ON ACTIVE DISTURBANCE REJECTION CONTROL: STABILITY .
ON ACTIVE DISTURBANCE REJECTION CONTROL:STABILITY ANALYSIS AND APPLICATIONS INDISTURBANCE DECOUPLING CONTROLQING ZHENGBachelor of Science in Electrical EngineeringNorth China University of TechnologyJuly, 1996Master of Engineering in Electrical EngineeringNational University of SingaporeJuly, 2003submitted in partial fulfillment of the requirements for the degreeDOCTOR OF ENGINEERINGat theCLEVELAND STATE UNIVERSITYJuly 2009
c Copyrightby Qing Zheng 2009
This dissertation has been approved for theDepartment of ELECTRICAL AND COMPUTER ENGINEERINGand the College of Graduate Studies byDissertation Committee Chairperson, Dr. Zhiqiang GaoDepartment/DateDr. Lili DongDepartment/DateDr. Paul LinDepartment/DateDr. Sally ShaoDepartment/DateDr. Dan SimonDepartment/DateDr. Sridhar UngaralaDepartment/Date
To my family.It is their anticipation that stimulates me to pursue a high education!
ACKNOWLEDGMENTSI would like to express my most sincere gratitude to my advisor, Dr. ZhiqiangGao, for his consistent guidance, patience, and support during the whole period ofmy study. Dr. Gao’s rigorous scientific approach and endless enthusiasm in researchhave influenced me greatly. His erudite knowledge as well as deep insights in thefield of control make this research an invaluable experience. Without his stimulatingdiscussions and continuous encouragement, this dissertation and many other resultswould have been impossible.Special thanks go to Dr. Lili Dong, Dr. Paul Lin, Dr. Sally Shao, Dr. DanSimon, and Dr. Sridhar Ungarala, who are on my dissertation committee, for theirtime in reviewing and evaluating this dissertation as well as for their guidance on myresearch.I am also thankful for the peer advice from my lab partners, especially GangTian, Shen Zhao, Jeff Csank, Classica Jain, and Zhongzhou Chen, but also includingFrank Goforth, Aaron Radke, Rob Miklosovic, Wankun Zhou, David Avanesov, andDae Hui Lee.In this dissertation, Some sections of some chapters are reprinted from myearlier publications during my doctoral study, with the permission from IEEE. ThankIEEE for granting this permission.My deepest gratitude is due to my family members. Without their love, patience, encouragement, and sacrifice, I would not have accomplished this. I wish todedicate this dissertation to all of them.
ON ACTIVE DISTURBANCE REJECTION CONTROL:STABILITY ANALYSIS AND APPLICATIONS INDISTURBANCE DECOUPLING CONTROLQING ZHENGABSTRACTOne main contribution of this dissertation is to analyze the stability characteristics of extended state observer (ESO) and active disturbance rejection control(ADRC). In particular, asymptotic stability of the dynamic system that describes theestimation error and the closed-loop system is established where the plant dynamicsis completely known. In the face of large dynamic uncertainties, the estimation error,the closed-loop tracking error, and its up to the (n 1)st order derivatives are shownto be bounded. Furthermore, it is demonstrated that the error upper bounds, ingeneral, monotonously decrease with the observer and control loop bandwidths. Thesecond contribution is to develop a dynamic disturbance decoupling control strategyfor square multivariable systems based on ADRC. The proposed method has beensuccessfully applied to chemical process problems and micro-electro-mechanical systems gyroscopes. It is shown that a largely unknown square multivariable systemcan be readily decoupled by actively estimating and rejecting the effects of both theinternal plant dynamics and external disturbances. By requiring little information onthe plant model, the intention is to make the new decoupling approach practical.vi
TABLE OF CONTENTSPageABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiACRONYMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiiiCHAPTERI.INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1II.PROBLEM FORMULATION AND LITERATURE SURVEY . . . . . .52.1Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . .52.1.1The Problem of ADRC Stability Characteristics . . . . .52.1.2The Disturbance Decoupling Control Problem. . . . . .7Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . .92.2.1A Survey of Disturbance Estimation . . . . . . . . . . . .92.2.2A Survey of Stability Analysis of Disturbance Estimators112.2.3A Survey of Decoupling Control . . . . . . . . . . . . . .13Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15ACTIVE DISTURBANCE REJECTION CONTROL . . . . . . . . . .162.22.3III.3.1Extended State Observer . . . . . . . . . . . . . . . . . . . . . .173.2Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . .183.3Simulation and Hardware Tests . . . . . . . . . . . . . . . . . . .193.3.1A Simulation Case Study . . . . . . . . . . . . . . . . . .203.3.2A Motion Control Hardware Test . . . . . . . . . . . . . .21Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243.4vii
IV.STABILITY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . .4.1Analysis of ESO Error Dynamics . . . . . . . . . . . . . . . . . .Convergence of the ESO with the Given Model of the Plant 284.1.2Convergence of the ESO with Plant Dynamics Largely Un. . . . . . . . . . . . . . . . . . . . . . . . . . . .31Stability Characteristics of ADRC . . . . . . . . . . . . . . . . .354.2.1Convergence of the ADRC with the Given Model of thePlant . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.24.3V.Unknown . . . . . . . . . . . . . . . . . . . . . . . . . . .39Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42A PRACTICAL APPROACH TO DISTURBANCE DECOUPLING CON-5.1Reformulation of Decoupling Control Problem5.2Multi-Loop Extended State Observer5.3Dynamic Disturbance Decoupling5.443. . . . . . . . . .44. . . . . . . . . . . . . . .46. . . . . . . . . . . . . . . . .47Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49DISTURBANCE DECOUPLING CONTROL IN CHEMICAL PROCESSES 506.1VII.35Convergence of the ADRC with Plant Dynamics LargelyTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VI.274.1.1known4.226A Linear Multivariable System . . . . . . . . . . . . . . . . . . .506.1.1Setpoint Tracking and Disturbance Rejection Performance526.1.2Control Signal Selection . . . . . . . . . . . . . . . . . . .536.2A Nonlinear Multivariable System . . . . . . . . . . . . . . . . .546.3Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57CONTROL AND RATE ESTIMATION OF MEMS GYROSCOPES .597.1Introduction to MEMS Gyroscopes . . . . . . . . . . . . . . . . .607.2Dynamics of MEMS Gyroscopes . . . . . . . . . . . . . . . . . .61viii
7.3DDC for MEMS Gyroscopes . . . . . . . . . . . . . . . . . . . .627.4Rotation Rate Estimation . . . . . . . . . . . . . . . . . . . . . .657.5Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .677.6Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . .758.1Findings and Conclusions . . . . . . . . . . . . . . . . . . . . . .758.2Remarks on Future Research . . . . . . . . . . . . . . . . . . . .76BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78VIII.ix
LIST OF TABLESTableIPageDescription of Variables for the CSTR Model [87] . . . . . . . . . . .x56
LIST OF FIGURESFigure1PageThe errors between actual and estimated information. (ESO1: withoutplant information; ESO2: with partial plant information; ESO3: withcomplete plant information. Note that ESO2 and ESO3 are almostoverlapped.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221The ADRC performance with different ESOs for the nonlinear system.(ESO1: without plant information; ESO2: with partial plant information; ESO3: with complete plant information. Note that ESO2 andESO3 are almost overlapped.) . . . . . . . . . . . . . . . . . . . . . .223A diagram for DC brushless servo system. . . . . . . . . . . . . . . .234The output comparison among an ideal double integrator, simulationtest, and hardware test. . . . . . . . . . . . . . . . . . . . . . . . . .245The performance for ECP Model 220 under the control of ADRC. . .256A simplified scheme of distillation column [29]. . . . . . . . . . . . . .517The comparison of disturbance rejection performance between the DDCand the MPC for Loop 1 of the distillation column. . . . . . . . . . .8The comparison of disturbance rejection performance between the DDCand the MPC for Loop 2 of the distillation column. . . . . . . . . . .95253The performance with non-dominant control signal selection for eachloop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5410The CSTR diagram [87]. . . . . . . . . . . . . . . . . . . . . . . . . .5511The output response of CSTR under the control of the DDC . . . . .5712The control signals of CSTR. . . . . . . . . . . . . . . . . . . . . . .58xi
13The tracking error of CSTR. . . . . . . . . . . . . . . . . . . . . . . .5814Block diagram of the ADRC and rate estimation. . . . . . . . . . . .6615The output of the drive axis. . . . . . . . . . . . . . . . . . . . . . . .6916The output of the sense axis. . . . . . . . . . . . . . . . . . . . . . . .7017The rotation rate estimation at frate 50 Hz. . . . . . . . . . . . . . .7118The control signals of the drive and sense axes. . . . . . . . . . . . .7219The output of the drive axis with parameter variations. . . . . . . . .7220The output of the sense axis with parameter variations. . . . . . . . .7321The rotation rate estimation at frate 50 Hz with parameter variations. 7322The rotation rate estimation at frate 100 Hz. . . . . . . . . . . . . .7423The rotation rate estimation at frate 200 Hz. . . . . . . . . . . . . .74xii
ACRONYMSUIO unknown input observerDOB disturbance observerPOB perturbation observerESO extended state observerADRC active disturbance rejection controlMIMO multiple-input multiple-outputMEMS micro-electro-mechanical systemsSISO single-input single-outputPID proportional, integral, and derivativeFLC fuzzy logic controlANN artificial neural networkGD generalized disturbanceHGO high gain observerECP Educational Control ProductsCSTR continuous stirred tank reactorMPC model predictive controlLPF low pass filterxiii
CHAPTER IINTRODUCTIONMost existing control design methods, such as Bode plot method in classicalcontrol and H2 /H control in modern control theory, are based on mathematicaldescription of plant dynamics. However, many physical plants in real world are notonly nonlinear and time-varying but also highly uncertain. Accurate mathematicaldescriptions of physical plants are usually not available in industrial control. Thiscreates a dilemma for control practitioners: the requirement of the plant mathematicalmodel from the theoretical side and the uncertainty of the plant dynamics in practice.Such dilemma caught much attention of many researchers. One solution is robustcontrol, where a small amount of uncertainty in physical plants can be tolerated.Another dilemma in control system design is how to handle disturbances. Inthe current modern control framework, disturbance attenuation is one of key controldesign objectives. A less known solution is to estimate and cancel disturbance directly.To this end, many disturbance estimators, such as unknown input observer (UIO) [1][8], disturbance observer (DOB) [9]-[16], perturbation observer (POB) [17]-[20], andextended state observer (ESO) [21]-[27], have been proposed. Based on the ESO,1
2an active disturbance rejection control (ADRC) algorithm has also been developed[21]-[27]. The ADRC offers a new and inherently robust controller building block thatrequires very little information of the plant. This control algorithm actively estimatesand compensates for the effects of the unknown dynamics and disturbances, forcingan otherwise unknown plant to behave like a nominal one. Such strategy offers analternative to the prevailing methods. That is, instead of depending on the model ofthe plant, the controller draws the information needed from the ESO to control theplant. This is achieved by using an ESO to estimate both internal plant dynamicsand external disturbances.In robust control, the stability analysis is mostly based on the small gain theorem, and the results tend to be quite conservative by nature. As to the class ofdisturbance estimators UIO, DOB, and POB, the rigorous stability proof or convergence is not established, although a few researchers speculated this could be done.Although the ESO and the ADRC have been applied to solve many different kinds ofproblems, their convergence and stability have not been proven. Furthermore, thereis a lack of understanding of the exact relationship between the control system tuningparameters and the performance requirements. In short, we know the ADRC works,but do not know why. In particular, the relationship between the tuning parameters,which include the observer and controller bandwidths, and the observer estimation error and the closed-loop tracking error is unknown. Without rigorous analytical study,the ADRC remains a trial and error method. Therefore the stability and convergenceanalysis for the ESO and the ADRC is essential.In this dissertation, one key objective is to provide an analytical insight on whyADRC achieves excellent performance, that is, to mathematically show that both theexternal disturbance and the plant dynamics can be estimated by ESO. Estimating theunknown parameters in plants has been studied in system identification. Estimating
3both the external disturbance and the plant dynamics crosses the boundary betweensystem identification and observer design. Whether the combined effect of plantdynamics and external disturbance can be estimated in real time or not is of greatimportance, because if the answer is yes, it could mean that the uncertainty problem,the adaptive control problem, and disturbance rejection problem, can all be handledin one single framework.The mathematical proof of the ESO convergence and the ADRC stability renders the theoretical support of why the ADRC can achieve high disturbance rejectionand robustness performance. It also explains why the ADRC has been successfullyapplied to many applications. In this dissertation, the ADRC is extended to anotherimportant class of practical problems, namely the decoupling control for multivariablesystems.The interactions or cross-couplings between the variables are the most significant characteristics with multivariable systems, i.e., systems with multiple inputsand multiple outputs, also known as multiple-input multiple-output (MIMO) systems. With the interactions or cross-couplings present, one input variable may affectall the output variables. In a vibrational micro-electro-mechanical systems (MEMS)gyroscope [28], the quadrature error is caused by coupling in the stiffness term between its drive and sense axes. In the Wood and Berry column example [29], theinputs reflux and vapor flow both affect the two outputs, which are the top andbottom composition. The interactions or cross-couplings among various inputs andoutputs of a system make design technologies in multivariable control systems fundamentally different from single-input single-output (SISO) control systems. Giventhat our understanding of the physics of MIMO systems usually helps us identify thedominant input-output pairs, one design strategy is to disentangle the interactionsamong various input-output pairs and reduce a multivariable system into a number
4of independent SISO systems. This strategy is usually known as decoupling.Although the interactions are present in most multivariable systems, the control engineers in industry frequently ignore the interactions between variables anddesign the controller for each loop independently. In most cases, proportional, integral, and derivative (PID) control is preferred. This is not because the controlengineers are not aware of the interactions but because the existing methods for solving the problem are difficult for engineers to understand and implement, either dueto their mathematical complexity or due to the unavailability of the necessary modelinformation. Therefore, it is important to find an approach that can address theinteraction problems and is practical and easy to be implemented in industry. Thisleads to the second key objective of this research, that is, to develop a new practicaldecoupling control approach for multivariable systems.In this dissertation, we first analyze the stability characteristics of the ESOand the associated ADRC, then discuss the proposed disturbance decoupling control(DDC) approach. The organization of the dissertation is as follows. The stabilityproblem of the ADRC and the decoupling problem are formulated in Chapter 2, wherethe literature survey for disturbance estimators, stability analysis of disturbance estimators, and decoupling control is also given. The idea of the ADRC is introduced andits effectiveness is demonstrated through simulation and hardware tests in Chapter3. The stability characteristics for the ESO and the ADRC are analyzed in Chapter4. A novel and practical DDC approach is proposed in Chapter 5. Simulation resultsobtained on two chemical process problems are shown in Chapter 6. The applicationsof DDC to MEMS gyroscopes are performed in Chapter 7. Finally, the highlightof the major contributions of this dissertation and the recommendations on possiblefuture research directions that may be pursued based on the insights gained from thisresearch are given in Chapter 8.
CHAPTER IIPROBLEM FORMULATION ANDLITERATURE SURVEYThis chapter begins with problem formulation of the ADRC stability analysisand decoupling control, followed by a literature survey on disturbance estimation,stability analysis of disturbance estimators, and decoupling control.2.1Problem FormulationThis section discusses what is the problem of the ADRC stability characteristicsand why a new decoupling control is needed.2.1.1The Problem of ADRC Stability CharacteristicsMost physical plants in real world are not just nonlinear and time-varying butalso highly uncertain. Control system design for such systems has been the focus ofmuch of the recent developments under the umbrella of robust, adaptive, and nonlinear control. Most of the existing results, however, are obtained presupposing that5
6a fairly detailed and accurate mathematical model of the plant is available. Thesmall gain theorem based robustness analysis does allow a small amount of uncertainties in plant dynamics, but not anywhere near the magnitude often encounteredin practice. As the well-known control theorist Roger Brockett puts it: “If there isno uncertainty in the system, the control, or the environment, feedback control islargely unnecessary” [30]. The assumption that a physical plant, without feedback,behaves rather closely as its mathematical model describes, as the point of departurein control system design, does not reflect either the intent of feedback control, or thephysical reality.The ADRC was proposed as an alternative paradigm to address this fundamental issue [21]. The main difference in the design concept pertains to the questionof how much model information is needed. Recognizing the vulnerability of the reliance on accurate mathematical model, there has been a gradual recognition overthe years that active disturbance estimation is a viable alternative to an accurateplant model. That is, if the disturbance, representing the discrepancy between theplant and its model, is estimated in real time, then the plant-model mismatch canbe effectively compensa
ON ACTIVE DISTURBANCE REJECTION CONTROL: STABILITY ANALYSIS AND APPLICATIONS IN DISTURBANCE DECOUPLING CONTROL QING ZHENG ABSTRACT One main contribution of this dissertation is to analyze the stability char-acteristics of extended state observer (ESO) and active disturbance rejection control (ADRC).
4. Active Disturbance Rejection Control . The active disturbance rejection control was proposed by Han [9][10][11][12]. It is designed to deal with systems having a large amount of uncertainty in both the internal dynamics and external disturbances. The particularity of the ADRC design is that the total disturbance is defined as
ing control technology, known as active disturbance rejection con-trol to this day. Three cornerstones toward building the founda-tion of active disturbance rejection control, namely, the track-ing di erentiator, the extended state observer, and the extended state observer based feedback are expounded separately. The paper
practical control strategy that is Active Disturbance Rejection Control (ADRC) [10-14] to the EPAS system. The controller treats the discrepancy between the real EPAS system and its mathematical model as the generalized disturbance and actively estimates and rejects in real time, the disturbance
shift. The active disturbance rejection concept, introduced in section III, could very well serve as the basis of the new paradigm, which is characterized in section IV. Also included in section III is the demonstration of the broad range applications of the active disturbance concept. Finally, the paper is concluded with a few remarks in .
The Active Disturbance Rejection Controller (ADRC), invented by Han Jing-Qing who serves in Chinese Academy of Science, is a new type of nonlinear controller which is based on active disturbance rejection control technology. It does not depend on system model andcan estimate and compensate the influences of all the
e recently developed active disturbance rejection con-trol (ADRC) is an e cient nonlinear digital control strategy which regards the unmodeled part and external disturbance as overall disturbance of the controlled system [ ] . e ADRC mainly consists of a tracking di erentiator (TD), an extended state observer (ESO), and a nonlinear state error
ADRC and inferential control has not been reported in the literature. The paper is organised as follows: Section 2 gives an overview of active disturbance rejection control and inferential control. An integrated ADRC and inferential control strategy for distillation columns is presented in Section 3.
7 Annual Book of ASTM Standards, Vol 14.02. 8 Discontinued 1996; see 1995 Annual Book of ASTM Standards, Vol 03.05. 9 Annual Book of ASTM Standards, Vol 03.03. 10 Available from American National Standards Institute, 11 West 42nd St., 13th Floor, New York, NY 10036. 11 Available from General Service Administration, Washington, DC 20405. 12 Available from Standardization Documents Order Desk .
