DESIGN AND APPLICATION OF SECOND ORDER SLIDING MODE CONTROL ALGORITHMS Thesis submitted for the degree of Doctor of Philosophy at the University of Leicester by Mohammad Khalid Khan Department of Engineering University of Leicester N o v e m b e r 2003 UMI Number: U180131 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U180131 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 To my grandfather IBRAHEEM KHAN for his efforts in my education A b st r a c t DESIGN AND APPLICATION OF SECOND ORDER SLIDING MODE CONTROL ALGORITHMS by Mohammad Khalid Khan The thesis considers the development and application of second order sliding mode control algorithms. Second order sliding mode control keeps the main advantages of standard slid­ ing modes and has the additional advantage that it can be used to remove chattering effect, providing smooth or at least piecewise smooth control. The method also provides better accuracy with respect to switching delays. A comparison has been made between dynamic sliding modes and higher order sliding modes. The application of dynamic sliding mode control has been demonstrated for sys­ tems not affine in the control in a MIMO case study for the control of an IC engine. The super-twisting algorithm has been implemented for robust speed control of a diesel engine available in the laboratory where the sliding variable has relative degree one with respect to the control input. A theoretical case has been made for the application of the algorithm and bounds on the controller parameters have been generated. The implementation results demonstrate the practical importance of higher order sliding mode control. A new second order-sliding algorithm has been developed to stabilize systems where the sliding variable has relative degree two with respect to the control input. More over, it does not require the derivative of the sliding variable to be measured or observed and hence reduces the number of sensors required for control implementation. Closed loop simulation of various systems has been carried out to validate the theory. The algorithm has been applied using dSPACE for position and speed control of a DC motor in SISO configuration. The algorithm has also been extended for a class of nonlinear uncertain MIMO systems. A MIMO case study for water level control in coupled twin-tanks system has been presented. The controller has been implemented in the laboratory to validate the theoretical assertions made in the thesis. A cknowledgements First of all I thank to Almighty ALLAH for blessing me with ability, courage and strength to complete my studies. I am grateful to my supervisor Professor Sarah K. Spurgeon for her support, supervision and invaluable advice throughout my research work. Her positive criticism and feedback were especially a great asset to me. I would like to thank Dr. Christopher Edwards for useful suggestions to improve the readability of the thesis and Dr. Matthew Turner for the valuable directives on anti-windup schemes. Further, I would also like to thank Dr. Arie Levant for discussions and critical suggestions over e-mails throughout my research. I further extend my regards to Professor Elbert Hen­ dricks for providing me with the Mean Value Engine Model (MVEM). I am grateful to Keng Boon for giving me introduction on dSPACE, and to Mr. Peter Barwell for helping me in debugging problems with the ‘twin-tank’ rig. I wish to thank to all my colleagues in the Control & Instrumentation Group at the Depart­ ment of Engineering, University of Leicester for their friendship and lively environment they provided. Dr. Liqun Yao was always available for consultation throughout my PhD studies. How can I forget to mention Mr. Bilal Haveliwala, Mr. Mukhtar Ahmad and their families for the help and friendship, which provided an enjoyable and comfortable homely stay at Leicester. I am grateful to my parents for their prayers, guidance and support throughout my educa­ tion. Their inspiration and encouragement has been invaluable. Finally, thanks to my wife, Zainab, for her love, support and patience especially during the write-up and viva preparation when she has to handle our daughter, Hiba, all day alone. Mohammad Khalid Khan Ta b l e o f C o n t e n t s Table of Contents 1 1 Introduction 5 1.1 Thesis contribution .................................................................................................. 7 1.2 Thesis stru ctu re ........................................................................................................ 9 2 Sliding Mode Control 11 2.1 Introduction ............................................................................................................... 11 2.2 Motivation for sliding mode co n tro l .................................................................... 11 2.3 Sliding mode control: literature survey ................................................................. 12 2.4 Sliding mode control: Basic concepts ................................................................. 14 2.4.1 Sliding surface d e s i g n ............................................................................. 15 2.4.2 Control law design .................................................................................... 16 2.5 An illustrative exam ple ........................................................................................... 18 2.5.1 Sliding surface d e s i g n ............................................................................. 19 2.5.2 Control law design .................................................................................... 19 2.5.3 Equivalent control ................................................................................... 20 2.5.4 Robustness property ................................................................................ 21 2.5.5 Order reduction .......................................................................................... 22 2.6 Chattering avoidance .............................................................................................. 23 2.7 Concluding Rem arks .............................................................................................. 26 3 Higher Order Sliding Modes 29 3.1 Introduction .............................................................................................................. 29 3.2 Literature survey .................................................................................................... 30 3.3 Higher order sliding modes (HOSM) ................................................................... 32 3.3.1 Sliding motion and sliding s e t ................................................................ 33 3.3.2 Real sliding ................................................................................................. 35 3.4 Dynamic sliding modes (DSM ) ............................................................................. 36 3.4.1 Equivalent control m e th o d ...................................................................... 38 3.4.2 Indirect sliding method ............................................................................. 38 1 CONTENTS 2 3.5 Finite time converging HOSM algorithm s ........................................................... 41 3.5.1 Terminal sliding mode c o n tro l ................................................................ 42 3.5.2 Second order sliding control algorithms ............................................... 44 3.6 HOSM vs. DSM .................................................................................................... 49 3.6.1 An illustrative exam ple ............................................................................ 50 3.7 HOSM reachability condition ................................................................................. 53 3.8 C o n c lu sio n .............................................................................................................. 54 4 HOSM Case Studies 56 4.1 Introduction ............................................................................................................... 56 4.2 Super-twisting algorithm ........................................................................................ 57 4.3 Diesel engine speed control .................................................................................... 58 4.3.1 Controller design and tuning .................................................................... 59 4.3.2 Speed response to load change ................................................................ 60 4.4 HOSM and unmodelled dynam ics ....................................................................... 61 4.5 IC engine speed control ........................................................................................... 64 4.5.1 The engine model ........................................................................................ 64 4.5.2 Controller design ........................................................................................ 67 4.5.3