急動度限制下之定位伺服及其干擾補償方法

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(1)National Taiwan Normal University Thesis. Advisor: Dr. Yu-Sheng Lu. A Jerk-Constrained Time-Optimal Servo with Disturbance Compensation 急動度限制下之定位伺服及其干擾補償方法. Student：Raymond Shieh JANUARY 2013.

(2) 國立臺灣師範大學機電科技學系. 碩士論文. 急動度限制下之定位伺服及其干擾補償方法. 謝立文. 撰. 中華民國一零二年.

(3) National Taiwan Normal University Thesis. Advisor: Dr. Yu-Sheng Lu A Jerk-Constrained Time-Optimal Servo with Disturbance Compensation 急動度限制下之定位伺服及其干擾補償方法. Student：Raymond Shieh JANUARY 2013.

(4) A Jerk-Constrained Time-Optimal Servo with Disturbance Compensation Student： Raymond Shieh. Advisor: Dr. Yu-Sheng Lu. Department of Mechatronic Technology, National Taiwan Normal University. ABSTRACT This paper presents an improved disturbance rejection method added to a model based time-optimal control method, in order to assure the correct system output response. In mechanical systems, the jerk (the time derivative of acceleration) may cause many unwanted results when too high. Thus, a jerkconstrained time-optimal control (JCTOC) method is proposed to not only constrain the jerk of the system, but also have a time-optimal characteristic. The JCTOC uses a time-optimal controller paired with an integrator and the system to directly constrain the jerk of the system. But, due to the JCTOC being a highly model based control method with switching characteristics, the system uncertainties and disturbance can have an adverse effect on the output response. Thus, a disturbance observer (DOB) is added for the compensation of the disturbance. The DOB used in this paper is in an integral form, thus called an integral disturbance observer (IDOB). The IDOB is further enhanced with a dynamic compensator to provide better noise immunity and asymptotic compensation for disturbances of various orders. This paper uses a symmetrically loaded servo motor as an experimental setup for the proposed control methods. The control kernel is a DSP/FPGA system separately programmed with C language and VHDL. Experiments of the disturbance rejection method have been conducted and proven to have better transient and steady-state responses than past approaches while fulfilling the timeoptimal characteristic of the JCTOC. Keyword: jerk-constraint, time-optimal control, disturbance observer. i.

(5) 急動度限制下之定位伺服及其干擾補償方法學生：謝立文. 指導教授：呂有勝. 國立台灣師範大學機電科技學系碩士班. 摘要本論文提出一種急動度限制下之定位控制及其干擾補償法，以改善系統輸出響應。在機械系統裡，急動度(加速度對時間的微分)若是太大可能會對系統造成不良的影響。故提出一急動度限制下之最佳時間定位控制法(Jerkconstrained time-optimal control, JCTOC)，此控制法不僅可限制最大急動度，又可達到最佳時間控制的效果。JCTOC 是利用最佳時間控制器結合一積分器與受控體，直接控制系統的急動度來做其限制。但由於提出的 JCTOC 是一種高度依賴受控體模型又俱備控制法則切換的控制法，故很容易受到系統不確定性以及系統干擾的影響。故在此加入一干擾補償器(Disturbance observer, DOB)以補償干擾並降低對系統的影響。本論文主要探討的干擾估測器以積分器的方式呈現，故在此稱為積分型干擾估測器(integral disturbance observer, IDOB)。在此 IDOB 加入一動態補償器或內部模型原理(Internal model principle, IMP)補償器，可讓干擾估測對於不同階數的干擾有更優越的抑制效果。本文實驗平台以一裝對稱性負載之無刷伺服馬達為受控體，並以一 DSP/FPGA 結合之單元為控制器，分別以 C 語言及 VHDL 撰寫。以此機台為實驗系統並實現本文提出之干擾補償及急動度限制下之最佳時間控制法，並由實驗結果證實其控制法的可行及實用性。關鍵詞：急動度、最佳時控制、干擾估測器. ii.

(6) ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Prof. Yu-Sheng Lu, for his guidance during my years in the Dynamic & Control Lab. If not for Prof. Lu, I may not have found the interest in control & systems or any other form of engineering. I would also like to thank my committee members, Dr. Ke-Han Su and Dr. Lien-Shing Hung, for the constructive comments and the effort of letting my thesis more complete. I would like to thank my family for the support during my years at NTNU. I would like to thank my mother for letting me pursue my own interests, my brother Richard for his support and help and my brother Robin for learning to take care of himself. I am also very grateful for the assistance of the relatives that have helped me and token care of my family members. I would like to express my appreciation to all the colleagues and friends that have helped me in my years at NTNU, from the cooperation during department events to the discussion of academic experiments, the valuable experiences are what make me a better team player. At last, I would like to thank my father and grandfather for their guidance. They have affected me much and let me know what kind of person I should be.. iii.

(7) 致謝在此，我得先感謝我的指導教授呂有勝教授。很感謝老師從我大三開始的培養及訓練，若是沒有老師的引導，我想我可能無法真的找到自己感興趣的方向吧。感謝蘇科翰學長以及洪聯馨處長擔任我的口試委員，幫我點出一些自己沒注意到的問題與錯誤。在讀碩士的這段期間，很感謝我家人的支持。感謝媽媽讓我找尋自己的興趣，感謝弟弟漸漸成熟，感謝二哥有良好的發展，也感謝我爸爸讓我更確定自己該走的路。很感謝叔叔，姑姑，嬸嬸及姑丈們照顧我的阿公阿嬤，也感謝我的舅舅阿姨及表哥表姐們對二哥的照顧，希望各位未來都順利。此外，很感謝陪我走過這段路的朋友們。特別是香菇，阿 GEE，郭翊，求文，金龍，阿良，阿公，北投貓，阿謙，阿碩，木馬，阿鵬等，很感謝你們在我人生的最低潮能有這樣的陪伴。特別感謝忠憲，家弘，景文及世杰學長給予的教導，感謝宗恆及倉軒學長帶給實驗室的無限歡樂，感謝岳豈及百恩陪著我修課及實驗內容的討論，也很感謝俊瑋，其錄及益頤給予實驗室的幫助。感謝我的大學專題夥伴秉儒以及我的帶領的專題成員祖昀及立明，感謝陪我一起努力的日子。很感謝我們實驗室的好鄰居們，微製造實驗室的小江，修茂，阿杜，其邁，俊華，士緯，阿皓等，樓上的煌緯，阿宗，威諭等，多次請你們幫忙加工及借工具，也常常光顧及偷吃零食，希望我們實驗室也有提供相對的溫暖。在機電系的這大段日子，過得很開心，不論從機電之夜有的沒的表演到帶助教的點點滴滴，經歷了非常多，也很感謝一路陪伴我的各位。最後要感謝機電系的全體老師及學生，希望機電系未來能更好。. iv.

(8) TABLE OF CONTENTS Page ABSTRACT 摘要 ACKNOWLEDGEMENTS 致謝 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS AND ABBREVIATIONS CHAPTER 1 1.1 Introduction 1.2 . Literature survey. i ii iii iv v vi vii x 1 1 2 . 1.2.1 Jerk-constraint 1.2.2 Disturbance observer 1.3 Summary. 2 3 6 . CHAPTER 2 2.1 System plant introduction. 7 7 . 2.2 . System hardware structure. 8 . 2.3 . System identification and modeling. CHAPTER 3 3.1 Revisit of time-optimal control of a triple integrator system. 10 14 14 . 3.2 . Jerk-constrained time-optimal control. 17 . 3.3 . JCTOC with IDOB compensation. 22 . 3.4 . JCTOC and PD comparison. 26 . CHAPTER 4 4.1 Revisit of integral DOB and its transfer function. 39 39 . 4.2 . Dynamically compensated integral DOB (DC-IDOB). 40 . 4.3 . Examples of dynamically compensated integral DOB. 41 . 4.4 . Internal model principal-based IDOB and its modification. 44 . 4.5 . Disturbance observer experiments. 46 . CHAPTER 5 APPENDIX A APPENDIX B REFERENCES. 59 60 63 65 . v.

(9) LIST OF TABLES Page Table 3- 1 Indexes of JCTOCs with different gains ......................................................... 36 Table 4- 1 Indexes of results of IDOB and T1O2-IDOB.................................................. 47 Table 4- 2 Indexes of results of T1O2-IDOB and T2O3-IDOB ....................................... 50 Table 4- 3 Indexes of results of T1O2-IDOB and IMP-IDOB ......................................... 53 Table 4- 4 Indexes of results of IMP-IDOB and MIMP-IDOB ........................................ 56 . vi.

(10) LIST OF FIGURES Page Fig. 1- 1 Block diagram of disturbance reduction controller .............................................. 4 Fig. 2- 1 Experimental setup of symmetrically loaded DC servo ....................................... 7 Fig. 2- 2 Block diagram of system hardware structure ....................................................... 8 Fig. 2- 3 DSP and FPGA controller core of system ............................................................ 9 Fig. 2- 4 Block diagram of swept-sine system identification method .............................. 10 Fig. 2- 5 Swept-sine and curve fitting results of DC servo ............................................... 11 Fig. 2- 6 TOC of a second-order plant with curve fit plant parameters ............................ 12 Fig. 2- 7 TOC of a second-order plant with adjusted pole................................................ 13 Fig. 3- 1 Block diagram of time-optimal control of third-order system ........................... 15 Fig. 3- 2 Trajectory of time-optimal control of third-order system .................................. 15 Fig. 3- 3 Block diagram of JCTOC applied to a servo-motor .......................................... 17 Fig. 3- 4 Block diagram of rearranged JCTOC................................................................. 18 Fig. 3- 5 Control method of the JCTOC ........................................................................... 20 Fig. 3- 6 Block diagram of the system control method..................................................... 22 Fig. 3- 7 Step response of JCTOC .................................................................................... 23 Fig. 3- 8 Control variables of JCTOC without IDOB compensation ............................... 23 Fig. 3- 9 Control variables of JCTOC with IDOB compensation..................................... 24 Fig. 3- 10 Mode switching characteristic of the JCTOC with IDOB compensation ........ 24 Fig. 3- 11 Response of JCTOC and PD controller with same settling time ( k  1024 , r f  2 (rad)) ...................................................................................................................... 27 Fig. 3- 12 Response of JCTOC and PD controller with same maximum nominal control effort ( k  1024 , r f  2 (rad)) .......................................................................................... 28 . vii.

(11) Fig. 3- 13 Response of JCTOC and PD controller with same settling time ( k  1024 , r f  4 (rad)) ...................................................................................................................... 29 . Fig. 3- 14 Response of JCTOC and PD controller with same maximum nominal control effort ( k  1024 , r f  4 (rad)) .......................................................................................... 30 Fig. 3- 15 Response of JCTOC and PD controller with same settling time ( k  2048 , r f  2 (rad)) ...................................................................................................................... 32 . Fig. 3- 16 Approximated acceleration signal of control methods..................................... 32 Fig. 3- 17 Response of JCTOC and PD controller with same maximum nominal control effort ( k  2048 , r f  2 (rad)) .......................................................................................... 33 Fig. 3- 18 Approximated acceleration signal of control methods..................................... 33 Fig. 3- 19 Response of JCTOCs with different jerk-constraints ....................................... 34 Fig. 3- 20 Nominal control effort of JCTOCs with different jerk-constraints .................. 35 Fig. 3- 21 Switching instants of the JCTOCs with different constraints .......................... 35 Fig. 3- 22 Response of JCTOC with maximum jerk-constraint ....................................... 37 Fig. 4- 1 Block diagram of simplified integral DOB ........................................................ 39 Fig. 4- 2 Block diagram of dynamic compensator added to the integral DOB ................ 40 Fig. 4- 3 Block diagram of DC-IDOB .............................................................................. 41 Fig. 4- 4 System response with IDOB and T1O2-IDOB compensation ........................... 48 Fig. 4- 5 Control effort of JCTOC with IDOB and T1O2-IDOB ..................................... 48 Fig. 4- 6 Frequency spectrum of error with IDOB and T1O2-IDOB compensation ........ 49 Fig. 4- 7 Theoretical frequency response of IDOB and T1O2-IDOB .............................. 49 Fig. 4- 8 System response with T1O2-IDOB and T2O3-IDOB compensation ................ 51 Fig. 4- 9 Control effort of JCTOC with T1O2-IDOB and T2O3-IDOB .......................... 51 Fig. 4- 10 Frequency spectrum of error with T1O2-IDOB and T2O3-IDOB compensation ........................................................................................................................................... 52 Fig. 4- 11 Theoretical frequency response of T1O2-IDOB and T2O3-IDOB .................. 52 viii.

(12) Fig. 4- 12 System response with T1O2-IDOB and IMP-IDOB compensation ................ 54 Fig. 4- 13 Control effort of JCTOC with T1O2-IDOB and IMP-IDOB ........................... 54 Fig. 4- 14 Frequency spectrum of error with T1O2-IDOB and IMP-IDOB compensation ........................................................................................................................................... 55 Fig. 4- 15 Theoretical frequency response of T1O2-IDOB and IMP-IDOB IDOB ......... 55 Fig. 4- 16 System response with IMP-IDOB and MIMP-IDOB compensation ............... 57 Fig. 4- 17 Control effort of JCTOC with IMP-IDOB and MIMP-IDOB ......................... 57 Fig. 4- 18 Frequency spectrum of error with IMP-IDOB and MIMP-IDOB compensation ........................................................................................................................................... 58 Fig. 4- 19 Theoretical frequency response of IMP-IDOB and MIMP-IDOB................... 58 . ix.

(13) LIST OF SYMBOLS AND ABBREVIATIONS . Angular position. . Angular velocity. . Angular acceleration. u. Control effort. u*. Nominal control effort. d. Disturbance. dˆ. Estimated disturbance. rf. Reference of position. e1. Error of position. e2. Error of velocity. e3. Error of acceleration. Kg. Gain constant of plant. k. Maximum admissible jerk of JCTOC. . Switching control signal of JCTOC. J. Parameter for  region of JCTOC. t s1. First switching instant of JCTOC. t s2. Second switching instant of JCTOC. tf. Settling time of JCTOC.  pd. Parameter for PD controller Gain of IDOB. K. x.

(14) n. CI. Parameter for dynamic compensator Zero-phase filter bandwidth for measuring chattering index. TOC. Time-optimal controller. JCTOC. Jerk-constrained time-optimal controller. DOB. Disturbance observer. IDOB DC-IDOB. Integral disturbance observer Dynamically compensated integral disturbance observer. T1O2-IDOB. Type-1 second-order disturbance observer. T2O3-IDOB. Type-2 third-order disturbance observer. IMP-IDOB MIMP-IDOB. dynamically. dynamically. compensated. integral. compensated. integral. Internal model principal-based integral disturbance observer Modified internal model principal-based integral disturbance observer. DSP. Digital signal processor. FPGA. Field-programmable gate array. ITSE. Integral of time multiplied by squared error. ITAE. Integral of time multiplied by absolute error. xi.

(15) CHAPTER 1 INTRODUCTION 1.1. Introduction. POSITIONING servos are widely employed in modern industry. While desiring to optimize the responding time, having a smooth response can also be an important aspect as well. The jerk, the time derivative of acceleration, plays an important role in servo motor control. Having a large jerk may cause unpleasant effects to the user, vibrations, excitation of un-modeled dynamics, and high wear to the mechanical parts. Hence, the maximum jerk should be limited to ensure smooth operation [1-11]. Many trajectory planning methods for jerk limitation or reduction have been made in the past. But, the trajectory planning methods lack a feedback of the system to ensure the jerk limitation or the response of the system states. Thus, a jerk-constrained timeoptimal control (JCTOC) method is introduced [1]. The JCTOC method ensures the jerk constraint by directly controlling the jerk of the system through the controller output. The JCTOC uses a TOC of a third-order system and adds an integrator to the servo motor system (originally a second-order system), letting the system act as that of third-order. But, due to the switching characteristics of the TOC, the control method can be very vulnerable to parametric errors and disturbance. Thus, a disturbance observer (DOB) is added to the system for correct mode switching and state behavior and additional robustness [28]. The DOB used in this study is of an integral form [19]. By adding a dynamic compensator to the integral DOB the characteristics of the DOB can be improved, and thus compensate for disturbances of various orders more efficiently.. 1.

(16) 1.2. Literature survey This section is divided into two sub-sections, one which discusses jerk-constraint. approaches, and the other which discusses disturbance observation and rejection methods.. 1.2.1 Jerk-constraint Many trajectory planning methods have been proposed to limit the maximum amount of jerk under the assumption of the perfect path tracking [3-9]. In the approach by Venkatesh et al. [3], the jerk of an elevator is limited due to the human sensitivity to the change of acceleration (jerk). Thus, a trajectory profile of position is constructed, on which jerk limitation is imposed. According to Osornio-Rios et al. [4], a high degree polynomial-based jerk limited profile is applied to the operation of computer numerical control (CNC) machinery. In contrast to the traditional trapezoidal velocity profile, the high degree polynomial-based jerk limited profile has no discontinuity problem. The small discontinuities in CNC controllers may produce undesired high-frequency harmonics on the position reference, which may cause excitation of natural frequencies or actuator saturation. In the approach by Gasparetto and Zanotto [5, 6], the execution time and integral of the squared jerk are weighted and minimized in the trajectory planning. With a larger weight on the jerk, the trajectory will be smooth but slow, and with a larger weight on time, the trajectory will be faster but less smooth. In the approach by Olabi et. al. [7], the method of trajectory planning is used in the control of a six-axis machining robot. The path of the machining tool on the robot arm is continuous rather than point-topoint, and jerk-constraint is taken into consideration during the motion command generation. With a jerk limited profile, endpoint vibration can be reduced, and residual vibration can be totally suppressed in some cases. In the approach by Hoshijima and Ikeda [10], the jerk of the intermediate mass of a triple massed rotational system is reduced to suppress the vibration of the head mass. This method can effectively reduce the vibration of the system through the systems coupling characteristics, but cannot. 2.

(17) directly constrain the jerk of the head mass. In the study by Osornio-Rios et al. [11], the peak jerk of a computer numerical control (CNC) machine is reduced through a polynomial profile generator with different orders of polynomials of acceleration. This method can reduce jerk, but cannot directly constrain it at a desired value. In practice, the states of the system may not be on the desired path or the system’s response may not follow that as planned. Unlike the path-planning approaches that design reference trajectories under jerk-constraint, the jerk-constrained time-optimal control method controls the jerk of the servo motor through a mode switching method [1]. While satisfying the constraint on the maximum amount of jerk, the proposed method guarantees a time-optimal motion. But due to parametric errors or additional disturbances, the system states may not respond ideally. Thus, an additional DOB structure is added for correct mode switching and state behavior.. 1.2.2. Disturbance observer Many forms of additional disturbance rejection methods have been proposed. The. PI-type closed-loop torque observer (PICTO) [12] structure can detect the abnormal loads added to servomechanisms. A PI structure based observer is applied to the system to find the abnormal load, which is viewed as a form of disturbance. The disturbance observer (DOB) is a well-known disturbance compensation structure which has been widely researched on and its’ applications have been applied to systems ranging from motor drives to grinding mills [13-24]. Modified versions of the DOB have been researched due to different aspects or characteristics of the disturbance [17-20]. The nonlinear DOB proposed by Back and Shim [17] introduces a saturation function for cancelling the peaking effect and a dead-zone function for stabilizing the fast dynamics under saturation, providing the DOB the infinite-gain property without peaking. The DOB structure proposed by Chen et al. [18] compensates the disturbance in the same order, leading to an integrator structure instead of the filter used in traditional DOBs. Although there is a. 3.

(18) sliding mode controller used in their research, it is used as a nominal controller and thus loses the benefits when another nominal controller is used. Disturbance observers have become widely used in various applications with different disturbance [13-24]. In the research by Kakinuma et al. [21], a disturbance observer is used to monitor the chatter vibration of a milling machine, leading to no additional sensors needed for vibration monitoring. The research by Kim et al. [22] introduces a disturbance observer for compensating the backlash problem in gear transmitted systems. Hace et al. [23] introduces a SMC with disturbance observer control method for a linear belt drive. The disturbance observer is designed for the compensation of the motor friction together with the load-side friction. Lin and Lee [24] proposed an observer-based controller designed for a gantry stage, in which a variable structure controller is used with a disturbance observer that compensates the frictional disturbance and modeling uncertainties. d (s ) R (s ).  . . . P (s ). dˆ ( s ). K s. Pˆ 1 ( s ).  . Pˆ ( s ) . Fig. 1- 1 Block diagram of disturbance reduction controller The disturbance reduction controller proposed in [19] extracts the disturbance in a slightly different way than the traditional DOB, the block diagram is shown in Fig. 1-1. The controller output is extracted before the addition of the estimated disturbance, letting the estimated disturbance dˆ have a direct subtractive property with the disturbance d . 4.

(19) In the disturbance reduction controller method, a simple integrator can be used with a gain of K instead of the low-pass filter used in the traditional DOB, this method can lead to an easier implementation. If Pˆ  P , then the disturbance transfer function. dˆ K .  d sK. (1). Thus, with a high gain K , the disturbance can be estimated and compensated. The DOBs that are introduced above can reduce the effect of disturbance to the systems. Although being able to compensate the disturbance, the past approaches cannot tune the gain of desired frequencies without tuning the bandwidth of the DOB, which may cause larger gain for high-frequency noise. In our approach, a dynamically compensated integral disturbance observer (DC-IDOB) is proposed. By adding a dynamic compensator to the DOB, it can gain characteristics of different types according to the compensator, such as increasing the order of the DOB or adding an internal model principal-based characteristic [29] to the DOB, thus letting the DOB compensate disturbances of specific forms more efficiently than the other DOBs.. 5.

(20) 1.3. Summary This thesis is consisted of 5 chapters. Aside from this chapter which contains the. introduction and past approach, the content of the other chapters are as follows: The second chapter introduces the experimental system used in the experiments of the following control methods, including the hardware structure of the system and identification of the plant. The third chapter introduces the jerk-constrained time optimal control method. The JCTOC is used as a nominal controller while different disturbance observers are compared when compensating the disturbance added to the system. In the fourth chapter, a dynamically compensated integral DOB is proposed, letting it estimate disturbances of different orders. In the fifth chapter, the conclusion of this thesis is presented.. 6.

(21) CHAPTER 2 EXPERIMENTAL SYSTEM 2.1. System plant introduction A Mitsubishi HC-KFS73 DC servo motor is used as the main actuator in the. system. The DC servo motor is symmetrical loaded as shown in Fig. 2-1. The servo plant is consisted of a servo driver and servo motor. While the load being symmetrical, the plant can be regarded as a second-order linear system with one pole of 0 if the output is position, first-order if the output is velocity. Letting the system dynamics be. x   ax  bu ,. (2). where a and b are plant parameters and u is the control input, x , x and x are system states being position, velocity and acceleration. The motor is paired with an encoder of 262144 p/rev resolution, the encoder signal is sent to the driver and is then output in an A/B phase form from the driver. The DC servo is used in torque command mode, with an input range of -5.5 ~ 5.5 V.. Driver. Symmetrically loaded motor. Fig. 2- 1 Experimental setup of symmetrically loaded DC servo. 7.

(22) 2.2. System hardware structure The experimental system’s hardware structure is shown in Fig. 2-2. Aside from. the plant, a controller core and personal computer are used. The controller core is mainly consisted of a Texas Instruments TMS320C6713 development starter kit and a FPGA daughter board. The FPGA daughter board is connected to the DSP through a data bus and has a D/A converter interface and differential line receiving interface for receiving the driver’s A/B phase signal. The control effort is calculated by the DSP and transferred to the FPGA and sent through the D/A interface into the driver, the driver receives the encoder data of the motor and outputs the A/B phase signal to the FPGA daughter board’s differential line receiver, the A/B phase data is separately counted for the position and velocity signal in the FPGA and then sent to the DSP, the DSP uses the feedback data for control to fulfill the closed loop. When the experiments are finished, the DSP sends the experiment data through a USB interface to the personal computer for analysis. Controller core Experiment data. Texas Instruments 6713DSP. Xilinx FPGA. D/A Diff. line receiver. Personal computer Plant. Control signal Driver. Motor. Encoder A/B phase signal. Fig. 2- 2 Block diagram of system hardware structure The hardware of the controller core is shown in Fig. 2-3, the bottom board is the Texas Instruments TMS320C6713 development starter kit and the top board is the FPGA daughter board. The DSP and FPGA have a synced clock of 90 MHz while operating. 8.

(23) with a sampling period of 9.102  10 5 s.. Fig. 2- 3 DSP and FPGA controller core of system. 9.

(24) 2.3. System identification and modeling The DC servo can be assumed as a linear system, thus by using a swept-sine. method, the plant model can be found. The swept-sine interface is shown in Fig. 2-4, a National Instruments PXI-4461 Dynamic signal analyzer (DSP) is used for the signal generation and receiving, the generated sinusoidal signal is directly input into the driver as a control signal and the A/B phase signal is sent to the controller core for velocity estimation, the estimated velocity is then output through a D/A converter on the controller core and sent back to the DSA as the output of the system.. National Instruments PXI-4461 Input den. Generated swept-sine output. num. Velocity signal Controller core. A/B phase. Plant (Driver + motor). Fig. 2- 4 Block diagram of swept-sine system identification method The swept-sine procedure is started from 0.02 Hz and ended at 50 Hz. A first order curve fit of the data is performed to find the transfer function of the system, the data points and curve fitting results are shown in Fig. 2-5. The transfer function of the curve fit result is.  (rad/s) u (V). . 180.5 . s  0.1642. 10. (3).

(25) But due to the pole of the system being very small and hard to find by a swept-sine method, a time-optimal control (TOC) of a second-order plant is used for further identification of the system.. magnitude (dB). 100 swept−sine data curve fitting result. 50 0 −50 −100. −1. 10. 0. 10. 1. 2. 10 ω (rad/sec). 10. 3. 10. phase (degree). 200 swept−sine data curve fitting result. 100 0 −100 −200. −1. 10. 0. 10. 1. 10 ω (rad/sec). 2. 10. 3. 10. Fig. 2- 5 Swept-sine and curve fitting results of DC servo The TOC of a second-order system as in [25] is used due to its model based properties and also easy to observe through the phase plane. The switching property of the control method is  k sgn(e~1  E * ) U (t )   SO k SO sgn(e~2 ). if e~1  E *  0 if e~  E *  0. (4). 1. where k SO is the gain of the TOC, U (t ) is the output of the TOC, ~ e1 is the estimated position error, ~ e1  E * is the switching surface of e2 is the estimated velocity error and ~. the TOC of a second-order system in which E *  e~2 e~2 / 2k SO .. 11. (5).

(26) By adjusting the control output of the controller to u (t )  b 1{U (t )  ax (t )} ,. (6). u (t ) being the output of the controller, the plant and controller can be viewed as TOC of second-order and a double integrator. Due to this method being very sensitive to plant parameters, the parameter adjustment can be done through experiments. Fig. 2-6 is the second-order TOC using the plant parameters of the curve fit results, it can be seen that the reaching near the origin of the phase plane isn’t ideal. Thus, by adjusting the parameters of the system to.  (rad/s) u (V). . 180.5 , s  0.001642. (7). making the pole of the plant 100 times smaller, the reaching near the origin of the phase plane is more ideal and the system states converge to zero more correctly as shown in Fig. 2-7. 8 system states TOC switching curve. 7. 6. error. dot. 5. 4. 3. 2. 1. 0 −2. −1.8. −1.6. −1.4. −1.2. −1 error. −0.8. −0.6. −0.4. −0.2. Fig. 2- 6 TOC of a second-order plant with curve fit plant parameters. 12. 0.

(27) 8 system states TOC switching curve. 7. 6. error. dot. 5. 4. 3. 2. 1. 0 −2. −1.8. −1.6. −1.4. −1.2. −1 error. −0.8. −0.6. Fig. 2- 7 TOC of a second-order plant with adjusted pole. 13. −0.4. −0.2. 0.

(28) CHAPTER 3 JERK-CONSTRAINED TIME-OPTIMAL CONTROL OF A POSITIONING SERVO 3.1. Revisit of time-optimal control of a triple integrator system. The time-optimal control method is generally a bang-bang type control approach. By switching the output state of the controller from one extreme to another, if switched under the precise conditions, time-optimization can be achieved. The ideal time-optimal controller for the triple integrator plant introduced by Kalyon [26], uses a control law of  sgn( x1  X 1 )  U ( x)  sgn( x2  X 2 )  sgn( x ) 3  *. if x1  X 1  0 if x1  X 1  0, x2  X 2  0 , if x1  X 1  0, x2  X 2  0. (8). where x1  x , x2  x1 and x3  x2  x which are states of the system and x3  K g u which is the controller output which K g is the gain of the triple integrator system. X 1 , X 2 are switching surface related functions described by 3  1 1 3 * *1 2 * 2  X1   x3  K g  x2 x3    x3  K g  x2   ,   2 K g 2 3  . X 2   x3 x3 /( 2 K g ) ,. (9). (10). in which *   sgn( x2  X 2 ) . The block diagram of TOC of a triple integrator system is. shown in Fig. 3-1, in which the control output of the TOC u  U * . In view of state space, according to the switching law (8), there are two switching surfaces of the TOC, being x1  X 1 and x2  X 2 , letting the states of the system initially move toward the first switching surface. When the system reaches the first switching surface, it moves along the first switching surface toward the second switching surface.. 14.

(29) After the system states reach the second switching surface, the representative point then moves along the switching curve which is the intersection of the two switching surfaces toward the desired system states.. u. TOC . Kg. ∫. x. x. ∫. x. ∫. Fig. 3- 1 Block diagram of time-optimal control of a third-order system. Fig. 3- 2 Trajectory of time-optimal control of a third-order system Fig. 3-2 is the trajectory of the switching of the TOC in state space. The x, y, z axis are e~1 (t )  rf (t )  x(t ) , ~ e2 (t )  rf (t )  x (t ) and e~3 (t )  rf (t )  x(t ) , where rf  2 ,. 15.

(30) e1 ~ e2 rf  0 and rf  0 . The trajectory starts off at with the system error ~.  2. ~ e3  . 0 0 , it then experiences two switching points before reaching the origin of state. space.. 16.

(31) 3.2. Jerk-constrained time-optimal control Consider a servomotor described by x   ax  bu. (11). where x(t ) is the displacement, u (t ) is the control input, and a and b are plant parameters.. The. tracking. error. variables. are. defined. as ~ e1 (t )  r f (t )  x(t ) ,. e~2 (t )  rf (t )  x (t ) and e~3 (t )  rf (t )  x(t ) in which r f (t ) is the position reference. The objective of the JCTOC design is to track the position reference, r f (t ) , under the following constraint on the maximum magnitude of jerk: e~3 (t )  k , where the constant k is the admissible maximum jerk.. JCTOC. a. rf. x TOC. . x. ∫.  . Plant. . b 1. u. b . ∫. ∫. . rf. a. Fig. 3- 3 Block diagram of JCTOC applied to a servo-motor Fig. 3-3 shows the block diagram of the JCTOC applied to the servomotor, in which the TOC denotes the non-linear time-optimal controller [26]. Here, the control input to the plant u (t ) generated by the JCTOC has an integral relationship to the output of the TOC,  (t ) ; that is,. 17.

(32) u (t )  b 1{ (t )dt  ax (t )  rf (t )} ,. (12). in which  (t ) is considered as a virtual control signal given by the TOC. Examining the error dynamics of the system gives e~1 (t )  e~2 (t ) ,. (13). e~2 (t )  e~3 (t ) ,. (14). e~3 (t )   (t ) ,. (15). from which it can be seen that   (t ) is equivalent to the jerk. The design of the JCTOC (11) becomes the TOC of a triple-integrator system (13)-(15) with the following control input.  (t )  k (t ) ,. (16). in which  (t )  1 . Thus, by controlling  (t ) we can directly control the jerk command of the plant. But unlike the TOC of a triple-integrator, the plant of the JCTOC only consists of two integrators. The extra integrator is contained in the controller, thus alleviating the bang-bang characteristics of the TOC. Thus, by rearranging the block diagram, the JCTOC method can be simplified to the form of Fig. 3-4. In this block diagram, the disturbance is neglected. The JCTOC with the addition of disturbance is discussed in chapter 3.3.. JCTOC. Plant.  TOC . k. . e~3. ∫. Fig. 3- 4 Block diagram of rearranged JCTOC. 18. ∫. e~2. ∫. ~ e1.

(33) According to system response time-optimal control [27], E1 and E2 are functions related to the time-optimal switching planes, which are defined as E2 (e~3 )  e~3 e~3 /(2k ) ,. 1 E1 (e~2 , e~3 )  2 k. (17). 3   2 1 1 ~ 3   2 *~ ~ * *~ ~  e3  k e2 e3    e3  k e2   , 2    3 . (18). in which *   sgn(e~2  E2 ) . The surface switching instants are calculated from the situations of ~ e1  E1  0 , e~2  E2  0 and ~e  e~1. e~2. ~ e3   0 0 0 , thus letting the. control approach become  sgn(e~1  E1 )    sgn(e~2  E2 ) sgn(e~ ) 3 . e1  E1  0 if ~ ~ ~ if e1  E1  0, e2  E2  0 . if ~ e  E  0, e~  E  0 1. 1. 2. (19). 2. The switching instants of the system are t s1  (. rf. 2k. 1. )3 ,. t s 2  3t s1  3(. t f  4t s1  4(. rf. 2k. rf 2k. (20) 1 3. ) , 1 3. ) ,. (21). (22). where the first switching occurs at t s1 , the second occurs at t s 2 , and t f is the settling time, the calculation of the switching instants are shown in Appendix A. In order to avoid the singularity problem around the origin of the state space, an additional switching plane is introduced for t  t f . The switching plane, S  0 , is defined as S  e~3  c~2 e~2  c~1e~1  0 ,. (23). in which c~1 and c~2 are constants determined through pole assignment. Here the assigned. 19.

(34) poles of the S-plane dynamics are both   J , yielding c~2  2 J and c~1   J 2 . The switching region of the S-plane is called the Ω region, which is defined as. k (3  3 ) 3  e~1   6 J 3   k (3  3 ) 2  ~ e2  , 2 J 2  k (3  3 )  e~3   J .      Ω  ~e     which e is the vector of e  e~1 ~ e2. (24). ~ e3  . When the phase space trajectory enters the Ω. region, the control method moves the trajectory toward the S-plane. The borders of the Ω region are set to let the trajectory be switching on the S-plane while changing from the former control scheme, the calculation of the borders of the Ω region are shown in Appendix B. The phase trajectory behaves as it does after the second switching point but without the singularity problem near the origin due to the sliding surface properties. 100 e1 e2 e3. variables. 50 0 −50 −100. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. variables. 10 ρ e1−E1 e2−E2. 5. 0. −5. 0. 0.1. 0.2. 0.3 time (s). Fig. 3- 5 Control method of the JCTOC. 20. 0.4. 0.5. 0.6.

(35) By combining the control laws above, the switching law of the JCTOC becomes  sgn(e~1  E1 )  sgn(e~  E )  2 2   ~ sgn(e3 ) sgn( S ). if e~1  E1  0 e1  E1  0, ~ e2  E2  0 if ~ . if ~ e1  E1  0, e~2  E2  0, ~e  Ω if ~ e1  E1  0, e~2  E2  0, ~e  Ω. (25). Fig. 3-5 shows the control method of the JCTOC, in which the top figure shows the errors e2 and ~ of the system e~1 , ~ e3 . The bottom figure shows the surface switching variables. e~1  E1 , e~2  E2 and  .. 21.

(36) 3.3. JCTOC with IDOB compensation The IDOB can help compensate the disturbance that is applied to the system,. which may be system uncertainties, external or internal disturbance…etc., the IDOB is added to the system as shown in figure 3-6 [28], which u * is the output of the nominal controller, d is the disturbance of the system and dˆ is the disturbance estimate of the IDOB. When the disturbance of the system is considered, with the IDOB added to the JCTOC, the switching characteristics can then be less affected by the disturbance and more correct.. d. r.  . JCTOC or PD. u*.  u   . dˆ. Plant. x. x. IDOB. Fig. 3- 6 Block diagram of the system control method The JCTOC is compared with a JCTOC added with an IDOB in the following experiment. The JCTOCs both have a jerk limit of k  1024 and Ω region pole of.  J  10 . The IDOB used in the experiment has an integral gain of K  100 . The reference is set at r f  2 (rad). Fig. 3-7 is the response of the JCTOC with and without the IDOB, it is obvious that the JCTOC without disturbance compensation does not have an ideal response which stops too early and moves slowly toward e  0 due to the sliding property of the Ω region, while the JCTOC with disturbance compensation has an ideal convergence 22.

(37) characteristic. Fig. 3-8 and Fig. 3-9 are figures of the system states and switching surface variables of the two methods. It can be seen that in Fig. 3-8, the switching functions do not response as smooth as in the simulation, while in Fig. 3-9 with the IDOB added, the response is better.. error (rad). 3 JCTOC+IDOB JCTOC. 2 1 0 −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. nominal control effort (V). 1 JCTOC+IDOB JCTOC. 0.5 0 −0.5 −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. Fig. 3- 7 Step response of JCTOC 100. e1 e2 e3. variables. 50 0 −50 −100 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. variables. 5 ρ e1−E1 e2−E2. 0 −5 −10 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. Fig. 3- 8 Control variables of JCTOC without IDOB compensation 23. 0.8.

(38) 100. e1 e2 e3. variables. 50 0 −50 −100 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. variables. 5 ρ e1−E1 e2−E2. 0 −5 −10 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. Fig. 3- 9 Control variables of JCTOC with IDOB compensation. 1. JCTOC (k=1024). 0.8 0.6 0.4. ρ. 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. Fig. 3- 10 Mode switching characteristic of the JCTOC with IDOB compensation. 24.

(39) The switching characteristic of the JCTOC is shown in Fig. 3-10. The ideal switching instants of the JCTOC of r f  2 (rad) and k  1024 can be calculated from (20) - (22), which are t s1  0.0992 (s), t s 2  0.298 (s) and t f  0.397 (s). In the experiment results of the JCTOC with IDOB, switching instants are t s1  0.1001 (s), t s 2  0.2995 (s) and t f  0.3978 (s), which are very close to the ideal switching instants.. 25.

(40) 3.4. JCTOC and PD comparison The JCTOC is compared with the PD controller in the following experiments. The. PD controller’s gains are tuned through pole assignment, with the system dynamics of e  (a  bK d )e  bK p e  0 ,. (26). where K p and K d are the proportional and differential gains of the PD controller, a and b are plant parameters. Letting the system have a damping ratio of 1, the gains of the PD. controller will then become K p   pd / b ,. (27). K d  (2 pd  a ) / b ,. (28). in which the system poles are set at   pd . When compared, the JCTOC and PD controller both are added with an IDOB for disturbance compensation, in which the integral gain of the IDOB is K  100 . Several experiments are conducted with different conditions for comparison: Exp.3-1 Same settling time: JCTOC of k  1024 compared with PD controller with r f  2 (rad). The JCTOC of k  1024 and r f  2 (rad) has a settling time of t  0.33 (s). Thus, if the PD controller is set at having the same settling time, the pole of the controller is then set at  pd  18.5 . The response and control effort of the JCTOC and PD controller are shown in Fig. 3-11. While having the same settling time (  2% of reference) of 0.33 (s), the JCTOC has a maximum nominal control effort of 0.564 V while the PD controller has a much greater maximum nominal control effort of 3.79 V. Thus, with the results of this experiment, it can be clear that the JCTOC needs a much less maximum nominal control effort than the PD controller, which can lead to less possibility of plant input saturation.. 26.

(41) error (rad). 3 JCTOC PD (ω =18.5). 2. pd. 1 0. nominal control effort (V). −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. 4 JCTOC PD (ω =18.5). 3. pd. 2 1 0 −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. Fig. 3- 11 Response of JCTOC and PD controller with same settling time ( k  1024 , r f  2 (rad)). 27.

(42) Exp.3-2 Same maximum nominal control effort: JCTOC of k  1024 compared with PD controller with r f  2 (rad). The JCTOC of k  1024 and r f  2 (rad) has a maximum nominal control effort of 0.564 (V). Thus, if the PD controller is set at having the same maximum nominal control effort, the pole of the controller is then set at  pd  7.13 . The response and control effort of the JCTOC and PD controller are shown in Fig. 3-12. The JCTOC and PD controller have the same maximum nominal control effort of 0.564 V. The JCTOC has a settling time of 0.33 (s), while the PD controller has a settling time of 0.93 (s). Thus, it can easily be seen that with the same maximum nominal control effort, the JCTOC has a much faster response than the PD controller.. error (rad). 3 JCTOC PD (ω =7.13). 2. pd. 1 0 −1. 0. 0.2. 0.4. 0.6. 0.8. 1. nominal control effort (V). time (s) 1 JCTOC PD (ω =7.13). 0.5. pd. 0 −0.5 −1. 0. 0.2. 0.4. 0.6. 0.8. 1. time (s). Fig. 3- 12 Response of JCTOC and PD controller with same maximum nominal control effort ( k  1024 , r f  2 (rad)). 28.

(43) Exp.3-3 Double reference: JCTOC of k  1024 compared with PD controller of same settling time and PD controller of same maximum nominal control effort, r f  4 (rad). The comparison of the JCTOC and PD controller are continued with a double reference of r f  4 (rad), the jerk limit of the JCTOC is still k  1024 . The comparison objectives are the same as the previous two experiments: the same settling time and same maximum nominal control effort. The poles of the PD controllers with the same settling time and maximum nominal control effort are respectively  pd  16 and  pd  5.67 . The comparison of the JCTOC and PD controller with same settling time is shown in Fig. 3-13. Both control methods have a settling time of 0.43 (s), the JCTOC has a maximum nominal control effort of 0.711 V while the PD controller has already caused saturation of the plant, having a maximum nominal control effort of 5.5 V. The comparison of the JCTOC and PD controller with maximum nominal control effort is shown in Fig. 3-14. With both controllers having the same maximum nominal control effort of 0.711 (V), the JCTOC has a settling time of 0.43 (s) while the PD controller has a much larger settling time of 1.4 (s).. error (rad). 6 JCTOC PD (ω =16). 4. pd. 2 0. nominal control effort (V). −2. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. 6 JCTOC PD (ωpd=16). 4 2 0 −2. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. Fig. 3- 13 Response of JCTOC and PD controller with same settling time ( k  1024 , r f  4 (rad)) 29.

(44) error (rad). 6 JCTOC PD (ω =5.67). 4. pd. 2 0 −2. 0. 0.2. 0.4. 0.6. 0.8. 1. nominal control effort (V). time (s) 1 JCTOC PD (ω =5.67). 0.5. pd. 0 −0.5 −1. 0. 0.2. 0.4. 0.6. 0.8. 1. time (s). Fig. 3- 14 Response of JCTOC and PD controller with same maximum nominal control effort ( k  1024 , r f  4 (rad)). 30.

(45) Exp.3-4 Double admissible jerk of JCTOC: JCTOC of k  2048 compared with PD controller of same settling time and PD controller of same maximum nominal control effort, r f  2 (rad). If the jerk-constraint of the JCTOC is doubled and set to k  2048 , the settling time and maximum control effort will both be different. The JCTOC is compared with the PD controller with the same comparison objectives: the same settling time and same maximum nominal control effort, the poles of the PD controllers with the same settling time and maximum nominal control effort are respectively  pd  23.6 and.  pd  9.01 . The velocity signals of the experiments are differentiated to examine the approximated acceleration, which have a direct relation with the torque and command input of the plant. The approximated acceleration signal can also be observed to examine the jerk of the plant. The JCTOC and PD controller with the poles of  pd  23.6 have the same settling time of 0.256 (s), the response of the plant and control effort of the controllers are shown in Fig. 3-15, the approximated acceleration is shown in Fig. 3-16, which is acquired by differentiating the velocity signal. With the same settling time, the PD controller has caused the plant to saturate for a certain amount of time while the JCTOC only has a maximum nominal control effort of 0.9 V. By observing the approximated acceleration signal, it can be seen that it is very similar to the nominal control effort of the controller due to the plant dynamics. The change of acceleration of the PD controller can be very steep in the beginning, leading to large jerk of the plant, while the change of acceleration of the JCTOC is constant due to the jerk-constraint. The JCTOC and PD controller with the poles of  pd  9.01 have the same maximum nominal control effort of 0.9V, the response of the plant and nominal control effort of the controllers are shown in Fig. 3-17, the approximated acceleration is shown in Fig. 3-18. With the same maximum nominal control effort, the JCTOC has a settling time of 0.256 (s) while the PD controller has a settling time of 0.71 (s), which is much larger. 31.

(46) Although the acceleration of the PD controller is smoother than the PD controller with same settling time, the initial nominal control effort is not continuous.. error (rad). 3 JCTOC PD (ω =23.6). 2. pd. 1 0. nominal control effort (V). −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. 6 JCTOC PD (ωpd=23.6). 4 2 0 −2. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. Fig. 3- 15 Response of JCTOC and PD controller with same settling time ( k  2048 , r f  2 (rad)) 400 JCTOC PD (ωpd=23.6). 200. 2. acceleration (rad/s ). 0. −200. −400. −600. −800. −1000. 0. 0.1. 0.2. 0.3 time (s). 0.4. Fig. 3- 16 Approximated acceleration signal of control methods 32. 0.5. 0.6.

(47) error (rad). 3 JCTOC PD (ω =9.01). 2. pd. 1 0. nominal control effort (V). −1. 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. 1 JCTOC PD (ω =9.01). 0.5. pd. 0 −0.5 −1. 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. 0.7. 0.8. Fig. 3- 17 Response of JCTOC and PD controller with same maximum nominal control effort ( k  2048 , r f  2 (rad)) 300 JCTOC PD (ω =9.01) pd. acceleration (rad/s2). 200. 100. 0. −100. −200. −300. 0. 0.1. 0.2. 0.3. 0.4 time (s). 0.5. 0.6. Fig. 3- 18 Approximated acceleration signal of control methods. 33. 0.7. 0.8.

(48) Exp.3-5 Comparison of JCTOC with different constraints: k  512 , k  1024 and k  2048 , with r f  2 (rad). The following experiment compares the JCTOC of 3 different jerk-constraints, k  512 , k  1024 and k  2048 . The 3 JCTOCs have a separate settling time and. maximum control effort, the response of the plant and control effort of the controllers are shown in Fig. 3-19 and Fig. 3-20. The JCTOC with the jerk-constraint of k  512 has a settling time of 0.418 (s) and maximum nominal control effort of 0.357 V. The JCTOC with the jerk-constraint of k  1024 has a settling time of 0.33 (s) and maximum nominal control effort of 0.564 V. The JCTOC with the jerk-constraint of k  2048 has a settling time of 0.256 (s) and maximum nominal control effort of 0.9 V. The JCTOC with a larger jerk-constraint has a shorter settling time, but a larger jerk would lead to a larger maximum control effort.. 2.5 JCTOC (k=512) JCTOC (k=1024) JCTOC (k=2048). 2. error (rad). 1.5. 1. 0.5. 0. −0.5. 0. 0.1. 0.2. 0.3 time (s). 0.4. Fig. 3- 19 Response of JCTOCs with different jerk-constraints. 34. 0.5. 0.6.

(49) 1 JCTOC (k=512) JCTOC (k=1024) JCTOC (k=2048). 0.8. nominal control effort (V). 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1. 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. Fig. 3- 20 Nominal control effort of JCTOCs with different jerk-constraints. 1. ρ. JCTOC (k=512) 0 −1 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. 1. ρ. JCTOC (k=1024) 0 −1 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. 0.6. 1. ρ. JCTOC (k=2048) 0 −1 0. 0.1. 0.2. 0.3 time (s). 0.4. 0.5. Fig. 3- 21 Switching instants of the JCTOCs with different constraints. 35. 0.6.

(50) The switching instants of the JCTOCs are shown in Fig. 3-21. The ideal and experimental switching instants of the different controller gains are shown in Table. 3-1. Experiment results are all very similar to the ideal switching instants, which assure the correct switching of the TOC at different gains.. Table 3- 1 Indexes of JCTOCs with different jerk constraints Ideal t s1 (s). t s1. experiment result (s). Ideal t s 2 (s). ts2. experiment result (s). Ideal t f (s). tf. experiment result (s). Max. u* (V). JCTOC ( k  512 ). 0.125. 0.1247. 0.375. 0.3768. 0.5. 0.5106. 0.357. JCTOC ( k  1024 ). 0.0992. 0.1001. 0.298. 0.2995. 0.397. 0.3978. 0.564. JCTOC ( k  2048 ). 0.0787. 0.7919. 0.236. 0.2367. 0.315. 0.3186. 0.9. 36.

(51) Exp.3-6 Maximum jerk-constraint of JCTOC while avoiding plant saturation, r f  2 (rad). While the previous experiments of the JCTOC have a small maximum nominal control effort, the following experiment raises the constraint of the JCTOC to let the maximum nominal control effort almost exceed the saturation limit of the plant. The maximum jerk-constraint can be found by a simple method, due to the maximum nominal control effort appearing at the first two switching instants of the JCTOC and the slope of the nominal control effort having a direct relation to the jerk-constraint k , the limit can be found by the function. kt s1  5.5 ,. (29). where  is a plant constant that can be experimentally found. Thus, by using this method, the maximum jerk limit can be found. 3. error (rad). JCTOC (k=30000) 2 1 0. nominal control effort (V). −1. 0. 0.05. 0.1 time (s). 0.15. 0.2. 6 JCTOC (k=30000). 4 2 0 −2 −4 −6. 0. 0.05. 0.1 time (s). 0.15. Fig. 3- 22 Response of JCTOC with maximum jerk-constraint. 37. 0.2.

(52) The system response and nominal control effort can be seen in Fig. 3-22. The JCTOC with jerk-constraint of k  30000 has a maximum nominal control effort of 5.4 V, which is very close to the saturation 5.5 V. The settling time of the JCTOC is 0.103 (s), which is very fast and unable to be accomplished with a normal PD controller. The nominal control effort is still continuous regardless of the high gain of the controller due to the integrator in the JCTOC.. 38.

(53) CHAPTER 4 DYNAMICALLY COMPENSATED INTEGRAL DOB 4.1. Revisit of integral DOB and its transfer function The integral DOB introduced in chapter 1.2.2 [19] can be simplified to a form of. Fig. 4-1, which can be easier to implement, in which u * is the output of the nominal controller, d is the disturbance of the system, dˆ is the estimated disturbance of the IDOB, K is the gain of the IDOB, P is the system plant and Pˆ 1 is the inverse model of the plant. d (s ) u* ( s).  . . . P (s ). dˆ ( s ) Pˆ 1 ( s ) K s. IDOB.  . Fig. 4- 1 Block diagram of simplified integral DOB If the system model is identical to the system plant (i.e., Pˆ  P ), the transfer function of the disturbance can then be written as dˆ K ,  d sK which is in a form of a first-order low-pass system and thus may have its limits.. 39. (30).

(54) 4.2. Dynamically compensated integral DOB (DC-IDOB) A dynamic compensator is added to the IDOB to improve the frequency response. of the disturbance observer and gain additional disturbance rejection characteristics. The dynamically compensated IDOB (DC-IDOB) adds an additional compensator to the IDOB as shown in Fig. 4-2.. d (s) u* ( s ).  . . . P(s). dˆ ( s ). Pˆ 1 ( s ) 1 s. C (s).  . DC-IDOB. Fig. 4- 2 Block diagram of dynamic compensator added to the integral DOB Consider a dynamic compensator described by. b s k  bk 1s k 1    b1s  b0 , C ( s)  k s m  cm 1s m 1    c1s  c0. (31). in which m  k , and bi and ci are coefficients of the compensator . If the dynamic compensator is added to the integral DOB as in Fig. 4-3, the transfer function of the DOB becomes bk s k  bk 1s k 1    b1 s  b0 dˆ GC   d 1  GC s m1  cm1 s m    c1 s 2  c0 s  bk s k  bk 1s k 1    b1 s  b0 . bk s k  bk 1s k 1    b1s  b0 , s m 1  am s m    a1s  a0. 40. (32).

(55) in which G ( s ) . 1 , and ai are coefficients of the generalized transfer function. s. G (s) d. . . 1 s. C (s). dˆ. Fig. 4- 3 Block diagram of DC-IDOB With the different order of the dynamic compensator, the integral DOB can gain the characteristics to reject various orders of disturbances. This can be accomplished by setting the values of m and k, while the bandwidth of the DOB can be set through the coefficients of the dynamic compensator bi and ci . The DC-IDOB is defined as a type-(k+1) IDOB by the order of the numerator. A type-(k+1) IDOB can assure a zero steady-state error to a k-degree polynomial disturbance.. 4.3. Examples of dynamically compensated integral DOB In this section, special cases of the dynamically compensated integral DOB are. introduced. Case Ⅰ(conventional IDOB, which is a special case of DC-IDOB): While letting m  k  0 , the transfer function becomes. b dˆ  0 , d s  b0. (33). and the dynamic compensator will then result in C ( s )  b0 , which is equivalent to the conventional integral disturbance observer introduced in chapter 1.2.2 equation (1). This is referred to as the type-1 first-order DC-IDOB (T1O1-IDOB).. 41.

(56) Case Ⅱ(General type-1 DC-IDOB): Let the desired transfer function. n 2 dˆ  , d s 2  2n s  n 2. (34). in which m  1 and k  0 , the coefficients b0   n , c1  3 n and c0  3 n , thus 3. 2. resulting in a compensator of C (s) . n 2 , s  2n. (35). which is referred to as the type-1 second-order DC-IDOB (T1O2-IDOB).. n 3 dˆ If the desired transfer function  3 , m  2 , k  0 and the d s  3 n s 2  3 n 2 s   n 3 compensator will result in C ( s ) . n3. s 2  3n s  n 2. , which is referred to as the type-1. third-order DC-IDOB (T1O3-IDOB). Case Ⅲ(Type-2 DC-IDOB): If the desired transfer function. 2n s  n 2 dˆ ,  d s 2  2n s  n 2. (36). in which m  1 and k  1 , the coefficients b0   n , b1  3n , c1  3n and c0  0 . The 3. 2. compensator is C (s) . 2n  n 2 , s  2n. and referred to as the type-2 second-order DC-IDOB (T2O2-IDOB).. 42. (37).

(57) If the desired transfer function. compensator will result in C ( s ) . 2 3 3 n s   n dˆ , m  2 , k  1 and the  3 d s  3 n s 2  3 n 2 s   n 3. 3n 2 s  n 3. s 2  3n s. , which is referred to as the type-2 third-. order DC-IDOB (T2O3-IDOB). Case Ⅳ(Type-3 DC-IDOB): Let the desired transfer function 3 n s 2  3 n 2 s   n 3 dˆ  , d s 3  3 n s 2  3 n 2 s   n 3. (38). in which m  2 and k  2 , the coefficients b0   n , b1  3 n , b2  3 n and c0  c1  0 , 3. 2. thus resulting in a compensator of C (s) . 3n s 2  3n 2 s  n 3 s2. ,. which is referred to as the type-3 third-order DC-IDOB (T3O3-IDOB).. 43. (39).

(58) 4.4. Internal model principal-based IDOB and its modification. Internal model principal-based IDOB (IMP-IDOB): In the dynamic compensator approach, a stable compensator is added to the IDOB for different characteristics of different ordered compensators. In this section, an internal model principal-based (IMP) method [29] is used in the compensator design. By using the IMP method, the estimated disturbance can asymptotically reject the disturbance of a certain frequency, thus letting the output to track the reference. Consider a disturbance model of d   2 d   1d   0 d  0 , which  i are the coefficients of the disturbance model, and let the transfer function of an IMP-IDOB be dˆ H (s) ,  3 2 d s   2 s  1 s   0  H ( s). (40). where H ( s )  h2 s 2  h1 s  h0 and is chosen so that s 3   2 s 2  1s   0  H ( s ) is. Hurwitz. The compensator is then. C ( s) . H ( s) 2. s   2 s  1 . 0. .. (41). s. dˆ When s3   2 s 2  1s   0  0 , the transfer function of the DOB will then become  1 , d which guarantees asymptotic disturbance estimation. For example, if the disturbance model is d   2 d  0 , in which  is the frequency of the disturbance, the transfer function then becomes 3 s 2  (3n 2   2 ) s  n 3 dˆ H ( s)   n , d s3   2 s  H ( s) s 3  3n s 2  3n 2 s  n 3 resulting in a compensator of. 44. (42).

(59) C ( s) . 3n s 2  (3n 2   2 ) s  n 3. .. s2  2. (43). With the compensator of this form, the gain of the compensator will be large at the frequency of  , thus letting the compensator effectively track the disturbance. Modified internal model principal-based IDOB (MIMP-IDOB): By adding a pole to the internal model, the transfer function then becomes dˆ H (s) ,  d ( s   0 )( s 3   3 s 2   2 s  1 )  H ( s ). (44). in which  i are the coefficients of the same disturbance model as the IMP-DOB, in which ( s   0 )( s 3   3 s 2   2 s  1 )  H ( s ) is chosen to be Hurwitz, giving the compensator C (s) . H (s) 2. ( s   0 )( s   3s   2 . 1 s. .. (45). ). For example, if the disturbance model is d   2 d  0 , the transfer function then becomes dˆ H (s) H ( s) ,   2 3 4 4 3 2 d s  4n s  6n s  4n s  n ( s  4n )( s 3   2 s )  H ( s ). (46). resulting in a compensator of. C ( s) . (6n 2   2 ) s 2  4n (n 2   2 ) s  n 4. s 3  4n s 2   2 s  4n  2. .. (47). This method increases the relative order of the system, having a smaller gain at high frequencies, and letting the control signal to have less chatter than the IMP-IDOB method.. 45.

(60) 4.5. Disturbance observer experiments The IDOB, DC-IDOBs, IMP-IDOB and MIMP-IDOB are compared in the. following experiments in this section. The disturbance added to the system is matched and spans as same space as the control input. The disturbance is directly added to the control effort output after the control effort is calculated by the DSP, then sent to the servo motor. The disturbance added in the following experiments is a triangular wave of 10 Hz, i.e., 20 rad/s with the amplitude of 0.2 V and offset value of -0.2 V. The DOBs are added to the system as in Fig 3-6, which means the nominal controller of the system is the JCTOC which admissible jerk (gain) k  1024 and r f  2 (rad). In the following experiments, the following indexes are analyzed. The chattering index [30] of the control effort is examined in the following experiments, which is.  (Tr ) . 1 Tr  u (t )  u (t ) dt , Tr 0. (48). where  is the chattering index, Tr is the duration of time, u is the unfiltered control effort and u is the filtered control effort. Zero-phase filters of different bandwidths are used and implemented with the MATLAB function “filtfilt”, in which CI denotes the bandwidth of the filter. The ITSE and ITAE [31] of the error, the steady-state maximum error and the spectrum of the steady-state error is also observed. The maximum steadystate error is chosen for that of t  1.2 (s). The error data of t  1.2 (s) is filtered with a Gaussian window function for less low-frequency leakage for a frequency spectrum with emphasis on the response of the 10-Hz disturbance. The prime index is the ITSE of the DOBs. The more advanced DOBs will not only have a better or similar ITSE than that of the DOB being compared with, but also better performance of the other indexes. Several experiments are conducted for comparison:. 46.

(61) Exp.4-1 Comparison of conventional IDOB and type-1 second-order DC-IDOB (T1O2IDOB). The conventional IDOB is set to have an integral gain of K  n  100 , while the T1O2-IDOB is set to n  650 for a similar ITSE of the conventional IDOB. The performance indexes are shown in Table 4-1, it can be seen that the T1O2-IDOB has a similar ITSE index to the conventional IDOB. But, the chattering index (CI) of the control effort of the T1O2-IDOB is much smaller, meaning that the control effort of the T1O2-IDOB is cleaner than the conventional IDOB, as observed through the control effort in Fig. 4-5. In Fig. 4-4, it can be seen that the T1O2-IDOB has a much faster settling time than the conventional IDOB, which intends that it has a faster transient characteristic. The frequency spectrum of the error Fig. 4-6 shows that although with a similar amplitude of error at 10 (Hz), the T1O2-IDOB has almost no DC elements. The frequency response of the systems Fig. 4-7 shows that the T1O2-IDOB has a larger bandwidth and also a lower gain of high-frequency signals, which may be noise. Table 4- 1 Indexes of results of conventional IDOB and T1O2-IDOB ITSE. CI CI  500. CI CI  1000. CI CI  2000. ITAE. Max. e. Conv. IDOB K  100. 1.418e-9. 0.04172. 0.02893. 0.01901. 1.305e-6. 0.001726. T1O2  n  650. 1.271e-9. 0.02349. 0.01386. 0.007147. 1.240e-6. 0.001342. 47.

(62) error (rad). 2. conv. IDOB (ωn=100) T1O2 (ω =650). 1.5. n. 1 0.5 0 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. 0.04 conv. IDOB (ω =100) n. T1O2 (ω =650). error (rad). 0.02. n. 0 −0.02 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 4 System response with IDOB and T1O2-IDOB compensation. control effort (V). 1. conv. IDOB (ωn=100). 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. control effort (V). 1. 1.2. T1O2 (ω =650) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. Fig. 4- 5 Control effort of JCTOC with IDOB and T1O2-IDOB. 48. 1.2.

(63) −3. 1.5. x 10. conv. IDOB (ωn=100). Amplitude of error (rad). T1O2 (ωn=650). 1. 0.5. 0. 0. 5. 10. 15. 20 25 Frequency (Hz). 30. 35. 40. Fig. 4- 6 Frequency spectrum of error with IDOB and T1O2-IDOB compensation Bode Diagram 0. Magnitude (dB). −10 −20 −30 −40 −50 0 conv. IDOB T1O2. Phase (deg). −45 −90 −135 −180 0. 10. 1. 10. 2. 10 Frequency (rad/sec). 3. 10. Fig. 4- 7 Theoretical frequency response of IDOB and T1O2-IDOB. 49. 4. 10.

(64) Exp.4-2 Comparison of type-1 second-order DC-IDOB (T1O2-IDOB) and Type-2 thirdorder DC-IDOB (T2O3-IDOB). The T1O2-IDOB is set to the same gain as Exp.4-1 n  650 , while the T2O3IDOB is set to n  265 for a better ITSE than the T1O2-IDOB. The performance indexes are shown in Table 4-2, it can be seen that the T2O3-IDOB not only has a better ITSE index than the T1O2-IDOB, but also a smaller chattering index (CI) of the control effort, which can be observed in Fig. 4-9, it can also be seen that high frequencies of the T2O3-IDOB have a smaller gain through the frequency response of the systems in Fig. 411, which may be the cause of less chattering. In Fig. 4-8, although the T2O3-IDOB has a slower settling time, the steady-state response is better than the T1O2-IDOB, which can be observed more easily in the frequency spectrum of the error Fig. 4-10, which shows a smaller amplitude of error at 10 Hz of the T2O3-IDOB. Table 4- 2 Indexes of results of T1O2-IDOB and T2O3-IDOB ITSE. CI CI  500. CI CI  1000. CI CI  2000. ITAE. Max. e. T1O2  n  650. 1.271e-9. 0.02349. 0.01386. 0.007147. 1.240e-6. 0.001342. T2O3 n  265. 1.318e-10. 0.02055. 0.009260. 0.004177. 3.543e-7. 0.001391. 50.

(65) error (rad). 2. T1O2 (ωn=650). 1.5. T2O3 (ω =265) n. 1 0.5 0 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. 0.04 T1O2 (ω =650) n. T2O3 (ω =265). error (rad). 0.02. n. 0 −0.02 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 8 System response with T1O2-IDOB and T2O3-IDOB compensation. control effort (V). 1. T1O2 (ω =650) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. control effort (V). 1. 1.2. T2O3 (ω =265) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 9 Control effort of JCTOC with T1O2-IDOB and T2O3-IDOB. 51.

(66) −3. 1.5. x 10. T1O2 (ωn=650). Amplitude of error (rad). T2O3 (ωn=265). 1. 0.5. 0. 0. 5. 10. 15. 20 25 Frequency (Hz). 30. 35. 40. Fig. 4- 10 Frequency spectrum of error with T1O2-IDOB and T2O3-IDOB compensation Bode Diagram. Magnitude (dB). 20. 0. −20. −40. −60 0 T1O2 T2O3. Phase (deg). −45 −90 −135 −180 0. 10. 1. 10. 2. 10 Frequency (rad/sec). 3. 10. Fig. 4- 11 Theoretical frequency response of T1O2-IDOB and T2O3-IDOB. 52. 4. 10.

(67) Exp.4-3 Comparison of T1O2-IDOB and IMP-IDOB. The T1O2-IDOB is set to have the same parameters as Exp.4-2 of n  650 , while the IMP-IDOB is set to n  45 and   20 for a better ITSE than the T1O2IDOB. The performance indexes are shown in Table 4-3, it can be seen that the IMPIDOB not only has a better ITSE index than the T1O2-IDOB, but also a smaller chattering index (CI) of the control effort, as shown in Fig. 4-13. The chattering of the IMP-IDOB is larger than the T1O2-IDOB due to a higher gain of higher frequencies (larger than 2000 rad/s), which can be seen in the frequency response of the systems in Fig. 4-15. In Fig. 4-12, it can be seen that although the IMP-IDOB has a slower settling time than the T1O2-IDOB, the steady-state response is much better. The frequency spectrum of the error Fig. 4-14 shows that there is almost no 10 Hz frequency in the error signal of the IMP-IDOB. The frequency response of the systems Fig. 4-15 shows that the IMP-IDOB has a lower gain at certain frequencies but enough gain at 10Hz, i.e., 20 rad/s, with almost 0 phase lag.. Table 4- 3 Indexes of results of T1O2-IDOB and IMP-IDOB ITSE. CI CI  500. CI CI  1000. CI CI  2000. ITAE. Max. e. T1O2  n  650. 1.271e-9. 0.02349. 0.01386. 0.007147. 1.240e-6. 0.001342. IMP-IDOB n  45. 5.801e-11. 0.01478. 0.01058. 0.007299. 2.462e-7. 8.880e-4. 53.

(68) error (rad). 2. T1O2 (ωn=650) IMP (ω =45). 1.5. n. 1 0.5 0 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. 0.04 T1O2 (ω =650) n. IMP (ω =45). error (rad). 0.02. n. 0 −0.02 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 12 System response with T1O2-IDOB and IMP-IDOB compensation. control effort (V). 1. T1O2 (ω =650) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. control effort (V). 1. 1.2. IMP (ω =45) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 13 Control effort of JCTOC with T1O2-IDOB and IMP-IDOB. 54.

(69) −3. 1.5. x 10. T1O2 (ωn=650). Amplitude of error (rad). IMP (ωn=45). 1. 0.5. 0. 0. 5. 10. 15. 20 25 Frequency (Hz). 30. 35. 40. Fig. 4- 14 Frequency spectrum of error with T1O2-IDOB and IMP-IDOB compensation Bode Diagram 10. Magnitude (dB). 0 −10 −20 −30 −40 −50 45 T1O2 IMP. Phase (deg). 0 −45 −90 −135 −180 0. 10. 1. 10. 2. 10 Frequency (rad/sec). 3. 10. Fig. 4- 15 Theoretical frequency response of T1O2-IDOB and IMP-IDOB. 55. 4. 10.

(70) Exp.4-4 Comparison of IMP-IDOB and MIMP-IDOB. The IMP-IDOB is set to have the same gain as Exp. 4-3 n  45 and   20 , while the MIMP-IDOB is set to n  82 and   20 for a similar ITSE to the IMPIDOB. The performance indexes are shown in Table 4-4, it can be seen that the MIMPIDOB has a similar ITSE index to the IMP-IDOB. But, the chattering index (CI) of the control effort of the MIMP-IDOB is much smaller, meaning that the control effort of the MIMP-IDOB is smoother than the IMP-IDOB, as shown in Fig. 4-17. In Fig. 4-16, it can be seen that the MIMP-IDOB has a faster settling time than the IMP-IDOB. The frequency spectrum of the error Fig. 4-18 shows that both DOBs both have almost no 10 Hz components, and the MIMP-IDOB has less DC leakage. The frequency response of the systems Fig. 4-19 shows that the MIMP-IDOB has a larger bandwidth and also a lower gain of high-frequency signals, thus letting the control signal to be cleaner, as shown in Fig. 4-17. Table 4- 4 Indexes of results of IMP-IDOB and MIMP-IDOB ITSE. CI CI  500. CI CI  1000. CI CI  2000. ITAE. Max. e. IMP-IDOB n  45. 5.801e-11. 0.01478. 0.01058. 0.007299. 2.462e-7. 8.880e-4. MIMP-IDOB n  82. 2.764e-11. 0.005499. 0.002317. 0.001039. 1.805e-7. 7.900e-4. 56.

(71) error (rad). 2. IMP (ωn=45) MIMP (ω =82). 1.5. n. 1 0.5 0 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. 0.04 IMP (ω =45) n. MIMP (ω =82). error (rad). 0.02. n. 0 −0.02 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 16 System response with IMP-IDOB and MIMP-IDOB compensation. control effort (V). 1. IMP (ω =45) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. control effort (V). 1. 1.2. MIMP (ω =82) n. 0.5 0 −0.5 0. 0.2. 0.4. 0.6 0.8 time (s). 1. 1.2. Fig. 4- 17 Control effort of JCTOC with IMP-IDOB and MIMP-IDOB. 57.

(72) −3. 1.5. x 10. IMP (ωn=45). Amplitude of error (rad). MIMP (ωn=82). 1. 0.5. 0. 0. 5. 10. 15. 20 25 Frequency (Hz). 30. 35. 40. Fig. 4- 18 Frequency spectrum of error with IMP-IDOB and MIMP-IDOB compensation Bode Diagram. Magnitude (dB). 20. 0. −20. −40. −60 45 IMP MIMP. Phase (deg). 0 −45 −90 −135 −180 0. 10. 1. 10. 2. 10 Frequency (rad/sec). 3. 10. Fig. 4- 19 Theoretical frequency response of IMP-IDOB and MIMP-IDOB. 58. 4. 10.