Intelligent control of mechatronic systems

653.14 Adaptive parameter ψ1 under AVSC with coupled backstepping andlarge control gains for CSTR... 3.15 Adaptive parameter ψ2 under AVSC with coupled backstepping andlarge control gain

Pey Yuen Tao

B Eng (Hons.), The National University of Singapore

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

NUS GRADUATE SCHOOL FOR INTEGRATIVE SCIENCES &

ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2008

First of all, to my thesis supervisor, Professor Shuzhi Sam Ge, for all the learningopportunities, his unwavering support, inspiring guidance, and for his time and effortthat he has spent on my training and education I would like to thank Dr Wei Lin and DrXiaoqi Chen, my current and former thesis co-supervisors, for their guidance and help

on all matters concerning my research I would also like to thank Professor Jianxin Xu,for his insightful advice and guidance

To fellow co-workers and friends in the Mechatronics and Automation Lab, and theEdutainment Robotics Lab Special thanks to Dr Cheng Heng Fua and Mr Keng PengTee, for the interesting discussions and time we spent working together To the TechXteam, in particular, Mr Aswin Thomas Abraham, Mr Brice Rebsamen, Mr ChenguangYang, Ms Bahareh Ghotbi, Mr Dong Huang, Mr Hooman Samani, Dr Qinghua Xiaand many others that have been part of the team, for the stressful but exciting time

we have spent developing and testing the system To Professor Hongbin Du from EastChina University of Science and Technology, Professor Tianping Zhang from YangzhouUniversity and Professor Zhijun Li from Shanghai Jiao Tong University, for the manyenlightening discussions and help they have provided in my research To Dr Feng Guan,

Dr Xuecheng Lai, Dr Zhuping Wang, Mr Voon Ee How, Mr Sie Chyuan Lau, Ms BeibeiRen, Ms Yaozhang Pan, and many more, for their friendship and help

To my family and close friends for their support and encouragement They havealways been there for me, stood by me through the good times and the bad

Finally, I am very grateful to the Agency of Science, Technology and Research(A*STAR) and the NUS Graduate School of Integrative Sciences and Engineering(NGS), for their funding and support

This thesis considers the theoretical aspects of adaptive neural network (NN) trol and its applications Traditional adaptive controls are limited to systems where thedynamics can be expressed in the linear-in-parameters form, while NNs can approx-imate, to an arbitrary degree of accuracy, any real continuous function on a compactset Through exploiting the approximation capabilities of the NNs, a NN based controlcan be developed where NNs are used to compensate for the functional and parametricuncertainties in the system model.

con-In this thesis, the focus is on the control problem of a class of uncertain ear pure-feedback single-input single-output (SISO) systems The non-affine nonlinearcontrol problem is challenging due to the relatively fewer mathematical tools available

nonlin-in comparison with that for affnonlin-ine nonlnonlin-inear systems In particular, it is difficult toconstruct control laws because pure-feedback systems have no affine appearance of thevariables which can be used as virtual controls In this thesis, an Adaptive VariableStructure Control (AVSC) is proposed, based on NN parametrization, for the class ofuncertain nonlinear pure-feedback single-input single-output (SISO) systems

For the design of the AVSC, three backstepping approaches are explored in thisthesis where the comparative advantages are highlighted during the design stages andbased on the simulation studies First, decoupled backstepping is proposed due to thepossibility of a circular control design if traditional backstepping methods are used

to cancel the coupled terms since the virtual control coefficients are still non-affine.Subsequently, a coupled backstepping approach is adopted where the coupled terms arehandled in the next step in order to remove an assumption made on the system that wasrequired in decoupled backstepping Finally, the design of the AVSC with dynamic

focus is on the control of marine shafting systems A case study on the application of telligent control in marine shafting systems modeled as a chained multiple mass-springsystem is presented where the control objectives include the tracking of the desired tra-jectory and the reduction of torsional vibrations within the shafting system In the ap-plication study, the control design is closely coupled with the system model Therefore,the system model for the marine shafting system is first presented where the systemmodel contains functional and parametric uncertainties The functional uncertaintiesinclude the hydrodynamic forces acting on the propeller, which are highly nonlinearand subjected to variations due to the diverse operating conditions such as air suction,cavitation and partial/full emergence of the propellers, and the frictional forces acting

in-on the shafting system The parametric uncertainties include the unknown moment ofinertia of each mass unit and torsional stiffness of the massless springs connecting themass units

Considering the dynamic model of the system and the uncertainties, an adaptive NNcontrol is proposed where NNs are used to approximate the functional and parametricuncertainties in the system In this application study, two adaptive NN controls are de-veloped for the marine shafting system, where the first control is uses the decoupledbackstepping approach followed by a second control design where the DSC approach

is adopted A simulation study is conducted to illustrate the effects of the two posed controls and to analyze the difference in the two approaches in particular on theimplementation and performance issues

pro-Acknowledgments ii

1.1 Adaptive Variable Structure Control 2

1.2 Adaptive NN Control of Marine Shafting System 6

1.3 Thesis Outline 8

2 AVSC with Decoupled Backstepping 10 2.1 Introduction 10

2.2 Problem Formulation and Preliminaries 11

2.3 Neural Networks and Parametrization 15

2.4 Control Design 17

2.5 Stability Analysis 27

2.6 Simulation 34

3 AVSC with Coupled Backstepping 45 3.1 Introduction 45

3.2 Problem Formulation and Preliminaries 46

3.3 Control Design 47

3.4 Stability Analysis 53

3.5 Simulation 55

4 AVSC with Dynamic Surface Control 71 4.1 Introduction 71

4.2 Problem Formulation and Preliminaries 72

4.3 Neural Networks for Dynamic Surface Control 73

4.4 Control Design 74

4.5 Stability Analysis 87

4.6 Simulation 91

5 Adaptive NN Control of Marine Shafting System 106

5.1 Introduction 106

5.2 Dynamic Modeling 107

5.3 Adaptive NN Control with Decoupled Backstepping 108

5.4 Adaptive NN Control with DSC 119

5.5 Simulation 134

5.5.1 Adaptive NN Control with Decoupled Backstepping 135

5.5.2 Adaptive NN Control with DSC 139

6 Conclusion and Future Work 145 6.1 Summary and Contributions 145

6.1.1 Adaptive Variable Structure Control 146

6.1.2 Adaptive NN Control of Marine Shafting System 147

6.2 Future Work 148

2.1 Neural network compact sets 332.2 State variables x1, x2, and x3under AVSC with decoupled backstepping 382.3 Control effort u under AVSC with decoupled backstepping 382.4 Adaptive parameters ψ1, ψ2, and ψ3under AVSC with decoupled back-stepping 392.5 System output y = x1 and desired trajectory yd under AVSC with de-coupled backstepping for CSTR 422.6 System state x2under AVSC with decoupled backstepping for CSTR 432.7 Tracking error y − yd under AVSC with decoupled backstepping forCSTR 432.8 Control effort u under AVSC with decoupled backstepping for CSTR 442.9 Adaptive parameters ψ1and ψ2 under AVSC with decoupled backstep-ping for CSTR 443.1 State variables x1, x2, and x3under AVSC with coupled backstepping 583.2 Control effort u under AVSC with coupled backstepping 583.3 Adaptive parameters ψ1, ψ2, and ψ3 under AVSC with coupled back-stepping 593.4 System output y = x1 and desired trajectory ydunder AVSC with cou-pled backstepping for CSTR 613.5 System state x2under AVSC with coupled backstepping for CSTR 613.6 Tracking error y − ydunder AVSC with coupled backstepping for CSTR 623.7 Control effort u under AVSC with coupled backstepping for CSTR 623.8 Parameter ψ1 under AVSC with coupled backstepping for CSTR 633.9 Parameter ψ2 under AVSC with coupled backstepping for CSTR 633.10 System output y = x1 and desired trajectory ydunder AVSC with cou-pled backstepping and large control gains for CSTR 643.11 System state x2 under AVSC with coupled backstepping and large con-trol gains for CSTR 643.12 Tracking error y − ydunder AVSC with coupled backstepping and largecontrol gains for CSTR 653.13 Control effort u under AVSC with coupled backstepping and large con-trol gains for CSTR 653.14 Adaptive parameter ψ1 under AVSC with coupled backstepping andlarge control gains for CSTR 66

3.15 Adaptive parameter ψ2 under AVSC with coupled backstepping and

large control gains for CSTR 66

3.16 System output y = x1 and desired trajectory ydunder AVSC with cou-pled backstepping and integral action for CSTR 68

3.17 System state x2 under AVSC with coupled backstepping and integral action for CSTR 69

3.18 Tracking error y − yd under AVSC with coupled backstepping and in-tegral action for CSTR 69

3.19 Control effort u under AVSC with coupled backstepping and integral action for CSTR 70

3.20 Adaptive parameters ψ1and ψ2under AVSC with coupled backstepping and integral action for CSTR 70

4.1 State variables x1, x2, and x3under AVSC with dynamic surface control 93 4.2 Control effort u under AVSC with dynamic surface control 94

4.3 Adaptive parameters ψ1, ψ2, and ψ3under AVSC with dynamic surface control 94

4.4 Error signal y1 = ω1− α1 under AVSC with dynamic surface control 95 4.5 Error signal y2 = ω2− α2 under AVSC with dynamic surface control 95 4.6 System output y = x1and desired trajectory ydunder AVSC with DSC for CSTR 97

4.7 System state x2under AVSC with DSC for CSTR 97

4.8 Tracking error y − ydunder AVSC with DSC for CSTR 98

4.9 Control effort u under AVSC with DSC for CSTR 98

4.10 Adaptive parameter ψ1 under AVSC with DSC for CSTR 99

4.11 Adaptive parameter ψ2 under AVSC with DSC for CSTR 99

4.12 System output y = x1and desired trajectory ydunder AVSC with DSC and integral action for CSTR 103

4.13 System state x2under AVSC with DSC and integral action for CSTR 103 4.14 Tracking error y − yd under AVSC with DSC and integral action for CSTR 104

4.15 Control effort u under AVSC with DSC and integral action for CSTR 104 4.16 Adaptive parameters ψ1 and ψ2 under AVSC with DSC and integral action for CSTR 105

5.1 Propeller velocity tracking performance under Adaptive NN control with decoupled backstepping 137

5.2 Tracking Error under Adaptive NN control with decoupled backstep-ping 137

5.3 Torsional vibrations in the first 2 seconds under Adaptive NN control with decoupled backstepping 138

5.4 Control effort u under Adaptive NN control with decoupled backstep-ping 138

5.5 Torsional vibrations in shafting system with constant control effort u = 30 139

5.6 Propeller velocity tracking performance under Adaptive NN controlwith dynamic surface control 1415.7 Tracking Error under Adaptive NN control with dynamic surface con-trol 1415.8 Torsional vibrations in the first 2 seconds under Adaptive NN controlwith dynamic surface control 1425.9 Control effort u under Adaptive NN control with dynamic surface con-trol 1425.10 Error signal y1 = ω1 − α1 under Adaptive NN control with dynamicsurface control 1435.11 Error signal y2 = ω2 − α2 under Adaptive NN control with dynamicsurface control 1435.12 Error signal y3 = ω3 − α3 under Adaptive NN control with dynamicsurface control 144

Demand for improved control performance and the prevalence of increasingly complexsystems have led to extensive research in the field of control theory to develop novelcontrol methodologies or to extend existing control designs for a larger class of systemsand to compensate for the uncertainties in the system model The control design iscritical to the performance of the closed-loop system where poor design can lead topoor performance or instability Therefore, rigorous stability analysis of the closed-loop system is essential to ensure that all the signals in the system remain bounded.The development of a suitable system model is also essential in practical applica-tions and the performance of the closed-loop system is highly dependent on the systemmodel and control approach adopted Ignoring modeling uncertainties during controldesign can result in poor performance and may even lead to instability of the closed-loop system While many control methods have been developed, practical application

of these controls have to be adapted based on the target system and the control designhas to be closely coupled with the system model in order to achieve the desired per-formance In addition, by drawing on engineering insights and physical properties ofthe system, it is possible to develop a more elegant control design for very complexsystems

In this chapter, an overview of the motivation and background of the research ducted on intelligent control is first presented, followed by an outline of the thesis

con-1.1 Adaptive Variable Structure Control

Adaptive control has been the subject of much research for control theorists over thepast half a century and under the efforts of many researchers, adaptive control has beenextended for larger classes of nonlinear systems and issues such as overparametriza-tion and robustness of the closed-loop system has been dealt with Robustness issuesassociated with adaptive control has been studied [1–4] where it is observed that theclosed-loop system under adaptive control can become unstable under bounded distur-bances or due to unmodeled dynamics of the system A number of approaches wereproposed where modifications to the adaptation law were made to enhance the robust-ness of the adaptive control system including normalization techniques [5–8] where anormalization signal is used in the adaptation law, introduction of dead zones [1, 9, 10]where the adaptation is suppressed when the error signals are within a predefined thresh-old, application of projection methods where the growth of the adaptive parameters arelimited when the parameters exceeds the known bounds [11–13], σ-modification [3, 14]where the adaptation law for the parameter vector θ includes an additional term −σθand e1-modification [15, 16] where the constant σ in σ-modification is replaced by aterm that is proportional to the absolute value of the output tracking error

Traditional adaptive control was initially targeted at linear systems and subsequentlyextended to nonlinear systems Motivated by the results in feedback linearization tech-niques [17], adaptive control was developed for a class of nonlinear systems [18–21].Robustness issues associated with adaptive nonlinear control has also been studied forsystems with bounded external disturbances or unmodeled dynamics [12, 20, 22] In theearly approaches, the nonlinear systems considered are restricted to a class of systemssatisfying matching conditions [20] or extended matching condition [23] In [20], thestrict matching condition is relaxed and adaptive feedback linearization technique waspresented for a class of systems with nonmatching conditions where the reduced-ordermodels satisfy matching conditions while the unmatched terms are treated as unmod-eled dynamics The concept of extended matching condition was introduced in [23],where the unknown constant parameters allowed to be separated from the control vari-

ables by one integration, and subsequently an adaptive nonlinear control was presentedfor the class of nonlinear systems satisfying the extended matching condition.

With the introduction of adaptive backstepping design [24–28], adaptive controlcan be applied in nonlinear systems with nonmatching conditions With these tech-niques, useful results have been obtained for a class of nonlinear systems with triangularstructure [29–31] Early results in adaptive backstepping suffers from the problem ofoverparametrization, where the same parameter vector has to be approximated multipletimes The issue of overparametrization was subsequently handled by the introduction

of tuning functions in [26, 32]

Adaptive output feedback control for nonlinear systems has been presented [12, 13,

33, 34] where high gain observers are developed to estimate the unmeasurable states

In [13], the effects of the peaking phenomenon in high gain observers on the closed-loopsystem is reduced by designing a bounded state feedback control, therefore the effects

of the peaking phenomenon will not affect the stability of the closed-loop system sincethe control will be saturated In [27], adaptive output feedback control was developedfor a class of nonlinear systems in output feedback form

In the early developments of adaptive nonlinear control, the uncertain systems areassumed to be linearly parametrization, where the unknown parameters appear linearly

in the system and the regressors are known Adaptive control of a class of early parametrized systems is presented in [35, 36] where the stability analysis for theclosed loop system is developed based on a suitable choice of the Lyapunov function.Neural networks (NN) have been proposed to approximate uncertain general nonlinearfunctions in the system model [37–50] Since the 1990s, NNs have been found to beuseful for modeling and controlling systems which are highly uncertain, nonlinear, andcomplex, as a result of their ability to approximate any continuous real valued functionwithin a compact set [51–53] Early works on adaptive neural network control rely onoff-line tuning of the NNs [37, 38, 40] In [41, 42], adaptive NN control for robotic sys-tems was proposed where the NN weights are tuned on-line Combining approximationcapabilities of NNs and adaptive backstepping design, more general uncertain nonlin-ear strict-feedback systems have been investigated [44, 45, 49, 54–57] The systems

nonlin-considered are generally affine systems with either known or unknown virtual controlcoefficients For the unknown case, the controller singularity problem was addressed

in [49, 54–57]

Neural network control of nonlinear strict-feedback systems is well documented inthe literature However, results for general nonlinear pure-feedback systems are rela-tively fewer than those for strict-feedback systems In addition, the systems consideredare often in special forms [58–62] The pure-feedback system represents a more gen-eral class of nonlinear systems than its strict-feedback counterpart, with the importantfeature being that the virtual or practical controls are non-affine In practice, manyphysical systems such as chemical reactions, pH neutralization and distillation columnsare inherently non-affine and nonlinear

In recent years, control of non-affine nonlinear systems have captured the attention

of researchers and poses a challenge to control theorists The main impediment in ing this control problem directly is that even if the inverse is known to exist, it may beimpossible to construct it analytically Consequently, no control system design is possi-ble along the lines of conventional model based control Fundamental research is calledupon for this class of nonlinear systems because of the relatively fewer tools available

solv-in comparison with that for affsolv-ine nonlsolv-inear system In [58], solv-inverse dynamic controlwas applied to deal with the non-affine problem under contraction mapping condition.For the same class of systems, a different approach using the Implicit Function The-orem and the Mean Value Theorem, was employed in [61], and then extended to thecase with zero dynamics in [62] In [59], a special class of pure-feedback systems wasconsidered, wherein the n order system is assumed to be affine in the control and inthe xn state variable for the ˙xn−1 equation to avoid a circular argument in the controldesign and stability analysis In [60], the system considered has the first n − 1 equationsnon-affine, and the main result heavily relied on the assumption that 1 − ∂αn−1

∂x n 6= 0,which is only effective when the input gain functions are known

For the control of completely non-affine pufeedback systems, however, few sults are available in the literature In [63], small gain theorem was combined withinput-to-state stability analysis for control design In [64], Nussbaum-Gain function

re-was utilized along with Mean Value Theorem to develop an adaptive NN control fornon-affine pure-feedback systems For such systems, the main difficulty is in dealingwith non-affine functions, particularly in the final step of backstepping, where circularargument of control may appear.

In this thesis, an Adaptive Variable Structure Control (AVSC) is presented, whichcombines the ideas of Variable Structure Control (VSC), Mean Value Theorem, neu-ral network parametrization to solve the control problem of non-affine pure-feedbacksystems In general, VSC cannot be applied in backstepping due to the discontinuoussignum functions in the virtual control In AVSC, the discontinuous function is approx-imated with a smooth C1 function For the design of the AVSC, three backsteppingapproaches are presented in this thesis First, decoupled backstepping is proposed due

to the possibility of a circular control design when traditional backstepping methods areused to cancel the coupled terms since the virtual control coefficients are still non-affine.Subsequently, a coupled backstepping approach is adopted where the coupled terms arehandled in the next step Finally, with the objective to reduce the size of the NNs re-quired in the design of the AVSC, a dynamic surface approach is adopted In dynamicsurface control (DSC) [65–67], the derivative of the virtual controls are estimated usingfirst order filters thus avoiding the problem of “explosion of terms”

One of the first uses of the technical term dynamic surface control in the literaturewas in [65] DSC was introduced as an extension of the multiple surface sliding (MSS)approach [68] where variable structure control is developed for triangular systems, inaddition, in the paper [65] the errors between the system states and the filter outputs aredefined as error surfaces and hence the “surface” term in dynamic surface control DSCwas meant as a dynamic extension of the MSS control where the derivative of the virtualcontrols are obtained using numerical differentiation compared with dynamic filters inDSC and hence the name dynamic surface control Although some of the subsequentpapers extending the work in [65] do not use variable structure control in the controldesign, the technique of using filters to approximate the virtual controls is now termed

as dynamic surface control [66, 69, 70]

Dynamic surface control for a class of uncertain systems with the uncertainties

bounded by a known function is developed in [65] The incorporation of Radial BasisFunctions (RBF) Neural Networks (NN) in dynamic surface control is presented in [66].Simulation studies are conducted for the three approaches where the results illustratethe effectiveness of the proposed controls and the differences between the approachesare analyzed base on the design considerations and simulation results.

1.2 Adaptive NN Control of Marine Shafting System

The problem of torsional vibrations within the marine shafting system, in the presence

of parametric and functional uncertainties, poses a challenge to both marine engineeringpractitioners and control theorists Torsional vibrations within the propulsive shaftingsystem of a marine vessel can be induced by the hydrodynamic forces acting on the pro-peller and inertia forces of the crank mechanism Excessive torsional vibrations withinthe shafting system will lead to failure of the drive shaft Furthermore, the propulsionsystem is connected to the ship’s hull which facilitates the transfer of vibrational energywithin the shafting system to other sections of the vessel leading to excessive noise andcompromising the structural integrity A common approach to tackle the problem of ex-cessive vibrations is to design the shafting system such that vibration response is withinthe tolerable limits Several methods have been proposed for the modelling of a marineshafting system [80–82] In this chapter, the marine shafting system is modeled as achained multiple mass-spring-damper system [82]

In recent years, research on control of marine vessels has been extended to includethe thruster dynamics in the control design The main focus of marine propulsion con-trol is on achieving the desired position and velocity for the marine vessel through thecontrol of shaft speed for fixed pitch (FP) propellers and pitch for controllable pitch(CP) propellers Nonlinear output feedback control of propeller shaft speed was de-veloped in [83] for underwater vehicles where variations in the propeller thrusts due tounsteady flow effects were compensated for through the estimation of the propeller ax-ial flow velocity Recent developments in marine propulsion control have shifted fromthe control of shaft speed or propeller pitch to the control of power from the drive sys-

tem in order to achieve the desired thrust [84] through a mapping from propeller force totorque/power This method has been shown to achieve improved performance in moder-ate seas over conventional shaft speed controls Although the marine propulsion controlhas been an active area of research, the design of an effective control strategy to activelyminimize torsional vibrations within the shafting systems have received relatively littleattention.

Hydrodynamic forces acting on the propeller are highly nonlinear and subjected tovariations due to the diverse operating conditions such as air suction, cavitation andpartial/full emergence of the propellers [83–85] Furthermore, torsional stiffness of thedrive shaft will be subjected to changes during operation due to the mechanical wearand tear of the shafting system Traditional adaptive controls are generally useful whendealing with systems whose dynamics can be expressed in the linear-in-parametersform, for which the regressor is exactly known and the uncertainty is parametric andtime-invariant Such a restriction is clearly ill-suited for this system To overcomethis limitation, approximation-based control techniques is adopted to compensate forfunctional uncertainties in the dynamic model of the shafting system

In this case study, an adaptive NN control using decoupled backstepping is firstpresented followed by an adaptive NN control using DSC to address the issues associ-ated with the control of marine shafting systems highlighted above The control objec-tive for both approaches is to track a desired propeller trajectory while simultaneouslyminimizing torsional vibrations within the shafting system, in the presence of paramet-ric/functional uncertainties and disturbances NNs are utilized to compensate for theparametric uncertainties in the system model, the unknown hydrodynamic forces acting

on the propeller as well as the frictional forces present in the shafting system In theimplementation of the adaptive NN control using decoupled backstepping, it is notedthat the complexities involved in the control design increases with each subsequent step

of the backstepping due to the required computation of the partial derivatives for theNNs Moreover, when applying RBF NNs in the backstepping process, this problemtranslates into more NN inputs required which leads to an exponential increase in thesize of the NNs for each additional element considered in the model When modelling

the shafting system as a multiple mass-spring-damper system, a large number of ments in the model may be required to achieve an accurate representation of the phys-ical system, thus compounding the problems associated with traditional and decoupledbackstepping Comparatively, when applying DSC, the “explosion of terms” problemexperienced in backstepping control can be avoided In addition, when applying back-stepping in the presence of functional uncertainties in the system model leads to theuncertainties appearing in the time derivatives of the virtual controls In the decou-pled backstepping approach, due to the structure of the dynamic model considered andthe approach adopted, the NNs in every alternate step of the backstepping are requiredexclusively for the approximation of the time derivatives of the virtual controls In con-trast, the time derivatives of the virtual controls are approximated by first-order filters

ele-in dynamic surface control, thus removele-ing the need of NNs for every alternate step ele-inthe backstepping which effectively reduces the number of NNs required in the system

by half Therefore, through the application of dynamic surface control technique, it ispossible to reduce the number of NNs required as well as reducing the size of the NNs

in the control

1.3 Thesis Outline

The remainder of this thesis is organized as follows:

In Chapter 2, an AVSC using decoupled backstepping approach is presented for aclass of uncertain, non-affine and nonlinear systems with the control objective to track

a desired trajectory In this chapter, the system model and some mathematical naries are presented, followed by the control design Subsequently, a stability analysis

prelimi-of the closed-loop system is presented where all the closed-loop signals are shown to

be bounded Finally, a simulation study is presented to illustrate the effectiveness of theproposed control

In Chapter 3, an extension of the work in Chapter 2 is presented An AVSC isdeveloped for the same class of uncertain, non-affine and nonlinear systems using amodified coupled backstepping approach where the coupling terms are handled in the

next step of the backstepping without resulting in circular design In this approach, anassumption that was required in the design of the decoupled backstepping approach can

be relaxed Simulation results shows the effectiveness of the proposed approach andthe differences, in terms of the performance and implementation issues, between thedecoupled and coupled approaches are highlighted

In Chapter 4, an extension of the work in Chapters 2 and 3 is presented An AVSC

is developed for the same class of uncertain, non-affine and nonlinear systems using adynamic surface approach where the derivatives of the virtual controls are approximatedwith first order filters In this approach, the derivatives of the virtual controls need not

be approximated by the NNs, thus reducing the number of inputs to the NNs which

in turn reduces the size of the NNs Simulation results are presented to illustrate theeffectiveness of the proposed approach

In Chapter 5, an application study is conducted where the system considered is

a marine shafting system The model of the marine shafting system is first presentedwhere the system model contains parametric/functional uncertainties and disturbances.The control objective is to achieve trajectory tracking while simultaneously minimiz-ing torsional vibrations within the shafting system In an extension on the study ofdecoupled backstepping and DSC is presented in previous chapters, two adaptive NNcontrols are developed where the first control is designed using decoupled backstep-ping Subsequently, an adaptive NN control using DSC is presented with the samecontrol objective Finally, a simulation study for for both approaches is conducted and

a comparative study on the two approaches is presented where the performance andimplementation issues are highlighted

Finally, a summary of the work presented in this thesis is given in Chapter 6 gether with a review of the contributions made in this thesis and a recommendation forfuture work

to-AVSC with Decoupled Backstepping

2.1 Introduction

In this chapter, adaptive variable structure control (AVSC) is investigated for a class

of nonlinear pure-feedback single-input single-output (SISO) systems with unknownnonlinear functions based on neural network parametrization The non-affine nonlinearcontrol problem is challenging due to the relatively fewer mathematical tools available

in comparison with that for affine nonlinear systems In particular, it is difficult toconstruct control laws because pure-feedback systems have no affine appearance of thevariables which can be used as virtual controls

This chapter is organized as follows: In Section 2.2 the system considered is sented followed by some definitions, Lemmas and assumptions that are required in thesubsequent design and stability analysis Thereafter, the structure and definition of the

pre-NN that will be used in the control design is presented Subsequently, in Section 2.4, thedesign procedures for the proposed AVCS utilizing decoupled backstepping is presentedand followed by the stability analysis which is presented in Section 2.5 A simulationstudy is presented in Section 2.6 to demonstrate the effectiveness of the control

2.2 Problem Formulation and Preliminaries

Consider the following class of uncertain nonlinear non-affine pure-feedback input single-output (SISO) systems:

single-˙xi = fi(¯xi, xi+1) for i = 1, , n − 1

˙xn = fn(¯xn, u)

where fi(·) is an unknown smooth function; and ¯xi = [x1, , xi]T ∈ Ri, y ∈ R and

u ∈ R, (i = 1, , n) are the state variables, system output and input, respectively.The control objective is to design an adaptive controller for system (2.1) such thatthe output y follows a desired trajectory yd, while all the signals in the closed-loopsystems are bounded

Assumption 2.2.1 The desired trajectory ydis known, bounded andCncontinuouslydifferentiable

Definition 2.2.1 [86] The solution of (2.1) is Semi-Globally Uniformly UltimatelyBounded (SGUUB) if, for any compact setΩ0, there exists anS > 0 and T = T (S, X(t0))such thatkX(t)k ≤ S for all X(t0) ∈ Ω0 andt ≥ t0+ T

Lemma 2.2.1 Mean Value Theorem [87]: Assume that f (x, y) : Rn× R → R has aderivative at each point of an open setRn× (a, b), and assume also that it is continuous

at both endpoints y = a and y = b Then, there is a point ξ ∈ (a, b) such that

f (x, b) − f (x, a) = f0(x, ξ)(b − a)

From Lemma 2.2.1, the non-affine functions in (2.1) can be written as

fi(¯xi, xi+1) − fi(¯xi, x0i+1) = ∂fi(¯xi, xi+1)

∂xi+1

x i+1 =xθii+1xi+1

fn(¯xn, u) − fn(¯xn, u0) = ∂fn(¯xn, u)

where x0i for i = 2, · · · , n and u0 are design constants For a more effective control

design, the values for x0i and u0 should be chosen such that x0i and u0 lies within theircorresponding operating region, for example, the value x0

i should lie within the ating region of the state xi For convenience, the control u is denoted as xn+1, and

oper-xθi

i+1 = θixi+1+ (1 − θi)x0

i+1 is some unknown point between xi+1and x0

i+1in which

θi is a time-varying parameter satisfying 0 < θi(t) < 1 (i = 1, 2, , n) ∀ t ≥ 0

Substituting (2.2) into (2.1), the system dynamics can be rewritten as

x i+1 =xθii+1 (i = 1, , n − 1)

gn(¯xn, uθn) = ∂fn(¯xn, u)

Remark 2.2.1 By applying Mean Value Theorem, the system dynamics (2.1) can berewritten in the form of (2.3), which can be viewed as another pure-feedback systemsince the input gain functions of (2.3) are still partially non-affine due to the presence

ofxθi

i+1ingi

Remark 2.2.2 The key distinction between the Mean Value Theorem and the Taylorseries expansion is that the former is valid globally, while the latter is only valid locallyaround a specified point In particular, with Taylor series expansion, a smooth functionwould be approximated at one point with a high-order approximating error On theother hand, with Mean Value Theorem, this function would be expanded betweentwopointswithout a high-order approximating error Unlike the Taylor series expansion,which usually yields local stability, the Mean Value Theorem, as used in our approach,yields semi-global stability, as will be shown in our main results in Theorem 2.5.1.Remark 2.2.3 If coupled backstepping design [27] was applied for the control of equa-

tion (2.3), it would result in a circular control design since the control gain functionsare still non-affine in control In the following, decoupled backstepping design [88, 89]will be employed for this problem.

Lemma 2.2.2 [90] For bounded initial conditions, if there exists a C1 continuous andpositive definite Lyapunov function V (x, t) satisfying V (kxk) ≤ V (x, t) ≤ ¯V (kxk),such that ˙V (x, t) ≤ −ρ1V (x, t) + ρ2, whereV , ¯V : Rn→ R are class K functions and

ρ1,ρ2 are positive constants, then the solutionx = 0 is uniformly bounded

Assumption 2.2.2 The signs of gi(·) ∈ R in (2.3) are known, and there exist constants

gmin > 0 and gmax > 0 such that ∀ ¯xn ∈ Ωx ⊂ Rn, we have gmax ≥ |gi(·)| (i =

1, 2, , n − 1) and ∀ ¯xn ∈ Rn, u ∈ R, |gi(·)| ≥ gmin (i = 1, 2, , n) where Ωx is

a compact set Without loss of generality, we assume that gi(·) > 0, ∀ ¯xn ∈ Rn and

u ∈ R

Remark 2.2.4 The smooth function gi(·) is strictly either positive or negative ing to Assumption 2.2.2 Since xθi

accord-i+1 is not available, gi(¯xi, xθi

i+1) cannot be mated by NNs However, the bounds ofgi(¯xi, xθi

approxi-i+1) will be used in the control systemdesign rather than gi(¯xi, xθi

i+1) It should be emphasized that gi(¯xi, xθi

i+1) is only quired for analytical purposes, and will not be approximated by neural networks atall

re-The following technical lemmas are required in the subsequent control design andstability analysis

Lemma 2.2.3 For a, b ∈ R+, the following inequality holds

ab

Lemma 2.2.4 Consider a first-order dynamical system

whereφ(t) is a bounded and continuous nonnegative function and λ is a positive stant If a nonnegative initial values(0) is chosen, then the solution s(t) is nonnegativefor allt > 0.

con-Lemma 2.2.5 For uniformly continuous z(t) ∈ R, t ∈ [0, +∞), if

z(t) + k

Z t 0

z(τ )dτ

... element of the ideal weight vector W1∗T isestimated, rather than the entire set of parameters

The advantage of this technique is that regardless of the number of nodes... the control of equa-

tion (2.3), it would result in a circular control design since the control. .. berewritten in the form of (2.3), which can be viewed as another pure-feedback systemsince the input gain functions of (2.3) are still partially non-affine due to the presence

ofxθi