BOUNDED BACK-STEPPING CONTROLLER FOR NONLINEAR SYSTEMS

MUHAMMAD NIZAM KAMARUDIN

UNIVERSITI TEKNOLOGI MALAYSIA

BOUNDED BACK-STEPPING CONTROLLER FOR NONLINEAR SYSTEMS

MUHAMMAD NIZAM KAMARUDIN

A thesis submitted in fulfilment of therequirements for the award of the degree of

Doctor of Philosophy (Electrical Engineering)

Faculty of Electrical EngineeringUniversiti Teknologi Malaysia

JULY 2015

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For my wife Sahazati..

For my children Huwaida and Muhammad Hadif..

For my parents Zainon Hashim and Kamarudin Abd. Razak..

iv

ACKNOWLEDGEMENT

I am grateful to ALLAH, our Lord and Cherisher, for guiding me to develop andcomplete this thesis. Verily, there is neither might nor any power except from Allah.Salutation to our beloved prophet MUHAMMAD (Sallallahu a’alaihi wassalam) and tohis companion.

My sincere appreciation goes to my supervisor PROF. MADYA DR. ABDUL

RASHID HUSAIN, whose guidance, advice, assistance and constructive commentswas valuable. I am also deeply indebted to my co-supervisors PROF. MADYA DR.MOHAMAD NOH AHMAD and PROF. MADYA DR. ZAHARUDDIN MOHAMED fortheir invaluable advice.

I would like to thank my wife DR. SAHAZATI MD. ROZALI for beingpatient and supportive during all these years. To my daughter HUWAIDA and my sonMUHAMMAD HADIF, your constant love and laughing has putting me in momentumto complete this thesis. To my parents PN. ZAINON HASSIM and EN. KAMARUDIN

ABDUL RAZAK for their encouragements. To my research colleague and friends forkindness and willingness to help. I would also like to thank the developers of the utmthesis LATEX project for preparing this thesis template.

I am eternally indebted to the MINISTER OF EDUCATION MALAYSIA and theUNIVERSITI TEKNIKAL MALAYSIA MELAKA (UTeM) for providing me funding andopportunity for this PhD study. Special appreciation to the member of BAHAGIAN

CUTI BELAJAR UTeM for their constant support, especially to former assistantregistrar PN. SITI SALWAH AHMAD. To EN. MOHD NAZRUL MOHD SHAFRI andERDALIA ERNA DAUD, may Allah give you all the best in return.

Muhammad Nizam Kamarudin, Melaka

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ABSTRACT

Back-stepping controller is a recursive design approach that offers flexibledesign steps to stabilize nonlinear systems. However, the well known back-steppingcontrol technique is a full state feedback that is highly dependent to system parametersand system dynamics. As such, back-stepping approach normally produces largemagnitude control signal which at times implausible that may lead to actuatorsaturation. In order to overcome this drawback, this thesis proposes a new boundedback-stepping controller technique. The design is based on a classical Lyapunovwith LaSalle invariance set principle. LaSalle invariance set principle relaxes thenegative definiteness of the derivative of a Lyapunov function while deducing theasymptotic stability of closed loop system. Hence, system trajectories are confinedinside the stability region. In the controller design, the universal Sontag’s formulais improved and merged with the back-stepping technique. To handle with theuncertainties and exogenous disturbances, a pseudo function is utilized during theLyapunov redesign phase. In order to observe the efficacy of the proposed method,a strict feedback numerical nonlinear system with time varying exogenous disturbanceis stabilized. The effectiveness of the proposed method is shown through control signalwhich can be bounded without impact to the closed loop stability and robustness.That is, the proposed control method guarantees global asymptotic stability uponperturbation in initial states with invariant set of solution. The proposed controlmethod also guarantees the asymptotic disturbance rejection and it also robust towardsuncertainties. In addition, the control law is smooth and continuous. The proposedmethod is used to develop a fixed pitch variable speed control for a numericalrepresentation of a two-mass wind turbine system that focuses on the nacelle.Simulation results show that the proposed approach requires less control energy toguarantee the asymptotic tracking of turbine rotor speed for optimum tip-speed-ratio.Thus, it produces a maximum power output from the wind turbine while preservingrobustness towards wind intermittent.

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ABSTRAK

Pengawal langkah-belakang adalah satu pendekatan reka bentuk rekursi yangmenawarkan langkah-langkah reka bentuk yang fleksibel bagi menstabilkan sistemtak lelurus. Walau bagaimanapun, teknik kawalan langkah-belakang adalah kaedahsuapbalik penuh yang amat bergantung kepada pemalar dan dinamik sesuatu sistem.Justeru itu, teknik kawalan langkah-belakang kebiasaannya menghasilkan magnitudisyarat kawalan yang besar yang kadang kala tidak munasabah dan boleh membawakepada ketepuan penggerak. Untuk mengatasi kelemahan ini, tesis ini mencadangkanteknik baru pengawal langkah-belakang yang disempadani. Reka bentuknya adalahberdasarkan kaedah klasik Lyapunov dengan prinsip set tak berubah LaSalle. Prinsipset tak berubah LaSalle melegakan kepastian negatif daripada terbitan fungsi Lyapunovketika menyimpulkan kestabilan asimptot sistem gelung tertutup. Oleh itu, trajektorisistem adalah terhad di dalam rantau kestabilan. Dalam reka bentuk pengawal,formula Sontag sejagat dipertingkatkan dan digabungkan dengan teknik kawalanlangkah-belakang. Untuk mengatasi ketidakpastian dan gangguan luaran, fungsi palsudigunakan semasa fasa reka bentuk semula Lyapunov. Untuk melihat keberkesanankaedah yang dicadangkan, sistem kawalan ini digunakan untuk menstabilkan sistemberangka tak linear dengan gangguan luar yang berubah terhadap masa. Kaedahyang dicadangkan ini menunjukkan keberkesanannya apabila isyarat kawalan bolehdisempadani tanpa memberi kesan buruk kepada keteguhan dan kestabilan gelungtertutup. Iaitu, kaedah kawalan yang dicadangkan menjamin kestabilan asimptotglobal apabila terdapat gangguan di keadaan awal dengan penyelesaian set tak berubah.Kaedah kawalan yang dicadangkan juga menjanjikan asimptot penolakan gangguanyang mantap terhadap ketidakpastian. Di samping itu, undang-undang kawalannyaadalah lancar dan berterusan. Kaedah kawalan yang dicadangkan digunakan untukmembangunkan kawalan pit-tetap kelajuan boleh ubah untuk sistem numerik turbinangin dua jisim yang memberi tumpuan kepada nasel sahaja. Keputusan simulasimenunjukkan bahawa kaedah kawalan yang dicadangkan memerlukan tenaga yangrendah untuk menjamin pengesanan kelajuan pemutar yang asimptot bagi tip-nisbahkelajuan yang optimum. Oleh itu, kuasa keluaran maksimum dapat dihasilkandisamping ianya teguh terhadap ketidakpastian angin.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION iiDEDICATION iiiACKNOWLEDGEMENT ivABSTRACT vABSTRAK viTABLE OF CONTENTS viiLIST OF TABLES xLIST OF FIGURES xiLIST OF ABBREVIATIONS xivLIST OF SYMBOLS xvLIST OF APPENDICES xvii

1 INTRODUCTION 11.1 Introduction 11.2 Problem Statement 21.3 Research Objectives 41.4 Scopes of Thesis 5

1.4.1 Purely numerical system 61.4.2 Numerical representation of a dynamical

system 61.5 Contributions of the Research Works 71.6 Thesis Organization 8

2 LITERATURE REVIEW 102.1 Introduction 102.2 Uncertainties and Disturbances 102.3 Nonlinear Systems with Multifarious Control

Techniques 12

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2.4 The Existing Control Strategies for BoundedControl Problems 15

2.5 Summary 20

3 METHODOLOGY FOR BOUNDED BACK-STEPPINGCONTROLLER DESIGN 233.1 Introduction 233.2 Preliminary Study: Stabilization by using Direct

Lyapunov Method 243.3 Development of Bounded Control Law 25

3.3.1 Bounded Control Algorithm Based onSontag 26

3.3.2 Realization of Proposition 3.1 283.3.3 Effectiveness of Bounded Control Algo-

rithm: A preliminary view 293.4 Mixed Back-stepping and Lyapunov Redesign

Control Technique 303.4.1 Back-stepping control strategies 313.4.2 Back-stepping with Lyapunov Redesign 333.4.3 Designing a Final Control Law 36

3.5 Numerical Case: Robust Bounded Control forNonlinear System with Exogenous Disturbances 393.5.1 Conventional Back-stepping and Lya-

punov Redesign 403.5.2 Bounded Back-stepping and Lyapunov

Redesign 443.6 Summary 48

4 SIMULATION RESULTS AND DISCUSSION 494.1 Introduction 494.2 Results 49

4.2.1 Regulation 504.2.2 Control Signal Energy and Power 51

4.3 Analysis of Invariance Set of Solutions: AComparison Between Bounded Back-stepping andConventional Back-stepping Controller 53

4.4 Significance of Control Parameters for AsymptoticStability 55

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4.5 Significance of Bounded Control Parameters d0 634.6 Summary 65

5 APPLICATION OF BOUNDED BACK-STEPPINGCONTROLLER TO A VARIABLE SPEED WINDTURBINE SYSTEM 675.1 Introduction 675.2 Design Assumption and Scope 705.3 Two-mass Wind Turbine System 70

5.3.1 Rotor Model 715.3.2 Aero-turbine Model 71

5.4 Variable Speed Control Design 745.4.1 Variable Speed Control using Conven-

tional Back-stepping 745.4.2 Variable Speed Control using Bounded

Back-stepping 765.5 Results 79

5.5.1 Testing Condition 795.5.2 Free Running 82

5.6 Summary 85

6 CONCLUSION AND SUGGESTIONS 866.1 Conclusion 866.2 Suggestions and Recommendations of Future

Works 87

REFERENCES 90Appendices A – F 131 – 143

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LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Summary on selected literature 214.1 Initial control parameters 504.2 Sum of squared error and steady state error 504.3 Power and energy produced by all control laws 524.4 Results for various C1 when C2 = 10 (10,000 seconds

simulation time) 584.5 Results for various C2 when C1 = 30 (10,000 seconds

simulation time) 604.6 Effect of varying d0 towards SSE and control signal power 645.1 Nomenclature for two-mass wind turbine system 705.2 SSRSE for rotor speed in 100 seconds run time 815.3 Betz limit 84C.1 Fuzzy rules for tuning C1 and C2 113F.1 Wind turbine parameters [1, 2, 3, 4, 3, 5, 6] 118

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Regulated x1 31.2 Control signal, u 31.3 Summarized problem of back-stepping technique 31.4 Set of solutions 41.5 Conceptual block diagram for system introduced by Choi in

[22] 61.6 Two-mass wind turbine structure 71.7 Typical power coefficient characteristic for fixed pitch angle 72.1 Summary on control techniques for uncertain systems 122.2 Fuzzy adaptive control systems having input saturation anti-

windup scheme (courtesy from [134]) 162.3 The LQR Design Method (courtesy from [134]) 173.1 Stabilized x when perturbed by initial condition x(0) = 20 293.2 Control signal for trajectory starting from x(0) = 20 293.3 History of D0(t) = d0 + |G|

|G|+e−α0t when d0 = 1 303.4 Conceptual block diagram for system in equation (3.21)-

(3.22) 313.5 Conceptual block diagram for a manipulated system in

equation (3.24) 333.6 Conceptual block diagram for back-stepped system in

equation (3.27)-(3.28) 333.7 Conceptual block diagram for bounded back-stepped system

in equation (3.27)-(3.28) with Proposition 3.1 - A nominalsystem 33

3.8 Conceptual block diagram for system in equations (3.54)-(3.55) 40

4.1 History of the stabilized x1 514.2 Phase portrait (trajectory) for initial condition X = [1− 1]T 514.3 Control signals produced by a variable structure control in

Choi’s 51

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4.4 Control signals produced a normal back-stepping control lawin equation (3.73) 51

4.5 Control signals produced a bounded back-stepping controllaw in equations (3.90), (3.91), (3.92) 52

4.6 Magnification of control signals - A comparison betweennormal back-stepping and bounded back-stepping controllaws 52

4.7 Surface of Lyapunov function and its derivative 544.8 Compact set / asymptotic stability region created by the

largest initial condition in equation (4.1) 544.9 Phase portrait of system (3.54)-(3.55) using normal back-

stepping 544.10 Phase portrait of system (3.54)-(3.55) using bounded back-

stepping 554.11 Relationship between SSE and C1, when C2 = 10 584.12 Relationship between control signal power and C1, when

C2 = 10 584.13 Regulated x1 for various C1, when C2 = 10 594.14 Relationship between SSE and C2, when C1 = 30 614.15 Relationship between control signal power and C2, when

C1 = 30 614.16 Regulated x1 for various C2, when C1 = 30 624.17 Magnification of Figure 4.16(f) 624.18 Systematic heuristic tuning approach for C1 and C2 634.19 Relationship between control signal power and bounded

parameter d0 654.20 Relationship between SSE and bounded parameter d0 654.21 Regulated x1 for various d0 654.22 Control signal for various d0 654.23 The effect of function D0(t) = d0 + |G|

|G|+e−α0t (for various d0)towards the regulated x1 and the overall control signal 65

4.24 How function D0(t) = d0 + |G||G|+e−α0t decays with respect to

various d0 655.1 Main research area in wind energy conversion system 685.2 Summary of control strategies for wind turbine system 685.3 The control strategy proposed by Inthamoussou [131]. Tg is

the control signal (generator torque), and Ω is the rotor speed 695.4 The control strategy proposed by Beltran et al. [179]. Tg is

the control signal (generator torque) 69

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5.5 The control strategy proposed by Boukhezzar and Siguerdid-jane [181]. Tem is the control signal (generator torque) 69

5.6 Power coefficient characteristic 725.7 Demanded rotor speed ω∗r as in equation (5.64) 805.8 Actual rotor speed ωr 805.9 Rotor speed error - using normal back-stepping control law 805.10 Rotor speed error - using bounded back-stepping control law 805.11 Trajectory, y1 versus y2 - Using normal back-stepping 805.12 Trajectory, y1 versus y2 - Using bounded back-stepping 805.13 Control signal for normal back-stepping controller 815.14 Control signal for bounded back-stepping controller 815.15 Wind speed profile 835.16 Demanded rotor speed ω∗r = λopt

Rυ 83

5.17 ωr versus ω∗r - Using bounded back-stepping control law 835.18 ωr versus ω∗r - Using normal back-stepping control law 835.19 Tip-speed-ratio, λ - Using bounded back-stepping control law 835.20 Tip-speed-ratio, λ - Using normal back-stepping control law 835.21 Power coefficient, Cp - Using bounded back-stepping control

law 835.22 Power coefficient, Cp - Using normal back-stepping control

law 835.23 Initial control signal - Bounded back-stepping control law 835.24 Initial control signal - Conventional back-stepping control

law 835.25 Power distribution 845.26 Power flow diagram 85B.1 Nonlinear system and linearized system trajectory 111C.1 (a) - Membership functions for z and dz

dt, (b) - Membership

functions for C1 and C2 (c) - Surface of C1, (d) - Surface ofC2 114

C.2 Regulated x1 by bounded back-stepping control law 114C.3 History of C1 and C2 during regulation 114D.1 Eulers approximation 115F.1 Two-mass wind turbine structure 118

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LIST OF ABBREVIATIONS

ANN - Artificial Neural Network

ASR - Asymptotic Stability Region

FL - Fuzzy Logic

GA - Genetic Algorithm

GSA - Gravitational Search Algorithm

LMI - Linear Matrix Inequality

LQR - Linear Quadratic Regulator

LQG - Linear Quadratic Gaussian

LUT - Look-up-table

MPC - Model Predictive Control

MRAC - Model Reference Adaptive Control

NMRAC - Nonlinear Model Reference Adaptive Control

N-PID - Nonlinear Proportional-Integral-Derivative

PD - Proportional-Derivative

PI - Proportional-Integral

PID - Proportional-Integral-Derivative

PSO - Particle Swarm Optimization

QFT - Quantitative Feedback Theory

SMC - Sliding Mode Control

SSE - Sum of Squared Error

SSRSE - Sum of Squared Rotor Speed Error

VSC - Variable Structure Control

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LIST OF SYMBOLS

R - Rotor blade radius

υ - Wind speed

ρ - Air density

Cp(λ, β) - Power coefficient

λ - Tip speed ratio

β - Pitch angle

γ - Gearing ratio

ωr - Rotor speed

ωg - Generator speed

Jr - Rotor inertia

Jg - Generator inertia

Kr - Rotor external damping

Kg - Generator external damping

Br - Rotor stiffness

Bg - Generator stiffness

Tm - Aerodynamic torque

Tg - Generator torque / Electromagnetic torque

Ths - High-speed shaft torque

Tls - Low-speed shaft torque

θg - Generator-side angular deviation

θr - Rotor-side angular deviation

Pcapt - Captured power

Tcapt - Captured torque

<n - Real number vector with n-size

C1, C2 - Control parameters for back-stepping controller

C1upper , C2upper - Maximum control parameter for back-stepping controller

C1lower , C2lower - Minimum control parameter for back-stepping controller

α0 - Decaying parameter for bounded controller

α1, α2 - Decaying parameters for pseudo function

ξ - Sum of uncertainties and disturbances

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ε - Parameter for pseudo function

A - Antecedent linguistic term for fuzzy logic controller

B - Consequent linguistic term for fuzzy logic controller

P - Dimension of the input space for fuzzy logic controller

Ni - The number of linguistic terms of the ith antecedentvariable

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LIST OF APPENDICES

APPENDIX TITLE PAGE

A List of Publications 107B Linearization Technique for Nonlinear System 109C On-line Tuning for C1 and C2 112D Euler’s Approximation 115E Sum of Squared Error 117F Wind Turbine Parameters 118

CHAPTER 1

INTRODUCTION

1.1 Introduction

The field of control engineering is greatly advanced in order to fulfils greatindustrial demand. Industrial sectors such as manufacturing, aerospace [7, 8], robotics[9], transportation [10, 11, 12], traffic flow [13], maritime [14], wind turbine system[15], flight control design [16, 17] and many more are expanding rapidly. Mostof the systems are nonlinear and to gain the asymptotic stability and robustness ofthese systems requires advanced control techniques. Nonlinear systems do not fulfilsuperposition principle as linear systems do. Nonlinear systems absorb nonlinearphenomena such as chaos, saturation, limit cycle, finite escape time, having multipleisolated equilibrium points, and unpredictable. Thus, solving nonlinear systemsrequires advance control techniques. The presence of uncertainties and exogenousdisturbances in the nonlinear systems dynamic is sometimes inevitable, and givecatastrophic effect to the stability and robustness of closed loop systems. As such,developing a robust control for nonlinear systems with uncertainties and exogenousdisturbances offer challenge to control research community.

The need to limit the magnitude and energy of a control signal is due tomultifarious causes. One of the causes is to avoid actuator saturation, and to respectthe actuator’s allowable input ranges. As quoted in [18] ‘saturation is probably themost encountered nonlinearity in control engineering’. In practice, the requirementto bound the control signal is a must because of the electrical constraints and themechanical constraints of the system under controlled. In fact, stabilizing unstablenonlinear system with fast settling time often requires large control energy. Moreover,the control signal might reach unreasonable magnitude when the system is perturbedwith large initial condition. This problem becomes unambiguous when a full statefeedback control technique such as back-stepping is used. In a family of nonlinear

2

control techniques, back-stepping is a new approach that has been developed in 1990by Petar V. Kokotovic [19]. Back-stepping is known as a full state feedback approachbased on systematic Lyapunov control technique [20]. As such, back-stepping dependshighly on system parameters and dynamics. For that reason, back-stepping controllernormally produces high magnitude control signal with large power depending on thesystem order and the location of the perturbed initial state.

1.2 Problem Statement

In this thesis, two main problem statements are outlined. These statementsexplain the shortcomings of the ideal back-stepping controller in term of the controlsignal magnitudes, control signal energy and the asymptotic stability condition basedon ideal Lyapunov function.

Statement 1

Quite often, control law primarily aims at the asymptotic stability andasymptotic disturbance rejection of the closed loop system. However, very littleresearch focuses on the control signal magnitude and the power produced by thecontroller while achieving the asymptotic stability and the asymptotic disturbancerejection [21, 22]. Hence, bounded control problem has become an incessant researchin control engineering field. For illustration, it is easy to stabilize unstable system byforcing their poles to the left-hand-side of the S-plane so that the closed-loop systemstable. Theoretically, placing the closed-loop poles near to −∞ may result in fastregulation rate but require high energy as a trade-off. For example, consider a linearsystem

x = Ax+Bu (1.1)

where the state x ∈ <n, system matrix A ∈ <n×n, input matrix B ∈ <n×m and controlinput u ∈ <m. By pole placement approach, states x can be regulated to equilibriumx = 0 by a simple state feedback control law u = −Kx [23]. Without considering themagnitude and the amount of energy in the control signal u, the designer will ponderhow to find K such that the closed loop system x = (A−BK)x stable. For thismotive, designer may simply choose any K so that the eigenvalues of a new systemmatrix (A−BK) positioned at the left-hand-side of the S-plane. For instance, let

3

consider numerical values for system (1.1) as

A =

[1 2

3 4

]and B =

[1

0

](1.2)

Then, placing the closed loop poles at−5 and−6 yieldsK = [16 32]T . Figure 1.1 andFigure 1.2 depict the regulated x1 and the control signal respectively. Placing closed

0 0.5 1 1.5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Time (Sec.)

x 1

poles at [-5 -6], K=[16 32]poles at [-7 -8], K=[20 46]poles at [-9 -10], K=[24 62.7]poles at [-11 -12], K=[28 82]poles at [-13 -14], K=[32 104]

Figure 1.1: Regulated x1

loop poles more toward −∞ will result in faster regulation rate (Figure 1.1), but willincrease the feedback gain K and hence, increase the magnitude of the initial controlsignal (Figure 1.2). This observation shows that avoiding excessive control signal iscrucial in the control design yet really important in practice.

This thesis treats back-stepping controller as its core nonlinear controltechnique. Nevertheless, normal back-stepping is a full-state feedback which dependshighly on system parameters and dynamics [21, 22]. Therefore, back-steppingcontroller normally produces high magnitude control signal with large signal powerdepending on the system dimension and the location of the perturbed initial state.Large control magnitude disrespects the actuator constraint and implausible in practice.This phenomena may result in singularity error to the closed-loop system and givescomputational burden to the processor. Hence, this thesis proposes bounded back-stepping controller that limits the magnitude of the control signal and reduces thecontrol signal power, namely a bounded back-stepping controller. Figure 1.3 shows