Adaptive controller design directly from plant data

ADAPTIVE CONTROLLER DESIGN DIRECTLY FROM PLANT DATA YAN LI (B. Eng., National University of Singapore, Singapore) (M. Sc., Mines ParisTech, France) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my research supervisor, Dr. Min-Sen Chiu, for his excellent guidance and valuable suggestions during my studies in the National University of Singapore. I am also thankful to Dr. Lakshiminarayanan for his valuable advices to my research work. Special thanks and appreciation are due to my lab mates, Martin Wijaya Hermanto, Xin Yang, Qinglin Su and Vamsi Krishna Kamaraju, for the stimulating discussions that we have had and the helps that they have rendered to me. I would also wish to thank technical and administrative staffs in the Chemical and Biomolecular Engineering Department for the efficient and prompt help. I cannot find any words to thank my parents for their unconditional support, affection and encouragement, without which this research work would not have been possible. i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY iv LIST OF TABLES v LIST OF FIGURES vi NOMENCLATURE x CHAPTER 1. INTRODUCTION 1.1 Motivations 1.2 Contributions 1.3 Thesis Organization CHAPTER 2. LITERATURE REVIEW 2.1 Adaptive Control for Nonlinear Processes 2.2 Direct Data-based Controller Design Methods 11 2.2.1 The VRFT Design Framework 12 2.2.2 Adaptive VRFT design 14 2.3 Nonlinear Internal Model Control (NIMC) 15 CHAPTER 3. ADAPTIVE PID CONTROLLER DESIGN USING EVRFT METHOD 19 3.1 Introduction 19 ii 3.2 PID Controller Design by VRFT method 20 3.3 Enhanced VRFT Design Method 23 3.4 Examples 24 3.5 Conclusion 42 CHAPTER 4. ADAPTIVE INTERNAL MODEL CONTROLLER DESIGN USING EVRFT METHOD 43 4.1 Introduction 43 4.2 IMC Controller Design Using VRFT Method 44 4.3 Enhanced VRFT Design Method 47 4.4 Examples 48 4.5 Conclusion 62 CHAPTER 5. CONCLUSIONS AND FURTHER WORK 63 5.1 Conclusions 63 5.2 Suggestions for Further Work 64 APPENDIX A. DERIVATION OF THE 2ND-ORDER REFERENCE MODEL 65 REFERENCES 66 iii SUMMARY Controller design for nonlinear dynamic processes has been of great interest in the chemical industry. Various nonlinear controller design strategies have been studied in the literature. Among them, adaptive controller is a well-established solution for this issue. In this thesis, a new adaptive controller design method is proposed based on the virtual reference feedback tuning (VRFT) method which was originally developed for linear controller design. This new method is termed as enhanced VRFT (EVRFT) design to account for the difference from the linear VRFT method. In the proposed method, not only a second-order reference model is employed instead of the first-order reference model commonly used in the linear VRFT design, but also the parameters of reference model are updated at each sampling instance to ensure the adaptive nature of the design strategy. In addition, to complete the on-line adaptation process, the database is updated at each sampling instance by adding the current process data into it and a relevant dataset is selected from the current database according to the k-nearest neighborhood criterion. Two different adaptive controllers are developed implementing the EVRFT strategy, i.e. an adaptive PID controller and an adaptive Internal Model Controller. Simulation results show that both proposed controllers give improved control performance than the linear PID controller designed using VRFT method. They are also shown to be quite robust in the presence of modeling error and can tolerate reasonable process noise through simulation studies. iv LIST OF TABLES Table 3.1 Model parameters for polymerization reactor 26 Table 3.2 Steady-state operating condition of polymerization reactor 26 Table 3.3 Tracking errors of servo responses obtained by various 28 design methods Table 3.4 Tracking errors of servo responses obtained by various 35 design methods Table 3.5 Tracking errors for EVRFT and VRFT designs for time 40 delay case Table 4.1 Tracking errors obtained by various design methods 49 Table 4.2 Tracking errors obtained by various design methods 56 Table 4.3 Tracking errors of VRFT and EVRFT designs for time 58 delay case v LIST OF FIGURES Figure 2.1 Diagram of adaptive control scheme Figure 2.2 Reference model 13 Figure 2.3 Feedback control system 13 Figure 3.1 Polymerization reactor 25 Figure 3.2 Input-output data used for constructing the database (example 1) Figure 3.3 27 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2ndorder model (A = 0.77); dashed line: VRFT using 1st-order model (A = 0.95) Figure 3.4 29 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2ndorder model (A = 0.3); dashed line: VRFT using 1st-order model (A = 0.5) Figure 3.5 29 Updating of tuning parameters in EVRFT design (+50% set-point change) Figure 3.6 30 Updating of tuning parameters in EVRFT design (-50% set-point change) Figure 3.7 30 Responses for +50% (top) and -50% (bottom) set-point changes by EVRFT design in the presence of modeling error Figure 3.8 31 Responses for +50% (top) and -50% (bottom) set-point changes by EVRFT design in the presence of process noise vi 32 Figure 3.9 Steady-state curve of van de Vusse reactor Figure 3.10 Input-output data used for constructing the database (example 2) Figure 3.11 33 34 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom). Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A = 0.65); dashed line: VRFT using1st-order model (A = 0.65) Figure 3.12 36 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom). Solid line: EVRFT; dotted line: VRF using 2nd-order model (A = 0.35); dashed line: VRFT using 1st-order model (A = 0.37) Figure 3.13 Updating of controller parameters in EVRFT design for set-point change to 1.25 Figure 3.14 38 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom) in the presence of process noise Figure 3.17 37 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom) in the presence of modeling error Figure 3.16 37 Updating of controller parameters in EVRFT design for set-point change to 0.62 Figure 3.15 36 39 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom) for time delay case. Solid line: EVRFT; dotted line: VRFT Figure 3.18 Figure 3.19 40 Updating of controller parameters in EVRFT design for set-point change to 1.25 in the presence of time delay 41 Updating of controller parameters in EVRFT design for 41 vii set-point change to 0.62 in the presence of time delay Figure 4.1 Block diagram of IMC structure Figure 4.2 Responses for +50% (top) and -50% (bottom) set-point 45 changes. Solid line: EVRFT; dotted line: VRFT using 2ndorder model (A=0.77); dashed line: VRFT using 1st-order model (A=0.95) Figure 4.3 50 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2ndorder model (A = 0.3); dashed line: VRFT using 1st-order model (A = 0.5) Figure 4.4 50 Updating of parameter A and IMC parameters (+50% setpoint change) Figure 4.5 51 Updating of parameter A and IMC parameters (-50% setpoint change) Figure 4.6 51 Responses for 50% (top) and -50% (bottom) set-point changes in the presence of modeling error Figure 4.7 52 Responses for 50% (top) and -50% (bottom) set-point changes in the presence of process noise 53 Figure 4.8 The catalytic continuous stirred tank reactor 54 Figure 4.9 Steady-state curve of catalytic CSTR 55 Figure 4.10 Input-output data used for constructing the database (catalytic CSTR) Figure 4.11 55 Closed-loop responses for set-point changes to 21.5 (top) and to 19.5 (bottom). Solid line: EVRFT; dotted line: VRFT using 2nd-order model; dashed line: VRFT using viii 57 1st-order model Figure 4.12 Updating of parameter A and IMC parameters for set-point change to 21.5 Figure 4.13 59 Updating of parameter A and IMC parameters for set-point change to 19.5 Figure 4.14 59 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of modeling error Figure 4.15 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of process noise Figure 4.16 60 60 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of time delay. Solid line: EVRFT; dotted line: VRFT Figure 4.17 61 Updating of parameter A and IMC parameters set-point change to 21.5 in the presence of time delay Figure 4.18 61 Updating of parameter A and IMC parameters for set-point change to 19.5 in the presence of time delay ix 62 Chapter Adaptive Internal Model Controller Design Using EVRFT Method To proceed with the EVRFT design of adaptive IMC controller, the tuning parameters specified are as follows: the IMC filter parameter is fixed as λ = 0.01 , the initial value of parameter A = 0.4, initial learning rate η = 2.5 × 10−3 , weight parameter used in the objective function Eq. (3.18) given by w = 0.7 , the updating parameters for learning rate linc = 1.02 and ldec = 0.55 , and the number of data points in the relevant dataset is set to 1200. To evaluate the performance of the adaptive IMC designed using EVRFT method, two respective set-point changes from the nominal operating condition to 21.5 and to 19.5 are conducted, as illustrated in Figure 4.11. The corresponding updating of tuning parameter A and IMC parameters in the above servo responses is shown in Figures 4.12 and 4.13. For the purpose of comparison, PID controllers designed using the VRFT method with both first-order and second-order reference models are developed. The tuning parameter A = 0.96 is selected for the first-order case and A = 0.74 is selected for the second order case to achieve the minimum tracking errors for the two set-point changes. It is evident from Figure 4.11 that the adaptive IMC controller designed using EVRFT method gives faster response for both set-point changes than the two PID controllers designed by VRFT method. Moreover, the PID controller designed using second-order reference model shows better performance than that with a firstorder reference model. The tracking errors for all methods are summarized in Table 4.2. Table 4.2 Tracking errors obtained by various design methods Set-point change to 21.5 19.5 VRFT design VRFT design using 1st-order using 2ndmodel order model 0.1385 0.0977 0.2270 0.1753 56 EVRFT based IMC design 0.0754 0.1384 % improvement (based on VRFT design using 2nd-order model) 22.8% 21.1% Chapter Adaptive Internal Model Controller Design Using EVRFT Method Figure 4.11 Closed-loop responses for set-point changes to 21.5 (top) and to 19.5 (bottom). Solid line: EVRFT; dotted line: VRFT using 2nd-order model; dashed line: VRFT using 1st-order model Next, to evaluate the robustness of the proposed control strategy, 10% modeling errors in the kinetic parameter k1 and k are considered. Figure 4.14 illustrates that the proposed EVRFT design gives satisfactory performance in the presence of modeling error. Furthermore, to study the sensitivity of the proposed method with respect to noise, both process input and output data are corrupted with 1% Gaussian white noise. It can be seen from Figure 4.15 that the proposed controller can yield reasonably good control performance in the presence of process noise. Finally, to further evaluate the performance of the proposed method, it is assumed that there exists time delay in the output measurement of five sampling time. In this case, most of tuning parameters for EVRFT design are identical to those used for the case in the absence of time delay, except that the initial values of A and 57 Chapter Adaptive Internal Model Controller Design Using EVRFT Method learning rate η are changed learning rate η are changed as A = 0.5 and η = 8.3 × 10−4 , while the IMC filter parameter is fixed as λ = 0.5 . Figure 4.16 shows the resulting performances of ERFT design for the same set-point changes described previously. The corresponding updating of controller parameters is shown in Figures 4.17 and 4.18. For the purpose of comparison, a PID controller designed using VRFT method with second-order reference model is developed. The tuning parameter A = 0.788 is selected to obtain minimum racking error for the two set-point changes. The tracking errors for both EVRFT and VRFT designs are summarized in Table 4.3. It can be seen from Figure 4.16 that adaptive IMC controller designed using EVRFT method gives better overall performance than that obtained by the PID controller designed using VRFT with second-order reference model. Table 4.3 Tracking errors of VRFT and EVRFT designs for time delay case Set-point change to VRFT 21.5 19.5 0.1279 0.2318 EVRFT based IMC design 0.1062 0.2058 58 % improvement 17.0% 11.2% Chapter Adaptive Internal Model Controller Design Using EVRFT Method Figure 4.12 Updating of parameter A and IMC parameters for set-point change to 21.5 Figure 4.13 Updating of parameter A and IMC parameters for set-point change to 19.5 59 Chapter Adaptive Internal Model Controller Design Using EVRFT Method Figure 4.14 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of modeling error Figure 4.15 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of process noise 60 Chapter Adaptive Internal Model Controller Design Using EVRFT Method Figure 4.16 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of time delay. Solid line: EVRFT; dotted line: VRFT Figure 4.17 Updating of parameter A and IMC parameters set-point change to 21.5 in the presence of time delay 61 Chapter Adaptive Internal Model Controller Design Using EVRFT Method Figure 4.18 Updating of parameter A and IMC parameters for set-point change to 19.5 in the presence of time delay 4.5 Conclusion In this chapter, an adaptive IMC design method is proposed. The proposed method makes use of the VRFT design and achieves the adaptive nature by updating the database, selecting a relevant dataset and most importantly, updating the reference model parameter A at each sampling instance. Moreover, a second-order reference model is employed for the VRFT design instead of the commonly used first-order reference model to achieve better control performance. The simulation results demonstrate that the proposed method gives improved performance over the conventional VRFT design. 62 Chapter Conclusions and Further Work 5.1 Conclusions In this thesis, the Enhanced Virtual Reference Feedback Tuning (EVRT) method is proposed and two adaptive controllers, namely adaptive PID controller and adaptive IMC controller, are developed using the EVRFT method. This method make use of information extracted from the process input and output database to obtain controller parameters at each sampling instance. The adaptive nature of this method is achieved by: (i) update the database by adding the current process data into the database; (ii) using the relevant dataset selected from the current database based on knearest neighborhood criterion; (iii) update the parameters in the reference model. Moreover, a second-order reference model is used in both designs instead of the firstorder reference model normally used in the literature. Simulation results are presented for both nominal cases and cases in the presence of modeling error and process noise. Both adaptive PID and IMC designs by the proposed EVRFT design are observed give faster and less oscillatory responses 63 Chapter Conclusions and Further Work than the PID controller designed using VRFT method. In summary, the EVRFT method is a useful strategy for adaptive controller design for nonlinear processes. 5.2 Suggestions for Further Work There are few open questions that need to be further studied. Some possible topics for future research are listed below. The present study focuses on the servo performance in the controller design. It is thus of practical importance to extend the EVRFT design to address disturbance rejection performance. Furthermore, the EVRFT method may be applied to more complicated control systems such as cascade control. Finally, the application of the EVRFT method to the multivariable systems may also be an interesting topic. 64 Appendix A Derivation of the 2nd-order Reference Model Consider the following second-order transfer function with a zero-order hold: − e − Δts ⋅ ⋅ e − NΔts s (λs + 1) (A.1) where N ⋅ Δt is the process time delay, Δt is the sampling period and λ is the time constant. Applying Z-transform of Eq. (A.1) obtains ⎫ ⎧1 − e −Δts Z⎨ ⋅ ⋅ e − NΔts ⎬ = z − N − z −1 (1 + λs ) ⎭ ⎩ s ⎡ 1 ΔtAz −1 = z − N − z −1 ⎢ − − −1 − Az −1 λ − Az −1 ⎢⎣1 − z ( ( where A = e ) − Δt z −N ⋅ λ ( and hence ln A = − [ Δt λ )⋅ Z ⎧⎨1s − λsλ+ − ⎩ ) ⎫ ⎬ (λs + 1) ⎭ λ ⎤ ⎥ ⎥⎦ (A.2) . Equation (A.2) can thus be simplified as: ) ] = (α + βz )z ( z −1 (1 − A + A ln A) + A − A − A ln A z −1 (1 − Az ) −1 − N −1 − Az −1 + A z −2 −1 where α = − A + A ln A and β = A − A − A ln A . 65 (A.3) References Alpbaz, M., Hapoglu, H., Ozkan, G., and Altuntas, S. (2006). 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Review 2.2 Direct Data- based Controller Design Methods Designing controllers directly based on a set of measured process input and output data, without resorting to the identification of a process model, is an attractive option for process control application Such ‘direct’ data- based design techniques are conceptually more natural than model-based designs where the controller is designed on the basis... application to adaptive PID controller and adaptive Internal Model Control (IMC) designs The main contributions of this thesis are as follows (1) Adaptive PID controller design using EVRFT method: In the proposed EVRFT design, a second-order reference model is employed instead of the first-order reference model commonly used in the literature In addition, other than the update of database and relevant dataset,... adaptive VRFT design, the off-line database employed in the conventional VRFT design is continuously updated by adding the current process data into the database Furthermore, PID parameters are determined by the VRFT design at each sampling instance using the relevant dataset selected from the current database based on k-nearest neighborhood criterion In this chapter, an enhanced version of the adaptive. .. addition to the update of database and relevant dataset 1.3 Thesis Organization The thesis is organized as follows Chapter 2 comprises the literature review of nonlinear process control In Chapter 3, an enhanced version of the VRFT method is developed to design an adaptive PID controller Moreover, by incorporating the enhanced VRFT design framework into IMC design, an adaptive IMC controller for nonlinear... processes Thus how to extract relevant information from data and to use this information for controller design becomes a significant research topic for chemical industry Toward this end, several data- based methods for controller design were developed in the last fifteen years Spall and Cristion (1998) has developed a stochastic design framework in which the controller is represented by a function approximator... extended this controller design technique to multivariable systems and showed a chemical process application An adaptive version of the VRFT method (Kansha et al., 2008) is proposed to extend its application to nonlinear processes In this adaptive VRFT design, the offline database employed in the conventional VRFT design is continuously updated by adding the current process data into the database Furthermore,... step is to design the controller based on the model thus obtained Another interesting alternative for controller design is the data- based methods For example, the Virtual Reference Feedback Tuning (VRFT) method developed by Campi et al (2000, 2002) is a direct data- based method that determines the 19 Chapter 3 Adaptive PID Controller Design Using EVRFT Method parameters of a controller by using a set... control performance (2) Adaptive IMC controller design using EVRFT method: IMC is a powerful controller design strategy for the open-loop stable dynamic systems However, the performance of IMC controller will degrade or become unstable when it is applied to nonlinear processes with a range of operating conditions In the proposed IMC design, the EVRFT design is applied to update the IMC controller at each... η (k +1) = l incη (k ) 23 Chapter 3 Adaptive PID Controller Design Using EVRFT Method The following outlines the computational algorithm for the proposed EVRFT design of adaptive PID controller Step 1 The process input (u(k)) and output (y(k)) data that characterize the dynamics of nonlinear system are assumed to be available and the off-line database for EVRFT design is constructed as (xi )i =1~ n... the nonlinear controller design for the nonlinear dynamic processes Adaptive controller is a well-established solution for nonlinear process control and its concept will be used throughout this thesis Figure 2.1 Diagram of adaptive control scheme Research in adaptive control has a long and vigorous history The development of adaptive control started in the 1950’s with the aim of developing adaptive flight . ADAPTIVE CONTROLLER DESIGN DIRECTLY FROM PLANT DATA YAN LI (B. Eng., National University of Singapore,. LITERATURE REVIEW 5 2.1 Adaptive Control for Nonlinear Processes 5 2.2 Direct Data- based Controller Design Methods 11 2.2.1 The VRFT Design Framework 12 2.2.2 Adaptive VRFT design 14 2.3 Nonlinear. different adaptive controllers are developed implementing the EVRFT strategy, i.e. an adaptive PID controller and an adaptive Internal Model Controller. Simulation results show that both proposed controllers