Mechatronics and robotics engineering for advanced and intelligent manufacturing

Lecture Notes in Mechanical Engineering Dan Zhang Bin Wei Editors Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing Tai ngay!!! Ban co the xoa dong chu nay!!! Lecture Notes in Mechanical Engineering About this Series Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality Original research reported in proceedings and post-proceedings represents the core of LNME Also considered for publication are monographs, contributed volumes and lecture notes of exceptionally high quality and interest Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering Topics in the series include: • • • • • • • • • • • • • • • • • Engineering Design Machinery and Machine Elements Mechanical Structures and Stress Analysis Automotive Engineering Engine Technology Aerospace Technology and Astronautics Nanotechnology and Microengineering Control, Robotics, Mechatronics MEMS Theoretical and Applied Mechanics Dynamical Systems, Control Fluid Mechanics Engineering Thermodynamics, Heat and Mass Transfer Manufacturing Precision Engineering, Instrumentation, Measurement Materials Engineering Tribology and Surface Technology More information about this series at http://www.springer.com/series/11236 Dan Zhang Bin Wei • Editors Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing 123 Editors Dan Zhang Department of Mechanical Engineering, Lassonde School of Engineering York University Toronto, ON Canada Bin Wei Faculty of Engineering and Applied Science University of Ontario Institute of Technology Oshawa, ON Canada ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-3-319-33580-3 ISBN 978-3-319-33581-0 (eBook) DOI 10.1007/978-3-319-33581-0 Library of Congress Control Number: 2016943792 © Springer International Publishing Switzerland 2017 This work is subject to copyright All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface The 2nd International Conference on Mechatronics and Robotics Engineering, ICMRE 2016, was held in Nice, France, during February 18–22, 2016 The aim of ICMRE 2016 is to provide a platform for researchers, engineers, academics as well as industry professionals from all over the world to present their research results and development activities in the area of mechatronics and robotics engineering This book introduces recent advances and state-of-the-art technologies in the field of robotics engineering and mechatronics for the advanced and intelligent manufacturing This systematic and carefully detailed collection provides a valuable reference source for mechanical engineering researchers who want to learn about the latest developments in advanced manufacturing and automation, readers from industry seeking potential solutions for their own applications, and those involved in the robotics and mechatronics industry This proceedings volume contains 36 papers that have been selected after review for oral presentation These papers cover several aspects of the wide field of advanced mechatronics and robotics concerning theory and practice for advanced and intelligent manufacturing The book contains three parts, the first part focuses on the Design and Manufacturing of the Robot, the second part deals with the Mechanical Engineering and Power System, and the third part investigates the Automation and Control Engineering We would like to express grateful thanks to our Program Committee members and Organization Committee members of the 2nd International Conference on Mechatronics and Robotics Engineering, special thanks to the keynote speakers: Prof Alexander Balinsky, Cardiff University, UK, Prof Farouk Yalaoui, Université de Technologie de Troyes, France, Prof Dan Zhang, York University, Canada, and Prof Elmar Bollin, Offenburg University of Applied Sciences, Germany We would like to express our deep appreciation to all the authors for their significant contributions to the book Their commitment, enthusiasm, and technical expertise are what made this book possible We are also grateful to the publisher for supporting this project and would especially like to thank Arumugam Deivasigamani, Anthony Doyle, and Janet Sterritt for their constructive assistance and cooperation, v vi Preface both with the publishing venture in general and the editorial details We hope that the readers find this book informative and useful Finally, the editors would like to sincerely acknowledge all the friends and colleagues who have contributed to this book Toronto, Canada Oshawa, Canada February 2016 Dan Zhang Bin Wei Contents Part I Design and Manufacturing of the Robot Critical Review and Progress of Adaptive Controller Design for Robot Arms Dan Zhang and Bin Wei Stiffness Analysis of a Planar 3-RPS Parallel Manipulator Bo Hu, Chunxiao Song and Bo Li 13 Overview of an Engineering Teaching Module on Robotics Safety Dan Zhang, Bin Wei and Marc Rosen 29 Mobile Robot Applied to QR Landmark Localization Based on the Keystone Effect Vibekananda Dutta A Collective Behaviour Framework for Multi-agent Systems Mehmet Serdar Güzel and Hakan Kayakökü 45 61 Kinematic Performance Analysis of a Hybrid-Driven Waist Rehabilitation Robot Bin Zi, Guangcai Yin, Yuan Li and Dan Zhang 73 Admittance Filter Parameter Adjustment of a Robot-Assisted Rehabilitation System (RehabRoby) Fatih Ozkul, Duygun Erol Barkana and Engin Maşazade 87 Continuum Robot Surfaces: Smart Saddles and Seats Ian D Walker 97 Structural Parameter Identification of a Small Robotic Underwater Vehicle 107 Martin Langmajer and Lukáš Bláha vii viii Contents Using Online Modelled Spatial Constraints for Pose Estimation in an Industrial Setting 123 Kenneth Korsgaard Meyer, Adam Wolniakowski, Frederik Hagelskjær, Lilita Kiforenko, Anders Glent Buch, Norbert Krüger, Jimmy Jørgensen and Leon Bodenhagen Comparison Study of Industrial Robots for High-Speed Machining 135 Alexandr Klimchik, Alexandre Ambiehl, Sebastien Garnier, Benoit Furet and Anatol Pashkevich Adaptive Robust Control and Fuzzy-Based Optimization for Flexible Serial Robot 151 Fangfang Dong, Jiang Han and Lian Xia Wired Autonomous Vacuum Cleaner 167 Emin Faruk Kececi and Fatih Kendir Human Safety Index Based on Impact Severity and Human Behavior Estimation 177 Gustavo Alfonso Garcia Ricardez, Akihiko Yamaguchi, Jun Takamatsu and Tsukasa Ogasawara Swarm Robots’ Communication and Cooperation in Motion Planning 191 Khiem N Doan, An T Le, Than D Le and Nauth Peter Indoor Localization for Swarm Robotics with Communication Metrics Without Initial Position Information 207 Türker Türkoral, Özgür Tamer, Suat Yetiş and Levent Çetin Multi-objective Optimization of a Parallel Fine-tuning Manipulator for Segment Assembly Robots in Shield Tunneling Machines 217 Guohua Cui, Haidong Zhou, Yanwei Zhang and Haibin Zhou An Imitation Framework for Social Robots Based on Visual Input, Motion Sensation, and Instruction 241 Mohsen Falahi, Faraz Shamshirdar, Mohammad Hosein Heydari and Taher Abbas Shangari Part II Mechanical Engineering and Power System New Reactionless Spatial Grasper Design and Analysis 257 Dan Zhang and Bin Wei Tracking and Vibration Control of a Carbon Nanotube Reinforced Composite Robotic Arm 265 Mohammad Azadi and Behzad Hasanshahi Contents ix Synthesis and Analysis of Pneumatic Muscle Driven Parallel Platforms Imitating Human Shoulder 275 Xingwei Zhao, Bin Zi and Haitao Liu Conceptual Design of Energy Efficient Lower Extremity Exoskeleton for Human Motion Enhancement and Medical Assistance 289 Nazim Mir-Nasiri A New Algorithm for Analyzing Method of Electrical Faults of Three-Phase Induction Motors Using Duty Ratios of Half-Period Frequencies According to Phase Angle Changes 303 YoungJin Go, Myoung-Hyun Song, Jun-Young Kim, Wangrim Choi, Buhm Lee and Kyoung-Min Kim Mathematical Foundations and Software Simulation of Stress-Strain State of the Plate Container Ship 319 Anatoliy Nyrkov, Sergei Sokolov, Valery Maltsev and Sergei Chernyi Kalman Filtering for Precise Mass Flow Estimation on a Conveyor Belt Weigh System 329 Tauseef Rehman, Waleed Tahir and Wansoo Lim Part III Automation and Control Engineering Stiffness Analysis and Optimization for a Bio-inspired 3-DOF Hybrid Manipulator 341 Dan Zhang and Bin Wei Robust Gust Rejection on a Micro-air Vehicle Using Bio-inspired Sensing 351 William A Dean, Badri N Ranganathan, Ivan Penskiy, Sarah Bergbreiter and J Sean Humbert Development of Guidance, Navigation and Control System Using FPGA Technology for an UAV Tricopter 363 Arturo Cadena, Ronald Ponguillo and Daniel Ochoa Fault Recoverability Analysis via Cross-Gramian 377 Hamid Reza Shaker Implementation of RFID-Based Car Ignition System (CIS) in Kazakhstan 387 Nurbek Saparkhojayev, Askar Kurymbayev and Azret Akhmetov Design and Development of a Self-adaptive, Reconfigurable and Low-Cost Robotic Arm 395 Kemal Oltun Evliyaoğlu and Meltem Elitaş Depth Control of AUV Using a Buoyancy Control Device 439 Fig Depth control process PID control which is used to freeze the robot there In this paper, the TcTr control is mainly focused CTD sensor enables to measure the depth (D) and the vertical speed (S) of the robot At every moment, it is possible to have an estimation of the time needed to reach the target depth (Tr) using Eq Tr ẳ Dt  Dị=Sị 4ị The buoyancy variation model established in Sect of this paper enables to estimate the time (Tc) needed for changing the robot buoyancy from it is current value to the neutral buoyancy The first step is based on the continuous estimation of the time to reach (Tr) and the time to change (Tc) while adjusting decreasing the buoyancy value, till the stop condition is reached • If the estimation of the Tr > Tc it means that it is possible to increase the vertical speed of the robot since we have enough time to change the buoyancy to its neutral level Hence, we decrease the target buoyancy value • In the case where the Tr = Tc or less, then it means we have just enough time to change the buoyancy to the neutral level before the robot reaches its target depth Hence, we start to increase the buoyancy of the robot progressively We introduce Tm, which corresponds to the time error margin used to compensate eventual inaccuracies in the buoyancy device model The condition for stopping to increase the robot vertical speed becomes Tr ≤ Tc + Tm Figure shows the TcTr control inputs and outputs diagram and Fig illustrates it flow chart Fig TcTr input/output diagram 440 M Choyekh et al Fig TcTr control flowchart Table TcTr parameters configuration 4.2 Parameter Value Target depth (Dt) Depth margin (Dm) Neutral buoyancy (Bn) Buoyancy margin (Bm) Time margin (Tm) 400 m 1m 51.0 % 2.0 % 30 s Experiments Results Sea experiments were conducted in Toyama Bay, Japan on June 11th, 2015 on the board of the Wakashio-maru of the National Institute of Technology, Toyama College The experimental site was located at 36° 52′N, 137° 11′E with a water depth of around 560 m In this sea experiment, the depth control with time estimation scheme was deployed, with a target depth of 400 m The configuration of the depth control parameters is listed in Table As shown in Fig 8, the TcTr control succeeded to bring the robot to the target depth smoothly with a vertical speed near m/s The buoyancy device was able to adjust the buoyancy so that it could achieve neutral buoyancy when reaching the target depth Progressive Depth Control One of SOTAB-I operating modes is the photograph mode To take pictures of the blow out position, SOTAB-I needs to approach the seabed around m altitude Hence, an altitude control is needed One way is to use thrusters to control the altitude However, there is a risk that they mix up the sediments on the seabed which influences the transparency of the water In addition, it will disturb the water flow, causing some inaccuracies in the water current measurement In this section we suggest a second method that consists in the adaptation of the depth control algorithm detailed in the previous section to altitude control This is possible by combining CTD sensor depth information with the DVL altitude measurement Depth Control of AUV Using a Buoyancy Control Device 441 Fig At-sea experiments results of the depth control Fig Altitude control flowchart when bottom tracking is active enabling the measurement of water depth Figure illustrates the flow chart of the progressive depth control Step 1: The robot dives with a fast speed until the depth of the robot reaches the depth limit (Dcert) of the “certain zone” The “Certain Zone” is the zone 442 M Choyekh et al in which we are sure that the robot will not be within the bottom tracking altitude (Amax) equal to 24 m Dcert is input by the user on board before starting the descent In this first step, the depth control with time estimation scheme is used At first, the buoyancy control device will decrease the buoyancy After the buoyancy level becomes lower than the neutral buoyancy of the robot, SOTAB-I will start diving The buoyancy control device will continue to reduce its buoyancy level down to 20 %, which is set as the minimum buoyancy level of the buoyancy control device, with maximum speed Then it will increase again its buoyancy level close to the neutral buoyancy level The purpose of this strategy is that the robot should have enough time to change its buoyancy level to its neutral buoyancy when reaching the target depth In this step, the target depth Dt is set as a fixed value, which is equal to the certain zone limit plus the DVL range Amax, minus the target altitude (At) Step 2: When the robot reaches the certain zone limit, the variable target depth control is started After passing the certain zone limit, there is a chance that the DVL will detect the seabed and output its current altitude (A) Therefore, from this viewpoint, the buoyancy change is limited up to the time needed to reach the target depth Tr In this step, the depth control with time estimation is still being used However, the target depth Dt is set equal to the current depth D plus the DVL range Amax minus the target altitude At The target depth will continuously change as the depth D of the robot decreases Hence, it is a depth control with variable target depth At this point, the buoyancy level of SOTAB-I is already close to the neutral buoyancy Therefore, there will not be much change in the buoyancy level to ensure that the robot is able to stop when reaching the target depth, as shown in Step As a result, the robot will dive at a steady speed Step 3: If the seabed is detected by DVL, the target depth Dt is set equal to the water depth Dw minus the target altitude At The water depth Dw is defined as the sum of the depth D measured by CTD and the altitude A measured by DVL This step is also carried out by using the depth control with time estimation scheme Step 4: When the robot is within the range of the target depth plus or minus the depth margin Dm, the depth control method is switched from the depth control with time estimation to the PID depth control method The depth margin Dm is usually set around m as a compensation in the control mechanism of the buoyancy device SOTAB-I will stay within the target depth for a certain period of time, which has been set on the timer When the timer reaches zero, the robot will start ascending Simulation result of the algorithm (Fig 10) shows that that the robot could reach the target altitude from the seabed with a slow speed close to m/s and a buoyancy near the neutral value Depth Control of AUV Using a Buoyancy Control Device Fig 10 Simulation results of the altitude control 443 444 M Choyekh et al Conclusions A new method for depth control using the buoyancy control device was developed A model of the buoyancy variation with time was established It was built starting from the results obtained on tank and at-sea experiments The depth control algorithm is based on the comparison between the time estimated for the robot to change its buoyancy from its current value to the neutral value, and the time expected for the robot to reach the target depth The method was demonstrated at-sea experiments in Toyama Bay in Japan in June 2015 It showed the ability of the control algorithm to smoothly bring the robot to the target depth without a significant overshoot The algorithm is characterized by its flexibility and doesn’t require a strict determination neutral buoyancy value A margin of inaccuracy can be customized before performing the dive The method could be further adapted to perform an altitude control needed for the photograph mode A progressive depth control algorithm based on steps was established The results of the simulation showed that it worked properly Further tests at-sea are needed to practically confirm the obtained results References Choyekh, M., et al (2014) Vertical water column survey in the Gulf of Mexico using autonomous underwater vehicle SOTAB-I Marine Technology Society Journal, 88–101 Kato, N., et al (2015) Autonomous spilled oil and gas tracking buoy system and application to marine disaster prevention system In Interspill 2015, Amsterdam, Netherland Seymour, R J., & Geyer, R A (1992) Fates and effects of oil spills Annual Review of Energy and the Environment, 261–283 Short, R T., Toler, S K., Kibelka, G P G., Rueda Roa, D T., Bell, R J., & Byrne, R H (2006) Detection and quantification of chemical plumes using a portable underwater membrane introduction mass spectrometer Trends in Analytical Chemistry, 25(7), 637–646 DOB Tracking Control for Systems with Input Saturation and Exogenous Disturbances via T-S Disturbance Modelling Xiangxiang Fan, Yang Yi and Yangfei Ye Abstract In this paper, the anti-disturbance dynamical tracking problem is investigated for a class of systems subject to input saturation and unknown disturbances under the framework of disturbance-observer-based-control (DOBC) In order to expand the application scope of exogenous disturbances, T-S fuzzy models are employed to describe those complex nonlinear disturbances, and the corresponding disturbance observer is also well designed The PI-type composite controller with the estimates of disturbance is designed to ensure the system stability and the convergence of tracking error to zero Meanwhile, an estimation of domain of attraction can also be described by the level set of the Lyapunov function Finally, a simulation example for flight control systems with nonlinear disturbances is given to verify the effectiveness of the proposed schemes Keywords Disturbance-observer-based control model Tracking control

Input saturation
T-S fuzzy Introduction It is well known that exogenous disturbances widely exist in controlled systems, see for instance Yang and Tsubakihara (2008), Chen et al (2000), Yang et al (2013), Guo and Chen (2005) Thus, many advanced control approaches have been employed to handle disturbance attenuation and rejection problem, such as adaptive dynamical compensation (Marino and Tomei 1995), H1 control (Schaft 1992), output regulator theory (Isidori 1990), and so on On the other hand, disturbanceobserver-based control (DOBC) theory was studied in late 1980s and the basic idea X Fan
Y Yi (&)
Y Ye College of Information Engineering, Yangzhou University, Huayang West Road, Yangzhou, Hanjiang District, Jiangsu, China e-mail: yiyangcontrol@163.com © Springer International Publishing Switzerland 2017 D Zhang and B Wei (eds.), Mechatronics and Robotics Engineering for Advanced and Intelligent Manufacturing, Lecture Notes in Mechanical Engineering, DOI 10.1007/978-3-319-33581-0_35 445 446 X Fan et al of the DOBC scheme is to construct an observer to estimate the unknown disturbance, then a feed-forward compensator plus conventional control laws are applied to reject the disturbance However, in most of DOBC results, such as Yao and Guo (2013), Sun and Guo (2014), and Wei et al (2015), the exogenous disturbances are assumed to be generated by a linear exogenous system, which seriously limits the types of disturbances According to Takagi and Sugeno (1985), the Takagi-Sugeno (T-S) fuzzy model, as a powerful tool for approximating complex nonlinear systems, has been intensively investigated during the past decades By introducing a family of fuzzy IF-THEN rules, the traditional linear control systems are applied to approximate and analyze the nonlinear control systems Moreover, many typical nonlinear systems, such as descriptor systems (Zhang et al 2007a), networked control systems (Zhang et al (2007b), stochastic systems (Yi et al 2009) and time-delay systems (Qiu et al 2011), can be modelled and controlled by the designed T-S fuzzy models Meanwhile, some mature control methods including adaptive control (Tong and Li 2012; Tong et al 2012), and fault estimation (Zhang et al 2012), can also be considered based on T-S fuzzy models On the other hand, control input saturation is a common problem in a wide range of practical control systems, since the signal that an actuator can implement is impossible to be unlimited Due to the existence of input saturation, system control performance as well as stability will be greatly effected (Luo and Zhang 2008) Moreover, the problem of controller design for systems with input saturation is widely investigated in recent years, see Hu et al (2002), Zuo et al (2010) The control problem for systems with saturated input, which including control input and disturbance input, was developed in Wei et al (2015) Motivated by the above observations, this paper considers the disturbance observer based tracking control problem for the nonlinear systems with input saturation and the T-S fuzzy modelling problem for nonlinear irregular disturbances Following the disturbance observer design based on T-S disturbance models, the composite anti-disturbance controller for systems subject to input saturation is proposed by combining the estimation of disturbance with PI control algorithm, such that there exists an initial condition domain ensuring that for every initial condition from this domain, the stability and the favorable tracking performance of augmented systems can achieved by the designed optimization algorithm in present of disturbances Finally, simulations for a flight control system are given to show the efficiency of the proposed approach Problem Formulation In this paper, we study anti-disturbance control problem for the following uncertain system with input saturation: DOB Tracking Control for Systems with Input Saturation …
447 x_ ðtÞ ẳ A0 xtị ỵ B0 satẵutị ỵ dtị ỵ F01 f01 xtị; tị ytị ẳ C0 xtị ỵ F02 f02 ðxðtÞ; tÞ ð1Þ where xðtÞ Rn ; dðtÞ Rm ; uðtÞ Rm and yðtÞ Rp1 are the state, the unknown disturbance, control input and the measurement output, respectively A0 ; B0 ; C0 and F01 are coefficient matrices satð
Þ is the standard saturation function with satð
Þi ¼ sgnð
Þminf
; 1g f0i ðxðtÞ; tÞ is known nonlinear functions Assumption For any xj tị Rn j ẳ 1; 2Þ; f0i ðxðtÞ; tÞ satisfies
f0i ð0; tÞ ¼ kf0i ðx1 ; tÞ  f0i ðx2 ; tÞk  kUi ðx1  x2 Þk ð2Þ where Ui are known constant weighting matrices Different from common disturbances such as harmonics and constant disturbances, T-S fuzzy models are employed to model more complex disturbances The unknown nonlinear disturbance dðtÞ is supposed to be generated by the following T-S fuzzy system with r rules: Plant rule j: If h1 is A1j , and … and hn is Anj , then
_ wtị ẳ Wj wtị dtị ẳ Vj wtị ð3Þ where wðtÞ Rq , Wj , Vj are known coefficient matrices hl and Aij are the premise variables and the fuzzy sets, respectively n is the number of If-Then rules and the premise variables By fuzzy blending, the global fuzzy model are as follows: r P > > _ ẳ hj hịWj wtị < wtị jẳ1 4ị r P > > hj hịVj wtị : dtị ẳ jẳ1 where wj hị ẳ Pr jẳ1 hj hị ẳ Qn iẳ1 Aij hi ị; hj hị ẳ wj hị P r jẳ1 wj hị, hj hị  and Assumption ðA0 ; B0 Þ is controllable and ðWj ; B0 Vj Þ is observable In order to realize the tracking performance of system, we introduce an extended state variable: ztị :ẳ4xT tị; Zt 3T eT sịds5 5ị 448 X Fan et al where etị :ẳytị  yd and yd is the desired system output Furthermore, we can construct that z_ tị ẳ Aztị ỵ B1 satẵutị ỵ dtị ỵ Cyd ỵ F1 f01 ztị; tị ð6Þ         B0 F01 A0 ; B1 ¼ ;C ¼ ; F1 ¼ C0 0 I According to (6), PI-type control input is designed as follows: where A ẳ Zt uPI tị ẳ Kp xtị ỵ KI esịds; K ẳ ẵ KP KI  7ị where KP ; KI are controller gains to be determined later Lemma (Hu and Lin 2001) Given K and H in Rmn ỵ p1 ị For a z Rn þ p1 , if z LðHÞ, then   satKzị ẳ co Di Kz ỵ D i Hz; i Q ð8Þ where Di is a diagonal matrix with each element of the diagonal being either or 0, m Di ỵ D i ẳ I; Q ẳ f1; ; g, and coð
Þ denotes the convex hull of a set Let P1 Rn ỵ p1 ịn ỵ p1 ị be a positive denite matrix We denote   XP1 ị ẳ z Rn ỵ p1 : zT P1 z  ð9Þ and a symmetric polyhedron     LHị ẳ z Rn ỵ p1 : H l z  1; l Qm 10ị where Qm ẳ f1; ; mg and H l is the lth row of the matrix H Controller Design and Stability Analysis We construct the following disturbance observer to estimate disturbance dtị: ^ ẳ dtị r X ^ tị; w ^ tị ẳ vtị  Lztị hj hịVj w 11ị jẳ1 v_ tị ẳ r X hj hị  Wj ỵ LB1 Vj ẵvtị  Lztị  LẵAztị  B1 utị jẳ1  Cyd F1 f01 ðzðtÞ; tÞg ð12Þ DOB Tracking Control for Systems with Input Saturation … 449 ^ and w ^ ðtÞ are the estimations of dðtÞ and wðtÞ, respectively vðtÞ is the where dðtÞ auxiliary variable, L is the observer gain to be determined ^ ðtÞ and introduce variable Define ew tị :ẳ wtị  w rtị ẳ zT tị eTw ðtÞ T ð13Þ According to Lemmai 1, and combining (4), (8) with (10), for 8rðtÞ LðHÞ with h Pr H ẳ H1 ; jẳ1 hj hịVj , the saturation can be expressed as 2m r X X  di Di K ỵ D hj hịVj ew tị satutị ỵ dtịị ẳ i H1 ztị ỵ 14ị jẳ1 iẳ1 Combing (5), (11), (12) with (14), yields e_ w tị ẳ r X hj hịWj ỵ LB1 Vj ịew tị ỵ jẳ1 2m X di LB1 D i H1  Kịztị 15ị iẳ1 By integrating the estimation of disturbance with PI-type control input, the composite controller is inferred as ^ ỵ Kztị; utị ẳ dtị K ẳ ẵ KP KI  ð16Þ Integrating disturbance error (15) with system (6), we can obtain that   d ỵF 1 f01 rtị; tị ỵ F 2 f02 rtị; tị _ ẳ Artị rtị þ Cy ð17Þ where 2m P A þ di B1 Di K ỵ D i H1 ị iẳ1  A¼6 2m P di LB1 D i H1  Kị iẳ1  ẳ ẵC C  ¼ ½ F1 T ; F r P hj hịB1 Vj jẳ1 r P hj hịWj ỵ LB1 Vj ị 7 jẳ1 2 ẳ ẵ F2 T ; F T ; f0i ðrðtÞ; tị ẳ f0i xtị; tị Theorem For system (17), if there exist matrices Q1 ¼ P1 [ 0, P2 [ 0, R1 , R2 , R3 and scalars l1 [ 0, l2 [ 0, l3 [ satisfying 450 X Fan et al P 6 4  ẵR3 B1 di R2  R1 ịT ỵ B 1 Vj sym P2 Wj ỵ R3 B1 Vj   Q1 U1T l2 I  Q2 U2T 7\0 ð18Þ l3 I and 4  Rl2 Q1  Vjl 50 P2 ð19Þ where Rl2 and Vjl are the lth row of the matrix R2 and Vj , respectively, and  T T T P ẳ sym AQ1 ỵ B1 Di R1 þ B1 D i R2 þ l1 CC þ l2 F1 F1 ỵ l3 F2 F2 Then the system (17) under the composite controller (16) is stable and the tracking error 1 1 satises limt!1 ytị ẳ yd with K ¼ R1 Q1 ; L ¼ P2 R3 ; H1 ¼ R2 Q1 Meanwhile, an estimation of the domain of attraction of the system (17) is given by XPị; where P ẳ diagfQ1 ; P2 g: Proof Consider a Lyapunov function candidate as follows Vrtịị ẳ rT ðtÞPrðtÞ ð20Þ Using Assumption 1, it is easy to get that ( _ VðrðtÞÞ  max i2Q r X ) di r tịHij rtị ỵ l1 y2d   a1 krtịk2 ỵ l1 yd T 21ị jẳ1 where a1 [ and " T # Ni P1 B1 Vj þ P2 LB1 D ðH  KÞ i Hij ẳ ;  symP2 Wj ỵ P2 LB1 Vj ị Ni ẳ sym P1 A ỵ P1 B1 Di K ỵ D i H1 ị 1 ỵ P1 CC T P1 ỵ P1 F1 F1T P1 ỵ P1 F2 F2T P1 ỵ l2 U1T U1 ỵ l3 U2T U2 : l1 l2 l3 _ Obviously, if krðtÞk2 [ a1 l1 yd , thus V\0 It is noted that for any zðtÞ and ew ðtÞ, we have   rT ðtÞrðtÞ  max rT ð0Þrð0Þ; a1 l1 yd ð22Þ where rð0Þ is the initial value of rðtÞ, which implies that system (17) is stable DOB Tracking Control for Systems with Input Saturation … 451 Supposed that /1 ðtÞ and /2 ðtÞ are two trajectories of the closed-loop system (17) corresponding to a fixed initial condition Dening  ẳ / tị  / tị /tị 23ị Then we have _  ỵF  /ðtÞ 1 ðf01 ð/1 ; tÞ  f01 ð/2 ; tịị ỵ F 2 f02 /1 ; tị  f02 /2 ; tịị /tị ẳA 24ị  then  tị ẳ /  T P/, Choose Lyapunov function as Cð/;  tÞ  /  T W/\0  _ /; Cð ð25Þ Obviously, (25) can be guaranteed by (19) in Theorem It can be verified that  ! Thus, it follows that the system (24) is asymptotically stable and /tị limt!1 ytị ẳ yd On the other hand, by pre-multiplying and post-multiplying diagfI; Q1 ; Ig to both side of (19), we can obtain that 61 4  H1l P1  hj Vjl jẳ1 70 P2 r P 26ị According to Hu and Lin (2001), (26) implies that XðPÞ LðHÞ It means that for any rðtÞ XðPÞ; rðtÞ LðHÞ Simulation Examples Consider the A4D aircraft model in Guo and Chen (2005), which is subject to input saturation and known nonlinearity The longitudinal dynamics of A4D aircraft at a flight condition of 0.9 Mach and 15,000 ft altitude can be described as (1), where xtị ẳ ẵx1 ðtÞ; x2 ðtÞ; x3 ðtÞ; x4 ðtÞT ; uðtÞ represents elevator deflection (deg) The related matrices are given as 0:0605 0:00014 A0 ¼ 0:0111 32:37 1:475 34:72 2:793 32:2 3 0:1064 7 7 7; B0 ¼ 7; 33:8 5 0 452 X Fan et al F01 ¼ ½ 0 50 T ; F02 ¼ 0; C0 ẳ ẵ U1 ẳ diagẵ 1 ; ; f01 xtị; tị ẳ sinð2p5tÞx2 ðtÞ: The nonlinear exogenous disturbance is described by the following T-S fuzzy model: _ Rule j: If w1 is A1j j ẳ 1; 2ị, then wtị ẳ Wj wtị; d1 tị ẳ Vj wtị, with     1 6 ; V1 ẳ ẵ ; W2 ẳ ; V2 ẳ ẵ , and member W1 ¼ 5 functions A1j w1  aj ị2 ẳ exp 2r21 !," ! !# w1  1:2ị2 w1  1ị2 exp ỵ exp 2r21 2r22 where a1 ¼ 1:2; a2 ¼ 1; r21 ¼ 0:5; r22 ¼ Based on the results of Guo and Chen (2005), we can get a candidate value of R3 :  R3 ¼ 50:6065 368:9759 12:7545 5:5669 0  By solving inequalities (18), (19) with R3 , we get K ẳ ẵ 0:0003 0:4188 0:1418 0:3739   0:0368 0:0145 0 L¼ 0:4384 0:0059 0 0:0000  The initial values of system are chosen as w0ị ẳ ẵ2; 1T ; x0ị ẳ ẵ0:3; 1:1; 1:4; 0T and yd ẳ Figure displays the irregular nonlinear disturbance and its estimation as well as the estimation error, which illustrates the satisfactory tracking ability of the designed T-S fuzzy disturbance observer Figure shows the trajectories and stability of system states When the composite controller is applied, the good tracking performance of system output is shown in Fig It can be seen that the scheme proposed in this paper guarantees system with nonlinear disturbance and input saturation has the satisfactory disturbance tracking ability and the good tracking performance of output, simultaneously DOB Tracking Control for Systems with Input Saturation … Fig Nonlinear disturbance and its estimation value 453 d dˆ Disturbance estimation d − dˆ −2 −4 −6 10 15 20 25 30 Time (Sec) Fig The state trajectory of A4D systems 40 x1 x2 x3 x4 30 20 State 10 −10 −20 −30 −40 −50 −60 10 15 20 25 30 Time (Sec) Fig The trajectory of system output 40 yd Tracking performance 30 y 20 10 −10 −20 −30 −40 −50 −60 10 15 Time (Sec) 20 25 30