Shubhobrata Rudra · Ranjit Kumar BaraiMadhubanti Maitra

Block Backstepping Design of Nonlinear State Feedback Control Law for Underactuated Mechanical Systems

Block Backstepping Design of Nonlinear StateFeedback Control Law for UnderactuatedMechanical Systems

Shubhobrata Rudra • Ranjit Kumar BaraiMadhubanti Maitra

Block BacksteppingDesign of Nonlinear StateFeedback Control Lawfor UnderactuatedMechanical Systems

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Shubhobrata RudraDepartment of Electrical EngineeringTechno India Salt LakeKolkata, West BengalIndia

Ranjit Kumar BaraiDepartment of Electrical EngineeringJadavpur UniversityKolkata, West BengalIndia

Madhubanti MaitraDepartment of Electrical EngineeringJadavpur UniversityKolkata, West BengalIndia

ISBN 978-981-10-1955-5 ISBN 978-981-10-1956-2 (eBook)DOI 10.1007/978-981-10-1956-2

Library of Congress Control Number: 2016947781

© Springer Science+Business Media Singapore 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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I dedicate this book to Sudeshna, my friend,who helped me a lot and encouragedme to believe that I too can be an authorof a reference book.

Shubhobrata Rudra

Preface

The purpose of this book is to provide a detailed presentation of a novel blockbackstepping based control law that can address the control problems of a largeclass of underactuated mechanical systems in a generalized manner. Study ofcontrol problems of the underactuated systems and thereafter devising appropriatecontrol actuations for those are presently being pursued with utmost vigor by thecontrol research groups. The reason behind this is the appeal of the dynamic natureof the systems along with the challenges they pose. Such systems belong to a classthat has a wide spectrum in reality not even restricting their domains related toRobotics, Aerospace, Industrial Processes, Marine systems etc. Consequently, a lotof theoretical propositions as well as practical application oriented approaches havebeen reported in the literature during the last few decades. Undoubtedly, thoseresearch contributions have enriched the literature of control engineering to a greatextent, but quite often they fail to compromise between the theory and practice.Theoretical approaches can ensure global asymptotic stability during analysis,while their complicated mathematical formulations restrict their implementations onthe practical systems in real-time. On the other hand, application-orientedapproaches are absolutely system specific, and eventually they fail to answer thecontrol problems of the other systems of similar nature. Therefore, design of ageneralized control law based on block backstepping has been presented in thisbook in a structured manner.

The theory and further employment of Block backstepping based control law hasbeen developed in a versatile manner, so that the control action generated, thereby,can ensure global asymptotic stability for most of the underactuated mechanicalsystems and at the same time can guarantee a desired performance during real-timeimplementations.

Indeed complicated state model of underactuated mechanical system restricts thestraightforward applications of naïve integrator backstepping method. Therefore,the authors of this book have endeavored to establish an algebraic transformation

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that could be used to reconfigure the original system model into a reduced orderblock-strict-feedback form. To be more precise, through this generic matrix-algebraic transformation, a higher order nonlinear underactuated system can betransfigured as a system having two cascaded blocks; the reduced order systemmodelled and depicted in strict feedback form followed by a dynamic block thatactually represents the unobservable internal dynamics of the transformed system.Thereafter, integrator backstepping has been explored to devise a control law for thereduced order system as well as to assess and ensure the stability of internaldynamics of the entire system. Further, applications of the so devised control law onten different systems (having completely different system dynamics) have beendemonstrated in the simulation and real-time environment to corroborate the the-oretical findings.

We reiterate that this book is intended to be used as a reference book by thegraduate students and incumbent researchers in the areas of nonlinear controlsystems, mechanical systems, mechatronics, and robotics.

The book originates from the doctoral thesis prepared by the first author at theJadavpur University of India and supervised by the second and third authors.Chapter 2 has been incorporated to provide with the prerequisite knowledge ofnonlinear control theory that might be helpful to the new yet potential researchers inthis domain. Thereafter, Chap. 3 has systematically presented the derivation ofBlock backstepping control law. At the onset, stepwise formulation of the controllaw has been described for the 2-DOF underactuated mechanical systems. Afterthat, a generalized version of the control law has been presented to address thecontrol problems of n-DOF underactuated mechanical systems. Chapter 4 has beendevoted to demonstrate the applications of the control law on seven differentunderactuated systems, namely, inertia wheel pendulum, TORA, Furuta pendulum,acrobot, pendubot, inverted pendulum, and single dimensional Granty crane. Forthe first five systems, mentioned above, performances of the control law have beenevaluated in the Matlab® Simulation environment. Moreover, the block backstep-ping control law has been employed for the inverted pendulum and overhead cranein real-time environment in a benchmark laboratory environment. For both thecases, it has been observed that the experimental outcomes have nearly corrobo-rated with the theoretical findings. In Chap. 5, applications of the control law onhigher-order systems have been portrayed, and have been substantiated withdetailed simulation results. The authors hope that this reference book would be ableto impart subtle yet sufficient knowledge about mathematical and implementation-oriented nuances towards control of under actuated nonlinear systems. Theexperiences gained by the authors while pursuing the work, are being conveyed tothe research community of the future through this book.

viii Preface

Authors are very much grateful to Prof. Subir Kumar Saha (MechanicalEngineering Department, IIT Delhi, India) for his valuable suggestions, especiallyregarding presentation of the contents. Professor Subhasis Bhaumik (IIEST,Shibpur, India) has also put forward some important modifications that haveenriched the contents of this book.

The first author would also like to express his deepest gratitude to Mr. BedadiptaBain for his continuous support in all endeavors. At last, very special thanks go toMs. Sumita Goswami for her continuous support during the final compilation of thisbook.

Kolkata, India Shubhobrata RudraKolkata, India Ranjit Kumar BaraiKolkata, India Madhubanti Maitra

Preface ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Underactuated Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief State-of-the-Art on the UMSs Control . . . . . . . . . . . . . . . . . . 3

1.2.1 Linear Control Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Nonlinear Control: Present Day Approaches . . . . . . . . . . . . 41.2.3 Block Backstepping Approach . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Motivations of Designing an Advanced Control Law . . . . . . . . . . . 71.4 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Theoretical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Input-State Feedback Linearization . . . . . . . . . . . . . . . . . . . 112.1.2 Output Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Partial Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Control Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Block Backstepping Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Block Backstepping Control of the Underactuated MechanicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Formulation of Generalized Block Backstepping Control Law

for Underactuated Systems with Two Degrees of Freedom(2-DOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Derivation of the Control Algorithm for 2-DOF

Underactuated Mechanical Systems. . . . . . . . . . . . . . . . . . . 333.1.3 Zero Dynamics Analysis of 2-DOF Underactuated

Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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3.2 Formulation of Generalized Block Backstepping Control Lawfor Underactuated Systems with N Degrees of Freedom. . . . . . . . . 393.2.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Derivation of the Control Law for n-DOF Underactuated

Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.3 Zero Dynamics Analysis of n-DOF Underactuated

Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Analysis of Global Diffeomorphism of the Control Law . . . . . . . . 483.4 Stability Analysis of the Proposed Controller . . . . . . . . . . . . . . . . . 493.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Applications of the Block Backstepping Algorithm on 2-DOFUnderactuated Mechanical Systems: Some Case Studies . . . . . . . . . . 534.1 Application on the Inertia Wheel Pendulum . . . . . . . . . . . . . . . . . . 54

4.1.1 Derivation of the Control Law for Inertia WheelPendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.2 Simulation Results and Performance Analysis. . . . . . . . . . . 584.2 Application on the TORA System . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Derivation of the Control Law for TORA System . . . . . . . 634.2.2 Simulation Results and Performance Analysis. . . . . . . . . . . 66

4.3 Application on the Furuta Pendulum . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 Derivation of the Control Law for Furuta Pendulum

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.2 Simulation Results and Performance Analysis. . . . . . . . . . . 74

4.4 Application on the Acrobot System . . . . . . . . . . . . . . . . . . . . . . . . 774.4.1 Derivation of the Control Law for Acrobot System . . . . . . 784.4.2 Simulation Results and Performance Analysis. . . . . . . . . . . 81

4.5 Application on the Pendubot System . . . . . . . . . . . . . . . . . . . . . . . 844.5.1 Derivation of the Control Law for Pendubot System . . . . . 864.5.2 Simulation Results and Performance Analysis. . . . . . . . . . . 89

4.6 Application on the Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . 914.6.1 Derivation of the Control Law for Inverted Pendulum . . . . 934.6.2 Results Obtained from Real-Time Experiments. . . . . . . . . . 96

4.7 Application on the Single Dimension Granty Crane . . . . . . . . . . . . 994.7.1 Derivation of the Control Law for Granty

Crane System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.7.2 Results Obtained from Real-Time Experiments. . . . . . . . . . 103

4.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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5 Applications of the Block Backstepping Algorithmon Underactuated Mechanical Systems with Higher Degreesof Freedom: Some Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1 Application on the VTOL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.1 Derivation of the Control Law for VTOL . . . . . . . . . . . . . . 1125.1.2 Simulation Results and Performance Analysis. . . . . . . . . . . 117

5.2 Application on the USV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.1 Derivation of the Control Law for USV . . . . . . . . . . . . . . . 1225.2.2 Simulation Results and Performance Analysis. . . . . . . . . . . 127

5.3 Application on Three Degree of Freedom RedundantManipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.1 Derivation of the Control Law for 3-DOF Robotic

Manipulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.3.2 Simulation Results and Performance Analysis. . . . . . . . . . . 139

5.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Challenges and New Frontiers in the Field of UnderactuatedMechanical Systems Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.1 Different Aspects of the Proposed Control Law . . . . . . . . . . . . . . . 1456.2 Major Inferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.3 Scope of the Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.1 Robust Adaptive Block Backstepping Design forUnderactuated Mechanical Systems. . . . . . . . . . . . . . . . . . . 148

6.3.2 Industrial Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3.3 High-DOF Complex UMS . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3.4 Fault Tolerant Deduction and Control . . . . . . . . . . . . . . . . . 1506.3.5 Networked UMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Appendix: Modeling of Different UnderactuatedMechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Contents xiii

About the Authors

Dr. Shubhobrata Rudra received his Bachelor’s degree in Electrical Engineeringin 2007 from West Bengal University of Technology, India, and Master ofElectrical Engineering in 2010 and Ph.D. in Electrical Engineering 2015 fromJadavpur University, India. He has secured the University gold medal for standingfirst in the order of merit at the Master of Engineering examination. He pursued hisdoctoral research with Inspire Fellowship, awarded by the Department of Scienceand Technology (Government of India). He has published a few research articles inreputed international journals and presented several papers in different internationalconferences. His fields of interest include nonlinear control engineering, underac-tuated mechanical systems and motion control systems.

Dr. Ranjit Kumar Barai graduated in Bachelor of Electrical Engineering in 1993and Master of Electrical Engineering in 1995 from Jadavpur University, India, andPh.D. in Artificial Systems Science in 2007 from Chiba University, Japan. He hasmore than 20 years of working experience in industry, research, and teaching atgraduate and postgraduate levels. He has supervised several masters and Ph.D.theses in the areas of mechatronics, robotics, and control systems. His researchinterests include mechatronics, robotics, control systems, machine learning andsoft-computing, modelling and system identification, and real-time systems.

Prof. Madhubanti Maitra graduated in Bachelor of Electrical Engineering in1989, Master of Electrical Engineering in 1991 and Ph.D. in Electrical Engineeringin 2005 from Jadavpur University, India. She has more than 25 years of workingexperience in research and teaching at graduate and postgraduate levels. She hassupervised several masters and Ph.D. theses in the areas of mobile communication,robotics, and control systems. Her research interests include mobile communica-tion, robotics, control systems, machine learning and soft computing, modelling andsystem identification, and real-time systems.

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Acronyms

DOF Degree of FreedomGAS Global Asymptotic StabilityIDA Inter connection and Damping AssignmentsIWP Inertia Wheel PendulumPBC Passivity Based ControlRP Rotating PendulumSMC Sliding Mode ControlTORA Translational Oscillator and Rotating ActuatorUMS Underactuated Mechanical SystemUSV Underactuated Surface VesselsVTOL Vertical Take off and Landing Aircraft

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Chapter 1Introduction

Abstract It was early 1970s, when researchers have observed that a large number ofmodernmechanical systems were using fewer control inputs (actuating inputs) than thenumber of output variables. Further inspections have revealed that those systems wereexhibiting some of the common dynamical properties, and thereby could be catego-rized as a different subclass of the mechanical systems. Since those systems use fewernumbers of actuating inputs, they have been classified as the Underactuatedmechanical systems (UMSs). A few examples of such systems are as follows differentrobotic systems, spacecrafts, underwater vehicles, surface vessel, helicopter, spacerobots, underactuated manipulators, etc. It has also been observed that this class ofsystems generates interesting control problems towhich the naive design approaches oforthodox control theory could not be applicable. Inspired by their increasing demand indiverse industrial applications, control of the UMSs started to gain its popularity as oneof the most active research fields among the control systems community.

1.1 Underactuated Mechanical System

As stated earlier, Underactuated mechanical systems (UMSs) are a special class ofmechanical systems that have fewer actuating inputs (control inputs) than config-uration variables (outputs from the system).

A cart-pole system, which is a typical example of the two degrees of freedomUMS, is shown in Fig. 1.1. In the cart-pole system, the actuating force is applied onthe cart in a horizontal direction, and it regulates the motion of the cart.A free-moving pendulum is pivoted to the cart, whose motion is indirectly con-trolled by the back and forth movement of cart. The two outputs from the cart-polesystem are defined as follows: horizontal deflection of the cart from the center of therail (denoted by x), and angular displacement of the pendulum bob from its invertedposition (denoted by h). Both these outputs from the system are controlled by

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application of a horizontal force along the direction of the cart motion. Thus, acart-pole system uses only one actuating input (F) to control the motion of both theconfiguration variables (x and h), and therefore, justifies such categorization.

Underactuated systems possess some unique constructional features, whichmake them more apt for practical applications. From this perspective, there exist atleast two distinct advantages in the design of controllers for underactuated systems.First, underactuated systems require less numbers of control inputs than that of thefully actuated MIMO systems, as an ordinary MIMO system is generally equippedwith more actuators to generate the necessary control actions. Accordingly, thoseadditional devices increase the cost and weight of such systems [14]. Thus, findinga way to control an underactuated version of the system would eliminate some ofthe actuating devices, and would either improve the overall performance or wouldreduce the cost of operation or would be able to achieve both the goals. Second, theunderactuation provides a backup control strategy for an ordinary MIMO system asin case of actuator failure of a fully actuated MIMO system, availability of anunderactuated controller may save the system from catastrophic consequences. Onecan think of hard real-time systems like aircraft or spacecraft, where actuatorfailures can be cataclysmic to the vehicle or its mission.

It is apparent that to synthesize the control objectives for such kind of systems,first the state models of these are to be derived, which are complicated enough andhence those often make the controller design tasks more difficult than that of theordinary MIMO systems. Obtaining such a state model for devising appropriatecontrol algorithms, a proper understanding about the reason of underactuationcomes out to be an essential prerequisite.

Root causes that could make a system an underactuated one are

(i) Inherent dynamics of the system as one can find in spacecraft, helicopter,underwater vehicle, wheeled mobile robot (this type of underactuation oftentermed as natural underactuation).

Fig. 1.1 A cart-pole system: a typical example of 2-DOF underactuated system

2 1 Introduction

(ii) Deliberate design of the system for some practical purposes. Examples ofsuch systems are satellites with two thrusters, flexible-link robot (this type ofunderactuation often termed as intentional underactuation).

(iii) Accidental failure of an actuator in a MIMO system. In this case, the systemautomatically starts behaving like an underactuated system for the sameoutput configuration.

(iv) Thoughtful use of low-order complex underactuated system. This is oftendone to gain the physical insights of the higher order mechanical systems’control aspects, e.g., the cart-pole system, the beam-and-ball system, theAcrobot, and the Pendubot.

Underactuation itself poses severe challenges on controller design. This is truefor regulator kind of problem as well as for tracking kind of problem. Moreover,lack of controllability of UMS makes the controller design task even more com-plicated. These compel the researchers to have some indepth knowledge about thecontrollers and associated controls algorithms devised so far.

1.2 Brief State-of-the-Art on the UMSs Control

Over the last two decades, different algorithms have been proposed to address thestabilization problem of the underactuated systems. Joint research endeavors fromacademia and industry have led to the propositions of several advanced controlalgorithms. Nonetheless, among those numerous control strategies only a few couldstand out as strong contenders. The followings are such a few relevant controltechniques employed for the UMSs.

1.2.1 Linear Control Approaches

Linear control theory offers a very simple and easily implementable control designapproach for real-world systems. In the early days of research, several linear controldesign techniques were proposed to address the control problem of underactuatedsystems. Although linear techniques are able to provide plausible solution for aparticular application, even then the complicated nonlinear dynamics of such sys-tems severely limits the generalized applications of the control laws. As a result,several redesign of the same algorithm became necessary to address the controlproblem of the similar types of systems.

In addition, linear approximation of the UMSs, often results in uncontrollablesystems, which are not amenable to orthodox linear control algorithms for stabi-lization and tracking purposes. Furthermore, approximate linearization (Jacobeanlinearization) of a complicated nonlinear system only provides accurate linearapproximation of the original system at a closed proximity of the equilibrium

1.1 Underactuated Mechanical System 3

points. Therefore, in case of the trajectory-tracking problem, where the desiredequilibrium point is continuously moving along the reference trajectory, severallinear models of the system must be obtained in order to acquire a true physicalinsight of the actual nonlinear system. However, the intricate mathematicalmanipulations make the design task extremely tedious and time consuming.Moreover, linear approximate model for the original nonlinear system often reducesthe speed of response. All these shortcomings of linear control approaches havegreatly motivated the design of several nonlinear control algorithms for UMSs.

1.2.2 Nonlinear Control: Present Day Approaches

To overcome the shortcomings of linear design techniques, several nonlinearcontrol algorithms have been evolved in the last few decades. Feedback lin-earization is a well-known nonlinear design tool, which is widely used to addressthe control problem of the UMSs. Feedback Linearization is mainly alie-algebra-based control design algorithm that transforms a nonlinear system into alinear system by a state diffeomorphism and a feedback transformation. Conversely,the complicated state model of an under actuated mechanical system restricts thedirect application of feedback linearization on its control problem. Consequently, aspecial form of feedback linearization, namely partial feedback linearization isbeing used to address the control problem for such class of systems. The mainadvantage of this method is that it decouples the actuated and unactuated dynamicsof the original underactuated systems. Therefore, from a given n-dimensionalLagrangian model of a system, it produces two decoupled state model of order n1and n2. This decoupling property of partial feedback linearization simplifies thecontrol design tasks for underactuated system. Another salient advantage of thepartial feedback linearization is that it is applicable for all types of the underactu-ated systems irrespective of their system dynamics. The two main genres of partialfeedback linearization are collocated feedback linearization and non-collocatedfeedback linearization. Collocated linearization refers to a feedback law, whichlinearizes the nonlinear equations of the underactuated systems’ associated with theactuated degrees of freedom. On the other hand, non-collocated linearization refersto a special class of partial feedback linearization method that linearizes the systemwith respect to the passive degrees of freedom. Nonetheless, the new controlappears in the both actuated and unactuated parts of the original system, and thishighly complicates the controller design exercise for the decoupled system.Furthermore, like the ordinary feedback linearization approach, partial feedbacklinearization also suffers from the problem of lack of robustness.

Since uncertainty is a common yet intractable issue for the control of theunderactuated systems, sliding mode approach-based robust controller design couldbe treated as a reasonable solution for the control problem of such class of systems.The behavior of the sliding mode depends only on the switching surface. Thus,sliding mode controller becomes insensitive to the parameter variations and external

4 1 Introduction

disturbances. The basic idea of sliding mode design is to alter the dynamics of thesystem by applying a discontinuous feedback control input that forces the system to“slide” along a predefined state surface, and the system produces a desired behaviorby restricting its state to this surface.

A large numbers of underactuated systems fail to satisfy the Brockett’s conditionof smooth feedback stabilization. Hence, those systems cannot be stabilized usingsmooth control input. Therefore, the non-smooth control input generated by thesliding mode control could be opted for as a natural choice to stabilize theunderactuated system. Accordingly, sliding mode control finds its wide range ofapplication on several underactuated systems like underactuated satellite, surfacevessel, helicopter, TORA, ball and beam, biped robot, etc.

However, the discontinuous nature of sliding mode control often suffers with theproblem of chattering, which in turn reduces the longevity of the actuators due towear and tear of the mechanical parts. Another drawback of the sliding mode is thatmost of the time sliding mode controller assumes a very high value of disturbancebound. Consequently, most of the time sliding mode controller yields overcon-servative design approach. In order to solve the drawback of the conventionalsliding mode several modifications have already been proposed. Proposed modi-fications, though have theoretical appeals, are too complicated for real-timeapplications.

Another popular nonlinear control algorithm, the Passivity-based controlapproach is also commonly used to address the stabilization problems of theunderactuated system. The said method exploits the concept of passivity to design astabilizing control law for complicated nonlinear systems. More precisely, thecontrol objective is to passivize the system with a storage function, which has aminimum at the desired equilibrium point. Simply stated, this passivity-basedapproach is just a modified version of the orthodox energy-based control approach.This method is particularly well suited for simple mechanical systems, where thesystem can be stabilized by shaping the potential energy of the system. Nonetheless,the main shortcoming of the passivity-based method is that they can only be appliedto address the control problems of nonlinear systems with relative degree one.A modification of the conventional passivity-based design has been proposed in thework of Ortega et al. to extend the application of passivity-based control approachfor more complicated nonlinear systems. They have proposed a method of dampingassignment in the passivity design to extend its application in the control problem ofthe higher order systems. However, similar to the complicated sliding mode designtechnique, this method also results in a very complex control law, which is inapt forreal-time applications.

Another energy-based method, commonly known as backstepping, evolved inthe early 1990s, has become an effective design tool to address the control problemof the nonlinear systems. Backstepping is a Lyapunov method-based versatilecontrol design approach that ensures the convergence of the regulated variables tozero for nonlinear systems. The main advantage of the backstepping is that this isthe only nonlinear design tool, which allows the designers to treat the controlproblem of an nth order system as a control problem of n numbers of first order

1.2 Brief State-of-the-Art on the UMSs Control 5

interconnected system. In the last two decades, backstepping has gained immensepopularity as a nonlinear control design algorithm.

Backstepping is a nonlinear control design method that provides an alternative tofeedback linearization. The main idea of the backstepping method is to partition theentire system into n numbers of cascaded subsystems. As a result, the states of thefirst subsystem acted as the control variables for the next subsystem. In backstep-ping approach, at first, the ideal input that can stimulate the desired output from thefirst subsystem is computed. Since the subsystems are connected in cascade, inputto the first subsystem automatically comes from the output of the second subsystem.Ideal input to the second subsystem is computed in a similar manner to that of firstsubsystem. The desired inputs for all the subsystems are calculated in a sequentialmanner until the last subsystem is arrived. As a case in point, the desired input forthe last subsystem gives the expression for the actual control input, and accordingly,the designers can implement a state feedback law for the entire nonlinear system.This recursive approach of the backstepping often serves as an advantageous fea-ture during the design of control law for complicated nonlinear dynamic systems.Since control of UMSs is considered as a challenging design problem,backstepping-based design technique could be thought of as one the most suitableoption to address the regulation problem for such class of systems.

However, the ordinary integral backstepping relies on the fact that the systemunder consideration is in strict feedback form. Generally, single-input single-output(SISO) nonlinear systems satisfy this condition under some simplifying assump-tions. Nonetheless, in case of multivariable control problem, quite often the systemstructure is not available in the strict feedback or semi-strict feedback (i.e., lowertriangular) form. Consequently, it is not possible to apply the integral backsteppingtechnique in the usual manner to devise a control law for MIMO systems. Now as acase in point, the state model of an underactuated system typically resembles thestructure of MIMO system, and it never possesses the strict feedback form. As aresult, direct application of the ordinary integral backstepping algorithm on thestabilization problem for underactuated system is not possible. Subsequently,several research endeavors have been made in the last few years to extend theapplications of backstepping control algorithm on the field of underactuated controlsystems. Nevertheless, most of these proposed control algorithms are either toocomplicated for real-time applications or they are subjected to several simplifyingassumptions that make the proposed algorithm inapt for real-time applications.

1.2.3 Block Backstepping Approach

Recently, several research endeavors have been directed towards arriving at a moregeneralized backstepping algorithm that can efficiently address the stabilizationproblems of complex nonlinear systems. Block backstepping technique hasemerged as one of the most efficient backstepping-based algorithm, which canaddress the control problem of different nonlinear multiple-input multiple-output

6 1 Introduction