On multi zone tracking and non gaussian noise filtering for model predictive control

ON MULTIZONE TRACKING AND NON-GAUSSIAN NOISE FILTERING FOR THE MODEL PREDICTIVE CONTROL WANG XIAOQIONG (B.Eng.(Hons.),NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 DECLARATION I here by declare that this thesis is my original work and it has been written by me in its entirely. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. WANG XIAOQIONG 30 Mar 2014 ON MULTIZONE TRACKING AND NON-GAUSSIAN NOISE FILTERING FOR THE MODEL PREDICTIVE CONTROL Copyright 2014 by WANG XIAOQIONG Acknowledgments I would like to thank several people who have guided, helped, assisted, accompanied, or supported me throughout my PhD course. Foremost, I would like to express my deepest gratitude to my supervisors Prof. Ho Weng Khuen and Prof. Ling Keck Voon for the continuous support of my Ph.D study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. Without their guidance and persistent help, this dissertation would not have been possible. My sincere thanks also goes to Prof. Tan Kok Kiong and Prof. Arthur Tay Ee Beng, for their time and efforts in assessing my research work, the valuable suggestions and critical questions during my qualification examination. I would like to thank my colleagues and lab mates, Jose Vu, Qu Yifan, Yu Chao, and Vathi for the stimulating discussions, for the accompany when we were working together, and all the fun we have had in the last four years. Many thanks also goes to my dearest friends, Xie Yanxi, Sun Wen, and Li suchun, accompanied me through the happiness and sadness. Finally, I am deeply indebted to my parents and my sister, for their love and support, which provide me the motivation for everything. i Contents Contents ii List of Figures iv List of Tables vii Introduction 1.1 An Overview of Model Predictive Control 1.2 Motivation of the Thesis . . . . . . . . . . 1.3 Contribution of the Thesis . . . . . . . . . 1.4 Scope of the Thesis . . . . . . . . . . . . . . . . . 1 10 . . . . . . . . . . 12 13 15 18 20 24 25 28 33 40 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control for Uniform Output 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Algorithm of UMPC . . . . . . . . . . . . . . . . . . . . 2.2.1 Formulation of UMPC . . . . . . . . . . . . . . . 2.2.2 Control Law of UMPC . . . . . . . . . . . . . . . 2.2.3 Cost Function Comparison of UMPC and SMPC 2.3 Bake Plate Thermal Modeling . . . . . . . . . . . . . . . 2.4 Experimental Results . . . . . . . . . . . . . . . . . . . . 2.4.1 UMPC Uniformity Validation Experiments . . . . 2.4.2 UMPC Robustness Experiments . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering of the ARMAX process with Generalized tDistribution Noise: The Influence Function Approach 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Maximum Likelihood Estimation of the ARMAX Process with GT Noise . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The ARMAX Process . . . . . . . . . . . . . . . . 3.2.2 The Diophantine Equation . . . . . . . . . . . . . . 3.2.3 Maximum Likelihood Estimation . . . . . . . . . . 3.3 Influence Function Approximation . . . . . . . . . . . . . . 3.3.1 The Recursive Algorithm . . . . . . . . . . . . . . . ii 57 . 57 . . . . . . 60 60 61 64 65 66 3.4 3.5 3.3.2 Mean, Variance and Outlier . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Example 1: The Kalman Filter Connection . . . . 3.4.2 Example 2: Variance . . . . . . . . . . . . . . . . 3.4.3 Example 3: Outlier . . . . . . . . . . . . . . . . . 3.4.4 Example 4: Liquid Level Estimation Experiment . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . MPC Closed-Loop Filter 4.1 Introduction . . 4.2 MPC Examples 4.2.1 Outlier . 4.2.2 Variance 4.3 Conclusion . . . . . . . . . . . . . . . . . 68 69 70 80 84 87 95 . . . . . 97 97 98 100 104 107 . . . . . 108 109 111 114 116 122 Control with ARMAX and Kalman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Load Comparison and Standard MPC 5.1 Introduction . . . . . . . . . . . . 5.2 The Experimental Setup . . . . . 5.3 Controller Design . . . . . . . . . 5.4 Experimental Results . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of Multiplexed MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Future Works 124 Bibliography 129 A Derivation of Equation (3.14) 141 iii List of Figures 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Receding-horizon control implementation of Model Predictive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control structure. . . . . . . . . . . . . . . . . The maximum likelihood criterion was used to fit a Gaussian distribution (dotted-line, µ = 0, σ = 28.5nm) and GT distribution (solid-line, p = q = 2, σ = 29.5nm) to the thickness measurement distribution. . . . . . . . . . . . . . . . . . . . . . Thickness measurements on 24 semiconductor wafers after Chemical Mechanical Polishing. . . . . . . . . . . . . . . . . . . . . . SMPC Temperature response of 3-zone bake-plate with room temperature wafer placed on. . . . . . . . . . . . . . . . . . . UMPC Temperature response of 3-zone bake-plate with room temperature wafer placed on. . . . . . . . . . . . . . . . . . . A photograph of the multizone bake plate. . . . . . . . . . . . UMPC Uniformity ISE trend with q1 = 1. . . . . . . . . . . . Output and Control singals of SMPC (left) with q1 = q2 = q3 = and UMPC (right) with q1 = 1, q2 = 3, q3 = 3. . . . . . . . . Zone block diagram of bake-plate. . . . . . . . . . . . . . . . Output performance and input Signals of UMPC with q1 = 1: upper from left to right are with q2 = q3 = 1, q2 = q3 = respectively; lower from left to right are with q2 = q3 = 10, q2 = q3 = 20 respectively. . . . . . . . . . . . . . . . . . . . . . Output performance and input Signals of SMPC (left) with q1 = q2 = q3 = and UMPC (right) with q1 = 1, q2 = 20, q3 = 20. . Output performance and input Signals when the gain of the identified plant model is artificially increased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . Output performance and input Signals when the gain of the identified plant model is artificially increased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 6 . 15 . 16 . 26 . 35 . 36 . 39 . 42 . 44 . 45 . 46 2.11 Output performance and input Signals when the gain of the identified plant model is artificially decreased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Output performance and input Signals when the gain of the identified plant model is artificially decreased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Output performance and input Signals when the time constant of the identified plant model is artificially increased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . 2.14 Output performance and input Signals when the time constant of the identified plant model is artificially increased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . 2.15 Output performance and input Signals when the time constant of the identified plant model is artificially decreased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . 2.16 Output performance and input Signals when the time constant of the identified plant model is artificially decreased by times: SMPC with q1 = q2 = q3 = (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right). . . . . . . . . . . . . . . . . . . . . . . 2.17 Comparison of Uniformity ISE of UMPC(left) and SMPC(right) when modeling error is presence. . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4.1 Different choices of the GT distribution shape parameters p and q can give different well-known distributions. . . . . . . . . . . Simulation results of Example 2. . . . . . . . . . . . . . . . . ARMAX filter output yˆ(N ). . . . . . . . . . . . . . . . . . . . Kalman filter output yˆ(N ). . . . . . . . . . . . . . . . . . . . Photo of the coupled-tank. . . . . . . . . . . . . . . . . . . . . Measurement y(N ) for the liquid level estimation experiment. The maximum likelihood criterion was used to fit a GT distribution (solid-line) and Gaussian distribution (dashed-line) to the noise distribution. . . . . . . . . . . . . . . . . . . . . . . . . . ARMAX filter estimate yˆ(N ). . . . . . . . . . . . . . . . . . . Kalman filter estimate yˆ(N ). . . . . . . . . . . . . . . . . . . . Outlier analysis. (x) measurement; (—)ARMAX filter estimate; (- - -) Kalman filter estimate. . . . . . . . . . . . . . . . . . . . 47 . 48 . 50 . 51 . 52 . 53 . 54 . . . . . . 61 81 85 85 87 88 . 89 . 91 . 91 . 95 Block diagram of the closed-loop system. . . . . . . . . . . . . . 98 v 4.2 4.3 Silumation results of MPC Outlier Example. . . . . . . . . . . . 101 Simulation results of MPC Variance Example. . . . . . . . . . . 106 5.1 Patterns of input moves for Standard MPC (solid), and for the Multiplexed MPC (dashed). . . . . . . . . . . . . . . . . . . . Zone Disturbance impulse response of experiment and estimated model respectively. . . . . . . . . . . . . . . . . . . . . Plot of MMPC and SMPC maximum computational time v.s. control horizon by experiment. . . . . . . . . . . . . . . . . . . Comparison of the computational time of MMPC (dashed) and SMPC (solid) with similar performance, Nu = 5. . . . . . . . . Comparison of the computational time of MMPC (dashed) and SMPC (solid) with similar performance, Nu = 20. . . . . . . . Comparison of the computational time of MMPC (dashed) and SMPC (solid) with similar performance, Nu = 25. . . . . . . . 5.2 5.3 5.4 5.5 5.6 vi . 109 . 113 . 117 . 118 . 119 . 120 List of Tables 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 Tuning results: ISE of uniformity (sum of ||y1 − y2 ||2 and etc.), q1 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ISE of uniformity (sum of ||y1 − y2 ||2 and etc.) when modeling error is presence . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Parameters of the ARMAX process and ARMAX Examples . . . . . . . . . . . . . . . . . . . . . . Mean and Variance of yˆ(N ) in Figure 3.2 . . . . . Variance (×10−3 ) of yˆ(N ) in Figures 3.8 and 3.9. filter in . . . . . . . . . . . . . . . the . . . 70 . . . 81 . . . 90 Parameters for ARMAX models and ARMAX filters in Examples 4.2.1 and 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 99 Mean and Variance of the simulation output in Figure 4.2 . . . 102 Mean and Variance of the simulation output in Figure 4.3 . . . 105 vii hood problem given the most recent N measurement samples instead of all the measurements. Given the most recent N measurements {yk }TT −N +1 , the state estimation at time T , xT , could be estimated by minimizing the following moving horizon maximum likelihood cost function. J =− T ∑ ln f (ε(k)) k=T −N +1 Likewise, the IF will be employed to give an approximate solution to the maximum likelihood estimation problem based on the most N recent measurements. The recursive moving horizon maximum likelihood estimation algorithm could make the ARMAX filter suitable for on-line and real-time implementation. In the loss function of Equation 3.17, all data points are given the same weight. It is common to reduce the influence of old data. This can be done by using a loss function with exponential weighting, that is, V = N ∑ ( )2 λN −k z(k) − HΦk−1 x ˆ(1|N ) k=1 The forgetting factor, , is less than one and is a measure of how fast old data are forgotten. Hence, this alternative approach to reduce the influence of the old data could also be studied in the future. Besides the computational comparison, the output performances of MMPC and SMPC could also be compared by simulation and experiment in the future study. 127 Author’s Publications [1] W. K. Ho, K. V. Ling, H. D. Vu, and X. Q. Wang, Filtering of the ARMAX process with Generalized t-Distribution noise: The Influence Function Approach, published in Industrial & Engineering Chemistry Research, 2014. 128 Bibliography [1] Eduardo Camponogara, Dong Jia, Bruce H Krogh, and Sarosh Talukdar. Distributed model predictive control. 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Mathematical Problems in Engineering, 2012, 2012. 140 Appendix A Derivation of Equation (3.14) By taking expectation, Equation (3.12) can be written as ∫ +∞ ψ(e)f (ε)dε = (A.1) −∞ To study the change in x ˆ(1) when the distribution changes from f (ε) to a new distribution f1 (ε), replace f (ε) in Equation (A.1) by (1 − h)f (ε) + hf1 (ε) where ≤ h ≤ 1, giving ∫ +∞ −∞ ψ(ε)((1 − h)f (ε) + hf1 (ε))dε = (A.2) Differentiating Equation (A.2) with respect to h gives (∫ +∞ ) ∂ ψ(ε)((1 − h)f (ε) + hf1 (ε))dε = ∂h −∞  +∞  +∞ ∫ ∫ ∂ψ(ε) ∂ θ¯ ψ(ε)(−f (ε) + f1 (ε))dε +  =0 ((1 − h)f (ε) + hf1 (ε))dε ¯ ∂h ∂θ −∞ −∞ (A.3) 141 Let h = and using Equation (A.1), Equation (A.3) reduces to ∂x ˆ(1) ∂h (∫ =− h=0 +∞ −∞ )−1 ∫ +∞ ∂ψ(ε) f (ε)dε ψ(ε)(f1 (ε))dε ∂x ˆ(1) −∞ (A.4) x(1)=0 When h = 0, the associated probability density function of x ˆ(1) is f (ε) and the usual assumption of zero initial condition for the ARMAX transfer function is made i.e. x(1) = 0. Let f1 (ε) = δ(ε) an impulse function at ε and Equation (A.4) reduces to the influence function IF(ε) = ∂x ˆ(1) ∂h (∫ =− h=0 ∞ −∞ )−1 ∂ψ(ε) f (ε)dε ψ(ε) ∂x ˆ(1) which is Equation (3.14). 142 x(1)=0 [...]... helicopter [19], and building cooling systems [20] However, MPC for multi- zone tracking 3 is not fully studied Some attempts have been done for temperature uniformity control [4–6] But these studies on temperature uniformity usually focused on the set-point tracking uniformity from batch to batch, not the uniformity of the zone- to -zone temperature trajectories Moreover, most of MPC control designs use...Abstract Model Predictive Control (MPC) has been widely studied and adopted in industrial applications because the actual control objectives and operating constraints can be represented explicitly in the optimization problem that is solved at each control instant [1–3] Some attempts have been done for temperature uniformity control [4–6] But these studies on temperature uniformity usually focused on the... Chapter 2 Model Predictive Control for Uniform Output For applications where uniformity among the outputs are critical, this chapter demonstrates that UMPC formulation is a possible candidate In this chapter, we compare the load disturbance performance of UMPC and SMPC We show that when the plant modeling error exists, the UMPC maintains the uniformity performance whereas the SMPC does not We formulate... guaranteeing that constraints are satisfied In determining the MPC control, one needs a process model to predict the future plant outputs, and an optimization criterion which is the cost function MPC could well handle the highly complex, non- linear, uncertain, and constrained dynamics The multi- variable cases can be easily dealt with by MPC The resulting controller is an easy-to-implement control law With... set-point tracking uniformity from batch to batch, not the uniformity of the zone- to -zone temperature trajectories In this thesis, we proposed a method called Uniformity Model Predictive Control (UMPC) to achieve output uniformity The idea of UMPC is to reconstruct the cost function of the Standard MPC Simulations and bakeplate experiments were carried out to show that UMPC gives better outputs uniformity... the receding-horizon control implementation of MPC The control horizon represents the number of parameters used to capture the future control trajectory The predictive horizon represents the number of samples we want to predict Although the optimal trajectory of future control signal is completely described within the length of control horizon, the actual control input to the plant only takes the first... function focuses on tracking the pre-set reference Model Predictive Control (MPC) operates by solving a constrained optimization problem on- line, in real-time, in order to decide how to update the control inputs (manipulated variables) at the next update instant The 15 UMPC, q1=1, q2=q3=20 90.5 zone1 zone2 zone3 Temperature 90 89.5 89 88.5 88 0 200 400 600 800 1000 Time Figure 2.2: UMPC Temperature response... one output and its reference, and errors between this output and the other outputs In order to have good output uniformity, only one output follows the setpoint, while this output becomes the references of the other outputs Work on applications of MPC as a feedback controller for bake-plate temperature control can be found in [38], and feed-forward control was reported in [39] In addition, a Linear... uniformity In contrast, as a strategy to ensure the good output uniformity, Uniformity Model Predictive Control (UMPC) is proposed UMPC solves the optimization problem with a different cost function The cost function of UMPC minimizes the error between one output and the pre-set reference, and also the errors between this output and the other outputs In order to have good output uniformity, only one... instead of the usual Gaussian distribution Moreover, the computational load is also a problem when applying the MPC designs to the industrial viii applications We provide one of the first experimental verification of the computational load reduction property of MMPC ix Chapter 1 Introduction 1.1 An Overview of Model Predictive Control The general design objective of Model Predictive Control (MPC) is to . ON MULTIZONE TRACKING AND NON- GAUSSIAN NOISE FILTERING FOR THE MODEL PREDICTIVE CONTROL WANG XIAOQIONG (B.Eng.(Hons.),NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR. information which have b een used in the thesis. This thesis has also not been submitted for any degree in any university previously. WANG XIAOQIONG 30 Mar 2014 ON MULTIZONE TRACKING AND NON- GAUSSIAN. . . 122 6 Conclusion and Future Works 124 Bibliography 129 A Derivation of Equation (3.14) 141 iii List of Figures 1.1 Receding-horizon control implementation of Model Predictive Control. . .