=1 2

Vxp =

(10.14)

with ηxi is a positive number. Using the time-derivative of s7 (t) defined in (10.1) and p using the AFONFTSMC in (10.13), the time-derivative of Vx can be written as V˙xp = s7 (t)˙s7 (t) + 

2  1   bi7 b˙ i7 η xi i=1

dx (t) − x¨d (t) + D γx1 cx1 e7 (t) px /qx m   2 1   γx2 ϕx −1 + I cx2 ϕx |e7 (t)| e8 (t) + bi7 b˙ i7 η xi i=1  = s7 (t) − x1 sgn(s7 (t)) − x2 s7 (t) − (bˆ07 + bˆ17 e7 (t) + bˆ27 e8 (t) = s7 (t) Vx1 + ρ X X8 +

+ μ)sgn(s7 (t)) +

  2 1   dx (t) + bi7 b˙ i7 m η i=1 xi

≤ −x1 |s7 (t)| − x2 s72 (t) − (bˆ07 + bˆ17 e7 (t) + bˆ27 e8 (t) + μ) |s7 (t)| 2  1   + |dx | s7 (t) + bi7 b˙ i7 η xi i=1

≤ −(x1 + μ) |s7 (t)| − x2 s72 (t) − (bˆ07 + bˆ17 e7 (t) + bˆ27 e8 (t)) |s7 (t)| 1 ˆ (b07 − b07 )b˙ˆ07 + (b07 + b17 e11 (t) + b27 e12 (t)) |s7 (t)| + ηx0 1 ˆ 1 ˆ + (b17 − b17 )b˙ˆ17 + (b27 − b27 )b˙ˆ27 (10.15) ηx1 ηx2 Adaptive rules (10.10) are being replaced in (10.15), it yields V˙xp ≤ −(x1 + μ) |s7 (t)| − x2 s72 (t)

(10.16)

The differential inequality can be rewritten as: √ V˙xp ≤ − 2(x1 + μ)Vxp 0.5 − 2x2 Vxp

(10.17)

√ Define λx1 = 2(x1 + μ) and λx2 = 2x2 . The differential inequality can be expressed in the same way as [1], resulting in V˙xp ≤ −λx1 Vxp 0.5 − λx2 Vxp

(10.18)

10.2 Controller Design

217

Both tracking errors of the x-subsystem will enter the sliding mode in a finite time p tx , as shown by the differential inequality above. tsxp ≤ tx0 +

  2 p 0.5 ln λx1 Vx λ(tx0 )+λx2 x2 λx1

(10.19)

Hence, the finite-time stability of the x-subsystem is achieved. Theorem 10.2 Consider the QUAV system under disturbances with Assumption 2.3. The tracking errors of the outer-loop converge to zero in finite time using the AFONFTSMC method in (10.9) and their adaptation laws (10.10)–(10.12). Proof To prove the second theorem, consider the Lyapunov function as follows: Vp =

 2  2  

 bi7 b2 b2 1 2 2 s7 (t) + s92 (t) + s11 (t) + + i9 + i11 2 2ηxi 2η yi 2ηzi i=1

(10.20)

Following that, the time-derivative of V p satisfies the following inequality after a simple calculus. V˙ p ≤ −(x1 + μ) |s7 (t)| − x2 s72 (t) − ( y1 + μ) |s9 (t)| 2 (t) −  y2 s92 (t) − (z1 + μ) |s11 (t)| − z2 s11

(10.21)

Positive gains should be used to pick control parameters; after that, V˙ p is negative.

10.2.3 Rotational Subsystem The proposed controller will be used to control rotational trajectory-tracking control in this segment. This section’s goal is to produce the yawing, pitching, and rolling signal torques that stabilize the QUAV system in the face of random disturbances and uncertainties. The following are the FONFT sliding manifolds for the attitude subsystem: s1 (t) = e2 (t) + D γφ1 −1 cφ1 e1 (t) pφ /qφ + I γφ2 cφ2 e1 (t)ϕφ sgn(e1 (t)) s3 (t) = e4 (t) + D

γθ 1 −1

s5 (t) = e6 (t) + D

γψ1 −1

cθ1 e3 (t)

pθ /qθ

cψ1 e5 (t)

+I

pψ /qψ

γθ 2

+I

ϕθ

cθ2 e3 (t) sgn(e3 (t))

γψ2

ϕψ

cψ2 e5 (t) sgn(e5 (t))

(10.22) (10.23) (10.24)

where c j1 , c j2 , and ϕ j for i = φ, θ, ψ are positive parameters. The parameters p j and q j satisfied that 1 < p j /q j < 2. The fractional operators γ j1 and γ j2 are 0 < γ j1 , γ j2 < 1. The e j (t) is the tracking error and its derivative e˙ j (t) = e j+1 (t)| j=φ,θ,ψ . To design FOANFTSMC laws for path following regulation of roll, pitch, and yaw angles, follow the same design procedure as in the previous section.

218

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

The rotational subsystem’s corresponding signal inputs are developed as follows:

1 − X4 X6 + ρ2 X4 + ρ3 X22 + φ¨ d (t) − D γφ1 cx1 e1 (t) pφ /qφ ρ1 − I γφ2 cφ2 ϕφ |e1 (t)|ϕφ −1 e2 (t) − φ1 sgn(s1 (t)) − φ2 s1 (t)  + (bˆ01 + bˆ11 e1 (t) + bˆ21 e22 (t) + μ)sgn(s1 (t)) (10.25)

u =

1 − ρ1 X2 X6 + ρ2 X2 + ρ3 X24 + θ¨d (t) − D γθ 1 cθ1 e3 (t) pθ /qθ ρ2 − I γθ 2 cθ2 ϕθ |e3 (t)|ϕθ −1 e4 (t) − θ1 sgn(s3 (t)) − θ2 s3 (t)  − (bˆ03 + bˆ13 e3 (t) + bˆ23 e4 (t) + μ)sgn(s3 (t)) (10.26)

u =

1 − ρ1 X2 X4 + ρ2 X26 + ψ¨ d (t) − D γψ1 cψ1 e5 (t) pψ /qψ ρ2 − I γψ2 cψ2 ϕψ |e5 (t)|ϕψ −1 e6 (t) − ψ1 sgn(s5 (t)) − ψ2 s5 (t)  − (bˆ05 + bˆ15 e5 (t) + bˆ25 e6 (t) + μ)sgn(s5 (t)) (10.27)

u =

with  j1 and  j2 are the control parameters. bˆ0k , bˆ1k , and bˆ2k for k = 1, 3, 5 are updated by the adaptive laws formulated as: b˙ˆ01 = ηφ0 |s1 (t)| , b˙ˆ11 = ηφ1 e1 (t) |s1 (t)| , b˙ˆ21 = ηφ2 e2 (t) |s1 (t)| b˙ˆ03 = ηθ0 |s3 (t)| , b˙ˆ13 = ηθ1 e3 (t) |s3 (t)| , b˙ˆ23 = ηθ2 e4 (t) |s3 (t)| b˙ˆ = η |s (t)| , b˙ˆ = η e (t) |s (t)| , b˙ˆ = η e (t) |s (t)| 05

ψ0

5

15

ψ1 5

5

25

ψ2 6

5

(10.28) (10.29) (10.30)

where ηi0 , η j1 , and η j2 for j = φ, θ, ψ to be designed. Theorem 10.3 Consider the rotational subsystem, where the control signals u , u  and u are formed as in (10.25)–(10.27), with the adaptation laws in (10.28)–(10.30). With the proposed FONFTS surfaces provided in (10.22)–(10.24), the closed-loop attitude subsystem will converge in a finite time for any initial condition. Proof The Lyapunov function can be used to demonstrate the stability of the attitude subsystem. V

ηϕ

  2 2 2 2   

 bi5 bi3 bi1 1 2 2 2 = s1 (t) + s3 (t) + s5 (t) + + + 2 2ηφi 2ηθi 2ηψi i=1

(10.31)

10.2 Controller Design

219

The differential inequality of the V ηϕ can be obtained using the same calculus as in the previous section for the translational subsystem. The time-derivative of V ηϕ can then be calculated using the differential inequality below. V˙ ηϕ ≤ −(φ1 + μ) |s1 (t)| − x2 s12 (t) − (θ1 + μ) |s3 (t)| − θ2 s32 (t) − (ψ1 + μ) |s5 (t)| − ψ2 s52 (t)

(10.32)

Positive gains should be used to pick control parameters; otherwise, V˙ ηϕ would be negative. In the presence of external disturbances, the rotational stability is then ensured by (10.25)–(10.27). Theorem 10.4 The ultimate control signals (10.13), (10.25), (10.26) applied to the QUAV dynamics, together with the ultimate adaptive laws built in (10.10)–(10.12) and (10.28)–(10.30) ensure the overall closed-loop QUAV system stability. Proof The overall QUAV system’s Lyapunov function is defined as: VQAV = V ηϕ + V p

(10.33)

The time-derivative of VQAV is, ˙ Q AV = V˙ ηϕ + V˙ p V

(10.34)

Using the Eqs. (10.21) and (10.32), we have, ˙ Q AV ≤ −(x1 + μ) |s7 (t)| − x2 s72 (t) − ( y1 + μ) |s9 (t)| V 2 (t) −  y2 s92 (t) − (z1 + μ) |s11 (t)| − z2 s11

− (φ1 + μ) |s1 (t)| − x2 s3 (t)2 − (θ1 + μ) |s3 (t)| − θ2 s32 (t) − (ψ1 + μ) |s5 (t)| − ψ2 s5 (t)2 ≤0

(10.35)

The overall stability of the QUAV system is shown by the above study. Remark 10.2 As can be shown, in the presence of random uncertainties/disturbances, the FO control method proposed in this paper achieves quick position and attitude regulation with high parameter accuracy. The computational cost of the FONFTSMC technique is comparable to most recent established works in Refs. [2–6]. As a result, the proposed method’s complexity is comparable.

10.3 Simulation Results Simulation experiments under a variety of scenarios are presented in the following section to demonstrate the superiority of the AFONFTSMC scheme proposed in this paper. Furthermore, detailed comparisons of feedback linear (FL) [8], backstepping

220

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

Fig. 10.1 The imitated drag coefficients

sliding mode control (BSMC) [9, 10], fractional-order BSMC [11], and nonsingular fast terminal sliding mode control [6] met the criteria. The initial conditions [0.05, 0.01, 0]m and [0.01, 0.01, 0.1]rad have been considered in all scenarios. Remark 10.3 The designed parameters of the controllers should be tuned in order to achieve satisfactory quadrotor trajectory-tracking output in the presence of random disturbances/uncertainties. We picked the best values for those parameters using the MATLAB optimization toolbox (see Ref. [12]). The position and attitude control parameters are taken into account in all proposed situations. cx1 = c y1 = 0.1797, cx2 = c y2 = 0.1769, cz1 = 4.4265, cz2 = 3.7049, cφ1 = cθ1 = cψ1 = 35.5118, cφ2 = cθ2 = cψ2 = 3.9926, x1 =  y1 = 2.6044, z1 = 16.1251, φ1 = θ1 = ψ1 = 156.4373, x2 =  y2 = 0.2204, z2 = 0.8876, φ2 = θ2 = ψ2 = 5.8670, ϕi = ϕ j = 3, pi = p j = 11, qi = q j = 10, ηi0 = η j0 = 0.01, ηi1 = η j1 = 0.01, ηi2 = η j2 = 0.01, γi1 = γ j1 = 0.99, and γi2 = γ j2 = 0.01. Three operating scenarios are illustrated to completely verify the performance and robustness of the AFONFTSMC methodology, as discussed below. In Scenario 1, the random variation of the rotational and translational drag coefficients is considered. This effect is presented in Fig. 10.1. The random variance of the rotational and translational drag coefficients is taken into account in Scenario 1. Figure 10.1 illustrates this effect. External disruptions are also introduced in the simulation to demonstrate the capability of the proposed controller. Scenario 2 adds random Gaussian disturbances to the influence of the drag coefficients considered in Scenario 1. I11 = I 11 (1 + Random[0, 30%]); I22 = I 22 (1 + Random[0, 30%]); I33 = I 33 (1 + Random[0, 30%]), m = m(1 + Random[0, 30%]); to further demonstrate

10.3 Simulation Results

221

Fig. 10.2 Gaussian random disturbances

the superiority of the proposed AFONFTSMC, external disturbances are also taking into account (Fig. 10.2). Remark 10.4 In order to approximate the fractional dynamics of the proposed controller, the CRONE method was used in numerical simulations. A transfer function of order 10 with a frequency range of 0.01 to 100 rad = s replaces the fractional expression.

222

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

10.3.1 With Drag Coefficients Uncertainties and Stochastic Disturbances The following are the external disruptions that are taken into account in this scenario: dx (t) = −(0.08 sin(0.11t − 3.0403) + 0.04 sin(0.5t − 13.5) + 0.008 sin(1.6t − 15π ) + 0.006 sin(0.3t − 8.6) m/s2 t ∈ [10, 30] = 0 m/s2 otherwise d y (t) = 0.05 sin(0.4t) + 0.05 cos(0.7t) m/s2 t ∈ [10, 50] = 0 m/s2 otherwise dz (t) = 0.05 cos(0.7t) + 0.07 sin(0.3) m/s2 t ∈ [0, 80] d (t) = 0.5 cos(0.4t) + 1 rad/s2 t ∈ [0, 80] d (t) = 0.5 sin(0.5t) + 1 rad/s2 t ∈ [0, 80] d (t) = 0.05sin(0.7 ∗ t) + 0.1 rad/s2 t ∈ [0, 80]

(10.36)

In this part, the QUAV is ordered to follow the reference yaw angle and position as follows:     π π m, yd (t) = 0.5 sin m (10.37) xd (t) = 0.5 cos 20 20   π m, d (t) = 0.5 rad z d (t) = 2 − 2 cos (10.38) 2 Figures 10.3, 10.4, 10.5, 10.6, 10.7, 10.8, 10.9 and 10.10 show the simulation results obtained in this scenario using various control methods and the proposed FONFTSMC method. Figures 10.3 and 10.4 show the position/attitude tracking control output. The AFONFTSMC can precisely monitor the ideal QUAV states in the presence of drag coefficient influences and major disturbances. In Fig. 10.5, the corresponding signal inputs are shown. These findings support the proposed controller’s effectiveness.

10.3 Simulation Results Fig. 10.3 Position tracking

Fig. 10.4 Attitude tracking

223

224 Fig. 10.5 Control inputs

Fig. 10.6 Tracking errors

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

10.3 Simulation Results

225

Fig. 10.7 3D trajectory tracking

Fig. 10.8 2D trajectory tracking

Both position and velocity tracking errors converge to zero in finite time, as shown in Fig. 10.6. The direction following results in 3-D and 2-D spaces are shown in Figs. 10.7 and 10.8, respectively. It can be shown that in the presence of uncertainties in drag coefficients/perturbation, all control approaches can successfully monitor trajectories. The position and attitude parameter adaptations are depicted in Figs. 10.9 and 10.10 respectively. As a result, the AFONFTSMC outperforms other compared techniques in terms of overshoot, settling/rising time, and steady-state error during the first phase of flight. Furthermore, the proposed approach will achieve better robustness results in the face of external disruptions and uncertainties.

10.3.2 With Drag Coefficients Uncertainties and Random Disturbances The negative effect of random disturbances on the dynamics of the QUAV is taken into account in this case to illustrate the efficacy of the control strategy suggested in this paper. Figure 10.11 shows the responses of the attitude and position of all controllers with extra random disturbances applied, while Fig. 10.12 shows the tracking efficiency of the QUAV using the five techniques (Figs. 10.13 and 10.14). Figure 10.11

226

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

Fig. 10.9 Parameters position

shows the responses of the attitude and position of all controllers with extra random disturbances applied, while Fig. 10.12 shows the tracking efficiency of the QUAV using the five techniques. The proposed controller’s 3-D and 2-D trajectories are shown in Figs. 10.15 and 10.16. It has been discovered that even in the presence of random disruptions, the QUAV’s path following can rapidly trace the desired ones. Figures 10.17 and 10.18 display the response of parameter estimations for attitude and position, respectively. As a result, the AFONFTSMC method outperforms other methods in terms of convergence speed, trajectory monitoring, and random disturbance rejection in comparative simulations.

10.3.3 With Random Uncertainties (Random Uncertainty 30% Added in Mass and Rotary Inertia) and External Disturbances To further check the suggested control strategy, the Random uncertainty 30% is applied to the nominal parameters of mass and rotary inertia in this case. External disruptions are also taken into account, which are expressed as:

10.3 Simulation Results Fig. 10.10 Parameters attitude

Fig. 10.11 Position tracking

227

228

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

Fig. 10.12 Attitude tracking

Fig. 10.13 Control inputs

dx (t) = 0.3 sin(2t) + 0.1 sin(3t) m/s2 d y (t) = 0.2 cos(2t) + 0.2 sin(2t) m/s2 dz (t) = 0.5 cos(2t) + 0.7 sin(3t) m/s2 d (t) = 0.9 cos(0.4t) + 2.2 sin(2t) rad/s2 d (t) = 0.8 sin(t) + 1.4 cos(2t) rad/s2 d (t) = 1.3 sin(5t) + 1.5 cos(3t) rad/s2 The following is the reference trajectory:

(10.39)

10.3 Simulation Results

Fig. 10.14 Tracking errors

Fig. 10.15 3D trajectory tracking

229

230 Fig. 10.16 2D trajectory tracking

Fig. 10.17 Parameters position

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

10.3 Simulation Results

231

Fig. 10.18 Parameters attitude

⎧ ⎨ 0 m t ∈ [0, 7] 

xd (t) = m t ∈ [7, 50] ⎩ 10 1 − cos 2π(t−7) 23  0 m t ∈ [0, 7] yd (t) = m t ∈ [7, 50] 5 sin 4π(t−7) 23 z d (t) = 7(1 − exp(−0.3t) m d (t) = 0.5 rad

(10.40)

Figures 10.19, 10.20, 10.21, 10.22, 10.23, 10.24, 10.25 and 10.26, display simulation results for the entire control techniques used in the contrast and the proposed control scheme. Position and attitude tracking, signal control inputs, tracking errors, path-following in 3-D and 2-D spaces, and parameter estimates can all be obtained

232

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

Fig. 10.19 Position tracking

Fig. 10.20 Attitude tracking

10.3 Simulation Results

233

Fig. 10.21 Control inputs

Fig. 10.22 Tracking errors

with high precision using Figs. 10.19, 10.20, 10.21, 10.22, 10.23, 10.24, 10.25 and 10.26. It can be shown that the random uncertainties have a significant effect on other approaches when the aforementioned perturbations influence the QUAV dynamics. Comparing Figs. 10.19 and 10.22, we can see that the proposed AFONFTSMC scheme can achieve more precise trajectory monitoring, stronger random uncertainty rejection, and disturbance rejection all at the same time. The properties of the proposed solution were further validated by the findings in 3D and 2-D spaces shown in Figs. 10.23 and 10.24. In conclusion, the AFONFTSMC approach proposed in this paper can achieve precise trajectory tracking in the presence of random uncertainties and/or disturbances.

234

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

Fig. 10.23 3D trajectory tracking

Fig. 10.24 2D trajectory tracking

10.4 Conclusions An adaptive fractional-order nonsingular fast terminal sliding mode control (AFONFTSMC) scheme for the QUAV system is proposed in this chapter to protect against random/multiple parametric uncertainties and external disturbances while improving path following efficiency. The proposed control solution based on adaptive laws compensates for the effects of drag coefficients, mass/inertia-moment variations, and spontaneous time-varying disturbances. The FONFTESMC used in the QUAV provided rapid FT convergence, improved trajectory tracking accuracy, and avoided chattering/singularity issues. Three separate examples are used to explore the robustness of the FO control method introduced in this paper in terms of random uncertainties/disturbances. Finally, three simulation results in various cases have shown that the AFONFTSMC controller outperforms recently published controllers in terms of quick finite-time convergence, smaller tracking errors, and rejection of uncertainties/disturbances.

10.4 Conclusions Fig. 10.25 Parameters position

Fig. 10.26 Parameters attitude

235

236

10 Robust FO Adaptive Nonsingular FTSMC for Uncertain QUAV …

References 1. Labbadi, M., Cherkaoui, M.: Adaptive fractional-order nonsingular fast terminal sliding mode based robust tracking control of quadrotor UAV with Gaussian random disturbances and uncertainties. IEEE Trans. Aerosp. Electron. Syst. 57(4), 2265–2277 (2021). https://doi.org/10.1109/ TAES.2021.3053109 2. Chen, S.Y., Chiang, H.H., Liu, T.S., Chang, C.H.: Precision motion control of permanent magnet linear synchronous motors using adaptive fuzzy fractional-order sliding-mode control. IEEE/ASME Trans. Mechatron. 24(2), 741–752 (2019) 3. Nojavanzadeh, D., Badamchizadeh, M.: Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators. IET Control Theory Appl. 10(13), 1565–1572 (2016) 4. Xu, S., Sun, G., Ma, Z., Li, X.: Fractional-order fuzzy sliding mode control for the deployment of tethered satellite system under input saturation. IEEE Trans. Aerosp. Electron. Syst. 55(2), 747–756 (2019) 5. Wang, Y., Chen, J., Yan, F., Zhu, K., Chen, B.: Adaptive super-twisting fractional-order nonsingular terminal sliding mode control of cable-driven manipulators. ISA Trans. 86, 163–180 (2019) 6. Labbadi, M., Cherkaoui, M.: Robust adaptive nonsingular fast terminal sliding-mode tracking control for an uncertain quadrotor UAV subjected to disturbances. ISA Trans. 99, 290–304 (2020) 7. Ahmed, S., Wang, H., Tian, Y.: Model-free control using time delay estimation and fractionalorder nonsingular fast terminal sliding mode for uncertain lower-limb exoskeleton. J. Vib. Control 24(22), 5273–5290 (2018) 8. Voos, H.: Nonlinear control of a quadrotor micro-UAV using feedback-linearization. In: 2009 IEEE International Conference on Mechatronics (2009) 9. Chen, F., Jiang, R., Zhang, K., Jiang, B., Tao, G.: Robust backstepping sliding-mode control and observer-based fault estimation for a quadrotor UAV. IEEE Trans. Ind. Electron. 63(8), 5044–5056 (2016) 10. Basri, M.A.M.: Design and application of an adaptive backstepping sliding mode controller for a six-DOF quadrotor aerial robot. Robotica 36(11), 1701–1727 (2018) 11. Shi, X., Cheng, Y., Yin, C., Dadras, S., Huang, X.: Design of fractional-order backstepping sliding mode control for quadrotor UAV. Asian J. Control 21(1), 156–171 (2018) 12. Freire, F.P., Martins, N.A., Splendor, F.: A simple optimization method for tuning the gains of PID controllers for the autopilot of Cessna 182 aircraft using model-in-the-loop platform. J. Control. Autom. Electr. Syst. 29(4), 441–450 (2018)

Chapter 11

Summary and Scope

11.1 Summary of Full Text Due to their outstanding combat capabilities in remote areas or hostile conditions, QUAVs have been widely used in civilian and military fields. However, since QUAVs structures are complex, modeling errors between the accurate model and the built model can occur, resulting in system uncertainties in QUAV systems. At the same time, since the flight environment of QUAV is constantly changing, the flight control system’s reliability is susceptible to external disturbances. The above-mentioned system uncertainties and external disruptions will not only reduce the flight control system’s control performance, but will also cause the QUAV systems to become unstable and insecure. Furthermore, the sliding mode control, backstepping and its related control approaches are suitable schemes for controlling QUAV system, finitetime control techniques are gaining in popularity. Therefore, studying the high order sliding mode control approaches with strong robustness to uncertainties and external disturbances on the flight control performance of QUAVs, so as to enhance the tracking performance, has become the key issues to be considered in the research of flight control techniques for QUAV systems. Also, combining the fractional calculus and sliding mode control techniques leading to improve the speed of the response and enhance the convergence of the states. In addition, the random variation of the external disturbances and uncertainties are considered, and the change of drag coefficient problems are less considered in the traditional design of flight control schemes for QUAV systems. Therefore, for the research on the advanced flight control methods of QUAV systems, robust fractional-order control is also one of the important problems in the design of QUAVs control techniques. In addition, studies on transient performance of QUAV systems such as convergence speed and overshooting are still relatively few. Excessive overshooting can cause QUAV actuators to exceed their physical limits, resulting in closed-loop QUAV system instability. As a result, when designing a stable advanced flight controller, the closed-loop system’s steadystate and transient output must be assured, leading to the problem of prescribed © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6_11

237

238

11 Summary and Scope

performance regulation. External disturbances, system uncertainties, and noise measurement are studied in the QUAV system, as well as robust advanced flight control methods under external disturbances in this book. The following are the book’s key research findings: 1. This book covers the research history and importance of QUAV control, as well as the research status of advanced flight control methods, adaptive flight control methods, and flight control based on fractional-order theory. 2. In order to better represent the impact of the wind field on mini-UAVs, dynamic modeling of multirotors was provided. A six DOF rigid body model is presented using the Newton-Euler formalism. Then, the main moments and forces acting on this vehicle have been included. In addition, the effect of wind was discussed by presenting a model for wind gusts and its effect was incorporated into the main multirotor model. The resulting equations are nonlinear and thus their direct application for the synthesis of control and estimation algorithms is complicated. To overcome this problem, some simplifications were considered in order to develop relatively simple control laws for application purposes by adopting the hierarchical control architecture. The overall system of the QUAV is rewritten in state space form to show the controls and wind influences on the QUAV. Using this state space representation, the control laws and algorithm estimation are calculated. 3. The development of flight control methods based on the super-twisting algorithm with an optimization method. This reformulation has increased the performance with respect to disturbances. Since the use of a discontinuous function in the ISMC leads to an undesirable reticence phenomenon, we introduced the modified super twisting control as a possible solution to overcome this drawback. This combination of ISMC and super twisting leads to a good robustness/performance trade-off. 4. The development of a hybrid control strategy composed of three different parts: the control of an altitude, a horizontal position and an attitude subsystem. First, the altitude subsystem is addressed using the adaptive backstepping technique in order to correctly estimate the perturbations on the vertical axis. In addition, the backstepping technique is designed to control the horizontal position. Second, a new robust full terminal slip mode controller has been constructed to stabilize the attitude subsystem. In addition, the proposed controller can quickly and accurately track the QUAV trajectory, the asymptotic stability of the closed-loop system is ensured. This nonlinear strategy guarantees the convergence of the tracking errors to zero in finite time when the sliding mode is reached. This type of control is known for its robustness to external disturbances. 5. The design of two new controllers, combined backstepping with adaptive fast terminal sliding mode and adaptive backstepping to control a QUAV. Adaptive laws were designed to adaptively estimate the upper bound of the cumulative uncertainty and adjust those control parameters that can improve the performance of the online system. Scenarios were considered with different inputs and external disturbances, and the performance of the designed controllers was

11.1 Summary of Full Text

6.

7.

8.

9.

239

studied in terms of the position and attitude tracking problem. The simulation results demonstrated the superior performance of the proposed controllers. From the simulation results, it can be concluded that the use of a fast terminal sliding surface converges the system states and the position tracking error to the equilibrium point and to the zero point. Furthermore, the simulation results show that the approaches have very high resistance to perturbations. This is due to the fact that a combination of two of the robust controllers, i.e., AFTSMC and backstepping control, were used in the design procedure. The design of a new RNFTSMC technique to solve the problem of trajectory control of an uncertainty QUAV under disturbances. In addition, the fast convergence of all state variables was achieved and the influence of the reluctance effect in the SMC was eliminated, while the online parameter estimation was presented and the singularity problem of the TSMC was avoided. The effectiveness of the proposed RNFTSMC approach has been demonstrated in multiple test scenarios (constant external disturbances, parametric uncertainties, measurement noise and time-varying external disturbances). The results presented in this section show that the RNFTSMC approach proposed in this work has good tracking of the desired trajectory, fast finite-time convergences of the slip surfaces with higher accuracy, zero steady state of tracking errors, and a high level of robustness against external disturbances compared to several control methods presented in the literature. In this study, a new scheme definition for controlling a QUAV that is subjected to random disturbances and uncertainties is presented. To push the QUAV states to equilibrium in a given time of convergence, the proposed solution uses a robust adaptive global time-varying sliding mode control algorithm. As a result, the reaching step is avoided by applying two unknown coefficients to the QUAV system’s sliding surfaces. Adaptive algorithms are proposed to reject the upper bound of random disruptions and uncertainties that influence QUAV dynamics. The RNFTSMC proposed in this paper provides online calculation of these upper bounds. The obtained results show that the proposed controller can monitor the desired trajectory even when random disturbances are present. The development of robust fractional-order controllers for the QUAV control system allows for improved flight path tracking under complex conditions. Fractional computation combined with sliding mode theory and Lyapunov stability to design efficient control schemes for the QUAV. Two fractional order control approaches based on sliding mode control have been designed. In addition, the problems of load variation and time variation of drag coefficients are solved using the fractional order super-twisting PID sliding mode control algorithm. The use of FO surfaces leads to a fast convergence of the system state variables in the presence of stochastic disturbances. An original nonlinear fractional order controller is proposed for the QUAV system. A new global fractional order sliding mode control method is proposed to stabilize the system globally in closed loop. The proportional action depends on a nonlinear transformation of the sliding variable, while the fractional order integral action rejects non-differentiable disturbances, such as turbulence effects and

240

11 Summary and Scope

wind gusts, by imposing a finite-time sliding motion. Finally, simulation results show that the proposed controllers are capable of performing trajectory tracking tasks under conditions of model uncertainty and wind disturbance with accuracy and speed, compared to existing robust approaches. 10. In order to protect the QUAV copter from random/multiple parametric uncertainties and external disruptions, an adaptive fractional-order nonsingular fast terminal sliding mode control (AFONFTSMC) scheme is proposed. The proposed control solution based on adaptive laws compensates for the effects of drag coefficients, mass/inertia-moment fluctuations, and spontaneous time varying disturbances. The FONFTESMC used in the QUAV provided rapid FT convergence, improved trajectory tracking accuracy, and avoided chattering/singularity issues. Three separate examples are used to explore the robustness of the FO control method introduced in this part in terms of random uncertainties/disturbances. Finally, three simulation results in various cases have shown that the AFONFTSMC controller outperforms recently published controllers in terms of fast finite-time convergence and smaller tracking errors, and rejection of the uncertainties/disturbances.

11.2 Future Research Prospect The advanced flight control of a QUAV has been studied in this book, with a focus on the control issue of a dynamics model of a QUAV with external disturbances, and some research results have been obtained. However, due to time and capacity constraints, there is still much to be discovered and mastered in the advanced finitetime flight control system of QUAVs: 1. Regarding the multi-rotor model, the model formulation can be further improved by adding some neglected elements of the terms that are not tested or validated this research. In addition, the four motors that drive the four rotors, have some difference due to the production process. 2. All the algorithms introduced in this work have been validated by simulations. Therefore, an experimental validation must be performed to confirm the results of the simulations. 3. In this work, all state variables are assumed to be measurable, which could not be the case in real systems. It then becomes necessary to develop a state observer to overcome this problem. 4. In this work, we proposed fractional order controllers based on the SMC and the super-twisting algorithm to develop adaptive controllers with fast nonlinear end surfaces to increase the performance of flight trajectory tracking. 5. We have proposed adaptive controls based on NFTSMC, another possible extension of this work could be the design of a time-varying NFTSM control technique to eliminate the reaching phase and optimize the control effort. 6. In addition, future work will focus on fault diagnosis and fault tolerant control of multirotor systems with actuator and sensor faults.

Appendix A

Simulation Parameters

A.1 Additions to the Simulation Parameters of the Model See Table A.1. Table A.1 QUAV parameters Parameter

Value

Parameter

Value

Parameter

Value

Parameter

Value

g(m/s 2 )

9.81

k2 (N /m/s)

5.5670e-4

m(kg )

0.486

k3 (N /m/s)

5.5670e-4

I x x (kg.m 2 )

3.827e-3

k4 (N /m/s)

5.5670e-4

I yy (kg.m 2 )

3.827e-3

k5 (N /m/s)

5.5670e-4

(kg.m 2 )

7.6566e-3

k6 (N /m/s)

5.5670e-4

Ir (kg.m 2 )

2.8385e-5

k p (N .s 2 )

2.9842e-3

k1 (N /m/s)

5.5670e-4

cd (N .m.s 2 )

3.2320e-2

–

–

–

–

Izz

A.2 Complements to the Control Law Simulation Parameters See Tables A.2, A.3, A.4, A.5, A.6, A.7, A.8, A.9, A.10.

Table A.2 Control System Parameters (ST-PID-ISMC) Parameters

Values

Parameters

Values

Parameters

Values

Parameters

K p , K p , K p

6.89

K i , K i , K i

0.003

K d , K d , K d

0.32

K px , K py , K pz

2.47

Ki x , Ki y , Ki z

0.002

K d x , K yd , K dz

0.53

K 1 , K 1 , K 1

6.92

K 2 , K 2 , K 2

42.16

K s , K s , K s

6.6037

K 1x , K 1y , K 1z

1.134

K 2x , K 2y , K 2z

1.2059

K sx , K sd , K sz

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6

Values

1.1894

241

242

Appendix A: Simulation Parameters

Table A.3 Control System Parameters (B-ITSMC) Parameter

Value

Parameter

Parameter

Value

Parameter

Value

Parameter

Value

cx1

1.68

cx2

Value 2.34

c y1

1

c y2

2.25

cz1

6.4

cz2

20

αi

59.78

βi

974.57

pi

54

qi

52

Table A.4 Control System Parameters (FO-ST-PID-SMC) Paramtre Valeur Paramtre Valeur Paramtre ki0 ki2 εi2 λi bi1

0.7474 0.1374 1 0.9 0.6617

k j0 k j2 ε j2 λj b j1

7.6041 0.2133 1 0.9 0.11

ki1 εi1 αi qi bi2

Valeur

Paramtre

Valeur

0.0144 0.6617 0.9 0.1 1

k j1 ε j1 αi qj b j2

0.0375 71.2189 0.9 0.1 0.1

Table A.5 Control System Parameters (FOGSMC) Parameter

Value

Parameter

σ,,

0.8

σx,y,y

Value 0.6

αx,y,y,,,

0.99

β,,

0.1124

βx,y,y

0.0171

q,,

0.7

qx,y,y

0.3640

ε,,

35.6089

εx,y,y

3.3689

γx,y,y

γ,,

39.1590

16.3958

Table A.6 Control System Parameters Parameters Values c , c , c ϒ , ϒ , ϒ q , q , q γ7 , γ9 , γ11

0.0527 9.3414 4.3864 2.0000

Parameter

8.7406 11.7180 0.5 6.2942e-04 1.2 2

b j+1 j μ0 j μ2 j μ0n μ2n

Parameter

Parameters

Values

 ,  ,  p , p , p γ , γ , γ cx1 , c y1 , cz1

0.2146 2.4429 17.4513 3.2270

Table A.7 Control System Parameters (RANFTSMC) Paramtre Valeur Paramtre Valeur bj cj hj μ1 j ϒj ϒn

Value

0.4838 1.0885 0.0118 0.01 0.5 0.01

Value

Paramtre

Valeur

bn bn+1 cn n μ1n hn

0.03 0.8 1.21 5/3 0.001 0.5

Appendix A: Simulation Parameters

243

Table A.8 Control system parameters MSTFTSMC Parameters Value Parameters K pφ , K pθ , K pψ K dφ , K dθ , K dψ μ7 , μ9 , μ11 K 1φ , K 1θ , K 1ψ K sφ , K sθ , K sψ K 2x , K 2y , K 2z γ1 , γ3 , γ5 C1i , C2i

6.89 0.32 2 6.92 6.6037 1.2059 10 26

μ1 , μ3 , μ5 K px , K py , K pz K d x , K yd , K dz K 2φ , K 2θ , K 2ψ K 1x , K 1y , K 1z K sx , K sd , K sz γ7 , γ9 , γ11 C3i , C4i

Value 2 2.47 0.53 42.16 1.134 1.1894 5 0.5

Table A.9 Control system parameters AGNSMC and PID super-twisting Parameters Value Parameters Value λ1 , λ3 , λ5 β1 , β3 , β5 ξ1 , ξ3 , ξ5 H1 , H3 , H5 1 , 3 , 5

30 10 3 1.9 0.001

Table A.10 Control system parameters Parameter Value σ,, γx,y,y,,, βx,y,y qx,y,y εx,y,y γ,,

0.8 0.99 0.0171 0.3640 3.3689 39.1590

λ7 , λ9 , λ11 β7 , β9 , β11 ξ7 , ξ9 , ξ11 H7 , H9 , H11 7 , 9 , 11

4 5 1 1.1 0.1

Parameter

Value

σx,y,y β,, q,, ε,, γx,y,y μi

0.6 0.1124 0.7 35.6089 16.3958 0.05

Appendix B

Simulations of Non-integer Systems and Stability in the Lyapunov Sense

B.1 Simulations of Non-integer Order Systems In the literature, two approaches are most commonly used to simulate non-integer order systems in the time domain. The first approach is based on a discrete numerical approximation derived from the Grünwald definition in order to obtain a discrete integer model. The second approach is based on the replacement of the non-integer derivation operator by a distribution of recursive poles and zeros in order to obtain a continuous integer model, which is the “CRONE” approximation OLMN:00. In this book, numerical examples are simulated using a “Toolbox” based on the “CRONE” approximation, which is available on the laboratory website of the CRONE research team. This Toolbox allows to add a predefined non-integer order integrator or derivator block in the editor library, where the user has the possibility to choose the desired parameters such as the frequency interval, the number of steps as well as the approximation approach as it can be seen in the Fig. B.1. This block can be combined and linked with other blocks in the editor’s libraries using lines that represent signals to establish mathematical relationships between the system components, thus creating a new model representative of the system in pseudo-state space. This flexibility has an advantage in the simulation of delayed non-integer order systems, as it is sufficient to use the delay block, in order to obtain a delayed state that will be looped later to the input of the non-integer order integrator −α. Figure B.2 shows a typical SISO model of the realization of a delayed non-integer order system in the Simulink-Matlab environment. The same principle can be applied for MIMO systems of order greater than or equal to 2.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Labbadi et al., Advanced Robust Nonlinear Control Approaches for Quadrotor Unmanned Aerial Vehicle, Studies in Systems, Decision and Control 384, https://doi.org/10.1007/978-3-030-81014-6

245

246

Appendix B: Simulations of Non-integer Systems and Stability in the Lyapunov Sense

Fig. B.1 Non-integer order integrator or derivator block α

Fig. B.2 Representation of a non-integer order SISO system with time varying delay

B.2 Stability in the Lyapunov Sense In the literature, we can quote different approaches concerning the analysis of the stability of dynamical systems of integer order, and in particular the one based on the theory developed by Lyapunov. Without loss of generality, we consider in the following x ∗ = 0 as the equilibrium point. Lyapunov’s theory is an important tool in automatic control, allowing to conclude on the stability of an equilibrium point of

Appendix B: Simulations of Non-integer Systems and Stability in the Lyapunov Sense

247

the system. It is based on the existence of functions, verifying certain criteria, and which represent in a certain way the energy of the system. The following gives more details on these functions and on the stability in the Lyapunov sense. These results, now classical, can be found in the references [6–8]. For example. In this section, we recall some concepts on the stability of dynamical systems. The analysis of the stability of a dynamical system consists in studying its behavior near the equilibrium point x ∗ = 0. This is done by analyzing the trajectory of the state of the system when its initial state is close to an equilibrium point or trajectory [1–3]. The stability of nonlinear dynamical systems of non-integer order is very complex and different from the linear system of non-integer order. The difference is that nonlinear systems may have several equilibrium points or even limit cycles. For nonlinear systems of non-integer order, there are many definitions of stability. As mentioned in Mat:96, exponential stability cannot be used to characterize the asymptotic stability of non-integer order systems. We consider the set of non-linear systems described by the following dynamic equation: x˙ = f (t, x)

(B.1)

where x ∈ Rn , we denote x = 0 as the equilibrium point ( f (0) = 0). The function f (t, x) is Lipschitzian with respect to x. Let t0 = 0 be the initial time. Let V (x, t): Rn × R+ −→ R+ a continuous function. The derivative of V (t, x) along the trajectory of the system (B.1) is ∂ V (t, x) ∂ V (t, x) + f (t, x) V˙ (t, x) = ∂t ∂x

(B.2)

Definition B.1 [2, 5] The equilibrium point x = 0 of the system (B.1) is stable si et seulement si     ∀ε > 0, ∃ϒ(ε) > 0 x ∗  t.q ⇒ (t, x ∗ ) < ε, ∀t > 0

(B.3)

Definition B.2 [2, 5] The equilibrium point x = 0 of the system (B.1) is attractive if ∀ϒ > 0 t.q. lim x(t, x ∗ ) = 0 t→+∞

(B.4)

Definition B.3 [2, 5] The equilibrium point x = 0 of the system (B.1) is asymptotically stable if it is stable and attractive. Definition B.4 [2, 5] The equilibrium point x = 0 of the system (B.1) is a locally exponentially stable equilibrium point if there exist two strictly positive constants j and ι such that:

248

Appendix B: Simulations of Non-integer Systems and Stability in the Lyapunov Sense

  (t, x ∗ ) ≤ j exp (−ιt), ∀t ≥ 0, ∀x ∗ ∈ Br

(B.5)

When Br = Rn , we speak of global exponential stability. Definition B.5 Let V (x, t): mathbb R n × R+ → R+ be a continuous function. V is called proper and positive definite if: • ∀ t ∈ R+ , ∀t ∈ Rn , x = 0 V (t, x) • ∀ t ∈ R+ , V (t, x) = 0 =⇒ x = 0; • ∀ t ∈ R+ , lim V (x, t) = ∞;

>

0;

t→+∞

• there exists a constant d > 0 and a function var phi of class mathb f K such that ϕ( x )  V (t, x), ∀t > 0, ∀ x  d.

(B.6)

Definition B.6 A function V (t, x) of class C 1 is a local (resp. global) Lyapunov function in the broad sense for the system (B.1) if it is positive definite proper and if there exists a neighborhood of the origin V0 such that ∀x ∈ V0 , (resp. ∀x ∈ Rn ) ∂ V (t, x) ∂ V (t, x) + f (t, x) ≤ 0 V˙ (t, x) = ∂t ∂x

(B.7)

If dot V (t, x) < 0, then V is called the Lyapunov function in the strict sense for the system (B.1). Definition B.7 If the system (B.1) admits a local Lyapunov function in the broad sense (resp. in the strict sense) then the origin is a locally stable (resp. asymptotically stable). If the Lyapunov function is global, then we speak of global stability (resp. global asymptotic stability). Definition B.8 The equilibrium point x = 0 of the system (B.1) is a locally exponentially stable equilibrium point if there are constants alpha, β, γ , an integer p > 0 and a function V (t, x): V0 × R+ → of class C1 such that, ∀x ∈ V0 : • α x p ≤ V (x, t) ≥ β x p • V˙ (t, x) < −γ V (x, t) If V0 = Rn , then the origin of (B.1) is globally exponentially stable. There is, in general (except in the linear case), no systematic method to find a candidate Lyapunov function. It is however quite classical to use a quadratic Lyapunov function, of the type V (t, x) = x(t)T P x(t), where the matrix P is symmetric and positive definite. This choice has the advantage of being simple to implement. However, it generally induces very conservative conditions. Moreover, if no suitable function is found, one cannot say anything about the eve.

Appendix B: Simulations of Non-integer Systems and Stability in the Lyapunov Sense

249

References 1. Vidyasagar, M.: On matrix measures and convex Liapunov functions. J. Math. Anal. Appl. 62, 90–103 (1969) 2. Vidyasagar, M.: Nonlinear Systems Analysis, 2nd edn. Prentice Hall, Englewood Cliffs, New Jersey (1993) 3. Khalil, H.K.: Nonlinear Systems. Macmillan Publishing Company, New York (1992) 4. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of the IEEE-IMACS Systems, Man, and Cybernetics Conference, Lille, France 5. Sastry, S.: Nonlinear Systems: Analysis Stability and Control. Springer, New York (1999) 6. Lyapunov, A.M.: The general problem of stability of motion. Ph.D. thesis, Kharkov Mathematical Society (1892) 7. Lyapunov A.M.: The general problem of stability of motion. Ph.D. thesis, Kharkov Mathematical Society, 1892. Publie dans IJC, Lyapinov Centenary Issue, vol. 55, no. 3, March (1992) 8. Lyapunov, A.M.: The general problem of stability of motion. Int. J. Contr. 55, 531–773 (1992). Lyapunov Centenary Issue