Robust Repetitive Control with an Application to a Linear Actuator

Transcription

1 Robust Repetitive Control with an Application to a Linear Actuator Submitted for the completion of the degree Doctor of Philosophy Maria Mitrevska 2018 Faculty of Science, Engineering and Technology Swinburne University of Technology Melbourne, Australia

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3 Abstract Linear actuator (LA) based servo systems are used in many applications of a repetitive nature where they are required to track periodic references in the presence of plant uncertainties and nonlinearities such as nonlinear friction. Although many robust and adaptive control methods have been successfully applied to uncertain LA based servo systems, these methods cannot fully eliminate the periodic uncertainties and nonlinearities inherent in many LA systems used in repetitive applications. Achieving the desired tracking accuracies in such uncertain LA systems requires a control method which is capable of achieving perfect cancellation of the periodic uncertainties. One such control technique is the repetitive control (RC) method. The RC method is a simple learning control paradigm which can achieve perfect tracking and rejection of periodic signals. However, despite the simple structure and implementation, the pure RC structure lacks robustness against non-periodic uncertainties and is often integrated with other robust methods for improved robustness. Motivated by the challenges of designing RC structures for achieving the desired tracking precision in LA based servo applications, this thesis proposes the design of three types of robust RC structures. The first group of robust RC structures consists of two optimal phase lead repetitive control (PLRC) methods. Firstly an optimal robust PLRC is proposed for reducing the large phase lead angle provided by conventional phase lead compensators which can often overcompensate the phase lag caused by the phase uncertainties and thus lead to degraded tracking performance. Secondly an optimal robust PLRC with flexible phase lag compensation is designed in the frequency domain. Although the optimal PLRC methods proposed in this thesis ensure improved tracking performance and robustness to phase uncertainties, they cause a sluggish transient response and offer limited robustness to other types of uncertainties and large disturbances. To overcome the limitations of optimal PLRC, the second type of robust RC structures proposed in this thesis includes two new discrete sliding mode control (SMC) based RC schemes with improved transient characteristics and robustness. i

4 In the proposed sliding mode repetitive control (SMRC) structures the SMC part is used to improve the robustness and transient capabilities of the system while the RC part is added to compensate for the repeatable uncertainties. Moreover, firstly a discrete SMRC is designed in the frequency domain for achieving optimal tracking performance. Then a new discrete terminal sliding mode repetitive control (TSMRC) method is proposed for fast transient response and improved tracking performance. Despite the improved transient and robustness properties offered by the SMC based structures, variable structure control methods such as SMC are model based and require an accurate model of the plant dynamics which is undesirable in many LA systems where the model of the nonlinear friction cannot be accurately measured. To cater for the requirements of many LA systems with unknown nonlinearities and uncertainties, the third type of robust RC represents two new discrete observerbased terminal sliding mode control (TSMC) structures where the observer structure is applied to estimate the unknown plant uncertainties and nonlinearities. Moreover, a discrete nonlinear extended state observer (NLESO) based TSMC is first proposed for improved tracking performance in LA systems with unknown uncertainties and nonlinear friction. In the second control structure a PLRC is added to the NLESO based TSMC structure to improve the rejection of the periodic components of the unknown nonlinear friction which cannot be compensated by the NLESO alone. The efficacy of the control methods proposed in this thesis has been verified through both simulation and experimental testing performed on a LA experimental setup. The results obtained indicate that the proposed controllers offer improved robust tracking performance when compared to conventional control methods. ii

5 Acknowledgment I would like to express my gratitude to my principal supervisor Dr. Zhenwei Cao for her unconditional support and guidance throughout my candidature. I am indebted to her for her insightful opinions and guidance which helped me shape the research direction and achieve my research goals. I would also like to thank my coordinating supervisor Dr. Jinchuan Zheng for all of his help and advice. I am very grateful for all of the support that he provided in resolving hardware related issues. Without his valuable input and help, this thesis wouldn t have been possible. I would also like to express my appreciation to Edi and Sayem for their unselfish support throughout the research journey. Also, a very big thank you to all of my friends and colleagues for making this PhD experience more pleasant and enjoyable. At the end I would like to express my deepest gratitude to my parents Ivan and Danica and my brother Louie for their love and constant support in the last 4 years. This thesis wouldn t have been possible without their encouragement throughout this PhD journey. iii

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9 Contents Contents vii List of Figures xiii List of Tables xix Nomenclature xxi 1 Introduction Background Motivation Contribution of Research Organisation of Thesis Literature Review Introduction Repetitive Control Fundamentals Robust Repetitive Control Robust Repetitive Control for Time-Varying Frequency Signals Robust Phase Lead Repetitive Control Sliding Mode Repetitive Control vii

10 CONTENTS 2.5 Extended State Observer Based Repetitive Control ESO Based Repetitive Control ESO Based Sliding Mode Control Summary of ESO Based RC and ESO Based SMC Summary Experimental Setup and System Modelling Introduction System Overview System Hardware dspace 1103 Controller and I/O module Voltage-to-Current Power Amplifier Linear Actuator Plant and Uncertainty System Software DC Motor Driver Software Software for Controlling the Plant Summary Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Introduction Robust Phase Lead Repetitive Controller for Optimal Tracking Performance Uncertainty Modelling Conventional Phase Lead Repetitive Control Robust Phase Lead Repetitive Control viii

11 CONTENTS Robust Phase Lead Repetitive Control with an Application to a Linear Actuator Simulation Testing and Analysis Experimental Validation Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain Uncertainty Modelling Stability of Repetitive Control Robust Phase Lead Repetitive Controller Design in the Frequency Domain Robust Phase Lead Repetitive Control with an Application to a Linear Actuator Simulation Testing and Analysis Experimental Validation Summary Discrete Sliding Mode Repetitive Control Introduction Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain Design Methodology Design of the Repetitive Controller Design of the Nonlinear Controller Application to a Linear Actuator Simulation Testing Experimental Validation ix

12 CONTENTS 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator Plant Modelling Design of Discrete Terminal Sliding Mode Repetitive Control Robust Stability Analysis Tuning of the Repetitive Control Application to a Linear Actuator Simulation Testing Experimental Validation Summary Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control Introduction Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction Design of Discrete NLESO Based TSMC Stability Analysis Simulation Testing Experimental Validation Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller Design of Discrete NLESO Based TSMRC Robust Stability Analysis Tuning of the Repetitive Controller Simulation Testing x

13 CONTENTS Experimental Validation Summary Conclusions and Future Work Conclusion Future Work Bibliography 165 Appendix 183 xi

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15 List of Figures 2.1 Continuous Internal Model Structure Modified Continuous Internal Model Discrete Internal Model Modified Discrete Internal Model Odd-harmonic Internal Model Discrete Plug-in RC Structure Dual-mode Repetitive Controller Phase Lead Repetitive Controller Modified RC structure for improved performance in scanning devices Robust RC Structure based on µ- synthesis Two Parameter Robust RC New DOB based RC Scheme Continuous ESO based RC ADRC based SMESO structure Linear Actuator System Block Diagram of the Linear Actuator Experimental Setup Magnitude and phase plots of the measured additive uncertainty ABB Mint Workbench Autotune Software xiii

16 LIST OF FIGURES 3.5 Control Desk Layout Magnitude and phase plots of the nominal and perturbed closed loop system Magnitude and phase plots of the measured additive uncertainty Discrete plug-in RC structure Phase response of G c (z)f (z) of the perturbed system with conventional PLRC Phase response of G C (z)f (z) of the system with the proposed PLRC Phase response of G C (z)f (z) of the perturbed system Tracking error of perturbed system (30kg load, Amp=±1000µm) Magnitude plots of S(z) Magnitude plots of T (z) Tracking error (2.3kg load, A = ±500µm) Tracking error for (2.3kg load, A = ±750µm) Tracking error for (2.3kg load, A = ±1000µm) Frequency plots of the LA system at different payloads Magnitude plot of the measured additive uncertainty Discrete plug-in RC structure Phase plot of the nominal system (G c (z)f (z) = 1) Phase plot of G c (z)f (z) after the first design iteration Phase plot of G c (z)f (z) after the second design iteration Phase plot of G c (z)f (z) after the third design iteration Phase plot of G c (z)f (z) after the final design iteration Frequency response of G f (z) Phase response of G f (z)g c (z) for the perturbed system xiv

17 LIST OF FIGURES 4.23 Tracking error of the conventional and proposed PLRC (30kg load, Amp=±1000µm) Magnitude plots of S(z) Magnitude plots of R(z) for the perturbed system Periodic reference trajectory Tracking error of the conventional and proposed PLRC (-3kg payload) Tracking error (3kg load) Proposed SMRC structure Modified RC structure Block diagram of the linear SMRC system Phase response of z m G c (z) Nyquist plot of H(jω) and 1/N(A) (no load) Nyquist plot of H(jω) and 1/N(A) (15kg load) Nyquist plot of H(jω) and 1/N(A) (30kg load) Tracking output and error of SMC and SMRC (no load) Tracking output and error of SMC and SMRC (30kg load) Tracking output and error at steady state of ZPETC and SMRC (30kg load) Tracking error of SMC and SMRC (0kg load, Amp =±500µm) Tracking error of SMC and SMRC (6kg load, Amp =±500µm) Tracking error of ZPETC and SMRC(6kg load, Amp =±500µm) Discrete TSMRC Modified RC structure Block diagram of the linear TSMRC system Phase response of z m G c (z) xv

18 LIST OF FIGURES 5.18 Tracking output and error of TSMC and TSMRC (30kg load) Tracking output and error at steady state of ZPETC and TSMRC (30kg load) Tracking position of TSMC and TSMRC(6kg load, Amp =±500µm) Steady state tracking error of ZPETC and TSMRC(6kg load, Amp =±500µm) NLESO based TSMC structure Tracking output and error of TSMC and NLESO based TSMC (no load, step input) Tracking output and error of TSMC and NLESO based TSMC (15kg load, step input) Tracking output and error of TSMC and NLESO based TSMC (30kg load, step input) Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (no load, step input) Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (3kg load, step input) Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (6kg load, step input) Discrete NLESO based TSMRC structure Modified RC structure Block diagram of the linear TSMRC system Phase response of z m G c (z) Tracking output and error of TSMC and NLESO based TSMRC (no load, step input) Tracking output and error of TSMC and NLESO based TSMRC (30kg load, step input) xvi

19 LIST OF FIGURES 6.14 Tracking output and error of TSMC and NLESO based TSMRC (no load, periodic reference) Tracking output and error (30kg load, periodic reference,) Tracking output and error for ZPETC and NLESO based TSMRC (30kg load, periodic reference) Tracking error of TSMC and NLESO based TSMRC(no load, step input) Tracking error of TSMC and NLESO based TSMRC (3kg load, step input) Tracking error of TSMC and NLESO based TSMRC (6kg load, periodic reference) Steady state tracking error of ZPETC and NLESO based TSMRC (6kg load, periodic reference) xvii

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21 List of Tables 3.1 LA System Parameters Steady State Error Analysis of conventional and proposed PLRC (2.3kg load) Steady State Error Analysis of conventional and proposed PLRC Steady State Error Analysis of SMC and SMRC Steady State Error Analysis of ZPETC and SMRC Steady State Error Analysis of TSMC and TSMRC Steady State Error Analysis of ZPETC and TSMRC Steady State Error Analysis of TSMC and NLESO based TSMC Steady State Error Analysis of TSMC + FF and NLESO based TSMC Steady State Error Analysis of ZPETC and NLESO based TSMRC Summary of the proposed control methods xix

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23 Nomenclature ADRC AFM BIBO DC DF DOB DSMC EID ESO FIR GESO HDD I/O active disturbance rejection control atomic force microscope bounded-input bounded-output direct current describing function disturbance observer discrete sliding mode control equivalent input disturbance extended state observer finite impulse response generalized extended state observer hard disk drive input output IIR IM IMP infinite impulse response internal model internal model principle xxi

24 Nomenclature LA LESO LM LMI MIMO NLESO NLSEF P-P PID PLC PLRC PMDC PMSM PRC PWM QSM QSMB RC RMS RTI SISO linear actuator linear extended state observer linear motor linear matrix inequalities multiple input multiple output nonlinear extended state observer nonlinear state error feedback peak-to-peak proportional integral derivative phase lead compensator phase lead repetitive control permanent magnet direct current permanent-magnet synchronous motor prototype repetitive controller pulse width modulation quasi-sliding mode quasi-sliding mode band repetitive control root mean square real-time interface single-input-single-output xxii

25 Nomenclature SMC sliding mode control SMESO sliding mode extended state observer SMRC sliding mode repetitive control SOSM second order sliding mode SPM Scanning Probe Microscope TD tracking differentiator TPRRC two parameter robust repetitive control TSMC terminal sliding mode control TSMRC terminal sliding mode repetitive control VSC variable structure control ZPETC zero phase error tracking control xxiii

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27 1 Introduction 1.1 Background High precision positioning systems are of enormous importance for the future of many industries and technologies. The recent advances in high precision positioning systems allow manipulation and assembly of materials and objects at the micro/nanometer scale [1]. Likewise, the invention and development of high precision positioning tools such as the scanning probe microscope (SPM) and the atomic force microscope (AFM) have changed the way research is conducted in many areas such as biology, medicine, material science and physics [1]. These devices allow precise control of the interaction between a probe and a surface as a specimen is scanned, resulting in fast and high resolution scanning. Additional improvements in the positional accuracy of such systems is not only crucial for the future development of lithography tools [2] but also for the next generation of space telescopes [3], high density storage devices [4 8] and hard disk drives (HDD) [9 13]. Furthermore, the increasing demands of fast growing technological areas such as nano-metric machining also require high precision multi-axis positioning devices and tools that will allow for accurate removal of material and high dimensional accuracy in the end products. These advances in the resolution and accuracy of servo systems applied to vast robotic platforms will allow for the manipulation and assembly of materials and objects with high precision and reliability. Such systems are the basis for many high precision robotic based applications - robotic surgery systems [14, 15], robotic assembly of micro structures [16], high precision robot welding [17] and many others [16, 18 22]. Due to the extent and significance of these fast growing areas, the next generation of technologies and systems in these application domains will require positioning systems with increased resolution, bandwidth, stability and fast response. This can be achieved through the design of sophisticated hardware (sensors and actuators), as well as through the implementation of advanced controllers which will be specifically designed to overcome the limitations of the available hardware. Design of advanced 1

28 1 Introduction controllers has been of great interest to a large research community and is the main motivation for this thesis. 1.2 Motivation The emerging need for increased bandwidth, resolution and robustness in high precision positioning systems such as the ones applied to SPMs [23, 24], memory storage devices [5, 25], machine tools [26], XY positioning tables [27], and robot manipulators [28, 29] requires the use of actuators which are capable of providing the desired tracking accuracy in the presence of various nonlinearities, plant uncertainties and external disturbances. Linear actuators (LA) are widely used in many high precision positioning applications where they are required to provide fine scale displacement with high precision [30 33]. In many high-precision LA based positioning applications the LA systems are required to perfectly follow periodic reference trajectories, often ones with high frequency components such as the triangular or sine trajectories [24]. Nonlinearities such as nonlinear friction are commonly found in many servo systems [31, 34]. Moreover, in applications of a repetitive character the nonlinear friction consists of repeatable and non-repeatable components which could significantly degrade the system s tracking accuracy if not compensated for by the overall control structure. Plant uncertainties caused by payload variations are also common in many servo applications such as work-piece positioning in machining processes [35, 36], pick and place tasks using robotic manipulators [37, 38], and many others [39, 40]. Achieving the desired tracking accuracies in such uncertain LA based servo systems requires the design of a control structure which is capable of eliminating the periodic error introduced into the system by the nonlinear friction, as well as compensating for the adverse effects of the non-repeatable uncertainties present in the system. A variety of different robust and adaptive control methods have been applied to LA based servo systems in order to achieve improved tracking performance and robustness to plant uncertainties and disturbances [33, 41 43]. Although these methods are effective when dealing with plant uncertainties and non-periodic disturbances, these structures cannot fully eliminate the periodic nonlinearities and uncertainties which are intrinsic to many LA systems used in repetitive applications. To satisfy the high performance requirements of LA based servo systems used in repetitive applications, a large part of the research community has also focused their research efforts on understanding learning based control methods which are 2

29 1.2 Motivation capable of eliminating periodic signals. One popular control method for perfect tracking and elimination of periodic signals is the repetitive control (RC) method [44]. Repetitive control is a well-known learning control technique used for the perfect tracking/rejecting of repetitive signals. It requires the period of the repetitive signal to be known and fixed [44]. Perfect tracking and elimination of periodic signals using RC is achieved by the inclusion into the feedback loop of a periodic signal generator which introduces infinite gains at the fundamental and harmonic frequencies of the periodic signal [45]. This periodic signal generator structure is known as the internal model (IM). Although the infinite gains introduced by the IM can perfectly eliminate the tracking error at steady state, the inclusion of the IM also introduces infinite gains at frequencies other than the fundamental and harmonic frequencies of the periodic signal [46, 47]. This causes amplification of the undesired non-periodic disturbances and plant uncertainties which have frequency components different to those of the periodic signal and could also lead to degraded tracking performance and instability. Furthermore, a RC structure consisting of the IM alone has limited tracking bandwidth which causes a sluggish transient response, especially in systems with plant uncertainties, nonlinearities and external disturbances [48]. A number of robust RC methods have been developed to improve the robustness of RC against non-periodic disturbances and uncertainties. The robustness of the RC system to unmodelled system dynamics and noise at high frequencies can be improved by inserting a low pass filter into the IM structure [49] but this comes at the cost of reduced tracking bandwidth. Robust RC structures based on the D-K technique are proposed in [50 52]. Although compensators designed using this technique can achieve robust stability and improved robust tracking performance, the resulting compensator is often of a high order and difficult to realise in software. In many other robust RC structures a stabilising RC compensator with phase lead characteristics is designed to eliminate the phase lag introduced in the closed loop by phase uncertainties. The PLRC methods which are based on the inverse of the plant require an accurate model of the plant and this is not always stable and/or possible to obtain [53 55]. Moreover, the pure phase lead compensator z m used in the phase lead RC structures in [38, 56 59] can overcompensate for the phase lag caused by the phase uncertainties and hence degrade the system s tracking performance. To overcome these limitations of existing PLRC, the design of an optimal low order PLRC with improved phase lag compensation is often desirable. Furthermore, many existing PLRC are often designed to compensate for phase uncertainties at a certain frequency range, predominately at high frequencies. However, different phase uncertainties can introduce phase lag at different frequencies 3

30 1 Introduction along the RC bandwidth. For example, phase uncertainties caused by load variations are dominant in the low to middle frequency range. This necessitates the design of a flexible phase lead RC which is capable of compensating for phase lag at any frequency, as well as at multiple frequencies along the RC bandwidth. Although many phase lead RC structures can improve the robustness and tracking performance of the RC system in the presence of phase uncertainties, they are only effective for eliminating uncertainties with small bounds. To overcome these limitations of PLRC and to further improve the transient characteristics and robustness to plant uncertainties, nonlinearities and large disturbances, the RC paradigm is often coupled with nonlinear methodologies such as the sliding mode control (SMC) method [5, 60 68]. Though SMC based RC can achieve satisfactory transient and tracking performance properties, it is well known that the inclusion of the RC structure decreases the convergence speed of the system [48]. So to achieve optimal tracking performance and fast transient response one would need to design an optimal SMC based RC and this is difficult to realise using the conventional time domain SMC design approaches used in [5, 60 68]. To avoid such design complexities, an optimal SMC based RC can be easily designed in the frequency domain using the describing function (DF) stability analysis approach together with conventional RC frequency domain design techniques. Another effective approach for achieving fast transient response in variable structure control (VSC) based RC is through the application of VSC methods with fast transient characteristics such as the terminal sliding mode control (TSMC) approach. Even though the inclusion of SMC can improve the robustness and transient properties of the RC system, these methods are model-based and require an accurate estimate of the plant dynamics, which is not always available. In applications where the uncertainties and nonlinearities present in the system are unknown and cannot be easily measured, observer-based structures such as the extended state observer (ESO) are applied [69 74]. Although the ESO structure can effectively estimate and eliminate the total lumped plant uncertainties present in the system, it can only ensure robustness to a limited range of plant uncertainties. In order to further improve the overall robustness of the system ESO can be integrated with VSC methods such as SMC and TSMC [75 90]. Whilst ESO based SMC structures can ensure improved robustness in applications of a repetitive nature, an ESO based structure cannot completely eliminate periodic error [91]. To improve the tracking of periodic references and rejection of periodic error signals the ESO structure can be coupled with the RC method as described in [73, 92 97]. Although these hybrid structures are effective in eliminating periodic uncertainties, they have limited robustness. Therefore, an ESO based VSC scheme could be designed for fast transient 4

31 1.3 Contribution of Research response and improved robustness. 1.3 Contribution of Research To address the limitations and the challenges involved in designing robust RC for LA systems as described in the previous section, the research presented in this thesis focuses on developing robust RC methods for fast transient response and improved tracking performance in uncertain LA systems used in repetitive applications. The control methodologies proposed in this thesis aim to improve the tracking performance in LA systems by eliminating the adverse effects of both the repeatable and non-repeatable uncertainties present in the system. Chapter 4 proposes the design of two optimal robust PLRC methods for improved tracking performance. First is a modified optimal robust PLRC structure which corrects the overcompensation caused by the pure phase lead compensator z m. In this control structure a low pass filter is added in series to the conventional repetitive compensator to reduce the fixed phase lead angle provided by the z m term. The second is an optimal robust PLRC in the frequency domain. Unlike the existing RC methods implemented in the frequency domain, this controller offers the flexibility to compensate for phase lag in the closed loop system at both single and multiple frequencies within the RC bandwidth. As discussed in Chapter 4, the improved phase lag compensation properties offered by the proposed controllers increase the RC tracking bandwidth, which consequently improves the overall tracking accuracy of the system. However, although the optimal PLRC methods proposed in this thesis are effective, they cause a sluggish transient response and offer limited robustness to other types of uncertainties and large disturbances. To address the limitations of the optimal PLRC put forward in Chapter 4, Chapter 5 proposes the design of two SMC based RC controllers for fast transient response and improved robustness. In these control methods the SMC component is used to ensure fast transient response and robustness to plant uncertainties, while the RC part eliminates the repeatable components of the nonlinear friction and improves the tracking of periodic trajectories. First a new discrete SMRC is designed in the frequency domain for achieving optimal tracking performance. Then a new discrete TSMRC is proposed for improved tracking performance and fast convergence. Although the control methods proposed in Chapter 5 have improved convergence and robustness properties, model based control methods such as SMC and TSMC do require an accurate model of the plant dynamics which is undesirable in many 5

32 1 Introduction LA applications where the nonlinearities such as the nonlinear friction cannot be accurately modelled. To accommodate applications where the uncertainties and nonlinearities present in the LA system are unknown and cannot be accurately measured, Chapter 6 proposes two NLESO based controllers in which an observer framework is used to estimate the unknown plant uncertainties. The first one is a discrete NLESO based TSMC which compensates for the effects of unknown plant uncertainties and nonlinear friction present in the LA system. In this control structure the NLESO component is used to estimate the total lumped uncertainty present in the system which is then cancelled by the TSMC control law. The second controller in Chapter 6 is a new discrete NLESO based TSMRC that overcomes the limitations of NLESO based TSMC in applications which require tracking of periodic references. In this control method the RC component is coupled with the NLESO based TSMC to eliminate the periodic error in the system caused by the nonlinear friction which cannot be fully compensated by the NLESO based TSMC structure alone. The resulting control structure has improved robustness to uncertainties and disturbances, faster transient response and improved tracking accuracy. The effectiveness of the proposed robust RC controllers is verified through simulation and experimental results. The experimental testing is performed on a LA system with nonlinear friction and plant uncertainties. The main contributions of this thesis can be summarised as follows: The design of a discrete robust PLRC for optimal tracking performance in LA systems with bounded uncertainties. The design of a discrete robust PLRC in the frequency domain which offers flexible phase lag compensation at multiple frequencies along the RC bandwidth. The design of a new optimal discrete SMRC in the frequency domain for a LA system with plant uncertainties and unknown nonlinear friction. The design of a new discrete TSMRC for a LA system with plant uncertainties and unknown nonlinear friction. The design of a discrete NLESO based TSMC for a LA system with unknown plant uncertainties and nonlinear friction. The design of a new discrete NLESO based TSMRC for a LA system with unknown uncertainties for improved tracking of periodic and aperiodic references. 6

33 1.4 Organisation of Thesis 1.4 Organisation of Thesis The structure of this thesis is as follows: Chapter 2: Literature Review This chapter provides a literature survey of basic RC, robust RC, SMC based RC, ESO based RC and ESO based SMC methods. Chapter 3: Experimental Setup This chapter presents an overview of the LA experimental setup with a focus on both the hardware and software components of the system. Chapter 4: Design of Discrete Robust Phase Lead Repetitive Control for Optimal Tracking Performance In Chapter 4, two discrete robust PLRC are proposed for optimal robust performance in LA systems in the presence of bounded plant uncertainties and nonlinear friction. Chapter 5: Discrete Sliding Mode Repetitive Control This chapter presents the design of an optimal SMRC in the frequency domain. Furthermore, a new discrete TSMRC structure is proposed for improved tracking performance in LA systems with plant uncertainties and unknown nonlinear friction. Chapter 6: Discrete Nonlinear Extended State Observer Based Terminal Sliding Mode Repetitive Control This chapter discusses the design of discrete NLESO structures for improved tracking performance. First a discrete NLESO based TSMC is proposed for a LA system with unknown uncertainties and nonlinear friction. Then a PLRC is added to the NLESO based TSMC for improved tracking performance of periodic and aperiodic references. Chapter 7: Conclusions and Future Work The conclusions from this research are summarised in Chapter 7 followed by recommendations for future improvements. Appendix: The Appendix lists other publications produced from this research. 7

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35 2 Literature Review 2.1 Introduction In many practical applications of servo systems such as industrial manipulators for pick and place operations [18], machine tools [26, 98] and storage devices [99, 100], the tasks performed or the disturbances acting on the system are of a repetitive nature. Achieving the desired high positioning accuracy in such applications entails the need for a control structure capable of providing perfect tracking of periodic reference signals or rejection of periodic disturbances. One such control technique is the RC method [44]. Repetitive control is one of the most well-known learning control methods for asymptotic tracking or rejection of periodic signals [44]. The RC concept is based on the internal model principle (IMP) [45] which states that asymptotic tracking or rejection of a periodic signal can be achieved if a periodic signal generator is incorporated into the structure of a stable closed loop system. The periodic signal generator in the continuous internal model (IM) structure introduces infinite gains at all frequencies in order to achieve perfect reconstruction of the periodic signal to be tracked or rejected [44], while the discrete IM provides infinite gains at the fundamental and harmonic frequencies of the periodic signal [101]. Although these properties of RC allow for high tracking accuracies, the inclusion of the continuous or discrete IM introduces infinite or N number of poles on the imaginary axis respectively. Consequently, a system with RC based on the pure IM is marginally stable and has limited applicability in many practical systems. To make RC based on the pure IM more stable a low pass filter is often inserted into the IM in order to improve the robustness of the system to unmodelled system dynamics and noise at high frequencies [49]. However, this is achieved at the expense of degraded tracking accuracies at high frequencies. The inclusion of the IM structure into the closed loop system also amplifies the loop gains at the dominant frequencies of the uncertainties and disturbances present 9

36 2 Literature Review in the system. This causes amplification of the adverse effects of the uncertainties which could lead to degraded tracking performance, sluggish transient response and loss of stability [46]. To overcome these limitations of conventional RC a stabilising phase lead compensator is often inserted into the RC structure to ensure overall system stability as well as to improve the tracking performance of the system in the presence of phase uncertainties [38, ]. Moreover, the transient and stability robustness properties of the RC structure can also be improved by integrating RC with other robust techniques such as the variable structure control (VSC) methods [5, 60 68] and vast observer frameworks [73, 92 97]. To broaden our knowledge of the advantages and limitations of RC this chapter provides an extensive literature survey of the RC method with a special focus on robust RC methods and applications. Section 2 gives an overview of RC fundamentals, Section 3 presents a review of robust PLRC structures, Section 4 presents a review of SMC based RC methods and Section 5 is a literature survey of ESO based RC and ESO based SMC methods. A summary of the reviewed literature is presented in Section Repetitive Control Fundamentals Repetitive control (RC) is a popular learning control technique which is used in many dynamic systems to eliminate periodic error at steady state [44]. The RC method is based on the concept of the IMP where a periodic signal generator is added to the stable closed loop system to ensure asymptotic tracking or rejection of repetitive signals [45]. The asymptotic tracking and rejection of periodic signals is accomplished by the periodic signal generator which introduces infinite gains to achieve perfect reconstruction of the periodic signal. The high precision, simple implementation and model-free nature of RC is why this control structure has been extensively used in many applications of a repetitive character. So far RC has been successfully applied and used in many systems - memory storage devices [99, 100], robot manipulators [18], machining tools [26, 98], power converters [38, 56, 57] and many others [20, 32, ]. The IM based RC method was first introduced by Inoue [44] for high accuracy tracking of periodic reference trajectories in continuous single-input-single-output (SISO) linear time-invariant (LTI) systems. The proposed RC structure was applied to a power supply system consisting of a proton synchrotron magnet for achieving high tracking accuracy, which was difficult to achieve using existing control methods. 10

37 2.2 Repetitive Control Fundamentals The RC method developed by Inoue [44] is based on the IMP which was proposed by Wonham and Francis in [45]. According to the IMP the inclusion of a signal generator into a stable closed loop system can achieve perfect tracking or rejection of that signal. Moreover, since a periodic signal of a known period T can be created using a time delay positive feedback structure, the RC structure initially proposed by Inoue [44] is based on an internal model signal generator as shown in Figure 2.1, where a continuous delay is inserted into a positive feedback system. Figure 2.1: Continuous Internal Model Structure The transfer function of the pure IM based RC structure depicted in Figure 2.1 can be represented as follows I(s) = e st 1 e st (2.1) where e st signal. represents the continuous delay, while T is the period of the periodic The main objective of the IM signal generator is to reconstruct the periodic signal with period T by providing infinite loop gains at all frequencies. In theory the infinite gains provided by the IM allow for perfect reconstruction of the periodic signal which then leads to perfect tracking or rejection of that periodic signal. However, despite the high accuracy and simple implementation characteristics of the initial RC, the RC structure based on the IM is infinite-dimensional, which means it has an infinite number of poles on the imaginary axis. This makes the RC system marginally stable and increases the difficulties of designing a stable controller [47]. The stability of RC systems has been studied in [49], where the BIBO stability conditions for the system in Figure 2.1 are given. Furthermore, the small gain theorem is used in [28] to evaluate the stability of RC systems based on the IM. According to the stability analysis provided in [28], the stability of the RC system can only be guaranteed when the plants are proper, but not strictly proper. This limited the application of the initial continuous IM based RC to plants whose relative degrees are zero. Moreover, as stated in [28], this limitation of the pure RC structure 11

38 2 Literature Review is due to the unrealistic demands for perfect tracking of periodic signals, including the high frequency harmonics of the signal. Motivated by this limitation of the initial RC, Inoue proposed a modified RC in [49]. In the improved design a low pass filter is added in series to the continuous time-delay term as shown in Figure 2.2. The inclusion of the low pass filter aims to improve the robustness of the RC structure to unmodelled system dynamics and noise at high frequencies, at the cost, however, of degraded tracking performance at high frequencies. Figure 2.2: Modified Continuous Internal Model follows The transfer function of the modified IM in Figure 2.2 can be represented as I(s) = Q(s)e st 1 Q(s)e st (2.2) As seen from Figure 2.2, the modified RC is also based on an infinite IM but the addition of the low pass filter pushes the poles of the RC system to the left half plane, thus improving the stability robustness of the RC. As stated in [28], any low pass filter that satisfies the condition Q(s) < 1 can be used to improve the robustness of minimum phase systems. However, for nonminimum phase plants the same condition imposes bandwidth restrictions [108]. The introduction of digital controllers increased the need for the theoretical development and practical applications of discrete RC. The first digital RC was proposed in [101]. Unlike RC for continuous systems, the discrete RC structure is finite-dimensional which means that the inclusion of the discrete IM introduces N number of poles on the imaginary axis. Furthermore, unlike the continuous IM, the discrete IM provides infinite loop gains only at the fundamental and harmonic frequencies of the periodic signal. 12

39 2.2 Repetitive Control Fundamentals For discrete RC structures, the periodic signal generator given in Figure 2.3 can be represented as I(z) = z N 1 z N (2.3) Figure 2.3: Discrete Internal Model where N = T T s N is a fixed integer number of samples per period (N also indicates the order of the internal model), T is the period of the periodic signal and T s is the sampling period. Although the discrete RC is finite-dimensional, the inclusion of the discrete IM also has the effect of amplifying the non-periodic uncertainties present in the system. To improve the robustness of discrete RC, a low pass filter Q(z) can also be added in series to the discrete-time delay term z N as shown in Figure 2.4. Similarly as for continuous RC, the resulting discrete RC structure has improved robustness although this is achieved at the cost of reduced tracking accuracy at high frequencies. I(z) = Q(z)z N 1 Q(z)z N (2.4) Figure 2.4: Modified Discrete Internal Model Many different low pass filter designs have been proposed in the discrete RC literature [10, 56, 105]. In [56], Q(z) is designed as a moving average filter with zero phase contribution and unity gain at low frequencies. The proposed zero phase low pass filter can be represented as follows Q(z) = m i=0 α iz i + m i=0 α iz i (2.5) 13

40 2 Literature Review where α 0 +2 m i=0 α i = 1 and α i > 0. Furthermore, the phase angle of Q(z), θ q (e jω ) = 0. Several modified IM structures have also been proposed to deal with the challenges present in a variety of different RC applications. In applications such as power systems the reference signals used and the disturbances acting on the system often consist of odd-harmonic frequencies. To improve the tracking accuracies in such systems an odd-harmonic IM as shown in Figure 2.5 is proposed in [109]. Figure 2.5: Odd-harmonic Internal Model Unlike the conventional discrete IM based RC structure, the odd-harmonic IM given in Figure 2.5 consists of a negative loop with only N/2 number of integrators. The transfer function of the odd-harmonic IM is as follows I(z) = z N/2 1 + z N/2 (2.6) Another equally important aspect of the design of the discrete RC is the design of the stabilising compensator F (z) which is added to the RC structure to ensure overall stability of the closed-loop system. Numerous different stabilising PLRC structures have been proposed in the literature, many of which are based on the zero phase error tracking control (ZPETC) structure proposed by Tomizuka [102]. One such controller is the prototype repetitive controller (PRC) [110] where the proposed compensator is designed to cancel the phase shift of the unstable zeros. For minimum phase systems the ZPETC is equivalent to the inverse of the stable plant and can be represented as follows F (z) = k r G(z) 1 = k r A(z) B(z) (2.7) where B(z) and A(z) represent the numerator and denominator of the plant G(z) respectively and k r is the RC gain which directly controls the convergence rate. For non-minimum phase systems, the ZPETC is a close approximation of the inverse plant and can be represented as follows 14

41 2.2 Repetitive Control Fundamentals F (z) = k r G(z) 1 = k r b A(z)B (z 1 ) B + (z) (2.8) where B + (z) and B(z 1 ) represent the cancellable and uncancellable zeros of B(z) respectively, B (z 1 ) represents B(z 1 ) with a backward shift operator z 1 and b is a scalar value. The stable poles and zeros of the plant G(z) are cancelled by A(z) and B + (z) respectively, while B (z 1 )/b cancels the phase of the unstable zeros. Furthermore, b max B (e jω ) 2 for ω [0, π] (2.9) Asymptotic stability of the proposed structure can be guaranteed if the RC gain is selected as follows k r (0, 2) (2.10) Yamanda et al. [111] replaced the repetitive gain in (2.8) with a zero-phase low pass filter in order to eliminate the error amplitude caused by the unstable zeros, which results in a faster convergence rate of the tracking error. Furthermore, to improve the stability robustness of the PRC, Chew et al. [112] modified the original structure by adding a low pass filter in series to the discrete-time delay component. This improved the robustness of the system, but yet again at the cost of degraded tracking performance. Due to the simple structure and implementation of PRC, this method represents the basis for a number of improved and modified PRC structures [53 55]. A zero phase odd-harmonic RC structure which combines an odd-harmonic periodic signal generator and a zero phase compensator is proposed for fast convergence and low tracking error in pulse width modulation (PWM) inverters in [113]. Furthermore in [114], Jeong et al. extended the PRC to discrete-time multiple input multiple output (MIMO) systems. The plug-in RC configuration shown in Figure 2.6 was first introduced in [115] for a SCARA type robotic manipulator. In the proposed RC structure, the RC compensator denoted by F (z) is designed to compensate the closed loop system G c (z) which can be represented as G c (z) = C(z)P (z) 1 + C(z)G(z) (2.11) 15

42 2 Literature Review where C(z) is a stabilising feedback controller which is designed to stabilise the system without the RC. Figure 2.6: Discrete Plug-in RC Structure Using the plug-in structure depicted in Figure 2.6, the RC design problem consists of selecting a low pass filter Q(z) and designing a stabilising RC compensator F (z) for achieving overall system stability. Moreover, overall stability of the RC structure given in Figure 2.6 can be achieved if the following sufficient stability condition is satisfied at all frequencies Q(z)(1 F (z)g c (z)) < 1 (2.12) In [116] a flexible dual-mode repetitive control structure as shown in Figure 2.7 is proposed for dealing with both odd- and even-harmonic periodic signals. The proposed control framework can achieve faster convergence of the tracking error when compared to the conventional RC structures. Figure 2.7: Dual-mode Repetitive Controller 16

43 2.2 Repetitive Control Fundamentals Many other repetitive applications require accurate tracking of periodic signals with multiple dominant frequencies and their harmonics. To cater for such applications a multi-periodic RC for a MIMO system is proposed in [117]. This section has provided an overview of the RC concept and the initial RC structures. The RC method is a popular technique used for perfect tracking/rejecting of repetitive signals which requires the period of the signal to be known in advance and fixed [44]. The perfect tracking and elimination of periodic signals using RC is achieved by the inclusion of a periodic signal generator into the feedback loop which introduces infinite gains at the fundamental and harmonic frequencies of the periodic signal. Although the infinite gains introduced by the signal generator eliminate the tracking error at steady state, the signal generator also amplifies the gains at frequencies other than the fundamental and harmonic frequencies of the periodic signal [46]. This causes amplification of the undesired non-periodic disturbances and plant uncertainties which have frequency components different to those of the periodic signal and could degrade the system s tracking performance or even cause instability [46, 47]. Furthermore, the RC structure based on the pure IM lacks robustness and has limited tracking bandwidth, which causes a sluggish transient response [48]. The robustness of the RC system to unmodelled system dynamics and noise at high frequencies can be improved by inserting a low pass filter into the IM structure [49]. However, this reduces the tracking accuracies at high frequencies. Another important aspect of the design of RC is the design of the stabilising compensator which is required to ensure improved tracking performance and overall system stability. A PRC is proposed in [110] where a ZPETC structure is used to achieve good tracking performance. Despite the simple structure and implementation of the PRC based methods [53 55], these structures are based on the inverse of the plant, which means their tracking accuracy is dependent on the accuracy of the available system model. Since in many practical applications the system could be subjected to various nonlinearities, uncertainties and external disturbances, an accurate estimate of the inverse of the plant is usually unavailable, unstable or difficult to accurately measure. This limits the use of these RC structures to systems with known dynamics. Furthermore, the design of a discrete RC also requires the period of the periodic signal to be known in advance and fixed. However, in many RC applications the period of the repetitive signal is time-varying. In such circumstances the signal generator becomes only an approximate model of the periodic signal to be tracked/rejected, which could lead to degraded tracking performance. Numerous robust RC methods have been proposed to deal with the limitations 17

44 2 Literature Review of RC in regards to robust performance and stability. The following section reviews the literature on robust RC with an emphasis on robust PLRC methods for improved tracking performance. 2.3 Robust Repetitive Control In many practical applications of the RC method the system could be subjected to various plant uncertainties, nonlinearities and non-periodic disturbances. Furthermore, in many situations the period of the periodic signal is unknown or varies over time [50, 104, ]. All of these factors could degrade the tracking accuracy of the system if not considered in the design of the overall RC structure. Motivated by these limitations, a vast number of robust RC methods have been proposed in the literature to overcome the stability robustness limitations of conventional RC. This section presents an overview of robust RC methods developed to cater for the needs of the different RC applications Robust Repetitive Control for Time-Varying Frequency Signals Several approaches have been proposed to solve the time-varying signal frequency problem in RC structures subjected to time-varying reference signals or uncertainties. This is especially important for discrete RC structures where the number of samples per period N is a known fixed integer value. The approaches can be grouped into two different frameworks. The first framework deals with methodologies that aim to preserve the sampling rate by either designing a high order RC [50, ] or by introducing a fictitious sampler that operates at variable sampling frequencies [121]. The second framework consists of approaches that try to adaptively change the sampling rate either by adjusting the sampling period T s or by adjusting the number of samples per period N [123, 124]. In [104] a robust repetitive compensator is proposed for compensating the parametric uncertainties caused by the variation in sampling time within known bounds. The proposed compensator design aims to compensate for the parameter variations when the system sampling time is adjusted in order to maintain the fixed ratio of samples per period N as required for perfect tracking of the repetitive controller. 18

45 2.3 Robust Repetitive Control The design of the robust compensator first requires the design of a nominal causal infinite impulse response (IIR) compensator F n (z) for guaranteeing stability of the RC system at the nominal sampling time. The robust compensator F r (z) is then designed by solving the following nonlinear optimisation problem min f r f n 2 (2.13) r 1r,...r 2r,q or,...q mr where the vectors f n and f r represent the coefficients of the nominal IIR controller F n (z) and the coefficients of the proposed robust compensator F r (z) respectively. Moreover, fn and f r can be represented as f n = [r 1 r 2... r m q 1 q 2 q m ] T (2.14) f r = [r 1r r 2r... r mr q 1r q 2r q mr ] T (2.15) Furthermore, the nonlinear constraints of the optimisation problem in (2.13) are defined as 1 + δ. 1 + δ < p 1R. p mr < 1 δ. 1 δ (2.16) h li < 1 0 < ω i < π T s (2.17) h ui < 1 0 < ω i < π T s (2.18) where h li and h ui represent the cost objective functions of the system at lower and upper sampling times, δ represents a small positive constant and p 1r,... p mr represent the real poles of the robust compensator F r (z) Robust Phase Lead Repetitive Control Non-periodic disturbances, unmodelled dynamics and nonlinearities are common in many high precision positioning systems where the tasks or disturbances 19

46 2 Literature Review are of a repetitive nature. Nonlinearities such as nonlinear hysteresis lead to inaccuracies in many piezoactuator based positioning mechanisms [125]. Furthermore, nonlinear friction is frequently found in many mechanical systems where there is relative motion between the mechanical components of the system [31, 34]. In high precision positioning systems the nonlinear friction nonlinearity is undesirable as it can limit the performance of the system, especially during low speed movements and reverse motions. External disturbances due to payload variations are also common in many repetitive applications, which include workpiece positioning in machining tasks [35, 36], pick and place operations performed by robotic manipulators [37, 38] and many others [39, 40]. The presence of such permutations could significantly degrade the tracking accuracy of servo systems if not accounted for by the design of the RC structure. A variety of different robust RC methods have been developed to address the stability robustness limitations of conventional RC. A modified continuous-time repetitive controller is proposed for eliminating harmonics in PWM voltage-source inverter systems in [126]. The modified repetitive controller consists of a zero magnitude and phase compensator and a lead-lag compensator to improve the stability and tracking accuracy at different load conditions. The structure of the proposed compensator is as follows G f (s) = 1 H(s) p s + p (2.19) where H(s) is the closed loop transfer function of the system and p represents the pole of the lead-lag compensator. Other continuous RC structures try to achieve an optimal trade-off between system stability and tracking performance by selecting a low pass filter and designing an optimal phase lead compensator to achieve the desired tracking bandwidth and stability robustness [127]. The low pass filter is designed as follows Q(s) = ω s + ω c (2.20) where ω c represents the cut-off frequency of the first-order low pass filter. The proposed phase lead compensator can be represented as G f (s) = k g(s + ω c ) αs + ω f (2.21) 20

47 2.3 Robust Repetitive Control where α is a small positive design parameter, while k g and ω f are the optimal RC gain and compensator cut-off frequency satisfying the RC stability conditions. Several discrete robust RC structures have also been designed for achieving improved robust tracking performance. A low-order causal discrete robust phase lead compensator is proposed in [104], where the stabilising robust RC compensator is developed as follows F (z) = q 0z m + q 1 z m q m z m + r 1 z m r m (2.22) The proposed causal RC compensator F (z) is obtained by solving the optimisation problem defined as min (h total ) (2.23) r 1,...r 2,q o,...q m while subject to the following nonlinear constraints 1 + δ. 1 + δ < p 1. p m < 1 δ. 1 δ (2.24) h i < 1 τ ω i = 2π i T s (2.25) where δ and τ are small positive constants, while p 1,... p m represent the real poles of the proposed causal compensator F (z). Furthermore, the cost objective function used to solve the optimization problem and obtain a stable compensator is defined as h total = L h i=0 i ω i = 2π i where i = 1, 2, 3..., L (2.26) T s where h i is a scalar function which represents the magnitude of the stability condition (2.27) at frequency ω, L = N/2 for even N, and L = (N 1)/2 for odd N. The robust stability of the proposed causal RC compensator is ensured through tuning the two positive constant parameters δ and τ. The positive constant δ defines the distance of the compensator poles from the unit circle. This parameter ensures 21

48 2 Literature Review that the compensator poles are within the unit circle and defines the safe stability margin. The second design parameter τ defines the margin between the closed loop poles and the unit circle. This parameter guarantees that the closed loop system will be stable within a pre-defined stability margin. (1 G c (z)f (s))q(z) < 1 τ (2.27) In the discrete PLRC control structures proposed in [38, 56 59] a pure phase lead compensator is used to improve the tracking accuracy and convergence rate in PWM inverter systems, as well as to widen the stability margins of the system. The PLRC structure is shown in Figure 2.8, where the stabilising phase lead compensator is designed as F (z) = k r z m (2.28) where z m is a discrete-time pure phase shift compensator and k r is the learning gain. Figure 2.8: Phase Lead Repetitive Controller In this compensator design the pure phase shift component z m, which provides a fixed phase lead angle θ F = m( ω ω N )180 where the maximum phase lead angle θ F = m 180 occurs at the Nyquist frequency ω N, is used to compensate for the phase lag in the closed loop system caused by the nonlinear load disturbances and parameter uncertainties. The improved phase compensation achieved by this compensator design results in improved tracking accuracy, faster convergence rate and a wider stability region. A robust stability condition for the control structures proposed in [38, 56 58] is derived as follows 0 < k r < 2 cos (θ G c (jω) + θ m (jω)) M Gc (jω) (1 + δ) (2.29) θ Gc (jω) + θ m (jω) < π 2 ε (2.30) 22

49 2.3 Robust Repetitive Control where M Gc and θ Gc are the magnitude and phase of the closed loop system G c respectively, θ m is the phase of z m, δ is a positive constant representing the upper bounds of the multiplicative uncertainty present in the system and ε is a positive robustness constant. Furthermore, in the control structure proposed in [128] the pure phase shift filter z m is also added in series to the low pass filter Q(z) to improve the tracking accuracy of scanning systems. As shown in Figure 2.9, the additional phase shift filter is included in the proposed control structure to increase the magnitudes of the loop gains at the harmonic frequencies of the periodic signal. This has the effect of improving the overall tracking accuracy of the system. Figure 2.9: Modified RC structure for improved performance in scanning devices Since conventional RC structures use a time delay consisting of one fundamental period T, they provide high loop gains at both the odd and even harmonic frequencies of the periodic signals. However, in many power system applications the periodic reference signal consists predominantly of odd harmonic components. In such applications the use of a conventional RC causes a sluggish transient response without improving the system s tracking accuracy. To improve the dynamic response of power systems with nonlinear loads a new discrete RC is proposed for compensating the (6n±1)th (n=1,2,3... ) harmonic components as follows [57] G rc (z) = k rc z N 6 z m 1 Q(z)z N 6 z m (2.31) where z N/6 is the odd-harmonic delay term, z m is a phase lead filter for improving robust performance, k rc is the RC gain and Q(z) is a low pass filter. Furthermore, a new robust repetitive control design method is proposed in [103]. In this control method the stabilising compensator is modified to ensure robust stability and zero tracking error. The proposed repetitive controller can be represented as 23

50 2 Literature Review R(z) = k gz m B(z) (2.32) 1 z N where k g is the learning gain, z m is a phase lead filter and B(z) is the proposed stabilizing controller. The transfer function of the proposed stabilising controller B(z) is given as B(z) = mi=1 1 a i z a i T c (z) (2.33) where T c (z) represents the transfer function of the closed loop system and a i represents the location of the i th compensator pole. RC structures based on the H robust design approach and the µ-synthesis technique have been proposed for improved stability robustness and improved robust tracking performance in [50 52]. In [52] a method which simultaneously optimizes the parameters of the low pass filter and the state feedback controller is developed. In this approach two LMI based robust stability conditions are derived to find the optimal cut-off frequency of the low pass filter and the state feedback gains. The first LMI based robust stability condition is used to obtain the maximum cut-off frequency of the low pass filter, the second to calculate the H state feedback gains. The proposed control scheme inherits the advantages of the optimal feedback controller and the RC and provides fast transient response and improved robust tracking performance. The repetitive controller is designed as G rep (z) = k gφ(z)q(z)z N+m 1 Q(z)z N (2.34) where Φ(z) is a stabilizing compensator, k g is the RC gain, Q(z) is a low pass filter, m is the number of plant delays, and N is the period of the fundamental frequency. A block diagram of the proposed scheme is given in Figure

51 2.3 Robust Repetitive Control Figure 2.10: Robust RC Structure based on µ- synthesis where K(z) represents the optimal feedback controller obtained using the µ-synthesis approach. Furthermore, a two parameter robust repetitive control (TPRRC) method is proposed for improved robust tracking performance in [100, 129, 130]. A block diagram of their proposed control structure is given in Figure 2.11 Figure 2.11: Two Parameter Robust RC where G(z) is the plant transfer function, q(z, z 1 ) is the low pass filter, while K 1 and K 2 are filters designed using the discrete-time µ-synthesis technique. 25

52 2 Literature Review A new RC scheme based on the disturbance observer (DOB) structure with improved loop-shaping properties is proposed in [10]. This RC configuration reduces the gain amplifications at the non-repetitive frequencies which results in improved robust performance in the presence of large non-periodic disturbances. A block diagram of this proposed RC configuration is given in Figure 2.12 Figure 2.12: New DOB based RC Scheme where P (z 1 ) is the discrete-time plant, C(z 1 ) is a stabilizing feedback controller, z m represents the relative degree of P (z 1 ), Pn 1 (z 1 ) is a nominal model of z m P 1 (z 1 ), and Q(z 1 ) is a zero phase low pass filter designed as ( 1 α N ) z (N m nq) Q(z 1 ) = 1 α N z N z nq q ( z, z 1) (2.35) where q(z, z 1 ) denotes the product of j number of zero-phase low pass filters q(z, z 1 ) = q j ( z, z 1 ) (2.36) Furthermore, n q represents the highest order of z in q(z, z 1 ), and α [0, 1] is a tunable parameter which allows flexible loop shape design as well as satisfactory transient performance. A non-causal phase lead finite impulse response (FIR) filter is proposed to compensate for the plant s phase lag and achieve an approximate zero-phase effect in [131]. The proposed compensator is designed as follows G f (z) = h 0 + h 1 z + h 2 z h n z n (2.37) where n is the order and h o,... h n are the coefficients of the proposed compensator. 26

53 2.3 Robust Repetitive Control A variety of different robust RC methods have been proposed in the RC literature. In [104] a robust RC structure is proposed for dealing with parametric uncertainties caused by sampling time variations. Although a robust compensator can be designed to deal with parametric uncertainties within known bounds, this proposed control method doesn t guarantee robust tracking performance. A low order robust IIR compensator can also be designed for dealing with plant uncertainties as described in [104]. Whilst this proposed control method can guarantee robust stability, the relationship between the robustness parameter τ and the system uncertainty is not clearly stated. Several robust RC structures based on the D-K technique [50 52] have been designed for achieving optimal robust tracking performance. Despite the optimal tracking properties of these controllers, obtaining a stable compensator using this design approach is not always feasible for the given uncertainty weighting functions. Adjustments to the initially selected performance weighting functions are often required in order to obtain a stable compensator. However, this could lead to degraded robust performance in many systems with substantial plant uncertainties. Furthermore, in many instances the resulting compensator is of a high order which makes the implementation of such a controller difficult in practice. Moreover, the D-K algorithms are complex and often difficult to realise in software. To overcome the limitations of RC methods that are based on the inverse of the plant, the pure PLRC concept has been developed [38, 56 59]. Unlike methods based on the inverse of the plant, the proposed compensator is model free and consists of a pure phase lead compensator z m for dealing with phase uncertainties in different RC applications. Although the phase lag present in the closed loop system caused by the phase uncertainties can be effectively compensated with the selection of an appropriate value for m, the phase lead angle provided by the pure phase lead compensator z m is a fixed value. In many cases this can overcompensate the phase lag introduced by the phase uncertainties. This can significantly reduce the ±90 deg crossover frequency, decrease the stability margin of the RC system and consequently degrade the tracking performance of the system or cause instability. Due to these limitations of the pure PLRC, this method is often considered to be inflexible and limited to applications that require fixed phase shift compensation. Motivated by these limitations in existing robust RC structures, Chapter 4 of this thesis proposes the design of two optimal robust PLRC methods. Firstly an optimal discrete-time PLRC structure is proposed to correct the overcompensation caused by a pure phase lead compensator [38, 56 59]. Secondly a simple and effective design of a robust phase lead compensator is presented in the frequency domain for a 27

54 2 Literature Review more flexible and effective compensation of phase lag present at different frequencies in the RC bandwidth. 28

55 2.4 Sliding Mode Repetitive Control 2.4 Sliding Mode Repetitive Control The standard RC structure based on the IM is rarely used alone as it limits the closed loop bandwidth of the system, causing a sluggish system response [48]. Unmodelled system dynamics, nonlinearities and various uncertainties which are common in many systems used in repetitive applications are some additional factors which can limit the tracking performance and reduce the convergence rate in RC systems. To achieve improved tracking performance and robustness in the presence of large bounded matched uncertainties, as well as fast dynamic response, RC is often coupled with variable structure control (VSC) methods such as the sliding mode control (SMC) and the terminal sliding mode control (TSMC) for applications that require fast dynamic response. An overview of the SMC fundamentals and existing SMC based RC structures is provided in the rest of this section. The fundamentals of the discrete-time VSC and SMC method were first introduced by Gao et al. in [132], where the design steps and stability analysis of discrete sliding mode control (DSMC) were also presented. The design of a discrete SMC structure is similar to that of continuous-time sliding mode and can be summarised in two steps. In the first step a sliding surface is designed so that the sliding mode exhibits the desired system dynamics. A linear sliding mode surface is often chosen for discrete-time systems as follows s(k) = c T x(k) (2.38) where c = [c 1 c 2 c 3... c n 1 1] are constants that determine the sliding surface dynamics. In the second step a continuous or discontinuous control action is selected to force the system states to reach and stay within a small region around the sliding surface known as the quasi sliding mode band (QSMB). The typical exponential reaching law for discrete-time systems can be represented as s(k + 1) s(k) = qt s(k) αt sgn[s(k)] (2.39) where α > 0, q > 0, 1 qt > 0, T represents the sampling period, while α and q determine the speed of convergence to the sliding surface and the exponential of convergence respectively. 29

56 2 Literature Review follows Meanwhile the discrete-time terminal sliding mode surface in [133] is selected as s(k + 1) s(k) = αx(k) + βx q/p (k 1) (2.40) where x(k) represents the system states, α > 0, β > 0, while p and q are positive odd constants. Furthermore p > q. The stability of discrete-time SMC can also be analysed using the Lyapunov stability theory where the Lyapunov function is usually selected as follows V (k) = 1 2 s2 (k) (2.41) According to the Lyapunov stability theory for discrete-time systems, asymptotic convergence of the system states to the QSMB is achieved when V (k) = s 2 (k + 1) s 2 (k) < 0 (2.42) From the above condition, the discrete-time sliding-mode condition is derived as follows s(k + 1) < s(k) (2.43) Moreover, s(k + 1) < s(k), s(k) > 0 (2.44) s(k + 1) > s(k), s(k) < 0 (2.45) represent the sliding-mode conditions that must be satisfied for achieving sliding mode. Despite the design similarities, and unlike continuous SMC, the control law in discrete SMC is updated at regular intervals of time as defined by the sampling frequency [134, 135]. Moreover, the switching in discrete SMC occurs at every 30

57 2.4 Sliding Mode Repetitive Control sampling interval which constrains the system trajectories to stay in a close region around the sliding surface instead of allowing them to slide on the sliding surface. Therefore, discrete-time SMC systems may not exhibit the desired invariance and robustness properties of continuous-time SMC structures. Although discrete sliding mode controllers can be designed to achieve the desired invariance and robustness properties of continuous SMC, achieving good tracking performance is also a very important design aspect which needs to be considered in the design of discrete SMC for high precision applications [ ]. Various system nonlinearities could also reduce the tracking bandwidth and hence limit the tracking accuracy of discrete SMC based systems, especially if they are unknown or cannot be accurately measured. It is well known that feedforward controllers such as the repetitive controller can beused to improve the tracking accuracy of the system at steady state. Whilst the RC method provides perfect tracking/rejection of periodic signals, it is characterised by a slow convergence time [48], especially in the presence of uncertainties. To improve convergence time and robustness to plant uncertainties and large non-periodic disturbances, RC and SMC are usually coupled together to achieve fast transient response, improved tracking performance and robustness. Several discrete SMC based RC structures which have the merits of both RC and SMC have been proposed in the recent literature [5, 60 68]. Moreover, in the discrete-time SMRC strategy proposed in [11] a repetitive reaching law is proposed as follows s(k + 1) = (1 qt ) s(k N + 1) εt sat [ ] s(k N + 1) δ (2.46) where the existence of the quasi-sliding mode and robust stability is guaranteed by the following condition s(k + 1) = s(k N + 1) (2.47) Furthermore, a discrete-time VSC based RC structure for rejection of periodic disturbances that arise from the tracking of periodic reference signals is proposed in [60]. The following switching function is used in the design of their proposed controller s k = C(q 1 ) e k (2.48) 31

58 2 Literature Review where C(q 1 ) represents the coefficients of a Schur polynomial. Furthermore, the following decay reaching law is used s k+d = (1 ρ) s k εsat [ ] sk δ (2.49) while the total control law is derived as u k = u k d [ G ( q 1)] 1 [ F ( q 1) (e k e k d ) + ρs k + εsat [ ]] sk δ (2.50) where G(q 1 )and F (q 1 ) are matrices of appropriate dimensions. The robust stability of the quasi-sliding mode dynamics is guaranteed by the delay repetitive reaching condition which is defined as s k+d = s k (2.51) A discrete-time SMRC structure with a disturbance estimator is proposed for eliminating periodic disturbances in optical disk drives in [5]. This proposed control discrete-time SMRC law is equivalent to u(k) = u eq (k) + v(k) (2.52) where v(k) is an estimate of the disturbance d(k). Moreover v(k) = d(k). Since d(k) is a periodic disturbance signal, the value of d(k) is assumed to be equivalent to the value of its previous period d(k N). Moreover, v(k) = d(k) = d(k N) (2.53) where d(k N) = ( c T b ) 1 [ c T x (k N + 1) c T Ax (k N) ] u (k N) (2.54) A disturbance observer based SMRC scheme is also proposed in [61]. In this control structure a feedback controller based on pole placement is designed to achieve 32

59 2.4 Sliding Mode Repetitive Control the desired error dynamics. Moreover, a sliding mode disturbance observer is used to ensure robustness, while the RC component is added to improve the tracking of periodic reference signals. The total control law proposed is as follows u = u pa + u do + u i ff (2.55) where u do represents the disturbance observer compensation, u pa represents the poleplacement control law, while u i ff denotes the repetitive control law during the i th iteration. The repetitive control law is designed as follows u i+1 ff (s) = β(s)ui ff(s) + γ(s)u i sw(s) (2.56) where u i sw is a switching input at the i th iteration, and β(s) and γ(s) are low pass filters designed to improve the robustness against high-frequency unmodelled dynamics and noise. Another SMRC scheme is proposed for tracking periodic references and rejecting periodic disturbances in a brushless direct current (DC) motor used in a simulation turntable system [62]. This SMRC scheme is based on the integral SMC and has improved reference tracking and robustness capabilities under periodic disturbances and parameter uncertainties. A SMRC structure based on an integral sliding mode perturbation observer is also proposed in [63]. In the proposed control scheme, a pole-placement feedback control is used to achieve the desired error dynamics, the perturbation observer component guarantees the robustness, while the RC part is included to improve the tracking accuracy when tracking periodic signals. A repetitive VSC method is proposed for improving the accuracy of microactuators in [64]. In the proposed control structure the repetitive control law introduced to cancel the period disturbance d(t) is designed as follows u l (t) = û l (t T f ) + βσ(t) (2.57) where β is the repetitive learning rate, σ(t) is the sliding function, while û l (t T f ) is defined as 33

60 2 Literature Review û l (t T f ) = proj (u l (t T f )) (2.58) with û l (t T f ) U m, where U m denotes the upper limit of the disturbance d(t). A hybrid SMRC scheme is proposed for tracking periodic references in a PWM voltage inverter in [65]. Here the proposed control method is based on the sliding mode PID structure where an equivalent sliding mode control law is derived to control the inverter. A PLRC is also proposed to guarantee overall system stability as follows G f (s) = 1 H(s) p s + p (2.59) The conventional SMC control law is modified to deal with mismatching uncertainties in [66] where a quasi-sliding mode RC is proposed for improving the rejection of unmatched uncertainties and external periodic disturbances in a nonlinear continuous time MIMO system. The discrete SMRC concept has also been extended to multivariable systems [67, 68]. Moreover, in [67], a discrete multivariable SMRC structure is proposed for rejecting periodic and multi-periodic disturbances where a new reaching law is designed to drive the system states on the sliding surface and maintain them without chattering. Meanwhile, a discrete SMRC structure for dealing with periodic disturbances in non-decouplable multivariable systems is proposed in [68]. In this approach the dieophantine polynomial matrix equation is modified to include the repetitive components, allowing for effective cancellation of periodic disturbances. Several discrete SMRC methods have been proposed for fast transient response, improved robustness and improved steady state tracking performance [5, 60 68]. Although the proposed methods achieve the desired performance specifications, they are developed in time-domain which makes the design of an optimal SMRC very challenging. Furthermore, in repetitive applications which require fast transient response, a VSC method such as the TSMC is required in order to achieve the desired transient speed. Given these limitations and gaps in the existing SMRCs, Chapter 5 of this thesis presents the design of two new discrete SMRC methods. In our proposed control approach a SMC component is added to achieve the desired transient response and robustness to parameter uncertainties while the RC is used to eliminate the repeatable uncertainties as well as to provide asymptotic convergence of the tracking error 34

61 2.4 Sliding Mode Repetitive Control to the sliding surface. For the first design an optimal discrete SMRC is proposed in the frequency domain, where the standard RC frequency domain design methods and the describing function (DF) stability analysis approach are applied to the design of the optimal controller parameters. Secondly a new discrete TSMRC is designed for improved tracking performance and fast transient response. 35

62 2 Literature Review 2.5 Extended State Observer Based Repetitive Control Many servo systems used in repetitive applications are often exposed to different uncertainties which can significantly degrade the system s tracking performance. Several different robust control structures have been used in combination with the RC method to ensure the desired robust tracking performance of servo systems in the presence of uncertainties. Although robust control methods such as the SMC algorithms discussed in the previous section can be successfully applied for achieving the desired tracking performance, they require an accurate estimate of the plant dynamics which is not always possible to obtain. In applications where the plant dynamics, uncertainties or nonlinearities cannot be accurately modelled and measured, the RC scheme can be integrated with different observer based control structures such as the disturbance observer (DOB) [61], sliding mode observer [84, 139, 140], equivalent input disturbance (EID) estimator [ ] and others [145]. Whilst a variety of different type of observers can be used to estimate and eliminate the system uncertainties and disturbances, many of these observer structures are also model based. Furthermore, many of these observer methods are only effective in estimating the system uncertainties and cannot estimate the internal system states even though this may be desirable in many applications where the system states cannot be directly measured by sensors. In applications where the plant dynamics or nonlinearities cannot be accurately modelled and measured, the active disturbance rejection control (ADRC) method based on an extended state observer (ESO) is often applied [69]. The ADRC concept using a nonlinear extended state observer (NLESO) and a nonlinear feedback controller was first introduced by Han in [69]. To reduce the design and computational complexity of the original NLESO based ADRC proposed by Han, a linear ADRC structure consisting of a linear feedback controller and a linear extended state observer (LESO) was proposed by Gao et al [146]. Due to its great robustness and tracking properties the ESO based ADRC framework has many practical applications. In [147] an ESO based ADRC is applied to a DC motor used in motion control applications. A generalised extended state observer (GESO) method for dealing with skidding and slipping disturbances acting on a wheeled mobile robot is proposed in [148] while a NLESO is used for dynamic compensation of hysteresis in [149] and nonlinear friction in [72, 150, 151]. 36

63 2.5.1 ESO Based Repetitive Control 2.5 Extended State Observer Based Repetitive Control Although the ESO structure has good uncertainty estimation properties, it has limited disturbance rejection capabilities for periodic uncertainties and disturbances [91]. Consequently, in many high precision positioning applications used in repetitive mode the ESO framework alone cannot fully compensate for the periodic disturbances acting on the system. Achieving the desired tracking accuracies in such circumstances would require the design of an ESO based RC structure that is capable of achieving fast transient response and improved tracking performance in the presence of repeatable and non-repeatable uncertainties. Several ESO based RC structures have been proposed in the literature for solving the tracking problem in applications of a repetitive nature [73, 92 97]. In [92] a continuous-time LESO based RC structure with improved friction compensation is proposed for a flight simulation table. A block diagram of the proposed structure is shown in Figure Figure 2.13: Continuous ESO based RC Furthermore, a continuous repetitive control structure based on a periodic adaptive ESO has been developed for an uncertain permanent-magnet synchronous motor (PMSM) in [73]. This control scheme consists of two ESO structures, a normal and a refined ESO. The normal ESO is first designed to perform a rough estimation and rejection of the total disturbances acting on the system. In the second stage, the refined ESO is proposed to estimate and eliminate the remaining disturbances acting on the system. This control method also consists of a feedback and a feedforward component, where the feedback part is required to ensure system stability, while the feedforward component, which consists of a repetitive control law, ensures improved steady state tracking performance. Likewise in [93] a state feedback RC structure with a GESO is proposed. In the proposed control structure the GESO is used to estimate and eliminate the input- 37

64 2 Literature Review dead-zone nonlinearity and external disturbances acting on the system, while the RC component is used for tracking periodic reference. The LMI and pole placement approaches are used to obtain the state feedback and observer gains respectively. Several discrete-time ESO based RC structures have also been proposed. A nonlinear control structure based on a LESO based RC method is proposed for an active power filter in [94]. In the proposed control structure a discrete-time RC is coupled with the continuous nonlinear ADRC framework for improved static and dynamic performance. In [95] a continuous ESO based RC is proposed for dealing with parameter uncertainties and periodic and aperiodic disturbances. Again a model free discrete ESO based RC is proposed for improved rejection of periodic and aperiodic disturbance and compensation for parameter uncertainties in [96], while a discrete ESO RC based on a predictive discrete estimator [152] is proposed in [97]. Considering the discrete space model given below, x d (k + 1) = Φ d x d (k) + Γ d u d (k) (2.60) y d (k) = H d x d (k) + J d u d (k) (2.61) the discrete ESO in [97] which is based on the a predictive discrete estimator [153] can be represented as follows ˆx d (k + 1) = Φ dˆx d (k) + Γ d u d (k) + L d (y d (k) ŷ d (k)) (2.62) ŷ d (k) = H dˆx d (k) + J d u d (k) (2.63) From (2.62) given above it is evident that the present estimation error y d (k) ŷ d (k) is used to estimate the next ˆx d (k + 1). By representing the predictive estimator gain vector as, L d = Φ d L c (2.64) the estimated state in (2.62) can be represented as 38

65 2.5 Extended State Observer Based Repetitive Control ˆx d (k + 1) = Φ d x d (k) + Γ d u d (k) (2.65) where x d (k) = ˆx d (k) + L c (y d (k) ŷ d (k)) (2.66) Furthermore, the control law consisting of both the ESO and RC is described as follows u(k) = u 0(k) ˆx 3 (k) b 0 (2.67) where u 0 (k) is defined as u 0 (k) = K 1 c(k) + K 1 (r(k) ˆx 1 (k)) ˆx 3 (k) (2.68) while, K 1 is a feedforward gain and K 2 is a state feedback gain ESO Based Sliding Mode Control The ESO structure has been successfully coupled with many SMC methods for more effective estimation and elimination of unknown disturbances, uncertainties and nonlinearities [75 90]. In [75] an ESO based second order sliding mode (SOSM) controller is proposed for three-phase two level grid-connected power converters. The control structure consists of two loops, where the current control loop consists of a SOSM, while the voltage regulation loop consists of the ESO based SOSM structure. A LESO based SMC scheme coupled with a hysteresis compensator based on a Bouc-Wen model is proposed for improving the dynamic performance and robustness of a piezoactuator in [76]. An ESO based double loop integral SMC scheme is also proposed for estimating the throttle opening angle changes and disturbance in an electronic throttle valve with gear backlash torque in [77]. Here the proposed integral SMC structure consists of two loops. The inner loop is based on the throttle opening angle changes, while the outer loop is based on the throttle opening angle disturbance. 39

66 2 Literature Review In [78] the LESO is extended to a sliding mode extended state observer (SMESO) as shown in Figure The proposed SMESO is used to estimate the internal states and lumped disturbance in a permanent magnet DC (PMDC) motor. Moreover, chattering is reduced and the estimation accuracy of the proposed observer is improved by replacing the nonlinear f al(e, α, δ) component of the conventional NLESO with the switching term e sign(e). As shown in Figure 2.14, this control structure also consists of a nonlinear state error feedback (NLSEF) controller which is designed to ensure the desired tracking performance. A tracking differentiator (TD) is also added to provide smooth approximation of the system states. Figure 2.14: ADRC based SMESO structure An ESO based SMC with a new exponential reaching law as described in (2.69) is proposed for reducing chattering in a PMSM in [79]. s = f(x, s)sign(s)ẋ 1 + αx 1 + βx q/p 1 (2.69) where f(x, s) = k x ε + (1 ε)e η s (2.70) while, k > 0, η > 0, 0 < ε < 1, e is the tracking error and x is the system state. A LESO based SMC method for uncertain square multivariable nonlinear systems with unknown system states is proposed in [80]. Other applications of ESO based SMC structures extend to PMDC motors [81], differential-driving mobile robots [82], four wheel drive electric vehicles [83] and reusable launch vehicles [84]. Several continuous ESO based TSMC structures have also been developed for fast convergence. In [85] an ESO based TSMC structure is proposed for reducing chattering in PMSMs. Furthermore a non-singular fast TSMC with an ESO and a TD is proposed for a chaotic PMSM with uncertainties in [86], while a non-singular 40

67 2.5 Extended State Observer Based Repetitive Control fast TSMC with an ESO and a TD is proposed for a class of uncertain second-order SISO nonlinear systems for improved tracking accuracy and improved suppression of chattering in [87]. Despite the simple implementation and merits of ESO based ADRC structures, the standard ESO estimator, just like many other observer-based structures, is only effective when dealing with matched disturbances (i.e disturbances acting on the same channel as the control input) and is insensitive to mismatched disturbances. To improve the performance of a DC-DC buck converter system with both matched and mismatched disturbances an ESO based SMC structure is proposed in [88]. A sliding mode based ESO design for dealing with mismatched uncertainties is also proposed in [89], while an ESO based SMC for a MIMO system with multiple matched and mismatched disturbances is proposed in [90] Summary of ESO Based RC and ESO Based SMC This section provides an overview of a variety of existing ESO based RC structures [73, 92 97] which are designed to achieve satisfactory tracking performance in the presence of unknown repeatable and non-repeatable uncertainties. Although ESO based RC structures can provide satisfactory tracking performance in the presence of periodic and aperiodic uncertainties, it is well known that the ESO approach is only robust for a small range of parameter uncertainties. Furthermore, ESO based SMC schemes have also been developed for improved robustness in applications where the uncertainties and nonlinearities present in the system are unknown and cannot be easily measured [75 90]. Although the integration of ESO and SMC improves the robustness of the system, the ESO structure cannot fully eliminate the effects of the periodic disturbances acting on the system [91]. Because of these limitations in existing ESO based RC and ESO based SMC methods, Chapter 6 of this thesis proposes the design of two new NLESO based structures. The first structure is a discrete NLESO based TSMC structure for improved robust tracking performance in uncertain LA systems. Here the NLESO framework is used to estimate and reject the adverse effects of unknown plant uncertainties and nonlinear friction. In the second approach RC is added to the NLESO based TSMC to eliminate the periodic components of the nonlinear friction, which are commonly found in many LA servo applications of a repetitive nature. 41

68 2 Literature Review 2.6 Summary This chapter provides an overview of the literature on robust RC, SMRC and ESO based structures, covering a wide range of control methods designed for dealing with different uncertainties and disturbances in applications of a repetitive nature. The chapter is organised in four sections. Section 1 presents an overview of the RC method, with a special emphasis on the stability robustness limitations of the conventional RC. Section 2 provides an overview of existing robust RC structures with a special focus on robust PLRC for improved tracking performance. Numerous PLRC structures have been proposed for improved tracking performance in systems with phase uncertainties. The robust RC structures based on the D-K technique [50 52] can achieve stability robustness and improved robust tracking performance. However, the resulting compensator is often of a high order and difficult to realise in software. Likewise, the inverse of the plant based controllers in [38, 56 59] require an accurate model of the plant, which is not always stable and/or possible to obtain, while the robust PLRC methods based on the pure phase lead compensator z m are inflexible and can often overcompensate the phase lag caused by the phase uncertainties. To overcome these limitations of existing robust PLRC, Chapter 4 of this thesis proposes the design of two optimal robust PLRC methods. Initially a discrete-time robust PLRC structure is proposed to correct the overcompensation caused by the pure phase lead compensator z m. In the second approach a robust PLRC design is proposed in the frequency domain for a more flexible phase lag compensation. Section 3 provides an overview of SMRC methods [5, 60 68]. Although existing SMRC schemes can achieve the desired robustness and tracking performance requirements, they are developed in time domain which makes the task of designing an optimal SMRC quite challenging. Moreover, the design of a discrete TSMRC has not been adequately represented in the current literature. Motivated by these research gaps, Chapter 5 of this thesis proposes the design of two new discrete SMRC methods. Firstly an optimal discrete SMRC design is proposed in the frequency domain. Secondly a new discrete TSMRC method is proposed for fast transient response and improved tracking performance. Section 4 of this chapter presents the review of the ESO based ADRC framework with a special emphasis on ESO based RC and ESO based SMC methods. A number of ESO based RC structures have been proposed to eliminate the undesirable effects of periodic uncertainties [73, 92 97]. However, these control schemes have limited 42

69 2.6 Summary robustness against other plant uncertainties and disturbances. Numerous ESO based SMC methods have also been proposed [75 90]. Even though these structures can achieve the desired robustness properties in systems with unknown uncertainties, this framework has limited compensation capabilities when dealing with periodic uncertainties. Motivated by these limitations of the ESO based SMC and ESO based RC methods, Chapter 6 of this thesis presents the design of a discrete NLESO based TSMC and a new NLESO based TSMRC for improved tracking of periodic and aperiodic references. 43

70

71 3 Experimental Setup and System Modelling 3.1 Introduction The effectiveness of the robust RC methods proposed in this thesis is verified through both simulation and experimental testing. Initially, the efficacy of the proposed control methods was demonstrated through simulation. Although simulation software such as Matlab & Simulink can provide a quick validation of the proposed controllers, the real-time experimental conditions are often difficult to be reproduced using computer simulations. Therefore, the effectiveness of the controllers proposed in this thesis is also demonstrated through hardware testing using an experimental setup. Due to the importance of experimental testing, this chapter provides a detailed overview of the hardware and software components of the LA experimental setup used in this thesis. Moreover, Section 2 provides an overview of the overall LA experimental system, Section 3 describes the hardware components of the system, including the LA plant dynamics and uncertainties. While, the details of the software components used in the experimental testing are provided in Section 4. A short summary of this chapter is provided in Section System Overview The LA experimental setup used to verify the effectiveness of the controllers proposed in this thesis consists of the following components: 1. Host PC: the computer that hosts the software used to develop the models and generate the executable code, as well as the software used for real-time monitoring of the hardware. 45

72 3 Experimental Setup and System Modelling 2. dspace 1103 controller: this is the computer that runs the executable control code generated by the host PC and provides the control action to the controlled plant. 3. Voltage-to-current amplifier: this hardware is used to filter and amplify the control signal from the controller before it is transmitted to the controlled plant. 4. LA plant: this includes the hardware to be controlled. A figure of the linear drive system, as well as a block diagram describing the LA experiment setup are provided in Figure 3.1 [30] and Figure 3.2 respectively. As seen from Figure 3.1, the LA system consists of a DC motor which is driven through a voltage-to-current power amplifier. The travel range of the DC drive system is 50cm. An optical encoder is used to provide absolute position measurements of the system output with a resolution of 1µm. As depicted in the block diagram of the experimental setup given in Figure 3.2, a host PC that is coupled with a dspace controller is used to generate the control input for the LA system. The control signal is then transmitted to the linear drive system through an I/O module and a voltage-to-current amplifier. Real-time monitoring of the system output and tuning of the controller parameters can be achieved via the software running on the host PC. 46

73 3.2 System Overview Figure 3.1: Linear Actuator System Figure 3.2: Block Diagram of the Linear Actuator Experimental Setup 47

74 3 Experimental Setup and System Modelling 3.3 System Hardware dspace 1103 Controller and I/O module As shown in Figure 3.2, a dspace controller is used to provide the desired control signal to the controlled plant. To achieve this, the dspace DS1103 controller board is added to the PCI slot of the PC to allow for rapid control prototyping. The dspace based experimental system used in this research represents a hard real time system, which is ensured by the dspace microcontroller system. Moreover, the dspace system can guarantee sufficient sampling accuracy for the controllers designed in this thesis. This means that the timing latency and jitter of the dspace controller are negligible in comparison to the desired sampling period adopted for the designed controllers Voltage-to-Current Power Amplifier As depicted in Figure 3.2, the signal generated by the dspace 1103 controller is passed onto a voltage-to-current power amplifier (by Renishaw PLC) through the I/O module before it gets transmitted to the controlled plant. The task of the voltage-to-current power amplifier is to filter and amplify the control signal before it reaches the linear motor (LM) Linear Actuator Plant and Uncertainty A DC motor by Baldor Electric is used to drive the linear drive system on the bearing rail as shown in Figure 3.1. The dynamics of the second order LA plant can be represented as follows [30] ẍ = bu g(x) d(x, t) (3.1) where u represents the control force acting on the linear actuator, x is the position output, b represents the control gain, g is a nonlinear function representing the nonlinear friction present in the system and d represents the total lumped disturbance and uncertainty present in the system. 48

75 3.3 System Hardware The nonlinear friction force g present in the linear actuator can be represented as follows g(v) = k vf v + k cf sgn(v) + k sf sgn(v)e v σ sf + f (3.2) where v is the velocity state of the system, k vf is the viscous friction coefficient, k cf is the Coulomb friction coefficient, k sf is the static friction coefficient, σ sf is the static friction velocity constant and f represents the unmodelled friction force. The parameters of the LA plant in (3.1) and the nonlinear friction in (3.2) are as given in Table 3.1 [30]. Parameter Value b N/kgV k vf 8.6 N s/m k cf 11.5 N k sf 3.3 N σ sf m/s Table 3.1: LA System Parameters As seen from (3.2), the nonlinear friction uncertainty consists of both linear and nonlinear terms. Furthermore, as given in (3.2), this uncertainty is a function of the system velocity which is a derivative of the system position. This means that in the presence of periodic references the friction nonlinearity consists of both periodic and nonperiodic components, where the periodic components have the same period as the reference signal and can be effectively eliminated with the design of a repetitive controller. The LA plant in (3.4) can also be represented in the frequency domain using the nominal open loop transfer function G p (jω) and the total additive uncertainty a (jω) as described below G(jω) = G p (jω) + a (jω) (3.3) where a (jω) γ and G p (s) = 1/b (jω) 2. The upper additive uncertainty bound denoted by γ can be determined by plotting the magnitude of a (jω) which can be obtained by calculating the difference between the magnitudes of the perturbed and nominal open loop system. Similarly, the phase of a (jω) can be obtained by calculating the difference between 49

76 3 Experimental Setup and System Modelling the phases of the perturbed and the nominal open loop system. The magnitude and phase plots of the total additive uncertainty a (jω) caused by the nonlinear friction and a 6kg payload added to the LA system are shown in Figure 3.3, where the maximum uncertainty bound measured from the magnitude plot is Figure 3.3: Magnitude and phase plots of the measured additive uncertainty Furthermore, the continuous state space representation of the nonlinear LA plant in (3.1) can be expressed as follows ẋ(t) = Āx(t) + Bu(t) g(x) d(x, t) (3.4) y(t) = Cx(t) (3.5) where Ā, B and C are matrices of suitable dimensions. A discrete state space representation of the LA system can be obtained by digitising the plant (3.4) using a sampling time interval T s x(k + 1) = Ax(k) + Bu(k) g(k) d(x(k), k) (3.6) y(k) = Cx(k) (3.7) 50

77 3.4 System Software where A = eāts = T s (3.8) B = B Ts 0 eāτ u(k)dτ = bt s T s 2 1 (3.9) C = [ 1 0 ] (3.10) Since the desired closed-loop bandwidth of the LA system is approximately 500Hz, as a rule of thumb, the optimal sampling rate should be less than 10 times the closed-loop bandwidth ( 1 T s < 5kHz). Therefore, the sampling rate used in the design of the controllers proposed in this thesis is 2kHz. The sampling accuracy of the proposed controllers using the selected sampling period is guaranteed through simulation. 3.4 System Software DC Motor Driver Software The LM driver needs to be initialised before the start of the experimental testing. Furthermore, the DC motor parameters also need to be calibrated before the control system can be tested on the hardware. This can be achieved by running the initialisation (autotune) routine available in the ABD Motor Driver software (ABB Mint WorkBench) as shown in Figure 3.4. This routine needs to be executed at the start of each experiment to calibrate the LM parameters and position, as well as to set the correct control mode required by the motor driver. The control mode option can be set to either voltage or current mode. Furthermore, the feedback mode also needs to be selected to enable feedback measurements from the optical encoder. 51

78 3 Experimental Setup and System Modelling Figure 3.4: ABB Mint Workbench Autotune Software Software for Controlling the Plant The steps required for running the control code on the experimental setup are as follows: 1. Developing the Model: The Simulink environment is used to create the models of the different controllers proposed in this thesis. Each model is created by selecting the appropriate blocks, setting up the block parameters and connecting the blocks together. In addition to the standard Simulink blocks, a number of I/O blocks from the Real-Time Interface (RTI) toolbox are also used to connect the model with the controlled hardware and the encoders. 2. Building the Model : The Simulink build option is then used to build/compile the models and convert the generated C code into executable assembly code for the DS1103 controller board. This process is achieved through the Microtec C compiler which is used by Matlab/Simulink to generate executable code for dspace controllers. 3. Running the Real-Time Code: The dspace Control Desk software is used to establish a connection between the host PC and the controlled hardware. Once the connection is established, the compiled code can be executed on the controlled hardware in real time. Moreover, the graphical user interface capabilities of the dspace Control Desk software allow monitoring of the controlled system as well as the tuning of its parameters in real time. This software also allows capture of the system output for further processing and 52

79 3.4 System Software analysing. The dspace Control Desk layout used in the experimental testing is as shown in Figure 3.5. Figure 3.5: Control Desk Layout 53

80 3 Experimental Setup and System Modelling 3.5 Summary This chapter provides an overview of the LA plant as well as the hardware and software components of the LA experimental setup used for verifying the effectiveness of the controllers proposed in this thesis. The LA experimental setup consists of the following hardware components: a host PC that is coupled with a dspace DS1103 controller, an I/O module, a voltage-to-current amplifier, and a linear drive system consisting of a DC motor and an optical encoder for absolute position feedback. The models of the controllers proposed in this thesis are developed in Simulink using the RTI toolbox which allows the models to be connected with the controlled hardware and the sensory devices. The executable code generated from the Simulink models is then used to control the hardware via the dspace Control Desk software. The Control Desk software also allows real-time monitoring and tuning of the controller parameters. 54

81 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance 4.1 Introduction Linear actuators are used in many industrial high precision applications of a repetitive nature which include belt-drive systems [154], machine tool feed drives [155] and many others [98, 156]. The growing need for improved accuracy in many high-precision servo systems imposes the demand for LA stages with improved tracking performance and stability in the presence of nonlinearities such as nonlinear friction [157] and payload variations [35, 36] which are common in many high precision positioning applications. The RC method is a well-known control technique used for achieving improved tracking accuracy in applications of a repetitive nature [44]. The simple structure and implementation of RC is why this control paradigm has been successfully implemented in many applications of a repetitive nature. Today the RC method is used in electric power systems [38, 56, 57, 113], as well as in many high precision positioning systems such as the ones used in memory storage devices [99], robot manipulators [158, 159] and many others [98, 160]. Although the RC method can achieve good tracking accuracies at steady state, it is sensitive to non-periodic disturbances and uncertainties. A variety of different robust RC methodologies have been proposed to improve the stability robustness of the RC method, many of which are based on the inverse of the plant based structures such as the PRC [53 55, 110]. Although these structures have a simple implementation, they require a precise estimation of the inverse of the plant which sometimes cannot be easily obtained due to the existence of unknown uncertainties. Furthermore, the design of a compensator using this methodology could be infeasible 55

82 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance if the inverse of the discrete plant is unstable due to the close proximity of the plant zeros to the unit circle [161]. To overcome the limitations of RC methods based on the inverse of the plant a number of PLRC structures have been proposed for improved robust tracking performance and fast transient response [56, 57, 104]. These structures introduce an additional phase lead angle to compensate for the phase lag in the feedback system caused by the phase uncertainties. A causal RC structure for dealing with bounded parametric uncertainties due to sampling time variations is proposed in [104], while PLRC structures using a pure phase lead compensator are presented in [38, 56 59]. In the latter a phase lead compensator in the form k rc z m is used to reduce the phase lag in the feedback system caused by the plant uncertainties, predominantly at high frequencies. Furthermore, in the robust PLRC structures proposed in [105, 113, 128] the pure phase lead term k rc z m is added in series to the inverse of the plant transfer function to improve the overall phase compensation in the presence of uncertainties at high frequencies, as well as to further improve the tracking capabilities of the system. Although these control structures are easy to design and implement, the inclusion of z m adds a large fixed phase lead angle contribution at high frequencies which can overcompensate the phase lag in the closed loop system introduced by phase uncertainties at low to middle frequencies. This could lead to a reduced stability region and degraded tracking performance. Furthermore, these structures are inflexible and cannot effectively compensate phase lag in the feedback system at different as well as at multiple frequencies within the RC bandwidth. This contributes to a reduced stability margin and degraded tracking performance when the system is exposed to phase uncertainties distributed at multiple frequencies in the RC bandwidth. To address the limitations of the conventional discrete-time PLRC proposed in [57, 83, 105, 113, 128] this chapter presents the design of two optimal robust PLRC methods. The first optimal robust PLRC aims to correct the overcompensation caused by the conventional pure phase lead compensator z m to give good phase cancellation over a wider frequency range, leading to improved tracking performance of the system. To achieve this, an optimal PLRC is first designed following the optimisation technique presented in [104]. A robust stability condition is then derived to ensure robust stability and improved tracking performance when this controller is applied to a LA system with bounded nonlinear friction and load uncertainties. The second method proposed in this chapter presents the design of a robust PLRC for flexible phase lag compensation. The proposed PLRC is designed in the frequency domain and is based on the design principles of the conventional phase 56

83 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance lead compensator described in [162]. However, unlike the traditional design method presented in [162], a simple and easy-to-use iterative design approach is presented in this chapter to allow for flexible compensation of phase uncertainties present at different or multiple frequencies. The rest of this chapter is organised as follow. Section 2 presents the design of the robust PLRC for optimal phase lag compensation. Section 3 presents the details of the frequency domain robust PLRC design approach. Simulation and experimental testing results as well as a comparative study are presented in each of the sections respectively. While, a summary of the research contributions presented in this chapter is given in Section Robust Phase Lead Repetitive Controller for Optimal Tracking Performance Uncertainty Modelling The closed loop transfer function of the uncertain LA system in (3.1) can be represented in the frequency domain as follows G(jω) = G c (jω) + a (jω) (4.1) where G c (jω) is the nominal closed loop transfer function of the LA system and a (jω) is the total additive uncertainty present in the system caused by the payload variations and nonlinear friction. The total additive uncertainty is bounded as a (jω) γ, where γ denotes the upper bound of the additive uncertainty. Furthermore, G c (jω) is defined as follows G c (jω) = C(jω)P (jω) 1 + C(jω)P (jω) (4.2) where C(jω) is a stabilizing feedback controller designed to stabilize the LA system C(z) = z z (4.3) while, P (jω) is the transfer function of the LA system defined as 57

84 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance P (jω) = 1 b(jω) 2 (4.4) where b is the control gain. In the presence of bounded payload variations and nonlinear friction the upper bound of the total additive uncertainty denoted by γ can be obtained by calculating the difference between the magnitudes of the perturbed and nominal closed loop system. To illustrate the above, the magnitude and phase response of both the nominal system in (4.2) and the perturbed system given in (4.1) are first obtained as shown in Figure 4.1. From the phase plot given in Figure 4.1 it is evident that load uncertainties up to 3kg introduce a phase lag in the frequency range 20Hz 600Hz. Furthermore, as depicted in Figure 4.1, the effects of the nonlinear friction which typically occur at low frequencies are well compensated by the feedback controller and do not introduce any additional phase lag into the closed loop system. Given the frequency plots in Figure 4.1, the magnitude and phase of the total additive uncertainty a (jω) can be obtained by calculating the difference between the magnitudes and phases of the perturbed and nominal closed loop system as shown in Figure 4.2. Moreover, as seen from the uncertainty magnitude plot given in Figure 4.2, the maximum upper bound of the total additive uncertainty γ is and it occurs at 158.5Hz. While, from the phase uncertainty plot also given in Figure 4.2 we can see that payload variations up to 3kg cause a maximum phase lag of 28.86º at 158.5Hz. Figure 4.1: Magnitude and phase plots of the nominal and perturbed closed loop system 58

85 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance Figure 4.2: Magnitude and phase plots of the measured additive uncertainty Conventional Phase Lead Repetitive Control The plug-in RC structure given in Figure 4.3 [115] is proven to be an effective methodology for dealing with the tracking problem of periodic signals. As shown in Figure 4.3, a low pass filter Q(z) is added in series to the digital delay term of the IM to filter out the unmodelled dynamics and disturbances present at high frequencies. The selection of the cut-off frequency of Q(z) defines the RC tracking bandwidth and represents a trade-off between system robustness and tracking accuracy. Moreover, in many practical RC applications the selection of the cut-off frequency for the low pass filter Q(z) is limited by the maximum mechanical bandwidth of the system. Furthermore, as depicted in Figure 4.3, a stabilizing controller F (z) is cascaded in series to the IM to ensure overall system stability as well as good phase lag compensation within the RC bandwidth. The transfer function of the digital plugin RC can be represented as follows Q(z)z N G RC (z) = F (z) (4.5) 1 Q(z)z N where N = T T s N represents the number of samples per period, T is the period of the periodic signal and T s is the sampling period. Overall system stability of the discrete plug-in RC given in Figure 4.3 can be achieved if the following sufficient conditions are satisfied at all times [56, 115] 1. G c (z) is stable 2. Q(z) < 1 59

86 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Figure 4.3: Discrete plug-in RC structure 3. (1 G c (z)f (z)) Q(z) < 1 where the last condition can be extended to (1 G c (z)f (z))q(z) < 1 0 < ω < π T s (4.6) Assuming Q(z) is selected as a moving average low pass filter with zero phase contribution [56, 115], the stability condition (4.6) can be written in the frequency domain as follows [56, 59] 0 < M F (jω) < 2 cos (θ G c (jω) + θ F (jω)) M Gc (jω) (4.7) where the magnitude and phase of the nominal closed loop system are represented by M Gc (jω) and θ Gc (jω) respectively. While, the magnitude and phase of the stabilizing filter F (z) are given by M F (jω) and θ F (jω) respectively. The stability condition in (4.7) will be satisfied if the angle condition below holds in the frequency range up to the cut-off frequency of the low pass filter Q(z). θ Gc (jω) + θ F (jω) < π 2 (4.8) Moreover, to satisfy the angle condition in (4.8), the stabilizing compensator F (z) must be designed as a phase lead compensator in order to compensate the phase lag in the feedback system up to the cut-off frequency of Q(z). The stabilizing compensator F (z) in the conventional PLRC structures presented in [105, 113, 128] is designed as follows 60

87 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance F (z) = k rc z m 1 G c (z) (4.9) where the phase lead filter k rc z m is added in series to the inverse of the plant to improve the stability margin of the system in the presence of phase uncertainties. The pure phase lead filter z m which is incorporated into these structures provides an additional phase lead angle of m ω ω N π (where ω and ω N are the frequency and Nyquist frequency respectively) to compensate for the phase lag in the closed loop system caused by the phase uncertainties. Figure 4.4 below shows the phase response of the perturbed LA system with the conventional PLRC for different m values, in which the system crosses the ± π 2 crossover frequency at 250Hz and 165Hz for m equals to 2 and 3 respectively. This shows that the conventional PLRC causes overcompensation in LA systems with phase uncertainties. Furthermore, the inclusion of the z m term in these RC structures reduces the frequency range for which (4.8) holds. This reduces the RC tracking bandwidth and leads to reduced convergence speed and degraded tracking accuracies which is undesirable in many systems used in high-precision positioning applications. Figure 4.4: Phase response of G c (z)f (z) of the perturbed system with conventional PLRC Robust Phase Lead Repetitive Control To correct overcompensation in conventional PLRC (4.9) this section presents a modified PLRC as follows 61

88 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance F (z) = zm G c (z) p z m + p (4.10) where p z m +p is the proposed phase lag compensator, while p defines the location of the phase lag compensator poles. The total phase angle contribution of the lead-lag compensator pzm is as follows z m +p φ = m ω π tan 1 ω N sin ( m ω ω N π ) p + cos ( m ω ω N π ) (4.11) From the second term in (4.11) it is evident that the addition of the phase lag compensator p z m +p reduces the overcompensation caused by the pure phase lead filter z m. The frequency of the maximum phase lead angle that can be provided by (4.10) is determined by the location of the compensator poles denoted by p which are selected to satisfy the stability condition (4.6). However, in many instances where the system is exposed to bounded uncertainties such as bounded nonlinear friction and load uncertainties, a robust stability condition needs to be derived. To derive a robust stability condition we first need to express the the stability condition (4.6) in the frequency domain. Assuming Q(z) is selected as a moving average low pass filter with zero phase contribution as described in [56, 115], the stability condition (4.6) can be represented in the frequency domain as follows [104] M Qi (jω) [ 1 2M Ti cosθ Ti + M 2 T i ] 1 2 < 1 ω i = 2π i T s for i = 1, 2, 3..., L (4.12) where M Ti = M Gci M Fi, and θ Ti = θ Gci θ Fi. Since sin 2 (θ Ti ) + cos 2 (θ Ti ) = 1, the stability condition given in (4.12) can be rewritten as follows M Qi (jω) [ sin 2 (θ Ti ) + cos 2 (θ Ti ) 2M Ti cosθ Ti + M 2 T i ] 1 2 < 1 (4.13) Furthermore, by performing a simple manipulation of the condition given above we get the following equivalent condition 62

89 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance [ ] 1 MQ cos (θ Ti ) 2 sin 2 i (θ Ti ) M 2 Q i M Ti < 1 (4.14) In the presence of bounded multiplicative uncertainties, the stability condition in (4.14) can be written as follows [ ] 1 MQ cos (θ Ti ) 2 sin 2 i (θ Ti ) M 2 Q i M Pi < 1 (4.15) where M Pi can be represented as M Pi = M Gli M Fi (4.16) While, M Gl is the magnitude of the perturbed closed loop defined as M Gl (jω) M Gc (jω) (1 + δ) (4.17) where M Gc is the magnitude of the nominal system and δ represents the upper bound of the total multiplicative uncertainty. Using the upper bound of the multiplicative uncertainty denoted by δ, the magnitude in (4.16) can be rewritten as follows M Pi = M Gci M Fi (1 + δ) (4.18) Furthermore, substituting the expression for M Pi given in (4.18) into (4.15) gives the following equivalent condition [ ] 1 MQ cos (θ Ti ) 2 sin 2 i (θ Ti ) MQ 2 i < 1 M Ti 1 + δ (4.19) From (4.19) it is apparent that the stability condition given in (4.6) can be extended to ensure robust stability as follows 63

90 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance (1 G c (z)f (z)) Q(z) < δ (4.20) Given the robust stability condition in (4.20), an optimal p value for a given m can be obtained by solving the following optimisation problem [104] min(h total ) (4.21) h i < δ ω i = 2π i T s for i = 1, 2, 3..., L (4.22) where the cost objective function h total is defined as follows h total = L i=0 h i (4.23) and h i = M Qi (jω) [ 1 2M Ti cosθ Ti + M 2 T i ] 1 2 ω i = 2π i T s for i = 1, 2, 3..., L (4.24) represents the cost objective function at a given frequency ω i, with L = N/2 for even N, and L = (N 1)/2 for odd N. The optimisation problem (4.21) subject to (4.22) needs to be solved to obtain an optimal F (z) compensator for different values of the parameter m. The phase response plots of G c (z)f (z) for different m values can then be used to determine the optimal m which results in the best phase compensation of phase lag present in the frequency range up to the cut-off frequency of Q(z). An overview of the overall compensator design procedure is as follows: 1. Design a stable feedback controller C(z) so that G c (z) is stable. 2. Design a low pass filter Q(z) to achieve the desired tracking bandwidth as described in [56]. 3. Determine the upper bounds of the system s multiplicative uncertainty δ. 4. Solve the optimisation problem (4.21) subject to (4.22) for different m values. 64

91 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance 5. Plot the phase response plots of G c (z)f (z) using the different compensators obtained in Step 4; select the optimal F (z) which achieves the best phase compensation in the RC bandwidth Robust Phase Lead Repetitive Control with an Application to a Linear Actuator Following the compensator design procedure outlined in Section 4.2.3, this section presents the design of the proposed robust PLRC (4.10) for a LA system with payload uncertainties and nonlinear friction. Given the stabilizing feedback controller C(z) in (4.3), the stable closed loop transfer function of the LA system can be represented as G c (z) = 0.45z z 0.42 z z z (4.25) The next step of the design of the robust PLRC requires the selection of a low pass filter Q(z). To achieve the desired system robustness without adding a phase displacement to the system, the low pass filter Q(z) must be selected as a moving average low pass filter with zero phase contribution as described in [56]. Furthermore, the cut-off frequency of the selected low pass filter needs to be below the mechanical bandwidth of the available LA system which is estimated to be approximately 500Hz. In order to satisfy these conditions a first order low pass filter is selected to provide a tracking bandwidth of 350Hz as follows Q(z) = 0.25z+0.5 = 0.25z 1. The design of the robust PLRC also requires the upper bound of the multiplicative uncertainty δ. To obtain δ we first need to represent the perturbed system (3.1) in terms of multiplicative uncertainty m (s) as given below G(jω) = G c (jω) (1 + m (jω)) (4.26) where m (jω) δ. From (4.1) and (4.26) the relationship between the additive uncertainty a and the multiplicative uncertainty m can be represented as follows G c (jω) + a (jω) = G c (jω) (1 + m (jω)) (4.27) 65

92 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Simplifying the expression above gives the following equivalent relationship a (jω) = G c (jω) m (jω) (4.28) Assuming the bounds of the additive uncertainty γ are known, the bounds of the multiplicative uncertainty δ can be calculated using the following relationship δ γ G c (jω) (4.29) Using the relationship in (4.29), the upper bounds of the multiplicative uncertainty can be easily obtained if γ and G c (jω) are known. By replacing the values of γ and G c (jω) obtained in Section into the expression given in (4.29), the upper bound of the multiplicative uncertainty δ present in the LA system is calculated to be Given the design of the low pass filter Q(z) and the upper multiplicative uncertainty bound value denoted by δ, the Optimisation toolbox available in Matlab is next used to solve the optimisation problem (4.21) and obtain the optimal p values for different m. At the end of this iterative design step an optimal F (z) compensator is obtained for m = 1, m = 2 and m = 3 respectively. Figure 4.5 shows the phase response of G C (z)f (z) for the different m values in which for m = 2 the obtained compensator provides the most optimal phase lead angle in the frequency range 20Hz 500Hz which is sufficient for compensating the phase lag caused by the load uncertainties. To assess the effectiveness of this optimal compensator, Figure 4.6 shows the phase cancellation of the perturbed system in which the proposed robust PLRC with m = 2 achieves close to perfect phase cancellation when compared to the conventional PLRC for m = 1. The selected robust PLRC can be represented as F (z) = 1.595z z z z z z z (4.30) 66

93 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance Figure 4.5: Phase response of G C (z)f (z) of the system with the proposed PLRC Figure 4.6: Phase response of G C (z)f (z) of the perturbed system 67

94 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Simulation Testing and Analysis Simulation testing is carried out to investigate and compare the tracking performance and robustness between the proposed PLRC in (4.30) and the conventional PLRC (4.9) for k r = 0.9 and m = 1. The robustness to load uncertainties of the two controllers is investigated by applying large payloads to the LA plant. A triangular reference trajectory with a magnitude of ±500µm and a frequency of 1.67Hz is used during the testing. Furthermore, the simulation sampling time was set to s. Consequently, the order of the RC denoted by N, which is computed as the ratio of the period corresponding to the periodic reference trajectory and the sampling period, was selected to be Figure 4.7 below shows the tracking errors obtained from the simulation testing from which it is evident that the proposed robust PLRC has superior robustness characteristics and tracking performance when a large payload is applied to the LA system. Figure 4.7: Tracking error of perturbed system (30kg load, Amp=±1000µm) The robustness of the LA system is also analysed using the sensitivity S(z) and complimentary sensitivity T (z) functions. If we let L(z) denote the loop gain of the system, S(z) and T (z) can be represented as follows S(z) = L(z) (4.31) T (z) = L(z) 1 + L(z) (4.32) where 68

95 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance L(z) = (G RC (z) + 1) C(z)P (z) (4.33) To achieve good tracking performance S(z) is required to have a small magnitude ( S < 1) at low frequencies. Moreover, as seen from the frequency response of the uncertainty in Figure 4.2, good tracking performance in the presence of payload variations can be achieved if the magnitude of S(z) is small in the frequency region around the frequency of the maximum load disturbance (158Hz). Furthermore, T (z) is required to maintain small values ( T < 1) at high frequencies to ensure robustness to sensor noise, unmodelled dynamics and other uncertainties present at high frequencies. From the plots of S(z) given in Figure 4.8 we can see that the system with the proposed PLRC can attenuate disturbances at frequencies below 150Hz more effectively than the system with the conventional PLRC. This suggests that the proposed PLRC has improved tracking performance in the presence of payload variations. Furthermore, from Figure 4.9 it is also evident that the magnitude of T (z) for the system with the proposed PLRC is smaller at high frequencies when compared to the magnitude of T (z) for the system with the conventional PLRC. This suggests that the proposed PLRC structure also has improved robustness to noise and unmodelled dynamics present at high frequencies. Figure 4.8: Magnitude plots of S(z) 69

96 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Experimental Validation Figure 4.9: Magnitude plots of T (z) Experimental testing is also performed on the LA system shown in Figure 3.1 to verify the robustness of the proposed PLRC against payload variations and nonlinear friction. Figure 4.10, Figure 4.11 and Figure 4.12 show the tracking errors of the conventional and proposed PLRC when a payload of 2.3kg is applied to the LA system for input magnitudes ±500µm, ±750µm and ±1000µm respectively. From the obtained plots we can clearly see that the proposed PLRC shows improved tracking performance when the LA plant is subjected to payload variations for a wide range of input magnitudes. Since the adverse effects of the nonlinear friction force acting on the LA system increase as payload is added to the system as well as during slow movements, from these plots we can also infer that the proposed robust PLRC shows improved robustness and tracking performance in the presence of both phase uncertainties due to payload variations and nonlinear friction. This improvement in robust tracking performance is even more evident at high input magnitudes as shown in Figure 4.12 where the effects of both load disturbance and nonlinear friction are more significant. In addition to the error tracking plots, Table 4.1 shows a steady state error analysis summary for different input magnitudes in which both the root mean square (RMS) error and the peak-to-peak (P-P) error of the proposed PLRC are smaller than the conventional PLRC. Moreover, at input magnitudes of ±1000µm the RMS error of the proposed robust PLRC is 1.81µm, while the RMS error of the conventional PLRC is 30.77µm. This verifies the superior steady state tracking capabilities 70

97 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance offered by the proposed PLRC in the presence of uncertainties. The error P-P value for input magnitudes of ±1000µm is 13µm and 121µm for the proposed and conventional PLRC respectively, which also indicates that the proposed PLRC has higher bandwidth tracking performance when compared to the conventional PLRC. (a) Conventional PLRC (b) Proposed PLRC Figure 4.10: Tracking error (2.3kg load, A = ±500µm) 71

98 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance (a) Conventional PLRC (b) Proposed PLRC Figure 4.11: Tracking error for (2.3kg load, A = ±750µm) (a) Conventional PLRC (b) Proposed PLRC Figure 4.12: Tracking error for (2.3kg load, A = ±1000µm) 72

99 4.2 Robust Phase Lead Repetitive Controller for Optimal Tracking Performance Amp (P-P) Conventional PLRC Proposed PLRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) ±500µm ±750µm ±1000µm Table 4.1: Steady State Error Analysis of conventional and proposed PLRC (2.3kg load) 73

100 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain Uncertainty Modelling As described in Section 4.2.1, the closed loop transfer function of the perturbed LA system given in (3.1) can also be described using the additive uncertainty a (jω), where the magnitude of a can be obtained by calculating the difference between the magnitude of the perturbed closed loop system and the magnitude of the nominal closed loop system. The frequency plots of the nominal and the LA system at a -3kg and 3kg payload are given in Figure While, the magnitude plots of the lower and upper additive uncertainty a present in the LA system are given in Figure 4.14, where the maximum uncertainty bound γ is measured to be As seen from the phase plots given in Figure 4.13, the phase lag due to the load uncertainties in the LA system is visible in the frequency range 20Hz 600Hz. Moreover, as shown from the phase plots in Figure 4.13, an additional phase lag is also present in the frequency range 600Hz 1000Hz. In applications which require high tracking accuracies this necessitates the design of a PLRC that is capable of compensating the phase lag caused by the phase uncertainties as well as the additional phase lag present in the RC bandwidth. To achieve this, the reminder of this section discusses the design of a robust PLRC for flexible phase compensation. Figure 4.13: Frequency plots of the LA system at different payloads 74

101 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain Figure 4.14: Magnitude plot of the measured additive uncertainty Stability of Repetitive Control The discrete-time plug-in RC structure given in Figure 4.15 is considered in the design of the PLRC proposed in this section [115]. The transfer function of this discrete plug-in RC can be represented as follows Q(z)z N G RC (z) = G f (z) (4.34) 1 Q(z)z N where Q(z) is a low pass filter inserted in series to the digital delay of the IM to filter out and improve the robustness to noise and unmodelled system dynamics at high frequencies, N = T T s N is the number of samples per period, T represents the period of the periodic signal and T s is the sampling time. Furthermore, G f (z) = k rc F (z) represents the stabilizing compensator designed to achieve overall system stability, where k rc is the RC learning gain and F (z) is a phase lead compensator designed to compensate the phase lag in the closed loop system. Figure 4.15: Discrete plug-in RC structure The robustness and tracking performance of the RC system is limited by the maximum cut-off frequency of the low pass filter Q(z) which is defined by the maximum mechanical bandwidth of the system. To achieve the desired system robustness without adding an additional phase displacement to the system, Q(z) is often 75

102 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance chosen as a moving average low pass filter with zero phase contribution [56, 115]. Furthermore, the feedback controller C(z), the low pass filter Q(z), the phase lead compensator F (z) and the learning gain k rc are designed to satisfy the following stability conditions at all times 1. G c (z) is stable 2. Q(z) < 1 3. (1 k rc G c (z)f (z)) Q(z) < 1 where the last stability condition is equivalent to (1 k rc G c (z)f (z)) Q(z) < 1 0 < ω < π T s (4.35) The stability condition (4.35) can also be represented in the frequency domain as follows [56, 59, 115] ( 1 krc M Gc (jω)m F (jω)e ) j(θ Gc +θ F ) 1 < M Q (jω)e jθ Q (4.36) where M Gc (jω) and θ Gc (jω) represent the magnitude and phase of G c (z) respectively. Furthermore, M Q (jω) is the magnitude and θ Q (jω) is the phase of the low pass filter Q(z). Moreover, M F (jω) and θ F (jω) represent the magnitude and phase of F (z) respectively. Since θ Q (jω) = 0, the condition in (4.36) can be simplified to ( 1 krc M Gc (jω)m F (jω)e ) j(θ Gc +θ F ) 1 < M Q (jω) (4.37) Moreover, since the magnitudes M Gc (jω) and M F (jω) are positive, the stability condition in (4.37) can be extended to 0 < k rc M F (jω) < 1 MQ(jω) 2 M F (jω)mg 2 c (jω)mq(jω) + 2 cos (θ G c (jω) + θ F (jω)) 2 M Gc (jω) (4.38) As M Q (jω) 1 at low frequencies and M Q (jω) 0 at high frequencies, the condition given in (4.38) can be simplified as follows 76

103 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain 0 < k rc M F (jω) < 2 cos (θ G c (jω) + θ F (jω)) M Gc (jω) (4.39) Furthermore, 0 < k rc < 2 cos (θ G c (jω) + θ F (jω)) M F (jω)m Gc (jω) (4.40) where the magnitude condition (4.40) will be met if the angle condition below holds θ Gc (jω) + θ F (jω) < π 2 (4.41) From (4.41) it is evident that F (z) should be designed as a phase lead compensator in order to provide a positive phase lead angle θ F to cancel out the phase lag of the feedback system in the frequency range up to the cut-off frequency of Q(z). However, for a system with bounded uncertainties the magnitude condition (4.40) needs to be extended to ensure robust stability. Moreover, in the presence of bounded multiplicative uncertainties where m (z) δ, the transfer function of the closed loop system can be represented as G(jω) = G c (jω) (1 + m (jω)) (4.42) Consequently, the magnitude condition (4.40) can be rewritten as follows [56] 0 < k rc < 2 cos (θ G c (jω) + θ F (jω)) M F (jω)m Gc (jω) (1 + δ) (4.43) where the selection of k rc represents a trade-off between system robustness and convergence rate. Moreover, larger k rc values will result in a faster convergence rate; however, at the cost of reduced robustness and vice versa Robust Phase Lead Repetitive Controller Design in the Frequency Domain A robust PLRC with flexible phase lag compensation is proposed in this section as follows 77

104 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance G f (z) = k rc F (z) (4.44) where F (z) represents a phase lead network which consists of n first-order phase lead compensators (PLCs). The phase lead compensator network F (z) can be represented as n F (z) = G Li (4.45) i=1 where G Li (z) represents the i th discrete first-order PLC designed in the frequency domain to compensate a portion of the total phase lag caused by the phase uncertainties [162]. Each first-order PLC G Li (z) is capable of providing a maximum phase shift of 65 to the overall phase lead angle of F (z). The value of n is determined by the magnitude of the maximum phase lag angle present in the feedback system which needs to be compensated by G f (z). Furthermore, each of the discrete first-order PLCs denoted by G Li (z) can be represented as follows [162] G Li = k di z + z 0i z + z pi (4.46) where z 0i is the compensator zero, z pi represents the compensator pole, while k di is the compensator gain of the i th PLC. The parameters of (4.46) are designed to achieve the desired maximum phase lead angle φ max at a frequency ω max. Moreover, to increase the convergence rate of the system output the value of the maximum phase angle provided by each of the first-order PLC denoted by φ max at frequency ω max is recommended to be 30. The design of each first-order PLC G Li (z) is performed in the frequency domain using the continuous time domain design techniques defined for the s-plane as described in [162]. However, as presented in [162], these techniques apply only when the PLC is represented in the w-plane. The w-plane representation of the PLC in (4.46) is as follows G Li (w) = 1 α w + 1 τ i w + 1 α i τ i (4.47) 78

105 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain where α i is a design parameter, while τ i is the time constant. Furthermore, the relationship between the frequency in s-plane ω, and the frequency in w-plane v, can be represented as v = 2 ( ) ωts tan T s 2 (4.48) While, the w-plane frequency at the maximum phase lead angle φ maxi is also defined as v maxi = 1 τ i αi (4.49) For a maximum phase lead angle of φ maxi at frequency ω maxi, the value of α i can be calculated as follows α i = 1 sin (φ max) 1 + sin (φ max ) (4.50) While, the value of the parameter τ i can be calculated from (4.49) as follows τ i = 1 v maxi αi (4.51) where v maxi can be obtained using the relationship given in (4.48) for ω = ω max. Once α i and τ i are designed following the above w-plane relationships, the discrete first-order PLC in (4.47) can be represented in the z-plane by performing a bilinear transform from the w-plane to the z-plane. This can be done by replacing w in (4.47) by 2 T s z 1 z+1 as shown bellow G Li (z) = 1 α 2 z T s z+1 τ i 2 z (4.52) T s z+1 α i τ i Moreover, since the expressions given in (4.46) and (4.52) are equivalent, the gain and the zero/pole pair of each of the discrete first-order PLC in (4.46) can be represented in terms of α i and τ i as follows k di = T s 2 α i τ i T s 2 + α i τ i (4.53) 79

106 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance z 0i = T s 2 τ i T s 2 + τ i (4.54) z pi = T s 2 α i τ i T s 2 + α i τ i (4.55) Finally the learning gain k rc needs to be selected to satisfy the robust stability condition in (4.43). A summary of the simple iterative design procedure involved in the design of G f (z) is as follows: 1. Design a feedback controller C(z) so that G c (z) is stable 2. Design Q(z) to provide the desired tracking bandwidth. 3. Determine the upper bound of the system s multiplicative uncertainty δ. 4. Plot the phase of G c (z)f (z) and design the i th discrete-time phase lead compensator G Li (z) to provide a maximum phase lead angle φ max at a frequency ω max using the design procedure described above. Note that initially F (z) = Repeat Step 4 n times until the optimal phase cancellation is achieved in the frequency range up to the cut-off frequency of Q(z). 6. Select the largest value of k rc that satisfies the robust stability condition (4.43) Robust Phase Lead Repetitive Control with an Application to a Linear Actuator The proposed robust PLRC scheme is applied a LA positioning stage with plant dynamics as defined in (3.1). The implementation details are as provided below. A stabilizing controller C(z) is first selected as given in (4.3) to ensure overall closed loop stability. Moreover, for T s = s the closed loop transfer function denoted by G c (z) is as follows G c (z) = 0.23z z 0.22 z 3 1.3z z (4.56) 80

107 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain The next step of the design of the robust PLRC for the LA system in (3.1) requires the selection of the low pass filter Q(z) as described in [56, 115]. The following low pass filter is chosen to provide a tracking bandwidth of 350Hz: Q(z) = 0.25z = 0.25z 1. To calculate the maximum bound of the multiplicative uncertainty δ we first need to derive the relationship between the additive uncertainty a and the multiplicative uncertainty m. From (4.1) and (4.42), the relationship between a and m can be represented as follows G c (jω) + a (jω) = G c (jω) (1 + m (jω)) (4.57) By simplifying the expression in (4.57) we get a (jω) = G c (jω) m (jω) (4.58) Moreover, given the upper bound of a, the following expression for calculating δ can be derived δ γ G c (jω) (4.59) Using the value for γ obtained in Section 4.3.1, the value of δ calculated using (4.59) is The next step of the design of G f (z) for the LA system in (3.1) requires the design of F (z) for achieving optimal tracking performance. As depicted from the phase plots of the nominal and perturbed LA system given in Figure 4.16, the phase lag in the closed loop system up to the cut-off frequency of Q(z) (350Hz) is approximately 120. This necessitates the design of a phase lead network F (z) capable of achieving close to zero phase cancellation of the measured phase lag in the frequency range up to the cut-off frequency of Q(z). To achieve this, the proposed discrete time phase lead network F (z) is designed to consist of four first order PLCs. Firstly a first-order PLC with a maximum phase shift of 30 at 200Hz is designed. The phase response of G c (z)f (z) after the first design iteration is given in Figure In the second and third design iteration another two first-order PLCs are designed to provide a phase lead angle of 30 at 200Hz and 30 at 500Hz respectively. 81

108 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Figure 4.16: Phase plot of the nominal system (G c (z)f (z) = 1) Figure 4.17: Phase plot of G c (z)f (z) after the first design iteration 82

109 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain The phase response plots after the second and third design iteration are given in Figure 4.18 and Figure 4.19 respectively. Figure 4.18: Phase plot of G c (z)f (z) after the second design iteration Figure 4.19: Phase plot of G c (z)f (z) after the third design iteration Finally a first order PLC with a maximum angle contribution of 30 at 550Hz is designed to obtain the final discrete phase lead network which can be represented as F (z) = z4 1.56z z z z z z z (4.60) The phase plot showing the phase compensation capabilities of the final phase lead network is given in Figure 4.20 where it is evident that the proposed phase 83

110 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance lead network F (z) achieves close to zero phase cancellation in the frequency range 0Hz 500Hz. Figure 4.20: Phase plot of G c (z)f (z) after the final design iteration To ensure overall system stability the value of k rc is then selected to satisfy the condition in (4.43). Moreover, for F (z) given in (4.60), the maximum k rc value which satisfies (4.43) is selected to be 2. The final robust PLRC compensator can be represented as follows G f (z) = 2z4 3.12z z z z z z z (4.61) The frequency plots of the proposed PLRC compensator in (4.61) are given in Figure 4.21 where it can be seen that the proposed PLRC provides a phase lead angle of approximately 110 to successfully eliminate the phase lag in the feedback system Simulation Testing and Analysis This section presents simulation testing results, as well as an analysis of the robust tracking performance of the proposed controller. Furthermore, a comparison study comparing the performance of the proposed controller in (4.61) to that of the conventional PLRC k rc z m for k rc = 0.9 and m = 2 is also presented in this section. Firstly the robust tracking performance of the perturbed LA system with the proposed controller and the conventional PLRC for different m values is analysed 84

111 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain Figure 4.21: Frequency response of G f (z) by plotting the phase plot of G f (z)g c (z) as shown in Figure From the phase plots given in Figure 4.22 it can be seen that the proposed PLRC shows close to perfect phase cancellation in the frequency range up to the cut-off frequency of the Q(z), while the conventional PLRC overcompensates the phase lag in the closed loop system for m > 3. Figure 4.22: Phase response of G f (z)g c (z) for the perturbed system Simulation testing is then performed to investigate and compare the robust tracking performance between the proposed controller and the conventional PLRC. A triangular periodic reference trajectory with a frequency of 1.67Hz and an amplitude of ±1000µm is used in the simulation testing. While, the sampling period 85

112 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance T s is set to s and the value of N is selected to be The tracking error plots for a payload of 30kg given in Figure 4.23 show that the proposed robust RC structure has superior robustness characteristics and improved robust tracking performance when compared to the conventional PLRC. Figure 4.23: Tracking error of the conventional and proposed PLRC (30kg load, Amp=±1000µm) The robust performance of the LA system with the proposed and conventional PLRC under load disturbances can also be analysed using the sensitivity S(z) function which is defined as follows S(z) = L(z) (4.62) where L(z) is the loop gain of the system defined as L(z) = (G RC (z) + 1) C(z)P (z) (4.63) To achieve satisfactory robust tracking performance, the sensitivity function S(z) is required to have a small magnitude at low frequencies. As seen from the S(z) plots given in Figure 4.24, the magnitude of S(z) for the system with the proposed PLRC is smaller at frequencies different to those of the periodic signal s fundamental and harmonic frequencies. This indicates that the system with the proposed PLRC can attenuate non-periodic disturbances more effectively than the system with the conventional PLRC. Moreover, this suggests that the proposed PLRC has improved robustness against load disturbances. Furthermore, the plots of the magnitude of the stability condition (4.35) (also known as the regeneration spectrum function R(z)) given in Figure 4.25 indicate that the magnitude of R(z) of the system with the proposed PLRC remains less than 1 for a wider range of payload variations. This suggests that the proposed PLRC has improved stability robustness properties when compared to the conventional PLRC. 86

113 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain Figure 4.24: Magnitude plots of S(z) Figure 4.25: Magnitude plots of R(z) for the perturbed system 87

114 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Experimental Validation Experimental testing is also performed on the LA experimental setup given in Figure 3.1 to verify the effectiveness of the proposed controller. The obtained results as well as a comparison study are provided in this section to show and compare the robust tracking performance of the system with the proposed and the conventional PLRC. The robust tracking performance of the system with the proposed and conventional PLRC controllers is tested by varying the payload from 3kg to 3kg, where a payload of 0kg indicates nominal load conditions. A triangular reference signal with frequency of 1.67Hz and amplitude of ±1000µm as shown in Figure 4.26 is used in the experiments. The steady state error plots for payloads of 3kg and 3kg given in Figure 4.27 and Figure 4.27 respectively clearly show that the system with the proposed PLRC has improved tracking accuracy in the presence of nonlinear friction and load uncertainties. While, the numerical steady state error and error values for payloads of 3kg, 0kg and 3kg given in Table 4.2 strongly indicate that the system with the proposed PLRC is stable and has improved tracking performance within the entire uncertainty range. Moreover, as seen from Table 4.2, at payloads of 3kg and 3kg the system with the conventional robust PLRC is unstable. Figure 4.26: Periodic reference trajectory 88

115 4.3 Design of a Robust Phase Lead Repetitive Controller in the Frequency Domain (a) Conventional PLRC (b) Proposed PLRC Figure 4.27: Tracking error of the conventional and proposed PLRC (-3kg payload) (a) Conventional PLRC (b) Proposed PLRC Figure 4.28: Tracking error (3kg load) 89

116 4 Design of Robust Phase Lead Repetitive Control for Optimal Tracking Performance Payload (kg) Conventional PLRC Proposed PLRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) 3 unstable unstable unstable unstable unstable unstable unstable unstable Table 4.2: Steady State Error Analysis of conventional and proposed PLRC 4.4 Summary This chapter presents the design of two optimal PLRC for improved tracking performance in LA systems with bounded load uncertainties and nonlinear friction. The first section of this chapter proposes the design of an optimal robust PLRC for improved tracking performance and robustness in LA systems with bounded payload variations and nonlinear friction. In this section the conventional PLRC in [105, 113, 128] is modified to correct the overcompensation caused by the pure phase filter z m. Furthermore, a robust stability condition is derived to ensure robust stability. The proposed PLRC is compared with the conventional PLRC through simulation and experiments. Both analysis and results show that the proposed robust PLRC offers a significant improvement in tracking performance and robustness when the system is subjected to both nonlinear friction and payload uncertainties. The second section proposes a simple and effective design of a discrete PLRC in the frequency domain. In comparison to the conventional PLRC methods [56] which provide a fixed phase lead angle to cancel out phase lag in the feedback system present predominantly at high frequencies, the proposed PLRC can effectively compensate phase lag present at any frequency, as well as at multiple frequencies within the RC bandwidth. This ensures improved tracking performance and robustness in the presence of different phase uncertainties. The obtained simulation and experimental results show a significant improvement in robust stability and robust tracking performance of the proposed robust PLRC when compared to the conventional robust PLRC. Although the two control methods proposed in this chapter ensure improved robust tracking performance in the presence of phase uncertainties, they have limited robustness to other non-periodic uncertainties and large external disturbances. Furthermore, the two approaches proposed in this chapter incorporate the discrete plug-in RC structure which is based on the conventional IM, resulting in controllers with sluggish dynamic response. To overcome these limitations of the PLRC methods proposed in this chapter, the following chapter proposes the design of SMC 90

117 4.4 Summary based RC paradigms for fast transient response and improved robustness to large non-periodic disturbances. 91

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119 5 Discrete Sliding Mode Repetitive Control 5.1 Introduction Chapter 4 of this thesis presented the design of two optimal PLRC for improved robust tracking performance. Although the controllers designed in Chapter 4 can provide satisfactory robustness to phase uncertainties, the inclusion of the RC structure causes a sluggish transient response, especially in the presence of uncertainties. Furthermore, these methods are limited to applications where the system is exposed to phase uncertainties with small bounds. However, in many practical applications the system could be subjected to large disturbances and uncertainties which may not introduce a phase lag in the closed loop system. A number of nonlinear control structures such as the variable-structure control (VSC) methods have been proposed for fast transient response and improved robustness in many linear and nonlinear systems [137, ]. Although these methods can ensure fast transient response and improved robustness, they are model based and require accurate modelling of the plant dynamics which is not always possible to achieve. Furthermore, in applications of a repetitive nature the VSC methods such as the SMC and TSMC cannot effectively eliminate the repeatable uncertainties present in the system. Several time-domain based SMRC methods have been proposed for improved tracking accuracies in systems used in repetitive applications [5, 60 68]. Even though these methods can achieve improved tracking performance and robustness against periodic and non-periodic disturbances, the design and analysis of a SMRC for achieving optimal tracking performance is difficult to realise using time domain design techniques. While, in applications which require fast transient response, a VSC method for fast transient response such as the TSMC should be considered instead of the conventional SMC structure. 93

120 5 Discrete Sliding Mode Repetitive Control To address these limitations of the PLRC methods proposed in Chapter 4 as well as the conventional SMRC structures, this chapter presents the design of two new discrete SMRC methods for improved robust tracking performance in LA systems with repeatable and non-repeatable uncertainties. The first section of this chapter presents the design of a discrete SMRC in the frequency domain. The proposed method uses the describing function (DF) frequency design approach to design a stable and optimal SMRC structure for improved tracking performance. In the second part of this chapter a new discrete TSMRC structure is proposed for fast transient response and improved tracking performance. The controllers proposed in this chapter combine RC with SMC to achieve the desirable features of the two distinct control methodologies. Moreover, the SMC/TSMC component is first designed to achieve fast convergence and improved robustness against non-periodic nonlinearities and parameter uncertainties. Then the RC control part is added to learn and eliminate the periodic components of the friction as effectively as possible. A pure phase lead compensator is also incorporated into the RC component to further improve the high bandwidth tracking performance and convergence rate of the system in the presence of phase uncertainties [56]. This rest of this chapter is organised as follows. The design of the optimal robust SMRC structure in the frequency domain is presented in Section 2. The design details of the new discrete TSMRC structure are presented in Section 3. Finally the conclusions of this chapter are presented in Section Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain Design Methodology The robust SMRC method presented in this section considers the discrete nonlinear second order plant dynamics described in (3.6). However, the following assumption is made regarding the system uncertainty: Assumption 1: The total lumped uncertainty present in the system is bounded as follows d(x(k), k) g(k) γt s (5.1) 94

121 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain where T s is the sampling time and γ is a known positive constant obtained in Section A block diagram of the proposed sliding mode repetitive control (SMRC) structure is given in Figure 5.1. While, a block diagram of the proposed RC structure is given in Figure 5.2, where k r represents the learning gain and m is the order of digital pure phase lead filter k r z m [56] which is added to the RC structure to improve the tracking accuracy of the overall system in the presence of payload variations. The low pass filter Q(z) is added to the RC structure in order to improve the robustness of the RC system to unmodelled system dynamics and noise at high frequencies, while N is the order of the RC and is defined as the number of samples per period i.e. N = T T s N, where T is the period of the periodic signal and T s is the sampling time. Furthermore, u l (k) is the repetitive update law and e = y r represents the tracking error, while r(k) denotes the periodic reference signal applied to the system. Figure 5.1: Proposed SMRC structure Figure 5.2: Modified RC structure The repetitive control law u l (k) can be represented by the following expression u l (k) = Qu l (k N) + k r Qe(k N + m) (5.2) 95

122 5 Discrete Sliding Mode Repetitive Control The design of the overall SMRC structure given in Figure 5.1 consists of selecting a sliding mode control law and designing an optimal digital pure phase lead filter k r z m for achieving fast transient response and improved robustness to parameter uncertainties and periodic nonlinearities. Moreover, the design of the discrete SMC structure consists of two steps. Firstly the following sliding mode function σ(k) is proposed to ensure the system exhibits the desired dynamics on the sliding surface σ(k) = c T ẽ(k) (5.3) where ẽ(k) = [ ] T e 1 (k) e 2 (k) represents the system error states and c T R m n is a matrix representing the sliding mode function parameters. Furthermore, e 1 (k) = x 1 (k) r represents the position tracking error and e 2 (k) = x 2 (k) ṙ is the velocity error. The ideal switching surface is reached when σ(k) = c T ẽ(k) = 0 (5.4) Since the system states rarely reach the ideal switching plane they are forced into a quasi-band around the ideal switching surface when the following condition is satisfied. σ(k + 1) = σ(k) = 0 (5.5) The forward expression of (5.3) can be represented as follows σ(k + 1) = c T ẽ(k + 1) (5.6) while, the forward step of the error dynamics can be written as given below ẽ(k + 1) = x(k + 1) r(k + 1) (5.7) Assuming the disturbances and nonlinearities present in the system in (3.6) are negligible, (5.7) can be rewritten as ẽ(k + 1) = Ax(k) + Bu(k) r(k + 1) (5.8) 96

123 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain Substituting (5.8) into (5.6) gives the following equivalent relationship σ(k + 1) = c T Ax(k) + c T Bu(k) c T r(k + 1) (5.9) Moreover, by replacing x(k) with ẽ(k) + r(k), the equation in (5.9) can be rewritten as σ(k + 1) = c T Aẽ(k) + c T Bu(k) + c T Ar(k) c T r(k + 1) (5.10) Setting the above equation to be equal to zero and solving for u gives the following equivalent sliding mode control law u e (k) = ( c T B ) 1 c T Aẽ(k) + ( c T B ) 1 [ c T r(k + 1) c T Ar(k) ] (5.11) Furthermore, simplifying (5.11) gives the following equivalent control law u e (k) = ( c T B ) 1 [ c T Aẽ(k) + c T r(k + 1) c T Ar(k) ] (5.12) As seen from (5.12), the equivalent sliding mode control law consists of terms that are a function of the reference signal r(k). The sum of these terms can be represented as follows u r (k) = ( c T B ) 1 [ c T r(k + 1) c T Ar(k) ] (5.13) The inclusion of these terms into the control equivalent law has the objective to improve the tracking capabilities of the system. However, it is well known that the inclusion of (5.13) into the equivalent control law cannot fully eliminate the periodic error present in the system output. Therefore, in this section we replace the reference tracking component in (5.13) with the repetitive control law c T Au l (k). The resulting equivalent sliding mode control law can be rewritten as follows u e (k) = ( c T B ) 1 [ c T Aẽ(k) c T Au l (k) ] (5.14) The second step of the design of the SMC requires selecting a suitable reaching control law to force the error states into the small quasi sliding mode (QSM) region 97

124 5 Discrete Sliding Mode Repetitive Control around the sliding surface. The following reaching law is selected to guarantee the convergence of the error states to the QSM region σ(k + 1) σ(k) = qt s σ(k) k s T s sat (σ(k)) (5.15) where q and k s are positive design parameters that are selected to drive the error states to the QSM region. Furthermore, 1 qt s > 0. While, sat(σ(k)) is a saturation function which is defined as sat(σ) = 1 k sat σ 1 σ > δ σ δ σ < δ (5.16) where δ represents the boundary layer thickness, while k sat = 1 δ gain. is a linear feedback The left-hand side of the expression in (5.15) can also be represented as follows σ(k + 1) σ(k) = c T Aẽ(k) + c T Bu(k) + c T Au l (k) σ(k) (5.17) By equating (5.15) and (5.17) and solving for u we get the following total sliding mode control law u(k) = ( c T B ) 1 [ c T Aẽ(k) (1 qt s )σ(k) + k s T s sat (σ(k)) c T Au l (k) ] (5.18) As seen from the total control law in (5.18), the design of the discrete SMRC requires the selection of the optimal SMC and RC parameters so that the system errors reach the QSM region from where they will asymptotically converge to a smaller region around the sliding surface in the presence of uncertainties. To prove the convergence of the system errors to the reduced QSM we propose an easy-to-apply graphical frequency design approach which is based on the describing function (DF) method [ ] and the conventional RC frequency domain design techniques. In the proposed design approach the RC frequency domain design method is used to guide the design of the optimal k r z m. While, the DF stability analysis method is used to ensure stability robustness in the presence of uncertainties. 98

125 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain Design of the Repetitive Controller It is well known that the phase lag introduced in the closed loop system due to the payload variations and other uncertainties present in the system significantly reduces the tracking bandwidth of the system. To solve this problem a small phase lead compensator k r z m is added to the RC component to compensate the phase lag introduced by the payload variations and improve the high bandwidth tracking performance of the system [56]. The following section describes the steps involved in selecting the optimal order of the phase lead compensator k r z m which is determined by the parameter m. Firstly the SMC parameters c, q and k s are selected for achieving fast convergence of the system errors to the QSM region. Once the error states reach the bound defined by the parameter δ, the output of the saturation nonlinearity can be reduced to the linear term k s k sat T s σ(k). From this point the error states are forced to asymptotically converge to a smaller region around the sliding surface by the repetitive control law. Moreover, when the system error states reach the QSM region, the control law in (5.18) is linear and can be represented as follows u(k) = ( c T B ) 1 [ c T Aẽ(k) ls(k) c T Au l (k) ] (5.19) where l = 1 qt s k s k sat T s (5.20) The proposed SMRC can also be represented using the block diagram representation as shown in Figure 5.3, where G 1 (z) and G 2 (z) represent the transfer functions of the linearised sliding mode controller and are defined as G 1 (z) = (c 1T s + c 2 + lc 2 ) z + lc 1 T s lc 2 c 2 (c 1 T s + c 2 ) z c 2 (5.21) G 2 (z) = (c 1T s + c 2 ) z c 2 T s (5.22) Furthermore, P (z) is equivalent to: 99

126 5 Discrete Sliding Mode Repetitive Control P (z) = 2 bt s (c 1 T s + 2c 2 ) G p(z) (5.23) where G p (z) is the discrete transfer function of the LA plant. Figure 5.3: Block diagram of the linear SMRC system The transfer function of the repetitive controller G RC (z) is defined as follows G RC (z) = k r Q(z)z N+m 1 Q(z)z N (5.24) While, the closed loop transfer function of the system without the RC component once the system error states reach δ can be written as follows G C (z) = G 2 (z)p (z) 1 + G 1 (z)g 2 (z)p (z) (5.25) Once the system states enter the δ-band, the overall stability of the linearised SMRC system can be analysed using the small gain theorem for RC systems. Moreover, for achieving overall system stability the following conditions must be satisfied at all times [56, 59, 115]: 1. G c (z) is stable 2. Q(z) < 1 3. (1 k r z m G c (z)) Q(z) < 1 where the last stability condition can also be represented as given below (1 k r z m G c (z))q(z) < 1 0 < ω < π T s (5.26) 100

127 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain By using the frequency domain design method described in [56, 59], the stability condition in (5.26) can be analysed in the frequency domain. Moreover, by selecting the low pass filter Q(z) as a zero-phase moving average filter [56, 115], the following magnitude and angle conditions can be derived 0 < k r < 2 cos (θ G c (jω) + θ m (jω)) M Gc (jω) (5.27) θ Gc (jω) + θ m (jω) < π 2 (5.28) where M Gc (jω) and θ Gc (jω) are the magnitude and phase of G c (z) respectively, while θ m (jω) is the phase of the k r z m filter. As seen from the angle condition (5.28), improved robust performance of the RC system can be achieved if a large enough θ m is selected to cancel out the phase lag angle introduced in the closed loop system by the phase uncertainties. An effective phase lag cancellation will enlarge the frequency range that satisfies the angle condition in (5.28) and hence increase the tracking bandwidth of the system, leading to improved steady state tracking performance of the system in the presence of phase uncertainties such as payload variations. In order to determine the optimal m we need to plot the phase of k r z m G c (z) for m = 1, 2, 3... n and select the optimal compensator which provides a sufficient phase angle to eliminate the phase lag caused by the phase uncertainties. The details describing the selection of the optimal m value are given in Section The design of the SMRC structure also requires the selection of the optimal learning gain k r that satisfies (5.27) and the DF stability criterion described in the next section. It is important to note that the selection of the repetitive learning gain k r is a trade-off between robust stability and fast convergence of the tracking error. This means that selecting a small value for k r improves the stability margin of the system, while a larger value for k r gives a faster convergence rate and improved steady state tracking performance. However, this is accomplished at the cost of decreased stability margin and stability robustness Design of the Nonlinear Controller To achieve the desired convergence rate and robust stability margin, an optimal value for k r that satisfies (5.27) is obtained using the DF approach as described 101

128 5 Discrete Sliding Mode Repetitive Control below [ ]. A practical guide for the selection of the boundary layer thickness δ is also provided below. Assuming a sufficiently fast sampling rate, the describing function of the saturation function described in (5.16) can be expressed as [172] N(A) = 2k sat π k sat [ ] sin 1 δ + δ 1 δ2 A A A 2 for A 1 δ for A > 1 δ (5.29) where N(A) denotes the describing function of the saturation function and A is the amplitude of the sinusoidal input to the saturation function. In order to derive the DF stability criterion, the closed loop transfer function of the system with the saturation function also needs to be derived. Moreover, by replacing the saturation function with N(A), the closed loop transfer function of the entire system can be represented as G(z) = [G 1(z) + G RC (z)] G 2 (z)p (z) 1 + [G 1 (z) + G RC (z)] G 2 (z)p (z) (5.30) where the transfer function of G 1 (z) with the saturation function is equivalent to and G 1 (z) = (c 1T s + c 2 )z c 2 (c 1 T s + c 2 )z c 2 + l c 2 z + c 1 T s c 2 (c 1 T s + c 2 )z c 2 (5.31) l = 1 qt s k s T s N(A) (5.32) By denoting J(z) = (c 1T s+c 2 )z c 2 (c 1 T s+c 2 )z c 2 = 1 and K(z) = c 2z+c 1 T s c 2 (c 1 T s+c 2 )z c 2, the expression for G 1 (z) in (5.31) can be simplified to G 1 (z) = J(z) + l K(z) (5.33) Moreover, the characteristic equation of (5.30) can be rewritten as given bellow 1 + [J(z) + l K(z) + G RC (z)] G 2 (z)p (z) = 0 (5.34) 102

129 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain By letting z = e jωts, the characteristic equation in (5.34) above can be simplified as follows H(jω) = k s T s K(jω)G 2 (jω)p (jω) 1 + [J(jω) + G RC (jω) + (1 qt s )K(jω)] G 2 (jω)p (jω) = 1 N(A) (5.35) According to the DF stability analysis approach [ ], the closed loop system (5.30) is stable and there are no limit cycles if there is no intersection between the Nyquist plot of H(jω) and the inverse of the saturation describing function denoted by 1 N(A) within the entire uncertainty range. Moreover, for achieving robust stability, the value for k r needs to be selected so that to the DF stability condition described above is satisfied over the entire uncertainty range. The distance of 1 N(A) from the imaginary axis is determined directly by the boundary layer thickness δ, where a larger boundary layer thickness will lead to larger robust stability margins; however, at the cost of slow convergence of the error states. Consequently, the boundary layer thickness δ needs to be selected to satisfy the following inequality [170] η < δ < σ(k) max (5.36) where σ(k) max represents the maximum value of the sliding mode function defined in (5.3) and η is the critical value where 1 N(A) intersects H(jω) Application to a Linear Actuator The following section discusses the application of the proposed robust SMRC structure to a linear actuator (LA) plant with system dynamics as defined in (3.6). The nominal discrete transfer function of the LA plant is as follows G p (z) = z z 2 2z + 1 (5.37) Furthermore, the sliding mode error states are defined as follows ẽ(k) = ẽ(k) ẽ(k+1) ẽ(k) T s (5.38) 103

130 5 Discrete Sliding Mode Repetitive Control For achieving the desired system dynamics the sliding mode surface parameters are selected as follows c T = [ c 1 c 2 ] = [ ] (5.39) Moreover, the sliding mode switching parameter k s is selected as 3.8 and the value for q is chosen to be 200. While, the boundary layer thickness denoted by δ is chosen to be The phase plot of k r z m G c (z) for different m values is next plotted to guide the selection of an optimal compensastor which can effectively cancel the phase lag caused by the phase uncertainties. As shown in Figure 5.4, the best phase lag compensation over the entire uncertainty range can be achieved when the order of the pure phase lead compensator m is set to 1. Using the DF stability analysis approach described previously, the next step of the design involves selecting the maximum value for k r that satisfies (5.27) in the range from lower to upper bounds of uncertainty. By applying a trial and error approach, the optimal value for k r which satisfies both conditions over the entire uncertainty range is determined to be 0.2. Moreover, from the Nyquist plots of H(jω) and the inverse of the saturation describing function for no load, small load and maximum load conditions given in Figure 5.5, Figure 5.6 and Figure 5.7 respectively, we can clearly see that the locus of 1 1 (the line along the negative real axis which starts at point ( N(A) k sat, 0) for A > δ ) does not intersect the locus of H(jω) for k r = 0.2. This indicates that for the selected k r range. value, the LA system is stable within the entire payload uncertainty 104

131 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain (a) No payload (b) With payload Figure 5.4: Phase response of z m G c (z) 105

132 5 Discrete Sliding Mode Repetitive Control Figure 5.5: Nyquist plot of H(jω) and 1/N(A) (no load) Figure 5.6: Nyquist plot of H(jω) and 1/N(A) (15kg load) 106

133 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain Figure 5.7: Nyquist plot of H(jω) and 1/N(A) (30kg load) Simulation Testing Simulation testing is performed to verify the effectiveness of the proposed control structure. A comparison study is also presented to show the effectiveness of the proposed control structure and highlight the improved tracking properties of the proposed controller when compared to the conventional ZPETC structure [102], as well as the conventional SMC structure. The sampling time used in the simulation testing is set to T s = s, while a triangular reference of period T = 0.6s and amplitude of ±500µm is applied to the LA model. The value of N is chosen to be To verify the effectiveness of the proposed control structure in the presence of nonlinearities, the nonlinear friction model defined in (3.2) is simulated using Simulink. Furthermore, large payload variations are also simulated to test the robustness of the proposed compensator to payload variations. Figure 5.9 and Figure 5.10 below show the tracking position and error of the system with the proposed discrete SMRC co-plotted with those of the conventional SMC structure at no load conditions and in the presence of payload variations, respectively. As seen from the obtained plots, the proposed SMRC structure shows improved tracking performance in the presence of both nonlinear friction and payload variations when compared to the conventional SMC scheme. Moreover, these plots clearly show that the proposed controller ensures improved friction compensation when compared to the conventional SMC structure. Likewise, from the steady-state error plots at no load and load conditions given 107

134 5 Discrete Sliding Mode Repetitive Control in Figure 5.9 and Figure 5.10 respectively, it is also evident that the proposed discrete SMRC structure has improved robustness to plant uncertainties when compared to the conventional ZPETC structure. (a) Tracking output (b) Tracking error Figure 5.8: Tracking output and error of SMC and SMRC (no load) 108

135 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain (a) Tracking output (b) Tracking error Figure 5.9: Tracking output and error of SMC and SMRC (30kg load) (a) Tracking output (b) Tracking error Figure 5.10: Tracking output and error at steady state of ZPETC and SMRC (30kg load) 109

136 5 Discrete Sliding Mode Repetitive Control Experimental Validation The effectiveness of the control structure designed in Section is also verified through experimental testing which is performed on the LA experimental setup with plant dynamics as described in the (3.1). A comparison study is also performed and presented in this section to compare the tracking performance of the proposed controller to that of the conventional SMC and the conventional ZPETC. As seen from the obtained error plots at no load and at maximum load conditions given in Figure 5.11 and Figure 5.12 respectively, the proposed control structure again shows improved tracking performance in the presence of nonlinear friction when compared to the conventional SMC structure. These properties of the proposed SMRC at evident at zero load conditions as well as in the presence of large payload variations. Furthermore, the error plots given in Figure 5.13 indicate that the proposed discrete-time SMRC structure also has improved robustness when compared the conventional ZPETC structure. Moreover, as seen from Figure 5.13, when a 6kg load is applied to the system the ZPETC based system becomes unstable. An overview of the steady state error RMS and P-P values of the three control structures for different payload variations is provided in Table 5.1 and Table 5.2. The obtained numerical results in Table 5.1 and Table 5.2 also indicate the superior robustness properties of the proposed SMRC when compared to both the conventional SMC and the ZPETC structure. As seen from Table 5.1, at nominal load conditions the steady state error RMS of the proposed controller and the conventional SMC are 2.26µm and µm respectively. While, when a 6kg load is added to the system, the steady state error RMS values of the proposed controller and the conventional SMC are 2.31µm and µm respectively. This again highlights the superior friction compensation capabilities of the proposed SMRC when compared to the conventional SMC. 110

137 5.2 Design of a Robust Sliding Mode Repetitive Controller in the Frequency Domain (a) SMC (b) SMRC Figure 5.11: Tracking error of SMC and SMRC (0kg load, Amp =±500µm) (a) SMC (b) SMRC Figure 5.12: Tracking error of SMC and SMRC (6kg load, Amp =±500µm) 111

138 5 Discrete Sliding Mode Repetitive Control (a) ZPETC (b) SMRC Figure 5.13: Tracking error of ZPETC and SMRC(6kg load, Amp =±500µm) Payload (kg) SMC SMRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) Table 5.1: Steady State Error Analysis of SMC and SMRC Payload (kg) ZPETC SMRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) unstable unstable unstable unstable Table 5.2: Steady State Error Analysis of ZPETC and SMRC 112

139 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator Plant Modelling The dynamics of the second order nonlinear LA plant shown in Figure 3.1 can also be represented as follows ẍ = bu f(x) + l(x, t) (5.40) where u represents the control force acting on the linear actuator, x is the position output, b represents the control gain, f(x) is an unknown periodic function, and l(x, t) is the total lumped disturbance and uncertainty present in the system. Moreover, f(x) + l(x, t) γ, where γ represents the upper additive uncertainty bound obtained in Section Furthermore, the discrete state space model of the second order nonlinear LA plant in Figure 3.1 can be represented as follows x(k + 1) = Ax(k) + Bu(k) f(k) + l(x(k), k) (5.41) y(k) = Cx(k) (5.42) where A, B and C are matrices of suitable dimensions, u(k) represents the control input, x(k) is the position output, f(k) is an unknown periodic function and l(x(k), k) is the total lumped disturbance and uncertainty present in the system. Moreover f(k) + l(x(k), k)) γt s, where T s is the sampling period. Since the total uncertainty present in the LA system is bounded, the following assumption can also be made regarding the plant dynamics in (5.41): Assumption 1 : The unknown function f(k) is a function of the reference signal and is bounded by periodic functions as follows f min (k) f(k) f max (k) (5.43) 113

140 5 Discrete Sliding Mode Repetitive Control Design of Discrete Terminal Sliding Mode Repetitive Control As depicted in Figure 5.14, the proposed discrete TSMRC combines discrete TSMC with RC to improve the tracking of the periodic reference trajectory r in the presence of nonlinearities and parameter uncertainties. Since the desired trajectory signal r is periodic we have r(k T ) = r(k) (5.44) where T is the period of the periodic reference signal. Furthermore, the unknown periodic function f is also periodic. Hence, f(k T ) = f(k). Figure 5.14: Discrete TSMRC Let ˆf(k) denote the estimate of the periodic function f(k). Under Assumption 1, the following control update law can be applied to estimate f(k) ˆf(k) = B 1 A ˆϕ(k) (5.45) where ˆϕ(k) is a projection based repetitive control update law [174]. Furthermore, ˆϕ(k) can be represented as ˆϕ(k) = P roj (u l (k)) (5.46) where u l is the repetitive update rule and P roj( ) denotes the projection mapping defined as 114

141 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator P roj( ) = ϕ max ϕ min > ϕ max < ϕ min (5.47) ϕ min ϕ max where ϕ max and ϕ min are design constants. As seen from the modified RC structure given in Figure 5.15, the repetitive learning control law u l can be represented as follows u l (k) = Qu l (k N) + k r Qe(k N + m) (5.48) where e = r y represents the tracking error, Q is a low pass filter added to improve the robustness of the RC system to unmodelled dynamics and uncertainties present at high frequencies, k r is the learning gain, while m is the order of the digital phase lead filter z m [56] which is added to improve the high bandwidth tracking performance of the system in the presence of bounded parameter uncertainties. Furthermore, N = T T s N represents the number of samples per period. Figure 5.15: Modified RC structure The discrete-time sliding function considered in the design of the TSMRC is defined as follows σ(k) = cẽ(k) + βẽ p (k 1) (5.49) where σ(k) is a nonlinear sliding function, ẽ(k) = [ ] T e 1 (k) e 2 (k), e1 (k) = x 1 (k) r is the position tracking error and e 2 (k) = x 2 (k) ṙ is the velocity error. Furthermore, c = [ ] c 1 c 2 defines the parameters of the sliding function while β and p are positive design constants. Moreover, p < 1 and the value of p is selected as a rational number with an odd numerator and denominator. 115

142 5 Discrete Sliding Mode Repetitive Control To derive the total TSMRC control law using the equivalent control method described in [133] we need to consider the forward expression of (5.49) which can be represented as follows σ(k + 1) = cẽ(k + 1) + βẽ p (k) (5.50) Next the periodic function estimate in (5.45) is incorporated into the sliding mode reaching law to ensure robustness to periodic uncertainties. Moreover, the following reaching law is used to insure robustness of the system to both non-periodic and periodic uncertainties σ(k + 1) = (1 qt s ) σ(k) (ε + γ) T s sgn (σ) + ca ˆϕ(k) (5.51) where ε > 0 is a switching parameter, q > 0 is a converging parameter and 0 < 1 qt s < 1. Furthermore, the system dynamics need to be represented in terms of the error dynamics as given below ẽ(k + 1) = x(k + 1) r(k + 1) (5.52) Assuming the system uncertainties are negligible, by substituting (5.41) into (5.52) we get the following expression ẽ(k + 1) = Ax(k) + Bu(k) r(k + 1) (5.53) Moreover, as x(k) = ẽ(k) + r(k), the relationship in (5.53) can be extended to ẽ(k + 1) = Aẽ(k) + Ar(k) + Bu(k) r(k + 1) (5.54) Since the objective of the discrete-time TSMRC controller design is to obtain a control update law which is achieved when σ(k+1) = 0, we substitute the expression in (5.54) into (5.50) and equate the resulting expression to zero. The following total TSMRC control law is obtained 116

143 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator u(k) = (cb) 1 [cr(k + 1) car(k) caẽ(k) + (1 qt s )σ(k) (ε + γ) T s sgn (σ) + ca ˆϕ(k) βẽ p (k)] (5.55) By representing u l (k) in terms of ẽ(k) the TSMRC control law in (5.55) can be represented as u(k) = (cb) 1 [cr(k + 1) car(k) caẽ(k) + (1 qt s )σ(k) (ε + γ) T s sgn (σ) ca ˆϕ e (k) βẽ p (k)] (5.56) where ˆϕ e (k) = P roj (u le (k)) (5.57) and u le (k) = Qu l (k N) k r Qẽ(k N + m) (5.58) Robust Stability Analysis According to the discrete-time sliding mode reaching condition [132, 175], the system states will converge to the sliding surface if the following reaching condition is satisfied σ(k + 1) < σ(k) (5.59) where the condition given in (5.59) is equivalent to the following two conditions σ(k + 1) σ(k) sgn (σ(k)) < 0 (5.60) σ(k + 1) + σ(k) sgn (σ(k)) > 0 (5.61) 117

144 5 Discrete Sliding Mode Repetitive Control By substituting (5.51) into the conditions in (5.60) and (5.61) we obtain the following equivalent reaching conditions qt s σ(k) (ε + γ) T s sgn (σ) ca ˆϕ e (k) sgn (σ(k)) < 0 (5.62) (2 qt s ) σ(k) (ε + γ) T s sgn (σ) ca ˆϕ e (k) sgn (σ(k)) > 0 (5.63) Since ca ˆϕ e (k) < DT s, where D is a positive constant determined by the design, the inequalities in (5.62) and (5.63) can be simplified as follows σ(k) > (ε + γ) T s + DT s 2 qt s (5.64) σ(k) < (ε + γ) T s + DT s 2 qt s (5.65) The above stability conditions indicate that the system states will converge from their initial state to a region around the sliding surface in finite time [176]. This region around the sliding surface denoted by is known as the quasi-sliding mode band (QSMB) and can be further represented as ( (ε + γ)t s + DT s 2 qt s, ) (ε + γ)t s + DT s 2 qt s (5.66) Once the system states enter this band they will be driven into a smaller band around the sliding surface by the repetitive control law Tuning of the Repetitive Control Once the system states reach the initial QSMB described in (5.66) they are forced to asymptotically converge to a smaller region around the sliding surface by the repetitive update control law. Moreover, once the system state reach the initial QSMB, the fast switching components of the total control law in (5.55) which were added to force the error states in the QSMB region can be treated as a small additive uncertainty which can be effectively compensated by the repetitive update rule. 118

145 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator By treating the switching terms of u(k) as a negligible system uncertainties, the total control law given in (5.55) can be represented by the following linear control law approximation u(k) = (cb) 1 [cr(k + 1) car(k) caẽ(k) + ca ˆϕ(k)] (5.67) Moreover, since r(k + 1) = zr(k), the control law in (5.67) can also be represented as u(k) = (cb) 1 [czr(k) car(k) caẽ(k) + ca ˆϕ(k)] (5.68) Simplifying the relationship in (5.68) gives u(k) = (cb) 1 [(cz ca) r(k) ca (ẽ(k) ˆϕ(k))] (5.69) Since ẽ(k) = e(k), the expression in (5.69) can be represented as u(k) = (cb) 1 [(cz ca) r(k) + ca (e(k) + ˆϕ(k))] (5.70) The expression in (5.70) can also be represented in Z-domain as follows U(z) = (cb) 1 [ (cz ca) R(z) + ca ( E(z) + ˆΦ(z) )] (5.71) A block diagram of the system with a control law given in (5.71) is given in Figure Figure 5.16: Block diagram of the linear TSMRC system. 119

146 5 Discrete Sliding Mode Repetitive Control where G rc is the transfer function of the repetitive controller, while G 1 (z), G 2 (z) and P (z) represent the linear TSMC system and are defined as follows G 1 (z) = c 2z 2 2c 2 z + c 2 T s (5.72) G 2 (z) = (c 1T s + c 2 ) z c 2 T s (5.73) P (z) = 2 bt s (c 1 T s + 2c 2 ) G p(z) (5.74) and G p (z) represents the discrete transfer function of the linear LA plant. Furthermore, the closed loop transfer function of the system without the repetitive controller can be written as follows G C (z) = G 2(z)P (z) 1 + G 2 (z)p (z) (5.75) Since the TSMC control law in (5.71) is designed to ensure G c (z) is stable, the stability of the overall system as well as the asymptotic convergence of the system states to a smaller bound around the sliding surface can be achieved if the following small-gain conditions are satisfied at all frequencies [56, 115] Q(z) < 1 (5.76) (1 k r z m G c (z)) Q(z) < 1 (5.77) Where, to achieve the stability condition (5.77), the low pass filter Q(z) should be designed as a moving average filter with zero phase contribution [56]. A detailed analysis of the stability condition in (5.77) in the frequency domain is presented in [56, 59]. Assuming the system plant uncertainties are compensated by the TSMC component, the stability condition in (5.77) is equivalent to the following magnitude and angle conditions 0 < k r < 2 cos (θ G c (jω) + θ m (jω)) M Gc (jω) (5.78) 120

147 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator θ Gc (jω) + θ m (jω) < π 2 (5.79) where M Gc (jω) and θ Gc (jω) represent the magnitude and phase of G c (z) respectively. While, θ m (jω) represents the phase of the pure phase lead filter k r z m. The angle condition (5.79) suggests that the value of the parameter m should be selected so that z m minimizes the phase lag of the feedback system in the frequency range up to the cut-off frequency of Q(z). This will improve the tracking performance of the system, especially the tracking of reference signals with high frequency components such as the triangular periodic reference signal, and hence reduce the QSM bound defined in (5.66). Furthermore, the selection of k r should be done so that the magnitude condition (5.78) is satisfied for all frequencies up to the cut-off frequency of Q(z). It is important to note that the selection of k r provides a trade-off between overall robust stability and fast convergence. Moreover, a larger k r value ensures fast convergence of the system states to the reduced QSM region; however, this is achieved at the cost of decreased robust stability margin Application to a Linear Actuator The discretised second order nonlinear LA plant in (5.41) is considered in this section. The nominal discrete transfer function of the LA plant can be represented as follows G p (z) = z z 2 2z + 1 (5.80) First the design parameters of the proposed TSMRC are selected to achieve closed-loop stability and satisfactory tracking performance as follows: c = [ ] 15 3, p = 0.9, β = [ ] 0.6 0, q = 0.01, ε = 0.46 and γ = Furthermore, a low pass filter Q(z) is selected to achieve the required RC bandwidth i.e Q(z) = 0.25z +0.5 = 0.25z 1 [56, 115]. The optimal value of m is next selected by co-plotting and analysing the phase plots of z m G c (z) for different m. As seen from the plots of z m G c (z) given in Figure 5.17, close to perfect phase cancellation in the frequency range up to the cut-off frequency of Q(z) and thus improved high bandwidth tracking performance in the presence of phase lag due to plant uncertainties can be achieved when m =

148 5 Discrete Sliding Mode Repetitive Control Figure 5.17: Phase response of z m G c (z). Finally the learning gain k r is selected to satisfy the condition in (5.78). The optimal value of k r that satisfies (5.78) is selected as k r = Simulation Testing Simulation results are presented in this section to verify the feasibility and effectiveness of the proposed control structure when applied to a LA system with nonlinear friction and plant uncertainties. Moreover, the tracking performance of the proposed TSMRC to a periodic reference signal r of period T = 0.6s and magnitude ±500µm is analysed and compared to that of the conventional TSMC and ZPETC [102] structures. The sampling interval and the ratio N are selected as s and 1200 respectively. The tracking performance of the system with the proposed and the conventional TSMC and ZPETC structures is shown in Figure 5.18 and Figure As depicted in Figure 5.18, the proposed TSMRC has a faster convergence rate and improved tracking performance in the presence of nonlinear friction and load uncertainties when compared to the conventional TSMC structure. Furthermore, as seen from Figure 5.18, the inclusion of the RC component into the TSMC improves the friction compensation capabilities of the system. Additionally, as depicted in Figure 5.19, the proposed TSMRC also shows improved robustness when compared to the ZPETC structure. Moreover, when large loads are applied, the system with ZPETC structure becomes unstable while the system using the proposed TSMRC controller maintains satisfactory tracking performance. 122

149 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator (a) Tracking output (b) Tracking error Figure 5.18: Tracking output and error of TSMC and TSMRC (30kg load) (a) Tracking output (b) Tracking error Figure 5.19: Tracking output and error at steady state of ZPETC and TSMRC (30kg load) 123

150 5 Discrete Sliding Mode Repetitive Control Experimental Validation The efficacy of the proposed TSMRC is also verified through experimental testing using the LA system shown in Figure 3.1. During the experiments payloads ranging from 0kg-6kg were added to the available experimental setup to test the robustness and tracking properties of the proposed controller. A comparison analysis which compares the tracking performance of the proposed controller to that of the conventional TSMC and ZPETC structure is also presented in this section. The results from the experimental testing are presented in Figure 5.20 and Figure As seen from the error plots given in Figure 5.20, the system with the proposed TSMRC structure has a faster transient response and improved tracking performance in the presence of nonlinear friction and load uncertainties when compared to the system with the conventional TSMC structure. From this plot it is also evident that the RC based TSMC structure can effectively eliminate the periodic components of the nonlinear friction. From the error plots given in Figure 5.21 it can also be seen that the proposed TSMRC structure has improved robust tracking performance in the presence of both nonlinear friction and parametric uncertainties when compared to the ZPETC structure. Moreover, when a 6kg load is applied to the system the ZPETC based structure becomes unstable. Additionally, the numerical steady state error RMS and P-P values for different payloads summarised in Table 5.3 and Table 5.4 also show that the system with the proposed TSMRC has improved robust tracking performance when compared to the TSMC and ZPETC structures for a wide range of payload uncertainties. Moreover, at zero load conditions the steady state error RMS of the proposed TSMRC and the conventional TSMC are 1.49µm and µm respectively. While, the steady state error P-P values recorded for the TSMRC and TSMC are 10.67µm and 851.0µm respectively. Similar results are obtained when a 3kg and a 6kg load is applied to the LA system. This again shows that the proposed controller has improved friction compensation properties when compared to the conventional TSMC. 124

151 5.3 Discrete Terminal Sliding Mode Repetitive Controller for a Linear Actuator (a) TSMC (b) TSMRC Figure 5.20: Tracking position of TSMC and TSMRC(6kg load, Amp =±500µm) (a) ZPETC (b) TSMRC Figure 5.21: Steady state tracking error of ZPETC and TSMRC(6kg load, Amp =±500µm) 125

152 5 Discrete Sliding Mode Repetitive Control Payload (kg) TSMC TSMRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) Table 5.3: Steady State Error Analysis of TSMC and TSMRC Payload (kg) ZPETC TSMRC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) unstable unstable unstable unstable Table 5.4: Steady State Error Analysis of ZPETC and TSMRC 5.4 Summary This chapter presents the design of two new discrete SMRC methods where the SMC part is added to improve the convergence and robustness of the system against non-periodic uncertainties, while the RC component is added to eliminate the periodic uncertainties. The first section of this chapter proposes a simple and easy-to-apply design of a robust SMRC in the frequency domain. In the proposed design approach the conventional RC frequency domain design method is applied to obtain the optimal RC parameters, while the DF stability analysis technique is used to design the nonlinear controller and ensure robust stability. Simulation and experimental results are presented in this section to show the efficacy of the proposed controller when applied to a LA system with plant uncertainties and nonlinear friction. The obtained results show that the proposed controller has superior robustness properties and improved tracking accuracy in the presence of both nonlinear friction and parametric uncertainties when compared to SMC and ZPETC. The second section of this chapter proposes the design of a new discrete TSMRC for improved robust tracking performance and fast transient response in LA systems with parameter uncertainties and nonlinearities. The proposed controller incorporates RC and TSMC to achieve improved robust performance and fast dynamic response in the presence of bounded parameter uncertainties caused by payload variations, as well as periodic and non-periodic nonlinearities. Unlike existing TSMC structures, the proposed TSMRC controller incorporates the RC component to eliminate the effects of the periodic nonlinearities present in the system. A small discretetime phase lead filter is also added to the RC structure to improve the convergence 126

153 5.4 Summary rate and the high bandwidth tracking performance of the overall system in the presence of payload variations. Simulation and experimental testing is performed to verify the effectiveness of the proposed controller when applied to a LA system with payload variations and nonlinear friction. The obtained results show significant improvement in convergence speed, robustness and robust tracking performance of the proposed controller when compared to the conventional TSMC and the ZPETC structures for a wide range of payload variations. The experimental results also indicate that the proposed TSMRC controller has improved friction compensation capabilities when compared to the conventional TSMC structure. Despite the improved convergence and robustness properties of the two controllers proposed in this chapter, SMC based RC methods require an accurate knowledge of the plant dynamics which is not always possible to accurately measure and model. These constraints of SMC based methods limit their application to systems where the uncertainties, nonlinearities and disturbances are measurable or are known in advance. However, in many servo applications the nonlinearities in the system dynamics such as the nonlinear friction and hysteresis cannot be accurately measured and modelled. In such instances an observer based control method is required to estimate and eliminate the unknown uncertainties and nonlinearities which cannot be compensated by the SMC method, or the RC structure in applications of a repetitive nature. Motivated by these limitations of the SMC based RC methods proposed in this chapter, Chapter 6 proposes the design of an observer based TSMC and an observer based TSMRC for improved tracking performance and robustness in LA systems with unknown uncertainties and nonlinearities. 127

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155 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control 6.1 Introduction Chapter 5 of this thesis proposes the design of SMC based RC methods for improved robustness in LA systems subjected to payload uncertainties and nonlinear friction. Although the RC method can be used together with SMC methods to eliminate the repeatable components of the unknown nonlinear friction, these structures cannot fully eliminate the non-periodic components of the friction and thus are limited only to applications of a repetitive nature. Several observer-based variable structure control structures have been developed for estimating the unknown uncertainties in applications where the disturbances, plant uncertainties and the nonlinearities present in the system are unknown or cannot be accurately measured [139, 140]. Although different observer frameworks can be employed to estimate the unknown plant uncertainties and disturbances, many of these are also model-based and require an accurate estimate of the plant dynamics. Unlike model-based observer structures, the ESO framework [69] only requires the relative order of the plant. Moreover, in applications where the internal states of the system cannot be accurately measured, the ESO structure can also be used to estimate the unknown states of the system. Despite the effectiveness of ESO based SMC methods [75 90] in estimating and eliminating unknown uncertainties in applications which require tracking of aperiodic references, these structures alone cannot fully eliminate the periodic uncertainties which are common in many repetitive applications. To fully eliminate the periodic uncertainties and improve the tracking of periodic references, ESO/ NLESO based SMC methods can to be integrated together with learning control methods such as the repetitive control (RC). 129

156 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control This chapter proposes the design of two discrete NLESO based SMC methods. Firstly a NLESO based TSMC structure is proposed for dealing with unknown nonlinear friction and parameter uncertainties in LA systems used for tracking of nonperiodic references. Secondly a new discrete-time NLESO based TSMRC control structure is proposed for improved robustness, fast convergence rate and improved steady state tracking performance in LA systems which are required to track both periodic and aperiodic references. The rest of this chapter is organised as follows. The design and analysis of the discrete NLESO based TSMC is provided in Section 2. Section 3 presents the design and analysis of the discrete NLESO based TSMRC. A summary of the research contributions presented in this chapter are given in Section Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction Design of Discrete NLESO Based TSMC This section presents the design of a discrete NLESO based TSMC with an application to a LA system as described in (3.6). The total uncertainty present in the LA system in (3.6) can be represented as follows ω(x(k), k) = g(k) d(x(k), k) (6.1) where the following assumption can be made regarding the system dynamics: Assumption 1 : The total lumped uncertainty ω(x(k), k) and its derivative ρ(k) are bounded. Moreover, ω(x(k), k) < γt s (6.2) where T s is the sampling period and γ is an known positive constant obtained in Section A diagram of the proposed NLESO based TSMC structure is given in Figure 6.1. As depicted in Figure 6.1, the proposed control structure consists of a TSMC and a NLESO component. Firstly the TSMC component is designed to ensure fast 130

157 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction transient response. Then the NLESO is added to the TSMC structure to estimate and eliminate the effects of the unknown parameter uncertainties and nonlinearities. Figure 6.1: NLESO based TSMC structure [133] The proposed discrete TSMC can be described by the following sliding function σ(k) = ce(k) + ke p (k 1) (6.3) where σ(k) represents the nonlinear sliding function, e(k) = [ ] T e 1 (k) e 1 (k), e 1 (k) = x 1 (k) r is the position tracking error, e 2 (k) = x 2 (k) ṙ is the velocity error, r is the reference signal and ṙ is the first derivative of the reference trajectory. Furthermore, c = [ ] [ ] c 1 c 2 and k = k 2 represent the sliding function parameters, while p is a positive design constant selected as a rational number with an odd numerator and denominator. Moreover, p < 1. In order to design the TSMC law we need to satisfy σ(k +1) = 0, where σ(k +1) can be represented as follows k 1 σ(k + 1) = ce(k + 1) + ke p (k) (6.4) follows Furthermore, a reaching law is designed to ensure fast transient response as σ(k + 1) = (1 qt s ) σ(k) ηt s sgn (σ) c 1ˆω(x(k), k) (6.5) where η > 0 is a switching parameter, q > 0 is a converging parameter and 0 < 1 qt s < 1. Furthermore, ˆω(x(k), k) is an estimate of the total lumped uncertainty present in the system ω(x(k), k) which is defined in (6.1). 131

158 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control The error dynamics of the system in (3.6) can be represented as follows e(k + 1) = x(k + 1) r(k + 1) (6.6) Assuming the system uncertainties in (3.6) are negligible, by substituting (3.6) into (6.6) we get the following equivalent expression for the error dynamics e(k + 1) = Ax(k) + Bu(k) r(k + 1) (6.7) Since x(k) = e(k) + r(k), the expression (6.7) can be rewritten as e(k + 1) = Ae(k) + Ar(k) + Bu(k) r(k + 1) (6.8) Substituting the expression in (6.8) into (6.4) gives the following equivalent expression for σ(k + 1) σ(k + 1) = cae(k) + car(k) + cbu(k) cr(k + 1) + ke p (k) (6.9) Furthermore, by equating (6.9) and (6.5), the following TSMC control law can be derived u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) c 1ˆω(x(k), k) + (1 qt s )σ(k) ηt s sgn (σ) ke p (k)] (6.10) Given the TSMC control law in (6.10), the next step involves the design of a discrete NLESO to accurately estimate and eliminate the effects of ω(x(k), k). Firstly an additional state x 3 (k) = ω(x(k), k), where x 3(k+1) x 3 (k) T s = ρ(k), is added to the system dynamics in (3.6). The following discrete NLESO structure is then adopted to estimate ω(x(k), k) ε 1 (k) = x 1 (k) z 1 (k) z 1 (k + 1) = z 1 (k) + T s (z 2 (k) + β 1 ε 1 (k)) z 2 (k + 1) = z 2 (k) + T s (z 3 (k) + bu(k) + β 2 ε 1 (k)) z 3 (k + 1) = z 3 (k) + T s β 3 fal (ε 1, α, δ) (6.11) 132

159 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction where ε 1 represents the observer estimation error; z 1, z 2, and z 3 are the observer outputs, while β 1, β 2, and β 3 are the observer gains. Moreover, z 1 and z 2 are the estimated system position and velocity states respectively, while z 3 represents the estimate the total disturbance in the system ω(x(k), k). Furthermore, the f al function is a nonlinear function defined as ε δ fal(ε, α, δ) = 1 α ε α sign(ε) ε δ ε > δ (6.12) where ε is the error input to the fal function. While, α and δ are small positive design parameters. Given the estimate of the total uncertainty in the system denoted by z 3, the total TSMC control law can be represented as follows u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) c 1 z 3 + (1 qt s )σ(k) ηt s sgn (σ) ke p (k)] (6.13) Stability Analysis The stability of the closed loop system can be guaranteed by the following theorem [177]: Theorem 1 : Considering the system defined in (3.6), the ESO described in (6.11) and the sliding mode control law given in (6.13), there exist observer gains β 1, β 2, β 3, α and δ which will force the estimated states z 1, z 2, and z 3 to converge into a residual set of the ESO states x 1, x 2, and ω respectively, and the system states will be driven onto the sliding surface in finite time and converge into a region around the origin. To analyse the stability of the NLESO we first need to consider the observer error dynamics as given below ε 1 (k) = x 1 (k) z 1 (k) ε 2 (k) = x 2 (k) z 2 (k) ε 3 (k) = x 3 (k) z 3 (k) (6.14) 133

160 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control where the derivatives of the estimation errors can be represented as follows ε 1 (k+1) ε 1 (k) T s ε 2 (k+1) ε 2 (k) T s ε 3 (k+1) ε 3 (k) T s = x 1(k+1) x 1 (k) T s = x 2(k+1) x 2 (k) T s = x 3(k+1) x 3 (k) T s z 1(k+1) z 1 (k) T s z 2(k+1) z 2 (k) T s z 3(k+1) z 3 (k) T s (6.15) By simplifying (6.15) above we have ε 1 (k + 1) ε 1 (k) = x 2 (k) z 2 (k) β 1 ε 1 (k) T s ε 2 (k + 1) ε 2 (k) = x 3 (k) z 3 (k) β 2 ε 1 T s ε 3 (k + 1) ε 3 (k) = ρ(k) β 3 fal (ε 1, α, δ) T s (6.16) Furthermore, by simplifying (6.16) we get ε 1 (k + 1) ε 1 (k) = ε 2 (k) β 1 ε 1 (k) T s ε 2 (k + 1) ε 2 (k) = ε 3 (k) β 2 ε 1 (k) T s ε 3 (k + 1) ε 3 (k) = ρ(k) β 3 fal (ε 1, α, δ) T s (6.17) When the observer is stable, the vector consisting of the observer error derivatives is equal to zero. Moreover, ε 2 (k) β 1 ε 1 (k) = 0 ε 3 (k) β 2 ε 1 (k) = 0 ρ(k) β 3 fal (ε 1, α, δ) = 0 (6.18) By simplifying (6.18) we get the following expressions for the observer errors ( ) ρ ε 1 (k) = fal 1 β 3 ( ) ρ ε 2 (k) = β 1 fal 1 β 3 ( ) ρ ε 3 (k) = β 2 fal 1 β 3 (6.19) 134

161 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction If ε 1 (k) > δ, the estimation errors can be represented as ρ 1/α ε 1 (k) = β 3 ρ 1/α ε 2 (k) = β 1 β 3 ρ 1/α ε 3 (k) = β 2 β 3 (6.20) While, if ε 1 (k) δ, the estimation are ρδ 1 α ε 1 (k) = β 3 ρδ 1 α ε 2 (k) = β 1 β 3 ρδ 1 α ε 3 (k) = β 2 β 3 (6.21) As seen from (6.19), (6.20) and (6.21), the lumped uncertainty estimation error ε 3 is determined by the parameters β 2, β 3, α and δ. Moreover, by tuning these parameters as described in [70], the estimation error ε 3 can be reduced which indicates that the observed state z 3 will converge to a small region around the actual state ω. After proving that the estimated state z 3 converges into a residual set of ω, we are still left to prove that the system states converge to the sliding surface in finite time. To analyse the stability of the TSMC component we consider the well-known reaching condition for discrete VSC systems [132, 175]. According to the discretetime sliding mode reaching condition the system states will converge to the sliding surface if the following condition is satisfied σ(k + 1) < σ(k) (6.22) where the condition given in (6.22) is equivalent to the following two conditions σ(k + 1) σ(k) sgn (σ(k)) < 0 (6.23) 135

162 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control σ(k + 1) + σ(k) sgn (σ(k)) > 0 (6.24) By substituting (6.5) into (6.23) and (6.24) we can obtain the following equivalent reaching conditions qt s σ(k) ηt s sgn (σ) c 1ˆω(k) sgn (σ(k)) < 0 (6.25) (2 qt s ) σ(k) ηt s sgn (σ) c 1ˆω(k) sgn (σ(k)) > 0 (6.26) follows Moreover, since z 3 = ˆω(k), the conditions (6.25) and (6.26) can be rewritten as qt s σ(k) ηt s sgn (σ) c 1 z 3 (k) sgn (σ(k)) < 0 (6.27) (2 qt s ) σ(k) ηt s sgn (σ) c 1 z 3 (k) sgn (σ(k)) > 0 (6.28) Since previously we proved that z 3 converges into a residual set of ω which is determined by the selection of the parameters β 2, β 3, α and δ, we can also conclude that c 1 z 3 is also bounded i.e c 1 z 3 (k) < ΓT s, where Γ is a small positive constant. By selecting ηt s > ΓT s, the conditions in (6.27) and (6.28) can be simplified to σ(k) > ηt s + ΓT s 2 qt s (6.29) σ(k) < ηt s + ΓT s 2 qt s (6.30) From the conditions in (6.29) and (6.30) we can see that the system error states will converge from their initial state to the QSM region around the sliding surface which can be represented as follows [176]. ( ηt s + ΓT s 2 qt s, ) ηt s + ΓT s 2 qt s (6.31) 136

163 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction Simulation Testing Simulation testing is performed and the obtained results are presented in this section to demonstrate the effectiveness of the proposed controller. The discretised second order nonlinear LA plant defined in (3.6) is used in the simulation testing. Furthermore, a comparison study is also presented in this section to show the superiority of the proposed controller when compared to the conventional TSMC. Firstly the discrete NLESO based TSMC structure given in (6.13) is designed to achieve satisfactory tracking performance in the presence of plant uncertainties and nonlinear friction. The observer gains and design parameters are first selected to guarantee the convergence of the estimated states i.e β 1 = 1500, β 2 = , β 3 = , α = 0.25 and δ = 0.1. To ensure closed-loop stability and satisfactory tracking performance, the design parameters of the TSMC controller are then selected as follows: c = [ ] [ ] 15 3, p = 0.9, k = 0.6 0, q = 0.01 and η = 0.5. The tracking performance of the proposed controller to a non-periodic reference signal in the presence of nonlinear friction and different payloads is shown in the plots given in Figure 6.2, Figure 6.3. and Figure 6.4. As depicted in Figure 6.2, the proposed controller shows faster convergence and improved robustness in the presence of nonlinear friction when compared to the conventional TSMC structure. Likewise, the output and error plots for different payloads given in Figure 6.3 and Figure 6.4 clearly show that the proposed controller has improved robust tracking performance and faster convergence in the presence of both large payload variations and nonlinear friction. 137

164 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control (a) Tracking output (b) Tracking error Figure 6.2: Tracking output and error of TSMC and NLESO based TSMC (no load, step input) (a) Tracking output (b) Tracking error Figure 6.3: Tracking output and error of TSMC and NLESO based TSMC (15kg load, step input) 138

165 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction (a) Tracking output (b) Tracking error Figure 6.4: Tracking output and error of TSMC and NLESO based TSMC (30kg load, step input) Experimental Validation The results from the experimental validation are presented in this section to verify the superiority of the proposed controller when applied to systems with nonlinear friction and plant uncertainties. The experimental validation is performed using the LA plant shown in Figure 3.1 and the design parameters selected in the previous section. A comparison study is also presented in this section where the performance of the proposed structure is compared to that of the conventional TSMC and a TSMC structure with a feedforward friction compensator based on the inverse of the friction model given in (3.2). The obtained error plots are presented in Figure 6.5, Figure 6.6 and Figure 6.7. As shown from the obtained error plots for the proposed, conventional TSMC and the TSMC with feedforward friction compensation given in Figure 6.5, Figure 6.6 and Figure 6.7 respectively, the proposed controller outperforms the two TSMC structures in the presence of both nonlinear friction and payload variations. Moreover, as seen from Figure 6.5, the friction nonlinearity is effectively observed and compensated by the proposed NLESO structure. Furthermore, as observed from Figure 6.6 and Figure 6.7, when large load is added to the system the proposed control structure maintains satisfactory tracking performance. This is also evident from 139

166 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control the obtained error RMS and error P-P values summarised in Table 6.1 and Table 6.2 where it can be seen that the steady-state error values of the system with the proposed controller are significantly reduced when compared to those of the system with the TSMC structures. Moreover, when a 6kg load is added to the system the error RMS value of the proposed controller are 2µm, while the error RMS of the TSMC structures is larger than 300µm. This strongly indicates that the inclusion of the NLESO improves the friction compensation capabilities of the system. (a) TSMC (b) TSMC + FF (c) NLESO based TSMC Figure 6.5: Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (no load, step input) 140

167 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction (a) TSMC (b) TSMC + FF (c) NLESO based TSMC Figure 6.6: Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (3kg load, step input) 141

168 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control (a) TSMC (b) TSMC + FF (c) NLESO based TSMC Figure 6.7: Tracking error of TSMC, TSMC + FF, and NLESO based TSMC (6kg load, step input) 142

169 6.2 Discrete NLESO Based TSMC for a Linear Actuator with Nonlinear Friction Payload (kg) TSMC NLESO based TSMC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) Table 6.1: Steady State Error Analysis of TSMC and NLESO based TSMC Payload (kg) TSMC + FF NLESO based TSMC RMS(µm) RMS(%) P-P(µm) P-P(%) RMS(µm) RMS(%) P-P(µm) P-P(%) Table 6.2: Steady State Error Analysis of TSMC + FF and NLESO based TSMC 143

170 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller Design of Discrete NLESO Based TSMRC This section presents the design of a new discrete NLESO based TSMRC structure. As shown in Figure 6.8 the proposed discrete NLESO based TSMRC structure consists of two design components. Firstly a discrete NLESO based TSMC is designed to achieve the desired robustness against the unknown non-repeatable uncertainties as described in Section 6.2. Then a plug-in RC structure is integrated into the NLESO based TSMC to eliminate the repeatable uncertainties. Figure 6.8: Discrete NLESO based TSMRC structure. The TSMC control law derived in Section 6.2 can be represented as follows u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) c 1 z 3 + (1 qt s )σ(k) ηt s sgn (σ) ke p (k)] (6.32) As shown in (6.32), the uncertainty estimate z 3 is added to the TSMC control law to eliminate the total unknown uncertainty present in the system. Although the ESO parameters can be tuned to minimize the disturbance estimation error ω, where ω(x(k), k) = ω(x(k), k) z 3, the NLESO based TSMC cannot fully eliminate the periodic error caused by the repeatable uncertainties present in the system [91]. Motivated by these limitations of NLESO based TSMC the controller proposed in this section incorporates the RC structure as shown in Figure 6.8 in order to eliminate the periodic components of the estimation error ω(x(k), k) which are caused by the nonlinear friction. 144

171 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller The total disturbance estimation error ω(x(k), k) can be represented as follows ω(k) = f(k) µ(k) (6.33) where f(k) represents the periodic while µ(k) represents the non-periodic components of ω(k) which are assumed to be negligible. For a periodic reference signal r, namely, r(k T ) = r(k) (6.34) where T represents the period of the periodic reference signal, the unknown periodic function f(k) can be represented as f(k T ) = f(k). Furthermore, the following assumption can be made regarding f(k): Assumption 2: The unknown function f(k) is a periodic function which is bounded by periodic functions as follows f min (k) f(k) f max (k) (6.35) By letting ˆf(k) denote the estimate of the periodic function f(k), the following control update law can be used to estimate ˆf(k) ˆf(k) = B 1 A ˆϕ(k) (6.36) where ˆϕ(k) is a repetitive control estimate which is designed to eliminate the periodic uncertainties present in the system. The following projection based repetitive control update law is used to update the estimate ˆϕ(k) [174] ˆϕ(k) = P roj (u l (k)) (6.37) where u l is the repetitive update rule and P roj( ) is a projection mapping function which can be represented as 145

172 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control P roj( ) = ϕ max ϕ min > ϕ max < ϕ min (6.38) ϕ min ϕ max where ϕ max and ϕ min are known functions. Furthermore, as shown in Figure 6.9, the repetitive control law u l can be represented as follows u l (k) = Qu l (k N) + k r Qẽ(k N + m) (6.39) where ẽ = r y represents the system error, while Q is a low pass filter added to improve the robustness of the RC system at high frequencies. The parameter k r is the learning gain, while m is the order of the digital phase lead filter z m [56] which is added to the RC structure to improve the high bandwidth tracking performance. Furthermore, N = T T s N is the number of samples per period. Figure 6.9: Modified RC structure In order to eliminate the repeatable components of the disturbance estimation error defined in (6.33), the TSMC reaching law given in (6.5) is modified to incorporate the estimate in (6.36) as follows σ(k + 1) = (1 qt s ) σ(k) ηt s sgn (σ) c 1ˆω(x(k), k) + ca ˆϕ(k) (6.40) The total control law for the modified reaching law represented in (6.40) is updated as follows 146

173 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) c 1 z 3 + ca ˆϕ(k) + (1 qt s )σ(k) ηt s sgn (σ) ke p (k)] (6.41) By representing u l (k) in terms of e(k), the TSMRC control law in (6.41) can be represented as u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) c 1 z 3 ca ˆϕ e (k) + (1 qt s )σ(k) ηt s sgn (σ) ke p (k)] (6.42) where ˆϕ e (k) = P roj (u le (k)) (6.43) and u le (k) = Qu l (k N) k r Qe(k N + m) (6.44) Robust Stability Analysis The stability of the discrete NLESO is analysed in Section where it was proved that the estimated error ω(k) converges into a residual set of zero by selecting appropriate values for the parameters β 2, β 3, α and δ. The stability of the TSMRC can be analysed using the reaching condition for discrete-time VSC systems [132, 175]. Moreover, based on the discrete sliding mode reaching condition the system error states will converge to the sliding surface if the following condition is met σ(k + 1) < σ(k) (6.45) where the condition in (6.45) is equivalent to the following two inequalities 147

174 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control σ(k + 1) σ(k) sgn (σ(k)) < 0 (6.46) σ(k + 1) + σ(k) sgn (σ(k)) > 0 (6.47) Substituting (6.40) into (6.46) and (6.47) above gives the following equivalent reaching conditions qt s σ(k) ηt s sgn (σ) c 1 z 3 (k) ca ˆϕ e (k) sgn (σ(k)) < 0 (6.48) (2 qt s ) σ(k) ηt s sgn (σ) c 1 z 3 (k) ca ˆϕ e (k) sgn (σ(k)) > 0 (6.49) As previously we proved that ε 3 (k) converges into a residual set of zero, we can also conclude that c 1 z 3 (k) < ΓT s, where Γ is a small positive constant. By selecting ηt s > ΓT s and since we also have ca ˆϕ e (k) < DT s, where D is a positive design parameter, the conditions in (6.48) and (6.49) can be simplified to σ(k) > ηt s + ΓT s + DT s 2 qt s (6.50) σ(k) < ηt s + ΓT s + DT s 2 qt s (6.51) From (6.50) and (6.51) it is evident that the system error states will converge from their initial state to the following QSM region [176]. ( ηt s + ΓT s + DT s 2 qt s, ) ηt s + ΓT s + DT s 2 qt s (6.52) Tuning of the Repetitive Controller When the system error states reach the QSM region defined in (6.52) they are forced into a smaller region around the sliding surface by the repetitive update control law. Therefore, at steady state the fast switching terms in (6.41) can be 148

175 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller treated as a small negligible additive uncertainty acting at the input of the system which can be easily compensated by the repetitive update rule. The total control law (6.41) at steady state is equivalent to u(k) = (cb) 1 [cr(k + 1) car(k) cae(k) + ca ˆϕ(k)] (6.53) Since r(k + 1) = zr(k), the control law in (6.53) can be represented as u(k) = (cb) 1 [czr(k) car(k) caẽ(k) + ca ˆϕ(k)] (6.54) Simplifying (6.54) gives u(k) = (cb) 1 [(cz ca) r(k) ca (e(k) ˆϕ(k))] (6.55) Since ẽ(k) = e(k), we can represent the expression in (6.55) as follows u(k) = (cb) 1 [(cz ca) r(k) + ca (ẽ(k) + ˆϕ(k))] (6.56) Furthermore, the expression in (6.56) can be represented in Z-domain as follows U(z) = (cb) 1 [ (cz ca) R(z) + ca ( Ẽ(z) + ˆΦ(z) )] (6.57) A block diagram of the system with a control input defined in (6.57) is presented in Figure 6.10 below Figure 6.10: Block diagram of the linear TSMRC system. 149

176 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control where G rc is the transfer function of the repetitive controller, while G 1 (z), G 2 (z), and P (z) are transfer functions which can be represented as follows G 1 (z) = c 2z 2 2c 2 z + c 2 T s (6.58) G 2 (z) = (c 1T s + c 2 ) z c 2 T s (6.59) P (z) = 2 bt s (c 1 T s + 2c 2 ) G p(z) (6.60) Furthermore, G p (z) is the discrete transfer function of the linear LA plant. The closed loop transfer function of the system without the repetitive controller can be written as follows G C (z) = G 2(z)P (z) 1 + G 2 (z)p (z) (6.61) Once the TSMC control law is designed to ensure G c (z) is stable, the stability of the overall TSMRC system can be guaranteed if the following small-gain conditions are satisfied at all times [56, 115] 1. Q(z) < 1 2. (1 k r z m G c (z)) Q(z) < 1 By choosing a low pass filter Q(z) with zero phase contribution as given in [56], the second stability condition can be represented in the frequency domain by the magnitude and angle conditions as given below [56, 59] 0 < k r < 2 cos (θ G c (jω) + θ m (jω)) M Gc (jω) (6.62) θ Gc (jω) + θ m (jω) < π 2 (6.63) where M Gc (jω) and θ Gc (jω) represent the magnitude and phase of G c (z) respectively, while θ m (jω) is the phase of the pure phase lead filter k r z m. 150

177 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller As seen from the angle condition in (6.63), the value of m should be selected so that z m cancells the phase lag of the feedback system in the frequency range up to the cut-off frequency of Q(z). The compensation of the phase lag present in the feedback system will improve the tracking performance of the system, especially the tracking of reference signals with high frequency components. While, the selection of the learning gain k r should satisfy the magnitude condition (6.62) at all frequencies up to the cut-off frequency of the low pass filter Q(z). The selection of the k r parameter is a trade-off between robust stability and fast convergence which means that larger k r values will achieve fast convergence of the system errors to the reduced QSM region but at the cost of reduced robust stability margins Simulation Testing Simulation testing is performed to show the effectiveness of the proposed controller against unknown uncertainties and nonlinearities when tracking both aperiodic and periodic reference signals. A comparison study which compares the performance of the proposed controller to that of the conventional TSMC and ZPETC structures is also presented in this section. The discretised second order nonlinear LA plant defined in (3.6) is considered in this section. A periodic reference signal r of period T = 0.6s and magnitude ±500µm, as well as step reference with a magnitude of +500µm is used during the testing. In order to perform the testing, the NLESO and TSMC parameters are first designed as described in Section 6.2. The RC design parameters are then selected for achieving the desired tracking accuracy and ensuring overall system stability as described below. Firstly the parameters of the low pass filter Q(z) are selected to achieve the desired tracking bandwidth as described in [56] i.e. Q(z) = 0.25z +0.5 = 0.25z 1. While, the parameter m of the phase lead RC filter z m is determined by plotting the phase of z m G c (z). As seen from the phase plots given in Figure 6.11, close to zero phase cancellation in the frequency range up to the cut-off frequency of Q(z) can be achieved when m = 1. Hence, the value of m used in the simulation analysis is set to 1. Finally the value for k r which satisfies (6.62) is selected to be 0.3. After the selection of the design parameters simulation testing was performed to show the effectiveness of the proposed controller when tracking aperiodic and periodic references. The obtained output and error plots are presented in Figure

178 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control Figure 6.11: Phase response of z m G c (z) Figure 6.16 below. As depicted in Figure 6.12 and Figure 6.13, the proposed NLESO based TSMRC achieves a fast step response and has improved tracking performance in the presence of plant uncertainties and nonlinear friction when compared to the conventional TSMC structure. Furthermore, as shown in Figure 6.14 and Figure 6.15, the proposed controller also shows superior tracking performance compared to the TSMC when tracking periodic reference signals. All of these plots strongly indicate that the proposed controller has improved robustness against plant uncertainties and nonlinear friction. The improved friction compensation capabilities of the proposed controller are especially evident in Figure 6.12 and Figure 6.14 where the proposed controller outperforms the conventional TSMC in the absence of load uncertainties. From the output and error plots presented in Figure 6.16 we can also see that in the presence of both nonlinear friction and load uncertainties, the proposed NLESO based TSMRC structure has superior robustness when compared to the conventional ZPETC. Moreover, as shown in Figure 6.16, the system with the conventional ZPETC becomes unstable in the presence of large payloads. 152

179 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller (a) Tracking output (b) Tracking error Figure 6.12: Tracking output and error of TSMC and NLESO based TSMRC (no load, step input) (a) Tracking output (b) Tracking error Figure 6.13: Tracking output and error of TSMC and NLESO based TSMRC (30kg load, step input) 153

180 6 Discrete Nonlinear Extended State Observer based Terminal Sliding Mode Repetitive Control (a) Tracking output (b) Tracking error Figure 6.14: Tracking output and error of TSMC and NLESO based TSMRC (no load, periodic reference) (a) Tracking output (b) Tracking error Figure 6.15: Tracking output and error (30kg load, periodic reference,) 154

181 6.3 Design of a Discrete Nonlinear ESO Based Terminal Sliding Mode Repetitive Controller (a) Tracking output (b) Tracking error Figure 6.16: Tracking output and error for ZPETC and NLESO based TSMRC (30kg load, periodic reference) Experimental Validation Experimental validation is also performed to show the superiority of the proposed controller when compared to the TSMC and ZPETC structures. The experimental validation is performed on the LA system shown in Figure 3.1 using the design parameters selected in the previous section. The effectiveness of the proposed controller in the presence of nonlinear friction and plant uncertainties when tracking both aperiodic and periodic reference trajectories is presented and analysed in the rest of this section. Moreover, as depicted in the plots given in Figure 6.17 and Figure 6.18, when a step reference is applied to the system, the proposed controller outperforms the conventional TSMC structure in the presence of both nonlinear friction and payload variations. Similarly, as shown in Figure 6.19, when a periodic reference is used the proposed controller again shows superior friction compensation properties when compared to the TSMC structure. Furthermore, from the steady state error plots given in Figure 6.20 where the steady-state tracking performance of the proposed controller is compared to that of the conventional ZPETC, it is evident that the proposed controller not only can effectively eliminate the periodic friction compensation error, but also shows superior robustness to plant uncertainties when compared to the conventional ZPETC. 155